Math H TG by lighthousecurriculum

Level H

Teacher's Guide

Standard Edition

LEVEL H - TEACHER'S GUIDE

Lighthouse Math Program Directors Mrs. Zehava Kraitenberg M.S. Curriculum Advisor, Elementary School Principal Jane Chamberlain Master of Education, Curriculum and Instruction

Credits Curriculum Writers

Review Team

Layout & Design

Yehudis Leitner Curriculum Coordinator

Chaya Breindy Kenigsberg Curriculum and School Leadership Specialist Master in Education

Akiva Leitner Project Manager

Jane Chamberlain Middle School Math Instructor M.Ed. in Curriculum and Instruction Kelly Christensen 6th-7th Grade Math Teacher M. Ed in Administration and Leadership (K-12) Esther Aboud Curriculum Consultant M.Ed. in Special Education Lauren Noorparvar Middle School Math Educator Math Curriculum Coach K - 12 Master in Education Keely Franklin Curriculum Developer Cierra Henderson Curriculum Developer

Esther Aboud Curriculum Consultant M.Ed. in Special Education

Mirko Zunic Layout Director Freepik Illustrations

Lauren Noorparvar Middle School Math Educator Math Curriculum Coach K - 12 Master in Education Fraydel Sharf Content Editor Miriam Shulamis Eisemann Content Editor Yehuda Gartenhaus M.A. Elementary School Principal Zehava Kraitenberg M.S. Curriculum Advisor Elementary School Principal

Michal Davydov Curriculum Developer

Lighthouse Math Teacher's Guide Level H • ISBN 978-1-955773-89-8 ©Copyright 2026 Lighthouse Curriculum Inc. All rights reserved. Contact Lighthouse Curriculum: Call 718.285.7100 or email info@lighthousecurriculum.com For more information visit www.lighthousecurriculum.com Content developed in collaboration with The Reimagined Classroom No part of this publication may be reproduced, stored in a retrieval system, stored in a database and/or published in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher. To obtain permission to use portions of material from this publication, please contact Lighthouse Curriculum.

Welcome to the Lighthouse Math Curriculum! Here's what you'll find in every chapter:

Chapter Introduction Overview of the skills covered in the chapter

Chapter Checklist Review Skill

Practice prerequisite skills needed for the chapter

Brief review of each skill Self-evaluation for students

Lesson Pages Daily Review includes Prerequisite Skills for the coming lesson and a Spiral Review of the previous lesson. Learn and Connect clearly explains the lesson content. Apply Section provides scaffolded exercises designed for teacher-guided practice with students. Exercise problems include plenty of practice to develop the new skills and concepts. Challenge problems encourage higher-level thinking and enrich student learning.

Review Brief recap of key lessons in the chapter Practice exercises that reinforce and review the skills taught in the chapter

About the Curriculum The program builds in review and new concepts throughout each level as students step up through mastery of skills. By providing foundational skills and practice, students retain information. The stepwise approach is consistent as students work through 11 chapters of review, new skills, guided practice, and problem solving. All lessons include step-by-step instructions for clarity, giving all teachers the tools for success. Clear instruction presented in a visually pleasing way provide a vibrant learning experience. Illustrations complement math questions by including information directly tied to and used to solve the problem. The books are formatted in a way that each grade level can be completed successfully by the culmination of the school year. Lighthouse Math gives teachers the tools they need to teach and gives students everything they need to learn.

Intro

Lighthouse Math Level H Teacher's Guide

1-1 | Square R oo PRE REQ UIS ITE SKI LLS

Solve. 1.

DA ILY RE VIE W

SPI RAL REV IEW

3×

= 12

7×

= 49

9×

122 =

= 81

4×

82 =

= 16

Squaring numb

ers.

62 =

LE AR N AN D CO NN

102 =

EC T

A square is wh at you get when A perfect you multiply a nu inverse, or oppo mber by itself. Th site, of a square. square is a e square root is It tells you what the number was mult number whose iplied to make th To find the squa e square. square root re root, ask: “W hat number tim is an integer. es itself equals this number?” 9 = 3 because 3 ×3=9 25 = 5 because 5 × 5 = 25 81 = 9 because 9 × 9 = 81 When taking th e square root of a number, we usually write the answer as a positive number, but it can be negative, 9 can equal 3 or too! -3 because 3 × 3 = 9 and -3 × -3 = 9 Therefore, if we want the answer to be negative, put the negativ we e symbol in fro nt of the square 9=3 - 9 = -3 root.

AP PL Y

strictly prohibite

Circle the corr ect answer. 1.

Which expressio

riculum. Copying

n can you use to

solve 49?

n can you use to

solve 4?

7×7

8×8

solve 100?

4×4

3×3

502

9×9

11 × 11

102

Which expressio

n can you use to

Match the perfe

ct square with

64 1

49 2

its square root

16 3

100 4

Vocabulary

36 7

Square - the pro duct of a number times itself Integer - a wh ole number, wit hout any fractio Inverse - a ma ns or decimals; thematical opera can be positive tion that undoes Square root - the or negative another operatio inverse of squarin n, such as square g; a number tha Perfect square s and square roo t, when multiplied - a number wh ts ose square root by itse lf, gives the origin is an integer al number under

Level H

Chapter 1

Lighthouse Math Level H Teacher's Guide

Lesson 1

the square root

Lighthouse Math

Intro

How to Teach a Lesson 1. Daily review: Begin the lesson with a quick review.

3. Direct Instruction: Teach the target skill.

y Allow students to complete this section independently in the Student Book.

y As a class, learn from the clear and concise instruction found in the Student Book in the Learn and Connect section.

y The Spiral Review section covers the skills from the previous lesson. The Prerequisite Skills section covers skills that will be needed in the coming lesson.

y Step-by-step guidance in how to introduce the lesson is provided in the Introduce the Lesson section of the Teacher's Guide.

y Check students' answers. Make sure they are comfortable with the review skills before teaching the lesson.

y Vocabulary words used in the lesson can be referenced in the Student Book or Teacher's Guide.

2. Pre-Lesson Warm Up: Activate students’ prior knowledge to prepare for the lesson.

4. Guided Practice: Allow students to practice with teacher guidance.

y Look in the Teacher's Guide in the Prelesson Warm-up section for step-bystep guidance in leading an activity or discussion before teaching the lesson.

y Scaffolded tasks are found in the Student Book in the Apply section. They gradually release responsibility for the skill that was just taught.

y Guiding questions and answers are provided to promote discussion in the classroom.

y Provide guidance as students complete the Apply section. Look in the Teacher's Guide in the Apply and Develop skills section for pointers.

Intro

Lighthouse Math Level H Teacher's Guide

5. Activity: Reinforce the skill with a hands-on activity.

7. Challenge: End the lesson by extending student learning.

y A hands-on group activity is found in the Teacher's Guide in the Activity section.

y Complete the critical thinking and problem-solving questions found in the Challenge section of the Student Book.

y Encourage collaboration and discussion. Allow students to respectfully construct arguments and critique the reasoning of their peers.

y Discuss students' reasoning. Look in the Teacher's Guide in the Challenge section for discussion points and tips.

6. Independent practice: Students will build fluency and proficiency.

8. Assessment: Check student proficiency. y Look over students’ independent work. Suggested problems can be found in the Assess section of the Teacher's Guide.

y Allow students to complete the exercises in the Student Book independently, with a partner, or as a class.

y Assess retention and proficiency over time. Use the chapter assessment found at the end of the book.

y Use the Apply and Develop Skills section in the Teacher's Guide for guidance on how a particular exercise should be completed. y Look out for common errors mentioned in the Teacher's Guide. y Use the Struggling Learners section of the Teacher's Guide for tips and ideas to support struggling students. y Students who finish early can complete the activity in the Early Finishers section of the Teacher's Guide.

Lighthouse Math Level H Teacher's Guide

Intro

5-1 | Proportions

PREREQUISITE SKILLS

DA ILY REV IEW

Use the Daily Review to practice concepts from the previous lesson and review prerequisite skills for the coming lesson.

SPIRAL REVIEW

Begin the lesson with direct instruction. Support students' learning with a step-bystep guide for using the target skill and real-life example problems.

2x − 3 = 7

x + 10 = 8 2

(x + 7) = -2.5 4

2. 6.

3(x − 6) = 12

4x + 5 = 1

6(x + 2) = -12

(x − 3) =9 2

x ( )−4=0 5

Find the slope. 1.

(2,5) and (6,11)

Slope =

(1,-2) and (5,6)

Slope =

(-3,7) and (4,-2)

Slope =

(-1,-5) and (3,-9)

Slope =

LEA R N A N D CO N N ECT At the state fair, Nick wants to buy popcorn for the best value. Vendor A sells a 48-ounce bucket of popcorn for $9.60. Vendor B sells an 80-ounce bucket for $18.40. He wants to see which gives the better value. Nick can use the unit rate, or how much each costs per one unit, to compare the two vendors. To find the unit rate, divide the total by the number of units.

Vendor A

Vendor B

$9.60 = $0.20 per ounce 48 oz

$18.40 = $0.23 per ounce 80 oz

Unit rate

If Nick wants to know how much it would cost to buy 60 oz. of popcorn from Vendor A he can set up a proportion using Vendor A’s pricing: 48 oz 60 oz = $9.60 x © Lighthouse Curriculum. Copying strictly prohibited.

Use the Apply section as the students' first entry into the concept. These problems provide a gradual release to help guide students through processes in a stepwise fashion. They can also be completed together as class and are useful for evaluating student understanding and clarity before continuing with independent work.

Solve.

Cross multiply

Vendor A is the better buy.

48 oz 60 oz = $9.60 x

Proportion

48x = (60)(9.60)

48x = 576

x = $12

It would cost $12 for 60 oz of popcorn.

A PPLY Solve the proportion. 1.

5 miles x miles = 2 hours 6 hours

x pages 30 pages = 4 minutes 6 minutes

18 pencils 42 pencils = 3 boxes x boxes

Vocabulary Unit rate - a comparison of two quantities where one of the terms is 1; it tells how much of something there is per one unit Proportion - an equation that shows that two ratios or rates are equal Better buy - the item that gives you more for your money, found by comparing unit rates

Level H

Chapter 5

Lesson 1

Lighthouse Math

Introduce vocabulary during direct instruction. Reinforce the words by using them in the classroom. Teach students to reference the vocabulary section when encountering an unfamiliar word.

Intro

Lighthouse Math Level H Teacher's Guide

Multiple problems allow for differentiated instruction and practice. Choose odd-numbered problems to complete first, or choose some problems to assign.

Exercise | 5-1

Continue practice as a class or assign problems for independent work.

Name Find the unit rate. 1.

A printer produces pages at a steady rate. How many pages does it print per minute?

A machine bottles juice at a constant rate. How many bottles does it complete per minute?

Pages

Bottles

135

270

405

540

Minutes

Emma reads books at a constant pace. How many books does she read per week?

A gardener always uses the same fertilizer to water ratio. What is the unit rate for fertilizer per gallon of water?

Books

Ounces

Weeks

Gallons

Determine the better buy. 5.

A 16-oz bottle of juice for $4.80 or a 10-oz bottle for $3.00?

A 30-pack of markers for $12.00 or a 10pack for $4.50?

A 32-oz bottle of shampoo for $7.20 or an 18-oz bottle for $4.32?

A 1.5-lb box of cereal for $4.20 or a 2-lb box for $5.40?

Bonus Challenge Section Use Challenge problems as an enrichment opportunity for students who are ready to advance deeper into the concept.

A recipe uses 3 cups of flour to make 12 cookies. How many cups of flour are needed to make 20 cookies?

10. A car travels 180 miles in 3 hours. How far can it travel in 5 hours at the same speed?

11. A printer makes 60 copies in 4 minutes. How many copies can it make in 7 minutes?

12. Lisa bought 5 notebooks for $8.75. How much would 8 notebooks cost at the same rate?

Solve by writing a proportion.

C HAL L E NG E 13. A box of granola bars contains 6 bars and costs $3.60. Another store sells 10 bars for $6.50. Is this a proportional relationship? Why or why not?

Level H

Lighthouse Math

Lighthouse Math Level H Teacher's Guide

Intro

14. Emma types 135 words in 3 minutes. She wants to type a 675-word essay and plans to take one 5-minute break halfway through. About how many minutes will it take her to complete the entire task, including the break?

Chapter 5

Exercise 1

How to Use the Teacher's Guide

Prepare your lesson. y Use objectives to guide you through the main idea.

Our easy to use, colorcoded Teacher's Guide complements the student workbook. It provides an answer key, guidance to the teacher, ideas for activities, and ways to differentiate instruction.

y Quick reference for vocabulary covered in the lesson y Materials list helps you be prepared to lead activities and be ready with resources. Level H | 5-1 5-1 | Proportions

DA I LY R EV I EW

Objective and Learning Goals y Students will be able to find the unit rate of a ratio or set of ratios and use it to solve problems. y Students will be able to use the unit rate or cross multiplication to solve proportions.

PREREQUISITE SKILLS

SPIRAL REVIEW

(x + 7) = -2.5 x = -17 4

4x + 5 = 1

x + 10 = 8 x = -4 2

3(x − 6) = 12 x = 10

(x − 3) =9 2

(1,-2) and

(-1,-5) and

Find the slope. 1.

(2,5) and (6,11)

Slope =

(-3,7) and (4,-2)

Slope =

Write the following table on the board:

Vendor A

$9.60 = $0.20 per oun 48 oz

Cross multiply

Vendor A

48 oz 60 oz = $9.60 x

Proportion

48x = (60)(9

It would co

A P P LY Solve the proportion. 1.

5 miles x miles = 2 hours 6 hours

x = 15

x pages 30 pages = 4 minutes 6 minutes x = 20

Vocabulary

Pizzas

Cost

[$9]

$18

[$36]

[$90]

Ask students to work independently to find the values that go in the blank spaces in the table. Review as a class and ask students what methods they used to solve. Explain that this table uses a ratio to evaluate costs, and today’s lesson will review using unit rates and proportions to find and compare rates and find missing values. Guiding Questions: 1. How can we use ratios to find the missing values? [Find the unit rate and use a proportion to solve for a missing piece.] 2. Why is finding a unit rate important when solving? [It allows us to easily calculate any missing value since it is per 1 unit.]

Unit rate - a comparison of two quantities where one of the terms is 1; it tells how much of so Proportion - an equation that shows that two ratios or rates are equal Better buy - the item that gives you more for your money, found by comparing unit rates

Level H

Chapter 5

Lesson 1

INTRODUCE THE LESSON (Learn a

Begin the lesson by writing the key vocabulary term better buy, on the board. Ask students to define eac and discuss their meanings as a class. Next, read alo about Nick buying popcorn at the state fair and ask the two vendors’ prices. Guide students to recogniz allows a fair comparison. Then, work through settin vendors and demonstrate how to simplify them. Dis unit rates confirm that Vendor A offers the better bu part of the problem by asking students how to set u missing value. Review the steps of cross multiplying emphasize interpreting the answer in context. Conc is just one of many real-world situations where prop useful.

APPLY AND DEVELOP SKILLS (P

Work through problems 1-3 in the Apply section tog solving proportions. Emphasize how to correctly set the corresponding terms, and use cross multiplicati value. Work through problems 1, 5, and 9 in the Exer a clear example for each problem type. Then have s

Use teacher prompts to make lessons interactive. Questions and dialogue for discussions are provided with answers to help guide the lesson.

- 9

If Nick wants to know how much it would cost to buy 60 oz. of popcorn from Vendor A he can set up a proportion using Vendor A’s pricing:

Lighthouse MATH Level H | Teacher's Guide

Intro

3 2

Nick can use the unit rate, or how much each costs per one unit, to compare the two vendors. To find the unit rate, divide the total by the number of units.

48 oz 60 oz = $9.60 x

PRE-LESSON WARM-UP

Use these questions during the pre-lesson warm-up to guide the discussion and ready students for the new concept.

2x − 3 = 7 x = 5

Start off the class with a warm-up activity to activate prior knowledge and prepare students for the lesson of the day.

L EA R N A N D C ON N EC T

Vocabulary y Unit rate - a comparison of two quantities where one of the terms is 1; it tells how much of something there is per one unit y Proportion - an equation that shows that two ratios or rates are equal y Better buy - the item that gives you more for your money, found by comparing unit rates Materials y Poster board y Markers

Solve.

Lighthouse Math Level H Teacher's Guide

Easy to find references to help differentiate learning

Copy of the student book pages with answers available for quick answer checks and corrections

Tips and ideas to assist learners who may need more support

Exercise | 5-1

Do you have students who complete the work more quickly? Find activities or ideas to provide these students with useful work to deepen their thinking or to reinforce concepts without just adding extra problems.

Name Find the unit rate. 4.

6(x + 2) = -12 x = -4

9 x = 21

x ( ) − 4 = 0 x = 20 5

(5,6)

Slope =

-1

Pages

Bottles

Minutes

5 pages per minute

STRUGGLING LEARNERS

A machine bottles juice at a constant rate. How many bottles does it complete per minute?

Minutes

d (3,-9)

A printer produces pages at a steady rate. How many pages does it print per minute?

135

270

405

540

Provide students with a step-by-step guide to finding a unit rate. Have students simplify all ratios in a given problem to ensure understanding and have extended practice solving.

9 bottles per minute

Emma reads books at a constant pace. How many books does she read per week?

A gardener always uses the same fertilizer to water ratio. What is the unit rate for fertilizer per gallon of water?

Books

Ounces

Weeks

Gallons

2 books per week

EARLY FINISHERS

4 oz. per gallon

Ask students to create their own comparisons using products they actually shop for. They should make up or look up realistic prices for two sizes or two stores, compute unit rates for each option, and write a brief justification that names the better buy and explains why.

Determine the better buy. 5. Vendor B

48x = 576

A 30-pack of markers for $12.00 or a 10pack for $4.50?

Same value

Unit rate

A 32-oz bottle of shampoo for $7.20 or an 18-oz bottle for $4.32?

30-pack

32-oz bottle

A 1.5-lb box of cereal for $4.20 or a 2-lb box for $5.40? 2-lb box

Solve by writing a proportion.

A is the better buy.

9.60)

A 16-oz bottle of juice for $4.80 or a 10-oz bottle for $3.00?

10. A car travels 180 miles in 3 hours. How far can it travel in 5 hours at the same speed?

A recipe uses 3 cups of flour to make 12 cookies. How many cups of flour are needed to make 20 cookies? 5 cups

x = $12

CHALLENGE AND EXPLORE

300 miles

11. A printer makes 60 copies in 4 minutes. How many copies can it make in 7 minutes?

ost $12 for 60 oz of popcorn.

12. Lisa bought 5 notebooks for $8.75. How much would 8 notebooks cost at the same rate?

105 copies

nce

$18.40 = $0.23 per ounce 80 oz

$14.00

CHALLENGE 3.

18 pencils 42 pencils = 3 boxes x boxes

13. A box of granola bars contains 6 bars and costs $3.60. Another store sells 10 bars for $6.50. Is this a proportional relationship? Why or why not?

x=7

No, this is not a proportional relationship because the cost per bar is different at each store.

omething there is per one unit

Lighthouse Math

and Connect)

ms, unit rate, proportion, and ch word as a brief review oud the word problem how they could compare ze that finding the unit rate ng up proportions for both scuss how the resulting uy. Continue with the next up a proportion to find the g and dividing to solve, and clude by explaining that this portions and unit rates are

Practice)

gether as a class to review t up each ratio, identify ion to solve for the missing rcise section to give students students work in pairs to

Level H

Chapter 5

Exercise 1

Have students work with a partner to solve problem 13, then review as a class. Set up up both proportions clearly and simplify to reach the unit rate, highlighting how the unit rate supports the conclusion. For problem 14, ask students to draft the steps first, then work through those steps together as a class to evaluate, revise, and finalize a clear method.

solve the remaining problems, 2-4, 6-8, and 10-12, using the examples as a reference. Review as a class and walk through any problems students found challenging to ensure understanding and reinforce key steps.

ACTIVITY Carnival Stands Better Buys: This activity allows students to apply their understanding of proportions and unit rates in a real-world context. Divide the class into groups of two or three and assign each group a type of carnival stand, such as a popcorn booth, lemonade stand, or carnival game. Ensure that at least two groups are assigned the same type of stand to create opportunities for comparison later. Each group will design a poster advertising three different price levels for their product (for example, a carnival game might cost a certain amount for three tries, five tries, and ten tries). Remind students that their prices must remain proportional. After all posters are complete, display the similar stands side by side around the room. Have students walk around to compare the stands, calculate unit rates, and decide which offers the better buy. Students should record their findings and conclusions, then share and discuss as a class to reinforce how proportional reasoning helps identify the best deal.

COMMON ERRORS Students may set up the proportions incorrectly. Students may make calculation errors when simplifying.

ASSESS Check the odd-numbered problems in the Exercise section.

x = -1

Help guide students through problem solving or challenge questions. Extend the thought process by providing discussion points and tips for trickier problems. Have students share their ideas so you can see how they are thinking and they can work on communicating their thought processes. This can also be used as a formative assessment for some of your advanced learners.

Be on the lookout for common errors to help prevent mistakes or point out best practices.

Lighthouse MATH Level H | Teacher's Guide

Guide students to work in their student books. Start with a problem together, or have students work independently through problems. Find examples of work or tips for reading directions, showing work, and labeling answers correctly.

Prepare and lead group or partner activities to help reinforce the concepts in a hands-on, interactive way.

Lighthouse Math Level H Teacher's Guide

Intro

A quick check of the skills taught in the lesson. The suggested problems to look over or the exit ticket provided allow you to determine whether your students are ready to move on or need more review.

Table of Contents Chapter 1 1-0 1-1 1-2 1-3 1-4 1-5 1-6 1-7

Chapter 2 2-0 2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 2-10

Real Numbers Skill Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cube Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decimal Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rational and Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Understanding Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparing and Ordering Rational and Irrational Numbers . . . . . . . . . . . . . . . . Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 6 8 10 12 14 16 18

Exponents and Scientiﬁc Notation Skill Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiplication with Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Division with Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Negative and Zero Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Order of Operations with Exponents and Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Converting Numbers to Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparing Numbers in Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiplication and Division with Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . Addition and Subtraction with Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22 24 26 28 30 32 34 36 38 40 42

Table of Contents Table of Contents | Lighthouse | Level Math H Level | HLighthouse Teacher's Guide Math

Chapter 3 3-0 3-1 3-2 3-3 3-4 3-5 3-6 3-7

Equations Skill Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simplifying Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solving Multi-Step Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equations with Variables on Both Sides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . More Equations with Variables on Both Sides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identifying the Number of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equation Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46 48 50 52 54 56 58 60

Chapter 4 Linear Relations and Functions Skill Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1 Introduction to Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2 Functional Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3 Graphs of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4 Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5 Slope and y-intercept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6 Find Slope and y-intercept from a Set of Coordinates . . . . . . . . . . . . . . . . . . . . . . 4-7 Slope and Similar Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-8 Slope-Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9 Graphing Lines from Slope-Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10 Modeling and Interpreting Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11 Comparing Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-12 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-0

Chapter 5 5-0 5-1 5-2 5-3 5-4

64 66 68 70 72 74 76 78 80 82 84 86 88

Proportions Skill Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Graphs of Proportional Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Comparing Proportional Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Lighthouse Math Level | Level H Teacher's H | Guide Table of| Contents Table of Contents

Chapter 6 6-0 6-1 6-2 6-3 6-4 6-5 6-6 6-7 6-8

Chapter 7 7-0 7-1 7-2 7-3 7-4 7-5 7-6 7-7

Chapter 8 8-0 8-1 8-2 8-3 8-4 8-5 8-6 8-7

Linear Systems Skill Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solutions of Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solving Systems of Equations by Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isolating Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solving Systems of Equations by Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solving Systems of Equations by Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

104 106 108 110 112 114 116 118 120

Angles and Triangles Skill Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Understanding Angle Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parallel Lines and Transversals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Angle Relationships in Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to the Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distance on the Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applying the Pythagorean Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

124 126 128 130 132 134 136 138

Transformations and Congruence Skill Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Congruent and Similar Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dilations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combining Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

142 144 146 148 150 152 154 156

Table of Contents Table of Contents | Lighthouse | Level Math H Level | HLighthouse Teacher's Guide Math

Chapter 9 9-0 9-1 9-2 9-3 9-4 9-5

Volume Skill Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volume of Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volume of Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volume of Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solving Real-World Problems with Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

160 162 164 166 168 170

Chapter 10 Scatter Plots, Association, and Probability Skill Checklist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-1 Understanding Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2 Modeling Linear Associations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-3 Constructing Two-Way Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-4 Interpreting Two-Way Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-5 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-0

174 176 178 180 182 184

Chapter 11 Review Real Numbers Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-2 Exponents and Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-3 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-4 Linear Relationships and Functions Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-5 Proportional Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-6 Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-7 Angles, Triangles, and the Pythagorean Theorem Review . . . . . . . . . . . . . . . . 11-8 Transformations and Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-9 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-10 Scatter Plots, Association, and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-1

188 190 192 194 196 198 200 202 204 206

Glossary Assessments �� 208 208 Glossary �� 220 | Level H | Guide Table of| Contents Lighthouse Math Level H Teacher's Table of Contents

Chapter 1

In Chapter 1, we will deepen our understanding of

Real Numbers The real-number system divides numbers into different categories based on their features. • We will review squaring and cubing and investigate their inverses: square roots and cube roots. • We will learn about repeating and terminating decimals and changing these decimals into fractions. • We will learn about rational and irrational numbers and how to order them and place them on a number line.

Level H | 1-0 Chapter 1 | Skill Checklist

Skill 1: Operations with Rational Numbers Multiplication

Solve.

Same Sign: Positive

−

-5 × 8 = -40

-3 + 7 =

−

14 ÷ -2 =

-7

-8 − 4 = -12

-0.5 + -1.9 = -2.4

-2.3 × -0.2 = 0.46

−

-2.5 − (-1.2) = -1.3

-8.1 ÷ -9 = 0.9

−

Different Sign: Negative

−

Division Same Sign: Positive

−

Different Sign: Negative

−

Objective and Learning Goals

I can add, subtract, multiply, and divide positive and negative decimals and integers.

out 8 correct

Skill 2: Exponents

Students will review skills needed for Chapter 1:

Exponent

Match each exponent to the correct expression.

Base

Two to the sixth power

5×5×5

3×3×3×3×3

5×5×5×5

34 = 81

83 = 512

25 = 32

Skill 3: Converting Between Fractions and Decimals

0.75 =

75 ÷ 25 3 = 100 ÷ 25 4

3 = 3 ÷ 8 = 0.375 8

Lighthouse MATH Level H | Teacher's Guide

I can solve exponents.

out 4 correct

Convert the decimals to fractions in the simplest form. 1.

Level H

2.03 =

2100

0.46 =

23 50

20 200

20.015 =

8 0.32 = 25

Convert the fractions to decimals. 1 = 0.2 5

3 0.1875 = 16

I can convert decimals to fractions and fractions to decimals.

out 6 correct

y Review the rules for adding and subtracting positive and negative integers. y Review the rules for multiplying and dividing positive and negative integers using the gray box. y Have students complete problems 1-8 and review the answers and strategies used together as a class.

Solve.

Skill 1: Operations with Rational Numbers

53 4×4×4×4×4

2×2×2×2×2×2

y Operations with rational numbers y Exponents y Converting between fractions and decimals y Estimating quotients with compatible numbers y Understanding the equal sign y Solving one-step equations

Chapter 1

Skill Checklist

Skill 2: Exponents y Review the parts of a power: the base and the exponent and what they mean. y Review how to read 26 and how to calculate it. y Have students complete problems 1-4 and review the answers and strategies used together as a class.

Lighthouse Math

Skill 3: Converting Between Fractions and Decimals y Review how to change a decimal to a fraction in its simplest form. y Review how to change a fraction to a decimal using division. y Have students complete problems 1-6 and review the answers and strategies used together as a class.

Name

Skill 4: Estimating Quotients with Compatible Numbers Fill in the blanks. Use compatible numbers to estimate the quotients.

43 ÷ 4 = ? 4 × ? = 43

4 × 10 = 40 4 × 11 = 44 43 closer to 44, so 43 ÷ 4 will be a little less than 11. 43 ÷ 4 = about 11

64 ÷ 7 = ?

170 ÷ 3 = ?

481 ÷ 5 = ?

7 × ? = 64

64 is closer to 63 , so 64 ÷ 7

7 × 9 = 63 7 × 10 = 70

will be a little greater than 9 , 64 ÷ 7 = about 9

3 × ? = 170 3 × 50 = 150

170 is closer to 180 , so 170 ÷ 3

3 × 60 = 180

170 ÷ 3 = about 60

5 × ? = 481 5 × 90 = 450

481 is closer to 500, so 481 ÷ 5 will be a little less than 100.

5 × 100 = 500

481 ÷ 5 = about 100

will be a little less than 60 .

I can use compatible numbers to estimate quotients.

out 3 correct

Skill 5: Understanding the Equal Sign Circle the equation in which the same change was made to both sides.

a−3=b−3 4a = 4b a b = 2 2

3=x

3(4) = 4x

9 = 2y

9 − 2 = 2y + 2

9 + 2 = 2y + 2

a=b

ac = bc

ab = bc

8 = 6k

0 = 6k

0 = 6k − 8

I can keep an equation balanced by making the same change to both sides.

out 4 correct

Skill 6: Solving One-Step Equations Addition

Subtraction

x + 9 = -4 −9 −9 x = -13

x−3=6 +3 +3 x=9

Multiplication

5x = 25 ÷5 ÷5 x=5

Solve. 1.

m+4=7

y =5 2

Division

3 × x = -2 × 3 3 x = -6 out 4 correct

Lighthouse Math

Skill 4: Estimating Quotients with Compatible Numbers y Review how to use multiples to estimate quotients. y Using the example in the gray box, estimate 43 ÷ 4. y Have students complete problems 1-3 and review the answers and strategies used together as a class.

m=3

y = 10

x=9

9x = 81

p − 3 = 17 p = 20

I can solve one-step equations.

Level H

Chapter 1

Skill Checklist

Skill 5: Understanding the Equal Sign

Skill 6: Solving One-Step Equations

y Remind students that the expressions on either side of the equal sign have the same value. y Using the examples in the gray box, go through the logic of the equal sign and have students come up with more examples that work. y Have students complete problems 1-4 and review the answers and strategies used together as a class.

y Review which operations are inverses of each other. y Using the examples in the gray box, go through how to solve one example of each operation. y Have students complete problems 1-4 and review the answers and strategies used together as a class.

Lighthouse MATH Level H | Teacher's Guide

a+1=b+1

a=b

Level H | 1-1 1-1 | Square Roots

PREREQUISITE SKILLS

Objective and Learning Goals

SPIRAL REVIEW

Vocabulary

Ask students if they can come up with another riddle similar to the last three and explain that in today's lesson we will be solving problems like these riddles. Guiding Questions: 1. How do we solve these riddles? [work backwards, use division] 2. What is the same about the last three riddles? [The number that we found was the same as the number we multiplied by.] 3. What do we call numbers that are the product of multiplying a number by itself? [square numbers]

Lighthouse MATH Level H | Teacher's Guide

7 × 7 = 49

9 × 9 = 81

4 × 4 = 16

102 = 100

122 = 144

82 = 64

Squaring numbers. 62 = 36

A perfect square is a number whose square root is an integer.

A square is what you get when you multiply a number by itself. The square root is the inverse, or opposite, of a square. It tells you what number was multiplied to make the square. To find the square root, ask: “What number times itself equals this number?” 9 = 3 because 3 × 3 = 9

25 = 5 because 5 × 5 = 25

When taking the square root of a number, we usually write the answer as a positive number, but it can be negative, too!

81 = 9 because 9 × 9 = 81

9 can equal 3 or -3 because 3 × 3 = 9 and -3 × -3 = 9

Therefore, if we want the answer to be negative, we put the negative symbol in front of the square root.

9=3

- 9 = -3

A P P LY Circle the correct answer.

Read the following riddles out loud to students and have them answer: y I am the number that, when multiplied by 5, gives you 35 [7]. y I am the number that, when multiplied by 5, gives you 45 [9]. y I am the number that, when multiplied by 5, gives you 25 [5]. y I am the number that, when multiplied by 6, gives you 36 [6]. y I am the number that, when multiplied by 4, gives you 16 [4].

L E A R N A ND C O NNE C T

y Square - the product of a number times itself y Integer - a whole number, without any fractions or decimals, can be positive or negative y Inverse - a mathematical operation that undoes another operation y Square root - the inverse of squaring; a number that, when multiplied by itself, gives the original number under the square root y Perfect square - a number whose square root is a whole number or integer

PRE-LESSON WARM-UP

3 × 4 = 12

DAI LY REVI EW

y Students will understand square roots as the inverse of squares and will be able to simplify perfect squares.

Solve.

Which expression can you use to solve 49?

7×7

8×8

Which expression can you use to solve 4?

4×4

3×3

Which expression can you use to solve 100?

502

9×9

11 × 11

102

Match the perfect square with its square root. 4.

100

Vocabulary Square - the product of a number times itself Integer - a whole number, without any fractions or decimals; can be positive or negative Inverse - a mathematical operation that undoes another operation, such as squares and square roots Square root - the inverse of squaring; a number that, when multiplied by itself, gives the original number under the square root Perfect square - a number whose square root is an integer

Level H

Chapter 1

Lesson 1

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Read the first paragraph of the Learn and Connect section to introduce the vocabulary terms square, inverse, and square root. Read the examples and ask students if they can come up with more examples [ 4 = 2; 16 = 4 36 = 6]. Explain to students that these numbers are called perfect squares because they have a square root that is an integer. Not all square roots will be nice round numbers. In fact, most are not. Next, ask students if 9 has another answer besides 3. Ask: “What other number can be multiplied by itself to equal 9?” [-3] Read the yellow box in the Learn and Connect section to explain why this is and to teach how we usually give our answers when finding square roots. Caution students that when we put a negative with a square root, it must go outside the square root symbol. Have them think about why that might be. [You can’t do -9 because there is no number that can be multiplied by itself to equal -9.]

Exercise | 1-1 Name Circle the correct answer. What number multiplied by itself equals 144?

What number multiplied by itself equals 121?

Which expression is NOT equal to the others?

2×2

6×6

STRUGGLING LEARNERS Students who struggle should be encouraged to use a perfect square reference chart throughout the unit and should work on their understanding of square numbers using visual models such as arrays.

Simplify. 5.

25 =

-4

- 36 =

-6

13.

400 =

10. - 4 =

-2

11.

100 =

12. - 64 =

-8

14. - 9 =

-3

15.

49 =

16. - 121 =

-11

19. -

64 = -8

20. 62 = 36

23.

25 = 5

24. 10 2 = 100

- 16 =

81 =

EARLY FINISHERS

Fill in the missing numbers. 17.

9= 3

18.

21.

1,600 = 40

22. (-8)2 = 64

= 25

Students who finish early should investigate larger perfect squares to add to their perfect square charts.

CHALLENGE AND EXPLORE

Solve. 25. A square garden bed has an area of 36 square feet. Andrew wants to build a rock wall around the edges of the garden.

26. Can a square piece of glass with an area of 144 square inches fit into a square window with a side length of 1 foot? How do you know? Yes. The piece of glass has a side length of

A. What is the length of one wall?

12 inches, which equals 1 foot.

6 feet

B. What is the perimeter of the garden? 24 feet

CH AL L ENGE 27. Michael is trying to find the square root of 45, but he realizes that since it is not a perfect square, the answer will not be a whole number. How can Michael estimate the square root of 45?

For problem 27, encourage students to think about where the square root of 45 falls on a number line and what numbers it is close to.

Since 45 is close to 49 (a perfect square), he can estimate that 45 is close to 7 (the square root of 49) but a little less, since 45 is a little less than 49. Estimate: 6.7

Lighthouse Math

Level H

Chapter 1

Exercise 1

Have students work with a partner on the problems in the Apply section and then go over them as a class to ensure understanding. Next, have students work independently on problems 1-24 in the Exercise section, reminding them to pay attention to when there are negative signs in front of roots. Review the problems together. For problem 21, tell students that we can break down larger numbers to help us solve: 1,600 = (16 x 100) = 4 × 10 = 40. Work together to solve problems 25 and 26, reminding students that to find the area of a square, we multiply its side length by itself.

ACTIVITY Perfect Square Reference Chart: The purpose of this activity is to have students investigate the perfect squares up to 144 and lay them out in a chart to use as a reference during the school year. Have students create a chart of the perfect squares up to 144. Have them write 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, and so on, each on a different line. Then, next to each perfect square, have them write 1 = 1, 4 = 2, 9 = 3, and so on. Have students color code or decorate the chart.

COMMON ERRORS Students may confuse finding the square root with squaring. Students may divide by 2 to find a square root. Students may omit a negative in the answer to a negative square root.

ASSESS Check numbers 3, 4, 12, 18, and 22 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 1-2 1-2 | Cube Roots

PREREQUISITE SKILLS

Objective and Learning Goals

43 = 4 × 4 × 4 = 64

DAI LY REVI EW

y Students will understand that a cube root is the inverse of a cube and will be able to simplify cube roots of perfect cubes.

Solve.

SPIRAL REVIEW

23 = 2 × 2 × 2 = 8

49 =

53 = 5 × 5 × 5 = 125

81 =

Solve. 1.

Vocabulary

100 =

L E A R N A ND C O NNE C T

y Cubing - multiplying a number by itself and then by itself again; notated with the exponent 3 y Cube root - the inverse of cubing y Inverse operations - mathematical operations that undo one another, such as cubing and taking the cube root

A cube is what you get when you multiply a number by itself 3 three times. The cube root ( ) is the inverse, or opposite, of a cube. It tells you what number was multiplied to make the cube.

A box in the shape of a cube can hold a volume of 8 cubic feet. What is the length of one edge of the box?

To find the cube root, ask: “What number multiplied by itself and by itself again equals this number?” 3 3

64 = 4 because 4 × 4 × 4 = 64 1,000 = 10 because 10 × 10 × 10 = 1,000

The cube root of a negative number will be negative. 3 -27 = -3 because -3 × -3 × -3 = -27 3 -1 = -1 because -1 × -1 × -1 = -1

8=2

Each edge of the box is 2 feet long.

A P P LY

PRE-LESSON WARM-UP

5 × 5 × 5 = [125] Next, write on the board: 125 = ★ × ★ × ★

Tell students that the stars all have the same value. Ask them what the value of the star is and how they know. If students say to divide 125 by 3, ask them what strategy they are using to solve [working backwards] and if they used it correctly here. [No, because we are multiplying the stars, not adding them.] Tell students that they used the first problem to find the value of the star [5] and that today, we will be learning what this strategy is called. Guiding Questions: 1. What is it called when we multiply a number by itself and then by itself again? [cubing] 2. How can we find the value of a missing number if we are given the final answer? [By working backwards]

Write the following problem on the board for students to solve:

Choose the correct answer. 3

6×6

6×6×6

6×3

7×7×7

14 × 7

7×7

7×3

Which can be used to find 216?

Which can be used to find 343?

Match the perfect cubes with their cube roots. 3.

125

343

729

216

1,000

512

Vocabulary Cubing - multiplying a number by itself and then by itself again; notated with the exponent 3 Cube root - the inverse of cubing Inverse operations - mathematical operations that undo one another, such as cubing and taking the cube root

Level H

Chapter 1

Lesson 2

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Read the introductory paragraphs of the Learn and Connect section to review what a cube is and establish the idea that a cube root is the inverse operation of cubing. Go through the examples given using the question “What number multiplied by itself and then by itself again equals this number?” Emphasize that a negative can go under a cube root, and it will give a negative answer when simplified. Go through the examples. Finally, read through the box word problem on the right to apply the concept of cube roots to a real-life situation.

APPLY AND DEVELOP SKILLS (Practice) Have students work in pairs on the problems in the Apply section. Review the problems as a class to ensure that students got the correct answers and understand cube rooting. Then, have students work independently on problems 1-19 in the Exercise section. Review the answers, and then work on problems 20-21 together.

Lighthouse MATH Level H | Teacher's Guide

Exercise | 1-2 Name Circle the correct answer. 1. 2.

10 × 10 × 10

10 × 3

100 × 10

5×5×5

25 × 5

10 × 5

Which expression CANNOT be used to find 1,000? Which expression CANNOT be used to find 125?

STRUGGLING LEARNERS Student who struggle should be encouraged to use a perfect cube reference chart throughout the unit.

Simplify. 3.

-8

-512 = 1

1= 512 =

10.

- 729 =

-64 =

-4

125 =

11.

-9

27 =

216 =

EARLY FINISHERS Students who finish early should investigate larger perfect cubes to add to their perfect cube charts.

Fill in the missing numbers. 12.

16.

-27 = -3

13.

= 729

14.

1,000 = 10

17. (-4)3 = -64

18.

-512 = -8

15. 63 = 216

125 = 5

19.

CHALLENGE AND EXPLORE

Use cube roots to solve. 20. Ann wants to buy a container to fill with beads. She estimates that she has a volume of about 25 cubic inches of beads. She finds a container in the shape of a cube with side lengths all equal to 3 inches. Will the container be big enough for her beads? Explain.

21. A storage compartment is in the shape of cube. If the storage compartment can hold a total volume of 216 cubic feet, what is the length of one wall of the storage compartment?

Yes. The container has a volume of 33, which is 27

The storage compartment has a wall with a length

cubic inches, which is more than 25 cubic inches.

of 6 feet because 216 = 6. © Lighthouse Curriculum. Copying strictly prohibited.

CH AL L ENGE 22. There are two boxes in the shape of cubes. The first has a volume of 512 cubic centimeters, and the second has double the volume of the first. A. What is the side length of the first box? 8 centimeters B. What is the side length of the second box? Hint: Use a perfect cube you know to estimate the length. 10.1 cm C. Explain why the side length of the second cube is not double the side length of the first cube, even though its volume is double.

For problem 24, first have students find the volume of the second box. After students answer question a, have them think about question b. Remind them of the hint given - to use a perfect cube close to the volume of the box. When thinking about question c, students can be reminded about growth rates. What happens when you multiply by three versus multiplying a number three times? Which grows faster?

Volume increases a lot faster than the length. Increasing the length by one centimeter does not increase the volume by one square centimeter because the side length is cubed to find the volume. When the volume was doubled, the length only increased by about 25%.

Lighthouse Math

Level H

Chapter 1

Exercise 2

Perfect Cube Reference Chart: The purpose of this activity is to have students investigate the perfect cubes up to 1,000 and lay them out in a chart to use as a reference during the school year. Have students create a chart of the perfect squares up to 1,000. Have them write 1 × 1 × 1 = 1, 2 × 2 × 2 = 8, 3 × 3 × 3 = 27, and so on, each on a different line. Then, next to each perfect cube, have them write 1 = 1, 8 = 2, 27 = 3, and so on. Students may add any extra they want, and they may color code or decorate the chart.

COMMON ERRORS Students may confuse finding the cube root with cubing. Students may divide by three to find a cube root. Students may omit a negative in the answer to a negative cube root.

ASSESS Check problems 1, 2, 3, 11, 13, and 14 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

ACTIVITY

Level H | 1-3 1-3 | Decimal Patterns

PREREQUISITE SKILLS

Objective and Learning Goals DAI LY REVI EW

y Students will convert fractions to decimals and note if they repeat or don't. y Students will convert repeating decimals to fractions.

Write the place value of the underlined digit. 1.

0.245

SPIRAL REVIEW

23.6

hundredths

tenths

1 can be written as a decimal using 3 long division: 0.3333 3 1.0000 − 9 10 − 9 10 − 9 10 − 9 1

Remind students that we can always add a decimal and an infinite amount of zeros after a number when we us divide. This is especially helpful when we divide a smaller number by a bigger number. Also remind them to bring the decimal in the dividend straight up above the division symbol so that it is placed in the correct spot in the quotient. Guiding Questions: 1. Where does the decimal go in the answer to a long division problem? [Directly above the decimal in the dividend] 2. What can we do to help us divide a smaller number by a bigger number? [Add a decimal and any amount of zeros needed.] 3. How do we know we are done dividing in long division? [When we have divided every digit in the dividend AND we have a remainder of zero]

Lighthouse MATH Level H | Teacher's Guide

-3

- 27

0.6 can be written as a fraction using an equation:

The long division could continue forever. We will always get a remainder of 1.

1. Write an equation where x equals the decimal.

We use bar notation to show which digits are repeating.

3. Subtract x from both sides of the equation. Remember that x equals the decimal you started with.

x = 0.6 x = 0.6 × 10 × 10 10x = 6.6

2. Multiply both sides of the equation by 10.

1 = 0.3 3

10x = 6.6 − x − x or − 0.6 9x = 6 9x = 6 ÷9 ÷9 6 x= 9

4. Solve for x. 6 2 or 9 3

0.6 can be written as © Lighthouse Curriculum. Copying strictly prohibited.

1 ÷ 8 = [0.125]

hundreds

Some fractions convert to repeating decimals. Repeating decimals can be converted back to a fraction.

4.52 ÷ 4 = [1.13]

812.59

L E A R N A ND C O NNE C T

y Terminating decimal - a decimal with a finite number of decimal places (does not go on forever) y Repeating decimal - a decimal whose digits will repeat endlessly; notated with bar notation y Simplest form - a fraction written in its lowest terms; it is found by dividing both the numerator and denominator by the same factor

Review long division with decimals using the following problems.

1.275 thousandths

Find the cube root.

Vocabulary

PRE-LESSON WARM-UP

0.896

tens

2 . 3

A P P LY Tell whether the fraction is a terminating decimal (T) or a repeating decimal (R). 1.

2 5 T

2. R

1 11 T

3. R

19 4 T

4. R

21 10 T

5. R

5 12 T

6. R

7 8 T

7. R

1 6 T

Vocabulary Terminating decimal - a decimal with a finite number of decimal places (does not go on forever) Repeating decimal - a decimal whose digits will repeat endlessly; notated with bar notation Simplest form - a fraction written in its lowest terms; it is found by dividing both the numerator and denominator by the same factor

Level H

Chapter 1

Lesson 3

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Write the fraction 31 on the board and ask students how we can change this fraction into a decimal. [by dividing 1 by 3]. Write 1 ÷ 3 on the board set up as a long division problem and have students begin dividing. Ask students: What should we do to help us divide this smaller number (1) by this larger number (3)? [add a decimal after the 1 and a zero.]. Prompt students through the steps of long division if needed. Ask students: How do we know when we are done dividing? [When we have divided every digit in the dividend AND we have a remainder of zero] Help students notice that with this problem, we will never get a remainder of zero, and we will keep on adding 3s to the quotient forever. Show them how to write the answer with bar notation as 0.3. Next, tell students that since everything in math has an inverse, we must be able to change a repeating decimal back to a fraction. Tell them that to do this, we are going to use algebra. Using the number 0.6, set up the equation and go through the steps as they are laid out in the Learn and Connect section. In step 2, tell students that we can multiply both sides by any number, but we chose 10 because the repeating decimal is in the tenths place. In step 3, remind students that because we know what x is equal to, we can use that number when subtracting x. Finally, remind students to simplify their final answer.

Exercise | 1-3 Name Use division to change the following fractions into decimals. Use bar notation to show if there are any repeating digits. 1.

1 8

0.125

1 6

0.16

4 5

0.8

5 9

0.5

2 3

0.6

6 12

0.5

STRUGGLING LEARNERS Students who struggle with long division should be given a chart with the steps laid out for them. Students who struggle with changing a repeating decimal to a fraction should be given a chart with the steps laid out for them in an example.

Change the following decimals into fractions. Write in the simplest form. 0.2

2 9

0.5

5 9

11. 0.8

8 9

12. 3.3

1.4

10. 2.1

13. 0.6

2 3

14. 3.7

EARLY FINISHERS

Early finishers should continue looking into which denominators give a repeating decimal always, sometimes, or never. They can make a chart of their findings and include any patterns they discover. The chart can be shared for the benefit of the class.

Use the given information to find patterns. 16. Nick says that since 31 = 0.3, therefore 32 must be 0.6. Do you agree or disagree? Why? I agree with Nick because

I disagree because there is also a 3 in the hundredths place

2 1 is double , and 0.6 is double 3 3

0.3. Also, if you do the long division for

and also a three in the thousandths place and so on…

17. Complete the table, then describe any patterns you see.

2 , you get 0.6. 3

As the fractions go up by 1 in the numerator,

1 11

2 11

3 11

4 11

5 11

6 11

7 11

8 11

9 11

10 11

0.09

0.18

0.27

0.36

0.45

0.54

0.63

0.72

0.81

0.90

3 15. Ben says that 0.3 must be equal to 10 because the 3 is in the tenths place. Do you agree or disagree? Why?

the decimals go up by 0.09. The decimal part is equal to the numerator times 9.

CH AL L ENGE 18. Convert the repeating decimals to fractions. Hint: Instead of multiplying by 10, you will need to multiply by 100. A. 0.28

B. 0.17

C. 0.24

28 99

17 99

24 99

Lighthouse Math

Level H

Chapter 1

Exercise 3

CHALLENGE AND EXPLORE The Challenge problems involve changing repeating decimals to fractions where the repeating decimals occur in the hundredths place as well as the tenths place. Help students through the algebraic steps if needed, reminding them to use 100 instead of 10 when manipulating their equation.

Work together as a class on problems 1 and 2 in the Apply section by dividing the fractions using long division. Next, have students work in pairs or in groups to complete the Apply section. When they finish, they should continue on to problems 1-6 in the Exercise section. Review the answers as a class and make sure that repeating decimals were written with the bar notation. Work on problems 7-9 together, and have students complete 10-17 with a partner. Review the answers, using problems 15-17 as opportunity to discuss the concepts as a class, focusing on what patterns students have seen with repeating decimals.

ACTIVITY Pattern Investigation: The purpose of this activity is to have students figure out which denominators will result in a repeating decimal and find patterns that can help them predict what the repeating decimal will be (or vice versa). Have students work in partners to divide to find the decimal represented by the following fractions: 1 1 1 1 1 1 1 1 1 1 1 1 2 5 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 20, 3 , and 6 . Have them record their results in a clear way so that they can look for patterns as to what types of numbers give a terminating decimal and what types give a repeating decimal. Lead a discussion with the class about how only denominators with only 2s or 5s as prime factors give a terminating decimal, the rest are repeating decimals, and why this is. [Because decimals are based on tens, and 2 × 5 = 10.]

COMMON ERRORS Students may forget the steps to change a repeating decimal to a fraction using algebra. Students may forget to use bar notation when writing repeating decimals.

ASSESS Check problems 3, 4, 5, 10, 11, and 15 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 1-4 1-4 | Rational and Irrational Numbers

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to approximate square roots of non-perfect squares and locate them on a number line. y Students will approximate expressions with irrational numbers and locate them on a number line.

Find the square root.

SPIRAL REVIEW

-5

- 25 =

49 =

100 =

Change the fractions to decimals. Change the decimals to fractions in simplest form. 1.

1 = 0.5 2

1 = 0.25 4

1 = 0.1 9

0.4 =

2 5

1.7 =

1 10

L E A R N A ND C O NNE C T

Vocabulary

Rational numbers can be written in fraction form. This includes all terminating and repeating decimals.

y Rational numbers - numbers that can be written in fraction form, including terminating and repeating decimals y Irrational numbers - numbers that cannot be written in fraction form y Whole number - the counting numbers and 0 y Integer - whole numbers and their opposites

Rational Numbers

- 41 , 3.2, 8 21 , 0.6

Some rational numbers are integers, which are positive and negative numbers without any fractional parts or decimals. Some integers are whole numbers, which are always positive or 0.

Integers

Irrational Numbers

-2, 0, 1, 25, -47

2, π, 47

Whole Numbers

Irrational numbers cannot be written in fraction form. As a decimal, they go on forever with no predictable pattern.

0, 1, 2, 3, 4

Some square or cube roots will give an irrational number.

2 ≈ 1.4142135624…

Some irrational numbers have a special name. Pi (π) is used to calculate the area and circumference of a circle. It starts with 3.1415926536… and continues forever.

8 ≈ 2.8284271247… approximately equal to

A P P LY

Materials

PRE-LESSON WARM-UP Draw a number line on the board and have students draw one on their whiteboards, as well. Place the number zero in the center. Label the left end as -10 and the right end as 10. Ask students: What is the span [or length] of our number line? [20] Tell students to add tick marks along the number line so that we can label only the even numbers. Then, have them place the following numbers on their number line:

-3.5

3 31

-7.1

1 3

9 2

8 45

-5.9

Once students have labeled their number lines, fill in the number line on the board together, discussing strategies for how to place numbers as you go. Guiding Questions: 1. How can we find the value of each space between tick marks on a number line? [Divide the span - in this case, 20 - by the number of spaces: 20 ÷ 10 = 2.] 2. How do we know where to put a fraction or a decimal on a number line? [Figure out which two whole numbers it is between, and then figure out which it is closer to and how close.]

Lighthouse MATH Level H | Teacher's Guide

Put each number below in the box that describes it most specifically.

y Whiteboards

1. 4. 7.

0.5 1 3 4 5

-6

-2.7

4 5

1 3

3.6

-2.7

Integers

-6

11. π

12.

13. 3.6

14. -1

15. -4

10.

Rational numbers

0.5

-1

Irrational numbers

-4 7

Whole numbers

Vocabulary Rational numbers - numbers that can be written in fraction form, including terminating and repeating decimals Irrational numbers - numbers that cannot be written in fraction form Whole numbers - the counting numbers; always either positive or 0 Integers - whole numbers and their opposites

Level H

Chapter 1

Lesson 4

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Use the Learn and Connect section to introduce the new vocabulary for this chapter When discussing the terms, use the chart to give examples of each. Discuss how rational numbers include any positive or negative numbers that can be written in fraction form, even if they are often encountered as just a whole number or in decimal form. For example, repeating decimals can all be written in fraction form, so they fall into this category, and the number 5 can be written as 51 , so it also falls into this category. Numbers with no fractional parts are called integers. The term whole numbers is limited to positive integers and zero, sometimes also called “the counting numbers.” Irrational numbers will have decimals that go on forever and have no predictable pattern. Discuss rational numbers that students may have encountered before, such as pi, and new ones that they will be working with, such as square roots and cube roots of non-perfect squares and cubes. Inform students that today, we will focus on classifying, or sorting, numbers into the correct categories.

Exercise | 1-4 Name Write the term that most specifically describes the given number: whole number, integer, rational number or irrational number. whole number

-3

integer

-7

integer

whole number

4.5

rational number

7.25

rational number

irrational number

10.

π 2

STRUGGLING LEARNERS

whole number

Provide struggling learners with a notecard with all the vocabulary terms, their definitions, and examples written out. They may also benefit from a chart like the one in the Learn and Connect section.

whole number irrational number

Draw lines to all the terms that can describe each number. 11. 2π

12. - 16

13. 16 15. -8.5

3 4

21.

Students who finish early can create riddles for a friend to describe a number based on its category and characteristics. For example: “I am an integer less than 1 but more than -10. I am divisible by 3, but I am not 3 and not even.” [-9]

18. -2.75 20.

whole number

22. -6

integer

23. 2.3

EARLY FINISHERS

11 3

16. 3.1416

irrational number

17. -12 19.

14.

rational number

24.

Answer the questions about the rational and irrational numbers. 26. You are part of a team designing a new city park that will have a circular fountain and a square garden. Your job is to determine which measurements are rational and which are irrational so the construction team can plan accurately.

I disagree. The square root of 9 is equal

A. The circumference of the fountain is 8π.

to 3, so this number is a whole number.

B. The area of the square garden is 62.

CHALLENGE AND EXPLORE © Lighthouse Curriculum. Copying strictly prohibited.

25. Alex says that the number 9 should be categorized as an irrational number because it has a square root symbol. Do you agree or disagree with Alex? Explain.

irrational

rational

CH AL L ENGE 27. Euler's number is an irrational number represented by the letter e. It is used when calculating continuous growth of something. e ≈ 2.718 Tom is measuring the growth of a plant. The plant is 2 inches tall on day 1, and each day, it grows by a factor of e. How tall will it be on day 3? Day 1: 2 inches

Lighthouse Math

Day 2: 2 inches × e =

5.436 in

Level H

Chapter 1

Day 3:

5.436 in

Exercise 4

×e=

Read the Challenge problem together as a class to introduce another irrational number - Euler’s number (pronounced like “oilers”). Help students understand that to calculate the height of the plant, we need to use the height from the day before multiplied by Euler’s number.

14.775 in

Have students work in pairs on questions 1-15 in the Apply section. When finished, discuss as a class to ensure understanding. Next, students should work independently on questions 1-24 in the Exercise section. Before they begin, make sure students understand that for questions 11-24, they may need to draw more than one line from each number to the answer choices. Review the answers as a class, and then work together on questions 25 and 26.

ACTIVITY Number Sort: The purpose of this activity is to get students up and moving while simultaneously sorting numbers into their correct categories. Give students sticky notes with a number on them (be sure to include numbers from each category, with repeats if the numbers fall into more than one category). Post the names of the different categories on each classroom wall. Students may work together to place their number in the correct category. The catch is, if they see that their number was already placed in that category, they have to find another category that it also falls into to put their sticky note in. Guide the discussion so that students can see that some numbers fall into one category, while others fall into two or three!

COMMON ERRORS Students may categorize repeating decimals as irrational numbers because they go on forever. Students may not realize that numbers can fall into more than one category.

ASSESS Check the odd-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 1-5 1-5 | Understanding Irrational Numbers

PREREQUISITE SKILLS

Objective and Learning Goals

4.2

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 SPIRAL REVIEW

-1

-1 3

-7.4

DAI LY REVI EW

Place the following numbers in the correct spot on the numberline 1 3

2.75

4.2

2.75

-7.4

32 5

100 =

Write R if the number is rational and I if it is irrational. 1.

64 =

π=

25 =

L E A R N A ND C O NNE C T

Vocabulary y Irrational numbers - numbers that cannot be written in fraction form y Approximation - a good estimate for a number that can be used in its place y Perfect square/cube - a number whose square root/cube root is an integer

George has a square picture frame with an area of 50 square inches. Each side equals 50 inches. Since 50 is an irrational number, George cannot determine the exact length of each side. Instead, he can make an approximation. 1. He finds the two perfect squares that 50 lies between.

2. He finds the square root of both perfect squares.

50 is between 49 and 64.

49 = 7

3. He makes an approximation. 50 is between 7 and 8, but it is closer to 7. 50 ≈ 7.1

64 = 8 49 50

Each side of frame is about 7.1 inches.

PRE-LESSON WARM-UP Draw a number line on the board as shown: -2

-1

Ask a student to come to the board and draw a dot at the spot that shows 2. Ask, “Was that easy or hard to do?” Next, ask a student to come to the board and draw a dot at 4.5. Ask the student how they knew where to put it [halfway between 4 and 5 because 0.5 is half] and ask, “Was this easier or harder than placing the 2? Why?”. Next, ask another student to place 6.15 on the number line. Ask them how they knew where to put it [a little after the 6] and if it was easy or hard to do. Next, draw the following number line: 6

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7

Ask the same student to place the number 6.15 now and if it was easier or harder to do on this number line. How did they know where to put it? [halfway between 6.1 and 6.2]. Guiding Questions: 1. If a number isn’t written on a number line, does that mean it is not there? [No, it means it is in between two numbers.] 2. How can we find a number’s spot on a number line if it is not written on the number line? [Find the two numbers that it falls between and decide which it is closer to.]

Lighthouse MATH Level H | Teacher's Guide

A P P LY Use perfect squares to approximate the value of each irrational number. Then place them on the number line. 1.

80 64 and

20 is between 16 and 25 .

80 is between

20 is between 4 and 5 .

80 is between 8 and 9 .

20 is about 4.5 .

80 is about 8.9 .

81 .

-9

-8

-7

-6

-4

-3

-2

-1

36 .

- 33 is about -5.8. 20

-5

25 and -

- 33 is between -

- 33 is between -5 and -6 .

- 33 -10

- 33

80 6

Vocabulary Irrational numbers - numbers that cannot be written in fraction form Approximation - a number that is close to another number and is used in its place

Level H

Chapter 1

Lesson 5

Perfect square - a number whose square root is an integer

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Use the Learn and Connect scenario to introduce the idea of approximating numbers. Tell students that because irrational numbers have an infinite amount of decimal places, we need to use approximations to represent their values in the real world or on a number line. Ask students when we might use irrational numbers in the real world. [Area of a circle, circumference of a circle] Explain that in this example, George knows that his picture frame is a square, and he knows the area because the label told him the area, but he wants to be sure the frame will fit in the spot on his wall, so he needs to find its length. Because it is a square, he can take its square root to find the length. However, the result is an irrational number. Go through the steps of approximating the square root of a non-perfect square with students. Encourage them to use a chart with the perfect squares and their square roots in order to go about this process.

Exercise | 1-5 Name Use perfect squares to approximate the value of the irrational numbers to one decimal place. 1.

10 3.2

- 75 -8.7

- 140 -11.8

118 10.9

40 6.3

STRUGGLING LEARNERS

17 4.1

62 7.9

Students who struggle should be provided with a more detailed number line showing smaller decimal places to help with approximations, as well as a number line with the perfect squares written above the integer that they simplify to.

- 53 -7.3

Plot the numbers on the number line below. 9.

10. -3π

2π

-2

-3 -10

11. π

-9

-8

-7

-6

12. -2π

13. -π

-5

-4

14.

-2

-1

15. π2

-3

π 2

EARLY FINISHERS

Students who finish early should put the numbers in 1 - 8 on a number line.

Read the problems. Then, answer the questions. Approximate your answer to two decimal places. 16. An architect is designing a square pool with an area of 150 square feet.

B. Between which two whole numbers does the side length of the pool lie? 12

150 .

and

CHALLENGE AND EXPLORE

A. To calculate the length of the balloon arch in terms of pi: πd 2

B. Approximate the length of the balloon arch.

C. Approximate the length of the pool.

15.71 feet

A. The length of one side of the pool is the square root of

17. A balloon arch is being constructed for a party. The balloon arch is in the shape of a semicircle with a diameter of 10 feet.

12.25 feet

CH AL L ENGE 18. The Pythagorean theorem is a mathematical formula for calculating the longest side of a right triangle. If you know the other two sides of the triangle, you can find the third.

a2 + b2 = c2

The Challenge problem introduces the Pythagorean theorem, which students will learn in a future chapter. Help students understand that to solve this equation, they will need to take the square root of a non-perfect square and approximate the answer.

Given that side a = 5 and side b = 7, approximate the length of side c. 7

c = 74 or ≈ 8.6

Lighthouse Math

Level H

Chapter 1

Exercise 5

Work on the problems in the Apply section together as a class, filling in the blanks as you go. Have students work in pairs to complete problems 1-8 in the Exercise section. Review the answers together. Next, have students work in pairs to complete problems 9-15. Draw the number line on the board and go over it together as class. Finally, have students work on problems 16 - 17 in pairs. Check to ensure understanding.

ACTIVITY Human Number Line: The purpose of this activity is to have students approximate square roots of non-perfect squares in an interactive way. Call up students one by one to be a human on a number line. Call out a number, such as “Square root of 15.” Ask for two volunteers to be the humans on the number line that stand as the boundary for 15. Help students figure out that one should be 9 and the other 16 because 15 is between the these two perfect squares. Then, have a third student come up and stand somewhere between the other two, approximating 15. They should stand closer to 16 than 9. Finally, have the class determine together the exact value of 9, 16, and finally, the approximate value of 15. Repeat with different numbers and different students.

COMMON ERRORS Students may think that all square roots can be approximated to a number ending in .5 instead of trying to be more accurate to show which whole numbers they are closer to.

ASSESS Check problems 2, 6, and 9-15 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 1-6 1-6 | Comparing and Ordering Rational and Irrational Numbers

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to place rational and irrational numbers on a number line and compare them.

Choose the correct symbol: >, <, or =.

4. SPIRAL REVIEW

1.02

3.3 2.5 7 3

-0.346

1 3

1 4

-0.345

6.03

6.3

1,720

1,270

79 8.9

3π

15 3.9

9.4

A number line can be used to compare and order rational and irrational numbers. 6.23, 6 31 ,

36,

Irrational numbers do not have an exact location on the number line. Instead, an approximation is used.

48, and 2π

• 6.23 is between the numbers 6.2 and 6.25. • 6 31 is a little to the right of 6.3 because 6 31 is 6.3. • 36 is exactly 6 because 36 equals 6. • 48 is a little to the left of 7 because it is a little less than 49, which equals 7. • 2π is a little to the left of 6.3 because 2 × 3.14 is 6.28. 6.23 2π

36 6

6.1

6.2

6 31

6.3

48 6.4

6.5

6.6

6.7

6.8

6.9

A P P LY Compare the numbers with >, <, or =. © Lighthouse Curriculum. Copying strictly prohibited.

1.8

L E A R N A ND C O NNE C T

PRE-LESSON WARM-UP

2.9

1.002

5.3

y Rational numbers - numbers that can be written in fraction form, including terminating and repeating decimals y Irrational numbers - numbers that cannot be written in fraction form

2.1

7.45 >

Approximate the value of the irrational numbers to one decimal place.

Vocabulary

Tell students that you are going to play a game called “Who Is Closer?” They need to determine if the number called out is closer to 2 or 3. When you call out the number, they should hold up two fingers if they think it is closer to 2 or three fingers if they think it is closer to 3. Call out numbers such as:

7.5

1 6

3.14159

< > =

3.3

100

11. We can find the exact location of 16 on the number line.

12.

1 is an irrational number. 3

1 7 10 2

- 5

3.3 <

10.5

3.1

Write whether each statement is true or false. 10. We can find the exact location of 2 on the number line. false

true

false

Vocabulary Rational numbers - numbers that can be written in fraction form, including terminating and repeating decimals Irrational numbers - numbers that cannot be written in fraction form

Level H

Chapter 1

Lesson 6

Lighthouse Math

2 23 4 5

8 Guiding Questions: 1. What is the middle point between 2 and 3 on the number line? [2.5] 2. How can we find the location of a square root of a non-perfect square on a number line? [Find an approximation using the two perfect squares that it falls between.]

Lighthouse MATH Level H | Teacher's Guide

INTRODUCE THE LESSON (Learn and Connect) Tell students that in this lesson, we will be working with both rational numbers and irrational numbers, and review how these two terms are defined. Ask students for examples of each type of number. Tell students that we will be plotting the numbers on a number line and comparing numbers. Remind them that to plot an irrational number on a number line, we will need to use an approximation. Draw a number line on the board with the two end points marked as 6 and 7. Ask students to help you plot the following numbers on the number line: 6.23, 6 1 3 , 36, 48, and 2π. Allow students to come up to the board to plot where they think the number should go, but be sure they provide an explanation for their placement. Allow students who disagree with another student’s placement to change it and provide their own justification. Once all the numbers are placed on the number line, go through the Learn and Connect section to check their answers and show them how to justify the placement of a number on a number line.

Exercise | 1-6 Name Place the numbers on the given number line. 4

10 3 , 27 3

5, 4, 8, π,

3 48, 2π, 36, 6 , 35, 5.3 4

2.5

3.5

35 36

5.5

8.1 8

STRUGGLING LEARNERS

10 3

27 3

5.3

2 100 19 , 72, 8.1, 9 , 8.7323, 3 11 2

8.7323

8.5

4 3

6.5

100 11

19 2

9.5

Students who struggle may use number lines that are labeled in more detail to aid in the placement of numbers.

48 7

93 10

EARLY FINISHERS

Write the correct symbol: >, <, or =. 4. 7.

- 2

3π

1 13. 3 17 16. 10

> <

-5.4

11.

0.3

14. -3.81

1.75

1 17. 5 5

9 4

-1 <

10. - 16

-1

-5.3

12. 2π

-22.22

-3.18

15. -0.8

5.2

9 18. 10

-222.2

4 <

Students who finish early can plot the numbers from problems 4-18 on a number line. They will need to figure out the smallest and largest numbers in order to know what the ends of their number line should be.

4.1 >

3 4

0.99

CHALLENGE AND EXPLORE 20. Amy divides her stamp collection into 6 different books. She wants each book to hold the same number of stamps. If she has 200 stamps, how many stamps will go in each book? What might be a better way for Amy to divide her stamps?

The first table. The length of its sides is the square

33.333…, If she divides them among 5 books, she

root of 40, which must be greater than 6.

will have exactly 40 in each book.

Solve. 19. Jimmy builds two square tables. The first one needs to fill a space with an area of 40 square feet. The second one has side lengths of 6 feet. Which table is longer? Explain how you got your answer.

CH AL L ENGE 21. Daniel and Tom compare their running routes. Daniel ran once around a trail that borders a circular park with a diameter of 2 miles. Tom ran along a straight road for 3.3 miles. Who ran longer, and by about how much?

For problem 21, students need to remember that a circle with a diameter of 2 has a circumference that is actually more than 3 times 2. If needed, remind students that C = 2πr and that diameter is 2r. Once students find the circumference, they can clearly see that Daniel ran more, and they can subtract to find the approximate difference.

Daniel’s distance is 2π, or about 6.28 miles. Tom only ran 3.3 miles. Therefore, Daniel ran almost 3 more miles than Tom.

Lighthouse Math

Level H

Chapter 1

Exercise 6

Students should work in pairs on the problems in the Apply section. Review the problems together as a class to check for understanding. Students should continue to work in pairs on problems 1-3 in the Exercise section. Review these number lines as a class before allowing students to move on to problems 4-20. Discuss their answers to problems 19-20 as a class.

ACTIVITY Human Number Line: The purpose of this activity is to have students compare and order numbers in an interactive way. Divide students into teams. Each team is given the same set of notecards with one number on each (be sure to use a mixture of rational and irrational numbers), one notecard per person. Then, they work together to line up from least to greatest. The team to line up correctly first wins. Once both teams are done, have them compare their human number lines and explain their reasoning for placing each number where they did.

COMMON ERRORS Students may order negative numbers incorrectly. Students may ignore a square root symbol when comparing numbers.

ASSESS Check problems 3, 4, 5, 6, and 19 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 1-7 1-7 | Review

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will recall all concepts to do with rational numbers and irrational numbers, including classifying numbers, ordering and comparing numbers, square roots and cube roots, changing fractions to decimals, and changing repeating decimals to fractions.

Square Root

Cube Root

Types of Numbers

a2 = 100 a × a = 100 100 = a

b3 = 64 b × b × b = 64 3 64 = b

Can be written as a fraction

a = 10

b=4

Rational Numbers

- 51 , 1.3, 9 21 , .6 Integers

Fractions as Repeating Decimals

Do not have fractional parts

Repeating Decimals as Fractions

-4, 0, 5, 31, -17

0.1 2 3

PRE-LESSON WARM-UP

Jimmy and five of his friends go to a restaurant to celebrate his birthday. At the end, they receive a bill for $100.45. At first, the friends decide to split the bill so that each person will pay the same amount. Then, one friend says that it would be nice to pay for the birthday boy’s meal. Besides the fact that it is a nice thing to do, why else would splitting the bill the second way be a good idea?

[When splitting the first way, 100.45 is divided by 6, and the answer is a repeating decimal, 16.7416. When splitting among the five friends, 100.45 is divided by 5, and the answer is 20.09, a number that has just two decimal places, which makes sense for money.] Guiding Questions: 1. When splitting the bill the first way, do you get a rational number or an irrational number? How do you know? [A rational number. The result is a repeating decimal, which can be written as a fraction.] 2. What do we do in real life when a bill or anything to do with money has more than two decimal places? [We round to the nearest hundredth.]

Lighthouse MATH Level H | Teacher's Guide

Read the following scenario to the class and have them discuss:

0.6666 3 2.0000 − 18 20 − 18 20 − 18 20 − 18 2

Whole Numbers Are only positive or 0

x = 0.1

= 0.6

0, 1, 2, 3, 4

× 10 × 10 10x = 1.1 − x (or 0.1) 9x = 1 ÷9 ÷9

−x

Irrational Numbers

1 9 1 0.1 = 9 x=

Go on forever with no pattern

3, π, 45

A P P LY Write if the number is rational or irrational. 1.

6.23

2 3

rational

1 9

6.3

1 5

rational

Chapter 1

1 3

terminating

Lesson 7

repeating

10. terminating

3 11. 1,000

6.0003

Level H

8. repeating

rational

6π irrational

7. rational

irrational

Write if the fraction represents a terminating or repeating decimal.

3 2 terminating

100 12. 6 repeating

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Use the table in the Learn and Connect section to review the concepts from this chapter. First, discuss how taking the square root is the inverse of squaring and how we can use perfect squares we know to find square roots. Ask students what kind of number we simplify to when taking the square root of a perfect square [integer/whole number] and what kind of number we simplify to when taking the square root of a non-perfect square [irrational number]. Next, repeat this process with the box on cube roots. Then, move on to converting fractions to decimals using long division. Review the algebraic process for converting repeating decimals to fractions, and ask students if they recall any of the patterns they found when doing this process [Fractions with a denominator of 9 will always be a repeating decimal with the numerator of the fraction repeating.]. Finally, categorize all these numbers using the Types of Numbers chart. Be sure to review the differences between whole numbers and integers and rational numbers and irrational numbers.

Exercise | 1-7 Name Simplify. 1.

-4

- 16

10.

11.

12.

STRUGGLING LEARNERS

144

-10

216

-6

- 100

- 36

125

Provide struggling learners with anchor charts to help them remember the steps for long division and the steps for changing repeating decimals to fractions. Also provide a chart outlining the different classifications for the numbers.

Change the fraction to a decimal. 13.

5 3

1.6

14.

7 2

3.5

15.

1 6

0.16

16.

1 5

0.2

17.

4 12

0.3

18.

24 5

4.8

19. 2

3 10

2.3

20.

2 9

0.2

21.

4 11

0.36

22.

1 4

0.25

EARLY FINISHERS Early finishers can come up with their own riddles and trade with a partner to guess the number.

Change the decimal to a fraction. 2

23. 3.2

27. 1.6

24. 0.7

7 9

28. 1.4

25. 2.3

29. 2.1

26. 9.5

30. 0.8

8 9

CHALLENGE AND EXPLORE

Put the rational and irrational numbers on the number line.

18 2.4

8 2

2.4

4.45

3.74

9 2.5

3.5

4.45

4.5 © Lighthouse Curriculum. Copying strictly prohibited.

31.

CH AL L ENGE 32. Bobby wants to cover a square countertop with contact paper. He knows the countertop has an area of 900 square centimeters. 30 cm

A. What is the length of one side of the countertop?

B. The contact paper comes in a roll that is 10 centimeters wide. Explain how Bobby will need to cut the contact paper to fit the countertop so that the whole thing is covered.

For the Challenge problem, help students use the square root of 9 to find the square root of 900. Then, have them sketch out what it might look like to cover the counter top with contact paper that doesn’t fit the full width of the surface.

Bobby will need to cut three pieces that are each 30 cm long. By laying them side by side, the three 10 cm-wide pieces will create a width of 30 cm.

Lighthouse Math

Level H

Chapter 1

Exercise 7

Have students complete the problems in the Apply section independently, and then review the answers together. Make sure to address any misconceptions along the way. Next, model one problem from each section on the Exercise page, and then have students work independently on the rest of the problems. Review all the problems when complete.

ACTIVITY Clothesline Math: The purpose of this activity is to order numbers on a number line in an interactive way. Prepare several notecards with different types of numbers (integers, decimals, fractions, repeating decimals, square roots, cube roots, irrational numbers) between -10 and 10. Fasten a string from one side the room to the other, and put a notecard with the number 0 in the middle, -10 on the left end, and 10 on the right end. Have students work in groups to place their numbers in the correct place on the number line, encouraging discussion. Be sure to remind them about spacing and using approximations for irrational numbers. If any numbers are equivalent, provide clips so that they can be placed in the same spot.

COMMON ERRORS Students may forget the steps for changing a repeating decimal to a fraction. Students may classify a repeating decimal as an irrational number.

ASSESS Check the even-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Chapter 2

In Chapter 2, we will learn about

Exponents and Scientific Notation Exponents and scientific notation are methods of writing very large and very small numbers in more concise ways. • We will learn rules for multiplying and dividing numbers with exponents. • We will learn what negative and zero exponents mean. • We will learn how to follow the order of operations when there are exponents and roots. • We will learn how to write numbers in scientific notation and how to compare and order them. • We will learn how to multiply and divide numbers in scientific notation. • We will learn how to add and subtract numbers in scientific notation. • We will solve real-world problems with numbers written in scientific notation.

Level H | 2-0 Chapter 2 | Skill Checklist

Skill 1: Expression Vocabulary Parts of Expressions

Match each part of the expression to the words that describe it.

term

12t + 6 coefficient

operation

Lighthouse MATH Level H | Teacher's Guide

9b 6

Variable

Operation

Constant

4g 7h -h 4r

8a -a2 3b 3a3

3a2

Write the place of the underlined digit.

tth hth m

5 7 8 3 2 . 9 4

5,007,895,230 Ten millions

Millionths

Ten Thousandths

Hundred Thousandths

Tenths

Hundredths

Thousandths

Tens

Ones

Decimal Point

Hundreds

M Hth TTh Th

Millions

Thousands

Ten Thousands

Hundred Thousands

Place Value

104.00753

0.520893

Ten thousandths

hundredths

Write the value of the underlined digit. 4.

56.2069 0.006

985,540,000.9

7.058328

900,000,000

0.000008

I can name the place and value of a digit in a number.

out 6 correct

Skill 3: Multiplying and Dividing by Multiples of Ten 36 × 10 = 360 36 × 100 = 3,600 36 ÷ 10 = 3.6 36 ÷ 100 = 0.36

y Use the graphic in the gray box to show students that terms are units that are being added or subtracted in an expression. y Remind students that terms can have a variable or be just a number or a constant. y Remind students that the number in front of the variable, the number it is being multiplied by, is called the coefficient. y Have students complete problems 1-4 and review answers and strategies as a class.

Coefficient

Skill 2: Place Value

Solve. 1.

24 × 1,000 = 24,000

8.596 × 100 =

859.6

32.007 × 10 = 320.07

4,830 ÷ 10 =

483

7,820 ÷ 1,000 =

7.82

9.26 ÷ 100 = 0.0926

I can multiply and divide by multiples of ten.

out 6 correct

Skill 1: Expression Vocabulary

I can correctly label the parts of expressions.

out 4 correct

Students will review skills needed for Chapter 2: y Expression vocabulary y Place value y Multiplying and dividing by multiples of ten y Rounding numbers y Reciprocals y Order of operations

Term

Circle the like terms. 3.

Objective and Learning Goals

8a − 9b + 6

constant

variable

7y − 3

Level H

Chapter 2

Skill Checklist

Skill 2: Place Value y Review the concept of place value with students, reminding them that it is based on tens. y Use the example in the gray box to talk about the value of each digit in the number 57,832.94. y Have students complete problems 1-6 and review answers and strategies as a class.

Lighthouse Math

Skill 3: Multiplying and Dividing by Multiples of Ten y Building off of the previous section about place value, review with students what happens when you multiply by ten and what happens when you divide by ten. Remind students that multiplying by ten shifts the digits one place to the left, and dividing by ten shifts the digits one place to the right. y Using the example in the gray box, review with students what happens when you multiply or divide by another multiple, or power, of ten. y Have students complete problems 1-6 and review answers and strategies as a class.

Name

Skill 4: Rounding Numbers Round to the nearest hundredth.

Round each number to the given place value. 1.

1.382

2.458

1.382

2.458

1.38

2.46

Round 8,652,258.25 to the nearest hundred thousand: 8,700,000 45.7

Round 45.6592 to the nearest tenth:

Round 0.00782 to the nearest ten thousandth:

Round 1,521.036 to the nearest hundred:

Round 789,526.0268 to the nearest whole number:

0.0078

1,500

789,526

I can round numbers to a given place value.

out 5 correct

Skill 5: Reciprocals Circle the pairs of fractions that are reciprocals of each other. Draw an x over pairs that are not reciprocals. 3 7

7 3

Fraction

Reciprocal

1 and 2 2

1 and 17 9

5 1 and 1 6 5

Write the reciprocal of each number. 4 19

19 4

1 7

1 4

2 3

3 2 © Lighthouse Curriculum. Copying strictly prohibited.

I can write the reciprocal of a number.

out 7 correct

Skill 6: Order of Operations Parentheses Exponents Multiplication/Division

A/S

Addition/Subtraction

Use PEMDAS to solve. 1. 3.

out 4 correct

Lighthouse Math

3+2×7=

5 × 4 − (3 + 1) ÷ 2 =

7 + 82 × 5 =

3 + 3(9 − 3) = 3

327 45

I can solve using the order of operations.

Level H

Chapter 2

Skill Checklist

Skill 4: Rounding Numbers

Skill 5: Reciprocals

Skill 6: Order of Operations

y Review how, when we are asked to round to a given digit, we look to the digit to the right to tell us how to round. y Use the two examples in the gray box to show that when the digit to the right of the hundredths place is 5 or greater, we round up. When the digit to the right is less than 5, we round down, keeping the digit the same. y Have students complete problems 1-5 and review answers and strategies as a class.

y Using the example in the gray box, explain to students that a number’s reciprocal is found by switching its numerator and denominator. y Remind students that if a number does not have a denominator, the denominator is actually 1. y Have students complete problems 1-7 and review answers and strategies as a class.

y Review the accepted order of operations as listed in the gray box. y Remind students that multiplication and division, as well as addition and subtraction, are completed from left to right in an expression. y Have students complete problems 1-4 and review answers and strategies as a class.

Lighthouse MATH Level H | Teacher's Guide

P E M/D

Level H | 2-1 2-1 | Multiplication with Exponents

PREREQUISITE SKILLS

y Students will be able to apply the product of powers property to simplify expressions involving multiplication of terms with the same base.

DAI LY REVI EW

Objective and Learning Goals

Solve.

SPIRAL REVIEW

(7 − 4) + (-5)2 × 2 =

(4 − -3)2 + -14 ÷ 7 =

25 ÷ -5 + -4 × 3 =

-41

(9 × 2) ÷ 6 − -10 + 2 =

47 21

Compare the numbers with >,<, or =.

Vocabulary

1 3 7 5 8 3

L E A R N A ND C O NNE C T

y Term - a constant, or a variable with a coefficient y Exponent - a number that indicates how many times a base is multiplied by itself y Base - a number multiplied by itself in a power y Products of powers property - when multiplying two expressions that have the same base, keep the base the same and add the exponents y Factors - numbers that are multiplied

When multiplying terms with exponents that have the same base, keep the base the same and add the exponents. This is called the product of powers property. 32 × 34

32 + 4

Product of Powers Property am × an = am + n

Same base

(3 × 3) × (3 × 3 × 3 × 3)

If you expand the terms and then simplify, you get the same answer. 32 × 34 The product of powers property only applies to terms with the same base. If the base is different, the terms cannot be combined.

23 × 29 × 32 simplifies to 212 × 32, NOT 614.

A P P LY

y Colored pencils

PRE-LESSON WARM-UP Write the following problems on the board:

3 × 3 × 3 × 3 = [34 = 243] 5 × 5 × 5 = [53 = 125] 2 × 2 × 2 × 2 × 2 = [25 = 32] Ask students to rewrite each product using exponential notation and then calculate the result. After reviewing answers, tell students that these terms are called powers, and they serve as shorthand for repeated multiplication. Identify the base and the exponent in each example. Remind students that the base is the number being multiplied, and the exponent tells you how many times to multiply it. After reviewing, invite students to explore exponent patterns and share what they notice. Explain that today’s lesson will build on this understanding of exponents by introducing new rules for solving exponents. Guiding Questions: 1. When there is no exponent on a number, what exponent is hiding? [The exponent 1] 2. How many times greater than 53 is 54? [five times greater]

For each expression, circle the terms with the same base.

Materials

122 × 123

x2 × y3 × x4

v3 × y3 × v3

67 × 97

k3 × k3 × k8

r5 × r6 × r7

x24 × x9

42 × 4 × 49

Simplify the expression. 9.

92 × 93 =

14 10. 83 × 84 × 87 = 8

10 13. 35 × 33 × 32 = 3

18. 2 × 2 = 5

15 12. 57 × 58 = 5

17 15. 46 × 411 = 4

12 16. 62 × 6 × 69 = 6

14. 97 × 32 × 94 = 9 × 3

17. 5 × 5 × 6 = 5 × 6 2

11. 63 × 23 × 25 = 6 × 2 2

19. 7 × 7 × 7 = 7 10

20. 109 × 105 × 104 = 10

Vocabulary Term - a constant, or a variable with a coefficient Exponent - a number that indicates how many times a base is multiplied by itself Base - a number multiplied by itself in a power

Level H

Chapter 2

Products of powers property - when multiplying two expressions that have the same base, keep the base the same and add the exponents Factors - numbers that are multiplied

Lesson 1

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Begin by writing 32 × 34 on the board. Ask students to expand this expression so that they can see all the factors being multiplied. [3 × 3 × 3 × 3 × 3 × 3] Ask: How many threes are being multiplied in total? Can you rewrite this expression as one power of 3? [6 threes; 36] Next, ask the same question for the expression 43 × 45 [4 × 4 × 4 × 4 × 4 × 4 × 4 × 4; 8 fours; 48] Ask students what they notice about the exponents in each expression. [If you add the exponents in the original expression, you get the exponent in the new expression.] Introduce the product of powers property. This property tells us how to simplify expressions where terms with the same base are being multiplied. Tell students that when we multiply powers with the same base, we can simplify by adding the exponents together. Write the product of powers property formula on the board as it is written in the Learn and Connect section. Next, introduce an example that includes different bases: 23 × 29 × 32. Have students help you write out the expanded factors and then circle the twos as one group and the threes as another. Ask why the twos and threes cannot be combined into one term [because the bases are different, so they have a different value]. Finally, guide students to simplify the expression as 212 × 32. Emphasize that the product of powers property can only be applied when the bases are the same.

APPLY AND DEVELOP SKILLS (Practice) Have students complete problems 1-8 in the Apply section individually. When Lighthouse MATH Level H | Teacher's Guide

Exercise | 2-1 Name Simplify the expression. x8 × x2 =

x10

p3 × p2 × p5 =

p10

x5 × y2 × x3 =

x8 × y2

a ×a =

a14

g ×g ×g =

g30

a ×b ×b =

a6 × b9

b10 × b =

b11

w4 × w2 × w15 =

t5 × t3 × z7 =

t8 × z7

10. c9 × c8 =

w21 r

11. r8 × r10 × r15 =

STRUGGLING LEARNERS

12. r12 × s3 × s10 × r8 =

13. 2 × x × x =

22 × x6

16. 48 × y2 × 45 =

4 ×y

17. 93 × x2 × 910 × x14 =

9 ×x

10 8 6 18. a3 × b6 × 23 × 27 × a5 = 2 × a × b

19. 65 × a9 × a8 =

65 × a17

5 10 20. 45 × d7 × d × d2 = 4 × d

21. h7 × 74 × g6 × g10 × g2 = 7 × h × g

3 15 22. 10 × w × w = 10 × w

14. 8 × 8 × s × s =

810 × s16

11 18 11 15. 5 × v × y × 5 × v8 = 5 × v × y

312 × f23

23. f × 3 × f × f = 8

Provide students with a formula sheet that introduces the basic rules of exponents. Throughout the chapter, students will add new properties to this sheet, creating a personalized reference guide. For this lesson, have them include the product of powers property. When working through problems, encourage students to expand expressions into repeated factors so they can visually confirm how the exponents combine. For more challenging problems, guide them to circle terms with different bases using different colors to clearly separate what can and cannot be combined.

r ×s 20

8 7 15 24. 2 × p × s × p × s = 2 × p × s

Find the missing value. 25. 56 × 514 = 5 28. 4

34. 3

26. 94 × x3 × x 6 = 94 × x9 29. 5 × 5

×4 =4 4

31. 79 × 7 5

35. 8

× 38 = 313

× b10

× 59 × d10 = 516 × d18

30. 9 × p 12 × 93 × p2 = 97 × p14

×a =5 ×a 8

32. 77 × 74 × b10 = 7

= 715

27. 57 × d 4

33. 68 × a10 × a6 × a 36. 43 × y

× 85 × c10 = 817 × c10

= 68 × a25

× 47 × y = 410 × y10

Solve. 37. Sara simplifies the expression 62 × 34: “Since 6 × 3 = 18, then 62 × 34 = 186.”

38. Eric simplifies the expression x4 • x8: “Since 4 × 8 = 32, then x4 • x8 = x32.” Is Eric correct? Explain why or why not.

No. She multiplied the bases first, then added the

No. He multiplied the exponents, but the correct

exponents. Exponents can only be added when the

rule is to add the exponents when multiplying

bases are the same.

powers with the same base.

EARLY FINISHERS © Lighthouse Curriculum. Copying strictly prohibited.

Is Sara correct? Explain why or why not.

CH AL L ENGE 39. If x = 15, find three different values for m and n in the expression below:

40. 93 × 32 can also equal 38. Can you figure out why? 9 is the same as 3², so 9³ is 3² × 3² × 3² which is 36.

am • an = ax m = 11 and n = 4, m = 10 and n = 5, m = 8 and n = 7

Lighthouse Math

Level H

36 × 32 = 38

Chapter 2

Exercise 1

they finish, discuss which problems cannot be simplified, which problems will be simplified to one term, and which to two terms. Next, work as a class on problems 9-12, writing out the expanded form of each expression as you go. Have students work with a partner to complete problems 13-20 and review the answers to ensure understanding. Next, transition to independent practice, assigning problems 1-36 in the Exercise section. Once finished, review these as a class, clarifying any areas of confusion. To conclude, walk through problems 37-38 together. Invite students to share their reasoning, discuss where errors may have occurred, and then work collaboratively to correct the mistakes and arrive at the correct solutions.

ACTIVITY Exponent Team Relay: The purpose of this activity is to have students practice simplifying exponent multiplication problems. Divide the class into three teams of equal size and have students line up so that everyone will get a turn at the board. A representative from each team will step up to the board as the teacher reads an exponent multiplication problem aloud, such as 23 × 24. Each student will then write the simplified version of the expression. The fastest correct response earns 3 points, the second fastest earns 2 points, and the final team earns 1 point. After each round, students rotate so the next person in line is ready, and play continues until every student has had a chance to solve. At the end of the relay, the team with the highest score wins. To increase the challenge and keep students engaged, add in a few “bonus rounds” with trickier problems that involve multiple bases or more complex expressions.

Have students revisit problems 14 and 15 in the Apply section and rewrite the problems using 3 as the base for problem 14 and 2 as the base for problem 15. If they need some direction, ask them how they can rewrite the number 9 with a base of 3 [32], then tell them to rewrite it in expanded form with the given exponents.

CHALLENGE AND EXPLORE For problems 39 and 40, place students in small groups to discuss their strategies. Then, bring the class together to collect possible values for problem 39 on the board. For problem 40, have groups share their reasoning out loud, emphasizing not just the answer but the different ways they approached the problem.

COMMON ERRORS Students may multiply powers instead of adding them. Students may combine terms with different bases.

ASSESS Check questions 25-36 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

Level H | 2-2 2-2 | Division with Exponents

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to apply the quotient of powers property to simplify expressions involving division of terms with the same base.

Simplify.

SPIRAL REVIEW

Guiding Questions: 1. How do we simplify when we have repeated factors in a fraction? [Cancel out any factors that are same in the top and the bottom.] 2. What do you notice about the exponents when the expression is written in exponential notation, and how does this relate to the answer we got? [They are one apart, and we were left with one 5 at the end.]

34 × 36 × 35 =

315

5 ×5×5 =

511

55 52

Same base

21 = 35 42 = 60

3 5

7 10

50 = 65 35 = 56

10 13 5 8

11 3 25 × 53 × 26 = 2 × 5

3 9 73 × b5 × b4 = 7 × b

6 9 a ×6 ×6 ×a = 6 ×a

14 11 910 × 45 × 46 × 94 = 9 × 4

55 − 2

Quotient of Powers Property am = am - n an

If you expand the terms and then simplify, you get the same answer. The quotients of powers property only applies to terms with the same base. If the base is different, the terms cannot be simplified.

PRE-LESSON WARM-UP

55 52

5×5×5×5×5 5×5

66 ÷ 32 ÷ 62 simplifies to 66 ÷ 32, NOT 24.

A P P LY © Lighthouse Curriculum. Copying strictly prohibited.

Have students work with a partner to simplify the expression, and then have a student come up to the board to show how they solved it. If the student begins crossing off pairs of fives, introduce the term cancel out by asking, “What happens when we multiply by 5 and then divide by 5?” Next ask, “What is one set of 55 equal to?” [1] and “What happens when we multiply by 1?” [Nothing] Explain that this is the reason we can “cancel out” pairs of numbers in the top and bottom of a fraction that are the same. Once all the pairs of five cancel out, there should be one five left in the top without a partner. Therefore the answer is 5. Ask students what the answer would be if two fives were left in the top. [25 because 5 × 5 = 25] Finally, ask students how we can rewrite the original 6 expression using exponential notation. [ 555 ]

2 3

When dividing terms with exponents that have the same base, keep the base the same and subtract the exponents. This is called the quotient of powers property.

y Notecards y Colored pencils

5×5×5×5×5×5 5 × 5 × 5 × 5 × 5 [5]

2 3

18 = 27 24 = 36

L E A R N A ND C O NNE C T

Materials

Write the following on the board and ask students to simplify:

1 2

Simplify the expression.

Vocabulary y Quotient of powers property - when dividing expressions with the same base, subtract the exponents to simplify

3 4

12 = 16 9 = 18

Simplify the expression. 1.

4 37 =3 33

109 = 106

103

45 = 4

2 610 =6 68

911 = 97

710 = 75

3 5 65 × 38 =6 ×3 62 × 33

7 11 57 × 313 =5 ×3 32

3 3 7 95 × 77 × 810 =9 ×7 ×8 92 × 74 × 83

10.

9 9 89 × 714 = 8 ×7 75

11.

3 2 35 × 27 = 3 ×2 32 × 2 5

12.

3 2 57 × 38 × 23 = 5×3 ×2 56 × 35 × 2

Vocabulary Quotient of powers property - when dividing expressions with the same base, subtract the exponents to simplify

Level H

Chapter 2

Lesson 2

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Begin by writing the example 552 on the board and ask students to expand the expression. [ 5 × 5 ×5 ×5 ×5 5 × 5 ]. Next, ask students to simplify by canceling out pairs of fives. [5 × 5 × 5] Have them rewrite the expression using exponential notation. [53] Ask them what they notice about the exponents in the original expression and the new one. [5 − 2 = 3] Tell them that similar to how we add the exponents when we multiply powers with the same base, we can subtract the exponents when dividing powers with the same base. Write the formula for the quotient of powers property on the board as it is written in the Learn and Connect section. Emphasize that this property is only applied when the 6 bases are the same. If the bases are different, such as 632 , the exponents cannot be combined because the bases don’t have the same value. Tell students that a common error is simplifying this expression to 24. This expression actually cannot be simplified without first finding the values of the numerator and denominator. 5

APPLY AND DEVELOP SKILLS (Practice) Begin by working together as a class on problems 1, 4, 7, and 10 in the Apply section to scaffold understanding and model the process of simplifying expressions with exponents. Next, have students work with a partner to Lighthouse MATH Level H | Teacher's Guide

Exercise | 2-2 Name Simplify the expression.

x8 = x5 a11 = a4

9 b10 = b b 10 5 3x 10. = 3x x5 2 6 22a8 2 a 13. = a2

b8 = 53b5

a5 × b4 a3 × b4 = a2 4d9 4d5 11. = d4 3 4 65y7 6 y 14. 2 3 = 6 y

16.

w12 = w11 y5 = y3

17.

Support students by having them continue to build their reference sheet, adding the quotient of powers property as a new entry to guide their practice. Encourage students to write expressions in expanded form and physically cross out the pairs of factors to see the simplification process step by step. Provide colored pencils or markers so they can circle and group terms with the same base, making it visually clear which factors can be simplified and which cannot.

a10 × b3 × c7 a × b × c4 = a9 × b2 × c3 58 × t8 55 × t4 12. = 53 × t4 6 4 616 × g4 15. = 6 ×g 610

b3 53

STRUGGLING LEARNERS

2 t5 × s4 = t×s t4 × s2 11 14 11 3 f ×g = f ×g g11

5 3 45 × a6 = 4 ×a a3

18.

45 × a9 × b11 42 × a3 × b8 = 43 × a6 × b3

Find the missing value. x12 = x2 x 10 38 22. 4 = 34 3 a3b 11 25. = a3b6 b5

52d6 = 52d 4 d2 7 3 y9 23. 2 3 = 7y6 7 y 815 × a8 26. = 815 × a2 a6

19.

104 × t12 = 102 × t 7 102 × t5 54 × p5 = 5 3 × p5 24. 5 99 × a10 × b 5 27. = 93 × a6 × b3 9 6 × a4 × b2

20.

21.

Solve. 28. Danny simplifies the expression “Since 6 divided by 2 is 3,

46 : 42

29. Lisa simplifies the expression

4 = 43.” 42 6

58 : 53

EARLY FINISHERS

“Since 5 divided by 5 is 1, the answer is 15.”

Is Danny correct? Explain why or why not.

Is Lisa correct? Explain why or why not.

instead of subtracting them. When dividing powers

No. She divided first, when she should have just

with the same base, you subtract the exponents.

subtracted the exponents.

CH AL L ENGE 30. If x = 9, find three different values for m and n in the expression below: am = ax an

31. A student says that to simplify the expression 10x9 2x5 , first divide 10 by 2 to get 5, and then subtract the exponents for a final answer of 5x4. Is the student correct? Explain. Yes, the student is correct. They correctly said to divide

Answers vary: m = 12 and n = 3, m = 15 and n = 6,

the coefficients (10 ÷ 2 = 5) and subtract the exponents

m = 20 and n = 11

(9 – 5 = 4) using the quotient of powers property.

Lighthouse Math

Level H

Chapter 2

Exercise 2

complete the remaining problems, encouraging them to discuss their reasoning as they solve. Bring the class back together to review the answers, taking time to walk through any problems that students found especially challenging. Next, have students work independently on problems 1–29 in the Exercise section. Review the answers, allowing students to share their reasoning and compare strategies. Conclude with a class discussion of problems 28 and 29, modeling the correct solving process step by step.

No. Danny is incorrect. He divided the exponents

Have students return to problems 1 and 2 in the Apply section. Have them write the reciprocal of the fraction given and then simplify. Ask them to compare their original answers to their new answers and look for similarities and differences.

CHALLENGE AND EXPLORE Have students pair up to work through problems 30 and 31. Afterward, bring the class back together to discuss. For problem 30, record different combinations that students found on the board. When discussing problem 31, emphasize that this problem was completed correctly. Because 10 and 2 are coefficients without exponents, they can be divided to simplify.

ACTIVITY Exponent Division Match-Up: The goal of this activity is for students to strengthen their understanding of how to simplify division problems with exponents and recognize that the quotient of powers property only applies when the bases are the same. Prepare one deck of notecards with division 5 4 9 problems written with exponents (such as 772 , 332 , or 226 ) and another deck with their simplified equivalents (733, 32, 23). Also mix in a few problems that use different bases (for example, 252 ), so students are reminded that not all expressions can be simplified with this property. Working in pairs or small groups, students sort through the decks by matching each division problem with its correct answer and placing any “different base” problems in a separate pile. After the groups finish, bring the class together to review the matches, discuss common errors, and emphasize that the quotient of powers property only applies when the bases are the same.

COMMON ERRORS Students may divide exponents instead of subtracting them. Students may divide unlike bases.

ASSESS Check problems 19-27 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

Level H | 2-3 2-3 | Negative and Zero Exponents

PREREQUISITE SKILLS

Objective and Learning Goals DAI LY REVI EW

y Students will be able to define and apply zero and negative exponents to simplify expressions.

Find the reciprocal. 1. 5.

SPIRAL REVIEW

2 = 5 10 = 11

5 2

1 = 8 5 = 6

8 6 5

95 = 94

11 10

3 4

4 = 3 1 = 9

7 6

6 = 7 2 = 13

13 2

Simplify the expression. 1.

48 = 45

x11 = x6

a14 = a6

Vocabulary y Zero power property - a value raised to the zero power equals 1 y Negative power property - a value raised to a negative exponent is the reciprocal

L E A R N A ND C O NNE C T 50 = 1

Any base raised to the power of zero always equals 1. Follow the pattern in the examples on the right.

x0 = 1

÷5

51 = 5

÷x

Zero Power Property

÷x

a0 = 1

x1 = x ÷5

52 = 25

x2 = x • x

A negative exponent shows the reciprocal of the number that should be used. A negative exponent flips a whole number into a fraction with a positive exponent in the denominator.

PRE-LESSON WARM-UP Have students pair up and choose a base number, then write out its powers from 1 through 5 and their values. An example is shown below.

Once they have completed the sequence, ask each pair to look for patterns in how the values change as the exponents increase. Then, have them write a short statement explaining exponents in their own words, as if they were teaching the concept to someone who has never seen exponents before. After pairs have written their statements, invite different groups to share their explanations with the class. Conclude by discussing similarities across the various statements. Then, draw a general conclusion with the class about how exponents represent repeated multiplication and how the exponent tells us how many times the base is used as a factor. Guiding Questions: 1. What do exponents represent in mathematics? [They are a way to show repeated multiplication of the base number.] 2. What patterns do you notice as the exponent increases? [Each time the exponent goes up by 1, the value is multiplied by the base again.]

Lighthouse MATH Level H | Teacher's Guide

1 x-4

Negative Power Property

x4 1

a-m =

1 am

Some expressions require multiple steps to simplify. 43 • 45 • x-4 48 • x2

21 = 2 22 = 4 23 = 8 24 = 16 25 = 32

A negative exponent in a denominator of a fraction will flip to become a positive exponent in the numerator.

1 33

3-3

43 • 45 • x-4 48 • x2

48 • x-4 48 • x2

40 • x-6

Multiply.

Divide.

Apply zero and negative power properties.

1 x6

A P P LY Circle the correct answer. 1.

What is the value of 70? 0

Simplify (x2y3)0.

2 3

Which expression equals 1? 01

101

Evaluate 4 • 30.

1000

Simplify 2x0. 2

What is x0 + x0? 0

Which expression does not equal 1?

5a0

Evaluate (3b2)0. 1

Vocabulary Zero power property - a value raised to the zero power equals 1 Negative power property - a value raised to a negative exponent turns it into its reciprocal

Level H

Chapter 2

Lesson 3

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Begin by using a student’s work from the pre-lesson warm-up or creating the chart given in the pre-lesson warm-up on the board. With input from the students, draw arrows showing that as you move from the bottom value up the chart, each value is divided by the base. Next, ask students to use this pattern to figure out what (base)0 is equal to. Add (base)0 = 1 to the chart above (base)1. Use another student’s chart that shows values for a different base to confirm this pattern, and demonstrate that the value of any base to the 0 power is 1. Write the formula for the zero power property on the board. Then, ask students to think about what negative exponents might mean given this same pattern. With student input, add (base)-1 and its value above (base)0. Continue adding values with negative exponents above to reinforce the pattern. Discuss how negative exponents give the reciprocal of what the same positive exponent 1 (the reciprocal of 52 = 25). Write the formula would give. For example, 5-2 = 25 for the negative power property on the board. Next, give students an example where the negative exponent is in the bottom, such as x1-4 , and show them how to flip it over and make the exponent positive. Finally, use the example given in the Learn and Connect section to show how we can use all the rules we have learned so far to simplify an expression with many powers.

Exercise | 2-3 Name Simplify each expression and write the answer with positive exponents. 1.

3-2 =

1 = 4-7

9 = 910 4

13. 9-11 = 17.

1 = x-3

21.

y6 = y11

1 32

1 = 2-4

52 = 59

32.

38.

7 = 73

1 y5

22. c-5 =

3-1 × 36 = z4 × z-6

1 74

35z2

16. a-5 =

1 y8

20.

1 = b-4

24.

c5 = c12

19. y-8 =

1 c5

23.

2 1 = d d-2

27.

a2 = a5

20 1 = a a-20

8-10 × 86 = 8-4 × 80

2-4 × b5 = 27 × b9

28. b-14 =

94 b2

1 211b4

8-3 × d8 = 85 × d-7

Have students continue building their reference sheet for the properties of exponents by adding the zero power property and the negative power property. Encourage them to refer back to this sheet as they solve problems so they have a visual reminder of the rules. If needed, have them create the chart from the pre-lesson warm-up with a zero power and negative powers added to reinforce the pattern that occurs.

1 a5

b4 1 c7 1 b14 1 52b5

31. 40 × 5-2 × b-9 × b4 = 34.

36. 93 × 9-7 × 98 × b-2 = 39.

1 a3

y4 612

30. 6-5 × y4 × 6-7 × y0 =

1 72

1 x7

33. b

35. 50 × b-2 × b3 =

4 = 410

15.

x = x8

1 a3

12.

1 43

15 1 = 10 10-15

18.

a-1 × a-2 = a0

11.

1 a5

1 = 6-8

5-8 =

6 1 = 9 9-6

29. a-2 × a0 × a-3 =

14.

26.

1 58

1 57

1 911

11 10-11 = 10

63 = 65

10. 7-4 =

25. a-10 = a10

1 96

STRUGGLING LEARNERS

1 62

d15 88

EARLY FINISHERS

37. q0 × q-1 × 53 × 5-5 × 5-4 = 56 × q 40.

94 × 8-8 × x-3 × z-2 9 × 812 × x10 = 95 × 84 × x7 × z-6

42. Victor simplifies

2-3 : 2-1

“Since -2 + 0 = -2, then 4-2 × 40 = -16.”

“Since -3 − -1 = -2, then the answer is 2-2.”

Is Justin correct? Explain why or why not.

Is Victor correct? Explain why or why not.

No. Negative exponents indicate the reciprocal, not

Yes. When subtracting a negative number, you add,

1 1 a negative value. The correct answer is 42 or 16.

1 so the correct answer is 2 or 22 . -2

CH AL L ENGE 43. Simplify. Show your work.

Lighthouse Math

x2 × y-9 × z³ x-6 × y-3 × z² x-2 × y4 × z0

Level H

x8 × y-6 × z x-2 × y4 × z0

Chapter 2

x10 × y-10 × z1

Exercise 3

x10z y10

Solve. 41. Justin simplifies the expression 4-2 × 40:

Have students create a problem for a partner to solve that uses all four of the properties taught in the chapter: power of product, power of a quotient, zero power, and negative power properties.

CHALLENGE AND EXPLORE Have students work on problem 43 in pairs. Then, bring the class together to walk through each step of the simplification process. As students share, record the steps on the board, discuss reasoning, and confirm the correct answer.

Have students work in pairs to solve problems 1-8 in the Apply section. Then, review the problems as a class, pausing to discuss any that students found challenging. Next, have students complete problems 1-42 in the Exercise section independently. Conclude with a whole-class review. Select several problems from 29-40 for student volunteers to show their work on the board. Prompt them to explain both the steps they took and which properties were applied.

ACTIVITY Exponent Properties Stations: In this activity, students will rotate through different stations to practice the exponent properties they have learned so far. Set up one station for each property—product of powers, quotient of powers, zero exponents, and negative exponents—with a set of practice problems at each. Students work individually or with a partner to solve the problems, recording their solutions on a sheet as they go. To extend learning, include an optional challenge station with problems that combine multiple properties, requiring students to decide which rules to apply and in what order. After all groups have rotated through the stations, review as a class to highlight strategies, address common mistakes, and emphasize how the properties connect to one another.

COMMON ERRORS Students may think negative exponents mean the value of the power is negative. Students mean think zero powers equal zero.

ASSESS Check problems 21, 29, 32, and 38 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 2-4 2-4 | Order of Operations with Exponents and Roots

PREREQUISITE SKILLS

y Students will be able to use the order of operations correctly when simplifying expression that have square roots and complex fractions.

DAI LY REVI EW

Objective and Learning Goals

Solve.

SPIRAL REVIEW

(10 − 4) + (-1)2 × 2 =

45 ÷ -5 + -3 × 3 = 2

-36

(6 × 2) ÷ 3 − -6 =

b-6 =

1 b6

c =

1 c4

Rewrite with positive exponents. 1. 5.

4-3 =

1 43

8 =

1 85

-5

2. 6.

1 = b-2 1 -8 = a

(4 − 8)2 + -14 ÷ 7 =

-4

4. 8.

1 = 8-3 1 -5 = 6

83 65

Vocabulary y Order of operations - a standard order to follow when calculating with more than one operation

L E A R N A ND C O NNE C T When simplifying expressions, we follow the order of operations. Parentheses Exponents (and Roots) Multiplication Division Addition Subtraction

Materials

Square roots and fractions can act like parentheses. That means that everything under the square root symbol and in the numerator or denominator of a fraction should be simplified first.

y Colored pencils

1. Simplify what is under the square root. 2. Take the square root. 3. Continue the calculations outside the root.

9 + 8(5) − 6

PRE-LESSON WARM-UP 8−

y 24 − 8(3 + 6) [-48] y 60 12 + 3(8) [29] y 16 + 12 [16]

Have students explain their calculations and guide the discussion towards the need to follow the order of operations. Use the third expression to remind students about square roots. Let students know that today, they are going to learn about how the order of operations works with more complex expressions. Guiding Questions: 1. What should we do first when simplifying an expression with many operations? [Follow PEMDAS - Parentheses, Exponents, Multiplication/Division, Addition/ Subtraction] 2. How do we find a square root? [Ask: What number is multiplied by itself to get the number under the root?] 3. What operation do fractions tell us to do? [Divide]

Write the following expressions on the board and ask students to simplify them.

9 + 40 − 6 49 − 6 7−6=1

1. Simplify the top. 2. Simplify the bottom. 3. Divide the top by the bottom. 4. Continue the calculations in order.

5 + 30 13 − 6

8−

35 7

8−5=3

A P P LY Add parentheses around the parts of the expression that should be grouped. Then simplify. 1.

( 24 + 9 ) 33 = = ( 10 − (-1) ) 11

(8(3) + 12 ) = 36 =

(62 − 17) 45 ( 3(3)) = 9 =

(7 + 16 − (-2)) =

(-14 + -6) -20 ( 2(5) ) = 10 =

-2

( 2(4) − 7 ) =

25 =

1 =

Vocabulary Order of operations - a standard order to follow when calculating with more than one operation. PEMDAS - an acronym to help remember the order of operations: parentheses, exponents, multiplication, division, addition, subtraction

Level H

Chapter 2

Lesson 4

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Write out the order of operations on the board and add “and roots” after the word Exponents. Explain that because roots are the inverse of exponents, these two operations fall in the same category. Next, explain that radical signs (the name of the symbol used for roots) and fraction bars act as grouping symbols, much like parentheses. Everything under a square root and everything in the numerator or denominator should be grouped and evaluated before continuing with order of operations. Write the first example from the Learn and Connect section on the board, ( 9 + 8(5) − 6). Have a student draw parentheses under the radical to emphasize that the expression under it is one group. Walk the class through each step, asking students to calculate as you go. Then, walk through the second example of the fraction being subtracted from 8. Have students help you group the numerator and denominator with parentheses and walk through the steps to simplify. Emphasize that the last step is to subtract the simplified fraction from 8. Caution them that they must do this in the correct order to get the correct value.

APPLY AND DEVELOP SKILLS (Practice) Begin by working through problems 1 and 2 in the Apply section together as a class by first inserting parentheses around the parts that need to be grouped and then following PEMDAS. After modeling these problems, have students Lighthouse MATH Level H | Teacher's Guide

Exercise | 2-4 Name Simplify the expression. 1.

12 + 8(2) 5 − (-2)

18 − 30 −1 42

6(10) +5 8 − 12

-1.75

-1(9 − 25) + 3

10.

-8

11.

20 − 4 × (4 + 6) 32 − 10

Have students keep a simple reference sheet at their desks that lists PEMDAS with roots included next to exponents and a reminder to group anything under a radical sign and in the numerator and denominator of fractions. Have them draw in the parentheses to show this for each problem. Encourage students to write PEMDAS next to each problem and check off each step as they complete it.

64 + (32 − 2)

101 − 72 + 12 -7 − 6

STRUGGLING LEARNERS

121 × 52 − 3 11

272

5(15) + 2(12.5)

4.5

52 + 19 − 22 4−9

-10

10 − (-39) − 2.5

12.

(1 − 4)² + (6 − 2)²

-40

EARLY FINISHERS Write an expression and simplify.

20 10

(105 * 68) - 740 = 6,400 80 meters

The ratio is 2:1

24 - 4 18 - 2(4)

14. The length of Oak Field is 105 meters, and its width is 68 meters. Elm Field is a square, and its area is 740 square meters less than the area of Oak Field. What is the length of Elm Field?

CH AL L ENGE Simplify. 15.

16.

(12 − 8)2 + (45 − 48)2 (5 − 4(8))³ (4)2 + (-3)2 = (5 − 32)³

Lighthouse Math

-5 16 + 9 25 = = -27³ -19,683 19,683

72 + 9 + 1 14 − (-13) 3

Level H

81 +1 27

Chapter 2

9 +1=4 3

Exercise 4

complete problems 3–6 independently. Review the solutions as a class, walking through any problems that students found challenging. Next, have students work with a partner to solve problems 1–12 in the Exercise section. Bring the class back together to review, and invite volunteers to show their work on the board for problems 9–12. Conclude the lesson by working as a class to solve problems 13 and 14. First, create an expression based on the scenario, and then have students simplify it using the skills learned in this chapter: by grouping first, then following the order of operations.

Ask students to write a clean, stepby-step solution for problems 9-12 in the Exercise section. Have them write a brief explanation after each line, for example, “simplified inside parentheses” or “evaluated exponent.” When finished, have them swap papers with a partner and verify each step.

CHALLENGE AND EXPLORE Recommend to students that they use a colored pencil to draw in parentheses around each part that should be grouped for Challenge problems 15 and 16. Also recommend that they rewrite the problem each time they complete a step in the simplifying process to be sure they don’t mess up the order of terms.

ACTIVITY Order of Operations Relay: The purpose of this activity is to have students practice using the order of operations in a fun and interactive way. Divide students into teams. Write an expression on the board that involves many steps and following the order of operations to simplify. Students will stand in line and simplify the expression, one student per step. They will go to the back of the line once they’ve completed each step. Important rules to note: a student may choose to complete the next step or correct an error that was made previously. Each time, the student needs to write out the entirety of the expression that remains after they have completed their step so that the next person is able to look at the problem as a whole and make a decision about what to do next. The first team to finish with the correct answer wins.

COMMON ERRORS Students may simplify without first grouping and then following the order of operations.

ASSESS Check problems 9-12 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

13. Mrs. Smith’s class has 24 students. Mrs. Baker’s class has 18 students. On Monday, there are 4 students missing from Mrs. Smith’s class, and two times as many missing from Mrs. Baker’s class. What is the ratio of students in Mrs. Smith’s class to Mrs. Baker’s class on Monday? Hint: Write the ratio as a fraction and simplify.

Level H | 2-5 2-5 | Converting Numbers to Scientiﬁc Notation

PREREQUISITE SKILLS

y Students will be able to express large and small numbers in scientific notation and convert numbers in scientific notation to standard form.

DAI LY REVI EW

Objective and Learning Goals

SPIRAL REVIEW

4 + 12(2) − 3 + 2

1 + 52 − 3 -5 3 − 16

8(4) − 2 3−8

0.003 × 100 =

6 × 100 = 600

0.3

-6

- (-4) − (-8) + 10

Break down the power of ten into repeated multiplication.

Express the repeated multiplication as an exponent.

Large Number

50,000 = 5 × 10,000

50,000 = 5 × 10 × 10 × 10 × 10

50,000 = 5 × 104

Small Number

1 0.0013 = 1.3 × 1,000

1 1 1 0.0013 = 1.3 × 10 × 10 × 10

0.0013 = 1.3 × 10-3

The number in front of the power of 10, called the coefficient, must be between 1 and 10. You can only have one digit before the decimal point.

Correct: 1.5 × 108 2 × 10-4

Incorrect: 15 × 107 0.2 × 10-3

Converting from Scientific Notation to Standard Form

20,000

The decimal point moves to the right the same number of places as the positive exponent.

1.2 × 10-6

0.0000012

The decimal point moves to the left the same number of places as the negative exponent.

A P P LY Determine if the number is written in the correct scientific notation. Fix any incorrect values. Original Number

Scientific Notation

Correct?

85,000

8.5 × 104

Yes

0.0025

25 × 10-4

100,000

1 × 105

Yes

0.00009

9 × 105

Fix Incorrect Values

2.5 × 10-3 9 × 10-5

Vocabulary Scientific notation - a way of writing very large or very small numbers using multiplication by powers of ten Coefficient - a number that multiplies a variable, a term or powers of 10

0.2 × 10,000 = 2,000

Rewrite the number as a multiplication problem.

2 × 10

Lighthouse MATH Level H | Teacher's Guide

1.5 × 1000 = 1,500

Converting from Standard Form to Scientific Notation

Guiding Questions: 1. How can we use place value to simplify the expressions? [Place value is based on tens, so each time we increase by 10, we go up a place value.] 2. When can we not use the trick of “add a zero onto the number” when multiplying by ten, and what can we do instead? [When multiplying 10 by a decimal. For example 5.1 × 10 is not 5.10, it’s 51. Each digit shifts over by one place value to the left.]

Scientific notation is a way to rewrite very large or very small numbers using powers of ten.

y Whiteboards y Dry-erase markers

Ask students to tell you what patterns they see and if they could use them to tell you what the answer would be for 5 × 1,000,000. [5,000,000] Next, ask students to rewrite the numbers that 5 is being multiplied or divided by as powers of 10 (using exponential notation). [10 = 101, 100 = 102, 1,000 = 103] Explain that today’s lesson will build on these two skills so that we can learn how to write numbers in a different way.

18 × 100 = 1,800

L E A R N A ND C O NNE C T

Materials

y 5 × 10 [=50] y 5 × 100 [=500] y 5 × 1,000 [=5,000] y 5 ÷ 10 [=0.5] y 5 ÷ 100 [=0.05] y 5 ÷ 1,000 [=0.005]

0.61 × 100 =

y Scientific notation - a method of writing very large or very small numbers using multiplication by powers of ten y Coefficient - a number that multiplies a variable, term, or powers of 10

Write the following expressions on the board for students to simplify:

Simplify.

Vocabulary

PRE-LESSON WARM-UP

Multiply.

Level H

Chapter 2

Lesson 5

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Begin by writing the number 50,000 on the board and asking students how to break it apart into two factors. [5 × 10,000] Next, ask students how to write 10,000 as a power of ten. [104] Point out that this means 50,000 can also be written as 5 × 104. Explain to students that the number 50,000 is now written in scientific notation: a way to write very large or very small numbers using powers of ten. Next, demonstrate the process with a very small number, 0.003. This number may be harder for students to break down into factors. Ask them how to read this number using place value. [three one 1 . thousandths] This means that we can also write this number as 3 × 1,000 1 1 Ask students how we can write 1,000 as a power of 10. [ 103 or 10-3] Tell them that we can write this number in scientific notation as 3 × 10-3. Next, tell them the number being multiplied by the power of ten is called the coefficient and must be between 1 and 10, but cannot be 10. Use the second example in the Learn and Connect section to exemplify this. Explain that 0.0013 can be 1 because the first digit after the zeros is 1, and it is in the written as 1.3 × 1,000 thousandths place. Next, use the information in the Learn and Connect section to show students a shortcut for converting numbers from scientific notation to standard form by moving the decimal. Take this opportunity to show them how to use the shortcut to convert from standard form to scientific notation as well. Additionally, reinforce the idea that powers of ten with positive exponents make large numbers, while powers of ten with negative exponents make small numbers.

Exercise | 2-5 Name Rewrite each number in scientific notation. 3,200 =

3.2 × 103

0.005 =

5 × 10

78,000 =

7.8 × 104

10. 0.00041 =

4.1 × 10

13. 6,000,000 =

-3

-4

6 × 106

4.2 × 102

420 =

0.0000003 =

9,100 =

3. 3 × 10

-7

9.1 × 103 7.1 × 10

11. 0.71 =

-1

14. 1,000 =

0.000045 =

0.003 =

4.5 × 10-5

Students who struggle may benefit from using the shortcut method of moving the decimal to write numbers in scientific notation. Have students put their pencil on the decimal (or write a decimal on the far right side of the number if there isn’t one) and then draw arcs under each digit to count how many places it would need to be moved in order to create a number between 1 and 10. Tell students that this is the number that should be used for the exponent on the power of ten. The number they create by doing this is the coefficient.

3 × 10-3 8.9 × 10

12. 8,900,000 =

1 × 103

STRUGGLING LEARNERS

3.5 × 104

35,000 =

8.2 × 10-2

15. 0.082 =

Rewrite each number in standard form. 16. 3 × 103 =

3,000

17. 1.2 × 10 -4 =

0.00012

18. 9.8 × 105 =

19. 7.1 × 104 =

71,000

20. 5 × 106 = 5,000,000

21. 1.5 × 10-3 =

0.0015

22. 6 × 10 =

0.06

23. 4.3 × 10 =

0.43

24. 2.4 × 10 =

24,000

25. 2.5 × 105 =

250,000

26. 3.9 × 10-5 = 0.000039

28. 9 × 10 =

0.009

29. 6.7 × 10 =

-2

-3

-1

6,700

980,000

27. 7.75 × 10-6 = 0.00000775 30. 8 × 104 =

80,000

Solve. 31. A scientist counts about 7,200,000 bacteria in a petri dish. Write this number in scientific notation.

32. A factory produces 4.8 × 107 sheets of paper each month. How many sheets is that in standard form?

EARLY FINISHERS

48,000,000 sheets

7.2 × 106

33. A human hair is about 7 × 10-5 meters thick. What is the thickness in standard form?

34. One raindrop has a volume of about 0.000002 liters. Express this in scientific notation. © Lighthouse Curriculum. Copying strictly prohibited.

2 × 10-6 liters

0.00007 meters

CH AL L ENGE 35. Kevin writes the number 430,000 in scientific notation: 43 × 104 Is Kevin correct? Explain why or why not.

36. Sara writes the number 0.00042 into scientific notation: 4.2 × 10 -3 Is Sara correct? Explain why or why not. No. Sara moved the decimal the wrong number of

No. The coefficient must be a number between 1 and 10. He should have written it as 4.3 × 10 . He

places. She should have moved the decimal four

needed to move the decimal five places, not four.

places, not three. The correct answer is 4.2 × 10 -4.

Lighthouse Math

Level H

Chapter 2

Exercise 5

Students can look at each row in problems 1–30 in the Exercise section and circle the largest number in each row. Afterward, have them write a short explanation about their strategy for doing this in each section.

CHALLENGE AND EXPLORE In Challenge problems 35 and 36, students will need to find the error in the numbers written in scientific notation and explain how the person may have made a mistake.

Have students work in pairs on the table in the Apply section. Then, review these problems as a class, discussing how they knew the incorrectly written numbers were wrong. Next, work through problems 1-3 and 16-18 in the Exercise section together as a class to reinforce understanding and model the process of changing between the two forms of writing numbers. Once these have been completed, have students work in small groups to solve problems 4-15, 19-30, and 31-34. Conclude with a whole-class review, addressing any questions students have and clearing up any misconceptions.

ACTIVITY Scientific Notation Trivia: Students will strengthen their ability to write numbers in correct scientific notation and convert between scientific notation and standard form. Give each student a whiteboard and marker. In the first round, write a number in standard form on the board and have students convert it into scientific notation on their whiteboards. When you say “Show,” all students raise their whiteboards to show their answers. Call on a student who got the answer right to explain how they did it. Check in with students who got the answer wrong to see where they may have gotten confused. Points are awarded for correct answers and deducted for incorrect ones, with several problems given in this round. In the second round, present a number in scientific notation and have students convert it into standard form. The same scoring rules apply. In the final round, write a number in incorrect scientific notation and have students correct it on their boards. After all rounds are complete, points are tallied to determine the student with the highest score.

COMMON ERRORS Students may use coefficients greater than 10. Students may use the incorrect power of 10.

ASSESS Check problems 31-34 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 2-6 2-6 | Comparing Numbers in Scientiﬁc Notation

PREREQUISITE SKILLS

y Students will be able to compare and order numbers expressed in scientific notation and standard form.

DAI LY REVI EW

Objective and Learning Goals

Write the numbers in order from least to greatest.

SPIRAL REVIEW

Materials

0.001, 0.12, 0.0012 0.001, 0.0012, 0.12

1.801, 1.081, 1.81

0.45, 0.405, 0.054 0.054, 0.405, 0.45

0.002, 0.00022, 0.02 0.00022, 0.002, 0.02

0.62, 0.6, 0.602

0.541, 0.451, 0.154 0.154, 0.451, 0.541

0.6, 0.602, 0.62

Convert to scientific notation. -4 1. 0.00015 1.5 × 10

y Notecards

8,000,000,000

8 × 10

4 40,600 4.06 × 10

-2 0.099 9.9 × 10

0.034 3.4 × 10

5,800

Order these numbers from least to greatest: 2.5 × 10-2, 2,050, 2 × 103, 25,000 Strategy One: Convert All Numbers to Standard Form

y 45, 43, 413, 46 [43, 45, 46, 413] y 108, 1015, 104, 102 [102, 104, 108, 1015] y 3-2, 34, 3-1, 30, 3-8 [3-8, 3-2, 3-1, 30, 34]

Strategy Two: Convert All Numbers to Scientific Notation

2.5 × 10-2

2,050

2 × 103

25,000

2.5 × 10-2

2,050

2 × 103

25,000

0.025

2,050

2,000

25,000

2.5 × 10-2

2.05 × 103

2 × 103

2.5 × 104

2.5 × 10-2 < 2 × 103 < 2.05 × 103 < 2.5 × 104

0.025 < 2,000 < 2,050 < 25,000

5.8 × 103

To compare numbers in standard form and scientific notation, convert all of them to the same format. Use place value to compare numbers in standard form. Look at the power of 10 first when comparing numbers in scientific notation.

Write the following sets of numbers on the board and have students work in pairs to order each set of numbers from least to greatest.

Guiding Questions: 1. What does the sign of the exponent tell you about the size of the number? [Positive exponents indicate a large number, and negative exponents indicate a small number.] 2. How can we compare numbers that are not in the same form? [Rewrite them so that they are in the same form but still equivalent to their original value.]

-2

L E A R N A ND C O NNE C T

PRE-LESSON WARM-UP

The numbers ordered from least to greatest: 2.5 × 10 -2, 2 × 103, 2,050, 25,000 © Lighthouse Curriculum. Copying strictly prohibited.

Review as a class and ask students to share how they determined the order without actually calculating the exact values. Guide them to recognize that the exponent itself provides the key information for ordering when the base is the same. Emphasize that comparing powers gives a shortcut to understanding relative size, and explain that today’s lesson will build directly from this idea.

1.081, 1.801, 1.81

A P P LY Compare each value using <, >, or =. <

3.2 × 106

9 × 104

6.3 × 10 -3

6.3 × 10-2

1.5 × 108

7.2 × 10

9.1 × 10

2.4 × 10

2.3 × 107

5.01 × 10 -1

5.01 × 10-1

3.9 × 102

4.2 × 10 -4

4.21 × 10-4

10. 8 × 107

4.6 × 105

-5

-6

Level H

Chapter 2

8.1 × 104

< >

3.9 × 103 7.99 × 107

Lesson 6

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Write the numbers 2 × 10-2, 2,500, 2×103, and 25,000 on the board. Ask students to predict which number they believe is the greatest and which is the least, and have them explain their reasoning. Then introduce the lesson by explaining that today’s focus is on comparing and ordering numbers in scientific notation. Explain that to make it easier to compare the numbers, we must first rewrite them in the same form. Begin by asking students which form they would like to put all the numbers into. Convert all of the numbers into that form, then have them place the numbers in order before converting them back to their original forms. Next, guide students to convert all of the numbers into the other form and order them again, then put them back into their original form. Confirm that the order came out the same for both methods. Emphasize that when ordering numbers in scientific notation, the exponent should be considered first, and if that is the same, then the coefficients are compared. When ordering numbers in standard form, place value is used. Conclude by encouraging students to practice both methods so they can strengthen their understanding and flexibility when comparing numbers in scientific notation.

APPLY AND DEVELOP SKILLS (Practice) Walk through problems 1-4 in the Apply section together as a class to demonstrate how to compare exponents and coefficients for numbers in Lighthouse MATH Level H | Teacher's Guide

Exercise | 2-6 Name Compare each value using <, >, or =. 1.

5.2 × 103

5,000

7 × 102

700

3.4 × 104

8.1 × 10

8,000

2 × 10

200,000

4.5 × 10

9.9 × 105

999,000

6 × 102

1,000

1.2 × 102

1,200

10. 3.75 × 103

3,750

11. 7.4 × 106

740,000

12. 8.8 × 104

88,000

13. 6.01 × 10

601,000

14. 2.5 × 10

25,000

15. 9.1 × 10

910

STRUGGLING LEARNERS

34,000

Students who struggle should choose one form that they are more comfortable converting to and focus only on using that form to order numbers. Once they are comfortable with that method, they can begin to work on getting comfortable with the other method of comparison.

Convert to standard form and then order the numbers from least to greatest. 16. 5.1 × 102; 700; 4.8 × 102

17. 4.51 × 102; 500; 450; 4.7 × 102 450 < 451 < 470 < 500

480 < 510 < 700

18. 0.0003; 3.2 × 10 -4; 2.9 × 10-4

19. 0.007; 7.1 × 10-3; 6.9 × 10-3; 0.0072

0.00029 < 0.0003 < 0.00032

EARLY FINISHERS

0.0069 < 0.007 < 0.0071 < 0.0072

20. 6.5 × 103; 5,900; 7,100

21. 1.1 × 105; 99,900; 100,000; 9.8 × 104

5,900 < 6,500 < 7,100

98,000 < 99,900 < 100,000 < 110,000

Have students redo problems 16–21 in the Exercise section by converting the numbers to scientific notation and redo problems 22–27 by converting the numbers to standard form.

Convert to scientific notation and then order the numbers from least to greatest. 22. 3,100; 2.9 × 103; 3.3 × 103

23. 0.00065; 6.7 × 10 -4; 6.2 × 10-4; 0.00061

2.9 × 103 < 3.1 × 103 < 3.3 × 103

6.1 × 10-4 < 6.2 × 10-4 < 6.5 × 10-4 < 6.7 × 10-4

24. 0.006; 5.8 × 10-3; 0.005

25. 7000; 6.9 × 103; 6,500; 7.1 × 103

5 × 10-3 < 5.8 × 10-3 < 6 × 10-3

6.5 × 103 < 6.9 × 103 < 7 × 103 < 7.1 × 103

26. 920; 9.1 × 102; 9.3 × 102

CHALLENGE AND EXPLORE

27. 0.34; 3.3 × 10 -1; 3.5 × 10-1; 0.36 3.3 × 10-1 < 3.4 × 10-1 < 3.5 × 10-1 < 3.6 × 10-1 © Lighthouse Curriculum. Copying strictly prohibited.

9.1 × 102 < 9.2 × 102 < 9.3 × 102

CH AL L ENGE 28. A student says: "5.2 × 105 is less than 6.3 × 104 because 5.2 is less than 6.3."

29. Two students write the following expressions: Student A: 3.4 × 106 Student B: 34 × 105

Is the student correct? Explain why or why not using place value and powers of ten.

Are they equivalent? Explain your answer with a conversion or simplification.

No, the student is incorrect. While 5.2 is less than 6.3, the

Yes, they are equivalent. 34 × 105 = (3.4 × 101) × 105 =

exponent on the power of ten makes the number much larger.

3.4 × 106, so both expressions represent the same number.

Lighthouse Math

Level H

Chapter 2

Exercise 6

For Challenge problem 28, students will need to write a statement explaining why the error occurred using specific reasoning. For problem 29, students will need to compare numbers where one is not in proper scientific notation. Remind students that while it may not be proper, it does still have a value that we can find. We can compare the numbers by putting both in proper scientific notation or by changing both to standard form. When students finish working, discuss their answers together as a class.

ACTIVITY Human Number Line Race: The purpose of this activity is for students to strengthen their skills of ordering and comparing numbers in both scientific notation and standard form through group collaboration. Prepare two identical set of notecards, each with a number written either in scientific notation or standard form. Divide the class into two teams. Distribute the cards so that each student receives one. Students line up in order from least to greatest, racing to see which team finishes first and most accurately. When both teams are done lining up, they turn to face each other and display their cards. Both teams’ cards should be in the same order. If they aren’t, students with cards in different orders have to justify their placement and reposition themselves correctly. The team that finishes first wins, if they are completely accurate as well. Otherwise, the team with the least mistakes wins.

COMMON ERRORS Students may convert to standard form incorrectly. Students may convert to scientific notation incorrectly. Students may mistake larger negative exponents as larger values.

ASSESS Check the odd-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

scientific notation. Then, have students work with a partner to complete problems 5-10 before reviewing as a class and discussing any problems they found challenging. Next, have students work in groups to complete problems 1-27 in the Exercise section. Conclude by reviewing as a class and selecting a few problems for volunteers to walk through on the board.

Level H | 2-7 2-7 | Multiplication and Division with Scientiﬁc Notation

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to perform multiplication and division with numbers expressed in scientific notation and interpret results in context.

Solve.

SPIRAL REVIEW

45 ÷ 100 =

0.45

94 × 10 =

940

960 ÷ 10,000 =

0.096

0.5 × 1,000 =

89 ÷ 1,000 =

0.089

1.2 × 104

1,200

4.8 × 10-2

9.2 × 108

92,000,000

6 × 104

500

2 ÷ 100 =

0.02

Compare using >, <, or =. <

8 × 10-5

7.05 × 106

0.008 <

7,500,000

0.048 60,000

Materials y Whiteboards y Dry-erase markers y Colored pencils

L E A R N A ND C O NNE C T When multiplying or dividing numbers in scientific notation, separate the coefficients from the powers of ten. Group the coefficients together and the powers of ten together. Then, multiply or divide each part separately. After multiplying or dividing, the final answer may need to be adjusted to correct scientific notation. Move the decimal point until the coefficient is between 1 and 10 (excluding 10). Then, adjust the exponent on the power of ten.

PRE-LESSON WARM-UP

Multiplication:

(4 × 102) × (3 × 105)

Write the following problems on the board: y 3 × 3 = [3 ] 5

y 99-2 = [99] 7

y 7 7×3 7 = [71] -2

Have students solve the problems independently by simplifying using the rules of exponents rather than fully calculating the values. Afterward, review the answers as a class and go over the rules for multiplication and division with exponents to reinforce understanding. Explain that these properties will be useful when solving problems involving multiplication and division of numbers in scientific notation. Guiding Questions: 1. What shortcut do we use to multiply powers with the same base? [Add the exponents] 2. What shortcut do we use to divide powers with the same base? [Subtract the exponents]

1.2 × 108

Simplify.

Adjust.

(2.4 × 106) ÷ (4 × 103) 2.4 × 106 4 × 103

y 445 =[48]

12 × 107

Separate values.

Division:

y 125 × 124 = [129] y 56 × 5-3 × 57 = [510]

(4 × 3) × (10 × 105) 2

2.4 106 × 3 10 4

0.6 × 103

6 × 102

Separate values.

Simplify.

Adjust.

A P P LY Adjust each number to be in the correct scientific notation. 1.1 × 104

0.45 × 107 =

4.5 × 106

11 × 103 =

128 × 10 -4 =

1.28 × 10-2

0.04 × 108 =

4 × 106

54 × 103 =

5.4 × 10

280 × 105 =

2.8 × 107

0.1 × 10 -5 =

1 × 10-6

5. 7.

0.03 × 109 =

3 × 10

Level H

Chapter 2

Lesson 7

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Remind students that numbers in scientific notation have two parts: a coefficient and a power of ten. Let them know that splitting numbers into these two parts makes multiplication and division easier. Write the example from the Learn and Connect section on the board: (4 × 102) × (3 × 105). Ask students to identify the coefficients [4 and 3] and circle them in one color. Then, ask for the powers of ten [102 and 105] and circle them in another color. Show students how to separate and sort the parts and multiply each by their corresponding part. Have them simplify 4 × 3 and 102 × 105 to get 12 and 107, then put that back together to get the result, 12 × 107. Point out that this number is not in proper scientific notation. Emphasize that the final answer must always be rewritten in correct scientific notation, with a coefficient that is between 1 and 10. Show students how to adjust by changing 12 to 1.2 and 107 to 108. Repeat the process with the division example, (2.4 × 106) ÷ (4 × 103), walking through separating and sorting coefficients and powers, simplifying, and adjusting as needed.

APPLY AND DEVELOP SKILLS (Practice) Work through problems 1–2 in the Apply section as a class to review how to adjust a number so that it is in proper scientific notation. Tell students that when we make the coefficient smaller by one place value, we must adjust the exponent by increasing it by one to compensate for the shift. Next, have Lighthouse MATH Level H | Teacher's Guide

Exercise | 2-7 Name Multiply. (3.5 × 103) • (7 × 104)

2.45 × 108

(1.3 × 10 -5) • (4 × 10-3)

5.2 × 10-8

(8.4 × 10 ) • (5 × 10 )

4.2 × 10-4

(2.1 × 10 ) • (2 × 10 )

4.2 × 1010

(9.1 × 102) • (3 × 106)

2.73 × 109

(6.7 × 104) • (3 × 103)

2.01 × 108

(6.3 × 104) • (2 × 105)

1.26 × 10

(5.9 × 106) • (4 × 10-2)

2.36 × 10

(2.8 × 10 ) • (3 × 10 )

8.4 × 109

10. (4.1 × 10 ) • (7 × 10 )

2.87 × 108

11. (4.6 × 10-3) • (4 × 10-4)

1.84 × 10-6

12. (3.8 × 105) • (2.5 × 103)

9.5 × 108

13. (7.9 × 105) • (2 × 103)

1.58 × 109

14. (7.6 × 10 -2) • (1.2 × 1010)

9.12 × 108

15. (6 × 108) ÷ (2 × 104)

3 × 104

16. (7.2 × 105) ÷ ( 8 × 102)

9 × 102

17. (9 × 10−3) ÷ (3 × 10−5)

3 × 10

18. (6 × 104) ÷ (2 × 10−6)

3 × 1010

19. (2.4 × 107) ÷ (6 × 103)

4 × 103

20. (4.5 × 107) ÷ (1.5 × 103)

3 × 104

21. (8.1 × 10 ) ÷ (9 × 10 )

9 × 10−8

−5

22. (2.4 × 10 ) ÷ (6 × 10 )

4 × 102

23. (5 × 106) ÷ (2.5 × 103)

2 × 103

24. (2.7 × 107) ÷ (3 × 10−4)

9 × 10−12

25. (1.5 × 10−2) ÷ (3 × 10−6)

5 × 103

26. (5 × 105) ÷ (1.25 × 10 -3)

4 × 108

27. (3.6 × 109) ÷ (1.2 × 104)

3 × 105

28. (4.2 × 10−3) ÷ (2.1 × 10−6)

-8

STRUGGLING LEARNERS Encourage students to use different colors to separate the coefficients from the exponents and rewrite the problem so that they are sorted by type. When students need to convert back to proper scientific notation, remind them that if they make their coefficient smaller by one place value, that means the exponent has to go up by one to compensate for the shift.

Divide.

−6

−2

EARLY FINISHERS

2 × 103

Write an equation. Then, solve. 30. A lab sample contains 4.8 × 106 bacteria evenly spread across 6 × 103 milliliters of liquid. How many bacteria are there per milliliter? Equation: (4.8 × 10 ) ÷ (6 × 10 ) 6

Equation: (9 × 105) ÷ (6 × 107) Solution:

29. A satellite transmits data at a rate of 6 × 107 bytes per second. If a file is 9 × 105 bytes in size, how many seconds does it take to transmit? Hint: Use division.

1.5 × 10 seconds -2

Solution:

8 × 10 bacteria per milliliter 2

CH AL L ENGE 31. Machine A consumes 2.4 × 106 units of energy over 6 × 102 seconds. Machine B consumes 1.2 × 106 units of energy over 3 × 102 seconds. Which machine uses more energy per second, and by how much? Both machines use 4 × 10 units/second. 3

Lighthouse Math

Level H

Chapter 2

Exercise 7

Challenge students to create two multiplication or division problems, one where the answer will not need to be adjusted at the end (it will end up in proper scientific notation), and one where the coefficient will need to be adjusted by moving the decimal two places.

CHALLENGE AND EXPLORE Have students work with a partner to solve Challenge problem 31. Guide them by reminding them that they will need to find a unit rate to compare the machine usage.

students work with a partner to solve problems 3–8, followed by a class review to address any additional questions. Walk through problems 1-2 and 15-16 in the Exercise section together as a class, then have students work in pairs to complete problems 3-14 and 17-28. Conclude by reviewing as a class and walking through problems 29-30, asking students to help build the equations and solve them, with volunteers showing their work on the board.

ACTIVITY Scientific Notation Relay: The purpose of this activity is for students to practice the steps for solving multiplication and division problems with scientific notation through a collaborative relay game. Divide the class into groups of four and provide each group with a whiteboard. Within each group, students select one of four roles: Separator, Coefficient Solver, Exponent Solver, and Checker. Each student also chooses a different dry-erase color. Write a multiplication or division problem in scientific notation on the board. The whiteboard will be passed from one person to the next, beginning with the Separator, who rewrites the problem by separating the coefficients and exponents. Next, the Coefficient Solver multiplies or divides the coefficients. Then, the Exponent Solver simplifies the powers of ten. Finally, the Checker ensures the final answer is written in proper scientific notation, adjusting if necessary. Once the group has completed all steps, they present their final answer. After each round, students rotate roles.

COMMON ERRORS Students may make calculation errors when multiplying or dividing. Students may forget to convert their answers back to proper scientific notation.

ASSESS Check the even-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

Level H | 2-8 2-8 | Addition and Subtraction with Scientiﬁc Notation

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to add and subtract numbers in scientific notation.

Solve.

SPIRAL REVIEW

0.25 + 0.065 = 1.6 − 0.8 =

0.315

2.04 + 0.81 =

2.85

9.8 − 0.02 =

0.012 + 0.05 =

0.062

1.901 + 0.25 =

6 × 10-3 • 8 × 10-8 = 4.8 × 10

-10

1 1.8 × 107 ÷ 9 × 105 = 2 × 10

5 × 1010 • 3.2 × 10 -4 = 1.6 × 10

2.8 × 10-2 • 6 × 103 = 1.68 × 10

0.8

9.78 2.151

Solve.

Materials

6 4 × 1012 • 5 × 10-7 = 2 × 10

3.6 × 10 -5 ÷ 4 × 10-3 = 9 × 10

-3

y Notecards L E A R N A ND C O NNE C T When adding or subtracting numbers in scientific notation, both numbers must have the same power of 10.

PRE-LESSON WARM-UP

First, adjust one or both numbers so the exponents match. Convert to the higher exponent because it's more efficient, accurate, and prevents having to adjust the exponent after simplifying. The decimal can be moved over any number of places to do this.

Write the following problems on the board:

Next, add or subtract the coefficients. Keep the common power of 10. The final answer should be adjusted to proper scientific notation.

y 700,000 + 4,000 = [704,000] y 250,000 − 3,000 = [247,000] y 6,000 + 80,000 = [86,000] y 1,500,000 − 700,000 = [800,000]

Guiding Questions: 1. How does place value help us make sense of problems when we add and subtract? [It is important to line up place values so that numbers of the same place are added/ subtracted from like numbers.] 2. Why might scientific notation make it easier to add or subtract really large or really small numbers? [Numbers written in scientific notation are usually more compact than their equivalent number in standard form, and we are less likely to lose digits, especially when there are a lot of zeros floating around.]

Lighthouse MATH Level H | Teacher's Guide

(7 × 105) + (4 × 103) (7 × 10 ) + (0.04 × 10 )

(7 + 0.04) × 105

7.04 × 105

Change to the same power of ten.

Add the numbers.

Simplify.

Subtraction:

(1.4 × 10-2) − (2 × 10-3) (1.4 × 10 ) − (0.2 × 10-2)

(1.4 − 0.2) × 10 -2

Change to the same power of ten.

Add the numbers.

-2

Ask students to solve independently. Then, review as a class and highlight how they had to line up place values in order to add or subtract correctly. Transition by explaining that scientific notation works the same way: we must first “line up” the place values by matching exponents before adding or subtracting.

Addition:

1.2 × 10-2 Simplify.

A P P LY Solve. 1.

(5.2 × 104) + (3.1 × 104)

8.3 × 104

(7 × 106) − (2.5 × 106)

4.5 × 106

(1.6 × 10 -3) + (2.9 × 10-3)

4.5 × 10-3

(9.3 × 107) − (1.8 × 107)

7.5 × 107

(6.1 × 105) + (3.8 × 105)

9.9 × 10

(4.2 × 10 -6) − (1.2 × 10 -6)

3 × 10-6

(3.5 × 103) − (2 × 103)

5. 7.

(8 × 108) + (1.5 × 108)

9.5 × 10

Level H

Chapter 2

Lesson 8

1.5 × 103

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Start the lesson by explaining to students that to add or subtract numbers in scientific notation, the exponents on the powers of ten must be the same. Write the problem from the Learn and Connect section on the board, (7 × 105) + (4 × 103). Ask students why they can’t separate 7 and 4 and add like they would when solving multiplication problems. Point out that it is not possible to simply add 7 and 4 because the numbers 7 and 4 are not actually in the same place values when written in standard form (have a student write these numbers in standard form to prove this). Explain to students that we can temporarily change one of the numbers to improper scientific notation so that the powers of ten match. We can work with numbers that are not in proper scientific notation, as long as we put our final answer in the correct form. Tell students that it is recommended to change the number with the lower exponent so that it matches the higher exponent because it is more efficient, accurate, and prevents having to adjust the exponent after simplifying. Next, demonstrate how to add the coefficients while keeping the common power of ten in the final sum. Tell students that they should always check to make sure their answer is in proper scientific notation at the end. Repeat the process with a subtraction example, this time having students help you convert one number to the same power of ten before subtracting. Emphasize that matching the exponents first is the essential step before solving and that changing the numbers to match the higher of the two exponents will be helpful.

Exercise | 2-8 Name Add.

STRUGGLING LEARNERS

(6.3 × 106) + (2.4 × 104)

6.324 × 106

(6.6 × 107) + (4 × 105)

(8.1 × 10 ) + (9 × 10 )

8.19 × 105

(2.2 × 10 ) + (5.5 × 10 )

2.2055 × 106

(3.5 × 10-2) + (1.2 × 10 -4)

(5.1 × 104) + (9 × 102)

5.19 × 104

3.512 × 10-2 2.05 × 10

(2 × 107) + (5 × 105)

(4.4 × 10 ) + (3.1 × 10 ) 3

4.71 × 103

6.64 × 107 3

3.387 × 10

(3.3 × 10 -4) + (8.7 × 10-6)

9.72 × 105

10. (9.7 × 10 ) + (2 × 10 ) 5

4.218 × 10-3

7.86 × 106

12. (4.2 × 10 -5) + (1.8 × 10-3)

13. (7.5 × 105) − (2.5 × 103)

7.475 × 105

14. (7 × 105) − (2 × 102)

15. (3.2 × 10 ) − (1 × 10 )

3.19 × 106

16. (4.2 × 10 ) − (2.2 × 10 )

17. (6 × 10-3) − (2 × 10-4)

5.8 × 10-3

18. (9 × 106) − (1 × 104)

19. (9.1 × 10 ) − (1.1 × 10 )

9.089 × 104

20. (2.3 × 10 ) − (1.3 × 10 )

2.287 × 104

21. (8.3 × 105) − (4 × 103)

8.26 × 105

22. (1.6 × 105) − (5 × 103)

1.55 × 105

23. (1 × 108) − (7.5 × 105)

9.925 × 10

24. (6.2 × 10 -3) − (4.2 × 10-5)

11. (7.8 × 106) + (6 × 104)

Provide a step-by-step reference sheet for students to use when solving problems that involve adding and subtracting in scientific notation. Allow students to convert numbers to standard form to check their answers.

-4

Subtract.

-4

4.178 × 10-2

Have students select five addition and five subtraction problems from the Exercise section, convert them into standard form, and solve them in standard form to check the accuracy of their work.

8.99 × 106

EARLY FINISHERS

6.998 × 105

−2

6.158 × 10-3

Write an equation. Then, solve. 25. A research station collects 4.2 × 106 units of data in the morning and 3.8 × 105 units in the afternoon. How much total data was collected that day? Equation: (4.2 × 10 ) + (3.8 × 10 ) Solution:

Equation: (9.6 × 10 ) + (2.4 × 10 )

4.58 × 106 units

Solution:

CHALLENGE AND EXPLORE

26. A telescope spots 9.6 × 107 stars in one part of the sky and 2.4 × 106 stars in another. How many stars are seen in total?

9.84 × 107 stars

CH AL L ENGE 27. Two spacecraft are fueled for a long journey. • Ship A receives 6.4 × 106 liters.

• Ship B receives 6.75 × 106 liters.

Work as a class to solve problem 27. Ask students which operation needs to be used to solve and which number should be adjusted. Then, ask students to explain how they can compare the values.

Later, it’s found that Ship B's tank was overfilled by 2.5 × 105 liters and must be reduced. After correcting Ship B's fuel, which ship has more fuel and by how much? Ship B still has 1.0 × 105 liters more fuel than Ship A.

Lighthouse Math

Level H

Chapter 2

Exercise 8

Walk through problems 1-8 in the Apply section as a class to reinforce understanding of how to add and subtract numbers in scientific notation. Next, place students in groups to solve problems 1-24 in the Exercise section, followed by a whole-class review in which a few volunteers present their work on the board. Conclude by working through problems 25-26 as a class, emphasizing that these are real scenarios in which scientific notation is used in life. Create the equations together and invite a student volunteer to solve in front of the class.

ACTIVITY Addition and Subtraction Shuffle: Students will practice adding and subtracting numbers in scientific notation while reinforcing the importance of converting to the same powers of ten. Prepare a deck of notecards, each with a number written in scientific notation, and give one card to each student. Begin by pairing two rows of students together to form partners. Each pair adds and subtracts their two numbers and writes down the results. After completing the pair work, one row shuffles so that each student now works with a new partner, repeating the process. Continue for five to six rounds so that each student works with all students in the next row. To close, have students share a few of the problems they created and solved with the class. Discuss common pitfalls, such as forgetting to match exponents before combining, and emphasize strategies for avoiding these mistakes.

COMMON ERRORS Students may change the coefficient but forget to adjust the exponent accordingly or adjust it in the wrong way.

ASSESS Check the odd-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 2-9 2-9 | Applications of Scientiﬁc Notation

PREREQUISITE SKILLS

y Students will be able to solve realworld problems involving scientific notation, including problems comparing magnitudes.

DAI LY REVI EW

Objective and Learning Goals

SPIRAL REVIEW

Materials

Solve. 1.

(7 × 104) • (2 × 10-9) =

(4.5 × 10-2) ÷ (5 × 101) =

1.4 × 10-4

-3 (9 × 10-6) • (3 × 102) = 2.7 × 10

9 × 10-4

-3 (6 × 103) • (4 × 10-7) = 2.4 × 10

(2.5 × 10-4) + (4.5 × 10-6) = 2.545 × 10

3 (5.2 × 103) + (3.1 × 102) = 5.51 × 10

Solve. 7.3 × 106

(7 × 106) + (3 × 105) =

-7 (9.8 × 10-7) − (3 × 10-8) = 9.5 × 10

-4

y Highlighters L E A R N A ND C O NNE C T The Saturn V rocket, used during NASA’s Apollo missions, used about 3.2 × 106 liters of fuel during launch. During the spaceflight portion (after launch), it used about 2.0 × 103 liters of fuel per hour. The full mission lasted about 195 hours.

PRE-LESSON WARM-UP

Guiding Questions: 1. In what real-world fields might scientific notation be helpful and why? [Astronomy, sociology, biology, chemistry, economics, technology, etc. Scientific notation is helpful in these fields because they often work with very large or very small numbers.] 2. Why is it important to understand scientific notation? [It is widely used in real-world fields such as space science, medicine, technology, and finance, and we may come across these numbers when reading the news or seeing advertisements.]

How much total fuel did the spaceship use during the entire mission? Step 1: Multiply to find how much fuel is used during travel.

Step 2: Add the launch fuel.

2.0 × 103 liters/hour × 195 hours (2.0 × 195) × 103 = 390 × 103 = 3.9 × 105

(3.9 × 105) + (3.2 × 106) (0.39 × 106) + (3.2 × 106) = (0.39 + 3.2) × 106 = 3.59 × 106

Final Answer: 3.59 × 106 liters of fuel were used during the mission.

A P P LY © Lighthouse Curriculum. Copying strictly prohibited.

Tell students that the United States census is a survey given by the government to the citizens of the U.S. every ten years. The survey gathers information about the population in order to be able to better serve the population in each state. According to the U.S. Census, the population of the U.S. in the year 2000 was 281,421,906. Write this number on the board and tell students that often, large numbers such as these are rounded and then changed into scientific notation so that they will look cleaner in presentations, advertisements, or news articles. Ask students to round this number to the nearest ten millionth. [280,000,000] Next, ask them to change this number to scientific notation. [2.8 × 108]

Circle the larger quantity. 1.

The distance from Earth to Saturn is about 1.4 × 109 kilometers, while the distance to Jupiter is about 7.8 × 108 kilometers.

The distance to Saturn

The distance to Jupiter

The distance from Earth to Saturn is about 1.4 × 109 kilometers, while the distance to Jupiter is about 7.8 × 108 kilometers.

Star A

Star B

The mass of an ant is about 4 × 10-3 grams, while the mass of a mosquito is about 2.5 × 10-2 grams.

The mass of an ant

The mass of a mosquito

California’s economy produces 3.9 × 1012 dollars per year. Vermont’s economy produces 4.3 × 1010 dollars per year.

California’s economy

Vermont’s economy

Level H

Chapter 2

Lesson 9

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Inform students that today’s lesson will focus on real-life applications of scientific notation. Begin by reading the word problem in the Learn and Connect section aloud and asking students to identify the important pieces of information. [the amount of fuel used for launch, the rate of fuel used in travel, the length of the mission] Next, ask what the first step should be to solve this problem. [multiply to determine the amount of fuel used during travel] Have students help you work through the multiplication together. Then, ask what the next step is. [adding the launch fuel] Write out the addition statement and have students assist in solving it. Have students add a unit to the final answer and explain what the solution means in the context of the problem [in total, the mission used 3.59 × 106 liters of fuel].

APPLY AND DEVELOP SKILLS (Practice) Have students work in pairs to solve problems 1-4 in the Apply section. Review as a class and ask students to share their reasoning for each answer choice. Next, work through problems 1-4 in the Exercise section as a class, asking students to identify the key words in the problem that tell them which operation they will need to use to solve. As you solve, remind students of the rules for each operation. Then, have students work in pairs to solve questions 5-8, followed by a class review. Conclude by working through any problems students found especially challenging. Lighthouse MATH Level H | Teacher's Guide

Exercise | 2-9 Name Write an equation. Then, solve. 1.

One solar panel on a space station collects 3.2 × 103 watts per hour. The station uses 25 identical panels. How much energy is collected in one hour by all the panels? Equation:

3.2 × 103 × 25

Solution:

8.0 × 104 watts

7 Solution: 1.875 × 10 bytes per second

Equation: (1.5 × 10 ) + (5.5 × 10 ) Solution:

Have students highlight the values in the word problem and underline the key words that indicate which operation to use. Provide a reference key that outlines the steps for solving multiplication, division, addition, and subtraction problems in scientific notation.

12 4 Equation: (1.5 × 10 ) ÷ (8 × 10 )

In one year, Houston, Texas, collected about 5.5 × 108 dollars from business taxes and 1.5 × 109 dollars from residential taxes. What is the total amount of taxes Houston collected? 9

STRUGGLING LEARNERS

The Solar Dynamics Observatory (SDO) sends roughly 1.5 × 1012 bytes of data to Earth each day. How many bytes of data are sent each second? (24 hours ≈ 8 × 104 seconds)

2.05 × 10 dollars 9

Red light has a wavelength of 7 × 10-7 meters. Ultraviolet light has a shorter wavelength of 1 × 10-8 meters. How many times longer is red light’s wavelength compared to ultraviolet’s wavelength? Equation:

(7 × 10-7) ÷ (1 × 10-8)

Solution:

70 times longer

EARLY FINISHERS

Solve. The human body makes 2 × 10 new blood cells per second. How many red blood cells does the body make in a week? (1 week ≈ 6 × 105 seconds)

A satellite sends 6 × 10 bytes of data. Then, it sends an extra 2 × 106 bytes in a burst. How much data did it send? 8

6.02 × 108 bytes

1.2 × 1012 red blood cells

Ask students to create a fictional country and the populations of five different areas in the country. They should write these numbers in scientific notation and then add them together to find the total population of their country.

A colony of cells starts with 1.2 × 106 cells and ends up with 3.6 × 107. How many times its original size is the cell colony at the end?

A telescope collects 2.4 × 106 units of data every night for 5 nights. Then, 7.5 × 104 units are deleted. How much data remains?

30 times greater

1.1925 × 107 units of data

CH AL L ENGE 9.

A glacier loses ice at a rate of 3.5 × 107 kilograms per day. Over 2.4 × 103 days, this totals one part of its loss. Satellite pictures show an additional 1.2 × 1011 kilograms broke off suddenly because of a recent heatwave.

10. A clean water project costs 4.5 × 107 dollars in each region. It’s being set up in 2.2 × 102 regions. As part of this project, another 1.3 × 109 dollars are spent on other improvements.

A. How much ice has the glacier lost in total?

A. What is the total cost of the program?

2.04 × 1011 kilograms

1.12 × 1010 dollars

B. Is it reasonable for a scientist to say that more than half of the glacier's loss came from the sudden heatwave?

B. If the project provides clean water to 4 × 108 people, what is the cost per person? $28

Yes

Lighthouse Math

Level H

Chapter 2

Exercise 9

CHALLENGE AND EXPLORE In Challenge problem 9, students will need to find the amount of ice lost over a given period of time using a given rate. Then, they will need to decide if the amount of ice lost during another time period - a heatwave - is more than half the amount lost during the initial time period. Guide students towards using division to compare the two quantities since we are talking about a fraction ( 21 ).

Scientific Notation in the Real World Stations: This activity will allow students to see how scientific notation is applied in real-world contexts by practicing with word problems. Set up five different stations around the room, each focused on a specific context: biology, technology, astronomy, finance, and chemistry. At each station, provide a word problem for students to solve that highlights the use of scientific notation in that field. Students will rotate through the stations, working in pairs or small groups to complete the problems and record their solutions. After all groups have visited each station, bring the class together for a discussion. Ask students to share examples they solved, highlight differences between contexts, and reflect on why scientific notation is useful in handling very large and very small numbers.

COMMON ERRORS Students may choose the wrong operation when solving a word problem. Students may forget the rules of multiplication, division, addition, and subtraction of numbers in scientific notation.

ASSESS Check problems 5-8 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

ACTIVITY

Level H | 2-10 2-10 | Review

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will review concepts introduced in Chapter 2, including properties of exponents, order of operations with exponents and fractions, writing and comparing numbers in scientific notation, and calculating with numbers in scientific notation.

Product of Powers Property

Quotient of Powers Property

When multiplying, if the base is the same, add the exponents.

When dividing, if the base is the same, subtract the exponents.

x6 · x9

x6 + 9

Negative Exponents

Any number or variable with a zero exponent is equal to 1.

Negative exponents indicate a reciprocal. They can be written as a positive exponent by flipping the numerator to the denominator.

x0 1

Write the following on the board and have students work independently to solve:

Adding and Subtracting in Scientific Notation

Separate the coefficients and the powers of 10. Simplify. Put in proper scientific notation.

Change to the same power. Add the coefficients. Keep the power of 10. Simplify.

(4 × 10²)(6 × 10³)

y Simplify: bb2 [b5] 7

y Write in standard form: 3.2 × 10-3 [0.0032]

Review the answers as a class. Then, tell students that today’s lesson will review all of these skills and give them a chance to practice applying the properties together. Guiding Questions 1. Why do you think it is helpful to write numbers with exponents or in scientific notation? [Answers will vary but should touch on the idea that large and small numbers are represented more efficiently.] 2. What is important to remember about comparing numbers? [They should be written in the same form.]

24 × 105

y Convert to scientific notation: 45,000 [4.5 × 104]

1 x4

x-4

Multiplying and Dividing in Scientific Notation

y Simplify: a3 × a5 [a8]

y Rewrite with positive exponents: c-4 [ c14 ]

x7 − 2 x5

Zero Exponent Rule

PRE-LESSON WARM-UP

y Simplify: 70 [1]

x7 x2

x15

(4 × 6)(10² × 10³)

(3 × 10²) + (2 × 10³)

(0.3 × 10³) + (2 × 10³)

(0.3 + 2) × 10³

2.4 × 106

2.3 × 103

A P P LY Simplify each expression. 1.

34 × 38

312

95 × 92 × b4 × b8

97b12

84 × a7 83 × a4

8a3

76 × 74

710

y3 × y × 27 × 25

y4212

109 × y9 102 × y8

107y

x4 × x2

94 9

48 × 43 × x10 45 × x3 × x4

11.

z10 z5

12.

28 × a9 × y4 26 × a6 × y3 22 × a3 × y

10. 48 × 43 × 45

416

Level H

Chapter 2

Lesson 10

46x3

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Create a table on the board with a box for each topic in the chapter: Products of Powers Property, Quotient of Powers Property, Zero Exponent Rule, Negative Exponents, Multiplying and Dividing in Scientific Notation, and Adding and Subtracting in Scientific Notation. Work through the table one topic at a time, asking students to share what they remember about each concept. Record their responses in the corresponding box until the table is complete with all key ideas. Once finished, present the table in the Learn and Connect section and compare what the students brainstormed to what is in the book.

APPLY AND DEVELOP SKILLS (Practice) Have students work in pairs to complete problems 1-12 in the Apply section and then review the answers as a class by inviting volunteers to show their work for selected problems on the board. Next, have students work independently to complete problems 1-38 in the Exercise section. Conclude with a class review, addressing any problems students found challenging and clarifying misconceptions.

Lighthouse MATH Level H | Teacher's Guide

Exercise | 2-10 Name Simplify each expression and rewrite using only positive exponents. 1.

43 × 4-5

1 42

b-6 × b3 × c-8 × c8

1 b3

a ×a

28 311

y0 × y-7

-2

1 y7

-8

2 ×3×3 ×3

1 z0

-4

STRUGGLING LEARNERS

9-8 × t3 911 93 × t a-3 × b5 a2b2 a-5 × b3 s3 9-3 × r × s5 97r6 94 × r7 × s2

Advise students to use the table in the Learn and Connect section as well as their anchor charts from the chapter as a reference for how to solve the various problems while they work.

Convert to scientific notation. 10. 4,500 4.5 × 10

11. 0.00000044 4.4 × 10

13. 0.00081 8.1 × 10

12. 980,000 9.8 × 10

-7

9 14. 6,000,000,000 6 × 10

-4

17. 0.00000000032 3.2 × 10

16. 72,000,000 7.2 × 10

-3 15. 0.007 7 × 10

EARLY FINISHERS

18. 5,600 5.6 × 10

-10

Convert to standard form. 19. 5 × 108 500,000,000

20. 2.4 × 10 -2

0.024

21. 9.81 × 107

98,100,000

22. 3.2 × 104

32,000

23. 8.9 × 102

890

24. 1 × 102

100

25. 7.1 × 10-3

0.0071

26. 4.7 × 10-6

0.0000047

Have students create three additional problems for each section on the Exercise page. They should trade problems with a partner to solve.

27. 4.5 × 10-8 0.000000045

CHALLENGE AND EXPLORE

Simplify each expression. 28. (3.2 × 104) + (4.5 × 104) 7.7 × 10

31. (7.1 × 10-3) − (2.4 × 10-3) 4.7 × 10

-3

29. (7.2 × 10 -2) ÷ (8 × 102) 9 × 10

-5

32. (9 × 105) − (3.2 × 105) 5.8 × 10

30. (9.4 × 10-4) − (5.1 × 10-4) 4.3 × 10

-4

33. (1.2 × 103) × (4 × 102) 4.8 × 10

35. One star weighs 6.9 × 103 kg and another weighs 3.2 × 103 kg. If the smaller star loses 1.0 × 102 kg, what is the new difference in their weight? 3.8 × 103 kg

1.125 × 107 bytes

CH AL L ENGE 36. A researcher records 3.5 × 105 particles in one sample and 4.2 × 104 particles in another. He adds them as: 3.5 × 105 + 4.2 × 104 = 7.7 × 109. Explain the mistake in the calculation. The error was adding the exponents during addition.

Lighthouse Math

37. An astronomer observes two star clusters. The first contains 8.19 × 105 stars, and the second contains 4.21 × 103 stars. Using your knowledge of numbers written in scientific notation, about how many times greater is the number of stars in the first cluster than the second?

Solve. 34. A space probe sends 4.5 × 107 bytes of data. If the data is split into 4 memory banks, how much data goes into each memory bank?

Have students work in pairs to discuss Challenge problems 29-30. For problem 30, give students an example using smaller whole numbers as a hint to have them figure out that they will need to use division to find how many times greater the first star cluster is than the second. For example, if I have eight cookies and you have four, how many times more cookies do I have than you?

About 2 × 102 or 200

Level H

Chapter 2

Exercise 10

Choose any activity from this chapter and adapt it for review. For example, the activity in Chapter 2-6, Human Number Line Race, can be adapted to include numbers in exponential notation as well. Additionally, the activity in Chapter 2-7, Scientific Notation Relay, can be adjusted to address the steps for solving problems from any of the units.

COMMON ERRORS Students may multiply or divide exponents instead of adding or subtracting them. Students may forget to fully simplify expressions with powers. Students may use the rules for multiplication and division when adding and subtracting.

ASSESS Check one problem from each section on the Exercise page. Lighthouse MATH Level H | Teacher's Guide

ACTIVITY

Chapter 3

In Chapter 3, we will expand on

Equations Equations can help us model real-world situations to solve problems and make predictions about future events. • We will solve equations by simplifying them and applying the distributive property. • We will solve equations with variables on both sides. • We will identify the number of solutions in an equation as one solution, no solution, or infinitely many solutions. • We will create equations for real-world situations and use them to solve for an unknown value.

Level H | 3-0 Chapter 3 | Skill Checklist

Skill 1: Properties of Operations Commutative Property

Write = or ≠ based on the commutative property.

Changing the order of the numbers in addition or multiplication does not change the result. Associative Property

3+2 = 2+3

8÷2 ≠ 2÷8

7−6 ≠ 6−7

4×5 = 5×4

Write = or ≠ based on the associative property.

Changing the grouping of numbers in addition or multiplication does not change the result.

4 + (2 + 3) = (4 + 2) +3

(9 − 1) − 3 ≠ 9 − (1 − 3)

12 ÷ (6 ÷ 2) ≠ (12 ÷ 6) ÷ 2

(5 × 6) × 9 = 5 × (6 × 9)

Distributive Property

Identity Property

Students will review skills needed for Chapter 3:

Adding any number to 0, or multiplying any number by 1, keeps the number the same.

4 ( 6 )+ 4 ( 4 )

15 + 10 = 25

24 + 16 = 40

12. 32 × 1 = 32

13. 873 + 0 = 873

I understand and can use properties of operations.

out 13 correct

Skill 2: Inverse Operations Match equations that show inverse operations. Operation

Inverse

Use inverse operations to solve.

2×3=6

2+3=5

8÷4=2

2×4=8

2+4=6

6÷3=2

5−3=2

6−4=2

Level H

7 − 15 + 15 =

89 × 0.367 ÷ 0.367 = 89

I can identify and use inverse operations.

out 6 correct

5( 3 ) + 5( 2 )

Use the identity property to fill in the blanks.

Chapter 3

Skill Checklist

Skill 1: Properties of Operations y Review each of the properties of operations as listed in the gray box. y Remind students that the commutative and associative properties apply to addition and multiplication, but not to subtraction or division. y Have students complete problems 1-13 and review the answers and strategies as a class.

Lighthouse MATH Level H | Teacher's Guide

10. 4(6 + 4)

5(3 + 2)

11. 5 + 0 = 5

y Properties of Operations y Inverse Operations y Simplifying Expressions y Translating Expressions y Solving Two-Step Equations

Fill in the blanks. Simplify using the distributive property.

Multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.

Objective and Learning Goals

Lighthouse Math

Skill 2: Inverse Operations y Tell students that the word “inverse” means opposite. Using the gray box, discuss which operations are opposites or inverses of each other. y Discuss when inverse operations can be helpful: 1 - to check your work, 2 - if inverse operations are applied to the same number, they cancel each other out, 3 - to solve equations y Have students complete problems 1-7 and review the answers and strategies as a class.

Name

Skill 3: Simplifying Expressions

2x + 4 + -6x + 3

-4x

Combine like terms to simplify each expression. 1.

2(3x) + 2(4)

7h − 8g + 6h − 2g 13h − 10g

Apply the distributive property to each expression. 3.

2(3x + 4)

9 + 5y + y − 2 6y + 7

4(5x + 9) 20x + 36

-3(8j − 6) -24j + 18

-5(3n + 5) + 20 -15n − 5

Simplify each expression.

6x + 8

6(7p + 2) − 10p 32p + 12

I can combine like terms and use the distributive property to simplify expressions.

out 6 correct

Skill 4: Translating Expressions

• Difference • Less than • Minus • Subtracted from • Decreased by

Choose a variable to represent the unknown value. Then, write an expression for each scenario. 1.

Sam pays $3 for a tub of cream cheese plus $0.75 for each bagel

• Product • Times • Of • Twice • Double

• Half • Quotient • Over • Divided by • Separated into

3 degrees colder than half the temperature today

A pool has 5,000 gallons of water and each minute another 9 gallons are added 9m + 5,000

0.75b +3 t −3 2

I can translate words into expressions.

out 3 correct

Skill 5: Solving Two-Step Equations -3y − 7 = -10 +7 +7 -3y = -3 ÷3 ÷3 y=1

Solve. 1.

3m + 4 = 7

m=1

-9x − 9 = 81

x = -10

y −3=5 2

y = 16

5p + 2 = 17

p=3

out 4 correct

Lighthouse Math

I can solve two step equations.

Level H

Chapter 3

Skill Checklist

Skill 3: Simplifying Expressions

Skill 4: Translating Expressions

Skill 5: Solving Two-Step Equation

y Review vocabulary from the previous lesson. y Using the example in the gray box, review with students what it means to combine like terms. Remind students that if a term is subtracted in the expression, it can be written as addition with a negative number and added in any order. y Use the second example in the gray box to review the distributive property. Remind students to multiply both addends in the parentheses. y Have students complete problems 1-6 and review the answers and strategies as a class.

y Remind students that variables are used to represent an unknown number. y Using the gray box, brainstorm a list of hints that let you know which operation to use when translating expressions. y Remind students to read each problem carefully and to mark numbers, operations, and unknown values. y Have students complete problems 1-3 and review the answers and strategies as a class.

y Tell students than an equation always has an equal sign. Sometimes, there can be a variable on one side of the equation. To solve the equation, you find the value of that variable. y Using the example in the gray box, review how to use inverse operations to solve the equation. y Remind students to first undo addition and subtraction and then undo multiplication and division. y Have students complete problems 1-4 and review the answers and strategies as a class. Lighthouse MATH Level H | Teacher's Guide

• Total • Plus • Sum • More than • Increased by

+ × ÷

Level H | 3-1 3-1 | Simplifying Equations

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to simplify equations by combining like terms and using the distributive property.

Solve. x =7 7 x = 49

15y = 225 y = 15

SPIRAL REVIEW

0.2g = 64

-11.7 + v = -18.2

g = 320

v = -6.5

Simplify. 1.

2 6.3 × 107 ÷ 9 × 104 = 7 × 10

3.6 × 10 -3 ÷ 6 × 103 = 6 × 10

-7

Vocabulary y Equation - a math sentence with an equal sign y Like terms - terms that have the same variable with the same exponent y Distributive property - when a factor is multiplied by each term in an addition or subtraction expression

L E A R N A ND C O NNE C T

Apply the distributive property by multiplying a number found outside the parentheses by each term inside the parentheses. Combine the like terms, or terms that have the same variable with the same exponent.

20x – 10 – 19x = 20

x = 30

x – 10 = 20

Sort the equations into the table based on if you can combine like terms, do the distributive property, or both.

Guiding Questions: 1. What does an equal sign mean? [the same as] 2. What does it mean to simplify an equation? [to combine terms that are alike and rearrange them without changing the value of the equation in order to make it easier to solve]

Solve: x – 10 = 20 + 10 + 10

A P P LY

y Notecards

Write 1 + 2 + 4 = 7 on the board and ask students to tell you why that is called an equation [because it has an equal sign; because 1 and 2 and 4 combined have the same value as 7]. Ask students to write another equation that is equivalent to this one [such as 3 + 4 = 7]. Next, write 2x + 2x + 3 = 11 on the board. Ask students to tell you why that is also called an equation and how it is different from the first equation. [because it has an equal sign; it is different because it has x’s/numbers we don’t know] Ask students if they can tell you another equation that is equivalent to this one [such as 4x + 3 = 11]. Tell students that 4x + 3 = 11 is not just an equivalent equation; it’s also a simplified equation. Ask them what it means to simplify an equation. [To rewrite an equation as short and as clear as possible, without changing its value] Ask them what simplification rule was used. [Combining like terms] Tell them that today we will be simplifying more complex equations to make it easier for us to work with them.

Simplify: 5(4x – 2) – 19x = 20

Once the like terms are combined, solve the equation.

Materials

PRE-LESSON WARM-UP

5(4x – 2) – 19x = 20

An equation is a math sentence with an equal sign. Some equations need certain parts simplified before they can be solved.

-8(12x – 9) = 3

16 = 8(5x + 7)

-2x + 9x – 4 = 18

1 = 3x + 2(-3x – 9)

41 = -17x + 11x – 2

15x + 5 – 13 = 8

4(x + 2) – 9 = 8

-14(6x + 4) = 30

Combine Like Terms

Both

Distributive Property

B C G

D F

A E H

Follow the steps to solve each equation.

2. -31 = -23x + 32 + 24x

Simplify

Solve

Combine the like terms:

Subtract 32 from both sides:

-31 = x + 32

Apply the distributive property:

3. 36 = 2(2g – 6)

36 = 4g − 12

4. -27v – 35 + 53v = 251

Combine the like terms:

26v − 35 = 251

-63 = x

Add 12 to both sides: Divide by 4:

48 = 4g g = 12

Add 35 to both sides: 26v = 286 v = 11 Divide both sides by 26:

Vocabulary Equation - a math sentence with an equal sign Like terms - terms that have the same variable with the same exponent Distributive property - when a factor is multiplied by each term in an addition or subtraction expression

Level H

Chapter 3

Lesson 1

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Write the equation from the Learn and Connect section on the board: 5(4x − 2) − 19x = 20. Remind students that this is an algebraic equation because it has an equal sign and an unknown value represented by an x. Tell students that before we can solve this equation, we have to simplify it and many equations we will see will need to be simplified before they can be solved. Ask students why we simplify equations. [To make them easier to solve] Ask students which simplification rules could be applied to this equation. Tell students that we cannot combine like terms if some of these terms are inside parentheses and some are outside. You may need to remind students that like terms are terms that have the same variable with the same exponent. If a number is outside parentheses, we need to distribute it to the terms inside the parentheses using multiplication. This is simplifying using the distributive property. Walk through the simplifying steps in the Learn and Connect section to simplify the equation to x − 10 = 20. Finally, solve the equation to get a solution of x = 30. As students transition to practice, tell them to bear in mind as they work that like terms cannot be combined if one is inside and one is outside parentheses. Ask them what needs to get done first in order to be able to simplify an equation in such a scenario. [Distributive property]

APPLY AND DEVELOP SKILLS (Practice) Have students work on problem 1 in the Apply section independently, then Lighthouse MATH Level H | Teacher's Guide

Exercise | 3-1 Name Write a simplified equation. Then solve. 2.

-3x + 5 – 2x = 20

7y − 4 + 2y = 41

Equation:

9y - 4 = 41

-13 + 2a = 17

Solution:

x = -3

y=5

a = 15

-5(6x – 12) = 0

7(2s + 3) = -35

-30x + 60 = 0

14s + 21 = -35

-10r + 15 = 15

Solution:

x=2

s = -4

r=0

-8x – 4(2x + 3) = 28

-3(5y − 4) + 10y = -3

Students should draw boxes and circles around like terms. They should draw arrows to assist them with the distributive property and to remind them to multiply all the terms inside the parentheses by the number outside of them.

-2.5(4r – 6) = 15

Equation:

STRUGGLING LEARNERS

-10 + 4a – 3 – 2a = 17

-5x + 5 = 20

EARLY FINISHERS

-6p – 2(7p + 1) = 18

Equation:

-16x -12 = 28

-5y + 12 = -3

-20p - 2 = 18

Solution:

x = -2.5

y=3

p = -1

Students can write their own unsimplified equation that involves combining many like terms and applying the distributive property. They can then trade with a partner to simplify and solve. Challenge students to create an equation that has a nondecimal answer.

Write an equation for each word problem. Simplify, then solve. 10. Bill and Jack collect cans for recycling. Bill collects 5 times as many cans as Jack. Then, they find 14 more cans together. If the total number of cans they have now is 104, how many cans did Jack originally collect?

11. A square garden has the dimensions shown below. If the total perimeter of the garden is 60 feet, what is the width of the garden? 4(4x + 3) = 60; 4x + 3 ft

Width = 15 ft

5x + x + 14 = 104; 6x +14 = 104; x = 15 cans

CH AL L ENGE 12. Lily is organizing a school fundraiser, where she’s selling tickets for an event. The price per ticket is $12. She also needs to buy some supplies for the event, which will cost her $48 upfront. After selling tickets, Lily plans to donate 20% of her total earnings to charity. A. Write an expression for the amount of money Lily will donate to charity if she sells x tickets. 0.2(12x – 48) B. Write an equation if the total donation is $120.

0.2(12x – 48) = 120

C. Solve the equation to find out the total number of tickets sold.

Lighthouse Math

Level H

Chapter 3

54 tickets

Lesson 1

16x + 12 = 60: x = 3

CHALLENGE AND EXPLORE For problem 12, remind students that when we work with percentages, we use the decimal (or fraction) version of the percentage. For an added challenge, ask students to return to problems 4-6 in the Exercise section and solve without using the distributive property. If they need a hint, tell them the first step is division.

go over the answers as a class to ensure understanding. Next, go through problems 2-4 together as a class. Although students are asked to solve the equation, the emphasis in this lesson should be on the simplifying steps. They may need a reminder about how to solve an algebraic equation, including to use inverse operations to solve and to start with addition/subtraction first and then multiplication/division. Have students complete problems 1-9 independently. Go over problems 1, 4, and 7, discussing the steps to get a simplified equation and solve. Finally, work together to create equations for problems 10 and 11, then have students simplify and solve them on their own.

ACTIVITY Simplify Me: The purpose of this activity is to have students work together to practice combining like terms and using the distributive property. Create notecards each with a different (but “like”) term or expressions in which the distributive property can be applied. Shuffle the cards and pass them out to students. Each group should get several cards. They should write down all the terms from their cards and combine them by first applying the distributive property where possible, and then combining like terms until they’ve simplified the expression as much as possible. Give students a few minutes to work, then have them switch their cards with another group and do the process again with the new set of cards. As time permits, shuffle the cards and redistribute to groups to create brand-new expressions that can be simplified.

COMMON ERRORS Students may try to combine like terms before completing the distributive property. Students may forget to distribute to all the terms inside the parentheses.

ASSESS Exit ticket: List the steps needed to simplify and solve the following equation: 3(4x + 3) − 5 = 40 [distribute 3 to 4x and 3 to get 12x + 9, then subtract 5 from 9 to get 12x + 4 = 40. To solve, subtract 4: 12x = 36, divide by 12: x = 3] Lighthouse MATH Level H | Teacher's Guide

Level H | 3-2 3-2 | Solving Multi-Step Equations

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to solve multistep equations involving squares and square roots as well as complex fractions.

Simplify.

SPIRAL REVIEW

729

Solve for x. 1.

4x + 3x = 49

4(x − 2) = 4

x=7

2.5(6x + 1) + 1.5 = 40

-5(3x − 4) − 5x = 85

x = 2.4

x=3

x = -3.25

Vocabulary L E A R N A ND C O NNE C T

y Algebraic Equation - a math sentence with an equal sign that is solved by isolating the variable y Inverse operations - operations that undo one another, like addition with subtraction and multiplication with division

To solve an equation, isolate the variable (get it alone) by using inverse operations. Whatever is done to one side is done to the other side.

y Notecards

√(45+-9) + 8 [14]

Once students have finished working, ask a student to walk the class through the steps of simplifying the expression with the fraction. Then, have a student walk the class through the steps of simplifying the expression with a root. Tell students that we will be working with equations that have parts like these expressions and that it is important to understand where things are grouped in order to be able to simplify and solve them correctly. Guiding Questions: 1. What do the parentheses in these expressions show? [How parts of an expression are grouped] 2. Where are parentheses placed in the expressions above? [in the numerators and denominators of fractions, under a square root] 3. Why are the parentheses helpful in simplifying these expressions? Why might they be helpful in simplifying and solving algebraic equations? [Accept a range of answers.]

There will be two possible answers, one positive and one negative. © Lighthouse Curriculum. Copying strictly prohibited.

(6 − 18) [-12] [-4] (24 − 9) [15] = [5]

−

Remember that square roots and fractions can act like parentheses. To solve an equation with a variable that is squared, apply the square root to both sides.

Write the following expressions on the board and have students simplify them:

When isolating variables, follow the order of operations in reverse - SADMEP. First, addition/subtraction, then multiplication/division, and finally exponents/roots. Once parentheses are alone on one side of the equal sign, they can be removed. Then, start over using SADMEP to solve what is left.

Materials

PRE-LESSON WARM-UP

Inverse Operations

Check

x2 − 3 = 6 +3 +3 x2 = 9 x = 3 or -3

To solve an equation with a variable under a square root, square both sides.

(3)2 − 3 = 6 (-3)2 − 3 = 6

Then, you can use SADMEP to solve what was in the square root.

2x = 4 2 2 2x = 4 2x = 16 ÷2 ÷2 x=8

2(8) = 4

Check

A P P LY Draw parentheses inside square roots and to group the top of fractions. Then, write the first operation you would use to solve. 1.

(5 + x) = 15 9

Multiplication

(x) + 4 = 5

( 8 − x)

+ 7 = 20

Subtraction

( x − 6) = 8

Squaring

( x − 8)

= 12

Multiplication

Vocabulary Equation - a math sentence with an equal sign Inverse operations - operations that undo one another, like addition with subtraction and multiplication with division

Level H

Chapter 3

Lesson 2

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Tell students that they began simplifying and solving equations in the last lesson, and this lesson will focus on the skills needed to simplify and solve more complex equations involving squares, square roots, and fractions. Use the chart in the Learn and Connect section to go over which operations are the opposite or inverse of one another, and tell students that they will be using inverse operations to solve these equations. Explain that similarly, when solving, the reverse order of operations or SADMEP is used. It is important to remember that roots are in the same category as exponents and that fractions and roots can act like parentheses. Parentheses can be drawn in to remind us of what is grouped. Go over the two examples in the Learn and Connect section that show how to solve equations involving squares and square roots. Emphasize that these inverse operations undo another. Therefore, when they square a square root, all that is left is what was under the root to begin with. Then, they can use SADMEP to solve. Also remind students that there are two solutions to a square root. One is positive, the other negative.

APPLY AND DEVELOP SKILLS (Practice) Have students complete the problems in the Apply section, then review them as a class to ensure understanding. Make sure that their parentheses placement is correct and that they understand which operation they will need Lighthouse MATH Level H | Teacher's Guide

Exercise | 3-2 Name Solve. 1.

x – 3.5 = 12.6

g = -16 -8

x = 16.1

-5x – 2 = 13

3y + 7 = 22

x = -3

v2 = 144

x2 + 9 = 25

17.

x = 13

n–2 = -8 10

s = 2.5

13. -2d – 9.5 = 12.5

3( c) = 12

10. w2 – 5 = 20

c = 16

w = 5 or -5

14.

p + 8 = 10

19.

-2c = 10

20. 5x2 + 3 = 128

c = -50

x = 5 or -5

Students should write SADMEP at the top of their papers to remind them of the order for solving more complex equations. Students should add parentheses to numerators and denominators of fractions as well as inside of square roots to remind them to wait to solve those parts until they have dealt with everything else first.

15. 3(j2) = 27 j = 3 or -3

p=4

p=8

STRUGGLING LEARNERS

76 + x = 29 x = -47

d = -11

18. (p + 2)2 = 100

n = -78

s = 6.25

x = 4 or -4

k = 30

z = 30

x+3=4

v = 12 or -12

y=5

k – 7 = -5 12. 15

z 11. 18 + = 23 6

16.

g = 128

Study both equations. Then, answer the questions. 21.

x + 8 = 10

and

x + 8 = 10

22.

x +4=7 8

and

x+4 =7 8

EARLY FINISHERS

A. How are the equations different?

One equation has just x under the square root, and

One equation has just x in the numerator, and the

the other has x + 8 under the square root symbol.

other has x + 4 in the numerator.

B. In which equation would you first subtract 8? The second equation

B. In which equation would you first multiply by 8? The second equation

Students should solve problems 1-5 in the Apply section and then create their own problem that involves similar steps to problems 3 and 4.

Write and solve an equation for each word problem.

CHALLENGE AND EXPLORE

24. Pam can create 7 bracelets per hour. She gives 14 bracelets to friends. If she ends up with 21 bracelets, how many hours did she spend making bracelets?

x = 49; x = 7 feet

23. Levi is designing a square garden. The total area of the garden is 49 square feet. What is the length of one side of the garden?

7h – 14 = 21; h = 5 hours

CH AL L ENGE 25. A farmer is building a rectangular pen. Its dimensions are shown to the right. A. If the perimeter is equal to 106 ft, what is the value of x?

x=4

B. Using the value of x from part a, find the area of the pen.

672 ft2

x2 + 5 ft

For problem 25, remind students that terms with x2 are like and can be combined by adding their coefficients.

2x2 ft

C. If the pen is scaled by a scale factor of 2.5, what is the new area? 4,200 ft

Lighthouse Math

Level H

Chapter 3

Lesson 2

ACTIVITY Root of the Matter: The purpose of this activity is for students to refresh their memory of the perfect squares and their square roots. Prepare cards with square root expressions (such as √36, √49, √100) as well as cards with positive and negative integers that are the solutions to these roots. Students should walk around the room and find two people who hold cards that belong in their set (for example, people with √36, 6 and -6 will form a set). Once students have found their group, they should create a fraction that has addition and/or subtraction in the numerator and denominator that will simplify to either the positive or negative solution to the root in their set.

COMMON ERRORS Students may ignore things that should be grouped in problems with fractions or roots. Students may forget the negative solution when solving for the square root of a number.

ASSESS Check problems 10-11, 14-15, 19-20, and 21-22 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

to use first when solving the equation. Solve at least one of these problems with the students. Next, have students work with a partner on problems 1-22 in the Exercise section. Go over problems 8, 9, 12, 16, 17, 21, and 22. Finally, create equations as a class for problems 23 and 24, then have students solve independently.

Level H | 3-3 3-3 | Equations with Variables on Both Sides

PREREQUISITE SKILLS

y Students will be able to solve equations with variables on both sides and identify when simplifications are needed.

DAI LY REVI EW

Objective and Learning Goals

SPIRAL REVIEW

Evaluate. 1.

-72.2 + 61.9 =

-10.3

-4 21 × 2 41 =

-10

1 8

625.5 ÷ 4.5 =

139

-8 32 × 4 45 =

-41 5

Solve. 1.

-8x + 4 = 20 x = -2

2 x=3 3

-13 +

x = 24

x2 – 22 = 27

x = 7 or -7

5x + 11 = 6 x=5

Materials y Notecards

L E A R N A ND C O NNE C T -4x2 – 82 = -7x2 + 26 + 7x2 + 7x2

Some equations have variables on both sides of the equal sign. When you have a problem like this, follow the steps below:

PRE-LESSON WARM-UP

1. Use inverse operations to get the variable to one side of the equation and combine like terms.

Begin by giving students a riddle.

2. Isolate the variable using inverse operations and following the reverse order of operations - SADMEP.

Jimmy has twice as many apples as Brendan. Jimmy has six more apples than Brendan. How many apples does Brendan have?

Guiding Questions: 1. What is unknown in the riddle? [how many apples Brendan has] 2. What equation can we use to model this riddle? [2b = b + 6]

26 + 82

3x2 ÷3

108 ÷3

x2 x

= =

36 6 or -6

A P P LY Complete each step to solve the equations. 1. © Lighthouse Curriculum. Copying strictly prohibited.

Have students discuss the riddle in small groups and then hold a class discussion about what answers students found and what methods they used to solve the riddle.

3x2 – 82 = + 82

2m + 6 = -18

Subtract to isolate the variable.

m = -12

Divide to isolate the variable.

8x + 12 = 16 – 3x

4 3 1 1 z– = z+ 5 4 5 4 3 3 1 z– 4 = 4 5 3 z = 1 5 2 z=1 3

Subtract to combine the variables.

2m = -24

Subtract to combine the variables. Add to isolate the variable. Multiply by the reciprocal to isolate the variable.

2b + 20 = 9b – 22

11x + 12 = 16

Add to combine the variables.

20 = 7b – 22

11x = 4

Subtract to isolate the variable.

42 = 7b

Add to isolate the variable.

Divide to isolate the variable.

b=6

Divide to isolate the variable.

x = 11

12m + 6 = 10m – 18

Level H

Chapter 3

Lesson 3

Subtract to combine the variables.

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect)

Ask students: How do we solve an algebraic equation? [By isolating the variable] Write the algebraic equation from the Learn and Connect section on the board. Ask students what they notice about it that makes it different from other equations they have solved. [It has an x on both sides.] Ask students to turn to a partner and discuss, “What do you think we have to do to this equation in order to be able to solve it?” Have students share what they discussed. Help students come to the conclusion that in order to isolate x in this equation, we must combine the variables on one side. Tell students that we can do this by using inverse operations and remind them that whatever they do to one side, they must do to the other side. Additionally, when combining variables, always look for a way to make the coefficient in front of the variable positive. Ask: Which inverse operation can be used to combine -7x2 with -4x2? [addition] What should be added so that the result is positive? [7x2] Under the equation on the board, write out the step of adding 7x2 to both sides and the equivalent equation that results. At this point, the equation will have a variable only on one side. Ask students to walk you through the remaining steps of isolating the variable. [adding 82 to both sides, dividing by 3, taking the square root] Be sure to write out each step and the resulting equation. Remind students that when we take the square root of 36, there are two possible answers: 6 and -6.

Lighthouse MATH Level H | Teacher's Guide

Exercise | 3-3 Name Solve each equation. 1.

5x + 7 = 3x – 7

-6y – 5 = -3y + 10

b = -8

11g – 8 = 7g + 4

2x – 36 = x

x = 6 or -6

m=8

11. 5s – 54 = 2s + 138

y = 4 or -4

Have students circle and box pairs of like terms. Have students write SADMEP on the top of their page to remind them of the inverse order of operations.

3 1 m+5= m+9 4 4

v=2

10. 6y + 4 = 5y + 20 2

STRUGGLING LEARNERS

-10d – 21 = 5d + 9 d = -2

3.2v – 1.6 = 1.2v + 2.4

t = -2 2

y = -5

4.5t + 1.2 = 2.5t – 2.8

g=3 2

4b + 9 = 3b + 1

x = -7

s = 8 or -8

12. 7y – 3 = 9y 1

y = -1 2

Answer the error analysis question. 13. Diana solved an equation. Do you agree with her solution? Why or why not?

EARLY FINISHERS

27x + 56 = 13x + 42 − 13x − 13x No, I do not agree with her solution. She forgot to subtract

14x = 42 ÷ 14x ÷ 14x x=3

Have students return to a problem in the Exercise section and begin solving in a different way. For example, in problem 1, if they began by subtracting 3x from both sides, have them begin by subtracting 5x instead and note how that changes the course of solving.

56 from both sides, which would yield 14x = -14, so x = -1.

Write an equation for each word problem and solve. 14. John buys 5 backpacks and 2 binders for the same price that Ben pays for 3 backpacks and 6 binders. Each backpack costs x dollars, and each binder costs $4. What is the cost of one backpack?

15. Jack earns $14 per hour working at the bookstore. He also gets a $18 bonus each week. His friend Mike earns $17 per hour with no bonus. After how many hours of work will they earn the same amount?

5x + 8 = 3x + 24; x = $8

14x + 18 = 17x; x = 6 weeks

8x + 40 = 10x + 20; x = $10

20 + 20x = 40x; x = 1 month

CH AL L ENGE 18. Two engineers are hired to launch satellites for different companies: Company A pays a flat yearly salary of 1.2 × 105 dollars plus 3 × 104 dollars per satellite launched. Company B pays 6 × 104 dollars per satellite launched, with no base salary. A. After how many satellites will both engineers make the same total amount of money?

B. If they each launch 6 satellites, who earns more, and by how much?

1.2 × 105 + (3 × 104)x = (6 × 104)x; x = 4 satellites

Lighthouse Math

CHALLENGE AND EXPLORE

17. A gardener charges a one-time setup fee of $20 plus $20 per month. Another gardener charges only $40 per month with no setup fee. After how many months will the total cost be the same? © Lighthouse Curriculum. Copying strictly prohibited.

16. A vendor sells t-shirts at a local market. On Sunday, he sells 8 t-shirts at x dollars each and makes $40 in tips. On Monday, he sells 10 t-shirts at the same price but gets only $20 in tips. How much is each t-shirt if he makes the same total amount of money each day?

Level H

Company B’s engineer earns $60,000 more

Chapter 3

Lesson 3

If students struggle to set up the equation for problem 18, advise them to use smaller numbers first and replace them with the numbers written in scientific notation after. Then, remind students of the rules for adding and subtracting (the exponents on the powers of ten must match) as well as multiplying and dividing (multiply or divide the coefficients and either add or subtract the exponents on the power of ten) in scientific notation. For part B, students will need to use substitution to find out how much each company pays.

Have students solve problems 1 and 3 in the Apply section with a partner. Review the answers and strategies used as a class. Next, have students solve problems 2 and 4 independently. Then, have them swap their work with a friend to compare their answers. Review as a class to ensure understanding. Solve problem 6 from the Exercise section together as a class. Then, have students solve problems 1-5 and 7-12 independently. Solve problem 16 together as a class and then have students complete problems 13-15, and 17 independently. Review the answers, discussing the error made in problem 13, as well as strategies for writing equations for the word problems.

ACTIVITY Equation Walk-the-Room: Prepare notecards with equations that have variables on both sides. Each card should be labeled with a letter. Any card with the same letter has the same solution. Give each student a card to solve. Then, have them walk around the room to find another card with the same letter to see if they solved theirs correctly. If the pair solved correctly, their letters and answers should match. If someone solved incorrectly, they will need to work together to find and fix the error to discover which answer was correct.

COMMON ERRORS Students may forget to combine all like terms. Students may forget negative or positive sign rules when combining like terms.

ASSESS Check the even-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 3-4 3-4 | More Equations with Variables on Both Sides

PREREQUISITE SKILLS

y Students will be able to solve multistep linear equations, including the distributive property, with variables on both sides.

DAI LY REVI EW

Objective and Learning Goals

SPIRAL REVIEW

Simplify each expression using the distributive property. 2 ) 5

-5(3z +

6.7(2h – 6)

-15z – 2

18(-4b + 6)

-72b + 108

13.4h – 40.2

3x(x – 12)

3x2 – 36x

Solve each equation. 1.

5x + 7 = -2x + 56

Materials

t – 17 =

x=7

y Colored markers y Papers/ Posters

1 t – 11 3

2.1y – 4.1 = 18.3 – 1.1y

t=9

y=7

PRE-LESSON WARM-UP

Guiding Questions: 1. How do we simplify an equation when there is an x inside the parentheses? [use the distributive property] 2. When solving equations with variables on both sides, what do we need to do, and how do we do it? [get the variables on the same side using inverse operations, combine like terms]

When you have a problem like this, follow the steps below:

4(x + 3) = 8x – 3(2x – 4) 4x + 12 = 8x – 6x + 12 4x + 12 = 2x + 12 – 2x – 2x

1. Complete the distributive property.

2x + 12 = 12 – 12 – 12

2. Combine like terms. 3. Get the variable on one side of the equation.

2x = 0 ÷2 ÷2

4. Use inverse operations to isolate the variable.

Then write the problem 5(x + 2) − 3 = 2x + 16 on the board. Say: x = 3. How can I check my answer? [by substituting 3 for both “x”s] Have students walk you through the steps for plugging in 3 and simplifying.

k=7

L E A R N A ND C O NNE C T Some equations with variables on both sides of the equal sign need to be simplified before they can be solved.

Write the example 3(x + 2) = 18 on the board. Say: The answer to this problem is x = 4. How can I check my work? [by substituting 4 for x in the equation and seeing if it simplifies to a true statement] Ask students to walk you through the steps of plugging in 4 for x and simplifying. Emphasize the importance of following the order of operations.

k + 13 =k–2 4

x=0

A P P LY Complete each step to solve the equations. 1.

2(3x + 4) = 4x + 4(x + 6)

19 – 3(4x – 2) = 7x – 2(5x + 1)

6x + 8 = 4x + 4x + 24

Complete the distributive property.

19 – 12x + 6 = 7x – 10x – 2

Complete the distributive property.

6x + 8 = 8x + 24

Combine like terms.

-12x + 25 = -3x – 2

Combine like terms.

8 = 2x + 24

Subtract.

-9x + 25 = -2

-16 = 2x

Subtract.

-9x = -27

Subtract.

-8 = x

Divide.

x=3

Divide.

Level H

Chapter 3

Add.

Lesson 4

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect)

Write the problem from the Learn and Connect section on the board. Say: Sometimes, a problem needs to be simplified before solving. There are four steps to solving this problem. What do you think we should do first? [Complete the distributive property.] Have a student show this step on the board. Ask: What should we do next? [Combine like terms.] Have a student complete this step on the board. Ask what the next step should be. [move the variables to one side of the equation] Have a student complete this step on the board. Finally ask for the last step. [Isolate the variable.] Have a student complete this step on the board. Point out to students that these steps are all processes that they have learned, and today, we are combining them together.

APPLY AND DEVELOP SKILLS (Practice) Have students complete problems 1 and 2 in the Apply section independently. Review the answers and strategies used as a class. Next, have students complete problems 1-8 in the Exercise section independently. Have students check their answers with a partner. If there are any discrepancies in their answers, they should work together to find the error and fix it. Create equations as a class for problems 9 and 10, then have students solve the equations on their own.

Lighthouse MATH Level H | Teacher's Guide

Exercise | 3-4 Name Solve each equation. x=5

2(3x + 4) + 5 = 4(x + 5) + 3

3(x + 6) = 2(x + 2) + 10

25k − 3(5k − 8) = 40 + 2(6k − 7)

2.5(3x - 4) + 1.2 = 1.5(4x + 2) – 1.3

y = -4

k = -1

x=7

STRUGGLING LEARNERS

t=2

4(2t + 3) + 2t = 3(3t + 5) − 1

50g – 3(7g – 9) = 61 + 5(5g – 10)

40x − 12x + 10 = 32 + 20x − 24

5(g2 – 7) + 18 = 2g2 – (2g2 – 3)

Have students make arrows to remember to distribute to all numbers in the parentheses. Have students circle or box off like terms.

g = -4

x=-

1 4

EARLY FINISHERS

g = 2 or -2

Have students return to the second line of the problem in the Learn and Connect section. Ask students what they could divide the entire problem by to simplify it before continuing to solve. Tell them that sometimes, we can simplify equations in the same way that we do fractions. If all the terms have a common factor, we can divide all the terms by that common factor to make the equation easier to solve.

Solve each equation. 10. A landscaping company charges a $24 equipment fee plus $8 per hour of work. Another company charges $40 plus $6 per hour. After working the same number of hours, the total cost from each company is the same. How many hours did they work?

A local craft shop charges a membership fee of $30 per month. There is also a special promotion where you can pay $20 upfront for a one-time fee and then pay $25 per month. For how many months will the cost of both membership plans be the same? Write and solve an equation to find the number of months. 30m = 20 + 25m; m = 4 months

24 + 8h = 40 + 6h; h = 8 hours © Lighthouse Curriculum. Copying strictly prohibited.

CH AL L ENGE 11. A room in the school library is in the shape of a rectangle. Its dimensions are shown to the right. A. The perimeter of the space is 12(x + 2) ft. What is the value of x?

x=8

B. What is the value of the length and width? l = 40 ft; w = 20 ft C. What is the area of the space?

800 ft2

3x - 4 ft

D. The architects of the library made a scale drawing of the youth area before building it. If it was scaled 1in:5ft, what is the area of the scale drawing?

Lighthouse Math

4x + 8 ft

Level H

32 in2

Chapter 3

Lesson 4

CHALLENGE AND EXPLORE For problem 11, part D, remind students that they can set up a proportion with the scale ratio or use the scale factor (by dividing by 5) to find the new length. Then, they can use the new length to find the new area. Some students may remember that they can square the scale factor and divide it by the old area to find the new area without having to find a missing side length.

Equation Station Circuit: Place posters at stations around the room with a different multistep equation on each poster. Split the class into pairs. With each pair of students using a different colored marker, have students complete the first step for solving the equation on their paper. Tell students to complete only one step but to fully write out the new equation formed as a result of the process they completed. Next, students will rotate to another poster board to complete the next correct step on that equation. Tell students that they must check the previous pair's work before continuing with the next step at each station. At the end, each station should have a fully-solved problem. Review the answers with the class.

COMMON ERRORS Students may forget to distribute a coefficient to all terms in the parentheses. Students may forget negative signs when combining like terms.

ASSESS Check problems 4, 5, 8, and 10 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

ACTIVITY

Level H | 3-5 3-5 | Identifying the Number of Solutions

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to classify linear equations as having one solution, no solution, or infinitely many solutions.

Simplify.

SPIRAL REVIEW

31m – 22

15m – 13 + 16m – 9 5.7x + 19.6 – 2.3x – 4.2

3.4x + 15.4

2(8x + 3) = 3(-3x – 8)

5(2x + 3) = -3(4x – 2) + 10 1

p = 22

Until now, you have only seen equations with one or two solutions. Equations can have a variety of solution types. You may find equations have one solution, no solution, or infinitely many solutions. One Solution

No Solution

Infinitely Many Solutions

An equation has one solution if the isolated variable equals exactly one number.

An equation has no solution if the values at the end do not equal one another.

An equation has infinitely many solutions if the values at the end equal one another.

6p + 22 = 4p – 8

12m + 18 = 6(2m + 3)

10q + 4 = 2(5q + 2)

12m + 18 = 12m + 12 – 12m – 12m

10q + 4 = 10q + 4 – 10q – 10q

18 = 12

4=4

– 22 – 22 2p = -30 ÷2 ÷2

PRE-LESSON WARM-UP

x = 30

– 4p – 4p 2p + 22 = -8

Lighthouse MATH Level H | Teacher's Guide

-12g – 121 1

13t–15

L E A R N A ND C O NNE C T

y 20-30 equation cards

Guiding Questions: 1. Why was scenario A easy? [because they agreed on one time to meet] 2. Why was scenario B frustrating? [because they didn't actually agree on a time to meet] 3. What is the meeting time based off of scenario C? [There isn't one, because any time could work.]

2 4 2 3 t– + t– 3 5 3 5

6x + 18 – 3x = 4(x – 3)

x = -1.2

Materials

p = -15

Write three scenarios on the board. Imagine you and your friend are trying to meet after school. You say ‘Let’s meet at 4:00,’ and your friend says, ‘Okay, 4:00 works for me.’ You say, ‘Let’s meet at 4:00,’ and your friend says, ‘Sorry, I can only meet at 5:00.’ You and your friend both say, ‘I’m free any time today.’ Tell students to turn to their partner and study the scenarios. Have them discuss which sounds the easiest, which the most frustrating, and which one gives the most options. Then, ask them the Guiding Questions below to spark a discussion about the three different types of scenarios.

-59 – 36g – 62 + 24g

Solve each equation.

Vocabulary y One solution - the variable equals exactly one number y No solution - the variable does not have a number that it equals y Infinitely many solution - the variable can equal any value

A P P LY Choose if each equation has one solution, no solution, or infinitely many solutions. 3(x + 2) + 5 = 3(x – 1) + 8

4(x + 2) + 3 = 2(2x + 4) + 3

no solution

5(6x + 4) – 11 = 7(4x + 7)

infinitely many solutions

one solution

Vocabulary One solution - the variable equals exactly one number No solution - the variable does not have a number that it equals Infinitely many solutions - the variable can equal any value

Level H

Chapter 3

Lesson 5

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Begin by writing the problem 6p + 22 = 4p − 8 on the board and ask students to solve. Write the solution, p = -15. Ask: How many solutions are there to this equation? [one] Write “one solution” at the bottom. Next, write 12m + 18 = 6(2m + 2) on the board and ask students to solve. Ask: What do we need to do first? [Distribute] What should we do next? [Move our variables to one side] What are we left with? [18 = 12] Is 18 = 12 a true statement? [No] Say: Since 18 cannot equal 12, we know that this problem has no solution. An equation has no solution if the values at the end do not equal one another. Write “no solution” at the bottom. Next, write 10q + 4 = 2(5q + 2) on the board and have students solve. When students get the solution, 4 = 4, ask: Is this a true statement? [yes] Explain that when this happens, it means that if any number is inserted for x it will always be right. There are infinitely many solutions to this problem. Have students check by inserting 6, 9, 12, and 0 for x. Write “infinitely many solutions” at the bottom of this problem. Ask students if, looking back at their work, they can see why the second problem has no solution [Because the variable m on both sides has a coefficient of 12, but the constant is different. It can’t be true that you have 12 “m”s and 18 but also 12 “m”s and 12.] and why the third problem has infinitely many solutions. [Because the variable q has the same coefficient on both sides, and the constant is the same; it’s the same expression on both sides.]

Exercise | 3-5 Name Write if each equation has one solution, no solution, or infinitely many solutions. 12a + 18 = 6(2a + 3)

4(x + 2) = 2(3x + 1) + 6

infinitely many solutions

9(x − 2) = 3(3x + 4) + 3

4(2x − 5) = 8x − 8 – 12

10. 3(12x + 9) = 7(8x + 6) + 21

Have students circle or box like terms. Have them substitute real numbers for variables so that they see concretely if both sides are equal.

6x − 4 = 2(3x + 4) − 12 infinitely many solutions

5(2x − 3) + 10 = 3(4x − 2)

6(8x − 3) + 7 = 4(12x + 2)

one solution

no solution

11. 125(2x + 10) = 250x + 1,250

12. 9(4x − 8) – 12x = 3(8x + 10) + 20

infinitely many solutions

no solution

one solution

13. 1.2(4x + 5) = 5x + 16.8

no solution

infinitely many solutions

STRUGGLING LEARNERS

6(3x + 2) = 18x + 12 infinitely many solutions

10(y − 3) = 2(5y − 6) + 4

no solution

one solution

14.

1 3 (24x − 20) = (8x + 12) 4 4

one solution

15.

no solution

EARLY FINISHERS Provide an error analysis problem: Jenny solved the equation as follows: 5x + 3 = 5x + 7 − 5x − 5x 3=7 She says: “So x = 7 − 3, which is 4.” Where did Jenny go wrong? [Her algebra was correct, but 3 does not equal 7, so this equation has no solution.]

2 (10x + 10) = 4x + 4 5 infinitely many solutions

Solve each word problem. 16. Kevin wants to buy balloons for a party from Balloon Blast or Party Time, two stores with balloons for sale. Both charge $8 per balloon, but Balloon Blast charges $15 for helium, while Party Time charges $12 for helium. Is there an amount of balloons that he can buy at which the cost is the same for the two stores? Explain.

17. Tilly charges a flat fee of $15 for her artwork and $40 for each hour of work she puts into the piece. Lucy charges a total of $300 for any artwork. How many hours would Tilly need to work in order for the cost of her artwork to be the same as the cost of Lucy's? 1

40h + 15 = 300; 7 8 hours

No, the first store will always be cheaper.

CH AL L ENGE 18. Write a real-life word problem that results in one solution, no solution, and infinitely many solutions. Explain in the context of the situation why it has the solution type that it does. One Solution Answers vary; Sara charges a $5 flat fee for babysitting and $15 per hour. Judy charges $30 per hour. After how many hours will each girl earn the same? There’s one solution because h = 3 hours.

Lighthouse Math

No Solution

Infinitely Many Solutions

Answers vary; One art studio charges Answers vary; Jerry paid a base fee of a $10 appointment fee and $10 per 10 $5 and $0.20 per mile when taking a minutes of work completed. Another ride share. Joe paid a base fee of $5 art studio charges a $5 appointment and $0.20 per mile when taking a taxi. fee and $10 per 10 minutes of work completed. There’s no solution. Since There are infinitely many solutions they charge the same amount per because both will pay the same hour but a different amount for an amount, no matter how many miles appointment fee, the first art studio will are driven. always make more.

Level H

Chapter 3

Lesson 5

APPLY AND DEVELOP SKILLS (Practice) Have students simplify each equation in the Apply section without solving. Then, have students make a prediction as to whether each equation will have one solution, no solution, or infinitely many solutions. Then, solve each equation together. Have students look back at their work to see if they predicted correctly or not. Next, have students complete problems 1-12 in the Exercise section. Then, model problem 15 and have them go back and complete problems 13 and 14. Guide students through problem 16, and then have them complete problem 17 independently.

ACTIVITY Equation Relay Races: The purpose of this activity is for students to practice solving and recognizing equations with different types of solutions. Split students into two teams. Give each team a pile of equation cards. Draw a chart on the board for each team with sections titled: one solution, no solution, and infinitely many solutions. Have the teams race to complete an equation and place it under the correct answer category. The first team to complete all their cards correctly wins!

CHALLENGE AND EXPLORE For problem 18, students will need to create equations for each category. If they have not yet figured out what makes an equation end up with no solution or infinite solutions, give them the following example: 4x + 7 = 4x + 15. Ask: What do you notice is the same, and what is different about the two sides of this equation? [Equal coefficients will cause the variables to cancel out. Since the constants are different, we will be left with an untrue statement.] Then, use “What if” variations: Change one constant so that instead of no solution, the equation will have infinitely many solutions, and ask students to tell you why.

COMMON ERRORS Students may forget to follow the order of operations. Students may confuse no solution with completing the problem incorrectly.

ASSESS Check problems 2, 7, 14, and 17 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

Level H | 3-6 3-6 | Equation Applications

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to create and solve linear equations to solve realworld problems.

Translate each verbal model into an equation.

SPIRAL REVIEW

Materials

Eleven more than 11 + 3x three times a number Eighteen less than x2 – 18 a number squared

The quotient of a number and 5, plus four x ÷ 5 + 4 The product of eight 8x – 6 and a number, minus six

Solve each equation. 1.

5(2x – 3) = 7x + 3 + 3x

y Notecards

5x + 4 = 8(x + 9)

no solution

one solution

4x – 15 + 2x = 3(2x – 15) infinitely many solutions

L E A R N A ND C O NNE C T Harry wants to have a roller skating party with his friends. AstroRoll charges $8 per skate rental and a $43 rink rental fee. BlastRink charges $5 per skate rental and a $100 rink rental fee. How many skates will have to be rented for the cost to be the same at both rinks?

PRE-LESSON WARM-UP

Guiding Questions: What does x represent in an equation? [an unknown number] 1. What does an equal sign mean? [the same as]

Lighthouse MATH Level H | Teacher's Guide

1. Write the equation.

8s + 43 = 5s + 100

2. Solve the equation using inverse operations.

8s + 43 = 5s + 100 − 5s − 5s 3s + 43 = 100 − 43 − 43 3s = 57 ÷3 ÷3 s = 19

3. Understand the solution in the context of the problem.

Write the equation 3x + 2 = 11 on the board. Ask: What does x represent in an equation? [an unknown number we are trying to find] Say: This equation represents the following scenario: I went to the grocery store. I bought 3 bags of apples and also a $2 bag of chips. My total bill was $11. Each bag of apples costs the same amount. Which part of our equation represents the cost of 3 bags of apples? [3x] Which part represents the bag of chips? [2] Which part represents the total bill? [11] In this scenario, what does x represent? [The cost of one bag of apples] Why is there an equal sign? [Because the total cost (11) was the same as 3 times the cost of a bag of apples plus $2] Have students solve the equation. [x = 3] Have students explain the answer in the context of the problem [The price for one bag of apples is $3.] Say: This equation represented a story, a scenario. Each value in the scenario was represented using a number or variable. Today, we will be taking scenarios and translating them into equations in order to answer a question.

The variable s represents the number of skate rentals. There would need to be 19 skate rentals for the party to cost the same amount of money at each rink.

A P P LY Use the word problem to fill in the blanks and solve the equation. 1.

Sam and Mike are both saving their money. Sam already has $50 saved. He plans to save $15 each week. Mike already has $30 saved, and he plans to save $20 each week. After how many weeks will Sam and Mike have saved the same amount of money? Write and solve an equation to find the number of weeks. 50 $ saved already

$ saved per week

$ saved already

$ saved per week

Explain the solution in the context of the problem: At 4 weeks, Sam and Mike will have the same amount saved.

Level H

Chapter 3

Lesson 6

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Start by asking students to read the problem in the Learn and Connect section. Ask for a student to volunteer to paraphrase the problem. [Harry wants to have a roller skating party with his friends. He is comparing the costs of two roller skating rinks.] Ask: What does Harry want to find? [The amount of skates that need to be rented in both rinks for the cost to be the same] Tell students that we are looking to see when the cost of AstroRoll will be equal to the cost of BlastRink. On the board, write AstroRoll = BlastRink. Ask: How much does AstroRoll charge for each skate rental? [$8] How much does AstroRoll charge for their rink fee? [$43] Ask: What is unknown, and how can we represent it in an expression? [the number of skates that will be rented; use the variable s] Ask students to write an expression for the total cost at AstroRoll. [8s + 43] Write this on the board under AstroRoll. Ask: How much does BlastRink charge for each skate rental? [$5] Ask: How much does BlastRink charge for their rink fee? [$100] Ask students to write an expression using the same variable, s, for the total cost of BlastRink. [5s + 100] Write this on the board under BlastRink. Tell students that since we are looking for the costs to be the same, we can put an equal sign between these expressions. Have students solve the equation. [s = 19] Ask: What does 19 represent? [the number of skate rentals/ friends Harry must invite for the rinks to cost the same amount of money] Ask: Why might Harry want to know this? [To help him decide which rink to go with. He can use the number of skate rentals he needs to figure out which would be cheaper, and now he knows that if he needs 19, it will not matter which rink he uses.]

Exercise | 3-6 Name Write and solve an equation for each word problem. Explain the solution in context of the problem.

60 + 3f = 55 + 4f; f = 5 hours; After 5 hours, both

300 + 15s = 500 + 10s; s = 40; After 40 students,

cities will have the same temperatures.

both schools will be spending the same amount.

Mr. Randall sells apples at two different markets. On Sunday, he charges $0.10 per pound, but he has to pay $50 to rent a table at the market. On Monday, he charges $0.15 per pound but pays $55 for a table at the market. How many pounds of apples will he need to sell for his profit to be the same on both days?

EARLY FINISHERS

6h + 30 = 8h + 20; h = 5 hours; After 5 hours, Ed and

100 lbs of apples for the cost to be equal on both days.

Harry will have painted the same number of walls.

50 + 5x = 45 + 6x; x = 5 items; When shipping 5 items,

50x − 55 = 42x − 23; x = 4 days; The squirrels need to collect

the cost for both companies will be the same.

for 4 days for their quantity of acorns to be the same.

Spic and Span Cleaning charges $100 for cleaning supplies, plus $20 per hour worked. Another company charges $120 for supplies, plus $18 per hour. At what point would the cost for a house cleaning be the same?

CH AL L ENGE Emily is saving money to buy a new camera. She already has $100 saved. She plans to save $25 each week. Her sister, Nancy, already has $55 saved and plans to save $40 each week. After how many weeks will Nancy have saved more than Emily? Explain what this means in the context of the problem. 55 + 40w > 100 + 25w; w > 3 weeks; After the 3rd week, Nancy will have saved more than Emily.

Lighthouse Math

Level H

Have students choose a problem from the Exercise section and adjust one part of the scenario to create a new problem (i.e., the starting rate, the cost per item). Then have students solve and write a sentence explaining how their change influenced the result of the problem.

One squirrel buries 50 acorns per day but loses 55 of them. Another squirrel buries 42 acorns per day but loses 23. How many days will the squirrels need to collect acorns in order for them to end up with the same amount?

100 + 20h = 120 + 18h; h = 10; When working for 10 hours, both cleaning companies will cost the same.

Have students write down what they are comparing before creating an algebraic equation. For example: Class A = Class B. Have students highlight like terms for each problem.

Ed and Harry are painting walls. Ed paints 6 walls per hour, and Harry paints 8 walls per hour. Ed has already painted 30 walls, while Harry has painted 20 walls. After how many hours will Ed and Harry have painted the same number of walls?

0.1x − 50 = 0.15x − 55; x = 100 lbs; He will have to sell

Ship-It shipping company charges $50 for packaging, plus $5 per item shipped. Another company charges $45 for packaging, plus $6 per item. When is the cost the same for both companies?

STRUGGLING LEARNERS

Westland Middle School is organizing a field trip. The cost of the trip is $300 to rent the bus, plus $15 per student. Farlake Middle School is organizing a similar field trip. The cost is $500 to rent the bus, plus $10 per student. At what number of students will both schools have the same total cost for the trip?

Chapter 3

Lesson 6

APPLY AND DEVELOP SKILLS (Practice) Read the problem in the Apply section together with students. Then, ask them to fill in the blanks below to write an equation. Check with students to be sure they filled in the blanks correctly. Ask: What does w represent? [the number of weeks] Ask students to solve the equation independently, then review the answer and have a volunteer explain the solution in context of the problem. Next, have students work with a partner to solve each problem in the Exercise section. Remind them to first determine what is being compared as well as what the variable represents. Then, have them write and solve an equation. Review the answers together as a class and address any questions to ensure understanding.

ACTIVITY Creative Equations: The purpose of this activity is to have students work together with a partner to come up with a scenario that an equation could model. Provide a notecard with one equation for each pair of students. Students need to work together to create a scenario that matches the equation. They must write down what the variable represents, what question we are asking about the variable, and what is being done mathematically to that variable that matches the equation. Have students check with another pair to be sure their work is accurate, then have pairs share their scenarios with the class.

CHALLENGE AND EXPLORE For problem 8, point out to students that they are not looking for when Nancy and Emily will have equal amounts of money saved up; rather, they want to know when Nancy will have more savings than Emily. This means that they will not use an equal sign; they will use a greater than sign. Explain that the problem is solved the same way. However, the explanation changes in context. At three weeks, the girls have the same amount of savings, and once the three-week mark passes, Nancy’s savings are higher than Emily’s.

COMMON ERRORS Students may misinterpret what the variable represents in a given problem.

ASSESS Check the odd-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

In Westland City, the temperature at 8 a.m. is 60°F. Every hour, the temperature increases by 3°F. In Jersey City, the temperature at 8 a.m. is 55°F, but it increases by 4°F each hour. After how many hours will the temperature in both cities be the same?

Level H | 3-7 3-7 | Review

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will review concepts introduced in Chapter 3.

5(4y2 − 3) = 7y2 + 49 − 3y2 Apply the distributive property.

Materials y Notecards

Combine like terms.

5(4y − 3) = 7y + 49 − 3y

20y − 15 = 7y + 49 − 3y

20y2 − 15 = 7y2 + 49 − 3y2

20y2 − 15 = 4y2 + 49

Get the variable to one side of the equation.

20y − 15 = 4y2 + 49 − 4y2 − 4y2

16y2 − 15 =

Follow SADMEP to solve.

PRE-LESSON WARM-UP 2(1.5x − 4) + √9= 0.5x + 10

Exponents/Roots

16y − 15 = 49 + 15 + 15 16y2 = 64

16y = 64 ÷ 16 ÷ 16 y2 = 4

y2 = 4

y = 2 or -2

A P P LY

Tell students to write the steps needed to solve the problem without actually solving. Then, have students share what they wrote.

Choose if the equation has one solution, no solution, or infinitely many solutions. B

21x + 59 = 7(3x + 9) – 10

-60 + 16z = 4(4z – 15)

no solution

4x + 6 = 2x + 14

infinitely many solutions

one solution

Match each equation with its correct solution. © Lighthouse Curriculum. Copying strictly prohibited.

Guiding Questions: What is the process for solving for x in a multistep equation? [distribute, combine like terms, move the variable, SADMEP to solve] 1. Why can’t we combine like terms before we distribute? [Because the x inside the parentheses needs to be adjusted by the factor outside the parentheses.]

Multiplication/Division

Write the following problem on the board.

Say: Today, we will review what we have learned about simplifying and solving equations as well as solving real-world problems.

Addition/Subtraction

-3x = 6

x = -36

5x2 + 13 = 138

x = -6

x -30 = -26 + 9

x = 4 or x = -4

8(5x – 9) + 11 = 59

x = -15

7.3x + 2.9 = 4.8x – 12.1

x = -12

3x2 + 18 = 2x2 + 34

x = 3 or x = -3

10.

1 1 (12x + 30) + 9 = (15x + 21) + 2 2 3

x=3

11. 2(2x2 + 8) + 80 = 4(3x2 + 6)

x = 5 or x = -5

Level H

Chapter 3

Lesson 7

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect)

Hand each student a set of nine cards, each with one step for simplifying and solving equations on it: Distribute, combine like terms, move variables, and SADMEP. Write the equation 5(4y2 - 3) = 7y2 + 49 − 3y2 on the board. To solve the problem, ask: What is the first step? [Students raise distribute card] Have a student come up to the board to demonstrate. Ask: What is the next step? [students raise combine like terms card] Have a student come up to the board to circle all the like terms that will be combined. Point out that there are three here, but only the right side can be simply combined. Have the student combine the terms. Ask for the next step. [Students raise move variables card] Have a student demonstrate using inverse operations to move the variables. Ask for the next step. [Students raise SADMEP card] Have different students come to the board to complete the rest of the steps until the solution is found. [y = +/-2] Be sure that the students provide both the positive and negative solution to √4.

APPLY AND DEVELOP SKILLS (Practice) Have students work on the problems in the Apply section with a partner. Then, review the answers and address any questions or problems students encountered to ensure understanding. Have students complete problems 1-14 in the Exercise section independently. Review the answers as a class. Address any challenges students may have encountered. Lighthouse MATH Level H | Teacher's Guide

Exercise | 3-7 Name Write if each equation has one solution, no solution, or infinitely many solutions. 1.

14t – 22 = 2(7t + 9)

6(12x + 5) = 9(5x + 8) + 12

no solution

STRUGGLING LEARNERS

12(2g + 6) = 24g + 72

one solution

infinitely many solutions

Have students highlight like terms. Have students write SADMEP at the top of their page. For word problems, have students determine what the variable represents prior to writing out the equation.

Solve each equation. 4.

-7m + 3(4m + 5) = 50

3d + 6 = d + 26

-36 = -6(9x + 1) – 3

(11y + 24)² = 49y2

m=7

5(2x2 – 12) = 580

2 1 1 3 v–2 = v–3 3 4 3 4

x = 0.5

d = 10

10. 8(4x + 8) = 5x + 7(4x – 10)

x = 8 or -8

11. 6(3x + 5) = 5(3x + 3) – 6

x = 134

EARLY FINISHERS

v = -4 2

y = -6

12. 10(2x + 3) – 4x = 4(2x + 4)

Have students create an error analysis for each topic in this chapter. Have students choose one to share with the class.

x = -1.75

x = -7

Write and solve an equation for each word problem.

CHALLENGE AND EXPLORE

14. Ride With Us, a taxi company, charges a $5 booking fee and $2.50 per mile. Speedy Rides charges $20 flat for any ride. For how many miles traveled will the cost be the same? Explain what this solution means in the context of the problem.

30 + 50h = 80h; h = 1 hour; For 1 hour of tutoring

5 + 2.50m = 20; m = 6 miles; When traveling

the cost for both tutors is the same.

6 miles, both taxi companies will cost the same.

13. Charlie charges $30 to sign up and $50 per hour of tutoring. Arnold charges $80 per hour but no sign up fee. For how many hours will both Charlie and Arnold cost the same? Explain what this solution means in the context of the problem.

CH AL L ENGE 15. Ben and Dave are each hired to paint rooms at a local school. Ben charges a flat rate of $100 per room for painting, plus an additional $50 for materials. Dave, on the other hand, charges a flat rate of $100 per room and does not charge for materials. A. For how many rooms will the cost be the same for both Ben and Dave? There is no number of rooms where they will charge the same. The equation 100r + 50 = 100r yields no solution.

B. How much would Dave have to charge in order for the cost to be the same when 10 rooms were painted? When 5 rooms were painted? 1 room?

Lighthouse Math

Level H

Chapter 3

Lesson 7

$105; $110; $150

For part B of problem 15 in the Exercise section, point out that instead of solving for the number of rooms, they now need to solve for Dave's price per room. Students will need to calculate the amount that Ben would charge for 10 rooms (1000 + 50) and set that equal to 10x (x being Dave's unknown rate and 10 being the number of rooms) and solve for x. They will then do the same for 5 rooms (500 + 50 = 5x) and for 1 room (100 + 50 = x) and solve for x each time.

Tic-Tac-Toe: Split the class into two teams, Team X and Team O. Draw a tictac-toe board on the board. One student from Team X is called up and chooses a square. In order to claim the square, the student must solve an equation. If solved correctly, the student marks an X in the box for their team. If solved incorrectly, the team loses their turn to place and Team O gets a chance to solve. The first team to get three boxes in a row wins. Continue playing rounds until each student gets a turn. For an added bonus, have one or two mystery squares per tic-tac-toe board. If chosen, the student is given a word problem to solve. If solved correctly, the team gets a double turn. If solved incorrectly, the problem is given to the next team. If the next team solves it correctly, they get to fill in the square and go again.

COMMON ERRORS Students may forget to combine all like terms. Students may confuse “no solution” with solving the problem incorrectly. Students may forget positive or negative answers when solving problems with square roots.

ASSESS Check problems 3, 9, and 14 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

ACTIVITY

Chapter 4

In Chapter 4, we will discover

Linear Relations and Functions When two variables are connected to each other, we call that a relationship. In this chapter, we will see different kinds of relationships and how we can model them in different ways. • We will review graphing ordered pairs from a table onto a coordinate plane. • We will discover functions and how their inputs and outputs are related. • We will learn to interpret and draw graphs of functions. • We will discover linear functions and equations and their features. • We will learn about slope and y-intercept and how they can help us graph. • We will learn about how similar triangles relate to slope. • We will translate real-life scenarios into linear equations and compare them.

Level H | 4-0 Chapter 4 | Skill Checklist

Skill 1: Independent and Dependent Variables Match each variable to the correct type. 1. Independent Variable Something that changes in a situation, input The x variable in y = 3x

Every hour, Mark walks 4 miles. •

The number of miles (m) that Mark walks

Independent Variable

•

The number of hours (h) that Mark walks

Dependent Variable

Each bike costs $50. •

The number of bikes bought (b)

Independent Variable

•

The total cost (c)

Dependent Variable

Objective and Learning Goals

The thing that is affected by the change, output

Students will review skills needed for Chapter 4:

3+b=p

3+p=b

Ariel (a) eats two more cookies than Dana (d). d+2=a

2d = a

2a = d

I can identify independent and dependent variables.

out 4 correct

Skill 2: Input/Output Tables Complete the tables using the equations.

c = 1.53f # of flowers

Cost f

y = 3x

d=c+5

n 3

$1.53

$3.06

$4.59

$6.12

Level H

I can use an equation to complete an input/output table.

Chapter 4

Skill Checklist

Skill 1: Independent and Dependent Variables y Remind students that we use the word variable when we are talking about an unknown value (often, the letter x or y). y Remind students that variable means “changing,” and it refers to the things in a relationship that are changing. y Using the gray box, review how to identify the independent and dependent variables. y Have students complete problems 1-4 and review the answers and strategies as a class.

Lighthouse MATH Level H | Teacher's Guide

3p = b

2+a=d

out 3 correct

A teacher places 3 pencils (p) in each pencil box (b). 3b = p

y Independent and dependent variables y Input/output tables y The coordinate plane y Equivalent ratios y Absolute value

Choose the correct equation for each scenario.

The y variable in y = 3x

Lighthouse Math

Skill 2: Input/Output Tables y Remind students that a table can be used to keep track of values in a relationship. y When a relationship has a rule or formula, one side of the table shows the values that are being put in to the formula, and the other side shows the corresponding values that come out. y Using the gray box, review how to use a formula to fill an input/output table. y Remind students of the importance of identifying each variable in the context of the problem. y Have students complete problems 1-3 and review the answers and strategies as a class.

Name

Skill 3: The Coordinate Plane y-axis

Label the parts on the coordinate plane. 1. 4

(-2, 3)

Quadrant II

Quadrant I

2 1

-4

-3

-2

-1

x-axis

(1, 1) 1

x-axis y-axis origin Quadrant I Quadrant II Quadrant III Quadrant IV

y-axis

Quadrant IV

Quadrant I x-axis

origin -5

Quadrant III

Quadrant II -5

-1 -2

Quadrant III

-3

(2, -2)

Quadrant IV

-4

(-4, -4)

Write the coordinates of each point on the graph. Then, plot the rest of the points. 10

2. 4.

(3, 2)

c (2, 4)

(9, 7)

d (6, 9)

d b

I can find and plot points on the coordinate plane.

out 5 correct

Skill 4: Equivalent Ratios

$4.96 $3.72 = 16 oz 12 oz 4.96 * 12 = 16 * 3.72 59.52 = 59.52

Determine if the ratios are equivalent. Write = or ≠. 1.

1 8

2 16

18 24

8 12

≠

5 12

≠

15 45

For each ratio, write two equivalent ratios. 4.

5 = 7

10 14

20 28

100 10 = 100 = 1,000

1 10

I can identify and find equivalent ratios.

out 5 correct

Skill 5: Absolute Value

|-3| = 3

|3| = 3

Solve. 1.

out 3 correct

Lighthouse Math

|5| =

|-3| =

|-10| = 10

I can find the absolute value of a number.

Level H

Chapter 4

Skill Checklist

Skill 3: The Coordinate Plane

Skill 4: Equivalent Ratios

Skill 5: Absolute Value

y Remind students that the y-axis is the vertical axis on a graph, the x-axis is the horizontal axis, and that both are number lines that continue forever in two directions. y Tell students that the two axes cross at the point (0, 0), also called the origin. y Remind students that ordered pairs must be written as (x, y). y Using the image in the gray box, review the four quadrants of the coordinate plane and how the ordered pairs within each quadrant are similar to each other. y Have students complete problems 1-5 and review the answers and strategies as a class.

y Remind students that we can set up a proportion to compare ratios. y Remind students that units should be lined up in the numerator and denominator of a proportion. y Use the proportion in the gray box to review how to cross multiply to check for equality in a proportion. y Have students complete problems 1-5 and review the answers and strategies as a class.

y Remind students that absolute value means how far away a number is from 0 on the number line. y Remind students that the absolute value of a number is always positive. y Have students complete problems 1-3 and review the answers and strategies as a class.

Lighthouse MATH Level H | Teacher's Guide

Brand B

Brand A

Level H | 4-1 4-1 | Introduction to Graphing

PREREQUISITE SKILLS

y Students will identify features of a graph and coordinate plane (x- and y-axes, origin, coordinates, quadrants). y Students will create an input/output table and/or a list of coordinates for an equation and graph it.

DAI LY REVI EW

Objective and Learning Goals

Find the letter for the ordered pairs. 1.

(-4, -3)

(3, 2)

C B

(-3, 3)

(3, -3)

-5

5 D

C -5

SPIRAL REVIEW

Solve. 1.

x + 7 = -5 x=

y2 = 81

-12

-2a + 4a − 10 = 20

y = 9 or -9

2x + 6 = 4x − 8 7

L E A R N A ND C O NNE C T

Vocabulary

Materials y Notecards

y-axis

I (2, 5)

(1, 3)

Rule: y = 2x + 1 (x) input

Rule: y = 2x + 1

(y) output

Ordered pairs

-2

y = 2(-2) + 1

-3

(-2, -3)

-1

y = 2(-1) + 1

-1

(-1, -1)

y = 2(0) + 1

(0, 1)

y = 2(1) + 1

(1, 3)

y = 2(2) + 1

(2, 5)

(0, 1) -10

-5

x-axis 0

(-1, -1)

(-2, -3) -5

III

Did you know? -10 The ordered pair (0,0) is called the origin.

A P P LY Fill in the input/output table for each rule. © Lighthouse Curriculum. Copying strictly prohibited.

y Coordinate plane - a grid that helps to locate points using two number lines: x-axis and y-axis; divided into four quadrants labeled I, II, III, IV y Ordered pairs - two numbers that tell the location of a point on a coordinate plane (x, y) y Input - values that are chosen to put into a function, represented by the letter x y Output - values that depend on the input and the rule of the function, represented by the letter y y Origin - the point (0,0) on the coordinate plane

An input/output table shows x and y values for a particular rule. The output, or y value, will change based on the input, or x value. The x and y values from the table can be written as ordered pairs. The ordered pairs can be plotted on the graph.

Rule: y = 5x

Rule: y = 2x + 3

Rule: y = x + 1

-1

-5 0 5 10 15 20

-2

-1

-1 0

-3 -2 -1 0 1 2

-2 -1 0 1 2 3

0 1 2 3 4

2 3

5 7 9

Vocabulary

PRE-LESSON WARM-UP Ask students to imagine they have a machine that doubles the value of whatever is put into it. If they put in $5, they will get back $10. If they put in $20, they will get back $40.

Draw a blank table on the board and ask students to help you fill it in: Input $1.15 $2.41 $3.98

Output [$2.30] [$4.82] [$7.96]

Ask students to give you more examples of inputs and their corresponding outputs. After a few numbers, tell students that the machine broke, and now, every time a value is put in, instead of doubling its value, it lowers it by $0.10. Create a new table with a list of a few input and output values. Guiding Questions: 1. How can a table help us when we are looking at a relationship such as the moneydoubling machine? [helps us organize the values, helps us predict what values we can get in a future situation] 2. How did you find the output values in the first scenario? In the second? [multiplied the input by 2; subtracted 0.10 from the input] Lighthouse MATH Level H | Teacher's Guide

Coordinate plane - a grid that helps to locate points using two number lines: x-axis and y-axis; divided into 4 quadrants labeled I, II, III, IV Ordered pairs - two numbers that tell you the location of a point on a coordinate plane (x,y) Input - values that are chosen to put into a function, represented by the letter x Output - values that depend on the input and the rule of the function, represented by the letter y Origin - the point (0,0) on the coordinate plane

Level H

Chapter 4

Lesson 1

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Use the table in the Learn and Connect section to review how to find the input and output values when a rule is given. Tell students that the rule is really an equation because it tells us what to do with our input, or x-value, in order to get the output, or y-value that goes with it. When a number is put into a equation, we replace the x in that rule with that number and then follow the order of operations to find the corresponding output or y-value. Then, we can use the table to create a set of ordered pairs to graph the relationship described by the rule. Go through the examples in the table, asking students to help you calculate the output for each input. Next, help students compile the corresponding x’s and y’s into ordered pairs, reminding them that x always comes first. Then, show students how to graph an ordered pair by starting at the origin, finding the x-value on the x-axis, moving up or down based on the y-value, and then drawing a dot at that point. Ask students what shape we see on the graph after plotting all of the ordered pairs from the table. [a line]

APPLY AND DEVELOP SKILLS (Practice) Have students work with a partner to complete the tables in the Apply section. Tell students that in problem 2, they need to find some y-values and also some x-values. Remind them that they will need to replace the y-value in order to solve for x. In problem 3, students will need to pick their own x-values and find

Exercise | 4-1 Name Fill in the tables using the given equations. List the ordered pairs, then plot them on the graph below. 1.

Rule: y = x2

Rule: y = 2x

Ordered pairs

-2

(-2,4)

-2

-4

(-2, -4)

-2

(-2, 2)

-1

( -1,1)

-1

-2

( -1, -2)

-1

( -1, 1)

(0,0)

(0, 0)

( 1,1)

( 1, 2)

-1

( 1, -1)

( 2,4)

( 2, 4)

-2

( 2, -2)

-5

Students can benefit from creating tables such as the one in the Learn and Connect section where there is a column for work in between the input and output columns. This can help students organize and keep track of their work and remind them of what steps they need to take.

-5

STRUGGLING LEARNERS

Rule: y = -x

-5

EARLY FINISHERS

Use the graph to fill in the input/output table. 4.

-5

-2

-5

-1

-3

-1

-5

-1

Students can return to problems 1-3 in the Apply section and create a graph for each table.

CHALLENGE AND EXPLORE

A bike rental shop charges a $2.00 fee plus $2 per hour to rent a bike. A. Write an equation to model the cost (y) of renting a bike for x hours. y = 2x + 2 B. Create an input/output table. x

C. Graph the ordered pairs. Then, connect them with a line.

CH AL L ENGE Draw a line to connect the dots. Then, write the rule. 7.

(4, 9) (3, 7) (2, 5)

(-2, 11)

(-1, 8)

Rule:

(2, 11)

y = 5x + 1

(1, 6)

(0, 1) 0

(4, 21) (3, 16)

Rule: y = 2x + 1

(1, 3)

3 -3

-3

Lighthouse Math

Level H

Chapter 4

Exercise 1

Rule:

(0, 5) (1, 2)

For problems 7-9, students will need to figure out what rule created the ordered pairs that were graphed. Recommend to students that they create a table of values from the given ordered pairs so that they can look for patterns and figure out what was done to the input to create the corresponding output.

y = -3x + 5

3 (2, -1)

ACTIVITY Rule Match: The purpose of this activity is to have students practice recognizing how different rules will produce different outputs and form different patterns on a graph. Write several sets of equations, tables, and graphs that match each other on note cards and pass them out to students. Have students form groups once they have found the rule, table, or graph that matches their card. When they have found their partners, they should work together to write a new rule, table, and graph that go together.

COMMON ERRORS Students may forget to follow the order of operations when using a rule. Students may reverse the x- and y-coordinates when plotting ordered pairs.

ASSESS Give students a blank coordinate plane and the rule y = 2x +5. Have students create a input/output table with five numbers and plot the ordered pairs to create a line. Lighthouse MATH Level H | Teacher's Guide

the corresponding y-values. If they are unsure where to start, suggest beginning at -3 and working their way up. Review the answers and strategies and then have students work independently on problems 1-5 in the Exercise section. Review the answers as a class to ensure understanding and then read and solve problem 6 together as a class.

Level H | 4-2 4-2 | Functional Relationships

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to define a function as a rule that assigns exactly one output to each input and identify whether a given relation is a function.

Solve for y for each given x.

3. SPIRAL REVIEW

Tell students that today, we are going to be talking about a concept that is similar to a vending machine. Explain that a vending machine is loaded so that each lane is filled with one type of product. Ask students: After you give money to a vending machine, how do you choose your product? [You press the button that identifies the lane your desired product is in.] Then what happens? [The machine dispenses the product.] Use the Guiding Questions below to begin a discussion that will lead into the unit on functions. Guiding Questions: 1. What is the input and what is the output in the vending machine scenario? [input - the button(s) you press; output - the product you get] 2. What is the relationship between the input and output in this scenario? [The product you get (output) depends on what button you press (input).] 3. What happens if the person behind you in line presses the same button as you? [They will get the same product.]

Lighthouse MATH Level H | Teacher's Guide

y=x−1

x = -1, y =

-2

y = 5x

y = 3x − 1

x = 0, y =

-1

x = 0, y =

x -1 0 1 2

(-1,0)

y 0 1 2 3

(0,1) (1,2)

-3

-2

-1

(-3,-1) (-2,0) (-1,1) (0,2)

(2,3)

L E A R N A ND C O NNE C T A function is a relationship where each input (x) has exactly one output (y). Input values cannot repeat, but output values can. Think of a function like a machine: you put in one input and get one specific output.

Word form

Function

Not a Function

Pressing the letter R on a keyboard prints the letter R.

Pressing the letter R on a broken keyboard sometimes prints the letter R, sometimes the letter T.

Ordered pairs

(1, 2), (2, 3), (3, 4) x y

Table Equation

0 2

1 4

2 6

(1, 2), (1, 3), (2, 4) 3 8

x y

1 4

1 8

-10

3 16

One input (2) with infinite different outputs

When any vertical line is drawn on the graph of a function, it will touch the line or curve in only one place.

2 12

x=2

y = 2x + 1

Graph Use the Vertical Line Test

PRE-LESSON WARM-UP

Write four ordered pairs from each table.

Vocabulary y Relation - a set of ordered pairs (x, y) that shows how two variables are connected y Function - a relation where each input (x) has exactly one output (y) y Vertical line test - drawing a vertical line on a graph to help determine if a the graph is a function. If any vertical line passes through the line or curve of the graph one time only, the graph is a function. y Input - values that are chosen to put into a function, represented by the letter x y Output - values that depend on the input and the rule of the function, represented by the letter y

y = 2x + 3 x = 1, y =

-10

A P P LY Each of the following are NOT a function. Find an input value that gives more than one output. 1.

(-5,4) (-3,6) (-3,7) -3 gives 6 and 7

x y

8.5 2.4

8.5 2.5

8.6 2.6

8.5 gives 2.4 and 2.5

2 -2

-2 gives all values

Vocabulary Relation - a set of ordered pairs (x,y) that shows how two variables are connected Function - a relation where each input (x) has exactly one output (y) Vertical line test - drawing a vertical line on a graph to help determine if the graph is a function: if it is a function, it will only touch the line or curve in one place

Level H

Chapter 4

Lesson 2

Input - values that are chosen to put into a function, represented by the letter x Output - values that depend on the input and the rule of the function, represented by the letter y

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Read the definition of a function from the Learn and Connect section and ask students how a vending machine represents a real-life situation that is a function. [Each input - the button you press - has exactly one output - your desired product.] Use the table in the Learn and Connect section to teach students how to recognize a function in different forms and how to recognize when a relationship is not a function. Emphasize that in the broken keyboard example, sometimes the letter R will be typed when the R button is pressed, and sometimes the letter T will be typed. Since it could be either, this is not a function. Show students that the x-value 1 repeats in the list of coordinate pairs and in the table in the non-function column. Contrast the equations that are listed in the next row. Have students notice that when the equation says x = 2 (or any number), we are being told what the input is, and since there is no output listed, all the outputs will be valid, therefore the same input will have multiple outputs. Show students how the graph for x = 2 is a vertical line to reinforce this idea. Leave this graph on the board as you begin to explain about the graphs of functions. Tell students that there a special test we can do to see if a graph is truly a function. It is called the vertical line test. Any vertical line drawn anywhere on the coordinate plane will cross the graph of a function only once. Remind students that the graph of x = 2 was a vertical line, so it is automatically not a function. Show students the example and non-example from the graph row on the table in the Learn and Connect section.

Exercise | 4-2 Name Tell if the ordered pairs represent a function. Write YES for function or NO for not a function. Yes

(1, 2), (2, 3), (3, 4), (4, 5)

(1, 2), (1, 3), (2, 4)

(0, 0), (1, 1), (2, 2), (3, 3)

Yes

STRUGGLING LEARNERS

Yes

(1, 5), (2, 5), (3, 5), (4, 5)

(5, 1), (6, 2), (5, 3)

(7, 10), (8, 11), (9, 12), (10, 13)

Students who struggle to notice when an input repeats should be encouraged to write all inputs next to their corresponding outputs in a table to organize the information. Students who struggle with the vertical line test should use the vertical side of a piece of paper as their vertical line. They can slide their paper horizontally along the graph. If the piece of paper crosses the graph more than once, they will know it is not a function.

Yes

Tell if the tables are a function. Write YES for function or NO for not a function. If it is not a function, circle the ordered pairs that show that it is not a function. 7.

10.

-1

Yes

9. No

Yes

Tell if the following graphs show a function. Write YES for a function or NO for not a function. Draw the vertical line test on the graph. 25

11.

12.

20 15

13.

20 15

Yes

-5

EARLY FINISHERS

-5

Students can go back to any relation they marked as a non-function and fix it so that it is a valid function.

-5

Use the equation x = | y | to complete the table and answer the questions. x

-1

A. Is the equation a function? How do you know?

B. Find another example of an input that gives more than one output.

No; 1 gives -1 and 1

14.

3 gives 3 and -3

CH AL L ENGE 15. A vending machine gives out snacks based on a code you enter. The table shows what each person gets.

Person Jack

Chips

Does the relationship between the code entered and the snack received represent a function? Explain.

James

Chocolate

Mark

Chips

Billy

Dave

Candy

Not a function, because the same input ("B2") gave two different outputs ("Chocolate" and "Candy"). In a function, each input must have only one output.

Lighthouse Math

Level H

Chapter 4

Code Entered Snack Received

Exercise 2

CHALLENGE AND EXPLORE Problem 15 relates back to the original example of a vending machine, except this time, the vending machine does not represent a function.

Have students work in pairs to complete problems 1-3 in the Apply section, then review the answers as a class. Next, have them work independently on problems 1-14 in the Exercise section and review the answers to ensure understanding across all types of problems.

ACTIVITY Function Fix: The purpose of this activity is to have students become comfortable recognizing when something is not a function. Give students a list of non-functions in all the different forms and have them fix them in some way so that they will be valid functions. Using the examples of non-functions from the table in the Learn and Connect section, students could say, “Fix the broken keyboard so that only R will type, change (1, 3) to (3, 3) so that each x-value has exactly one unique y-value, change the output for the second 1 in the table so that it is also 4, change x = 2 to y = 2. Now it is a horizontal line. Eliminate the bottom curve on the graph so that a vertical line will only pass through once."

COMMON ERRORS Students may think that a function with a repeating output is not a function. Students may think that any graph with a curve is not a function.

ASSESS Have students choose one modality of representing a function and write a valid function and non-function example. They should write one sentence explaining why each is a function or not. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 4-3 4-3 | Graphs of Functions

PREREQUISITE SKILLS

y Students will be able to analyze and interpret graphs of functions to describe the qualitative behavior of a relationship.

DAI LY REVI EW

Objective and Learning Goals

SPIRAL REVIEW

Write ordered pairs using the table. 1.

(-1,-2)

(0,0)

(2,4)

(3,6)

x -1 0 1 2 3

(1,2)

y -2 0 2 4 6

Write yes or no to show if each of the following are a function. 1. 3.

Yes

y = 2x + 3 y=7

Yes

Materials y Graph paper y Whiteboard y Dry-erase markers

L E A R N A ND C O NNE C T Morris bikes from his house to the bakery. He then immediately turns around and comes back home. He bikes at the same speed the entire time. The graph shows how far Morris is from home as time passes.

Distance from Home (miles)

PRE-LESSON WARM-UP

Distance from Home Over Time

Until the point (3,18), Morris is moving at a constant speed farther away from home. After the point (3,18), Morris moves closer to home at a constant speed.

Sketch the following graph on the board without a title or labeling the axes and ask students to tell you if it represents a function and why. [Yes, it passes the vertical line test.]

That means that Morris turns around after 3 hours when he is 18 miles away from home.

18 15 12 9 6 3

Returns home

Starts at home

3 4 Time (hours)

A P P LY

0 1 2 3 4 5 6 7 8 9 10

Circle the graph that matches each scenario.

The recipe calls for 2 cups of flour for every cup of water.

John's Jog

3 2 1 0

10 15 20 Time (minutes)

Distance from Home (miles)

John jogs to the park at a steady pace for 15 minutes. He stays at the park for 10 minutes to rest. Then, he jogs home at the same pace.

3 2 1 0

Ratio of Flour to Water

Level H

Chapter 4

4 6 Cups of Water

Lesson 3

15 20 25 30 Time (minutes)

35 40

Ratio of Flour to Water

16 12 8 4 8

Cups of Flour

Distance from Home (miles)

10 9 8 7 6 5 4 3 2 1

Cups of Flour

Distance from Home (miles)

A Snail’s Pace

16 12 8 4 0

4 6 Cups of Water

Lighthouse Math

Time (minutes)

Next, add the title and labels to the axes. Ask for a student to describe what is happening in the graph based on the context added. [The snail takes 5 minutes to go 2 inches, then it doesn’t move for 2 minutes, then it takes 3 minutes for the snail to go 1 more inch.] Guiding Questions: 1. How do we know if a graph represents a function? [Each input will have only one output; a vertical line can be drawn anywhere on the graph, and it will only touch the line once.] 2. How does the value of y change as the value of x increases? [It increases except between 5 and 7; there it stays the same.]

Lighthouse MATH Level H | Teacher's Guide

INTRODUCE THE LESSON (Learn and Connect) Read aloud the scenario in the Learn and Connect section about Morris’s bike trip. Ask students to follow along with their finger on the graph as you read. Next, ask students to put their finger on the title of the graph. Explain to students that it is important to read the title of a graph because it helps us understand what is going on and interpret the story the graph is telling. In this case, we are graphing Morris’s distance from home over time. Next, ask students what the x-variable and y-variable are and how they are measured. [x: time measured in hours; y: distance from home measured in miles] Ask students to put their finger on the point that shows how far away from home Morris is when no time has passed. [0,0] Ask a student to interpret this point on the graph. [When no time has passed, Morris is zero miles away from home; he is at home.] Next, ask students to put their finger on the point where Morris turned around. [(3,18)] Ask a student to explain what this point means. [After 3 hours, Morris has gone 18 miles.] Ask students how they know that Morris turned around here. [Because after this point, the line starts to slope downward, meaning that as more time goes on, Morris is getting closer to home (the y-value is decreasing).] Point out to students that the two lines that form the graph are straight lines. This means that Morris is moving at a constant speed along each line.

Exercise | 4-3 Name Look at the graphs. Choose the real-life scenario that matches. 1.

Marks's Money 40

STRUGGLING LEARNERS

Number of Bunnies in the City 35,000 30,000

Remind students to read the title and labels on a graph before answering questions.

25,000 Bunnies

Dollars

30 20

20,000 15,000 10,000

5,000 0

4 6 Hours Tutoring

1,000

EARLY FINISHERS

Years

A. Mark has $20 and earns another $2 for every hour he tutors.

A. There are 1,000 bunnies in the city, and every year, the number of bunnies doubles.

B. Mark has $20 and doesn’t make any money.

B. There are 1,000 bunnies in the city, and every year, there are 90 more bunnies than the year before.

Ask students to return to the graph in the Learn and Connect section and figure out what speed Morris is traveling at during his bike trip. Remind them that Speed = distance/time. When completed, they should check with a partner and then do the same for other graphs in the chapter, where applicable.

Use the graphs to answer the questions. A family takes a car trip and tracks the number of miles they drive after each hour. Total Distance (miles)

Distance Traveled Over Time

A. After how many miles does the family take a break?

300 240

B. How long is their break?

180 120

C. How far do they drive in 5 hours?

60 0

2 3 Time (hours)

120 miles

1 hour 300 miles

CHALLENGE AND EXPLORE

A bucket is used to catch a drip from the ceiling.

Quarts of Water

Water in the Bucket

A. What most likely happened after 5 hours?

12 10 8 6 4 2

The bucket was emptied. © Lighthouse Curriculum. Copying strictly prohibited.

B. After 9 hours was the ceiling still dripping? How do you know? 0

Yes; the amount of water was still going up.

Hours

CH AL L ENGE Describe the scenario displayed by the graph.

Meters

Mike's Distance from Home

Mike walks for 4 minutes until he is 6 meters

28 24 20 16 12 8 4

away from home. Then, he stops for 2 minutes. He continues at a faster pace for 0

Minutes

Lighthouse Math

For problem 5, ask students to use specific language to describe the graph. For an additional challenge, ask them to calculate Mike’s speed during the three different portions of the graph.

Level H

Chapter 4

4 minutes until he is 19 meters farther away.

Exercise 3

Work together as a class to complete problems 1 and 2 in the Apply section. Discuss how students knew which graph to choose and how they knew which graph not to choose. Next, have students work in pairs to complete problems 1-4 in the Exercise section. Review all of the problems together as a class, asking students to explain their reasoning as you go through them.

ACTIVITY Graph Time: The purpose of this activity is for students to practice interpreting and graphing scenarios. Students will alternate describing a scenario and graphing it with a partner. For example, a student can say: Mikey starts at an altitude of 100 meters. He hikes for 12 minutes at a steady pace and reaches an altitude of 116 meters. He takes a break for five minutes and then continues to hike for 30 minutes, reaching a final altitude of 180 meters. The other student will sketch a graph on their whiteboard and then label the axes and give the graph a title. Their partner will check and then they will switch roles.

COMMON ERRORS Students may misinterpret a horizontal line on a graph as an increase instead of as something staying the same. Students may ignore labels on graphs that give them information about the context of the problem.

ASSESS Check problems 3 and 4 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 4-4 4-4 | Linear Functions

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

(-1,1)(0,0), (1,1), (2,4),(3,9) Yes

200

y=x+3

Yes

100

Yes 100

SPIRAL REVIEW

Label and sketch the graph described. 1.

200

Temperature

y Students will be able to distinguish between linear and nonlinear functions by creating an input/output table and graphing the points.

Is this relation a function? Circle Yes or No.

For the first three hours, the temperature increases at a constant rate. For the next four hours the temperature stays the same. For the last three hours, the temperature drops at a constant rate.

Vocabulary

time (hours)

y Linear function - a relation where the output increases or decreases by the same amount every time y Linear equation - a rule for a linear function y Nonlinear function - a relation where the output does NOT change by the same amount each time

L E A R N A ND C O NNE C T When you graph a linear function, you get a straight line. A linear equation is a rule for a linear function. It will not have any exponents or square roots. Linear Function 10

-5

-2

-8

-1

-4

5 (0, 5)

y Notecards

Write the following table of values on the board and ask students to plot the points on graph paper and then connect the points.

(2, 9) (1, 7)

Materials

PRE-LESSON WARM-UP

Nonlinear Function

(3, 11)

-10

The function is a straight line because the y-value always changes by the same amount as the x-value increases.

The function is a curve because square roots and exponents make the y-value change by different amounts as the x-value increases.

y = 2x + 5

y = x2

A P P LY Circle if the graph is linear or nonlinear. 1.

Linear

40 20 0

Nonlinear 20

Linear

-5

Nonlinear

Linear

20 10 0

Nonlinear 10

Vocabulary

Ask students to describe what is happening to the y-values as the x-values are increasing by 1. [They go up by 4.] Ask students what shape the points created once they were connected. [a straight line] Guiding Questions: 1. What patterns to you see in the table of values? [x-values go up by 1, y-values go up by 4] 2. Why do you think the graph forms a straight line? [because the points follow a pattern]

Lighthouse MATH Level H | Teacher's Guide

Linear function - a relation where the output increases or decreases by the same amount every time whose graph is a straight line Linear equation - a rule for a linear function Nonlinear function - a relation where the output does NOT change by the same amount each time whose graph is curved or not straight

Level H

Chapter 4

Lesson 4

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Tell students that some functions will form a straight line on a graph, and these functions are called linear functions. Any function that is curved or made up of more than one different straight line is called nonlinear. Show the students the two graphs in the Learn and Connect section to emphasize the difference between the graphs of linear and nonlinear functions. Have students compare the equations under each graph. Tell students that an equation for a linear function is called a linear equation, and it will not have any exponents, while an equation for a nonlinear function might have exponents. Next, have students create a table of values for the graph of the linear function based on the ordered pairs listed on the graph. Have them tell you what patterns they see in the table. [The x-values increase by 1, and the y-values increase by 2.] Explain that the reason the graph forms a straight line is that both the y-values and the xvalues follow a pattern. In all linear functions, as the x-values increase by 1, the y-values will always change by a given amount. The x- and y-values of nonlinear functions will not follow a pattern like this. Each time the x-values go up by 1, the y-values will change by a different amount. Students can create a table of values for the graph of the nonlinear function to prove this.

Exercise | 4-4 Name Graph the points to determine if the function is linear or nonlinear. 1.

-1

0 0

-3

Linear or Nonlinear

y -1

-1

-5

x 0

STRUGGLING LEARNERS 10

-10

Help students figure out how much the numbers in each column are increasing or decreasing by and guide them to write this amount and a + or a - between each number in a table. This strategy can help them to look for patterns and determine if a table of values represents a linear or nonlinear function.

Linear or Nonlinear

Create an input/output table and graph the points to determine if the function is linear or nonlinear. 3.

y=x+2

-2

0 1 2 3 4

-1 0 1 2

y = x2

-2

4 1 0 1 4

-1 0 1 2 0

EARLY FINISHERS

Linear or Nonlinear

-5

Students who finish early should write a table of values for the function y = x3. Then, they should graph several ordered pairs and connect the points to see what shape forms.

Write an equation, complete the table, and graph the points to determine if the function is linear or nonlinear. A designer gets a flat fee of $25 when she completes a project. She gets an additional $10 per revision. Let y be the amount of money earned and x be the number of revisions made. y = 10x + 25

Equation:

Linear or Nonlinear

Circle if the equation is a linear or nonlinear function. 6.

y=x−1

Linear or Nonlinear

y = 4x2 + 1

Linear or Nonlinear

y= x

Linear or Nonlinear

y = 3x + 1

Linear or Nonlinear

CH AL L ENGE

CHALLENGE AND EXPLORE For problem 10, students will need to come up with their own linear and nonlinear equation. As a hint, ask them what kinds of equations have variables with exponents [nonlinear].

10. Write a linear equation and a nonlinear equation. Linear equation:

Lighthouse Math

Answers will vary.

Nonlinear equation:

Level H

Chapter 4

Answers will vary.

Exercise 4

Have students complete problems 1-3 in the Apply section independently, then check the answers as a class. To ensure understanding, ask students to explain how they knew that the linear functions were linear, besides the fact that the graph is a straight line. Next, have students complete problems 1-4 in the Exercise section independently and review the answers and strategies as a class. Work together to complete problem 5 as a class by first creating an equation that serves as the rule for the function, then creating a table of values, and finally, plotting and connecting the points on the graph. Ask students at each step whether they think the function is linear or not and why. Finally, work together as a class to determine if the equations in problems 6-9 are linear or not.

ACTIVITY Match It: The purpose of this activity is for students to familiarize themselves with linear and nonlinear functions as they are represented in tables and graphs. Give each student a notecard with either a table or a graph that is either linear or nonlinear. Tell students to go to the right side of the room if their function is linear and the left if it is nonlinear. Then, have students find the table or graph that matches their card. Finally, once students find their match, have them write one sentence with their partner explaining why their function is linear or nonlinear.

COMMON ERRORS Students may graph ordered pairs in order to determine if a function is linear and forget that they can look for patterns in how the variables change. Students may think that the x- and y-variables need to increase in the same way in order for the function to be linear.

ASSESS Check problems 1-4 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 4-5 4-5 | Slope and y-intercept

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will identify the slope and y-intercept from a graph. y Students will describe the slope of a graph as increasing or decreasing.

Write each fraction in simplest form.

SPIRAL REVIEW

25 5

6 12

1 2

1 4

5 20

45 9

Circle the graph that shows a linear function. 60

20 0

Vocabulary y y-intercept - the point where the line crosses the y-axis y Slope - a number that tells how steep a line is; how much it rises or falls for each step y Positive slope - as the x-values increase, so do the y-values y Negative slope - as the x-values increase, the y-values decrease

-5

L E A R N A ND C O NNE C T All linear equations have two important features: a y-intercept and slope that stays the same.

7 6 5

The y-intercept is the y-value at the point where the line crosses the y-axis. The x value is always 0.

Run = 4

4 3

The slope is the steepness of the line. The slope can be calculated using any two points on the line. First, find the rise or difference in height between the two points. Then, find the run, or the difference in width between the two points. Write the slope as a fraction rise .

y-intercept: 1 at the point (0,1)

(4, 4)

Rise = 3

Slope:

1 (0, 1)

3 4

run

Materials

Positive Slope As the x-values increase, so do the y-values.

y Graph paper y Notecards

Negative slope As the x-values increase, the y-values decrease.

Have students plot the points from the two tables of values to create two lines on a graph. Then, ask students to tell you if they are functions, whether or not they are linear, and how they are the same and different.

A P P LY

PRE-LESSON WARM-UP

Find the slope and y- intercept. y

1. Rise 2

7 6 5 4 3 2 1

Run 5

0 1 2 3 4 5

slope = 5 y-intercept =

7 6 5 4 Run 1 3 Rise 1 2 1 0 1 2 3 4 5

slope = 1 y-intercept =

7 Run 2 6 5 4 Rise 6 3 2 1 0 1 2 3 4 5

slope = 3 y-intercept =

-5-4-3 -2 -1

Rise -2

0 -1 -2 -3 -4 -5 -6

Run 3

slope = 3 y-intercept = -6

Vocabulary Y-intercept - the point where the line crosses the y-axis. It tells the value of y when x is 0. Slope - a number that tells how steep a line is - how much it rises or falls for each step Positive slope - as the x-values increase, so do the y-values Negative slope - as the x-values increase, the y-values decrease

Level H

Chapter 4

Lesson 5

Lighthouse Math

-2

-1

INTRODUCE THE LESSON (Learn and Connect)

-2

-4

-6

Tell students that this lesson will cover two important features that all linear equations have: a y-intercept and a slope. Using the graph in the Learn and Connect section as a guide, ask students to put their finger on the y-intercept, the point where the line cross the y-axis. Ask: What are the coordinates of this point? [(0, 1)] Tell students that no matter what the y-intercept is, the x-value of the coordinate pair will always be 0. Ask them why that is. [Because for a point to be on the y-axis, it can’t be to the left or right at all. Therefore its x-value (which tells us how much to move horizontally) must be 0.] Next, ask students to navigate from the y-intercept to the point (4, 4) which is on the line, but tell them they can only move vertically and horizontally, not on a diagonal. Ask them how many boxes it takes to go up [3] and how many it takes to go right to get there. [4] Explain to students that this vertical movement is called the rise, and the horizontal movement is called the run. The ratio of the rise to the run tells us the slope, or steepness, of the line. Next, show students the graphs that show a positive slope and a negative slope and emphasize that we describe graphs as increasing or decreasing based on how their y-values change while the x-values increase.

Guiding Questions: 1. How do you know that the functions are linear from looking at the table? [The numbers in both the x and y columns follow a consistent pattern.] 2. How are the functions similar? How are they different? [Similar: straight lines, y-values change by 2 each time; Different: one goes up from left to right, one goes down from left to right.]

Lighthouse MATH Level H | Teacher's Guide

Exercise | 4-5 Name Choose any two points on the graph. Find the rise and the run. Then, find the slope. Point 1: Answers will vary.

Points Scored

Points

Run: Answers will vary.

2 2

4 6 8 Attempts

Slope:

Point 2: Answers will vary.

Run: Answers will vary.

20 0

To help students find the slope on a graph, have them use a piece of paper and a ruler to draw a vertical line and a horizontal line that goes from one point to another on a graph. After drawing in these lines, remind students to count how many boxes on the grid make up the vertical line and divide by how many boxes on the grid make up the horizontal line.

Rise: Answers will vary.

60 40

STRUGGLING LEARNERS

Point 1: Answers will vary.

Money Earned This Month 100

Rise: Answers will vary.

6 4

Point 2: Answers will vary.

Money Earned

8 12 16 20 Hours

Slope:

Write the slope and the y-intercept. Is the line increasing or decreasing?

Temperature

Tank Temperature After Power Loss

240

y-intercept =

increasing

30 0

slope = -10

Distance (miles)

2 Time (h)

Distance Traveled Based on Fuel Used

slope =

200

12 0

y-intercept =

160 120

increasing

80 40

decreasing

10 Liters

decreasing

EARLY FINISHERS

Circle the correct graph. 5.

Which graph has a slope of -2? Graph A

-5

Graph B

-5

Which graph has a slope of 4?

Graph B

-5

Graph B

-5

Use the graph to answer the question. Which scenario would best describe the slope of the graph? A. A taxi ride that gets more expensive as you travel B. A car losing value over time C. A balloon rising higher as time passes D. A savings account earning interest over time

CH AL L ENGE 8.

What happens to a graph if we change the y-intercept?

If a line goes through the origin, what is the y-intercept?

The line moves up or down.

Lighthouse Math

Level H

Chapter 4

Exercise 5

Students can create a graph of a linear function. They should give their graph a title and label the axes with the variables and how they are measured. Then, they should give their graph to another student to find the y-intercept and the slope and explain what the slope means in the context that is given on the graph.

CHALLENGE AND EXPLORE For problem 8, suggest to students that they draw a graph, identify the y-intercept, and then change it while keeping the slope the same. Have them describe what happens to the new graph.

Complete problems 1 and 4 in the Apply section together, then have students complete problems 2 and 3 with a partner. Review the the answers to ensure understanding. Next, have students independently complete problems 1-7 in the Exercise section. Review the answers and strategies as a class. When discussing the slopes for problems 1-4, use the context given on the graph to describe what the slope means in each situation. For problem 6, ask students how they know that the graph on the left is the one with the slope of 4 and how it differs from the graph on the right. [The one on the right has a slope of 41 . It goes up 1 over 4 instead of up 4 over 1.] Ask students which graph is steeper. [The one on the left] Finally, when going over problem 7, ask students to explain why the other answer choices do not work for that graph.

AACTIVITY Positive and Negative Sort: Students will work in groups to sort graphs based on whether their slope is positive or negative. Create a set of cards for each group. Have students find the slope based on the rise and run from one point to another, then have them divide them into separate groups based on if they are positive or negative. Finally, students should identify the y-intercept on each graph.

COMMON ERRORS Students may calculate slope as run/ rise. Students may forget a negative sign when writing a negative slope.

ASSESS Check problems 2, 3, and 6 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 4-6 4-6 | Find Slope and y-intercept from a Set of Coordinates

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will identify the slope and y-intercept from a set of coordinates. y Students will be able to find a missing coordinate located on a line by using the slope formula and another coordinate.

4. SPIRAL REVIEW

-7

-3 − 4 =

slope =

-7 + 9 =

-10 − (-5) =

-5

increasing 2

decreasing

If you know the coordinates of any two points on a line - (x1, y1) and (x2, y2) - you can use the slope formula to find the slope. Slope formula:

y2 − y1 x2 − x1

(1.7, 3.6)

(3.9, 1.4)

y2 − y1 1.4 − 3.6 -2.2 = = = -1 x2 − x1 3.9 − 1.7 2.2

(x1, y1) → (1.7, 3.6) (x2, y2) → (3.9, 1.4)

The slope is -1.

If you know the y-coordinate when x = 0, you can find the y-intercept. x

PRE-LESSON WARM-UP

Lighthouse MATH Level H | Teacher's Guide

-10

-4 + (-6) =

L E A R N A ND C O NNE C T

y Graph paper

Guiding Questions: 1. What is always the x-coordinate at the point where the line intercepts the y-axis? [x = 0] 2. How do you find the rise and run of a line? [Count how many boxes up and over from one point to another.]

6 − (-2) =

y-intercept =

Materials

When x is 0, y is 1. The y-intercept is 1.

A P P LY © Lighthouse Curriculum. Copying strictly prohibited.

Ask students to find the y-intercept and the slope and tell whether the graph is increasing or decreasing [y-intercept: 4; slope: 3; increasing, because it’s a positive slope]. Tell students that in this lesson, we will learn how to identify these features just by using the coordinates, without having to graph them.

2 8

y Slope - how the y-values change when the x-values increase by 1 y y-intercept - the point where a line crosses the y-axis on a graph

(-2, -2) (0, 4) (2, 10) (-1, 1) (1, 7)

5 + (-3) =

Find the slope and y-intercept. Is the line increasing or decreasing?

Vocabulary

List the following coordinates on the board. Ask students to plot them on a graph and connect the points with a line.

Solve.

Find y-intercept from the table. 1.

-2

-1

-2

y-intercept:

-1

y-intercept:

(x1, y1) → (3, 14) (x2, y2) → (4, 18)

y2 − y1 18 − 14 4 = = = 4 x2 − x1 4 − 3 1

Fill in the blanks to find the slope. 4.

(x1, y1) → (1, 3) (x2, y2) → (5, 15)

y2 − y1 12 15 − 3 = = = 3 x2 − x1 5 −1 4

Vocabulary Slope - how the y-values change when the x-values increase by 1. Can be positive (y-values increase), negative (y-values decrease), or 0 (no change) Y-intercept- the point where a line crosses the y-axis on a graph

Level H

Chapter 4

Lesson 6

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Explain to students that in the last lesson, we found the slope of a line by looking at its rise and run. Ask students what “rise” means. [How much you move in the vertical direction from one point to another] Ask students what “run” means. [How much you move in the horizontal direction from one point to another] Tell students that another way to find rise is to subtract the y-coordinate of one point from the y-coordinate of another point. The difference between the two y-coordinates tells us the how much the line rises from one point to the next. Tell students that another way to find run is to subtract the x-coordinate of one point from the x-coordinate of another point. The difference between the two x-coordinates is the how much the line runs from one point to the next. Tell students that we can divide these two values to find the slope. Next, present the slope formula to students with the example coordinates given in the Learn and Connect section. Have students calculate the slope, emphasizing the importance of the order of the coordinates. Tell them that it doesn’t matter if you do y2 − y1 or y1 − y2 as long as you subtract the x’s in the same order. Also emphasize that the y-values are always divided by the x-values for slope. Next, remind students that the x-coordinate for the y-intercept is always 0. Use the table in the Learn and Connect section to show students how they can easily identify the y-intercept using this knowledge.

Exercise | 4-6 Name Find the slope of a line passing through two points. 1.

(3, 7) and (5, 17) Label: x1 = 3 y1 =

Plug into formula:

x2 = 5 y2 = 17

17 − 7 10 = = 5 − 3 2

STRUGGLING LEARNERS

(-2, 4) and (2, -4) Label: x1 = -2 y1 = 4

Plug into formula: m=

x2 = 2 y2 = -4

Students can highlight the x- and y-values in different colors to help keep their work organized when calculating slope. Encourage students to check their work to be sure they didn’t miss any negative signs or subtract in the wrong order.

-4 − 4 -8 = = -2 2 −(-2) 4

Find the slope of the line that passes through the points. 3. 6.

(1, 3) (2, 7) (-7, 3) (-2, 5)

4. 2 5

(-2, -1) (-8, 2)

-2

(0,0) (-4,-1)

1 4

(-4, 0) (0, -3)

3 -4

(6, 8) (-3, 7)

1 9

Use the slope formula to find the missing y-value. 9.

10. slope =

slope = -3 Point 1: (2, y) Point 2: (6, 5) y=

1 2

EARLY FINISHERS

Point 1: (4, y) Point 2: (8, 6)

Have students plot the points in problems 3-5 in the Exercise section on a graph and connect the points. Then, have the students extend their lines to find the y-intercept of each.

Find the slope using the slope formula, then tell what it represents in the scenario. 11. The amount of water in a water tower during a typical weekday is tracked on the graph.

12. Jane tracks the growth of her plant each day on the graph.

400

10 8

(4, 230)

250

(6, 170)

200 150 100

CHALLENGE AND EXPLORE

(4, 6)

6 4

(2, 3)

50 1

Hours

Days 3 2

Slope = -30 Water is emptying at a rate of 30 gallons per hour.

Slope = The plant is growing at a rate of 1.5 inches per day.

CH AL L ENGE Use the slope formula to find the missing x-value. Hint: You can solve using a proportion. 13. Slope =

3 1

Lighthouse Math

Points: (x, 9) and (4, 18)

x=1

Level H

14. Slope =

2 1

Chapter 4

Exercise 6

Points: (3, 5) and (x, 9)

x=5

300

Height (in inches)

Gallons of Water

350

Students will need to set up algebraic equations to solve problems 13 and 14 in the same way they set up equations in problems 9 and 10, but this time, the unknown value will be in the denominator. If students need guidance to solve, remind them that the numerator and denominator of fractions act like parantheses, so each needs to stay together. The hint provided in the problem reminds students that they can set up a proportion to solve.

Have students complete problems 1-3 in the Apply section independently. Review the answers to ensure understanding. Next, complete problems 4-5 as a class, filling in the boxes to find the slope between the two points. Remind students to always subtract in the same order on the bottom as they did on the top. Next, have students complete problems 1-8 independently, checking their answers with a partner when complete. Review a few of these problems, then set up problem 9 together as a class. Have students solve their equation and complete problems 10-12 on their own. Review the answers as a class and be sure to discuss what the slope means in the context of problems 11 and 12.

ACTIVITY Sloping Chance: The purpose of this activity is to have students practice using the slope formula. Working in pairs, have students roll two pairs of dice. One person’s numbers represent (x1, y1) and the other (x2, y2). Next, the pair will calculate the slope between their two points. Have students repeat this process several times, then reconvene as a class to discuss what happened when their y-values were the same [slope of 0] and what kind of line that would make [horizontal line] and what happened when their x-values were the same [undefined slope] and what kind of line that makes. [vertical line]

COMMON ERRORS Students may subtract x’s and y’s in a different order. Students may forget negative signs.

ASSESS Check problems 2, 5, 7, 10, and 11 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 4-7 4-7 | Slope and Similar Triangles

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to explain why the slope is the same between any two points on a line using similar triangles.

Solve the proportions.

SPIRAL REVIEW

3 x = 4 8

x=6

x 3 x=12 = 4 8

6 3 = x 4

x=8

6 4 x=42 = x 3

Find the slope given the coordinates. 1.

(3, 2) and (4, 1) Slope =

-1

(12, 5) and (8, 6)

(1, 3) and (4, 9)

-4

Slope =

(-3, -2) and (-2, 2)

Slope =

Vocabulary L E A R N A ND C O NNE C T

y Similar triangles - triangles that are the same shape but not necessarily the same size y Slope - the rise/run of a line; a measure of how steep the line is y Proportion - two fractions that are set equal to each other

A right triangle can be drawn to connect any two points on a line. In the figure to the right, the orange triangle connects points A and B, and the blue triangle connects points B and C.

6 8

You can write a ratio of side lengths for each triangle. Put the vertical side length on top of the horizontal side length. If you simplify both ratios, you will find that they are equivalent.

y Graph paper y Scissors y Ruler

Ratio of sides:

Orange Triangle

Vertical Side Length Horizontal Side Length

3 1

B (2, 6)

Blue Triangle

3 4

6 2

A (1, 3)

Write the following proportions on the board and ask students which one(s) represent(s) a true statement (a proportional relationship). 1 6 3 = 18

3 6 5 = 10

Discuss as a class the answers and the strategies that students used. Guiding Questions: 1. How do we know if numbers are in a proportional relationship? [We can cross multiply. If the results are true, it is a proportional relationship; if they are false, it is not.] 2. How can we use equivalent fractions to determine if a relationship is proportional? [If you can multiply or divide the numerator and the denominator of the first fraction by the same factor and get the numerator and denominator of the second fraction, the relationship is proportional.]

Lighthouse MATH Level H | Teacher's Guide

rise Notice that the ratio between sides is also the run . Both sections of the line have a slope of 3. Since all triangles drawn on this line will be proportionate, the slope will be the same along the whole line.

PRE-LESSON WARM-UP

C (4, 12)

These triangles are similar triangles. They are proportionate, meaning that they are the same shape but not the same size.

Materials

4 5 10 = 15

A P P LY Circle the triangle with the same slope as the given triangle. 1.

2. 10

8 4

18 4

6 15

8 5

12 5

Vocabulary Similar triangles - triangles that are the same shape but not necessarily the same size. Their corresponding angle measures are equal, and their corresponding side lengths are proportional. Slope - the rise/run of a line. It is a measure of how steep the line is. Proportion - two fractions that are set equal to each other

Level H

Chapter 4

Lesson 7

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Replicate the graph from the Learn and Connect section on the board. Include the axes and line, as well as the labeled points, but not the triangles. Ask a student to come to the board and connect point A and point B (first horizontally and then vertically). Ask the class what shape was formed [triangle]. Ask another student to come up to the board and do the same to points B and C in another color. Define these triangles as similar triangles. Ask students why they are similar. [They are the same shape but not the same size.] Explain that triangles drawn this way along the same line will always be the same shape even if they are different sizes. (If students are up for a challenge, you can ask why they think this is the case. [Because the way the line is positioned creates two of the same angles for each triangle drawn on it, and the third angle is a right angle. When all the angles are the same, the shape will be the same, even if the line lengths are different.]) Ask students to find the lengths of both sides of the triangles using the graph boxes as units. Explain that the side lengths of both triangles are proportional and that this is true for any triangle of any size drawn anywhere along this line. Demonstrate this by creating a proportion using the side lengths of each triangle. For the sake of connecting this back to slope, put the y-values in the numerators and the x-values in the denominators. Have students put their finger at any point on the line and go up three boxes, then over one. Have them continue to do this as many times as possible. Their finger should always end up on the line after going up three and over one.

Exercise | 4-7 Name Set up a proportion to find the missing side. 1.

STRUGGLING LEARNERS

x 1

3 2

2 1 = 3 x

1 4 = x 4

x 4

5 2

4 2 = x 5

Remind students that we set up the proportions for similar triangles just like we set up proportions for ratio problems: by lining up our units. In this case, the units are corresponding sides. Encourage students to highlight numbers to differentiate which kind of side length it is.

x = 2.5

Find the missing side of the triangle. 4.

6. y

3 x

EARLY FINISHERS

Have students return to problems 7-9 and calculate the slope using the slope formula.

Find the missing coordinate. 7.

(4, y)

(x, 10) 6 7.5

(0, -2) (3, y)

CHALLENGE AND EXPLORE

3 (-1, -4)

(-5, -5)

(-20, -12.5)

(-4, -16)

-4

3 (-3, 0)

x = 25

CH AL L ENGE 10. Use the map to find the total distance of the red trail. 30 meters

Whiterock Park Trail Map West

Black Trail: 30 meters Blue Trail: 10 meters

Blue and red trails are 30 + 10 meters, and the black 4m

Red Trail: ? meters

trail is 30 meters, so they are 10 meters longer.

Level H

East South

How much longer are the blue and red trail together than the black trail?

Lighthouse Math

North

18 m

Chapter 4

For problem 10, students will need to use the information in the map key in order to set up a proportion for the two triangles and find the missing length of the red trail. Then, they will need to find the total length of the red trail. Next, they will need to compare the length of the blue and red trails combined to the length of the black trail.

Exercise 7

Have students complete the two problems in the Apply section on their own and then review the answers as a class. Next, have students work with a partner to complete problems 1-9, reminding them that they can set up a proportion for all the problems because any triangle drawn on a line will be proportionate to any other triangle drawn on the same line. Review the answers and strategies as a class to ensure understanding.

ACTIVITY Lining Up Triangles: The purpose of this activity is for students to see how similar triangles relate to slope in a hands-on way. Have students draw a horizontal line that is three boxes long on a piece of graph paper. Next, have them draw a vertical line from the end of their horizontal line that is five boxes long. Tell them to connect the remaining two endpoints to form a right triangle. Ask them to create three more triangles that are proportionate to this triangle. Remind students that they can use various scale factors to find the lengths of their additional triangles. After they have drawn all their triangles, have them cut out their triangles and then arrange them on a diagonal so that their third sides are all lined up to form a straight line. They can use a straight edge to assist them with this. Ask them to tell you the slope of the line that they created when lining up their triangles in this way. [ 53 ]

COMMON ERRORS Students may set up proportions incorrectly by not lining up y-values in the numerators and x-values in the denominators.

ASSESS Check problems 3, 6, and 9 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 4-8 4-8 | Slope-Intercept Form

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to use the equation y = mx + b to describe a line. y Students will be able to describe a line with the equation y = mx as a line that passes through the origin.

Find the slope and y-intercept from the graph. 2.

SPIRAL REVIEW

-5

y-intercept =

Circle the triangles with the same slope as the gray triangle. 8

Vocabulary

6 4

y Slope-intercept form (y = mx + b) - a formula for writing the equation of a linear function using its slope (m) and y-intercept (b) y y-intercept - the point where the line crosses the y-axis y Slope - the rise/run of a line; a measure of how steep the line is

-2

Slope:

y-intercept = -5

Slope:

3 12

12 48

L E A R N A ND C O NNE C T 10

All linear equations can be written in slope-intercept form. y = mx + b m is the slope.

b is the y-intercept. -10

The line on the graph has a slope of The equation of the line is y =

Materials

-5

1 and a y-intercept of -2. 3

5 3 -5

1 x − 2. 3 -10

y Notecards

A P P LY

PRE-LESSON WARM-UP Write the slope formula on the board as shown below. Explain to students that the m represents the slope. m=

y1 − y2 x2 − x1

Next, ask students to recall what the x-value is at the y-intercept. [0] Tell students that for this exercise, we will write the coordinate of the y-intercept as (0, b) since the x-value is always 0 and b represents any y-intercept. Next, have students place this coordinate in the slope formula for point (x1, y1). Write it on the board as follows: m=

y−b x−0

Next ask students: What if we wanted to figure out the value of y (for a given x)? What would the equation have to look like? Walk students through the process of rearranging the equation so that it says y = instead of m = by multiplying both sides by x and then adding b to both sides. Tell students that the resulting equation, y = mx + b, is the standard formula that is used for all linear equations. Guiding Questions: 1. Why is the x-coordinate of the y-intercept always 0? [because to get to that point on the graph, you don’t travel right or left at all] 2. How can we rearrange an equation? [Use inverse operations and always do the same thing to both sides.]

Lighthouse MATH Level H | Teacher's Guide

Write the slope (m) and the y-intercept (b) of each equation. 1.

y = 5x − 3 m=

2. b=

-3

-1

5. b=

8. b=

-2

3. b=

1 y= x 2 m=

y=x+1 m=

y = -2x + 10 m=

y = -x + 4 m=

y = 3x − 15 m=

-5

1 y=- x+2 8 1 -8 m= b=

y = -4x − 5 m=

9. b = -15

m= 6.

1 2

y = 7x + 8

-4

Vocabulary Slope-intercept form (y = mx + b) - a formula for writing the equation of a linear function using its slope (m) and y-intercept (b) Y-intercept - the point where the line crosses the y-axis. It tells the value of y when x is 0. Slope - the rise/run of a line. It is a measure of how steep the line is.

Level H

Chapter 4

Lesson 8

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Remind students that all lines that are graphed have linear equations. These equations describe the line and act like a formula for us to use to find more points along the line and make predictions about where the line, or the variables described, are headed. Use the Learn and Connect section to show students that the formula for a linear equation is y = mx + b. Tell them that this is called slope-intercept form because the m represents the slope of the line, and the b represents the y-intercept of the line. Use the graph in the Learn and Connect section to show students how to find the slope and the y-intercept and input them into the formula to find the equation of the line.

APPLY AND DEVELOP SKILLS (Practice) Have students complete the problems in the Apply section on their own and then review the answers as a class to ensure understanding. Next, complete problems 1, 6, 11, and 15 in the Exercise section together as a class to model the processes. Then, have students complete the rest of the problems on the page independently. Review some answers and strategies as a class and discuss how students knew which lines pass through the origin in problem 19. Remind students that the origin is (0,0) on a graph, therefore, the y-intercept would have to be 0, and the b-value would also need to be 0.

Exercise | 4-8 Name Write the equation given the slope (m) and the y-intercept (b). 1.

m = 2 b = -1

1 b=1 6 1 y=-6x+1

y = 2x − 1

m=3 b=4

m=-

1 b=2 3

y = 9x − 1

10. m =

m = -9 b = 2

STRUGGLING LEARNERS

m = 9 b = -1

y = -2x + 5

y= 3x+2

y = -x + 1

m = -2 b = 5

y = 3x

m = -1 b = 1

y = 3x + 4

m=3 b=0

Students should write down the slopeintercept formula on a notecard with the definitions of m and b as slope and y-intercept, respectively.

1 b = -10 4 1

y = 4 x − 10

y = -9x + 2

Use the graph to write the equation in slope-intercept form. 5

11.

-5

12.

-5

18.

-10

CHALLENGE AND EXPLORE

-5

-10

1 y= 2x+1

y = -2x − 2

y = 4x + 4

17.

-5

1 y= 4x−2

y=x+5

y = 3x − 1

y = 21 x − 0

y = 4x

Circle the lines that pass through the origin. 19.

y = 43 x + 5

y = 8x

CH AL L ENGE Use the given information to write the equations of the lines in the slope-intercept form. 20. A line passes through the points (-2, 5) and (0, 4). What is the equation of the line in slope-intercept form?

21. A line passes through the points (0, 7) and (3, 12). What is the equation of the line in slope-intercept form?

To assist students with problems 20 and 21, remind students that any point with an x-coordinate of 0 is actually the y-intercept. Then, remind them to use the slope formula to find the slope of the line.

y=-2x+4

Lighthouse Math

Students can write their own riddles modeled after problems 20 and 21, where they give a partner two points on the line and ask them to find the equation of the line.

-5

y = -2x + 1

16.

-5

EARLY FINISHERS

14.

-5

y= 3x−2

y = 2x

15.

13.

y= 3x+7

Level H

Chapter 4

Exercise 8

Graph Equation Match: The purpose of this activity is for students to practice recognizing which graphs go with which equation and vice versa. Create notecards with pictures of graphs and corresponding notecards with their equations. Shuffle the cards and pass out one to each student. Have the students walk around looking for the graph that matches their equation or the equation that matches their graph. Once students have found each other, have them identify the slope and the y-intercept of their line. Collect the cards, reshuffle them, and then play again.

COMMON ERRORS Students may confuse m and b when placing the slope and y-intercept into the slope-intercept formula.

ASSESS Check problems 5, 10, 14, 18, and 19 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

ACTIVITY

Level H | 4-9 4-9 | Graphing Lines from Slope-Intercept Form

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able graph a line given an equation in slope-intercept form. y Students will be able to name at least two coordinates on a line given its slope-intercept form.

Use the slope formula to find the slope (m) between the given points.

m= SPIRAL REVIEW

(3, 4) and (5, 9)

y Slope-intercept form (y = mx + b) - a formula for writing the equation of a linear function using its slope (m) and y-intercept (b)

y = 2x + 1

(6, -1) and (4,-5) 2

m = -1 b = 5 y = -x + 5

Step 1: Plot the y-intercept (b). b = 1; plot (0, 1)

Step 2: rise Use the run of the slope (m) to plot another point on the graph m=2=

Step 3: Connect the two points with a straight line and extend the line in both directions on the graph.

2 1

Start at (0, 1) move up 2, right 1, plot (1, 3)

Here’s how to graph a line using this equation:

To find another point on the line, plug in any x-value and solve to find the y-value.

Lighthouse MATH Level H | Teacher's Guide

1 5

Write the equation of the line given the slope (m) and the y-intercept (b). 1 b = -2 4. m = 6 b = -1 3. m = 1. m = 3 b = 1 2. m = -4 b = 9 2 1 y= 2x−2 y = 6x − 1 y = 3x + 1 y = -4x + 9

y Graph paper y Ruler

Guiding Questions: 1. How do we find the slope of a line using a graph? [We count the number of boxes up and over from one point to another on the line and then divide. Or we can find the coordinates of two points on the line and use the slope formula.] 2. What does the equation of a linear function tell us about the line itself? [It tells us how the y-values change as the x-values increase. It tells us where the line crosses the y-axis. It tells us what the slope of the line is is (m) and what the y-intercept of the line is (b).]

(-1, -2) and (-6, -3)

An equation written in the slope-intercept form gives all the information needed to graph a line and find coordinates on that line.

Materials

Have students draw a diagonal line on graph paper using a ruler. Then, have them draw in the y-axis and x-axis so that the line will have a y-intercept of 2. Ask students to identify the slope of their line and then create an equation for the line in slope-intercept form.

-2

L E A R N A ND C O NNE C T

Vocabulary

PRE-LESSON WARM-UP

(9, 7) and (15, 4)

5 2

1 2

-5

y = 2x + 1 y = 2(-2) + 1 y = -4 + 1 y = -3 The point (-2, -3) is on the line.

A P P LY Follow the directions to find a point on the given line. 1.

y = 3x + 5

1 x−3 2

y = -4x + 1

Plug in 2 for x: y = 3( 2 ) + 5

Plug in 6 for x: y = 21 ( 6

Solve to find y: y = 6 + 5 y = 11

Solve to find y: y = 3 − 3 y= 0

Solve to find y: y = -4 + 1 y = -3

Write the coordinates: (2, 11 )

Write the coordinates: (6, 0 )

Write the coordinates: ( 1 , -3 )

)−3

Plug in 1 for x: y = -4( 1 ) + 1

Vocabulary Slope-intercept form (y = mx + b) - a formula for writing the equation of a linear function using its slope (m) and y-intercept (b)

Level H

Chapter 4

Lesson 9

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Tell students that the equation for a linear function gives us all the information we need to graph the line. Use the steps in the Learn and Connect section to show students how to use this information to graph the line defined by the formula y = 2x + 1. The first step is to plot the y-intercept at (0, 1). Step two is to use the slope to find another point on the graph. Remind students here that slope is rise/run. If the slope is a whole number and not a fraction, recommend that students change the whole number to a fraction over 1. Now, the slope has a rise and a run. Show students how to start at the y-intercept (0, 1), go up two boxes (the rise), move over one box to the right (the run), and put a dot at (1, 3). At this point, students can use a ruler to connect the two dots and extend the line in either direction. It may be beneficial to show students how to work backwards with the slope, as well. Have them go to the point (0, 1) and go down two boxes, over one box, and plot a point at (-1, -1). This point should be on the line they already drew. Having them plot this extra point will help them to see how working backwards can also help them find a point on the line. Finally, use the example in the Learn and Connect section to show students how to choose an x-value to find another point on the line by plugging the x-value into the equation and solving for y. Have students plot this point on the graph as well. Ask: If we know a y-value, how can we find a corresponding x-value? [Plug in y and solve for x.]

Exercise | 4-9 Name Graph the linear equations on the coordinate plane. 1 y=- x+2 2

y = 3x − 1

-5

STRUGGLING LEARNERS

y = -x − 2

-5

y = 2x − 1

-5

1 y=- x+2 3

2 y= x 3

6. 5

-5

EARLY FINISHERS

-5

Students can use their notecard from the last lesson to assist them with graphing linear equations in slopeintercept form. Prompt students to use inverse operations when solving for a y-value given an x-value.

Have students return to the equations in the Apply section and graph them on a piece of graph paper. Be sure to have them plot the point they found during their exercise as well.

-5

CHALLENGE AND EXPLORE

Find two sets of coordinates (x, y) on each line.

Answers will vary. y = -4(1) + 1 y = -4 + 1 y = -3 (1, -3)

(

y = 8x − 2

)(

)

y = -4x + 1

)(

)

)(

)

Answers will vary. y = -2(1) + 7 y = -2 + 7 y = 5 (1, 5)

1 x − 10 5 Answers will vary. 1 y = 5 (10) − 10 y = 2 − 10 y = -8 (10, -8)

(

10. y = -2x + 7

)(

11. y =

)

)(

Answers will vary. y = 8(1) − 2 y=8−2 y = 6 (1, 6)

1 x−4 3 Answers will vary. 1 y = 3 (3) − 4 y=1−4 y = -3 (3, -3)

)

CH AL L ENGE 12. Mark and John are racing. Mark runs at a constant speed of 5 miles per hour. John is slower, so he gets a head start. He begins the race 1 mile ahead of Mark. He runs at a constant speed of 2 miles per hour.

5 4 3

A. Write an equation for each person’s distance Mark: y = 5x and John: y = 2x + 1 from the starting line (y) over time (x).

B. Graph each equation on the coordinate plane.

Lighthouse Math

Level H

Chapter 4

Exercise 9

1 1

For problem 12, students will need to create two equations, one for Mark and one for John, based on their running speeds and starting positions. Tell students that y represents the person’s distance from the starting line, and x represents how much time has passed. Then, tell them that their speed is the rate at which they are moving. Ask them if they think this is the slope or the y-intercept. [slope] Finally, ask them why the person’s starting point is the y-intercept. [Because the starting point occurs at a time of 0, so when x = 0, y is how far away they are from the starting point.]

Work as a class to fill in the blanks for problem 1 in the Apply section. Then, have students complete problems 2 and 3 independently. After reviewing the answers as a class, work together to complete problems 1 and 2 in the Exercise section. Once students have a solid understanding of the graphing process, have them complete problems 3-6 independently. Next, have students volunteer answers for problem 7. They will need to provide two coordinates on the given line. Then, have them complete problems 8-11 independently. Remind them that they need to plug in an x-value and solve for y and then write their answer as an ordered pair.

ACTIVITY Graphing Shift: The purpose of this activity is for students to practice graphing lines using an equation in slope-intercept form. Have students create an equation in slope-intercept form and graph it. Then, give them instructions for how to change their equation and have them graph the new equation. For example: Reduce your slope by 3, or multiply your y-intercept by 2. Give them a few of these directions, and then, hold a class discussion about how their lines changed with each equation change.

COMMON ERRORS Students may confuse the rise and the run of the slope, causing their line to have a different steepness.

ASSESS Check problems 3, 6, 8, and 9 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 4-10 4-10 | Modeling and Interpreting Linear Functions

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to write an equation in slope-intercept form from a real-life scenario and use it to answer questions.

Write the slope (m) and the y-intercept (b) of each equation. y= m=

SPIRAL REVIEW

1 x−5 2 1 2

2. b = -5

y=x+2 m= 1

3. b= 2

Find the point on each line when x = 4. Give your answer as an ordered pair. 1 1 y=x+6 2. y = -2x + 1 3. y = x + 3 4. y = - x − 2 4 2 (4, 10 ) (4, -7 ) (4, 4 ) (4, -4 )

L E A R N A ND C O NNE C T

Guiding Questions: 1. Which situation shows growth? [collecting money for a charity] 2. Which situation shows a decline? [going down a hill] 3. Which situation is more consistent? Why? [Walking down the hill, because it is at a steady pace. When collecting money for a charity, people can give different amounts each time.] 4. Which situation would more likely be graphed as a straight line? Why? [walking down the hill, because you are walking at a steady pace = consistent slope]

Dollars

Sam has $20 in his wallet. Every week, he uses $1.50 to buy a treat at the candy store. An equation can be written to show how much money he has over time. To write a linear equation that models a real-life scenario, you need to know four things:

2 3

b = -4

y = 5x − 1 (4, 19 )

Slope: 3 - 2 = -1.5

Number of Weeks

Input (x-variable)

Output (y-variable)

Slope

y-intercept

Something that changes that doesn't depend on the output

The thing that changes because of the input

The rate the output changes

The starting amount

The number of weeks change.

The amount of money Sam has changes based on the week.

The amount of money changes by -1.50 dollars per week.

Sam starts with $20.

y = -1.5x + 20

y Notecards

Using the Guiding Questions below, conduct a discussion in which students compare and contrast these two situations. For an added challenge, have students draw what a graph might look like for each situation.

Materials

Next, ask students to imagine that they are at the top of a tall hill and they begin to walk down the hill at a steady pace. Ask them to envision their progress of getting closer and closer to the bottom of the tall hill.

b= 0

2 x−4 3

20 (0, 20)

y Slope - the rise/run of a line; a measure of how steep the line is y y-intercept - the point where the line crosses the y-axis y Input - values that are chosen to put into a function, represented by the letter x y Output - values that depend on the input and the rule of the function, represented by the letter y

A P P LY © Lighthouse Curriculum. Copying strictly prohibited.

Ask students to imagine that they are collecting money for a charity. Ask them to envision what happens to the collection box each time a coin is added. [It gets fuller and fuller.]

m = -3

Vocabulary

PRE-LESSON WARM-UP

y = -3x

Fill in the table. Scenario

Output (y)

Input (x)

Slope (m)

Y-intercept (b)

A squirrel has 10 acorns. Each day, he collects 4 more.

Number of acorns

Time (days)

The water meter reads 36 L. Each hour, it decreases by 2 L.

Amount of water in liters

Time (hours)

-2

A tire receives 1 psi of air with each pump. It begins with 12 psi.

Tire pressure in psi

Number of pumps

Vocabulary Slope - the rise/run of a line. It is a measure of how steep the line is. Y-intercept - the point where the line crosses the y-axis. It tells the value of y when x is 0. Input - values that are chosen to put into a function, represented by the letter x Output - values that depend on the input and the rule of the function, represented by the letter y

Level H

Chapter 4

Lesson 10

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Use the example provided in the Learn and Connect section to model for students how to translate a word problem into a linear equation that can then be graphed. First, read aloud the word problem. Then, have students read it to themselves a second time. Next, explain to students the four things we need to identify to help us create a linear equation: the input (or what x represents), the output (or what y represents), the slope (the rate of change), and the y-intercept (the starting point). Ask students to identify each of these in the example problem, one by one. [Input (x) is the number of weeks, output (y) is the amount of money in Sam’s wallet, slope is $1.50, and y-intercept is $20.] Next, show students how to compile this into an equation, emphasizing that y and x remain as letters because they are variables; they change. [y = -1.5x + 20] Finally, have students sketch the graph of the scenario. Ask them whether the graph increases or decreases and why. [Decreases, because as time goes on, the amount of money in Sam's wallet goes down. Or, the line slopes down from left to right.]

APPLY AND DEVELOP SKILLS (Practice) Work together as a class to fill in the table in the Apply section. Ask for student participation as you go. Ensure that students understand the difference between the four features. Next, have students work in pairs to solve problems

Lighthouse MATH Level H | Teacher's Guide

Exercise | 4-10 Name Read the scenario and then answer the questions.

Students should be encouraged to highlight words in each problem that indicate the four features of a linear equation. Provide students with an anchor chart with the four features listed and described that they can check off as they go.

Weeks 500

Temperature (°F)

An oven cools down at a rate of 50 degrees per hour. The oven starts at a temperature of 350°F. y = -50x + 350 A. Write an equation B. What does x represent? Number of hours Temperature C. What does y represent? D. Graph the equation. Label the axes. 100 degrees E. What temperature will the oven be after 5 hours? F. Put a dot on the graph that represents the answer to E.

250

EARLY FINISHERS

Hours

Students can return to problems 1-3 in the Apply section and create a linear equation and a graph that models the scenario described.

Use the table to graph the line. Then, write the equation and answer the questions. Money Earned (y)

120

Equation:

y = 20x

A. What are the coordinates (0,0) of the y-intercept? B. Explain what the y-intercept means.

200

If 0 items are sold, $0 are earned.

C. How much money is earned for each item?

100

CHALLENGE AND EXPLORE

$20 per item sold

Number of Items Sold

D. How many items will need 9 items to be sold to earn $180? E. If 5 items are sold, how $100 much money is earned? © Lighthouse Curriculum. Copying strictly prohibited.

Number of Items Sold (x)

Money Earned ($)

CH AL L ENGE Read the scenario then, answer the questions. 4.

The number of miles driven per hour is tracked and graphed. Driving begins at mile-marker 6. After 3 hours, the vehicle is at mile-marker 51. A. The y-variable represents

miles traveled

B. What is the y-intercept of the line?

. The x variable represents

hours

C. What is the slope of the line?

. 15

D. What does the slope of the line mean given the context? The vehicle traveled at a speed of 15 miles per hour.

Lighthouse Math

Level H

Chapter 4

Exercise 10

For problem 4, students will need to calculate the rate at which the vehicle was traveling by first finding the distance it traveled [51 − 6 = 45] and dividing it by the time traveled [45 3 = 15]. They will need to identify the y-intercept as 6 and define the x and y variables as time in hours and distance traveled in miles. After interpreting the slope as the speed of the vehicle, challenge students to write the linear equation that models this scenario.

1-3 in the Exercise section. In problems 1 and 2, ensure that students complete all parts, including creating an equation, defining variables, graphing the line, labeling the axes, answering a question about the graph and plotting a point. In problem 3, students will need to recognize that the line will pass through (0, 0), meaning that it has a y-intercept of 0, and they will need to interpret that in the context of the problem.

ACTIVITY Identify and Define: The purpose of this activity is to have students practice identifying slope and y-intercept as well as variables in a real-world situation. Divide students into groups. Create a set of notecards for each group with different phrases such as “52 miles per hour,” “started 18 miles from home,” and “traveled along a road for a period of time.” Students should identify which card is the slope, which is the y-intercept, which is the input, and which is the output. Then they will write an equation that models the scenario and graph it. Finally, they should think of two questions they can ask, such as “How far from home is the car after three hours?” and “How long would it take to travel 20 miles?”

COMMON ERRORS Students may be unsure what x and y represent in a problem. Students may confuse slope and y-intercept.

ASSESS Check problems 1-3 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

y = 10x + 15 A. Write an equation B. What does x represent? Number of weeks Money saved C. What does y represent? D. Graph the equation. Label the axes. 3 weeks E. How long will it take Karen to save $45? F. Put a dot on the graph that represents the answer to E.

STRUGGLING LEARNERS

Karen has $15 in her bank account. She receives $10 each week for babysitting.

Money saved ($)

Level H | 4-11 4-11 | Comparing Linear Functions 5

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

1 3

b = -8 SPIRAL REVIEW

-1

m = -1

y Graph paper

Ann has 12 marbles. She collects 3 marbles each day. y = 3x + 12

-5

b= 1

m= 3 b= 1

-5

There are 14 gallons of gas in a tank. 1.5 gallons are used per day.

Mike reads 20 pages of his book each day. He begins on page 5.

y = -1.5x + 14

y = 20x + 5

L E A R N A ND C O NNE C T

PRE-LESSON WARM-UP

We can compare functions by looking at their slopes and y-intercepts. There are two buckets collecting water from two different leaky sinks.

Write the following equations on the board and ask students to compare and contrast them:

Bucket A x hours

y inches

of water

Bucket A fills at a rate of 2 inches per hour. Bucket B fills at a rate of 21 inch per hour.

Slope = 2

y-intercept = 0 Bucket B

Bucket A is filling faster than Bucket B.

b) y = -2x + 5

At 0 hours, Bucket A has 0 inches of water. At 0 hours, Bucket B has 2 inches of water. Bucket B started off with water in it already, while Bucket A started off empty.

Guiding Questions: 1. Which equation represents a linear function that is increasing? [equation a] 2. Which equation has a higher y-intercept? [equation b] 3. If the equations were graphed on the same coordinate plane, what would they look like? [They would cross over each other like an uneven “x.”]

Write an equation.

Materials

a) y = 3x − 7

1 y = ( )x − 8 3

Inches of water

y Students will be able to compare properties of two functions represented in different ways.

Find the slope (m) and y-intercept (b) for each.

Hours

Slope = 21

y-intercept = 2

A P P LY Find the slope (m) and y-intercept (b) of each function. Then, compare the slopes and y-intercepts. 1.

Function A 10

-10

Function B x

-1

Function C

Greatest Slope:

Least Slope:

Greatest y-intercept:

Least y-intercept:

Function C

y = 5x

Function A

Function B

1 2

-1

Level H

Chapter 4

Lesson 11

Function A

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect)

Ask students to tell you the four features of a linear function that help us write a linear equation and graph the function [input, output, slope, and y-intercept]. Explain to students that because all linear functions have these features, we can use them to compare different linear functions. Use the table and the graph in the Learn and Connect section to show this process. Explain that these show how much water is in two different buckets over time. Have one student explain how to find the slope from the table [The y-values increase by 2 as the x-values increase by 1, so the slope is 2], and have another student explain how to find the slope on the graph. [There is a rise of 1 for every 2 boxes on the run, so the slope is 21 .] Ask students what these slopes tell us about the buckets and how they know. [Since the slopes are positive, the buckets are filling with water.] Ask students to contrast the two functions given their slopes. [Bucket A is filling faster than bucket B.] Next, have a student explain how to identify the y-intercept from the table [when x = 0, y = 0]. Have a student explain how to find the y-intercept from the graph [the line crosses the y-axis at (0, 2)], and have another student explain what the y-intercepts mean. [Bucket A starts off empty, while bucket B starts off with two inches of water already in it.]

Lighthouse MATH Level H | Teacher's Guide

Exercise | 4-11 Name Find the slopes and y-intercepts of each function. Then, fill in the blanks with greater or less. Function A

Function B

STRUGGLING LEARNERS

Function C

-2

-1

-6

-3

-10

Have students use highlighters to color-code y-intercepts on a table, graph, and equation. Have them write the slope of the line next to the table or graph and circle it in the equation.

y = 4x − 2

-10

The slope of Function A is

greater

The slope of Function B is

less

The y-intercept of Function B is

than the slope of Function B. than the slope of Function C.

greater

b = -2

EARLY FINISHERS

than the y-intercept of Function A.

Three different functions show the drop in temperature over time. Compare them, then answer the questions.

8 6 4 2 0

Temperature Change in Boston

Temperature Change in Albany

-1

-6

-11

-16

y = -x − 2

The temperature is decreasing the fastest in Boston.

True

False

The temperature is decreasing faster in Albany than in Pittsburgh.

True

False

The temperature started off the highest in Pittsburgh.

True

False

10. Boston’s starting temperature was less than Albany’s.

True

False

Temperature Change in Pittsburgh

CH AL L ENGE 11. Anna has 50 stickers in her collection. Each day, she adds 5 more stickers to her collection. Sara has 20 stickers in her collection. Each day, she adds 10 more stickers to her collection. Whose sticker collection will reach 100 stickers first? Hint: Create an equation for each. Anna: y = 5x + 50 Sara: y = 10x + 20

(100) = 5x + 50 (100) = 10x + 20

x = 10 days to reach 100 x = 8 days to reach 100

Sara will reach 100 stickers first.

Lighthouse Math

Level H

Chapter 4

Exercise 11

Have students return to the example in the Learn and Connect section. Have them graph the data for Bucket A on the same coordinate plane that Bucket B is graphed on. Ask them to determine if the buckets will ever have the same amount of water at the same time, and if so, about how much water would be in the bucket at that time [Yes, almost 3 inches]. Then, have them compare how long it will take each bucket to fill up to six inches of water [bucket A - 3 minutes, bucket B, 7 minutes].

CHALLENGE AND EXPLORE For problem 11, students will need to create an equation for Anna and an equation for Sara to model how many stickers they each have in their collections. Then, they will need to use 100 as an output (y-value) to find how long it takes for each to reach 100 stickers in their collection.

Have students work in pairs to complete problems 1-7 in the Apply section. They will need to identify the slope and y-intercept of each function and then answer which has the greatest and least slope and y-intercept. Review answers as a class, and then have students work independently on problems 1-10 in the Exercise section. Review the answers as a class, focusing on problems 7-10, which compare different functions in different forms that all track temperature in different cities.

ACTIVITY Linear Function Investigation: The purpose of this activity is to have students explore what lines with the same slope but different y-intercepts look like on a graph. Ask students to graph y = 21 x + 3 on a piece of graph paper. Then, have them graph y = 21 x − 1 on the same coordinate plane. Ask students what they notice about both equations. [Their slopes are the same.] Ask them what they notice about the lines they graphed. [They are parallel.] Alternatively, ask students if the two lines they graphed will ever touch. [No.] Have students write two more equations with the same slope and different y-intercept to see if these lines will behave in the same way. Ask students to think about what would happen if two lines have different slopes but the same y-intercept? [The lines would cross at the y-axis.]

COMMON ERRORS Students may misinterpret negative numbers as larger than positive numbers.

ASSESS Check problems 7-10 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 4-12 4-12 | Review

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will review concepts introduced in Chapter four including identifying functions, linear functions and their slope and y-intercepts, linear equations in slope-intercept form, graphing linear functions, and comparing linear functions.

The table below represents the same scenario in different ways. The scenario is a function because the amount of water (y) in the tank depends on the amount of time (x) that passes. Each input has exactly one output. Table

A water tank is leaking water at a rate of 2 liters an hour. When it was full, the tank had 14 liters of water.

Materials y Notecards

Graph

Slope-Intercept Form

Amount of Water Left in the Tank 14

Liters of water

Word Form

y = -2x + 14 x represents time in hours

y represents liters of water 0

Time (hours)

PRE-LESSON WARM-UP

Slope: the rate the water is leaking

Guiding Questions: 1. Why is it important to define the x- and y-variable of a function? [We need to know what the input is and what the output is in order to understand the relationship.] 2. Why does a linear function form a straight line when graphed? [Because the rate of change stays the same; as x increases, y always changes in the same way] 3. How are similar triangles related to slope of a line? [Similar right triangles (which are proportionate) can be drawn all along the slope of a line. The ratio of their side lengths is the same as the slope of the line.]

Lighthouse MATH Level H | Teacher's Guide

Function [when two variables are related such that each input has exactly one output] Input [the x-variable, what you put into a function] Output [the y-variable, what comes out of the function after x is put in] Linear function [a function that, when graphed, forms a straight line because the rate of growth stays the same] Linear equation [a formula that defines a linear function by showing how the y changes based on the x] Similar triangles [triangles that have the same shape but are different sizes; their side lengths are proportional]

Slope: m

12 - 10 2 = =-2 1-2 -1

Rise: -2 = -2 Run: 1

-2

y-intercept: the amount of water the tank started with

y-intercept: the point where the x value is 0

y-intercept: the point where the line crosses the y-axis

y-intercept: b

It started with 14 liters.

When x is 0, y is 14.

(0,14)

It is losing water at a rate of 2 liters an hour.

Ask students to work in groups to define the following terms:

y2 - y1 x2 - x1

Slope: rise over run

Slope:

A P P LY Fill in the blanks to find the slope. Then, write an equation for each table. 1.

-2

-4

8− 5 = 1 −0

4− 6 = 0 − -2

-1

0 − -4 = 1 −0

m= b= 5

y= 3 x+ 5

b= 4

Level H

Chapter 4

y = -1 x + 4

Lesson 12

b = -4

y = 4 x + -4

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Use the table in the Learn and Connect section and the example of the function defined as a tank leaking over time as an in-depth review into how slope and y-intercept show up in different forms. Begin by reading the word form text in the column on the left. Then, ask students to pick out the words that tell us about the slope [rate] and y-intercept [full]. Be sure that students understand that the slope is always the rate of change, and the y-intercept is the starting point. Next, show students the table of values and have a student explain how to find the slope from the table. [The x-values consistently increase by 1, while the y-values decrease by 2.] Have another student choose two points from the table to use to calculate the slope with the slope formula. Ask another student to pick out the y-intercept from the table and explain their reasoning. Move to the graph next. Have students put their finger on the y-intercept, and then have them move their finger down two boxes to the right, one box at a time repeatedly, until they get to the bottom of the line. Confirm with students that they always end up with their finger on the line after going down 2 over 1. Finally, have students note the equation in the right hand column on the table. Ask them why the x-variable is defined as time, while the y-variable is defined as water in the tank. [The y-variable is the output, and it depends on the input. The amount of water in the tank depends on how much time has passed.]

Exercise | 4-12 Name Determine if the relationship is a function. 1.

-1

Function

Not a Function

Function

STRUGGLING LEARNERS

(0, 1) (1, 2) (1, 3) (4, 5)

Not a Function

Function

Students should use the table in the Learn and Connect section as a guide when answering questions about slope and y-intercept in the chapter.

Not a Function

Determine if the function is linear. 4.

5. x

-1

Linear

y = 6x − 14

EARLY FINISHERS

Non-linear

Linear

Non-linear

Linear

Have students graph problems 1, 4, 6, and 8 in the Exercise section.

Non-linear

Write the equation of the line. 10

8. -10

y -10

-4 4

y=-4x+3

y = 2x - 3

CHALLENGE AND EXPLORE

Danielle has 34 stickers. Each week, she receives 5 more stickers to add to her collection.

For problem 13, students will need to use the coordinates of the y-intercept to find the slope of the line. Then, they can put that information together to write the equation and graph the line.

y = 5x + 34

Find the slope (m), y-intercept (b), and equation of each function. Then, answer the questions.

b = -4 10

Hours

Equation:

m= 2

Degrees

m= 1

Degrees

11. Temperature Change on Tuesday

y=x-4

b= 0 Hours

y = 2x

Equation:

12. A. What was the starting temperature on Monday? -4 degrees B. What was the starting temperature on Tuesday? 0 degrees C. On which day did the temperature increase faster? Tuesday

10. Temperature Change on Monday

CH AL L ENGE 10

13. A line has a y-intercept of 2 and passes through the point (4, 5). Find the slope, write an equation, and graph the line. 3

-10

Slope = 4 3 y= 4x+2

Lighthouse Math

-10

Level H

Chapter 4

Exercise 12

Have students complete the problems in the Apply section with a partner. Then, review the answers and strategies for solving. Next, have students work independently on problems 1-12 in the Exercise section. For problems 1-3, remind students that to in order be a function, each input can have only one output. Review the answers to all the problems. Discuss the answers to problem 12 in depth.

ACTIVITY Different Forms: The purpose of this activity is for students to recognize linear functions in different forms. Write a linear function in word form, table, graph, and equation each on different notecards. Shuffle the cards, pass them out to different students, and have students find the three other cards that match with theirs. Once students have formed groups, have them define the variables, y-intercept and slope. Finally, have students present their findings to the class and explain how they could find this information from the different forms that each student had.

COMMON ERRORS Students may switch x- and y-values when calculating slope.

ASSESS Check the even-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Chapter 5

In Chapter 5, we will expand on

Proportions Proportional relationships are found in real-world situations such as baking, driving, and economics. • We will find unit rates and use them to find the better buy. • We will graph proportional relationships and identify their slopes and y-intercepts. • We will model proportional relationships using equations. • We will compare proportional relationships.

Level H | 5-0 Chapter 5 | Skill Checklist

Skill 1: Writing ratios Match the equivalent ratios. 1.

9:4 6 11

5 to 4 5:4

5:2

5 3

5 4

6:11

4:1

9 to 4

5 to 2

5 to 3

3 to 2

For every 5 stars, there are 4 smiles.

Objective and Learning Goals

5 stars 4 smiles

Write the rate. Make sure to label the units.

Students will review skills needed for Chapter 5:

Tina makes 20 sandwiches in 2 hours.

12 donuts cost $14.50.

Skill 2: Equivalent Fractions and Ratios

Determine if the fractions are equivalent. Write = or ≠. 1. 4 ÷ 4 1 = 16 ÷ 4 4 2 2 × 3 2

4 6

2 16

18 24

≠

8 12

5 15

15 45

5 = 7

10 14

20 28

10 100 = 100 = 1,000

1 10

Find the missing number that would make the fractions equivalent.

Level H

1 5 = 2 10

6 2 = 3 9

6 3 = 44 22

I can identify and find equivalent fractions and ratios.

out 8 correct

1 8

For each fraction, write two equivalent fractions.

Lighthouse MATH Level H | Teacher's Guide

$14.50 12 donuts

I can write ratios and rates.

out 3 correct

y Writing ratios y Equivalent fractions y Setting up a proportion y Finding the slope from a graph

20 sandwiches 2 hours

Chapter 5

Skill Checklist

Lighthouse Math

Skill 1: Writing Ratios

Skill 2: Equivalent Fractions

y Remind students that ratios compare two quantities using division. y Use the information in the gray box to review the different ways to write a ratio, including as a fraction. y Remind students that order matters when it comes to ratios. y Have students complete problems 1-3 and review the answers and strategies as a class.

y Remind students that to find equivalent fractions, we can either multiply or divide both the numerator and the denominator by the same factor. y Remind students that we can check if two fractions are equivalent using cross multiplication. If the products are equal, the fractions are equal. y Have students complete problems 1-8 and review the answers and strategies as a class.

Name

Skill 3: Setting up a Proportion Match each scenario to its proportion. A recipe uses 7 tablespoons (tbsp) lemon juice to make 3 4 gallon of lemonade. If Stacey wants to make 3 gallons of lemonade, how many tablespoons of lemon juice does she need?

Eric paints 4 walls in 15 minutes. How many minutes will it take him to paint 10 walls?

Jake mows 15 lawns in 4 hours. How many lawns can he mow in 10 hours?

Moses hikes 4 meters in 10 minutes. How long will it take him to hike 15 meters?

Lisa can write 10 sentences in 4 minutes. How many sentences can she write in 15 minutes?

7 tbsp = x tbsp 3 3 gallons 4 gallons

4 x 15 = 10

10 x 4 = 15

4 10 15 = x

4 15 10 = x

I can find a missing piece of a proportion.

out 4 correct

Skill 4: Finding the Slope From a Graph y 10 8 7 6 5

(4, 4)

Line A =

Line B =

Line C =

1 3

Line D = -3

(6, 5)

-1

6 4

1 0

9 10

2 -10 -8

-6

-4

-2

rise y − y1 = 2 Slope = x2 − x1 run

-2

1 5−4 = Slope = 2 6−4

-6

-4

Find the slope of each line on the graph.

-8 -10

I can find the slope from a graph.

Lighthouse Math

Level H

Chapter 5

Skill Checklist

Skill 3: Setting Up a Proportion

Skill 4: Finding the Slope from a Graph

y Use the example in the gray box to review how to set up the proportion for the given recipe. y Remind students that units must line up across the top and bottom of a proportion. y Remind students that we can use cross multiplication to solve for a missing piece of a proportion. y Have students complete problems 1-4 and review the answers and strategies as a class.

y Remind students that a slope is the rise (how much you go up or down from one point to the next) divided by the run (how much you go horizontally from one point to the next). y Remind students that they can also use the slope formula. y Use the information in the gray box to review how to find the slope both ways. y Have students complete problems 1-4 and review the answers and strategies as a class.

out 4 correct

Lighthouse MATH Level H | Teacher's Guide

Level H | 5-1 5-1 | Proportions

PREREQUISITE SKILLS

y Students will be able to find the unit rate of a ratio or set of ratios and use it to solve problems. y Students will be able to use the unit rate or cross multiplication to solve proportions.

DAI LY REVI EW

Objective and Learning Goals

Solve.

SPIRAL REVIEW

2x − 3 = 7 x = 5

x + 10 = 8 x = -4 2

(x + 7) = -2.5 x = -17 4

3(x − 6) = 12 x = 10

Find the slope. 1.

(2,5) and (6,11)

Slope =

(-3,7) and (4,-2)

Slope =

3 2

- 9 7

4x + 5 = 1 x = -1

6(x + 2) = -12 x = -4

(x − 3) = 9 x = 21 2

x ( ) − 4 = 0 x = 20 5

(1,-2) and (5,6)

Slope =

(-1,-5) and (3,-9)

Slope =

-1

L E A R N A ND C O NNE C T

At the state fair, Nick wants to buy popcorn for the best value. Vendor A sells a 48-ounce bucket of popcorn for $9.60. Vendor B sells an 80-ounce bucket for $18.40. He wants to see which gives the better value. Nick can use the unit rate, or how much each costs per one unit, to compare the two vendors. To find the unit rate, divide the total by the number of units.

Write the following table on the board:

48 oz 60 oz = $9.60 x

PRE-LESSON WARM-UP

Vendor B $18.40 = $0.23 per ounce 80 oz

Unit rate

If Nick wants to know how much it would cost to buy 60 oz. of popcorn from Vendor A he can set up a proportion using Vendor A’s pricing:

Materials y Poster board y Markers

Vendor A $9.60 = $0.20 per ounce 48 oz

Cross multiply

Vendor A is the better buy.

48 oz 60 oz = $9.60 x

Proportion

48x = (60)(9.60)

48x = 576

x = $12

It would cost $12 for 60 oz of popcorn.

A P P LY Solve the proportion. 1.

5 miles x miles = 2 hours 6 hours

x = 15

x pages 30 pages = 4 minutes 6 minutes x = 20

18 pencils 42 pencils = 3 boxes x boxes x=7

Vocabulary

Pizzas

Cost

[$9]

$18

[$36]

[$90]

Unit rate - a comparison of two quantities where one of the terms is 1; it tells how much of something there is per one unit Proportion - an equation that shows that two ratios or rates are equal Better buy - the item that gives you more for your money, found by comparing unit rates

Level H

Chapter 5

Lesson 1

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Begin the lesson by writing the key vocabulary terms, unit rate, proportion, and better buy, on the board. Ask students to define each word as a brief review and discuss their meanings as a class. Next, read aloud the word problem about Nick buying popcorn at the state fair and ask how they could compare the two vendors’ prices. Guide students to recognize that finding the unit rate allows a fair comparison. Then, work through setting up proportions for both vendors and demonstrate how to simplify them. Discuss how the resulting unit rates confirm that Vendor A offers the better buy. Continue with the next part of the problem by asking students how to set up a proportion to find the missing value. Review the steps of cross multiplying and dividing to solve, and emphasize interpreting the answer in context. Conclude by explaining that this is just one of many real-world situations where proportions and unit rates are useful.

APPLY AND DEVELOP SKILLS (Practice) Work through problems 1-3 in the Apply section together as a class to review solving proportions. Emphasize how to correctly set up each ratio, identify the corresponding terms, and use cross multiplication to solve for the missing value. Work through problems 1, 5, and 9 in the Exercise section to give students a clear example for each problem type. Then have students work in pairs to

Lighthouse MATH Level H | Teacher's Guide

Exercise | 5-1 Name Find the unit rate. 1.

A printer produces pages at a steady rate. How many pages does it print per minute? Pages

Bottles

135

270

405

540

Minutes

5 pages per minute

STRUGGLING LEARNERS

A machine bottles juice at a constant rate. How many bottles does it complete per minute?

Provide students with a step-by-step guide to finding a unit rate. Have students simplify all ratios in a given problem to ensure understanding and have extended practice solving.

9 bottles per minute

Emma reads books at a constant pace. How many books does she read per week?

A gardener always uses the same fertilizer to water ratio. What is the unit rate for fertilizer per gallon of water?

Books

Ounces

Weeks

Gallons

2 books per week

EARLY FINISHERS

4 oz. per gallon

Determine the better buy. 5.

A 16-oz bottle of juice for $4.80 or a 10-oz bottle for $3.00?

A 30-pack of markers for $12.00 or a 10pack for $4.50?

Same value

A 32-oz bottle of shampoo for $7.20 or an 18-oz bottle for $4.32?

30-pack

32-oz bottle

A 1.5-lb box of cereal for $4.20 or a 2-lb box for $5.40? 2-lb box

Solve by writing a proportion. 10. A car travels 180 miles in 3 hours. How far can it travel in 5 hours at the same speed?

A recipe uses 3 cups of flour to make 12 cookies. How many cups of flour are needed to make 20 cookies? 5 cups

CHALLENGE AND EXPLORE

300 miles

11. A printer makes 60 copies in 4 minutes. How many copies can it make in 7 minutes?

12. Lisa bought 5 notebooks for $8.75. How much would 8 notebooks cost at the same rate?

105 copies

$14.00

CH AL L ENGE 13. A box of granola bars contains 6 bars and costs $3.60. Another store sells 10 bars for $6.50. Is this a proportional relationship? Why or why not? No, this is not a proportional relationship because the cost per bar is different at each store.

Lighthouse Math

Level H

Chapter 5

Exercise 1

COMMON ERRORS Students may set up the proportions incorrectly. Students may make calculation errors when simplifying.

ASSESS Check the odd-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

Level H | 5-2 5-2 | Graphs of Proportional Relationships

PREREQUISITE SKILLS

y Students will be able to graph proportional relationships given one ratio. y Students will be able to identify the unit rate of a graph.

DAI LY REVI EW

Objective and Learning Goals

Find the unit rate.

SPIRAL REVIEW

A car travels 210 miles in 3 hours.

70 miles per hour

A pack of 6 markers costs $4.20.

$0.70 per marker

A recipe uses 12 eggs to make 4 cakes.

3 eggs per cake

Determine the better buy. 1.

A 32-oz bottle of juice costs $4.80, or a 20-oz bottle costs $3.00.

A 10-pack of pencils is $2.90, or a 15-pack is $4.20.

Same value

15 pack

Vocabulary y Proportional relationship - a relationship between two quantities where the ratio between them stays the same y Unit rate - a comparison of two quantities where one of the terms is 1; it tells how much of something there is per one unit

L E A R N A ND C O NNE C T Ratios and rates can be plotted as points on a graph. When two or more rates are in a proportional relationship, their points will form a straight line that passes through the origin. The unit rate or constant of proportionality shows up as the slope.

4-hour rental → $48 7-hour rental → $84

y Notecards

Unit rate:

Slope:

$24 $12 = 2 hours 1 hour

Rise: 24 = 12 Run: 2

100 (7, 84)

Cost ($)

2-hour rental → $24

Materials

Boat Rentals

Consider the costs for renting a boat. By plotting these rates as points on a graph, we can see that the costs are proportional. No matter how long a boat is rented for, the cost will be the same per hour.

Run = 2

Slope 12 (4, 48)

Rise = 24 20

(2, 24)

Hours Rented

(0, 0)

cups of sugar

cups of flour

Graph the two given points and determine if the relationship is proportional. 1.

A smoothie shop charges $8 for 2 smoothies and $10.50 for 3 smoothies.

10 5 0

Number of Smoothies

Not proportional

A student earns $30 for 2 hours and $45 for 3 hours.

Earnings ($)

Write the following tables on the board:

A P P LY

Cost ($)

PRE-LESSON WARM-UP

Renting a boat costs $12 per hour.

50 25 0

Hours Worked

Proportional

Vocabulary

Origin - the coordinate point (0,0) Proportional relationship - a relationship between two quantities where the ratio between them stays the same Unit rate - a comparison of two quantities where one of the terms is 1; it tells how much of something there is per one unit

miles

100

hour

height (in)

INTRODUCE THE LESSON (Learn and Connect)

years

Explain to students that when proportional relationships are graphed, they form a straight line that passes through the origin. Draw the price chart for renting a boat from the Learn and Connect section on the board. Then, ask a student to sketch the graph of these rates on the board while the other students check to see if they are in a proportional relationship. [Yes, the unit rate is $12 per hour.] Next, ask the students to find the slope of the graph. [12] Ask students if they think it's a coincidence that these two have the same value. Remind students that the slope of the graph is its rate of change. For a linear function, the slope is always the same. For a proportion, the unit rate is always the same. Therefore, the unit rate will show up as the slope of the graph. Have a student demonstrate that this unit rate is the same as the slope, using any of the points plotted to confirm consistency. Conclude by emphasizing that in proportional relationships, the slope and the unit rate represent the same constant of proportionality: the rate of change per one unit on the graph, in the equation, and in the real-world context.

Ask students to work with a partner to confirm which tables show a proportional relationship. [cups of sugar to cups of flour] Once students determine which tables show proportional relationships, ask them to explain how they know. Then ask students to predict how this table would look like when graphed. Have students assist with plotting the table on a graph on the board. Ask students what they notice about the graph. [Straight line through the origin] Inform students that today’s lesson is focused on reviewing graphs of proportional relationships. Guiding Questions: 1. How can we determine if a table has a proportional relationship? [All values simplify to the same ratio.] 2. How can we tell if a graph shows a proportional relationship? [Straight line, through the origin] Lighthouse MATH Level H | Teacher's Guide

Level H

Chapter 5

Lesson 2

Lighthouse Math

APPLY AND DEVELOP SKILLS (Practice) Work through problems 1 and 2 in the Apply section together as a class. Ask students to compare the non-proportional graph from problem 1 to the proportional graph from problem 2 and explain the differences they observe. Emphasize that proportional relationships must form a straight line that passes

Exercise | 5-2 Name Find the unit rate based on the graph.

7.2 5.4 3.6

(4, 3.60)

1.8

(2, 1.80)

330

(6, 330)

220 110

0.0 0 1 2 3 4 5 6 7 8 9 10

0 0 1 2 3 4 5 6 7 8 9 10

Number of Apples

Number of Lessons

$0.90 per apple

$55 per lesson

STRUGGLING LEARNERS

Time (mins)

(9, 495)

440

0 1 2 3 4 5 6 7 8 9 10 -2

Provide a reference sheet that outlines how to determine if a graph represents a proportional relationship and how to find the unit rate or slope. Include visuals and step-by-step examples showing that proportional graphs form a straight line through the origin. For problems involving graphing, have students first create a table of values to identify the pattern before plotting points. This helps them visualize the relationship and confirm proportionality before moving to the coordinate plane.

-4 -6 -8 -10

-1 degree per minute

Graph each proportional relationship. Determine the slope. 5.

A recipe uses 3 cups of flour for every 2 batches of cookies.

Cost of Notebooks Cups of Flour

Total Cost ($)

Zoo Ticket Pricing 100

25 20 15 10 5 0

3 4 5

9 10

12 9 6 3 0

Number of Notebooks

The temperature starts at 0 degrees and drops 8 degrees per minute.

3 4 5

-40 -60

9 10

480 360 240 120 3 4 5

3 4 5

9 10

Scuba Diver's Depth Over Time 1

-10 -20 -30 -40

9 10

-50 -60

Time (Hours)

-100

Slope =

-8

Slope = 60

Slope = -18

CH AL L ENGE 10. If you divide the y value by the x value in a proportional relationship, you get the slope of the line. Explain why this can't work for linear equations that are not proportional. They have a y-intercept, which shifts the y value.

Lighthouse Math

Level H

11. Can you think of three ways to find the unit rate from a graph of a proportional relationship? Use the slope, divide y by x, use the y value of x = 1

Chapter 5

EARLY FINISHERS

As a scuba diver begins at sea level and descends at a rate of 36 feet every 2 minutes.

Slope =

600

Number of Tickets

3 2

A delivery truck drives 180 miles in 3 hours.

0 0

-80

Distance Traveled by Delivery Truck Distance (Miles)

Temperature Dropping 0

3 4 5

Slope =

Time (Minutes)

80 60 40

Batches of Cookies

5 2

-20

Depth (Feet)

Slope =

Temperature (OF)

A zoo sells 4 tickets for $36.

Flour Needed for Cookie Batches

Each notebook costs $2.50.

Total Cost ($)

Exercise 2

through the origin, while non-proportional relationships do not. Next, model problems 1 and 4 in the Exercise section as examples for students to reference. Then have students work with a partner to solve problems 2-9, discussing their reasoning and checking their answers together. Conclude by reviewing the set as a class and selecting a few problems to solve on the board, reinforcing key ideas about identifying proportional relationships and interpreting slope.

ACTIVITY Proportional Graph Matching: The purpose of this activity is for students to practice connecting real-world scenarios to the graphs that represent them. Have students work in pairs to write a short real-world scenario that involves a proportional relationship on an index card (for example, the cost of renting a bike per hour or the number of miles traveled at a constant speed). On a second card, have them draw the corresponding graph of the relationship. Collect all the cards and organize them into four stations, keeping each matching pair together at the same station. Divide the class into four groups and assign each group to a station. At each station, students work together to match each scenario card with its corresponding graph card, recording their reasoning on a response sheet. After a set amount of time, have groups rotate through all four stations to complete each set. Conclude by reviewing the matches as a class and inviting groups to share their thinking. Emphasize that proportional graphs always form a straight line through the origin and that the steepness, or slope, represents the constant of proportionality that connects the situation to its graph.

Have students write a short reflection explaining how the unit rate and slope are connected in proportional relationships. Ask them to describe in their own words why a proportional graph must pass through the origin and what the slope tells us about the situation. Encourage them to include a real-world example in their explanation.

CHALLENGE AND EXPLORE Work through problems 10 and 11 together as a class. For problem 10, ask students to give examples of nonproportional linear equations and graphs and then discuss how these differ from the graphs and equations of proportional relationships (for instance, lines that do not pass through the origin or have an added constant). For problem 11, create a class list of different methods for finding the unit rate, such as using a table, an equation, or the slope from a graph, and discuss how each method connects to the others.

COMMON ERRORS Students may divide x by y instead of y by x to find the slope. Students may forget that graphs of proportional relationships must go through the origin.

ASSESS Check problems 4-9 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

Total Cost ($)

550

9.0

Temperature (°C)

Level H | 5-3 5-3 | Comparing Proportional Relationships

PREREQUISITE SKILLS

y Students will write an equation for a proportional relationship. y Students will find the unit rate from an equation. y Students will compare unit rates of proportional relationships presented in different forms.

DAI LY REVI EW

Objective and Learning Goals

Find the unit rate.

SPIRAL REVIEW

15 miles per hour

A cyclist rides 60 miles in 4 hours.

A grocery store sells 5 pounds of apples for $7.50.

A machine produces 300 parts in 12 minutes.

A student earns $84 after working 7 hours.

$1.50 per pound 25 parts per minute $12 per hour

Find the unit rate. 1.

-5

Unit rate: 1

-5

Unit rate: - 21

-5

Unit rate: 2

-5

Materials L E A R N A ND C O NNE C T

y Notecards

The equation for a proportional relationship is y = kx, where k is the slope. Since the slope is the unit rate, you can write an equation of any proportional relationship if you know the unit rate.

PRE-LESSON WARM-UP

Eitan and Mannie are working on their homework after school. They each record how much time it takes to solve math problems. Time Eitan Spends on Math Minutes (x)

Problems (y)

Ask students to work in groups to determine who paints faster [Sara] and come up with three ways to confirm their answer. Encourage them to consider strategies such as finding the unit rate (walls per minute), creating equivalent ratios, or setting up and comparing proportions. After the groups discuss it, review it as a class and highlight how using multiple comparison methods can help confirm the same result. Explain that today’s lesson will focus on exploring and comparing proportional relationships in several ways.

Guiding Questions: 1. How can we tell if a situation represents a proportional relationships? [If it has a constant unit rate] 2. How can we compare proportional relationships? [Find the unit rate, make a table, create a graph, or find the equation.]

Lighthouse MATH Level H | Teacher's Guide

Sara paints 4 walls in 30 minutes. Deborah paints 5 walls in 45 minutes.

Time Mannie Spends on Math 15

Unit Rate:

14 = 1.4 10

Equation: y = 1.4x

Problems

Write the following on the board:

Unit Rate:

Run = 2

Rise = 3

Equation: y = 1.5x

3 = 1.5 2

Minutes

Mannie solves problems at a faster rate. He solves 1.5 questions per minute while Eitan solves 1.4 questions per minute.

A P P LY Identify the unit rate. 1.

y = 4x

1 y = ( )x 3

y = 4.25x

Write an equation for each scenario. y = 3x

A lemonade stand earns $3 for every cup sold.

1 3

A car travels at a constant speed of 60 miles per hour.

4.25

Each zoo ticket costs $9.

Level H

Chapter 5

Lesson 3

y = 60x

y = 9x

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Start the lesson by asking students to use their prior knowledge to come up with an equation for a proportional relationship. Say: In the last chapter, we wrote equations for lines. What was the formula? [y = mx + b] What did m and b stand for? [m = slope, b = y-intercept] In this chapter, we’re talking about proportional relationships. What is something special about a line that is a proportional relationship? [It goes through the origin.] How does that affect the equation y = mx + b? [It makes the y-intercept 0, so there's no b; y = mx] How do you know? [When x = 0, it tells you what the y-intercept is, so it must always be 0 in a proportional relationship since all these lines go through (0,0)] Write the formula y = kx on the board. Ask students: What does the k represent? [The slope or unit rate] Read the word problem in the Learn and Connect section about Eitan and Mannie's homework times. Start with Eitan’s table and ask students how to find the unit rate. [Divide the y-values by the x-values.] Then, evaluate the unit rate using each data point to confirm consistency. Next, ask students to create the equation that represents the time it takes for Eitan to solve a math problem, reminding them that the unit rate is the slope. [y = 1.4x] Move on to Mannie’s graph and ask students how to find the unit rate from the graph. [Find the slope.] Ask students how we know that it remains constant. [It's a straight line.] Then, ask students to write the equation that represents Mannie’s rate. [y = 1.5x] Finally, ask students who solves problems faster and how they know. [Mannie, because his unit rate is higher.] Explain that unit rates and equations of proportional relationships can be used to compare proportional relationships and their rates of change.

Exercise | 5-3 Name Write the equation based on the graph. 2.

160 120 (5, 100)

80 40

(2, 40) 0

Cost of Pretzels 20

Total Cost ($)

Total Earnings ($)

Earnings for Mowing Lawns 200

16 12

(6, 12)

8 (3, 6)

4 0

Lawns Mowed

200 150

Provide a reference sheet that outlines how to identify and compare proportional relationships using tables, graphs, and equations. Include step-bystep examples showing how to find the unit rate or constant of proportionality in each format.

(6, 150)

100 (3, 75)

50 0

Number of Pretzels

Photo Sessions

y = 2x

y = 20x

STRUGGLING LEARNERS

Photographer's Rates 250

Total Cost ($)

y = 25x

Write the equation based on the table. y = 30x

Tickets (x)

Miles (y)

120

180

270

Cost $ (y)

21.75

36.25

43.50

65.25

y = 2.50x

y = 7.25x

Hours (x)

EARLY FINISHERS

y = 8x

Packs of Chips (x)

Boxes (x)

Cost $ (y)

5.00

12.50

17.50

25.00

Snacks (y)

Solve. Aaron earns $60 for mowing 4 lawns. Mike earns money weeding gardens, and his earnings are shown by the equation y = 14x. Who earns more per job?

Swift Car Rentals offers two pricing options. Plan A’s rate is based on the equation y = 0.50x, where x is the number of miles driven. Plan B’s prices are shown in the table. Which plan offers the lower rate per mile? Miles

100

140

Plan A

Cost ($)

11. Printer A prints 90 booklets in 6 minutes. Printer B is represented by the graph. Which printer prints faster?

Aaron

10. Team A bikes at a speed of 120 miles in 4 hours. Team B’s graph includes the point (2, 60). Which team is biking faster? They are the same.

30 25 15 10 5

Printer A

CH AL L ENGE 12. Imagine there are two graphs that show the distances two runners ran over time. Can you tell which runner is faster just by looking? How? Look at the slopes. The steeper the slope, the faster the runner.

Lighthouse Math

Level H

Chapter 5

Exercise 3

Have students select one problem from the Exercise section and write two new equations for the same scenario: one showing a greater unit rate and one showing a lesser unit rate. (For example, if the original equation is y = 2x, they might write y = 3x and y = 1.5x.) Ask them to explain how these new rates compare to the original and what that difference would mean in context (e.g., working faster, costing more, or traveling farther per unit).

CHALLENGE AND EXPLORE For problem 12, students will need to identify slope as a rate and therefore the correct place to look to compare the runners' speeds. Students will need to be able to describe a graph as steeper than another graph based on the angle of the line.

Have students work in pairs to solve problems 1-6 in the Apply section, then review the answers together as a class to ensure understanding. Next, have students work in small groups to complete problems 1-7 in the Exercise section, reminding them that the unit rate is the slope in the equation. Then, have students work with a partner to solve problems 8-11 reminding them to read through the whole problem carefully to find which two things they are comparing. Review the answers and strategies as a class to ensure understanding.

ACTIVITY Comparing Rates in the Real World: Students will extend their understanding of proportional relationships by comparing rates in real-world contexts. Prepare a set of notecards, each containing a short word problem that compares two proportional situations (for example, two people running, two machines producing items, or two stores with different pricing structures). Divide the class into groups of three or four and assign each group three to four word problems from the set. Have students work collaboratively to create a table, a graph, and an equation for each problem to represent and compare the two given rates. Encourage them to determine which scenario shows the greater rate of change and explain how they know. Once all groups have completed their assigned problems, have them present their findings to the class. Conclude with a discussion highlighting how proportional relationships can be represented in multiple ways and how unit rate (the constant of proportionality) allows meaningful comparisons across different real-world situations.

COMMON ERRORS Students may divide x by y when evaluating the unit rate.

ASSESS Check the even-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 5-4 5-4 | Review

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will review concepts introduced in Chapter 5.

What is a proportional relationship? When two quantities increase or decrease consistently, they are in a proportional relationship. The ratio of y to x, the unit rate, is always the same. When graphed, the line goes through the origin (0,0). Solving Proportions:

PRE-LESSON WARM-UP

Guiding Questions: 1. How does an equation show a proportional relationship? [It has the form y = kx, where k is constant and the slope.] 2. How does a table show a proportional relationship? [The ratio y ÷ x remains the same for all values.] 3. How does a graph show a proportional relationship? [It forms a straight line that passes through the origin.]

a×d=c×b

Example:

8 12 = x 18

8 × 18 = 12 × x

12 = x

144 = 12x

How to Find the Unit Rate in a Proportional Relationship From an equation

Use k in the equation y = kx.

From a table

Divide y by x in any column.

y = 3x → 3 is the unit rate. Miles (y)

Hour (x)

18 ÷ 2 = 9 miles per hour

From a graph

Total Cost ($)

Cost of Pencils

Find the slope.

10 8 6 4 2

(12, 9) (4, 3)

0 2 4 6 8 10 12 14 16 18 20

Number of Pencils

(12,9) → 9 ÷ 12 = $0.75 per pencil

Write or draw a mix of proportional and nonproportional relationships on the board using a variety of tables, graphs, and equations. Include an equation in the form y = kx and one with an added constant, a table with a constant ratio, and a graph that does not pass through the origin. Have students work in pairs to identify which examples represent proportional relationships. Then, ask them to prove their reasoning by showing the same relationship in another form: for example, creating a graph from a table or an equation from a graph. After the groups finish, review as a class to ensure all forms of representation (table, graph, and equation) are clearly connected and understood.

a c = b d

A P P LY Write a proportion. Then solve. Question

Proportion

Solution

A recipe uses 3 cups of sugar to make 24 cookies. How many cups of sugar are needed to make 40 cookies?

3 = x 24 40

5 cups

A car travels 180 miles in 4 hours. How far can it travel in 6 hours at the same speed?

180 = x 4 6

270 miles

A printer produces 120 pages in 8 minutes. How many pages will it print in 15 minutes?

120 = x 8 15

225 pages

It costs $7.50 for 3 pounds of apples. How much will 5 pounds of apples cost?

7.5 = x 3 5

$12.50

100

Level H

Chapter 5

Lesson 4

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect)

Write the term “proportional relationship” on the board and ask students to define it in their own words. As students share their ideas, record each definition on the board, continuing until all aspects of proportional relationships have been mentioned, including representations in a table, equation, and graph. Guide students to notice the common features, such as a constant ratio, the equation y = kx, and a straight line through the origin. Then, connect this discussion to the concept of unit rate. Ask students what a unit rate is and how it relates to proportional equations, tables, and graphs. Walk through the table in the Learn and Connect section showing how the unit rate appears as the constant ratio in a table, the coefficient k in an equation, and the slope of the line on a graph. Review the importance of understanding how these forms are connected and how they all describe the same relationship in different ways.

Lighthouse MATH Level H | Teacher's Guide

Exercise | 5-4 Name Find the unit rate to determine the better buy. Option B

STRUGGLING LEARNERS

Better Buy

12-oz bottle of shampoo for $4.80

18-oz bottle of shampoo for $6.75

Option B

3 notebooks for $6.60

5 notebooks for $10.50

Option B

5-pound bag of rice costs $6.25

2-pound bag of rice costs $2.80

Option A

Provide a comprehensive review sheet that summarizes all major topics from the unit. Include clear examples and visuals showing how to identify and compare proportional relationships using tables, graphs, and equations. The sheet should also review how to find the unit rate, recognize the constant of proportionality (k), and determine whether a graph is proportional. Encourage students to reference this sheet as they work through their review problems for additional support.

Graph the proportional relationship. Then write an equation for the graph. 4.

$30 is earned in 2 hours.

3 notebooks cost $7.50.

(5, 75) (4, 60) (3, 45) (2, 30)

60 30

Cost ($)

Money Earned ($)

y = 15x

(1, 15) 0

(3, 7.50) 5

y = 2.50x

(2, 5) (1, 2.50)

Time (hours)

Number of Notebooks

Solve. Explain your answer. 7.

Joseph’s earnings from his job are shown by the point (5, 65) on a graph. Sam is paid using the equation: y = 13.50x. Who earns more per hour?

Rachel types at a speed of 42 words per minute. Donna’s typing speed is shown in the table below. Who types faster?

Sam because he earns $13.50 per hour and Joseph

138

230

368

460

Ben’s bike ride is shown by the equation y = 15x, where x is time in hours and y is distance in miles. Sam’s biking speed is shown by the graph. Who is biking faster?

A fruit stand sells apples according to the table. A competitor’s prices are shown by the graph. Which stand charges more per apple?

Sam's Biking Distance Over Time

Pounds (x)

Cost $ (y)

18.75

10 8

Sam, because he is

Cost ($)

Distance (Miles)

54 36

biking at a rate of

18 0

Time (Hours)

because they charge

2 0

18 miles per hour.

The fruit stand

6 4

5 Pounds

$1.25 per pound.

CH AL L ENGE 10. Can you tell if a graph is in a proportional relationship if you know just one point? No, you need to know that it is straight line that goes through (0,0)

Lighthouse Math

EARLY FINISHERS

Donna, because her speed is 46 words per minute.

only earns $13 per hour.

Minutes (x) Words Typed (y)

Level H

Chapter 5

Exercise 4

101

APPLY AND DEVELOP SKILLS (Practice) Walk through problem 1 in the Apply section together as a class, modeling each step and explaining the reasoning behind setting up and solving the problem. Then, have students work with a partner to solve problems 2-4, discussing their strategies and comparing answers. Review these problems as a class, asking student volunteers to show their work on the board and explain their thinking. Next, have students work in small groups to complete problems 1-9 in the Exercise section, encouraging collaboration and discussion about proportional relationships. Conclude by reviewing the set as a class, selecting a few key problems to go over together in detail to reinforce understanding and address any common errors or misconceptions.

ACTIVITY Build off of or adjust any of the activities of this chapter for review. For example, create a matching activity similar to the one in Lesson 2, but include more concepts from the chapter.

Have students choose one topic from this chapter that they found challenging. Ask them to create two or three original problems on that topic, solve them, and write short explanations for each step.

CHALLENGE AND EXPLORE For problem 10, students will need to remember the two features of a proportional graph: that it is a straight line and that it passes through the origin (0,0). Also encourage students to remember that a unit rate remains constant in a proportional relationship and that this is why a straight line forms.

COMMON ERRORS Students may forget that proportional relationships must pass through the origin. Students may divide x by y when solving for the unit rate.

ASSESS Ask students to come up with an example of a proportional and a non-proportional relationship as an exit ticket. Lighthouse MATH Level H | Teacher's Guide

Option A

Chapter 6

102

In Chapter 6, we will discover

Linear Systems When we want to know the value of more than one unknown variable, we can use a system of equations to help us find the set of numbers that works for both. • We will learn how to use a graph to find the number of solutions that a system has. • We will learn how to isolate a variable and use substitution to solve linear systems. • We will learn how to use the elimination method to solve systems. • We will solve real-world problems using systems of equations.

103

Level H | 6-0 Chapter 6 | Skill Checklist

Skill 1: Graphing in Slope-Intercept Form Graph the lines.

y = mx + b

m = slope b = y-intercept

y = 2x − 5

1 x+4 2

-10

-5

m=2 b = -5

-10

10 5

2 y=- x−7 3

-10

Objective and Learning Goals

-5 -5 -10

-6

Students will review skills needed for Chapter 6:

y = -2x + 4 -10

-5 -5

-6

-10

I can graph linear equations in slope-intercept form.

out 3 correct

Skill 2: The Distributive Property and Factoring © Lighthouse Curriculum. Copying strictly prohibited.

y Graphing in slopeintercept form y The distributive property and factoring y Evaluating expressions using substitution y Solving equations y Zero pairs

The Distributive Property

Apply the distributive property. 1.

2(3x + 4) 2(3x) + 2(4) 6x + 8

3(4x + 7) 12x + 21

45x + 35 ÷5 ÷5 5(9x + 7)

30x − 40

8(8x + 9) 64x + 72

Factor each expression. 4.

18x + 10

104

Skill 1: Graphing in Slope-Intercept Form y Ask students to tell you the formula for slopeintercept form. [y = mx + b] y Remind students that m stands for the slope and b stands for the y-intercept. Define each term as a class. y Remind students that to graph, we first plot the y-intercept at (0,b) and then use the slope to find more points. Then we connect all the points with a straight edge or a ruler. y Have students complete problems 1-3 and review the answers and strategies as a class.

Level H

56x + 8 8(7x + 1)

100x − 36 4(25x − 9)

I can apply the distributive property and factor expressions.

out 6 correct

5(6x − 8)

Factoring

2(9x + 5)

Lighthouse MATH Level H | Teacher's Guide

Chapter 6

Skill Checklist

Lighthouse Math

Skill 2: The Distributive Property and Factoring y Remind students that the distributive property is applied when a factor is being multiplied by terms added or subtracted inside of parentheses. It is important to multiply all of the terms inside the parentheses by this factor. y Remind students that the inverse of the distributive property is factoring. y Remind students that we always want to factor fully, so we look for the greatest common factor of the terms in the expression, then we divide each term by this factor. y Remind students that we always write the factor we used to divide outside the parentheses. y Have students complete problems 1-4 and review the answers and strategies as a class.

Name

Skill 3: Evaluating Expressions Using Substitution x=4

5(3x + 8)

Evaluate. 1.

5(3(4) + 8) 5(12 + 8) 5(20)

100

9.2 + 5.3x when x = 1.1

-22

15.03

1 14x + 3 when x = 2

x + 5 when x = -9 3

2 − 4x when x = 6

I can evaluate expressions using substitution.

out 4 correct

Skill 4: Solving Equations Solve the equations using inverse operations. 1.

x + 8 = 15 2

x = 13

x = 14

-5x + 4 = 29

x − 12 = -15 4

3x − 7 = 32

x = -5

x = -12

I can solve equations using inverse operations.

out 4 correct

Skill 5: Zero Pairs 6 + -6 = 0

Fill in the blanks to make the statements true.

x + -x = 0

1. 3. out 4 correct

Lighthouse Math

Skill 3: Evaluating Expressions Using Substitution y Remind students that the value of an algebraic expression depends on the value of the variable. y Using the example in the gray box, remind students of how to plug in the value of x to find the value of the entire expression. Remind students that we must follow the order of operations. y Have students complete problems 1-4 and review the answers and strategies as a class.

-3

4x +

-4x

=0 =0

-15 +

1 - y+ 2

15 1 y 2

=0 =0

I can apply the additive inverse property.

Level H

Chapter 6

Skill Checklist

105

Skill 4: Solving Equations

Skill 5: Zero Pairs

y Use the table in the gray box to review the inverse operations. y Remind students that when solving, we follow the order of operations in reverse (SADMEP). y Remind students that whatever they do to one side of an equation, they must do to other side as well to keep it balanced. y Have students complete problems 1-4 and review the answers and strategies as a class.

y Remind students that every number has an inverse (sometimes called the number’s opposite). y Remind students that when a number is added to its inverse, the result is 0. y Tell students that terms with variables have an inverse too. y Have students complete problems 1-4 and review the answers and strategies as a class.

Lighthouse MATH Level H | Teacher's Guide

4x + 6 = -14 −6 −6 4x = -20 ÷4 ÷4 x = -5

Level H | 6-1 6- 1 | Introduction to Systems of Linear Equations

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to understand that the number of solutions to a system of linear equations is the number of places on a graph where lines overlap. They will use graphs, slope, and y-intercept to determine the number of solutions.

Graph each line given the slope and the y-intercept.

b=1

-5

m=-

1 4 -5

b=0

-5

Write an equation for each proportional relationship. 1.

4 cookies cost $12

y = 4x

-5

y= 2 x

y = 3x -5

L E A R N A ND C O NNE C T When two linear equations are plotted on a single graph, they form a system of equations. These lines may intersect, overlap, or not intersect at all. We can compare the slopes and the y-intercepts of the equations to tell us how lines meet or cross on a graph and how many solutions a system has. One solution

No solution

-5

The lines intersect at one point.

The lines do not intersect.

y = 2x + 1

2 y= x+4 3 2 y= x−2 3

y = -3x + 6 The lines have different slopes. © Lighthouse Curriculum. Copying strictly prohibited.

Infinite solutions

-5

y Notecards

Guiding Questions: 1. What does the slope tell us about the line? [It tells how steep it is and whether it slopes up or down.] 2. What about the slope tells you whether the line is increasing (slopes up) or decreasing (slopes down)? [If it is positive, it is increasing. If it is negative, it is decreasing.] 3. Which of these two lines are parallel? How do you know? [y = 3x + 4 is parallel to y = 3x + 1 because their slopes are the same.] 4. Which lines will cross? [y = -3x − 2 will cross both of the other lines.]

-5

m = -3

SPIRAL REVIEW

Materials

y = 3x + 4 [a steep line sloping up from left to right going through 4 on the y-axis] y = -3x − 2 [a steep line sloping down from left to right going through -2 on the y-axis] y = 3x + 1 [a steep line sloping up from left to right going through 1 on the y-axis]

-5

y Linear equation - an equation for a straight line y System of equations - a group of two or more equations that share the same variables y Solution - an ordered pair that satisfies an equation y Intersect - cross

Ask students to describe what the following equations would look like on a graph (without actually plotting it).

1 2

b=2

Vocabulary

PRE-LESSON WARM-UP

5 -5

The lines intersect at every point.

y = 3x + 2 y = 3x + 2

The lines have the same slope but different y-intercepts.

The lines have the same slope and y-intercept.

A P P LY Write the letter of the answer that best describes the lines on the graphs below. 5

1. -5

2. 5

-5

A -5

3. -5

C -5

B -5

They intersect at one point.

They do not intersect.

They intersect at every point.

Vocabulary Linear equation - an equation for a straight line Solution - an ordered pair that makes an equation true System of equations - a group of two or more equations that share the same variables Intersect - cross

106

Level H

Chapter 6

Lesson 1

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Explain to students that in this lesson, we will explore what two lines that are graphed on the same coordinate plane can tell us. This is called a system of equations. Just like we can find the solution to an equation, we can find the solution to a system of equations. Have students open their books and look at the table in the Learn and Connect section. Begin on the left with the graph with one solution. Point out the equations for each line and ask students to compare and contrast them. [They have different slopes and different y-intercepts.] Emphasize that when lines have different slopes, they will intersect on a graph. The point where the two lines cross is the solution to this system. Next, ask students what they notice about the lines in the second graph. [They don't cross; they are parallel.] Ask them what they notice about the equations for the two lines on this graph. [They have the same slope and different y-intercepts.] Emphasize that when lines have the same slope but different y-intercepts, they will be parallel and there will be no solution to the system. Ask students why they think that is. [Parallel lines will never cross, so there is no point of intersection.] Finally, have students notice the final graph on the right with two overlapping lines and the same equation. Explain that when this happens, it means there will be an infinite amount of solutions. Ask students to explain what that means. [Every point on the graph is an intersection because the lines were graphed one on top of the other.]

APPLY AND DEVELOP SKILLS (Practice) Have students complete problems 1-3 in the Apply section independently, then Lighthouse MATH Level H | Teacher's Guide

Exercise | 6-1 Name For each graph, write if the lines are intersecting, parallel, or the same line. 5

1. -5

2. 5

-5

3. 5

-5

same line

-5

intersecting

STRUGGLING LEARNERS

Have students highlight slopes in one color and y-intercepts in another color to help them compare them.

-5

intersecting

parallel

For each system of equations, write if the slope and y-intercept are the same or different. y = 6x + 4

y = 21 (18x – 14)

EARLY FINISHERS

y=x

y = -x

y = -2x + 1

y = -6x – 4

slope and y-intercept

slope is the same;

slope is different;

are the same

are different

y-intercept is different

y-intercept is the same

Have students go back to problems 5-8 in the Exercise section and write whether there is one solution, no solution, or infinite solutions. Have students go back to problems 1316 and change one part of one equation in each so there is a different number of solutions. Have students go back to problem 9 and write the ordered pair at which the lines cross.

Determine how many solutions exist for each system of equations. 5

9. -5

-5

13.

10.

11. 5

-5

one solution

no solution

one solution

y = 13x – 6

y = 12x – 9

y = 3x

14.

12.

y = -5x + 4

y = 3(4x – 2)

one solution

no solution

15.

infinitely many solutions

16.

y = 1.5(2x + 6) infinitely many solutions

y = 32 (15x – 9) y=

8x – 6

one solution

Determine how many solutions exist for each system of equations. Explain why. 18. y = -5x + 3 and y = -5x + 7

One solution because the slopes are different, so on a graph, they would intersect at one point

No solution because the slopes are the same, while the y-intercepts are different; the two lines would be parallel and never intersect

CH AL L ENGE 19. Two friends, Alex and Sam, are planning bike trips. Each of them tracks their total cost. Alex rents a bike for a flat $40 and pays $0.20 per mile. Sam brings his own bike but needs a tune-up before the trip, costing $60. Then, he pays $0.10 per mile for supplies and snacks. Write an equation for the total cost per mile driven for each boy. Alex: y = 0.2x + 40 Sam: y = 0.1x + 60 Will Alex and Sam ever spend the same amount of money on their trips? If so, after how many miles? Explain. Yes, because their equations have different slopes. After 200 miles, Alex and Sam will have spent the same amount of money on their trips.

Lighthouse Math

Level H

Chapter 6

Exercise 1

107

go over the answers as a class to ensure understanding. For problem 1, point out to students that although we can’t see the lines intersecting on the graph, if they were extended, they would in fact cross each other, therefore, there is one solution. Have students continue to work independently on problems 1-8 in the Exercise section. Pause here to review the answers as a class. For problem 2, ask students at what point the lines cross. [(1,-2)] Explain that this is the numerical solution to the system. After going over problems 5-8, ask students why they are being asked to compare the slopes and y-intercepts of two lines. [because it can help us determine whether the lines intersect and how many solutions there are] Next, have students complete problems 9-18 and then review the answers as a class.

ACTIVITY Speed System: The purpose of this activity is for students to compare equations to see how many solutions the system has. Give each student a card with one equation written in slope-intercept form. Be sure that some of the equation cards are repeats of the same equation, that some have just the same slope as another card, and that some have different slopes and different y-intercepts than other cards. Have students form two circles, an inner circle and an outer circle, so that they pair up with one other student in the class. Have the students compare their equations and decide how many solutions their system has. Then call out, “Switch!” and have the inner circle rotate so that each person has a new partner. Have them complete the exercise again until the circle has made one complete turn. Conclude with a discussion of strategy, asking students what they looked at first to compare the equations. [slope]

17. y = -x + 3 and y = x + 3

CHALLENGE AND EXPLORE Explain to students before completing problem 19 that solutions to systems have a meaning based on the context of the problem. First, students will need to write an equation for each person, and then, they need to notice that the slopes and y-intercepts of these equations are different. This will help them conclude that there is one solution to the system. They will need to guess and check to find the number of miles at which the friends will have spent the same amount of money.

COMMON ERRORS Students may think there is no solution on a graph where the lines do not appear to intersect but will eventually intersect somewhere off of the graph. Students may think that if the y-intercepts are the same, there is no solution even if the slopes are different.

ASSESS Exit ticket: Have students describe the difference between no solution and infinite solutions based on the slopes and y-intercepts of the equations of the lines in a system. [No solution: same slope, different y-intercept; Infinite solutions: same slope, same y-intercept] Lighthouse MATH Level H | Teacher's Guide

y = -2x + 1

Level H | 6-2 6-2 | Solutions of Systems of Equations

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to understand that the solution to a system of equations is an x and y value that satisfies both equations.

Evaluate each expression. 2.

5x + 2 if x = -4

-8x – 3 if x = 5

-18 SPIRAL REVIEW

18x + 11 if x = 2

-43

y = -5x – 4

y = 21 x + 8

y = -3x + 2

no solution

one solution

infinite solutions

one solution

To determine if an ordered pair is a solution to a system, substitute the x and y values into both equations.

L E A R N A ND C O NNE C T

y Notecards

System of Equations y = 2x + 1 y = -3x + 6 Solution (1, 3) 3 = 2(1) + 1 3=3

Not a Solution (2, 5) 5 = 2(2) + 1 5=5

3 = -3(1) + 6 3=3

5 = -3(2) + 6 5=0

If the values on each side of the equation equal one another, then the ordered pair is a solution to that equation. If the ordered pair is a solution for both equations, then it is a solution to the system.

PRE-LESSON WARM-UP

y = 31 x + 2

y = 3x + 8

Materials

Guiding Questions: 1. When all we knew was the total number of chickens the farmer has, could we figure out how many of each type he has? [No, we didn’t have enough information.] 2. Once we were told that he has twice as many hens as roosters, how could we find the solution? [We found the spot on our table where the number of hens was double the number of roosters.]

y = 3x + 8

y = 11x – 7

The solution to a system of equations is an ordered pair that makes each equation true. That means that when you plug in the same x value to each equation, they will both equal the same y value.

3x + 4y = 5 The system has no solution 3x + 4y = 9 because 3x + 4y cannot equal both 5 and 9.

On a graph, this is the point where both lines meet.

A P P LY © Lighthouse Curriculum. Copying strictly prohibited.

Have students take a minute to discuss the answer with a partner, then have a discussion about possible answers. Make a table of values to show possible solutions and ask students what equation can be written from this scenario. [x + y = 9] Then, provide students with another bit of information: The farmer has twice as many hens as roosters. Have students give an equation that models this. [2x = y] Ask students to discuss with a partner, and then as a class, use the table of values to find the solution [6 hens and 3 roosters]. Use the Guiding Questions to lead a discussion about how we needed both equations to figure out how many of each type of chicken he had and how there was only one right answer; the one that worked for both equations.

y = -5x + 3

y System of equations - a group of two or more equations that share the same variables y Solution - an ordered pair that satisfies an equation

A farmer has hens and roosters. He has 9 chickens all together. How many of each type of chicken does he have?

9x + 5x if x = 3

State whether each system has one solution, no solution, or infinite solutions. 1.

Vocabulary

Pose the following question to students:

Match each system of equations with its solution. 1.

y = 3x + 1 y= x +7

y = -2x + 5 y = 4x − 7

y = 6x + 3 y = -3x − 3

A. (2, 1)

B. (- 32 , -1)

C. (3, 10)

Determine whether the ordered pair is or isn’t a solution of the system. 4.

(2, 8)

(4, 0)

(-1, 5)

(0, -1)

(-2, -2)

y = 2x + 1

y= x −4

y = -2x + 3

y = 3x − 1

y = -3x + 6

y = -2x + 8

y=x+6

y = 2x + 4

y= x y = 2x + 2

Yes or No

Vocabulary System of equations - a group of two or more equations that share the same variables Solution - an ordered pair that makes an equation true

108

Level H

Chapter 6

Lesson 2

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Ask students: When a system has one solution, where will you see it on a graph? [the ordered pair at the point where the lines cross] The solution is both an x-value and a y-value and it works for both equations in the system. We can test the ordered pair to see if it is a solution by plugging in both the x-value and the y-value to BOTH equations to see if we get two true statements. Use the table in the Learn and Connect section to show students what this process looks like for an ordered pair that is a solution and an ordered pair that is not a solution. Then, show the example of the system below the table that has no solution. Ask students if they can think of a combination of x- and y-values that would work in this system. [No. There is no pair of x- and y-values that would make both true because the left sides are both 3x + 4y, but the right sides are different values. It’s not possible for 3x + 4y to equal both 5 and 9.]

APPLY AND DEVELOP SKILLS (Practice) Have students work with a partner to complete problems 1-3 in the Apply section. They should work together to plug in each ordered pair to each equation until they find the one that works for a pair of equations. Next, have students continue to work with a partner to complete problems 4-8. Review the answers to problems 1-8 before moving on to ensure understanding. Next, have students work on problems 1-15 in the Exercise section independently. For

Lighthouse MATH Level H | Teacher's Guide

Exercise | 6-2 Name Determine if the given ordered pair is a solution to the system.

yes

(-2, 8)

(5, 5)

y = -4x

y=x

y = x + 10 yes

(0, 0)

yes

(4, -2)

(2, 3)

(2, 1)

y = 3x − 3

y = 2x − 3

y = 3x − 10

y = 2x − 6

y = -x + 7

y = -x + 5

yes

(3, -8)

(4, 6)

(6, -4)

10. (-1, 1)

STRUGGLING LEARNERS

y = -x + 6

Provide students with an anchor chart to make clear the connection between this lesson and the last. The anchor chart should have the graphs of two equations in a system. The point at which they intersect should be highlighted and labeled with its ordered pair. The two equations should be written, and the ordered pair should be plugged into both of them, resulting in a true statement.

y = 3x

-41 = 5x + 7y

x + y = 10

2x + 3y = 0

-4x − 9y = 13

0 = -4x − y

y = 2x − 14

y = 2x + 3

5x − 2y = 38

7x + 8y = 15

Complete the solution for each system. 1 13. (- , 1 ) 3 9x + 6y = 3

12. ( 2 , 4)

11. (10, 15 ) -x + y = 5

2x + 2y = 12

y = 2x − 5

y = 2x

14. ( 0 , 0)

-9x + 2y = 5

15. ( 2 , 17)

21x + -8y = 0

y = 8x + 1

15x + 4y = 0

4x + y = 25

Fill in the x/y tables. Circle the matching ordered pair. Write the solution to each system. 16.

x+y=3 y=x−1 Solution: ( 2

1 )

x+y=3

y=x-1

-1

x + y = 15

17.

y = 2x Solution: ( 5

, 10 )

x + y = 15

y = 2x

EARLY FINISHERS

-2x + 7y = 11 -2x + 7y = 2

-2x + 7y cannot equal both 11 and 2

CH AL L ENGE 19. Kelly and Emily are selling chocolate and popcorn. Kelly sells 4 chocolate bars and 6 bags of popcorn and collects $48. Emily sells 8 chocolate bars and 12 bags of popcorn and collects $96. Write a system of equations where x is the cost per chocolate bar and y is the cost per bag of popcorn. 4x + 6y = 48; 8x + 12y = 96

Simplify both equations. Does it have one solution, no solution, or infinitely many solutions? Explain how you know. There are infinite solutions. The equations can both be "simplified" to the same equation: 2x + 3y = 24

Lighthouse Math

Level H

Chapter 6

Exercise 2

109

Answer the question. 18. Explain why the following system has no solution.

Have students return to problem 3 in the Exercise section to graph the two equations listed on the same coordinate plane. Then, have them use this graph to help them find the correct solution.

CHALLENGE AND EXPLORE For problem 19, students will need to create two equations. Then, they will need to simplify these equations by dividing each by a common factor. Each equation can be simplified to the same equation, which means that they are actually the same equation. Thus there are infinite solutions.

problems 1-10, remind students to check that the ordered pair works in both equations. For problems 11-15, students will need to find the missing part of the ordered pair. Review the answers and discuss what strategies students used to solve. Next, have students complete problems 16-17 by filling in the x/y tables and finding the matching ordered pair to solve the system. Review the answers and have students complete problem 18, an error analysis.

ACTIVITY Let’s Table It: The purpose of this activity is for students to use reasoning to find the solution to a system and to practice checking their solutions for correctness. Create six cards, each with a blank x/y table and a unique equation. Use: y = 4x, y = -2x + 6, y = -x + 5, y = x − 3, y = - 21 x , and y = x + 3. Divide students into six groups and give each group a card. Instruct them to fill in the x/y table by plugging in x-values from -1 to 10 and writing their corresponding y-values. Then, each group will pair up with another group, and they will find the solution to the system they created by finding the ordered pair that matches. They will check their answer by plugging in the x- and y-values to their equations to check for a true statement. One set (y = x − 3 and y = x + 3) will not have a solution. Ask students to figure out why. [Because the lines have the same slope so they are parallel]

COMMON ERRORS Students may not check that an ordered pair works for both equations.

ASSESS Check problems 5, 9, 13, and 17 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

Level H | 6-3 6-3 | Solving Systems of Equations by Graphing

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to find the solution to a system of linear equations by graphing.

Graph each linear equation. 5

1 x–1 3

-5

y = -3

-5

Vocabulary

-5

Determine if each ordered pair is a solution of the system. 1.

(1, 3)

y = 2x + 1 y = -x + 4

y System of equations - a group of two or more equations that share the same variables y Solution - an ordered pair that satisfies an equation y Intersect - cross

y = -3x

-5

SPIRAL REVIEW

y = -3x + 6

(2, 0)

yes

y= x –2

(2, 4)

yes

y= x +2 y = -2x + 5

L E A R N A ND C O NNE C T 5

You can find the solution to a system of equations by graphing. 1. Graph each line.

Materials

2. Find where the two points intersect. Write the solution as an ordered pair.

y Graph paper

1 x−4 2

y = -x + 2

-5

When this ordered pair is substituted into each equation, they will both give a true statement.

Solution: (4,-2)

-5

PRE-LESSON WARM-UP

y = 31 x + 4 y = -2x + 1 y=x−5 Guiding Questions: 1. What is the first thing to look for when graphing an equation in slope-intercept form? [the y-intercept] 2. How can we find another point on the graph after plotting the y-intercept? [Use the slope to navigate to another point on the graph.]

A P P LY © Lighthouse Curriculum. Copying strictly prohibited.

Have students graph the following equations on graph paper.

Graph each system of equations if necessary. Then choose the correct solution. y = -x

y = - 32 x + 4

y=3

y = - 41 x − 1

(3, -3)

-5

5 -5

(-3, 3)

(3, 3)

D. (-3, -3)

-5

5 -5

y = 2x

y = -3x + 5

A. (-2, -4) B.

(-2, 4)

(4, 2)

(4, -2)

-5

5 -5

(1, 2)

(2, 1)

(-1, 2)

(-2, 1)

Vocabulary System of equations - a group of two or more equations that share the same variables Solution - an ordered pair that makes an equation true Intersect - cross

110

Level H

Chapter 6

Lesson 3

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect)

Remind students that the solution to a system of equations is the ordered pair at the point at which the equations cross on the graph. Ask students to plot the equations given in the Learn and Connect section: y = 21 x − 4 and y = -x + 2. Then, have students draw a dot at the point where the two lines intersect. Ask students to label the point with the correct ordered pair. [(4,-2)] Tell students: This is the solution to the system. Then ask: How can we check to be sure that our answer is correct? [plug in the ordered pair to the two equations and see if you get a true statement.] Ask students to plug in 4 for x and -2 for y in each equation to check.

APPLY AND DEVELOP SKILLS (Practice) Have students complete the Apply section independently, then review the answers as a class to ensure understanding. Have students complete problems 1-7 in the Exercise section independently, then review the answers and strategies used as a class. Remind students that they can check their answers to see if they graphed correctly by plugging in x and y to both equations to see if they give true statements. Read problems 8 and 9 together as a class, discussing where the equations for each system came from. Then, have students graph and find the solution. Conclude by discussing the meaning of each solution. Lighthouse MATH Level H | Teacher's Guide

Exercise | 6-3 Name Write the solution to each system of equations. 5

-5

STRUGGLING LEARNERS

4. 5

-5

(-1, 1)

(2, 2)

(3, 4)

(-4, -2)

Students who struggle to graph should be provided with an anchor chart to help them graph. Provide a step-bystep list of the process of graphing from an equation in slope-intercept form. Add in a prompt to look for the point of intersection to find the solution to the system.

Solve each system of equations by graphing. 5.

y = 2x

y = -2x + 1

y = - 21 x + 3

y = -2x + 4

y = -x – 3 -5

(-1, -2)

-5

no solution

-5

y= x -5

(2, 2)

-5

Solve each word problem by graphing. 9.

Liam and Manuel are saving money. Liam starts with $60 already saved and plans to save $10 every week. Manuel starts with $20 and plans to save $30 every week. This can be represented by the following system of equations:

When will Flora and Lena have the same amount of money?

100

(2, 80); After two weeks,

120

Money ($)

Money Saved ($)

When will both Liam and Manuel have earned the same amount of money? How much will they have earned?

160

both Liam and Manuel

80 40 0

will have saved $80.

CHALLENGE AND EXPLORE

(1, 70) After 1 week,

80 60

both Flora and Lena

40 20 0

Number of Weeks

will have $70.

Number of Weeks

10. A club sells 20 cookies. Some are chocolate chip cookies (x) and some are sandwich cookies (y). They sell 1.5 times as many sandwich than chocolate chip cookies. A. Write two equations to represent this scenario y = 20 − x; y = 1.5x B. Graph the equations. How many of each type of cookie did they sell? 8 chocolate chip and 12 sandwich

Level H

Chapter 6

Money Raised

CH AL L ENGE

Lighthouse Math

Have students go back to problems 1-4 in the Exercise section to figure out which equations were graphed in each system.

y = -10x + 80 y = 30x + 40

y = 10x + 60 y = 30x + 20

200

EARLY FINISHERS

Flora has $80 in her wallet but spends 10 each week on food. Lena has $40 but saves $30 each week. This can be represented by the following system of equations:

Exercise 3

20 16 12 8 4 0

# of Cookies

111

ACTIVITY Graph It: The purpose of this activity is to have students practice graphing and looking for the solution to a system on a graph. Give each pair of students a notecard with a system of equations on it. The systems should have one solution. Have the students work together to graph their system on a piece of paper, find the solution, and label it. Next, ask students to make a change to one of their equations so that the system will have no solution. Ask them to graph again to test their change. Next, ask them to make one more change so that the system has an infinite amount of solutions. Have them graph one more time to test their change. Conclude by holding a discussion about what makes systems have no solution or an infinite amount of solutions and what that looks like on a graph.

For problem 10, students will need to create two equations based on the scenario. Since each equation needs to be graphed, they will need to begin with y = and should be in slope-intercept form. Tell students that if a total of 20 cookies are sold, we can find the number of sandwich cookies (y) by subtracting the number of chocolate chip cookies (x) from 20. The equation they should write is y = 20 − x. Ask students: How can we rearrange this equation to be in proper slope-intercept form? [y = -x +20] What is the slope? [-1] What is the y-intercept? [20] What other equation can we write? [y = 1.5x] What fraction is equivalent to 1.5? [ 23 ]

COMMON ERRORS Students may graph inaccurately, which will lead to an incorrect solution.

ASSESS Exit Ticket: Ask students to find the solution to the system y = -2x + 3 and y = x − 6 by graphing. [3, -3] Lighthouse MATH Level H | Teacher's Guide

Level H | 6-4 6-4 | Isolating Variables

PREREQUISITE SKILLS

y Students will manipulate a linear equation to isolate certain variables.

DAI LY REVI EW

Objective and Learning Goals

SPIRAL REVIEW

Vocabulary

Lighthouse MATH Level H | Teacher's Guide

2x + 9 = x – 5

x = -14

3 x – 10 = -2 4 1 2 x – 6 = 2x + 3 2

x = 10 3 x = 18

Determine if each ordered pair is a solution of the system. 1.

(1, 5)

y = 2x + 3

(3, 1)

yes

y = 3x – 4 y = -2x + 7

y = - 21 x + 5 y = 3x – 9 yes

(4, 3)

6x – 2y = 6

What does x equal? What does y equal?

You can isolate a variable so that x or y is alone on one side of the equation. Then, you can see what x or y equals. Use inverse operations:

Solve for x

Solve for y

Step 1: Add or subtract an entire term from both sides so that x and y are on opposite sides of the equal sign.

6x − 2y = 6 + 2y + 2y

6x − 2y = 6 − 6x − 6x

Step 2: If your x or y is left with a coefficient, multiply or divide to isolate the variable.

6x 6 + 2y = 6 6

-2y 6 − 6x = -2 -2

Step 3: Simplify. Make sure to multiply or divide each term.

x=1+

PRE-LESSON WARM-UP

Guiding Questions: 1. What is our goal when solving algebraic equations? [to isolate x] 2. What principle do we use to get x by itself? [inverse operations] 3. If we perform an inverse operation to one side of an equation, what else do we need to make sure to do and why? [the same inverse operation to other side to keep the equation balanced/equivalent]

x=3

Look at the following equation:

y Scrap paper

1 y 3

y = 3 − 3x

A P P LY Follow the steps to solve for x and y. © Lighthouse Curriculum. Copying strictly prohibited.

When students have finished solving on their own, write a list of the steps they used to solve in the order the used to solve them on the board. [subtract 15 from both sides, divide by -4]

7x + 14 = 35

L E A R N A ND C O NNE C T

Materials

15 − 4x = 59 [x = -11]

y = -x + 6

y Isolating a variable - rewriting an equation so that one side of the equation has just one variable y Term - a constant or a variable with a coefficient y Coefficient - a number multiplied (or divided) by a variable

Write the following algebraic equation on the board and ask students to solve.

Solve each equation for x.

1. Circle a term to subtract from both sides. Subtract.

Circle a coefficient to divide both sides by. Divide.

4x + 3y = 12 Solve for x

Solve for y

4x + 3y = 12 − 3y − 3y

4x + 3y = 12 − 4x − 4x

4x = 12 − 3y

3y = 12 − 4y

4x = 12 – 3y 4 4

3y = 12 – 4x 3 3

x=− 4 y+3

Simplify.

y = −1 3 x + 4

Vocabulary Isolating a variable - rewriting an equation so that one side of the equation has just one variable Term - a constant or a variable with a coefficient Coefficient - a number multiplied (or divided) by a variable Inverse Operations - operations that undo each other like addition/subtraction and multiplication/division

112

Level H

Chapter 6

Lesson 4

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Write the equation and questions from the Learn and Connect section on the board: 6x − 2y = 6. What does x equal? What does y equal? Students may say, “We can’t figure it out without another equation.” Acknowledge that this is an excellent answer, and tell students that we can figure out what x and y are equal to by rearranging the equation. We can isolate a variable to find what it is equal to “in terms of” the other variable, meaning we would still need to know the value of the other variable to find the value of the isolated variable. Tell students that we can isolate a variable by using inverse operations just like we would in an algebraic equation with one variable. Begin by going through the steps to isolate the x-variable. Tell students that we can move the entire term -2y by adding 2y to both sides. Have a student tell you the resulting equivalent equation. [6x = 6 + 2y] Ask students what they think we should do next. [divide both sides by 6] Have students tell you the resulting equivalent equation. [x = (6 + 2y)/6] Tell students that the final step is to simplify and emphasize that just like with the distributive property, we need to make sure that all the terms are divided by 6. [x = 1 + 31 y] Say: Now we know what x is equal to. Ask: Why is this helpful even though we don’t know the actual value of x? [If we know the value of y, we can substitute it in to easily find the value of x.] Finally, explain that we could have chosen to isolate y, and have a student come to the board to go through the process again, solving for y.

Exercise | 6-4 Name What does x equal? Isolate the x variable to find out. 1.

x= 6.

y = 2x + 8 1 y–4 2

x= 7.

3x – 18y = 9 x=

y = -3x – 15

6y + 3

- 3 y+5

1 2 y+ 3 3

1 y+6 4

x= 8.

2y = 6x – 4 x=

4x – y = 24

3 1 4x+ 2 y=6 2 - 3 y+8 x=

-4x + 5y = 12 1

-1 4 y – 3

- 5 y+2

y = 21 x – 3

10. y = - 25 x + 4

2y + 6

-2 3 y + 10

STRUGGLING LEARNERS

5x + 2y = 10

Provide students with scrap paper to allow them to take the space they need to write out all the steps and keep their work organized. Have students draw a vertical line from the equal sign down to keep the two sides of their work separate.

What does y equal? Isolate the y variable to find out. 11. 2x + y = 7

12. 4y – x = 12

y = -2x + 7

16. 4x + 6y = -8 y=

17.

- 3 x–1 3

14. 21 x + 8y = 5

13. 6x – 2y = 10

1 x +3 4

1 1 3x+ 2 y=6 2 - 3 x + 12 y=

18. -18 = -3y + 4x y=

15. -5x + 3y = 27

1 5 - x+ 8 y = 16

y = 3x – 5

20. 35 x + 65 y = 4

19. 2x – 14 = 7y

1 3 x+6

2 x–2 7

1 3 x+9

EARLY FINISHERS

18 4 - x+4 5 y = 25

Have students choose four problems from among 1-20 in the Exercise section, and have them solve for the other variable that they did not already solve for.

Read the problems. Isolate a variable to solve. 21. You are planning a rectangular garden using 50 feet of fencing. The perimeter formula is 2L + 2W = P.

22. A bakery makes loaves of bread (b) and cakes (c). They have exactly 150 pounds of flour available. Each loaf of bread requires 1.5 pounds of flour, and each cake requires 0.9 pounds of flour.

2L + 2W = 50 Rewrite this equation so that it will you tell you the width you need to use for any given length.

1.5b + 0.9c = 150

b = -0.6c + 100 © Lighthouse Curriculum. Copying strictly prohibited.

W = -L + 25

CH AL L ENGE 2 4 1 2 x + y – 7 = x + y – 18 3 5 2 3

23. Use the following equation to perform each task. 4

A. Isolate the x variable.

CHALLENGE AND EXPLORE

Rewrite this equation so that it will tell the number of loaves of bread they can make with any given number of cakes.

x = -66 – 5 y

B. Isolate the y variable.

y = -1 4 x – 82 2

C. What is the x-intercept? y-intercept? (Hint: To find the x-intercept, plug in 0 for y. To find the y-intercept, plug in 0 for x.) x-intercept = -66 y-intercept = -82.5

For problem 23, students will need to simplify and then solve the given equation for both x and y, and they will need to use these equations to find the x- and y-intercepts. Finally, they will need to make the connection that the x- and y-intercepts are the constants in each equation.

D. What do you notice about the x-intercept and y-intercept in relation to the isolated equations? The x-intercept is the constant in the equation solved for x, and the y-intercept is the constant in the equation for y.

Lighthouse Math

Level H

Chapter 6

Exercise 4

113

Go through one column in the table in the Apply section together as a class, and have students go through the other column independently. Check in with students to be sure that they are subtracting the correct term and simplifying all the terms in the equation. When students are comfortable with the process, have them complete problems 1-20 in the Exercise section independently, then have them check their answers with a partner. Review the answers to the odd-numbered problems. Have students work with a partner on problems 21 and 22. Review the answers and strategies used as a class and then ask students why it may be helpful to rearrange each equation so that one variable is isolated given the context provided. [We can try out different lengths to see what the width would need to be; we can figure out how many loaves of bread we would be able to make given any number of cakes.]

ACTIVITY Science Application: The purpose of this activity is for students to see how isolating a variable is helpful in a real-life scenario. Write the formula for changing from degrees Celsius to degrees Fahrenheit on the board: F = 95 × C + 32. Tell students that this formula is helpful if you know the temperature in Celsius and you want to convert it to Fahrenheit, but it is less helpful if you want to convert Fahrenheit to Celsius. Ask them to work with a partner to rearrange the equation so that C is isolated to make it easier to change a temperature in Fahrenheit to Celsius. [C = 95 × (F - 32)]

COMMON ERRORS Students may use the wrong inverse operation when rearranging an equation. Students may forget to divide both terms when simplifying.

ASSESS Check problems 6, 7, 16, 17, and 21 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 6-5 6-5 | Solving Systems of Equations by Substitution

PREREQUISITE SKILLS

y Students will be able to solve systems of linear equations using substitution.

DAI LY REVI EW

Objective and Learning Goals

Evaluate each expression if a = -4, b = 7, and c = 31 . 1.

SPIRAL REVIEW

-9c + 5b

-ab

20 = 3x – 5y

Solve each equation for y. 1.

2x + y = 17

-8x – 4y = 36

y = -2x + 17

Vocabulary

-26

5a – 18c

-3

4 1 x + y = -4 5 2 3

y= 5 x–4

y = 2x – 9

12c – b

y = -1 5 x – 8

L E A R N A ND C O NNE C T

y Substitution - when one variable is solved for and substituted into the other equation. y Expressions - a combination of numbers, variables, and operations

Another way to solve systems of equations is with substitution. You can solve for either x or y by plugging in an expression instead of a number. Step 1: Isolate a variable. Check to make sure at least one equation has an isolated variable. If not, isolate a variable in one of the equations.

Materials

Step 2: Substitute. Now, you can substitute an expression into the other equation. When you simplify, you should get a number value for the other variable.

y Notecards

y – 2x = 3

We can solve for the y-variable.

y = 2x + 3 4x + 4y = 24 We can substitute 2x + 3 for y to find x.

Step 3: Solve for the other variable. The value of x is substituted into either equation to find the value of y. The solution to the system is written as an ordered pair (x, y).

PRE-LESSON WARM-UP

y – 2x = 3 + 2x + 2x y = 2x + 3

4x + 4y = 24

4x + 4y = 24 4x + 4(2x + 3) = 24 4x + 8x + 12 = 24 12x + 12 = 24 12x = 12 x=1

x=1

y = 2x + 3 y = 2(1) + 3 y=2+3 y=5

y = 2x + 3 We can substitute 1 for x to find y.

Solution: (1, 5)

3x + 4(2x + 6) − 15 [273] Ask a student to explain the steps they used to find the value. Ask the class to compare different strategies used, such as substituting 24 for x versus simplifying the expression first and then substituting. Emphasize the importance of following the order of operations in either case.

Guiding Questions: 1. What method helps us evaluate the expression when a value for x is given? [substitution] 2. What can we do to simplify the expression before substituting? [Use the distributive property, then combine like terms]

Lighthouse MATH Level H | Teacher's Guide

Ask students to evaluate the expression when x = 24.

A P P LY Fill in the blanks to solve the system by substitution. 1.

y=x+2 5x + y = -4 A. We know that y is equal to

B. We can substitute x + 2 for y in the equation 5x + y = -4.

x+2

5x + (x + 2) = -4 6 x + 2 = -4 6 x = -6 x = -1

C. Now we know that x = -1 .

D. We can substitute -1 for x in the equation y = x + 2. y = -1 + 2 y = 1

E. Now we know that y= 1 . Solution: ( -1 , 1 )

Vocabulary Substitution - when one variable is solved for and substituted into the other equation Expression - a combination of numbers, variables, and operations

114

Level H

Chapter 6

Lesson 5

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Remind students that in the last lesson, we learned how to isolate a variable so that we know what one variable is equal to. In this lesson, we will use this idea and substitution to solve systems. Write the system from the Learn and Connect section on the board (y − 2x = 3; 4x + 4y = 24) and ask students if any of the variables are isolated. [No] Ask students which equation we can more easily manipulate so that one variable is isolated. [y − 2x = 3] Ask: Which variable should we isolate? [y] Ask a student to walk you through the steps of isolating a variable. [add 2x to both sides to get y = 2x + 3] Ask: What is y equal to? [2x + 3] Tell students that we can now replace the y in the second equation with this expression because it is equal to y. Ask a student to help you rewrite the second equation with the substitution for y. [4x + 4(2x + 3) = 24] Emphasize the importance of using parentheses around the quantity that y is equal to. Ask a student to walk you through the steps of simplifying (including using the distributive property) and solving for x. [x = 1] Finally, ask students what the last step is for solving systems. [Substitute again to find the value of the other variable, y = 5.]

Exercise | 6-5 Name Solve each system of equations using substitution. 1.

y = 2x + 1

x=y–4

y = 5x

x = 3y

STRUGGLING LEARNERS

y = 3x + 6

3x + y = 16

2x + y = 10

x + y = 18

2x – y = 10

2x + y = -4

(3, 7)

(2, 6)

(3, 15)

(6, 2)

(-2, 0)

Provide students a step-by-step checklist to help them through the substitution process: y Isolate y Substitute y Solve y Substitute y Solve

First isolate the x variable, then solve each system of equations using substitution. 6.

x – y=2 3x – 2y = 10

3x – 3y = -6 2x + 4y = 16

(1 3 , 3 3 )

(6, 4)

4x – 2y = 12

8x + 16y = 24

-x + 5y = 30 6x + 10y = -20

10.

(8, 0)

(-10, 4)

(3, 0)

1 3 2 x– 4 y=4 4 -2x + 5 y = -16

EARLY FINISHERS

First isolate the y variable, then solve each system of equations using substitution. 11.

3x + y = 10 2x – y = 0

12.

(2, 4)

5x – 2y = 16 -3x + y = -9

13.

6x – y = 7

14.

2x + 3y = -1

7x – 2y = -9

15.

3 4 x + 2y = 7

x – 3y = -2

Have students revisit problems 1-15 in the Exercise section, pick four problems, and solve them again, this time isolating the other variable. Finally, have them compare how the choice of variable changes the steps and the difficulty.

(4, 2)

(-5, -13)

(1, -1)

(2, -3)

18x – 6y = -12

Use substitution to find the solution to each system of equations. Answer the questions.

y=x+3 200y + 100x = 2,100

17. A furniture store sold tables and chairs last month. Each table (x) sells for $120, and each chair (y) sells for $45. They sold a total of 60 items and made $4,350. How many tables and chairs did they sell? 120x + 45y = 4,350

(5, 8); Children’s tickets are $5 each.

x + y = 60

(22, 38); The store sold 22 chairs and 38 tables.

CH AL L ENGE 18. A bookstore sells fiction and nonfiction books. On Monday, they sold a total of 100 books. Fiction books cost $12 each, and nonfiction books cost $15 each. The total sales for Monday were $1,290. A. Write a system of linear equations representing the number of fiction and nonfiction books sold. B. Solve the system. How many fiction books did they sell? How many nonfiction?

Lighthouse Math

Level H

12x + 15y = 1,290; x + y = 100 (70, 30); 70 fiction, 30 nonfiction

Chapter 6

Exercise 5

16. An adult ticket to a museum (y) costs $3 more than a children’s ticket (x). When 200 adult tickets and 100 children’s tickets are sold, the total amount is $2,100. What is the cost of a children’s ticket?

CHALLENGE AND EXPLORE In problem 18, students will need to write two equations for the scenarios described. Guide students to define the x and y variables and create one equation that addresses cost and sales and another equation that addresses the number of books sold. Students will then need to solve the system they created.

115

Have students fill in the blanks in the Apply section to solve the system. Then, review each part with the class to ensure understanding. Have students complete problems 1-5 in the Exercise section independently, then review the answers as a class. Next, complete problem 6 together as a class, and then have students complete problems 7-15 independently. Review both the answers and strategies used. Read through problems 16 and 17 together, modeling how to write equations for the systems. Have students solve the systems independently and then review the answers.

ACTIVITY Substitution Solution: The purpose of this activity is to have students practice solving a system using substitution. Give each student a notecard with an equation that does not have any isolated variables. Vary the difficulty so that you can differentiate by student readiness. Students will need to isolate a variable and then find a partner to create a system with. They will each separately substitute what is on their card into the equation that is on their partner’s card. Then, they will solve the system and compare their answers with their partner’s. Have partners discuss how they began with different substitutions but ended with the same answers.

COMMON ERRORS Students may use the distributive property incorrectly when substituting an expression into an equation. Students may isolate the variable incorrectly.

ASSESS Ask students to list the steps for solving the system {4y + 10 = 2x; y + 3x = 29}. [isolate, substitute, solve, substitute, solve] Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 6-6 6-6 | Solving Systems of Equations by Elimination

PREREQUISITE SKILLS

y Students will be able to solve systems of linear equations using the elimination method.

DAI LY REVI EW

Objective and Learning Goals

Use the distributive property to simplify. 1.

SPIRAL REVIEW

15x − 40

5(3x – 8)

-9(7x + 3)

-63x – 27

-7(8x – 2y + 4) -56x + 14y – 28

Solve each system of equations using substitution. 1.

y=x+3

5x – y = 12

x = -2y – 2 (2, -2)

3x + 2y = 6 (0, 3)

x + 3y = -11

-3x – 3y = 9 (1, -4)

-9x – 3y = 3 4x + 2y = 2 (-2, 5)

Vocabulary y Elimination - the process of removing a variable using addition or subtraction

L E A R N A ND C O NNE C T

1) See if a variable can be canceled out. If not, multiply each equation so that the coefficients of one variable are the same.

PRE-LESSON WARM-UP

-9y + ___ = 0 Ask students what idea they used to help them solve. [zero pairs, opposites] Tell students that we will be using this idea to help us solve systems of equations differently today.

Write the following number sentences on the board and ask students to fill in the blanks.

5x + ___ = 0

3x + 6y = 6

2(3x + 6y = 6)

6x + 12y = 12

2x + 5y = 6

3(2x + 5y = 6)

6x + 15y = 18

We’ve chosen to cancel the x. A common multiple of 3x and 2x is 6x. We multiply the first equation by 2 and the second equation by 3.

6x and 6x can cancel each other out

-4y and 4y cancel each other out Now you can solve for x

2) Add or subtract like terms. Add if the signs are different. Subtract if they are the same.

3) Substitute to find the other value. Write the solution as an ordered pair.

6x + 12y = 12 − 6x + 15y = 18 -3y = -6 y=2

3x + 6(2) = 6 3x + 12 = 6 3x = -6 x = -2

You can substitute 2 for y in either equation.

Solution: (2, -2)

A P P LY Fill in the blanks to solve the system of equations. 1.

-4x + y = 8 2x – y = 6

A. We know that y and -y can cancel if you add the equations -4x + y = 8 + 2x – y = 6 -2x = 14

B. Now we can solve for x . -2x = 14 x = -7 C. Now we know that x equals -7 .

D. We can substitute -7 for x in either equation to find y. -4( -7 ) + y = 8 28 + y = 8 y = -20

E. Now we know that y equals -20 . Solution: ( -7 , -20 )

Vocabulary Elimination - the process of removing a variable using addition or subtraction

116

Guiding Questions: 1. What number do you add to get zero? [The number's opposite, the number with the same absolute value but different sign] 2. What do we call two numbers that add to zero? [A zero pair]

2x – 4y = 10 + 4x + 4y = 8 6x = 18

You can use elimination to solve systems of equations. Add or subtract the two equations so that one variable gets canceled out. You can then solve for the variable that is left.

y Graph paper

-15 + ___ = 0

What happens when you add them?

4x + 4y = 8

Materials

3 + ___ = 0

2x – 4y = 10

Look at the two equations:

Level H

Chapter 6

Lesson 6

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Write the system from the Learn and Connect section on the board. [2x − 4y = 10; 4x + 4y = 8] Set it up vertically so that the like terms are lined up, and ask students how to add these two equations together. [Add the like terms.] Have a student walk you through the process and discuss the resulting equation. [6x = 18. Since -4y and 4y are opposites, they cancel each other out, and now there’s only one variable left.] Tell students that now, we can solve for x. [x =3] Ask students how we can use this information to finish solving the system. [We can plug in 3 in the place of x in one of the equations to get y = 1.] Tell students that this process of solving systems is called elimination because one of the variables disappears when we add the two equations together. Tell them that we can add or subtract the equations, but in order to use this method, one pair of variables needs to be opposites so that they will cancel out. If we don’t have opposites, we can multiply one or both equations to make it so that the coefficients (the number in front of a variable) are opposites. Write the second example from the Learn and Connect section on the board. [3x + 6y = 6; 2x + 5y = 6] Tell students that the process is a bit like finding a common denominator in fractions. If we want to make the coefficients in front of the x the same, we can multiply the first equation by 2 and the second equation by 3 so that both will have a coefficient of 6. Remind students that we have to multiply the entire equation, so the resulting equations will be 6x + 12y = 12; 6x + 15y = 18. Ask students if adding these two equations together will make the x-variable disappear. [no] We need to subtract the two equations to eliminate the x-variable. Have students walk you through subtracting the like terms, solving for y, and using substitution to find x. Finally, remind students to write the solution as an ordered pair.

Exercise | 6-6 Name For system of equations, determine what number each equation needs to be multiplied by, so the variables can cancel out. 3x + 4y = 3 6x – y = -3

Multiply by 2 to cancel x

-4x – 3y = -7

x – y =4

Multiply by 3 (or 4) to cancel x (or y)

8x – 6y = -20

STRUGGLING LEARNERS Have students complete problems just 1-6 in the Exercise section. If students feel confident with the process, have them continue with problems 7-10. Have students write on the top of the page, "Same sign - subtract, Different signs - add" to remind them of whether they need to add or subtract the equations. If the two variables with the same coefficient have the same sign, they will subtract. If they have different signs, they will add.

Solve each system of equations using elimination. 3.

6x – 5y = -8

x + 5y = -19

7x + 7y = 105

3x + 5y = 26

3x + 5y = -27

-16x + 7y = 30

6x + y = 60

(2, 4)

(-4, -3)

(-1, 2)

(9, 6)

3x – 2y = 4

2x + 5y = -11

-2x + 3y = -4

10.

14x + 3y = 85

5x + 4y = 14

3x + 2y = 0

5x – y = -3

10x – 6y = 20

(2, 1)

(2, -3)

(-1, -2)

(5, 5)

Write a system of linear equations to represent the word problem. Then use elimination to solve the problem. 11. A pet shelter has 73 dogs (x) and cats (y) to be adopted. Each dog eats 2 cups of food and each cat eats a 21 cup of food, for a total of 68 cups of food consumed daily. How many of each animal is up for adoption?

x + y = 200

(21, 52); There are 21 dogs and 52 cats up for adoption.

4x + 2y = 520

(60, 140); The booth sold 60 cups of coffee and 140 cups of tea.

CH AL L ENGE 13. Students from three different grade levels, 6th, 7th, and 8th, entered projects at the science fair. • The total number of projects was 90. • The number of 6th-grade projects was twice the number of 8th-grade projects. • The number of 7th-grade projects was 6 more than the number of 8th-grade projects.

Lighthouse Math

A. Assign a variable to the number of projects from each grade level. x = 8th; y = 7th grade; z = 6th grade

B. Write a system of equations to represent this situation. Think: How many equations will you need in order to solve this system? x + y + z = 90; z = 2x; y = 6 + 2x

C. Find out how many projects came from each grade level. 8th grade = 21; 7th grade = 27; 6th grade = 42

Level H

Chapter 6

Exercise 6

117

APPLY AND DEVELOP SKILLS (Practice) Go through the Apply section together as a class, asking for student input as you fill in the blanks to solve the system. Discuss which variables make sense to cancel and whether you need to add or subtract to do so. Have students fill in the blanks for problems 1 and 2 in the Exercise section. Review the answers and discuss as a class. Next, assign problems 3-10 for independent pracice. Review all the answers as a class, and go over the process for problems 4 and 8 and any other problems that students had questions on. Work together as a class to create equations for problems 11 and 12, then have them solve the systems on their own. Review the answers and strategies used as a class.

ACTIVITY Connect to Graphs: The purpose of this activity is for students to understand the standard form of a linear equation (ax + by = c) and how it can be graphed on a coordinate plane. Write the equation 3x + 2y = 12 on the board. Sketch a coordinate plane and tell students that we are going to graph this equation using x- and y-intercepts. Ask students what the x-value is for a y-intercept. [0] Tell students that to find the y-intercept, we can plug in 0 for x and solve. [y = 6] Have a student draw a dot on the coordinate plane for this point. [(0,6)] Ask students what the y-value is for an x-intercept. [0] Tell students that to find the x-intercept, we can plug in 0 for y and solve. [x = 4] Have a student draw a dot on the coordinate plane for this point. [(4,0)] Tell students that once we have two points, we can connect them with a straight edge and extend the line in both directions. Ask students to walk you through this process to graph 2y − x = 4 and to find the solution to the system. [(2,3)]

x + y = 73 2x + 21 y = 68

EARLY FINISHERS

12. A booth at the farmer’s market sells cups of coffee (x) for $4 and cups of tea (y) for $2. On Monday, they sold a total of 200 beverages and earned $520. How many cups of coffee and tea did the booth sell?

Have students go back to problems 3-6 in the Exercise section and solve again, this time eliminating the other variable. Have them check their answers with their original answers to ensure they were done correctly.

CHALLENGE AND EXPLORE In problem 13, students will need to figure out that when there are three variables in a system, they will need three equations to solve. They will need to assign a variable to each grade level and then use the information given to write these equations. Tell students that they can use both substitution and elimination in order to solve. For an extra challenge, ask students to figure out what a system with no solutions or infinite solutions would look like when the equations are set up in standard form as they are in this chapter.

COMMON ERRORS Students may add the equations instead of subtracting, or vise versa. Students may add or subtract without multiplying the equations to make the coefficients match.

ASSESS Check problems 6 and 7 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

Level H | 6-7 6-7 | Applications of Systems of Equations

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to solve realworld problems using systems of linear equations.

Determine if the ordered pair is a solution to the system.

SPIRAL REVIEW

(1, 14) 15x + 21y = 8 x = 14y

yes

x + y = 45

2. (24, 21)

4x + 2y = 44

x + y = 34

7x + 3y = 71 (5, 12)

2x + 3y = 89

(11, 23)

On Monday, the zoo sold 100 tickets for a total of $720. Adult tickets cost $10, and children's tickets cost $5. How many adult and children's tickets were sold? Step 1: Write a system of equations to represent the situation. A. Assign variables to each unknown value. x = # of adult tickets y = # of child tickets

x + y = 100 10x + 5y = 720

10x + 5y = 720

Substitution Method 10x + 5y = 720 10x + 5(-x + 100) = 720 10x – 5x + 500 = 720 5x + 500 = 720 5x = 220 x = 44

x + y = 100 → y = -x + 100

C. Write one equation to represent the cost.

B. Write one equation to represent the number of tickets. x + y = 100

Step 2: Solve using any method.

-4y + 2x = 6 3y − x = 10

Guiding Questions: 1. How can we use graphing to solve the system? [Plot the x- and y-intercepts and draw in the lines. Then, look for the point of intersection.] 2. How can we use substitution to solve the system? [Solve for x in the second equation, and then plug in what x is equal to in the first equation and solve for y.] 3. How can we use elimination to solve the system? [Multiply the second equation by 2 and then add the equations to eliminate x.]

(0.1, 1.6) 2x + 3y = 5 y = 6x + 1

L E A R N A ND C O NNE C T

Write the following system of equations on the board. Ask students to explain which method (graphing, substitution, or elimination) they would use to solve the system and why. Encourage different students to defend different methods. If there is one method no one chooses to defend, ask them why it seems less useful, and challenge a student to explain how it could still be used to solve the system.

Wrap up the discussion by emphasizing that some systems are set up so that one method is more efficient, but students can choose the method they are most comfortable using to solve the system.

yes

Solve each system using elimination. 1.

x–y=3

PRE-LESSON WARM-UP

(2, 8) 9x + y = 26 x + y = 10

x + y = 100 44 + y = 100 y = 56

(44, 56) They sold 44 adult tickets and 56 children's tickets.

A P P LY Choose two equations that can represent each situation. 1.

118

Two bags together weigh 45 pounds. The heavier bag weighs 3 pounds more than the lighter one. How much does each bag weigh?

Emma bought 7 red spools of thread and 3 blue spools of thread for $75. Sonia bought 4 red spools and 3 blue spools for $48. How much does each red and blue spool cost?

x + y = 45

y = x + 45

7x + 3y = 48

x+y=3

x = y + 45

4x + 3y = 48

x + y = 75

x – y = 45

y=x+3

4x + 3y = 75

x + y = 48

Level H

Chapter 6

Lesson 7

7x + 3y = 75

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Read aloud the problem from the Learn and Connect section about adult and children’s tickets at the zoo. Ask students how many variables there are in this scenario. [two] Ask students how many equations we can write with the given information. [two] Assign the variable x to represent the number of adult tickets and the variable y to represent the number of children’s tickets. Ask students to help you write the two equations. Explain that we can add x and y to get the total number number of tickets, and since we know the total number of tickets sold was 100, we can set this equal to the sum of x and y. [x + y = 100] We also know the cost of each ticket. Ask students how to find the cost of all the adult tickets [Multiply 10, the cost of each adult ticket, by x, the number of adult tickets.] and how to find the cost of all the children’s tickets. [Multiply 5, the cost of each children’s ticket, by y, the number of children’s tickets.] Tell students that we can set the sum of these two expressions equal to 720, the total amount of money brought in by ticket sales. [10x + 5y = 720] Ask students how they would solve this system. [Answers vary: graphing, substitution, elimination] Have students walk you through solving, using any method. The substitution method is shown in the Learn and Connect section. Remind students to give their final answer with context from the problem.

Exercise | 6-7 Name Use the system of equations to answer the questions. 1.

A farmer with cows and chickens has 20 animals. The animals have 54 legs in total. This can be represented by the following system of equations

STRUGGLING LEARNERS

x + y = 20 2x + 4y = 54

Have students use different colored highlighters to mark off important information in the problems. Also provide them with a checklist for solving: y Define the variables y Create two different equations y Choose a method to solve (graphing, substitution, elimination) y Solve for one variable, then use substitution to solve for the second variable y Write the answer with context

A. Why is there a 2x in the second equation and only x in the first equation? The second equation has a 2x because we are counting legs, while the first equation has x because we are counting the number of chickens.

B. What do x and y represent? How do you know? x is the number of chickens, and y is the number of cows. I know because chickens have 2 legs, and cows have 4 legs, and x is multiplied by 2, and y is multiplied by 4.

C. How many of each type of animal does the farmer have?

13 chickens and 7 cows

Write a system of equations for each problem. Solve using any method. 3.

A museum sold 300 tickets in one day and earned $4,000. Adult tickets were $20, and children’s tickets were $10. How many of each type of ticket were sold?

Jamal took two tests. The sum of the two scores was 175. The second test score was 15 points higher than the first. What were Jamal’s two test scores?

x + y = 300 20x + 10y = 4,000

x + y = 175 y = x + 15

The museum sold 100 adult tickets and 200 children’s tickets.

Jamal scored 80 on the first test and 95 on the second test.

Jordan exercised for a total of 2 hours. She walked at 4 miles per hour and ran at 8 miles per hour, covering 12 miles in total. How much time did she spend walking and running?

EARLY FINISHERS

Alina and her sister Robin have a combined age of 42. Alina is 6 years younger than twice Robin’s age. How old is each sister?

x+y=2 4x + 8y = 12

x + y = 42 y = 2x – 6

She spent 1 hour walking and 1 hour running.

Alina is 16 years old, and Robin is 26 years old.

CH AL L ENGE 6.

Adam bought a total of 15 school supplies, which included notebooks, pens, and folders. The number of pens is twice the number of notebooks. The number of folders is 3 more than the number of notebooks. A. Write a system of equations where x represents the number of notebooks, y represents the number of pens, and z represents the number of folders. x + y + z = 15 y = 2x z=x+3

B. How many of each type of school supplies did Adam buy? He bought 3 notebooks, 6 pens, and 6 folders.

Lighthouse Math

Level H

Chapter 6

Exercise 7

119

Have students create their own word problem for another student to solve. Remind them that they have to give enough information to create two different equations.

CHALLENGE AND EXPLORE For problem 6, students will need to write 3 different equations using the given information, and then solve. For an extra challenge, ask students what a system with no solutions or infinite solutions might look like in a word problem.

Have students work with a partner on problems 1 and 2 in the Apply section. Then, go over the answers as a class, addressing any questions or misconceptions that students have. Next, have students continue to work with a partner on problem 1 in the Exercise section. Review the answers, focusing on students’ reasoning. Next, set up equations for problem 2 as a class. Then, have students solve the system and complete problems 3-5 individually. Have them check their answers with a partner, and then review the answers and strategies as a class.

ACTIVITY Fill the Formula: The purpose of this activity is to have students experiment with the types of information that are provided in word problems for systems of equations. Tell students that many word problems follow a formula where the given information provided will be in two categories: the number of a unit and the rate for that unit. Have students work with a partner to create a word problem that follows this formula. Have them define their variables, create equations, and solve their system. Then, have them switch questions with another group, keeping their answers to themselves until the other group solves and is ready to check.

COMMON ERRORS Students may mix up the x and y variables when writing their second equation.

ASSESS Check problems 3 and 4 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 6-8 6-8 | Review

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will review skills learned in Chapter 6.

A gardener planted 45 pots with flowers, some with roses and some with tulips. Each rose pot has 10 flowers, and each tulip pot has 5 flowers. There are 275 flowers in total. How many pots of each flower type are there?

PRE-LESSON WARM-UP

Write a system of equations: x + y = 45

Ask students to tell you what a system with one solution looks like on a graph and what must be true about each equation in the system. [Two lines lines that intersect; they must have different slopes.]

10x + 5y = 275 Solve by Graphing

Solve by Substitution

Solve by Elimination

x + y = 45 → y = -x + 45

x + y = 45 → 5(x + y = 45)

10x + 5y = 275 → y = -2x + 55

10x + 5y = 275

Guiding Questions: 1. What can the slopes of each equation tell us about a system? [Whether the lines will cross] 2. Sometimes, equations are not given slopeintercept form. How can we analyze them for the number of solutions? [Solve for y so that they are in slope-intercept form, or compare the equations to see if they are the same equation when simplified or if they are two contradicting equations.]

Lighthouse MATH Level H | Teacher's Guide

10x + 5(-x + 45) = 275 10x – 5x + 225 = 275 5x + 225 = 275 5x = 50 x = 10 10 + y = 45 y = 35 (10, 35)

40 30 20 10 0

5x + 5y = 225 – 10x + 5y = 275 -5x = -50 x = 10 10 + y = 45 y = 35 (10, 35)

# of Rose Pots

10 rose pots 35 tulip pots

Ask students to tell you what a system with infinite solutions would look like on a graph and what must be true about each equation in the system. [The lines will overlap; they are the same line. The equations will have the same slope and the same y-intercept when simplified.]

# of Tulip Pots

Ask students to tell you what a system with no solutions looks like on a graph and what must be true about each equation in the system. [Two parallel lines; their slopes will be the same, their y-intercepts will be different.]

10 rose pots 35 tulip pots

A P P LY Determine how many solutions exist for each system of equations. 5

1. -5

2. 5

-5

no solution

infinite solutions

120

-5

one solution

no solution

infinite solutions

Chapter 6

4. 5

-5

one solution

Level H

5 -5

one solution

no solution

infinite solutions

Lesson 8

one solution

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Divide students into three groups. Read aloud the question in the Learn and Connect section about the different types of flowers that were potted. First, as a class, define the x-variable as the number of roses and the y-variable as the number of tulips. Then, have students work together in their groups to create two equations from the given information. Be sure that all groups have the same equations. [x + y = 45; 10x + 5y = 275] Next, assign each group a task: solve by graphing, solve by substitution, or solve by elimination. When students finish, have each group present their process to the class, explaining all the steps that are needed to be done in order to solve this way. Be sure the final answer is given with context in each case.

Exercise | 6-8 Name Determine if the given ordered pair is a solution to the system. 1.

Is (4, 3) a solution to this system of equations? 2x + y = 11

yes

STRUGGLING LEARNERS

Is (-2, 6) a solution to this system of equations? x + y = 10

x – 2y = -2

Remind students that they can choose which method to solve with based on what they are most comfortable doing. Tell them that they can always play around with equations to write them in a different form by using inverse operations as we learned in Lesson 6-4. If struggling students prefer to be told which method to use because it's too overwhelming to for them to choose, write an E or and S above each problem to eliminate the step of having to make that decision.

2x – y = 1

Solve each system of equations by graphing. 3.

y = 3x

y = 21 x – 3

y = - 32 x – 4

-5

y=0 y = -5x – 5

no solution

y = - 31 x – 1

(1, 3) -5

y = 21 x + 1

y = -x + 4

(-3, -2)

-5

(-1, 0)

-5

5 -5

Solve each system of equations using substitution or elimination. 7.

y = 3x + 1

y=x+4

y = 3x

10.

x – 5y = 0

2x + y = 11

-2x + y = 0

x + y = 12

x + y = 18

(2, 7)

(4, 8)

(3, 9)

(15, 3)

11.

-x + y = 2 3x + 2y = 24

EARLY FINISHERS

(4, 6)

x + y = 30 2x + 3y = 72

13. An amusement park sold a total of 160 tickets in one day. Children's tickets cost $25, and adult tickets cost $40. The total amount collected was $5,200. How many children's and adult tickets were sold?

They sold 18 game tickets and 12 food tickets.

x + y = 160 25x + 40y = 5,200

They sold 80 of each type of ticket.

CH AL L ENGE 14. A middle school is selling four types of shirts. T-shirts cost $10, long-sleeve shirts cost $15, hoodies cost $20, and zip-up jackets cost $30. • A total of 120 items were sold. • The number of T-shirts sold was equal to the number of hoodies. • The number of long-sleeve shirts sold was three times the number of zip-up jackets. • The school made $1,950 from all sales. How many of each type of shirt were sold? 40 T-shirts, 40 hoodies, 30 long-sleeve shirts, and 10 zip-up jackets were sold.

Lighthouse Math

Level H

Chapter 6

Exercise 8

121

Write and solve a system of equations for each word problem. 12. At a carnival, the children sold 30 tickets. Some were $2 game tickets, and some were $3 food tickets. They earned $72 in total. How many game tickets and how many food tickets did they sell?

Have students return to problems 7-11 in the Exercise section and solve them using a different method. Then, have them change one term in each system for problems 3 and 4 so that there will be a different number of solutions.

CHALLENGE AND EXPLORE For problem 14, students will need to solve the riddle using the given information and their knowledge of solving systems. They do not need to create equations before solving, but can go step by step through the clues using the strategies they have learned.

Have students complete problems 1-4 in the Apply section, then review the answers and reasoning for each one as a class. Next, have students complete problems 1-13 in the Exercise section independently, checking with a partner as they go. Review the answers and strategies used for the even-numbered problems and answer any other questions that come up about other problems along the way.

ACTIVITY Review For You: Challenge students to create a system of equations for a partner to solve that will give round numbers for the answers. Tell students that they can use graphs to help them and that trial and error is a useful strategy here.

COMMON ERRORS Students may forget to substitute the first variable back into an equation to solve for the second variable.

ASSESS Check the odd-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Chapter 7

122

In Chapter 7, we will explore

Angles and Triangles Angles and triangles are found all around us, and they follow rules that help us design and build structures safely. • We will review angle relationships and learn about angle relationships on parallel lines crossed by a transversal. • We will learn new rules about the angles in and around triangles. • We will learn and apply the Pythagorean theorem for right triangles. • We will learn and apply the distance formula.

123

Level H | 7-0 Chapter 7 | Skill Checklist

Skill 1: Points, Lines, and Angles Point - an exact location

Match each figure to the correct term.

Line - a straight path that continues indefinitely

Point

Line Segment - part of a line with two endpoints

Line

Ray - part of a line with one endpoint and another side that extends indefinitely

Line Segment

Ray

Vertex

Angle

Angle - two rays that meet at a point Vertex - the point in an angle where the two rays meet

Objective and Learning Goals Students will review skills needed for Chapter 7:

Skill 2: Types of Angles Circle the type of angle shown. a right angle is 90°

a straight angle is 180°

right

acute

straight

obtuse

right

acute

straight

obtuse

out 4 correct

I can classify angles.

Level H

Skill Checklist

Chapter 7

Skill 1: Points, Lines, and Angles y Review with students the vocabulary terms in the gray box. y Have students complete problems 1-6 and review the answers and strategies as a class.

Lighthouse MATH Level H | Teacher's Guide

an obtuse angle is more than 90° but less than 180°

124

an acute angle is less than 90°

y Points, lines, and angles y Classifying triangles y Types of angles y Naming angles y Squares and square roots

I can identify points, lines, and angles.

out 6 correct

Lighthouse Math

Skill 2: Types of Angles y Remind students that angles can be right, acute, straight, or obtuse. y Right angles measure 90°, acute angles measure less than 90°, straight angles measure 180°, and obtuse angles measure less than 180° but more than 90°. y Have students complete problems 1-4 and review the answers and strategies as a class.

Name

Skill 3: Naming Angles A

Circle the correct name for the angle shown. 1.

I B

B is the vertex of this angle. It can be named:

ABC

CBA

DFE

JHK

DEF

YXZ

GHK

FDE

ZYX

JHI

KHG

I can name angles correctly.

out 3 correct

Skill 4: Classifying Shapes Label each triangle as acute, obtuse, or right. Then, add another label of equilateral, isosceles, or scalene.

Equilateral Triangle All sides equal

Acute Triangle All angles acute (<90°)

obtuse isosceles

4. Obtuse Triangle One obtuse angle (>90°)

Right Triangle One right angle (90°)

acute scalene

Right Scalene

right isosceles

Acute Isosceles

Acute Equilateral

I can name and classify shapes.

out 6 correct

Skill 5: Squares and Square Roots a2 = 100

Evaluate.

a × a = 100 100 = a a = 10

82 =

0.42 = 0.16

16 =

out 9 correct

Lighthouse Math

602 = 3,600

6,400 =

3 80

2.52 = 6.25

400 =

8,100 =

I can evaluate squares and square roots.

Level H

Chapter 7

Skill Checklist

125

Skill 3: Naming Angles

Skill 4: Classifying Triangles

y Remind students that we name angles based on the letters at each of their points, and we always put the vertex in the center. y Use the gray box to review the three ways an angle can be named with its letters. y Have students complete problems 1-3 and review the answers and strategies as a class.

y Review the different types of triangles in the gray box. y Emphasize that we classify triangles by their side lengths as a well as their angle measures. y The words scalene, isosceles, and equilateral refer to the side lengths, while the words obtuse, acute, and right refer to the angle size. y Have students complete problems 1-6 and review the answers and strategies as a class.

Skill 5: Squares and Square Roots y Remind students that squaring means to multiple a number by itself, while taking the square root means figuring out which number was multiplied by itself to get the number under the root. y Review the perfect squares and their square roots with students. y Have students complete problems 1-8 and review the answers and strategies as a class. y Take a minute to discuss problems 6 and 8 and how to take the square root of a number that is a multiple of two perfect squares. Lighthouse MATH Level H | Teacher's Guide

Isosceles Triangle Two sides equal

Scalene Triangle No sides equal

Level H | 7-1 7-1 | Understanding Angle Relationships

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to identify and use properties of supplementary, complementary, vertical, and adjacent angles to solve problems.

Write the name of the highlighted angle using three letters with the letter for the vertex in the middle.

SPIRAL REVIEW

[Acute]

[Straight]

GIJ

3x − 16 = 14

12 + 5x = 47

144 = 4x + 20

32 = 8x − 6 x = 4.75

Complementary Angles

Supplementary Angles

Angles that add up to 90°

Angles that add up to 180°

Vertical Angles

b = 90°

d = 180°

m x°

h i

Angles that share a vertex and side

e is adjacent to

AEB is supplementary to AEC. When added together, they equal 180°.

Adjacent Angles

Formed by intersecting lines, these angles are directly across from one another and equal

AEB = x°

m AEC = (x − 10)°

(x - 10)° E

m ABC “the measure of angle ABC”

x + (x − 10) = 180 2x − 10 = 180 2x = 190 x = 95

Therefore, m AEB is 95°, and m AEC is 85°.

A P P LY Write C if the angle pair is complementary. Write S if they are supplementary. 1.

60° and 30°

85° and 95°

158° and 22°

41° and 49°

Vocabulary Angle- the space between two lines that cross; measured in degrees Vertex - the point at which two or more lines meet Complementary angles - angles whose measures add up to 90° or form a right angle Supplementary angles - angles whose measures add up to 180° or form a straight line Adjacent angles - angles that have a common side and share a vertex Vertical angles - angles formed by two intersecting lines; they are opposite each other and have the same measure

Level H

Chapter 7

Lesson 1

Lighthouse Math

[Obtuse]

Ask students where they see angles that look like these around the room. Tell students that two angles can share a side. Ask students if they can find any angles that share a side. (If there are none, show how two angles can be combined so they share a side or draw a picture on the board.) Use the visual to help students to see that two acute angles can be combined to form a right angle and that an acute and obtuse angle or two right angles can be combined to form a straight angle. Guiding Questions: 1. Which type of angle can join with another one of its type to form a right angle? [an acute angle] 2. Which types of angles combine to form a straight angle? [obtuse plus acute or two right angles] 3. Where can we find angles in real life? [They form the structures of things such as the walls, desks, and chairs and help them be structurally sound.] Lighthouse MATH Level H | Teacher's Guide

Angles are formed by lines that meet at a vertex. Angle relationships can be used to find a missing value.

126

[Right]

L E A R N A ND C O NNE C T

Draw three angles on the board as shown. Ask students to classify them by type.

WXZ

Solve for x.

Vocabulary

PRE-LESSON WARM-UP

ABC

y Angle - the space between two lines that cross; measured in degrees y Vertex - the point at which two or more lines meet y Complementary angles - angles whose measures add up to 90° or form a right angle y Supplementary angles - angles whose measures add up to 180° or form a straight line y Adjacent angles - angles that have a common side and share a vertex y Vertical angles - angles formed by two intersecting lines; they are opposite each other and have the same measure

INTRODUCE THE LESSON (Learn and Connect) Remind students that angles are formed when two lines or rays meet at a vertex point. Tell students that today we are going to use what we know about how angles are related to each other to find missing pieces. Go through the chart in the Learn and Connect section outlining the angle relationships and use the examples shown to describe how to recognize each one. Ask students how some pairs of angles can fit more than one description. Discuss what complementary angles would look like if they were adjacent and what they would look like if they were not adjacent. Next, ask students to look at the figure below the chart and tell you what the relationship is between AEB and AEC. [They are adjacent; they form a straight line; they are supplementary; they add up to 180°] Go through the process of finding the measures of the missing angles as shown. Tell students that in this lesson, they will be writing equations using all the rules we covered. Explain that not all equations will be constructed the same. Ask: How would we set up an equation with vertical angles? [Set the value of the two angles equal to each other.] How would we set up an equation with complementary angles? [Add the value of the two angles together and set them equal to 90.]

Exercise | 7-1 Name Write whether the angles are complementary or supplementary. Then find the missing angle. 1.

STRUGGLING LEARNERS

x°

Have students use highlighters to color-code the angles in each diagram. Use the same color for vertical angles. If the angles are adjacent, have students use one color for each of the inside angles and another color for the combined angle. Then, have students write the measure of the combined angle.

80° x°

52°

x°

68°

supplementary

x = 128°

x°

complementary

x = 22°

x = 10°

124°

supplementary

x = 56°

Write whether the angles are adjacent or vertical. Then find the missing angle. 5.

x°

8. x°

121°

40° x°

34°

vertical

adjacent

x = 40°

76° x° 25°

75°

vertical

x = 87°

EARLY FINISHERS

adjacent

x = 75°

x = 51°

Have students return to problems 7 and 11 to find the measures of the other unlabeled angles in the diagram.

Write an equation. Then solve to find the value of x. 10.

11. x°

(x + 22)°

12.

112°

(3x - 6)° 120°

(3x)° x°

3x − 6 = 120

4x = 112

x°

2x + 22 = 180

x = 79°

2x = 90

x = 45°

x = 42°

CHALLENGE AND EXPLORE © Lighthouse Curriculum. Copying strictly prohibited.

x = 28°

CH AL L ENGE 13. In the diagram, line AB is a straight line. Write an equation and solve to find the value of x. Then find the measures of the listed angles below.

4x + 30 = 180 x = 37.5

Lighthouse Math

2x°

m ACD = 67.5°

m ACE = 142.5° m DCE = 75°

Level H

m ECB = 37.5°

Chapter 7

(x + 30)° A

Exercise 1

In problem 13, students will need to set up an equation using three different expressions that add to 180. Then, they will need to solve and use the value of x to find different angles in the diagram. Some of the angles will be a combination of two parts.

x° C

127

Have students work independently on problems 1-4 in the Apply section, then review the answers and strategies used as a class. Next, work together as a class on problems 1, 5, and 9 in the Exercise section. Once students show a good understanding of each type, have them complete problems 2-4, 6-8, and 10-12, and review the answers when complete.

ACTIVITY Angle Mash Up: The purpose of this activity is for students to practice creating equations based on angle relationships and use them to find missing pieces. Have each student write an expression with no more than two terms (such as 4x or 3x + 5). Then, assign partners and give them a category: either complementary angles, supplementary angles, vertical angles, or adjacent angles. (If they are assigned adjacent angles, give them a total amount that their angles add up to.) The pair will need to work together to draw a picture of their angles, label it with their expressions, and create an equation that befits their angle relationship using these two expressions. Have students solve for x and then find the measure of each angle. Students can share their work when complete.

COMMON ERRORS Students may use the wrong angle relationship when setting up equations. Students may use the wrong inverse operation when solving equations.

ASSESS Exit Ticket: Angles A and B are complementary. Angle A has a measure of 3x − 3, and angle B has a measure of 2x + 8. Write an equation and solve for x to find the measure of each angle. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 7-2 7-2 | Parallel Lines and Transversals

PREREQUISITE SKILLS

Objective and Learning Goals

2. 89°

DAI LY REVI EW

y Students will be able to identify and solve for missing angles made when parallel lines are intersected by a transversal.

Find the missing angle.

SPIRAL REVIEW

a = 91°

3. 165°

b = 15°

(15x - 11)° 41°

x = 10

42° c

c = 42°

105° x° 81°

x = 24

122°

d = 58°

Find the value of x. 1.

(3x + 1)° 115°

x = 38

4. x°

x = 45

x°

Vocabulary

PRE-LESSON WARM-UP

Draw the following diagram on the board and ask students to write down the names of all the angles they see. Then, ask them what they notice about the size and the relationships of the angles. E

A C

B D

H F [ AGE, EGB, AGB, AGH, CHG, CHD, HGB, GHD, CHF, CHD, FHD] [There are pairs of angles that are supplementary, like CHF and FHD, some angles are acute, like EGB, and some are obtuse, like CHG. There are vertical angles that are equal, like FHD and CHG, and some angles are adjacent, like AGE and EGB.] Guiding Questions: 1. How do we name angles? [with three letters and the vertex in the center] 2. What angle pairs do we see here? [supplementary, adjacent, vertical] Lighthouse MATH Level H | Teacher's Guide

L E A R N A ND C O NNE C T E

Parallel lines are lines that will never touch. A transversal is a line that cuts through them, forming angles. The angle pairs that are created by a transversal have special names: Name

y Parallel lines - lines that will never touch y Transversal - a line that crosses over two other lines y Alternate interior angles - equal angles on opposite sides of a transversal, but inside the parallel lines y Alternate exterior angles - equal angles on opposite sides of a transversal, but outside the parallel lines y Corresponding angles - equal angles in the same position on a transversal relative to different parallel lines y Same side interior angles - adding up to 180°, these angles are on the same side of the transversal, inside the parallel lines y Same side exterior angles - adding up to 180°, these angles are on the same side of the transversal, outside the parallel lines

Sketch

Example from diagram

Relationship

> symbol on lines shows that they are parallel

What is another example from the diagram?

CHG and

BGH

DHF

EGB and

CHF

Alternate Interior

equal

AGH and

GHD

Alternate Exterior

equal

AGE and

Corresponding Angles

equal

GHD and

EGB

CHG and AGE FHD and HGB

Same Side Interior

add to 180°

BGH and

GHD

AGH and

GHC

Same Side Exterior

add to 180°

EGB and

DHF

EGA and

CHF

A P P LY Name the angle relationship. 1.

2. Alternate exterior

3. Same side exterior

4. Alternate interior

Corresponding

Vocabulary Parallel lines - lines that will never touch Transversal - a line that crosses over two other lines Alternate interior angles - equal angles on opposite sides of a transversal but inside the parallel lines Alternate exterior angles - equal angles on opposite sides of a transversal but outside the parallel lines Corresponding angles - equal angles in the same position on a transversal relative to different parallel lines Same side interior angles - adding up to 180°, these angles are on the same side of the transversal, inside the parallel lines Same side exterior angles - adding up to 180°, these angles are on the same side of the transversal, outside the parallel lines

128

Level H

Chapter 7

Lesson 2

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Refer to the diagram from the Pre-lesson Warm-Up. Add > to lines AB and CD and explain that this symbol shows that the two lines are parallel. Ask students to show parallel lines using their arms. Ask them to explain what parallel means. [The two lines will never touch.] Ask students to name the line that crosses through lines AB and CD. [line EF] Tell them that this line is called a transversal, a line that crosses through a set of lines and forms angles. Remind students that we named these angles in the warm-up activity. Next, have students open their book and refer to the chart in the Learn and Connect section. Go through each angle, pairing one at a time, and explaining their relationship. Have the students fill in the last column of each line, naming another example of each pairing, and have them mark and color-code them on the diagram at the top of the page. Tell students that in this lesson, we will practice recognizing these relationships, and we will use them to set up equations and solve for unknowns.

Exercise | 7-2 Name Refer to the diagram. Name all the angle pairs from each category.

2 4

1. Alternate Interior

2. Alternate Exterior

3 and 4 and

2 and 1 and

3 5

6 8

5 6

3. Corresponding 2 and 6 1 and 5 4 and 8, 3 and 7

8 7

STRUGGLING LEARNERS

4. Same Side 5. Same Side Interior Exterior 3 and 4 and

6 5

Allow students to use the chart in the Learn and Connect section as a reference for the angle relationships while they work.

2 and 7 1 and 8

Name the angle relationship, then find the value of x. 6.

x°

35°

x° x°

20°

Alternate interior

Corresponding

Same side interior

150°

20°

75°

EARLY FINISHERS

105°

x°

150°

Alternate exterior

Have students return to problems 6-13 and label each angle with its measure.

35°

CHALLENGE AND EXPLORE

Use what you know about angle relationships to write an equation. Then solve. 10.

(3x)°

11.

12.

(4x + 1)°

(8x + 5)° (x + 4)°

25° 102°

(10x + 7)°

13.

(5 - 2x)°

3x = 102

4x + 1 = 25

8x + 5 + x + 4 = 180

10x + 7 + 5 − 2x = 180

x = 34

x=6

9x + 9 = 180; x = 19

8x + 12 = 180; x = 21

y°

(6x + 5)°

(2x + 3)°

A. Jonathan wrote the equation 6x + 5 = 2x + 3 to solve for x in the diagram. Do you agree with him? Why or why not? No, the two angles are same side interior angles, so they add up to 180°. He should have written 6x + 5 + 2x + 3 = 180.

B. Solve for y. Explain how you found your answer. 8x + 8 = 180 (same side interior angles), so x = 21.5. Therefore, the angle labeled 2x + 3 is equal to 46°. Y = 46 because these angles are alternate interior angles.

CH AL L ENGE 15. Explain why same side interior angles add up to 180° using what you know about angle relationships. Use the diagram shown as a reference. Angle 6 and angle 7 are supplementary angles; therefore, they add up to 180°. Angles 7 and 3 are corresponding angles, therefore, they are equal. Since angle 3 is equal to angle 7, and angle 7 and angle 6 add to 180°, angle 3 and angle 6 (same side interior angles) must also add to 180°.

Lighthouse Math

Level H

Chapter 7

Exercise 2

2 4

3 5

14. Refer to the diagram below to answer the questions.

For problem 15, students will need to reason why same side interior angles add up to 180° using the angle relationships that they know. Tell students to reference specific angles in their explanation and suggest they look for relationships such as vertical angles and supplementary angles to write their proof.

6 8

129

Have students complete problems 1-4 in the Apply section independently and then review the answers to ensure understanding. Next, complete problems 6 and 10 in the Exercise section together with students to model finding missing angle measures. Have students complete problems 1-5, 7-9, and 11-14 independently, checking with a partner as they go. As a class, review problems 7, 12, and 14 and address any other questions that came up along the way.

ACTIVITY Angle Chain: The purpose of this activity is for students to practice finding angle relationships. Divide students into groups. Each group gets a piece of paper with parallel lines crossed by a transversal drawn across the page. Each group also gets a set of cards with the names of the angle relationships. The first student to play will put a finger on an angle and then draw a card. The next student will need to put their finger on an angle that satisfies the relationship shown on the card. If done correctly, the first student only then removes their finger and the second student will draw a card next. The third student will put their finger on an angle that satisfies the relationship shown. The second person only will then remove their finger and the third student will draw a card for the next student and so on.

COMMON ERRORS Students may think that all angle pairings are equal. Students may think corresponding angles must be on opposite sides of the transversal.

ASSESS Exit Ticket: An angle with a measure of 4x and an angle with a measure of 2x + 58 are alternate interior angles. What is the measure of each angle? [both angles are 116°] Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 7-3 7-3 | Angle Relationships in Triangles

PREREQUISITE SKILLS

y Students will be able to solve for missing angles in triangles using equations, relate interior and exterior angles, and apply these concepts to real-life problems.

DAI LY REVI EW

Objective and Learning Goals

Write the letter that matches the type of triangle that is shown. 1.

SPIRAL REVIEW

A. Equilateral triangle B. Isosceles triangle C. Right triangle D. Scalene triangle

Write the name of the angle relationship. 1.

corresponding

alternate interior

same side interior

same side exterior

corresponding

Vocabulary y Exterior angles - angles formed when one side of a shape is extended outside the shape; in a triangle, an exterior angle will be supplementary to the adjacent angle and equal to the sum of the two remote interior angles y Remote interior angles - angles in a triangle that do not share a vertex with the given exterior angle

L E A R N A ND C O NNE C T

m a + m b + m c = 180°

Rule: Supplementary angles add up to 180°.

m c + m d = 180°

If A is equal to B, and B is equal to C, what other conclusion can we come to? [C must be equal to A.]

New rule m a+m b=m d

Tell students that today, we will use this kind of logic to find a new rule about triangles. Guiding Questions: 1. What rule do we know about angles in triangles? [All the angles in a triangle add up to 180°.] 2. What rule do we know about angles that form straight lines? [They add up to 180°.]

Lighthouse MATH Level H | Teacher's Guide

The measure of an exterior angle in a triangle is always equal to the sum of the two interior remote angles in the triangle.

A P P LY Find the measure of the missing angle.

x°

42°

38°

x°

63°

75°

40° 67°

73°

21°

x°

84°

52°

75°

101°

x°

45°

34°

Find the measure of the missing angle. 6.

24°

x°

31°

42° x°

114°

71°

33°

104°

57°

x°

99°

x°

82°

113°

10.

91° 25°

x°

116°

Vocabulary Exterior angles - angles formed when one side of a shape is extended outside the shape; in a triangle, an exterior angle will be supplementary to the adjacent angle and equal to the sum of the two remote interior angles Remote interior angles - angles in a triangle that do not share a vertex with the given exterior angle

130

Explain, or ask a student to volunteer to explain how this can be so using a real-life example such as, “The tallest person in school is 5 feet and 11 inches. Jimmy is 5 feet and 11 inches. Therefore, Jimmy is the tallest person in school.”

Since both (m a + m b) and m d equal 180° when added to m c, we can say:

y Ruler y Protractor

Write the following on the board:

Rule: All the angles in a triangle add up to 180°.

Materials

PRE-LESSON WARM-UP

We can use rules we already know to find another rule.

Level H

Chapter 7

Lesson 3

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Draw a triangle on the board and label the top angle a and the bottom two angles b and c. Then, extend the base out to one side to form a new exterior angle and label it d. See the diagram in the Learn and Connect section for reference. Ask students to tell you what rule we know about the angles inside a triangle. [They add up to 180°.] Write m a + m b + m c = 180° on the board. Then, ask what we know about angle c and angle d. [They form a straight line, so they add up to 180°.] Write m c + m d = 180° under the first equation. Ask students what they see in both equations. [m c and 180°] Help students use logic to derive a new rule. [Since both (m a + m b) and m d equal 180° when added to m c, we can say that m a + m b = m d.] Give students another way to understand where this idea comes from by asking them to find m d if m c is 50°. [130°] Next, ask students to find the combined measure of angles a and b. [130°] Tell students that since a triangle’s angles add up to 180° and since a straight line is also 180°, the measure of angle d and the combined measure of angles a and b will always be the same. Write m a + m b = m d on the board. Explain that we will be using this and the other rules we have learned and discussed to solve problems in this lesson.

Exercise | 7-3 Name Write an equation and then solve for x. 2.

24°

120°

2x° (7x + 5)°

x + 6°

3x°

x°

50°

37.5°

70°

(x - 20)° (x + 8)°

17°

x°

64°

STRUGGLING LEARNERS

52° 3x°

x°

26°

Help students highlight the exterior angle in a triangle in the same color as the remote interior angles, but a different color than the angle it is adjacent to. This can help reinforce what is equal and where each angle can be found. Also, compile a list of all the rules about angles and angle relationships that have been learned so far for use when solving problems.

Use the shapes and rules you know to answer the questions. 6.

A. Find the value of x.

x°

The hexagon below is composed of 6 identical equilateral triangles.

30°

(2x)°

A. What is the measure of each angle in an equilateral triangle? 60°

B. Mark all the angles that also have a value of 2x. Hint: There are 3.

B. What is the measure of each angle in the hexagon?

EARLY FINISHERS

120°

C. What is the sum of all the angles in a hexagon? 120 × 6 = 720°

Below are two copies of triangle ABC. One copy is flipped, and the two triangles form a parallelogram. c

A. How many angles does a parallelogram have?

b a

x°

C. What is true about angles that are opposite each other in a parallelogram?

They are equal.

CH AL L ENGE 9.

Find the value of w, x, y and z. A

74°

w° x°

w = 70°

10. The shape below is made from 3 identical triangles. Explain how we can use this picture to prove that the sum of all the angles in a triangle is 180°.

x = 36°

The triangle was rotated so that all three different angles (1 purple, 1 blue, 1 green), are lined up and form a straight line, which equals 180°. Therefore, their sum must be 180°.

36°

y = 70° y° B

Lighthouse Math

z° D

z = 74°

Level H

Chapter 7

Exercise 3

131

APPLY AND DEVELOP SKILLS (Practice) Complete problems 1 and 6 in the Apply section together, and then have students complete problems 2-5 and 7-10 independently. Review the answers to ensure understanding. Next, have students complete problems 1-5 in the Exercise section independently. Review the answers and then work together as a class to solve problems 6-8. As you go through each problem, give students a few minutes to reason through the logic in each step. They will need to combine all the rules they have learned in this chapter so far to answer the questions. For problem 6, redraw the shape on the board and erase any parts that are distracting when trying to solve. Then, redraw them as you progress through the problem.

ACTIVITY Exterior Angle Proof: The purpose of this activity is for students to prove the exterior angle theorem using a protractor. Have students draw any kind of triangle and extend one side using a ruler. Then, have them measure all the angles including the exterior angle that was formed when they extended the side. Have them show that the two angles that are not attached to the exterior angle (called the remote interior angles) have a sum equal to the exterior angle. Then, have them do this again by extending a second side, and then the third side. Have students share their data with other groups for extra proof.

B. Given what you know about angles in a triangle, what is the sum of the angles in a parallelogram? 360°

y°

Ask students to tell you what the measure of an angle in a triangle (x) and the measure of all the exterior space around its vertex (y) add up to. They can use logic and rules they know and/or a protractor and any triangle in this lesson to figure out that the sum of an angle and all the exterior space outside its vertex is always 360°.

CHALLENGE AND EXPLORE In problem 9, students will use many different rules to solve for the missing pieces, including vertical angles, sum of angles in a triangle, and alternate interior angles in parallel lines. In problem 10, students will need to see in the figure that all three different angles in the triangle are lined up side by side in the shape to form a straight line. Therefore, they must add up to 180°.

COMMON ERRORS Students may confuse which angles are the remote interior angles.

ASSESS Check problems 2 and 4 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

Level H | 7-4 7-4 | Introduction to the Pythagorean Theorem

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to explain and apply the Pythagorean theorem to find missing side lengths in right triangles.

Solve for x.

x= SPIRAL REVIEW

Guiding Questions: 1. How do you square a number? [Multiply it by itself.] 2. How do you take the square root of a number? [Find the number that, when multiplied by itself, gives you the number under the square root.] 3. How can we take the square root of 2,500? [2,500 = 25 x 100. √25 = 5 and √100 = 10 so √2,500 is 5 x 10 = 50.]

Lighthouse MATH Level H | Teacher's Guide

10 + x2 = 91 x=

x2 + 28 = 37 x=

x = 70

20°

60°

x = 120

60°

73°

x2 − 8 = 41

115°

4. 40°

x = 42

x = 35

3x°

x°

Right triangles are triangles with one 90° (right) angle. The two sides that make the right angle are called legs. The side opposite the right angle is called the hypotenuse.

a2 + b2 = c2

hy p

ote

leg (a)

We can find a missing side of a right triangle using a formula called the Pythagorean theorem.

(c)

leg (b)

a and b are the two legs. c is the hypotenuse.

To find the length of a missing side, plug in the given side lengths and solve for the missing side. a2 + b2 = c2 (0.6)2 + (0.8)2 = c2 0.36 + 0.64 = c2 1 = c2 1=c

0.6 meters

y Graph paper y Ruler

Tell students that today we will be learning another rule about a special kind of triangle, and we will be using squares and square roots to find missing sides.

L E A R N A ND C O NNE C T

Materials

42 [16] 92 [81] 1.22 [1.44] √36 [6] √64 [8] √2,500 [50]

x2 = 16 x=

x°

y Right triangle - a triangle with one 90° (right) angle y Legs - in a right triangle, the sides that form the right angle y Hypotenuse - in a right triangle, the side opposite the right angle y Pythagorean theorem - a2 + b2 = c2 a formula to find any side of a right triangle; a and b are the legs, c is the hypotenuse

0.8 meters

Have students evaluate the following expressions:

Find the value of x. 1.

Vocabulary

PRE-LESSON WARM-UP

x2 = 36

The hypotenuse is 1 meter long.

A P P LY Circle the hypotenuse of each right triangle. 1.

Vocabulary Right triangle - a triangle with one 90° (right) angle Legs - in a right triangle, the sides that form the right angle Hypotenuse - in a right triangle, the side opposite the right angle Pythagorean theorem - a2 + b2 = c2; a formula to find any side of a right triangle; a and b are the legs, c is the hypotenuse

132

Level H

Chapter 7

Lesson 4

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Draw a right triangle on the board. Say: This is a right triangle because it has a right angle in it. Have a student come and draw a right angle symbol in the spot where they think the right angle is. Ask: Do you see the two sides of the triangle that make up the right angle? Highlight each side with a different colored marker. Label one a and one b, and say: These are called the legs of the triangle. Ask: What do you notice about the third side of the triangle? [It does not form the right angle, it is opposite the right angle, it is the longest side in the triangle] Say: The side across from the right angle is called the hypotenuse. Label it c. Write the Pythagorean theorem on the board. Say: This rule called the Pythagorean theorem can help us find a missing side of a right triangle. The legs are a and b and the hypotenuse is c. Draw the example given in the Learn and Connect section on the board. Ask a student to help you fill in the equation for the Pythagorean theorem. Ask students if it matters which leg is a and which leg is b. [No. Since addition is commutative, it can be added in any order.] Ask students if it matters which side is c. [Yes, this must be the hypotenuse and not a leg of the triangle.] Ask a student to walk you through solving the equation. Remind students to follow the order of operations. After taking the square root, ask students if we want to list the answer c = -1 as a possibility. [No, because lengths of sides cannot be negative.] Tell students that in this chapter, we will only give the positive square root for an answer.

Exercise | 7-4 Name Use the Pythagorean theorem to find the hypotenuse of the right triangle. 1.

2. c

3. c

1.6

STRUGGLING LEARNERS

2.4

1.8

Have students label all sides in each triangle with a, b, and c before they begin to set up their equations to solve. Have them check that c is labeled correctly by checking the ends of c to see if either are attached to a right angle. If they are, it is mislabeled. Allow students to use a perfect squares chart to help them solve.

4 1.2

c = 10

Use the Pythagorean theorem to find the missing leg of the right triangle. 5.

1.25

0.75

b b

a = 40

EARLY FINISHERS

Use the Pythagorean theorem to solve for x. 9.

10.

11.

x 4.8

Ask students to find the hypotenuse of a triangle with legs that are 7 cm and 9 cm. They will need to approximate the square root to find the length of the hypotenuse. [11.4 cm]

3.6

3 x

32 + 72 = x2 9 + 49 = x2 58 = x2 7.6 = x

No. x is the leg of the triangle, so it should be either a or b in the Pythagorean theorem, not c.

CH AL L ENGE Label the diagram with measurements. Use the Pythagorean theorem to answer the questions. 13. Davis is using a ladder to paint. His 2.5-meter ladder leans against the wall, with the bottom of the ladder on the ground 1.5 meters away from the wall. A. How far up the wall will the ladder reach?

2 meters

B. The ladder can extend 2 more meters. About how high up the wall can it reach when extended?

Lighthouse Math

Level H

about 4 meters

Chapter 7

Exercise 4

12. Look at the triangle and Trevor’s work. Did he use the Pythagorean theorem correctly?

CHALLENGE AND EXPLORE For problem 13, students will need to apply the Pythagorean theorem to a real-world situation. They will need to understand that the ladder forms a right triangle with the wall and the ground, and they will need to determine which side of the triangle they are trying to solve for (one of the legs).

133

Have students complete problems 1-5 in the Apply section independently, then review the answers to ensure students know how to find the hypotenuse. Next, work together to set up problems 1 and 5 in the Exercise section together as a class. Then, have students solve and complete problems 2-4 and 6-12 independently. Review the answers and strategies used as a class.

ACTIVITY Picture Proof: The purpose of this activity is for students to show the Pythagorean theorem using a picture. Have students draw a right triangle on graph paper. Tell them to use a ruler to draw one side with a length of three boxes. Tell them to turn their ruler 90° at one end of the line and draw another line from there that is four boxes long. Finally, have them connect the two open ends with a third line to close the triangle. While they do this, have them measure the length of the third side. Next, ask students to draw two squares, each time using one leg of the triangle as one side of the square. Ask students what the areas of the squares they drew are. [9 square units and 16 square units] Finally, have students use their ruler to draw a square anywhere on the page along the gridlines with side lengths equal to the third side of the triangle. If measured and lined up with the grid correctly, the sides should all be 5 units long. Ask them to find the area of this square. [25 square units] Ask them to relate this back to the Pythagorean theorem they just learned. [If you add up the area of the two leg squares, you get 25, the area of the hypotenuse’s square.]

COMMON ERRORS Students may always solve for the hypotenuse, even when a leg is missing. Students may use the length of the hypotenuse for a or b in the Pythagorean theorem.

ASSESS Check problems 4, 8, and 9 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 7-5 7-5 | Distance on the Coordinate Plane

PREREQUISITE SKILLS

y Students will be able to use the Pythagorean theorem to find the distance between two points in the coordinate plane.

DAI LY REVI EW

Objective and Learning Goals

Solve. 1.

4 − 32 -5

SPIRAL REVIEW

82 + 18 82

c=5

a=5

3. 6

b=8

The Pythagorean theorem can also be used to find the length of a line segment on the coordinate plane. Step 1: Draw a right triangle with line segment AB as the hypotenuse.

(8, 8)

Step 2: Subtract the x or y values to find the lengths of the legs. Length of vertical distance: 8 − 5 = 3 Length of horizontal distance: 8 − 4 = 4

A (4, 5)

Step 3: Use the Pythagorean theorem to find the hypotenuse. a2 + b2 = c2 42 + 32 = c2 16 + 9 = c2 25 = c2 5=c

PRE-LESSON WARM-UP

a2 + b2 = c2

A P P LY © Lighthouse Curriculum. Copying strictly prohibited.

All of this can be done in one formula called the distance formula:

1. 2.

Draw two lines to make a right triangle. Use the coordinates to find the length of each leg of the triangle.

a = 48 b = 36 Use the Pythagorean theorem to find the length of c.

2304 + 1296 = c2

3600 = c2

-32

(24, 16)

-16

c = 60

Explain why c is the distance between the two points.

c = (y2 − y1)2 + (x2 − x1)2

Complete the steps to find the distance between the two points.

482 + 362 = c2

(8, 5)

The length of line segment AB is 5 units.

Write the Pythagorean theorem on the board and ask students to isolate the variable c.

Guiding Questions: 1. How can we isolate a variable? [By using inverse operations] 2. Which inverse operation will isolate c in the Pythagorean theorem? [square root] 3. How does this version of the equation make it easier to solve for c? [We only need to input a and b and then follow the order of operations on one side of the equation.]

45 − 52

L E A R N A ND C O NNE C T

y Hypotenuse - in a right triangle, it is the side opposite the right angle y Distance formula - a formula for using coordinates to find the length of a line segment on a graph

Tell students that we will be using this version of the Pythagorean theorem to find the distance between two points on the coordinate plane.

(22 + 12)

Use the Pythagorean theorem to find the missing side.

Vocabulary

2(42 + 9)

-16

(-24, -20)

C is the hypotenuse of the triangle, it is also the distance between the two points

-32

Vocabulary Hypotenuse - in a right triangle, the side opposite the right angle Distance formula - a formula for using coordinates to find the length of a line segment on a graph

134

Level H

Chapter 7

Lesson 5

c = √(y2 - y1)2 + (x2 − x1)2

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect)

Have students look at the graph in the Learn and Connect section and locate the line segment between point A (4,5) and point B (8,8). Ask: Can we find the length of AB just by looking at the graph? [No, because it does not line up with any gridlines.] Tell students that this lesson is going to focus on finding the length, or distance, of a diagonal line on the coordinate plane. Tell them that we can draw in a right triangle with AB as the hypotenuse and we can use the Pythagorean theorem to find its length. First, have students find the length of the vertical distance by subtracting the y-values of the endpoints of the vertical line [8 − 5 = 3] and the horizontal distance by subtracting the x-values of the endpoints of the horizontal line. [8 − 4 = 4] Next, have a student volunteer to use the Pythagorean theorem to find the hypotenuse. [c = 5] Tell students that we can sum up what we did in one big formula called the distance formula: c = √(y2 − y1)2 + (x2 − x1)2. Explain that the two differences being squared under the square root are the legs of the triangle. We subtract here just like we subtracted earlier to find the vertical and horizontal distances. Ask: When evaluating, what is the first step in order to solve? [Subtract in each parentheses.] What is the second step? [Square each difference.] What is the third step? [Add the squares.] What is the last step? [Take the square root.] Emphasize the importance of following the order of operations, and remind students that the square root acts as a grouping symbol, so everything inside must be done first before the square root is taken.

Lighthouse MATH Level H | Teacher's Guide

Exercise | 7-5 Name Fill in the blanks to find the length of the lines using the distance formula. 1.

c = (y2 − y1)2 + (x2 − x1)2

(-20, 35)

c = (35 − 5 )2 + ( -20 − 20)2 (20, 5)

-50

900

2500

Have students highlight x-values in one color and y-values in a different color to keep track of the different dimensions. Additionally, provide students with a reference sheet with the distance formula and the order of operations as it applies to an example such as the distance formula.

c = (6 − -2 )2 + ( 2 − 4)2

(2, 6)

c = ( 30 )2 + ( -40 )2 c=

STRUGGLING LEARNERS

c = (y2 − y1)2 + (x2 − x1)2

c = ( 8 )2 + ( 6 )2 -10

+ 1600

(-4, -2) -10

100

Find the distance between the two points on the coordinate plane. 3.

120

EARLY FINISHERS

-10

120

-120

Length =

Length = 100

-120

Find the distance between the two points. (0, 4) and (-8,-2)

Length =

(3.2, 1.3) and (5.6, 4.5)

Length =

(-3, -4) and (1.8, 2.4)

Length =

CH AL L ENGE 8.

A delivery drone is launched from coordinates (70ft, 260ft) and flies in a straight line towards its first destination at (100ft, 300ft). It then proceeds to (180ft, 360ft) and then returns to its original launch point. What is the total distance it travels?

400 300 200 100

50 feet + 100 feet + ~150 = ~300 feet total 100

Lighthouse Math

Level H

Chapter 7

Exercise 5

200 300 400

135

APPLY AND DEVELOP SKILLS (Practice) Have students work with a partner on problems 1-4 in the Apply section. Review all the answers as a class to ensure understanding. Then, work together as a class to fill in the blanks for problem 1 in the Exercise section. Have students complete problem 2 on their own, then review the answer. Work as a class to set up the distance formula for problem 3, then ask students to solve on their own and work on problem 4. Review the answer and then have students work independently on problems 5-7. Review all the answers to ensure understanding.

ACTIVITY Coordination: The purpose of this activity is for students to practice using the distance formula. Have students choose a point anywhere on the coordinate plane. Tell them they can choose any point in any quadrant but to stay within -10 to 10 on each axis. Next, randomly assign them a partner. Have them work together to find the distance between their two points on the coordinate plane using a right triangle and/or the distance formula. Students will likely have to estimate the value of a square root. Remind them to use perfect squares that they know to help them with this. Afterward, discuss as a class what about the formula allowed us to have all positive answers even though many of the points had negative coordinates. [Squaring a negative number always results in a positive number.]

Ask students to use the Pythagorean theorem to find the length of the diagonal in a square with a length of 1 and compare it to the length of each side of the square. Then, have them use their results to explain why we can find the length of a horizontal or vertical distance by counting the boxes, but not the length of a diagonal line. [We can find the length of a horizontal or vertical distance by counting the number of boxes it lines up with, but a diagonal line does not line up with boxes. The Pythagorean theorem proves that the length of a diagonal in a square is more than the length of each side of the square. Therefore, we cannot just count the boxes a diagonal line passes through to find its length.]

CHALLENGE AND EXPLORE For problem 8, students will need to plot several points on the graph as described. They will then need to complete the distance formula to find the lengths described. Tell students to take it one piece at a time so that they do not get confused when drawing the lines on the graph.

COMMON ERRORS Students may mix together x- and y-coordinates in the different parts of the distance formula. Students may neglect to follow the order of operations.

ASSESS Check problems 4-7 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

-10

Level H | 7-6 7-6 | Applying the Pythagorean Theorem

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to solve real-world problems using the Pythagorean theorem.

Circle H if the bolded side is a hypotenuse. Circle L if it is a leg. 2. H

SPIRAL REVIEW

Find the distance between the two points. 1.

Vocabulary

(-6,-2) and (0, 6)

(2,1) and (5, 5)

y Hypotenuse - in a right triangle, it is the side opposite the right angle y Pythagorean theorem - a2 + b2 = c2 a formula to find any side of a right triangle; a and b are the legs, c is the hypotenuse

(3,8) and (-2, 9) 5.1

L E A R N A ND C O NNE C T The map shows the trails in a rectangular park. Randall wants to know which trail is longer, the blue trail or the orange trail, and how much longer it is. Since the angles in a rectangle are always right angles, Randall uses the Pythagorean theorem to find the length of the blue trail.

Materials

30 yards

y Ruler

40 yards

302 + 402 = c2 900 + 1,600 = c2 2,500 = c2 50 = c The blue trail is 50 yards long. The orange trail is 40 yards + 30 yards = 70 yards.

PRE-LESSON WARM-UP

Guiding Questions: 1. What is the measure of a right angle? [90°] 2. What word can we use to describe the relationship between the two legs of a right angle? [The legs are perpendicular to each other.] 3. What is the side opposite a right angle called? [hypotenuse]

Lighthouse MATH Level H | Teacher's Guide

Ask students to find a right angle in the classroom. Ask them how they knew it was a right angle. [e.g. the sides of the desk are square, the wall is perpendicular to the floor and not slanted] Next, ask them to identify the rays or sides that make up the right angle. Ask them what we can find (what is c?) if we use the Pythagorean theorem with the lengths of their sides as a and b. [The diagonal length of the desk (distance from one corner to the opposite corner), the diagonal length of a piece of paper from one corner to the opposite corner] Tell students that today, we are going to see how the Pythagorean theorem can be used practically in real-world situations.

Randall finds the difference: 70 yards − 50 yards = 20 yards. The orange trail is longer than the blue trail by 20 yards.

A P P LY Circle the equation that will help solve the problem. 1.

A field has a rectangular layout. A coach needs to measure the distance from one corner of the field to the opposite corner (diagonally) to set up a practice drill. If the length of the field is 100 meters, and the width is 70 meters, what is the diagonal distance across the field? 70 + 100 = c

702 + 1002 = c2

The top of an 8-foot ladder is leaning against a wall. The bottom of the ladder is 3 feet from the wall. How high up the wall does the ladder reach? 82 + 32 = c2

a2 + b2 = 112

32 + b2 = 82

702 + b2 = 1002

Vocabulary Hypotenuse - in a right triangle, the side opposite the right angle Pythagorean theorem - a2 + b2 = c2; a formula to find any side of a right triangle; a and b are the legs, c is the hypotenuse

136

Level H

Chapter 7

Lesson 6

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) As a class, read the problem in the Learn and Connect section about the map with the blue trail and the orange trail. Be sure to pause to talk about each part of the problem. Ask students: What does Randall want to know? [Which trail is longer and by how much] Why does Randall choose the Pythagorean theorem to help him answer the question? [The map is rectangular; therefore, it has right angles. The orange trail takes up two edges of the map, bending on the right angle.] Ask students what information we are given on the map [the lengths of two parts of the orange trail] and what information is missing [the length of the blue trail]. Point out to students that the blue trail goes from one end of the orange trail to the other. Ask them what part of the right triangle the blue trail is. [the hypotenuse] Next, ask a student to help set up the equation, leaving the missing piece blank. [302 + 402 = c2] Ask students to walk you through solving, reminding them to follow the order of operations. [c = 50] Ask students if solving answered the question that was asked. [No, we need to know the lengths of both trails to compare them.] Finish off solving by finding the length of the orange trail [70 yards] and the difference between the lengths of the two trails. [20 yards]

Exercise | 7-6 Name Circle the equation that will help solve the problem. Then find the missing piece. 1.

c2 = 82 + 62

60 + 80 = c2

The diagonal is 2.

A rectangular book has a ribbon bookmark whose length spans from the top left corner to the bottom right corner of a page. The pages are 6 inches wide and 8 inches long. How long is the ribbon?

10 inches

STRUGGLING LEARNERS 8

Have students label all sides a, b, and c, and check to make sure that a and b form the right angle and that c is opposite the right angle.

62 + b2 = 82 long.

The back of a frame has two 13-inch wooden pieces criss-crossing. If one side of the frame is 12 inches, find a missing side.

13 12

a2 + 132 = 122

a2 + 122 = 132

The missing side is

5 inches

EARLY FINISHERS

122 + b2 = c2 long.

Have students create their own problem that involves using the Pythagorean theorem. The should trade problems with a partner, solve, and check each other's work.

Use the pictures and the Pythagorean theorem to answer the questions. 3.

1.6 km coastline

1.2 km

A ship's captain spots a lighthouse. The ship is 1.2 km away from the coast, and the lighthouse is 1.6 km up the coast. How far is the ship from the lighthouse?

From the control tower, a radar measures the distance to an airplane as 1.25 mile. The airplane is 0.75 mile horizontally from the tower. How high is the airplane above the ground?

5 1.2

ile

0.75 mile

CHALLENGE AND EXPLORE

1 mile high

2 km away

Draw a diagram. Then answer the question. 6.

Ariel is walking across a rectangular field to the opposite corner. The length of the field is 60 feet and the width is 80 feet. What is the distance that Ariel will walk?

Bella and Sofia are flying a kite. Sofia is holding the 15-foot string and is standing 12 feet away from Bella. The kite is directly above Bella's head. How high above her is the kite?

60 ft 9 feet high

100 feet 80 ft

15 fee t

12 feet

CH AL L ENGE 7.

A firefighter needs to set up his 10-foot ladder to reach a window that is 8 feet above the ground. A. How far away from the wall will the base of the 6 feet ladder need to placed so that he can reach the window? B. If he keeps the ladder in the same spot but extends it to its full height of 12 feet, will the ladder be able to reach a window that is 11 feet above the ground?

For part A of problem 7, students will need to understand that the missing side is one of the legs. For part B, one leg length remains the same, but the hypotenuse changes. They will need to evaluate how this changes the length of the other leg and if the ladder in question can reach a certain spot on the building given these lengths.

No, the ladder will reach a spot just above 10 feet.

Lighthouse Math

Level H

Chapter 7

Exercise 6

137

As a class, read problems 1 and 2 in the Apply section and discuss which equation is set up correctly to help answer the question being asked. For question 2, be sure that students understand that we are not looking for the hypotenuse because the ladder is the hypotenuse, and we already know its length. Therefore, the missing side is a or b. Have students work on problems 1-4 in the Exercise section independently, and then review the answers and strategies as a class. Continue to work together to draw a diagram and set up an equation for problems 5 and 6, and then have students finish solving on their own.

ACTIVITY Triangulation: The purpose of this activity is for students to use the Pythagorean theorem in a way that it would be used in the real world. Explain to students that sometimes we cannot measure a distance directly, but we can use a right triangle and the Pythagorean theorem to find the unknown distance. Tell students that their task is to find the distance from the top right side of the classroom to the bottom left side of the classroom without measuring it directly. They will find and measure the legs of a right triangle with their unknown distance as the hypotenuse. Then, they will input these measurements into the Pythagorean theorem and solve to find the unknown distance. They will likely have to estimate the length using perfect squares that they know, or they can use a calculator for a more accurate measure.

COMMON ERRORS Students may label the wrong side as the hypotenuse.

ASSESS Check problems 3-6 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 7-7 7-7 | Review

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will review concepts from Chapter 7.

Angle Relationships

Materials

Complementary Angles

Supplementary Angles

Adjacent Angles

Vertical Angles

Angles that add up to 90°

Angles that add up to 180°

Angles that share a vertex and a side

Formed by intersecting lines, these angles are directly across from one another and equal.

y Index cards

50°

70°

90° – 50° = a a = 40°

PRE-LESSON WARM-UP

Next, ask students to draw two parallel lines that are crossed by a transversal. Ask them to label the following. y One pair of alternate interior angles y One pair of alternate exterior angles y One pair of same side interior angles y One pair of same side exterior angles y One pair of corresponding angles y One pair of vertical angles y One pair of supplementary angles Guiding Questions: 1. How do we know angles are complementary? [They form a right angle.] 2. How do we know angles are supplementary? [They form a straight line.] 3. How can we form vertical angles? [by intersecting two straight lines]

180° – 70° = d d = 110°

e is adjacent to f

67°

113°

g = 67° i = 113°

Angle Relationships Formed by Parallel Lines and a Transversal Alternate Interior

Alternate Exterior

Corresponding

Same Side Interior

Same Side Exterior

Equal

Add to 180°

Angle Relationships in a Triangle

Right Triangles The Pythagorean Theorem

hy p

ote

leg (a) b

c m

a+m m

Ask students to draw a right angle, an acute angle, a straight angle, and an obtuse angle on a piece of paper. Then, ask them to add the following: y A line that divides the right angle into two complementary angles y A line coming out of the vertex of the acute angle to create adjacent angles y A line that crosses the straight angle to create vertical angles y A line coming out of the vertex of the obtuse angle to create supplementary angles

(x1, y1)

(c)

(x2, y2)

b+m

d=m

The Distance Formula

leg (b)

c = 180°

a+m

a2 + b2 = c2

distance = √(y2 - y1)2 + (x2 − x1)2

A P P LY Write the name of the relationship between angle a and angle b. 1.

b a

4. a

5. a

b a

supplementary angles or adjacent angles

138

vertical angles

Level H

alternate exterior angles

Chapter 7

corresponding angles

Lesson 7

complementary angles

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) As a class, review the chart in the Learn and Connect section, touching on the important points from this chapter. Be sure to remind students that we can use what we know about the different angle relationships to set up equations and solve for missing pieces. Next, review the angle relationships in a triangle, including that all the angles add up to 180° and that the measure of the exterior angle is equal to the sum of the two remote interior angles. Finally, review the Pythagorean theorem and the related distance formula, touching on the importance of identifying which side is which in each problem.

APPLY AND DEVELOP SKILLS (Practice) Have students work independently on problems 1-5 in the Apply section. Then, review all the answers to ensure understanding. Have students work on problems 1-8 in the Exercise section independently and then have them check their answers with a partner before continuing to work on problems 9-12. Review the answers as a class and discuss the strategies students used to solve.

Lighthouse MATH Level H | Teacher's Guide

Exercise | 7-7 Name Find the value of x. 1.

(x + 60)°

x°

x = 70

x = 30

(3x)°

x = 36

STRUGGLING LEARNERS

4. (x + 50)° (2x + 14)°

x° 2x°

Have students reference the chart in the Learn and Connect section to help them identify which angle pairing is present in each problem and which formula to use.

x = 18

x°

(20 + 2x)°

(7x)°

x°

(x + 80)°

(4x)°

x° (2.4x)°

72°

x = 50

EARLY FINISHERS

42°

x = 138

(x - 20)°

x = 64

Have students go back to problems 1-8 in the Exercise section and write the name of the angle relationship.

x = 20

Use the Pythagorean theorem or distance formula to find the missing length. 9.

10. 25 in

CHALLENGE AND EXPLORE

11. (6, 4)

x (0, -4)

15 in

x = 20 in

x= 5m

x = 10

Solve. 12. A 2.4-meter telephone pole has a steel wire attached to its top. The other end of the wire is fastened to the ground. If the wire is 2.6 meters long, what is the distance from the bottom of the pole to the bottom of the wire? 1 meter

CH AL L ENGE 13. Avenue A and Avenue B run parallel to each other in a city. Avenue C crosses through them at an angle of 60°. Avenue D crosses through them at an angle of 70°. At what angle will Avenue C and Avenue D eventually cross? Use the diagram to help you.

Lighthouse Math

Level H

Avenue A 60°

Avenue B

50°

Chapter 7

Avenue C

Exercise 7

For problem 13, students will need to use the diagram to determine an unknown angle. Once students solve, ask them to write down their reasoning and then look for a different way to prove that their answer is correct.

70°

Avenue D

139

Picture Card: The purpose of this activity is for students to review the vocabulary terms from the chapter in a visual way. Have students create flashcards with a picture on one side and the vocabulary term of the other side and then use those cards to test each other on the various vocabulary terms from the chapter.

COMMON ERRORS Students may mix up the angle relationships. Students may set up equations incorrectly.

ASSESS Check the even-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

ACTIVITY

Chapter 8

140

In Chapter 8, we will learn about

Transformations and Congruence Shapes can be drawn on the coordinate plane and then changed based on rules to create new shapes. • We will identify congruent and similar figures. • We will transform a figure by translating it. • We will transform a figure by reflecting it. • We will transform a figure by rotating it. • We will transform a figure by dilating it.

141

Level H | 8-0 Chapter 8 | Skill Checklist

Skill 1: Drawing Lines, Angles, and Shapes A

E F

EFG or

H J

ABC

LMN M

HIJ

I can understand line, angle, and shape notations.

out 4 correct

Skill 2: Lines of Symmetry

Objective and Learning Goals

There are two lines of symmetry in the shape. Each side of the line is a mirror image of the other side.

Draw all the lines of symmetry in each shape.

Students will review skills needed for Chapter 8:

I can draw lines of symmetry.

out 4 correct

y Drawing lines, angles, and shapes y Lines of symmetry y Multiplying by numbers greater than and less than one y Coordinate plane fluency y Equations of horizontal and vertical lines

Skill 3: Multiplying by Numbers Greater Than and Less Than One A number times a number greater than one gets bigger.

5 × 8 will be more than 5. A number times a number less than one gets smaller.

9 × 3 will be more

less

9×

1 will be 3

less

12 × 6

12 ×

5 × 83 will be less than 5.

out 4 correct

142

Level H

than 9.

2 will be more 3

less

than 12.

2 will be more 3

less

than 12.

I can estimate the size of products.

Chapter 8

Skill 1: Drawing Lines, Angles, and Shapes

Skill 2: Lines of Symmetry

y Use the gray box to review the difference between a line segment (AB) and a line (CD), as well as how to name them. y Review how to name angles and triangles. y Have students complete problems 1-4 and review the answers and strategies as a class.

y Remind students that a line of symmetry is a (often imaginary) line that divides a shape in half. The two halves will be mirror images of each other. y Use the image in the gray box to show students that a shape may have more than one line of symmetry. y Tell students that if they fold a shape along its line of symmetry, the two halves of the shape should line up exactly with each other. y Have students complete problems 1-4 and review the answers and strategies as a class.

Lighthouse MATH Level H | Teacher's Guide

than 9.

Skill Checklist

Lighthouse Math

Skill 3: Multiplying By Numbers Greater Than and Less Than One y Ask students what happens to a batch of dough if you double (× 2) the recipe. [it gets larger.] y Ask students what happens to a batch of dough if you halve (× 21 ) the recipe. [it gets smaller.] y Tell students that when we multiply a number by a number greater than one, the result will be an increase. y Tell students that when we multiply a number by a number less than one, the result will be a decrease. y Have students complete problems 1-4 and review the answers and strategies as a class.

Name

Skill 4: Coordinate Plane Fluency 10

Start at the origin. Then, follow the directions to plot the point. Write the coordinates of each point. 1.

-10

3 units to the right, 4 units up ( 3 , 4 )

7 units to the right, 2 units down ( 7 , -2 )

3 units to the left, 8 units up ( -3 , 8 )

6 units to the left, 9 units down ( -6 , -9 )

1 -10

-10

Point O is the origin: (0,0)

Point A is 5 units to the left of the origin and 4 units below it at (-5, -4).

-10

I can navigate the coordinate plane fluently.

out 4 correct

Skill 5: Equations of Horizontal and Vertical Lines Equations for horizontal lines start with y =.

Write the equation of the lines on the graph.

Equations for vertical lines start with x =.

A: y = 2

B: y = -5

C: x = -7

D: x = 1

-10 -10

x = -4

Draw the lines on the graph.

B C

y = -1

D -10

x=8

-10

I can draw vertical and horizontal lines from an equation.

out 8 correct

Lighthouse Math

Level H

Chapter 8

Skill Checklist

143

Skill 4: Coordinate Plane Fluency

Skill 5: Equations of Horizontal and Vertical Lines

y Use the graph in the gray box to remind students of where the origin is and that its coordinates are (0,0). y Tell students that all points are given coordinates or directions to get there via the origin. y Have students tell you how to get to the points (5, 2) and (-6, 4) on the graph. [Go right 5 and up 2; go left six and up four] y Have students complete problems 1-4 and review the answers and strategies as a class.

y Remind students of the slope-intercept formula y = mx + b. y Ask students what the slope (m) of a line is if the mx term is not there. [0] y Ask students what this looks like on a graph. [a flat/horizontal line] y Tell students that all horizontal lines will begin with y=. y Tell students that all vertical lines will begin with x =. y Tell students that the number in the equation tells you where to draw the line through the y-axis or the x-axis. y Have students complete problems 1-6 and review the answers and strategies as a class.

y=6

Lighthouse MATH Level H | Teacher's Guide

Level H | 8-1 8-1 | Congruent and Similar Figures

PREREQUISITE SKILLS

Objective and Learning Goals

3 x = 5 20

DAI LY REVI EW

y Students will be able to identify congruent and similar figures as well as missing side lengths and angles.

Solve each proportion.

SPIRAL REVIEW

Lighthouse MATH Level H | Teacher's Guide

x 15 = 20 25

x ft 15 ft

x=7

10 in

x=8 17 ft

x = 12

5.6 m

x=8 6 in

x in

L E A R N A ND C O NNE C T Congruent figures have the same shape and are the same size. Corresponding (matching) sides and angles have equal measurements.

Congruent Figures A

13 in 70°

Similar figures have the same shape but are different sizes. Their corresponding (matching) angles are equal, but their corresponding side lengths are different. They are proportional.

11 in

60°

50° 15 in

15 in

60° 11 in

50°

ABC

JKL

is congruent to

70° 13 in J

Corresponding angles are equivalent.

A and B and C and

M 6 in N 110° 110° 4 in 4 in 70° 70° P O 10 in

J K L

Corresponding lines are equivalent.

AB and JK BC and KL AC and JL

Similar Figures

You can use a proportion to find a missing side length in similar figures. NO OP = RS ST 4 10 = 8 x 4 10 8 = 10 × =x 8 x 4 © Lighthouse Curriculum. Copying strictly prohibited.

12 in 110°

110°

70°

M and N and O and P and

Corresponding angles are equivalent.

x = 20

8 in

MNOP ~ QRST

8 in

70° S

x in

Q R S T

is similar to

Corresponding lines are proportional.

MN and QR NO and RS OP and ST MP and QT

A P P LY Circle if each pair of figures are congruent or similar. C

15 m

10 m F

congruent similar

12 55° U

65°

Y 60°

55°

Guiding Questions: 1. What is similar about the shapes? [the number of sides, the number of angles, the size of the shapes] 2. What is different about the shapes? [the number of sides, the number of angles, the size of the shapes]

x=8

Divide students into groups and give each group a a collection of different sized shapes. Included in the collection should be triangles and quadrilaterals that are the same shape but a different size. Ask students to group them based on their shape and size and then hold a discussion about what was similar and different about the shapes.

7 14 = x 16

4.2 m

4 in

Materials

PRE-LESSON WARM-UP

x = 80

2. x=5

x in

Vocabulary y Congruent figures - figures that have the same shape and size, with equivalent angle and side measurements y Corresponding - sides or angles in congruent or similar shapes that match up with one another y Similar figures - figures that have congruent corresponding angles and proportional corresponding sides y Proportional - two sets of measurements that have the same ratio or rate

9 72 = 10 x

Find the missing side using the Pythagorean theorem.

3 in

y Shape cutouts y Graph paper y Colored highlighters y Index cards

x = 12

65°

60° W

congruent similar

Vocabulary Congruent figures - figures that have the same shape and size with equivalent angle and side measurements Corresponding - sides or angles in congruent or similar shapes that match up with one another Similar figures - figures that have congruent corresponding angles and proportional corresponding sides Proportional - two sets of measurements that have the same ratio or rate

144

Level H

Chapter 8

Lesson 1

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Write the words congruent and similar on the board. Tell students that when a shape is exactly the same as another shape, we call those two shapes congruent. Draw two congruent shapes under this word on the board. Then under the word similar, draw two triangles with the same shapes but different sizes. Tell students that these triangles are called similar because their shape is the same, but they are different sizes. Similar figures will have different side lengths than each other, but they will be proportionate, and their angle measures will be the same. Explain that we can check if two figures are congruent or similar by looking at their corresponding sides and angles. Have students look at the chart in the Learn and Connect section. Explain that ABC and JKL might be positioned differently, but each side in triangle ABC matches up with a side in JKL, and each matching or corresponding side is the same length. Have students highlight the corresponding sides and angles with colored highlighters. Explain to students that when looking for corresponding sides and angles, always start by looking for two features that stand out, like a marked angle, a right angle, or a side that is obviously longest or shortest. Match each of those features to the same features in the other triangle, and use them as a guide to find which are the corresponding angles and sides. In this triangle, point A corresponds with J because A and J have the same angle measure. Point B corresponds with point K, and point C corresponds with point L. Continue by explaining the symbols and comparisons in the Congruent Figures box, and then move to the Similar Figures box. Ask: What do you notice about the sides and angles of these two figures? [The sides are not the same length, but the corresponding angles are equal.] Have students highlight the corresponding sides and angles. Next, have students set up a proportion, lining up the corresponding sides. Explain that we can use a proportion to prove that the figures are similar by

Exercise | 8-1 Name Identify the corresponding angle or side. M

65°

18 ft

50° 50°

11 ft

58°

65°

NO and XY

M and

T 122°

B and

L F

18 ft R

CD and TU

11 ft

18 m K 115°

65°

26 m

M and

VU = 32 m

T = 115 °

KL and EF

42°

UV = 50 ft

44 ft

42° Q 50 ft

10 in

EARLY FINISHERS

Have students construct two similar triangles and label all side lengths. Then, have them write and solve a proportion that proves their triangles are similar.

60°

78°

60° V

24 m M

34 ft

32 m

Have students highlight and color-code all corresponding angles and sides.

10 in

58°

122° C

Determine the unknown measurements of the congruent figures. 4.

STRUGGLING LEARNERS

28 in

9 in

W = 78 °

XY =

5 in

Z = 90 °

Determine the unknown measurements of the similar figures. F

D 64 m C

76°

6 in

x in

x = 50 in

114°

12.

55°

80°

For problem 13, students will need to use the distance formula to prove that all the sides are the same length.

96 m

12 m

45°

x = 28 in

6 in 10 in

80°

64 m

CH AL L ENGE 13. Two triangles, ABC and DEF, are drawn on a coordinate plane. Triangle ABC has vertices A(1, 2), B(5, 2), and C(3, 6). Triangle DEF is drawn with vertices D(3, 4), E(7, 4), and F(5, 8). Are the triangles congruent, similar, or both? Explain. Hint: To find the distance between two points, use the distance formula d = √(x2 – x1)2 + (y2 – y1)2.

The triangles are congruent because they have the same side lengths and shape.

E C 5

D A

F B 5

Lighthouse Math

CHALLENGE AND EXPLORE

x in

11.

20 in

66°

8 in

7 in 66°

76° G

40 in 8 in

114°

16 m

65° 39° E 80 m

x = 20 m 10.

Level H

Chapter 8

Exercise 1

145

APPLY AND DEVELOP SKILLS (Practice) Have students complete problems 1-2 in the Apply section independently. Review the answers to ensure understanding. Model problems 1, 4, and 7 in the Exercise section for the class before having students work independently on the rest of the problems. Review the answers and strategies used.

ACTIVITY Figure Match up: The purpose of this activity is to reinforce students’ understanding of congruent and similar figures. Prepare sets of congruent and similar figures on index cards (with labeled angles and sides). Place one pair of each set face down on the board and hand out the other pair to students. One student comes up to the board and turns over a card. All students, including the one at the board, have 15 seconds to observe the card and determine if it is congruent or similar to theirs. If the student at the board has the similar figure, they must prove it mathematically by writing and solving a proportion using corresponding sides. If their figure is not similar, they turn the card back over as in a classic matching game. Continue in this fashion until all cards are claimed. For a competitive element, the class can be split into two teams, and the first team to have all teammates matches claimed is the winner.

COMMON ERRORS Students may match corresponding sides incorrectly. Students may set up inconsistent proportions by not lining up proportional sides.

ASSESS Check problems 2, 5, and 10 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

showing that the proportion is equal. Tell them that we can also use a proportion to find the missing side in a shape. Go through the example given in the Learn and Connect section to find the length of ST.

Level H | 8-2 8-2 | Translations

PREREQUISITE SKILLS

y Students will be able to describe and perform translations on the coordinate plane and understand their properties.

DAI LY REVI EW

Objective and Learning Goals

Find the distance between the ordered pairs using the distance formula.

SPIRAL REVIEW

(0, 0) and (6, 8)

(2, -2) and (3.8, 0.4)

(-1, 2) and (2, 6)

(7, 15) and (10.6, 19.8)

State whether each set of shapes is congruent or similar. 1.

18 cm

6 cm

2 cm

20 in

5 in

11 in

6 cm

16 in

4 in 12 in

5 in

similar

Vocabulary y Transformation - a change that occurs to a geometric figure y Translation - a type of transformation that moves a figure up, down, left, or right

congruent

3 in

similar

L E A R N A ND C O NNE C T A transformation is a change that occurs to a shape. A translation is a type of transformation that moves a figure up, down, left, or right.

Triangle ABC translated to become triangle A’B’C’. Each vertex moved down 6 units and to the right 6 units.

Materials y Graph paper y 2 paper figures y Index cards

C 5

-5

A’

A (-4, 4) → A’ (2, -2) B (-4, 1) → B’ (2, -5) C (-1, 1) → C’ (5, -5)

B’

-5

C’

A P P LY Fill in the blanks to describe the rule of the translation from the black shape to the purple shape.

Draw four graphs on the board. On each graph, draw a pair of capital As. One graph should show a translation, one a reflection, one a dilation, and one a rotation. Have students observe the graphs and discuss with a partner the Guiding Questions below. Reconvene to discuss as a class. Encourage students to use the words congruent and similar when describing the two figures. To conclude, say: In math, when we change a figure by moving it, flipping it, turning it, or resizing it, we call that a transformation. A transformation is a change. Circle the graph that shows a translation and say: Today, we will learn one kind of transformation: When a figure is moved from one place to another. Guiding Questions: 1. How did the figure change in size, direction, and location? [It moved, it was resized, it was flipped over, it was turned on its side.] 2. Are the two figures the same or different? [Some are the same, but their position is different. One is a different size.]

Lighthouse MATH Level H | Teacher's Guide

1. © Lighthouse Curriculum. Copying strictly prohibited.

PRE-LESSON WARM-UP

2. Q

X’

T -10

-5

Q’

R’

T’

S’

-10

Y’

-5

-10

Move up/down Circle one

Z’

-5

-10

Move up/down 10 units.

units.

Circle one

Move right/left 10 units.

Move right/left

Circle one

units.

Vocabulary Transformation - a change that occurs to a geometric figure Translation - a type of transformation that moves a figure up, down, left, or right

146

Level H

Chapter 8

Lesson 2

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Have each student place their workbook at the left side of their desk and shift it to the right. Ask: Did the size and direction of the workbook change? [no] Explain that in geometry, this movement of shifting is called a translation. Say: A translation is a transformation that moves every point of a figure the same distance in the same direction. It slides an object from one position to another without rotating, resizing, or deforming it. Ask: What are the four possible straight line directions that a figure can be moved on a coordinate plane? [right, left, up, down] Say: Notice, we can’t move diagonally, so when translating a figure, we may need to move in more than one direction to get it to our desired location. Next, have students observe how the blue triangle was moved to become the purple triangle in the Learn and Connect section. Be sure to emphasize that each point/vertex of the figure must be moved the exact same distance in order to keep the transformed figure congruent to the original. Also be sure to explain how to say the name of the the purple figure (A’B’C’: “A prime, B prime, C prime”) and that these ‘prime’ marks are used to let us know that it is the new transformed figure.

Exercise | 8-2 Name Write a rule to describe the translation of the black shape to the purple shape. A’

10 L’

M’

STRUGGLING LEARNERS

C’

-10 L

-10

O -6

J’ I’

-10

Move up 3 units. Move left 5 units.

G H 10

G’ H’

B C

Have students color-code each vertex to help them track movement.

N’

O’

B’ -6

EARLY FINISHERS

-10

Move up 11 units. Move right 11 units.

Move down 1 unit. Move left 7 units.

Have students create a translation problem with three to five translation steps. Then have them switch with a partner to find the new location of the shape.

Translate each shape using the given rule. Label each vertex. 4.

Move 5 units left and 9 units up. 10

D’

Move 5 units right and 3 units down.

Move 7 units right and 13 units up. U’

L M P Q

V’ 10 W’ X’

N O F’

E’

L’ M’ P’ Q’

D -10

N’ O’

Z’

S’

Y’

-10

R’

CHALLENGE AND EXPLORE

V WX

-10

Complete each step. 7.

A triangle ABC has vertices A(3, 4), B(5, 8), and C(5, 2) and then is translated 5 units left and 2 units down to create A’B’C’. Draw both triangles. B 8

A. Write the new coordinates A’(-2, 2), B’(0, 6), C’(0, 0) for the vertices A’, B’, and C’. B. Use the distance formula to calculate the lengths of sides AB and A’B’. Are they the same distance? 2

√(5 – 3) + (8 – 4) = √20 and √(6 – 2) + (0 – (-2)) = √20 Yes, they are the same distance. 2

B’

A’

C’

-6

CH AL L ENGE 8.

Quadrilateral PQRS was translated up 3 and right 3 and then quadrilateral P’Q’R’S was drawn at vertices P’(0,5), Q’(3,7), R’(5,4) and S’(2,2). A. Draw both quadrilaterals. B. What were the original coordinates of PQRS? P(-3,2), Q(0,4), R(2,1), and S(-1,-1) C. P’Q’R’S’ was then translated and point Q’’ was plotted at (1,-1). What are the directions for the translation? left 2, down 8

Lighthouse Math

Level H

Chapter 8

Q’ P’ Q

For problem 8, students will need to plot the translated quadrilateral first, then work backwards through the translation to find the coordinates of the original quadrilateral. For part C, students will need to figure out the directions for the translation using one new point and its corresponding point.

R’ P -6

Exercise 2

S’ R S

147

Have students complete problems 1 and 2 in the Apply section independently then, review as a class. Model problems 1 and 4 in the Exercise section and then have students complete problems 2-3 and 5-7 independently. When students are working on problem 7, ask them if they think that the distances of the sides of the two figures will be equal. [Yes, a translation does not change the size of a figure.] Have students use the distance formula to confirm.

ACTIVITY Translation Relay: The purpose of this activity is to practice translating figures on a coordinate plane. Give each student a piece of graph paper and a cutout of a shape that can be moved around on the graph to show translations. Prepare note cards with different translation tasks on each. The card should tell them where to place their cutout to start and how to translate it. For example: Place one vertex at (-3,4) and translate up 4 right 2. Give each student a card and then have them complete the translation task listed on their card. After a minute or two, have students switch cards with a partner and complete their new task. Then, have students check their two translations with their partner to see if they completed them correctly. Have the partners pass their cards to the next group so that everyone will have two new task cards to complete.

COMMON ERRORS Students may move vertices diagonally. Students may forget to move all vertices the same distances. Students may flip figures instead of sliding them.

ASSESS Check problems 2, 5, and 7 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 8-3 8-3 | Reﬂections

PREREQUISITE SKILLS

y Students will be able to describe and perform reflections across a line and understand their properties.

DAI LY REVI EW

Objective and Learning Goals

Draw the line(s) of symmetry in each shape. 1.

SPIRAL REVIEW

Describe the translation that occurs from one ordered pair to the other. 1.

(3, 2) → (5, -1)

(-4, 0) → (-7, 5)

Right 2, down 3

(5, -9) → (-3, -2)

left 3, up 5

left 8, up 7

(2, 4) → (6.5, -2.5) right 4.5, down 6.5

Vocabulary y Reflection - a transformation that flips a figure across a given line

L E A R N A ND C O NNE C T A reflection is a type of transformation that flips a figure across a given line. This will look like the original figure is looking at itself in a mirror.

Materials

You can think about the figure being folded over the line. If the two shapes would completely align, the original figure was reflected correctly.

y Graph paper

To reflect a shape over any line, observe the distance of each vertex from the given line. Place the new vertices the same distance in the opposite direction from the given line. 5

PRE-LESSON WARM-UP

C’

-5

Guiding Questions: 1. How are the two halves of the heart the same? [They are congruent in shape and size.] 2. How are they different? [The copied half is flipped/opposite/reversed/a mirror image.] 3. Are the two halves the same distance from the line of symmetry? [Yes. They were drawn from the fold/line of symmetry.]

B’

A’

-5

C C’

-5 B’

Reflected across the y-axis

Have students fold a piece of paper exactly in half vertically to create a line of symmetry. Then, have them open the paper and, using a dark marker, draw half a heart starting and ending at the centerfold. Next, have them close the paper and trace that half heart on the outside of the opposite fold. Next, have them open the paper to see a full symmetrical heart. Finally, have them turn to a partner to answer the Guiding Questions below.

A’

B 5

A’

C’

B’

-5

Reflected across the x-axis

Reflected across the line x = 1

A P P LY Draw the line that each figure was reflected across. Then, circle the choice that matches your line. 1.

-5

A. y-axis

B. x = 1

A. y-axis

B. x = 1

A. y-axis

B. x = 1

A. y-axis

B. x = 1

C. x-axis

D. y = 1

C. x-axis

D. y = 1

C. x-axis

D. y = 1

C. x-axis

D. y = 1

Vocabulary Reflection - a transformation that flips a figure across a given line

148

Level H

Chapter 8

Lesson 3

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect)

Start by asking students to explain what is seen when a person looks into a mirror. [The person sees themselves in reverse.] Say: In geometry, an image that has been reversed is called a reflection. Read and explain the definition for a reflection in the Learn and Connect section. Be sure to explain to students that it is important that when a figure is reflected, it is drawn the same distance away from the line of reflection as the original image is. Go through each graph in the Learn and Connect section. Ask students: How did the figure change when it was reflected over the y-axis? [It was flipped horizontally.] Next, ask students to contrast the x and y coordinates for the blue and purple images. Ask students to share what happens to the x and y coordinates when an image is reflected over the y-axis. [The x-coordinate flips/becomes its opposite, the y-coordinate stays the same.] Next, have students count the distance from point C to the y-axis in both figures to confirm that it is the same. Ask students to look at the middle graph to see how an image changes when it is reflected over the x-axis. [It is flipped vertically.] Ask students how the coordinates changed. [The y coordinate became its opposite.] Have students confirm that the original and new shape are the same distance from the x-axis. Finally, have students observe the graph on the right to see that any line can be used as a line of reflection. Have them confirm that both figures are equidistant from x = 1, the line of reflection.

Lighthouse MATH Level H | Teacher's Guide

Exercise | 8-3 Name Write the rule to describe the reflection of the black shape to the purple shape. 1.

D’ E’

E D

A’ A C’ B’

B C

-3

-5

J K

J’ K’

-5

-2

reflection across x = 2

M’

W’ X’

W X

Z’ Y’

-5

G’

F’

-5

L’

reflection across y-axis

Have students color-code each vertex. Have students label and highlight the axis or line that each image should be reflected over.

-5

reflection across y = 1

STRUGGLING LEARNERS

E’

reflection across x-axis

Graph each of the given reflections. Label each vertex. 5.

Reflection across y-axis T’

Q’

R S

M M’

I K

S’ R’

L L’ K’

-5

Reflection across y=2

H’

B’

Reflection across x=4

E’

A’

F’

Y’

Z’

J’

-1

I’

E’ -1

Have students create an incorrect reflection and swap with a friend for error analysis.

X’

-5

D’

EARLY FINISHERS

Reflection across x-axis

D Z

Complete each step. 9.

A quadrilateral ABCD has the following vertices: A(2,3), B(5,7), C(4,2), and D(1,−1) and is reflected over the y-axis to create A’B’C’D’. Draw both figures. A. Write the new coordinates for A’(-2, 3), B’(-5, 7), C’(-4, 2), D’(-1, -1) the vertices A’, B’, C’, and D’. B. Use the distance formula to calculate the length of side AB and A’B’. Are they the same distance?

B’

CHALLENGE AND EXPLORE

A’

C’

C 5

-5

D’

Yes, they are the same distance. The lengths of AB and A’B’ are both 5 units.

D -2

C. Use the slope formula to calculate the slope of side AB and A’B’. Do they have the same slope? 4

CH AL L ENGE 10. Quadrilateral WXYZ has the following vertices: W(0, 0), X(3, 0), Y(3, 2), Z(0, 2). It is translated up 2 units and left 5 units. A. Graph both quadrilaterals. B. What are the coordinate points of the transformed quadrilateral?

W’ (-5,2), X’ (-2,2), Y’ (-2,4), Z’ (-5, 4)

C. When a figure is reflected across the line y = x, the x- and y-coordinates swap places. Reflect quadrilateral W’X’Y’Z’ across the line y = x. D. List the new coordinate points for W’’X’’Y’’Z’’. W’’(2, -5), X’’(2, -2), Y’’(4, -2), Z’’(4, -5)

Lighthouse Math

Level H

Chapter 8

Exercise 3

Z’

Y’

W’

X’

-5

X X’’

-5

W’’

No, their slopes are opposites. AB has a slope of 3 , while A’B’ has a slope of - 3 .

In problem 10, students will need to plot the original quadrilateral and the translated quadrilateral, and then for part C, reflect the quadrilateral over the line y = x. If needed, explain that the line y = x is a diagonal line going through the origin with a slope of 1.

Y’’

Z’’

149

Complete problems 1-3 in the Apply section together as a class. Model problems 1 and 6 in the Exercise section, then have students complete problems 2-5 and 7-8 independently. Review the answers and strategies used together as a class. Have students work with a partner on problem 9 and then review the answers as a class.

ACTIVITY Reflect the Word: The purpose of this activity is for students to practice reflecting coordinates using abstract art. Hand out a sheet of graph paper to each student. Students should draw in the x and y axes. Give the students the following points to plot on their paper. (9, -9), (9, -8), (4, -9), (5, -8), (2, -8), (4, -2) and (0, 0), (1, -1), (1, 1), (2, -1), (1, -3), (-1, -4), (-3, -3). The result will be two separate images. Then tell students that the line of reflection for this art is y = x. When a figure is reflected over this line, the x and y coordinates flip. Have students flip each coordinate and plot it on the graph using a different color. The two images together should spell the word ‘wow’.

COMMON ERRORS Students may confuse the x- and y-axes. Students may move an image over a line of reflection but forget to flip its orientation. Students may not place the new image the proper distance from the line of reflection.

ASSESS Check problems 3 and 8 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 8-4 8-4 | Rotations

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to describe and perform rotations around a point and understand their properties.

Find the slope between each set of points.

3. SPIRAL REVIEW

(3, 6) and (5, 10)

(-1, -2) and (-8, -4)

2 7

(-5, 4) and (-2, 5)

1 3

(0, 3) and (10, -7)

-1

(5, 6) reflected across the x-axis

Vocabulary

Guiding Questions: 1. How can you describe the movement of a rotation? [a circular motion] 2. Where is the point that an object spins around and why? [It’s in its center because that’s the part that doesn’t move.] 3. What stays the same and what changes when an object is rotated? [The size and shape stay the same; the location and direction of the object change.]

Lighthouse MATH Level H | Teacher's Guide

(-1, -8) reflected across the x-axis

(2, -4)

-10

(-1, 8)

-10

A rotation is a transformation where a figure is turned around a point called the center of rotation. When the center of rotation is the origin, the signs and placement of the x- and y-coordinates may be flipped. Rotations can be clockwise, which is the direction a clock’s hands move, or counterclockwise, which is the opposite direction. 5

B’

B C’

C A’

A’

B’

-5

A’

C’

-5

90° clockwise or 270º counterclockwise: (x, y) → (y, -x)

C’

B -5

y Index cards y Graph paper

Place images of the following objects on the board: Earth, a steering wheel, the wheels of a car, a spinning top, and a clock with the hands highlighted. Have students work with a partner to figure out what all these objects have in common. [They spin/rotate/turn in circles.] As a class, discuss the movement of the objects. Ask students: If you were to mark the point that each object spins around, where would you put it, and why? [In the middle because it’s the point that does not move; everything else moves around it.] Next, ask students to think about what happens to the shape of an object when it spins. Does it change? [No, its shape and size stay constant.]

(-2, -4) reflected across the y-axis

L E A R N A ND C O NNE C T

Materials

PRE-LESSON WARM-UP

(5, -6)

y Rotation - a transformation where a figure is turned around a point y Center of rotation - the point that a figure rotates around y Origin - the point (0, 0) on the coordinate plane y Clockwise - the direction a clock’s hands move y Counterclockwise - the opposite direction a clock’s hands move

Use the coordinate plane to reflect each point. Write the new ordered pair.

-5

180° clockwise or counterclockwise: (x, y) → (-x, -y).

270° clockwise or 90º counterclockwise: (x, y) → (-y, x)

A P P LY Write the rule to describe the rotation around the origin. 1.

S T V -5

U’

T’ S’

A’ D’ 5

V’

B’ C’

-5

3. L’

90º counterclockwise or 270º clockwise rotation

K’

-5

180º rotation

J’

K L -5

270º clockwise or 90º counterclockwise rotation

-5

G’ H’

-5

F’

90º counterclockwise or 270º clockwise rotation

Vocabulary Rotation - a transformation where a figure is turned around a point Center of rotation - the point that a figure rotates around Origin - the point (0, 0) on the coordinate plane

150

Level H

Chapter 8

Clockwise - the direction a clock’s hands move Counterclockwise - the opposite direction that a clock’s hands move

Lesson 4

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Write the following vocabulary words on the board: Rotation, center of rotation, origin, clockwise, and counterclockwise. Ask for student input as to what they may mean based on the introduction. Start by explaining that a rotation means a turn. Explain that in order for a rotation to happen, there must be a specific point that the object spins around, much like a pivot turn (one foot remains in place while the body turns). Tell students that when we rotate figures, we rotate them around the origin. This specific point is known as the center of rotation. Draw a graph on the board and ask a student to come up and mark the origin/center of rotation on the graph on the board in red. Next, lift a paper and ask: If i were to rotate this paper on the graph, what are the only two directions I can rotate it in? [left and right] Say: In geometry, we call rotating to the right clockwise and rotating to the left counterclockwise. Review the four quadrants and say that we name them in a counterclockwise direction from I-IV. Say: We will rotate shapes 90°, 180°, or 270° in either direction. Have students go through each rotation in the Learn and Connect section, being sure to point out what happens to the coordinates once the rotation is complete. It may be helpful for students to use a paper cutout the same size as the triangle in the figure to physically demonstrate each rotation. Ask students what happens when they rotate an image 360° in either direction. [It ends up in the same place it started.] Have students notice that a 90° clockwise rotation is the same as a 270° counterclockwise rotation and ask them to think why this may be true. [They are the same distance from the starting point/360°/0°.]

Exercise | 8-4 Name Rewrite each set of points given the degree of rotation around the origin. 90° clockwise/ 270º counterclockwise (y, -x)

180° clockwise/ counterclockwise (-x, -y)

270° clockwise/ 90º counterclockwise (-y, x)

(-4, 7)

(7, 4)

(4, -7)

(-7, -4)

(3, 8)

(8, -3)

(-3, -8)

(-8, 3)

(-2, -3)

(-3, 2)

(2, 3)

(3, -2)

(6, -1)

(-1, -6)

(-6, 1)

(1, 6)

STRUGGLING LEARNERS Have students make a table with the rotation rules to reinforce them and to use as a reference throughout the chapter. Have students draw arrows on the top of their page as a key to remind them which way is clockwise and which way is counterclockwise.

Rotate each shape around the origin using the given angle of rotation. Label each vertex. 5.

180º rotation L

G H M

-5 N’

F J

90º clockwise rotation

5 F’

J’ I’

G’ H’

-5

L’

X -5

-5

270º clockwise

V’ W’

X’

-5

O’

M’

90º counterclockwise rotation

C’

D’

F’

E’

5F C

EARLY FINISHERS

-5

V W

Have students create a rotation error analysis problem that they can switch with a partner to solve.

-5

Complete each step. 9.

Triangle ABC has vertices A(3,4), B(6,8), and C(5,2). Rotate ABC 90° clockwise around the origin to create A’B’C’. A. Write the new coordinates A(4, -3), B(8, -6), C(2, -5) for the vertices A’, B’, and C’. B. Use the distance formula to calculate the lengths of sides AB and A’B’. Are they the same distance?

CHALLENGE AND EXPLORE

A C -10

A’

Yes, they both have a distance of 5. C’

No, AB has a slope of 3 , while A’B’ has a slope of - 4 .

B’

C. Use the slope formula to calculate the slopes of side AB and A’B’. Do they have the same slope?

-10

CH AL L ENGE 10. Triangle JKL has the following vertices: J(1, 1), K(2, 4), L(-2, 4). It undergoes three transformations. First, it’s translated down 3 units and left 3 units. Next, it is reflected across the x-axis. Finally, it is rotated 180º.

L’ K’

-5

A. Graph both triangles. B. What are the coordinate points of the transformed triangle?

Lighthouse Math

For problem 10, students will need to complete three transformations in order. Students should work on a separate piece of graph paper to keep their work neat and clear, then write their final answer on the graph in their book.

J’

J’(2,-2), K’(1, 1), L’(5, 1)

Level H

Chapter 8

-5

Exercise 4

151

Work together as a class to complete problems 1-4 in the Apply section. Have students reference the rotation rules in the Learn and Connect section above. Have students write down the original and new coordinates and determine together which rotation rule matches. Students should complete problems 1-8 in the Exercise section independently, then review the answers as a class to ensure understanding. Have students work with a partner on problem 9, then review the answers and strategies used.

ACTIVITY Rotation Challenge: The purpose of this activity is to have students practice completing rotations while also reinforcing rotation vocabulary. Set up four stations around the room, each with a pile of rotation problems for students to complete. Put the solutions on the back of each card so students can selfcheck. Split the class into groups, putting one group at each station. Each student chooses a problem at their station to complete. Then, at the teacher’s command, students are told to make a rotation of 90°, 180°, 270°, and 360° clockwise or counterclockwise to the next station. Students rotate to the corresponding station and choose another problem to solve.

COMMON ERRORS Students may confuse clockwise and counterclockwise. Students may confuse the 90° and 270° coordinate shift rules.

ASSESS Check problems 5-8 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 8-5 8-5 | Dilations

PREREQUISITE SKILLS

10 m

80°

SPIRAL REVIEW

(3, 1) rotated 90º counterclockwise

(-5, -8) rotated 180º

x=4

x cm 7 cm

(-4, 2) rotated 90º clockwise

(5, 8)

(6, 6) rotated 270º counterclockwise

(-2, -4)

(-6, 6)

To dilate a figure, you can use a scale factor, or a number that is multiplied by each vertex. Enlargement

Reduction

Scale factor is a whole number.

Scale factor is a fraction.

Scale Factor = 2

A’

A(-2, 2) → (-2 × 2, 2 × 2) → A’(-4, 4) B’

B(-2, 1) → (-2 × 2, 1 × 2) → B’(-4, 2) C(0, 1) → (0 × 2, 1 × 2) → C’(0, 2)

49 cm

A dilation is a transformation that changes the size of a figure. An enlargement is a dilation that makes a figure bigger, while a reduction is a dilation that makes a figure smaller.

y Graph paper y Rubber bands (2 for each student)

Guiding Questions: 1. How did the pupils change? How did they stay the same? [Their size changed, but they stayed the same shape.] 2. Is the pupil changing shape proportionally? How do you know? [Yes, because it becomes bigger or smaller but keeps the same circular shape.]

28 cm

L E A R N A ND C O NNE C T

Materials

-5

Scale Factor =

C’

1 2

A(-4, 4) → (-4 × 21 , 2 × 21 ) → A’(-2, 2) B(-4, 2) → (-4 × 21 , 2 × 21 ) → B’(-2, 1) C(0, 2) → (0 × 21 , 2 × 21 ) → C’(0, 1)

B -5

A’

B’

C’

A P P LY © Lighthouse Curriculum. Copying strictly prohibited.

Have students pair up and complete a pupil study. Students study what their peer’s pupil (the black circle in the center of their eye) looks like when the light in the classroom is on. Then, close the lights and have them study the change in their peer’s eyes. Have the pairs discuss the Guiding Questions below and then hold a class discussion about their conclusions.

x in

(1, -3)

y Dilation - a transformation that changes the size of a figure y Enlargement - a dilation that makes a figure bigger y Reduction- a dilation that makes a figure smaller

x=8

32 in

Write the new ordered pair given the rotation.

Vocabulary

PRE-LESSON WARM-UP

58°

3 in

x = 12

80° xm

58°

12 in

42°

8 m 42°

16 m

y Students will be able to describe and perform dilations with a given center and scale factor and understand their effects on figures.

DAI LY REVI EW

Objective and Learning Goals

Find the missing value in each set of similar shapes.

Circle whether the dilation of the black figure is an enlargement or a reduction. 1.

-10

A D

A’ D’

B’ C’

X’ 10

F’

G’

I Z’

Y’

I’

H’

-10

enlargement or reduction

H 10

enlargement or reduction

Vocabulary Dilation - a transformation that changes the size of a figure Enlargement - a dilation that makes a figure bigger

152

Level H

Chapter 8

Reduction - a dilation that makes a figure smaller Scale factor - the number multiplied by each vertex to dilate a figure

Lesson 5

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Start by asking students if they have ever gone to an eye doctor and had their eyes dilated. Have students share what occurred when their eyes were dilated. [Their pupils were made bigger so that the doctor could see inside their eyes]. Explain that just like a dilation of an eye makes the pupil bigger, a dilation of a figure is when a figure is changed in size. Emphasize that the shape and orientation are preserved, and only the size changes. Ask students: What are the two ways a figure can be changed in size? [bigger or smaller] Explain that the term in math for bigger is called an enlargement, while the term for smaller is called a reduction. Tell students that the important thing to remember about a dilation is that the figure is changed proportionally. Ask: How can I ensure that my dilated image will be in proportion? [by scaling the image with a scale factor] Explain that the scale factor is the number that is multiplied by the coordinates of each vertex to get the dilated image. Say: When making an enlargement, the scale factor must be above the number one. However, when making a reduction, the scale factor must be between zero and one. Work through the two problems in the Learn and Connect section on the board.

APPLY AND DEVELOP SKILLS (Practice) Use problems 1-3 in the Apply section to play a quick game of Showdown. Have students take out a blank sheet of paper. Then, give students 30 seconds to observe problem 1 and write on their sheet “Enlargement” or “Reduction.” When you say, “3, 2, 1 Showdown!”, students lift their papers to reveal their answers. This is a quick way to Lighthouse MATH Level H | Teacher's Guide

Exercise | 8-5 Name Identify the scale factor of each dilation from the black shape to the purple shape. 2.

-10 U U’

M -10

L’

STRUGGLING LEARNERS

W Y

V’

Have students use a table to organize their work. It should include a column for the original points, the calculations to change these points and the final dilated points. Also have students color code corresponding vertices.

O’

M’

V X

N’

X’

R’

W’ T

Z’

N -10

Reduction; Scale factor = 5

Y’

S T’

-10

S’ 1

Reduction; Scale factor = 2

Enlargement; Scale factor = 3

Graph the dilated image using the given scale factor. Label each vertex. 4.

Scale factor = 3 10

Scale factor = 2

A’

B’

E’

C’ D’ A

1 4

D’

EARLY FINISHERS

Have students go back to problem 6 in the Exercise section and find the lengths of each side in the two images. Next, have them find the area of each rectangle. Then have students describe numerically how the area of the original shape changed once it was 1 of its original size.] dilated. [It is 16

J’ K’ D

-10

M’ L’ M

C’ C

-10

Scale factor =

E’

-10

Complete each step. A triangle XYZ has the following vertices: X(2,3), Y(5,3), and Z(4,5). Perform a dilation of XYZ with a scale factor of 2 and draw X’Y’Z’ on the graph.

Z’

A. Write the coordinates of X’, Y’, and Z’ after the dilation. X’(4, 6), Y’(10, 6), Z’(8, 10)

X’ 5

B. Find the distance of XY and X'Y'. XY has a length of 3, and X’Y’ has a length of 6. C. Show that the side length of XY is proportional to X’Y’. 2

Y’

Z Y

The scale factor is 1 . The ratio of the lengths X’Y’:XY is 3 = 1 .

CH AL L ENGE 8.

Rectangle HIJK has the coordinate points H(-3, -4), I(-3, -6), J(-7, -6), and K(-7, -4). The rectangle undergoes multiple transformations. First, it is translated up 2 units and left 1 unit. Next, it is reflected across the x-axis. Then, it is rotated 90º clockwise. Finally, it is dilated by a scale factor of 21 . A. Graph both HIJK and H’I’J’K’. B. List the ordered pairs for H’I’J’K.

Lighthouse Math

H'(1,2), I'(2,2), J'(2,4), K'(1,4)

Level H

Chapter 8

Exercise 5

K’ J’ H’ I’ -10

CHALLENGE AND EXPLORE For problem 8, have students complete each transformation in order. Students can work on a separate piece of graph paper to keep their work neat and clear. They should create four different images, one for each transformation.

-10

153

observe if students need additional help to understand enlargements and reductions. Complete problem 2 in the Exercise section together as a class. Have students identify the coordinates of a pair of corresponding points and notice how they changed from the original figure to the next. Have them repeat with another pair of corresponding points and use this information to find the scale factor. Have students complete problems 1 and 3 independently. After reviewing the answers, complete problem 4 together as a class, then have students complete problems 5 - 6 independently and problem 7 in pairs. Review the answers and strategies used as a class.

ACTIVITY Rubber Band Dilation: The purpose of this activity is to have students explore dilations in a kinesthetic manner. Hand out a sheet of graph paper and two rubber bands to each student. Have students tie the rubber bands together, creating a tight knot. Next, have students draw a triangle with vertices A (4,6), B (6,6), and C (4,9). Have students also plot point (8,12) as the starting point for the dilation they are about to do. Tell students to place one end of the rubber band at the origin (0,0) and hold it down with their finger. Next, have them place their pencil in the other end of the rubber band and pull their pencil until the knot of the rubber band is directly over vertex A and their pencil is on point (8,12). Then, tell them to trace the original triangle with the knot. Tell them that their pencil is drawing the new dilated triangle. As students reach each vertex with their knot, they should plot the new points with their pencil on their graph. Follow up by asking: Is the new image an enlargement or reduction? What do you think the scale factor for this dilation is?

COMMON ERRORS Students may forget to multiply all the coordinates by the scale factor.

ASSESS Check problems 4-6 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

L 10

Level H | 8-6 8-6 | Combining Transformations 5

PREREQUISITE SKILLS

y Students will be able to describe sequences of transformations that map one figure onto another.

DAI LY REVI EW

Objective and Learning Goals

SPIRAL REVIEW

J(3, 5)

F(-1, -3)

R(0, 0)

C(4, -2)

Vocabulary

Path A y Translate 4 units right y Rotate 90° clockwise about the origin [A'(1, –5), B'(3, –5), C'(1, –7)]

Path B y Rotate 90° clockwise about the origin y Translate 4 units right [A'(5, –1), B'(7, –1), C'(5, –3)] Have students work in pairs to complete the paths then, answer the Guiding Questions below. Guiding Questions: 1. Are the two final images the same or different? How? [They are the same image, but in completely different positions.] 2. What might have caused the two results to differ? [The order. Path A had a translation done first, then a rotation, while Path B had a rotation and then a translation. A rotation does not just shift the image; it moves a point differently.]

Lighthouse MATH Level H | Teacher's Guide

-3

(-3, -3) dilated by a 3. scale factor of 31

(10, -2) dilated by a scale factor of 5

(-1, -1)

(0, 6) dilated by a scale factor of 41

(50, -10)

(0, 1.5)

L E A R N A ND C O NNE C T You can perform multiple transformations like translations, reflections, rotations, and dilations on one figure. Perform one transformation at a time. Translate ABC up 2 units and right 3 units. Then reflect ABC across the y-axis. A’’

Rotate ABC 180º. Then dilate ABC by a scale factor of 2.

A’

A C’’

B’’ B

-5

B’

C’

A B

C B’ B’’ C’ C’’ A’

-5

A’’

y Graph paper y Tracing paper

-5

Hand out a sheet of graph paper to each student. Have students plot a triangle with the following points: A (1,1), B (1,3), C (3,1). Then write the following transformation paths on the board:

(4, -2) dilated by a scale factor of 2 (8, -4)

Materials

PRE-LESSON WARM-UP

R -5

Write the new ordered pair given the scale factor. 1.

y Translation - a type of transformation that moves a figure up, down, left, or right y Reflection - a transformation that flips a figure across a given line y Rotation - a transformation where a figure is turned around a point y Dilation - a transformation that changes the size of a figure

Graph each ordered pair on the coordinate plane.

-5

A(0, 3) → A’(3, 5) → A’’(-3, 5)

A(-1, 2) → A’(1, -2) → A’’(2, -4)

B(0, 0) → B’(3, 2) → B’’(-3, 2)

B(-1, 0) → B’(1, 0) → B’’(2, 0)

C(2, 0) → C’(5, 2) → C’’(-5, 2)

C(0, 0) → C’(0, 0) → C’’(0, 0)

A P P LY Circle the two transformations that took place. X

1. Z -10

Y X’

Y’ Z’

translation reflection rotation dilation

2. A D

-10

B C

C’ A’

-10

translation reflection D’ rotation dilation B’

3. R’ -10

S’

U’ T’

T S

translation reflection rotation dilation

-10

Vocabulary Translation - a type of transformation that moves a figure up, down, left, or right Reflection - a transformation that flips a figure across a given line Rotation - a transformation where a figure is turned around a point

154

Level H

Chapter 8

Lesson 6

Dilation - a transformation that changes the size of a figure

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Begin by telling students that today they will explore how multiple transformations can be combined. Tell them that as they saw in the warm-up activity, the order of transformations affects the final image. Explain that when determining a sequence of transformations, they should look closely at the orientation, direction, and size of the figure to identify what is preserved and what has changed. Have students tell you which transformation will change the orientation of a figure. [Rotation] Have them tell you which will change the direction of a figure. [Reflection] Have them tell you which will change the position of a figure. [Translation] And finally, have them tell you which will change the size of the figure. [Dilation] Ask students what is always preserved no matter the transformation. [The shape] Have students look at the two figures in the Learn and Connect section. Students can use a paper cutout of the black triangle to model the movements of each transformation. Explain to students that the black figure is the original figure, the blue is the figure after the first transformation, and the purple is the figure after the second transformation. Refer to the image on the left. Ask students how we can tell from the coordinates that the black triangle was translated two units up and three units to the right to become the blue triangle. [The y-values increase by two and the x-values increase by three.] Then have them tell you how we know the blue triangle was reflected over the y-axis. [The x-coordinates become their opposites, and the figure moves from quadrant I to quadrant II and changes its direction.] Refer to the image on the right. Have students describe what happened to the triangle after the first transformation. [It moved from quadrant II to quadrant IV, and now it’s upside down.] Ask how we know that this is not a reflection. [If it were reflected, it would have been a mirror image of itself.] Ask students how they can tell that the next transformation was a dilation. [The triangle got larger.] Finally, ask students to tell you the scale factor based on the coordinates. [2]

Exercise | 8-6 Name Complete each transformation. Label each vertex. 1.

Translate down 2, left 4 1 Dilate by a scale factor of 2

-10

E’

V P

-10

P’ -10 Q’

-10

U’(0, 3), V’(1, 1), W’(-1, 1)

Reflect across x = -3 Rotate 270º counterclockwise X

Y’

R’

G I 10

1 Dilate by a scale factor of 5 Reflect across y = 4

K H’

G’

-10

D’(-3, 9), E’(-8, 9), F’(-7, 4), G’(-2, 4)

X’

-10

Translate down 13, right 6 Rotate 90º counterclockwise J

-10

G’ E

P’(2, -6), Q’(2,-9), R’(7,-9), S’(7, -6)

Q S’

N’

10 M’

K’

L’

I’ J’

-10

Have students use tracing paper to help them physically move and map the original shape onto the transformed image.

-10

Z’ W’

-10

W’(6, -4), X’(6, 0), Y’(3, 1), Z’(3, -4)

Give students a notecard with the following questions for them to use as a guide. 1) Did the figure flip? Reflection 2) Did the figure turn? Rotation 3) Is the figure bigger or smaller? Dilation 4) Did the figure shift? Translation

D’

F’ D 10

STRUGGLING LEARNERS

Translate up 7, right 10 Reflect across y-axis

U’ W W’ V’

Rotate 270º clockwise Reflect across the x-axis

-10

G’(4,2), H’(6,5), I’(8,2), J’(6,-1)

K’(-2, 6), L’(1, 6), M’(1, 9), N’(-2, 9)

The graphs show a figure that underwent two transformations. Draw the shape after the first transformation. Then, describe the second transformation that took place. Reflection: across the x-axis up 2, right 1 Translation:

A’’ A

Rotation: 180º rotation Translation: down 1, right 2

C’’ 10

D’’

-10

EARLY FINISHERS

Dilation: scale factor of 2 Reflection: across y = -2

P’’

B’’

O’’

Q 10

-10

R’’ S’’

M’’

N’’

-10

M N

Q’’

C -10

-10

CH AL L ENGE 10. A quadrilateral has vertices P(-1, 1), Q(0, 4), R(3, 3), and S(1, 0). The quadrilateral PQRS undergoes a series of transformations to become quadrilateral P’’Q’’R’’S’’ with these final coordinates: P’’(3,2), Q’’(6,1), R’’(5,−2), and S’’(2,0). Determine a possible sequence of 2 transformations (chosen from translation, rotation, reflection, dilation) that maps PQRS onto P’’Q’’R’’S’’. Describe each step clearly.

Q R

-3

Level H

Chapter 8

Q’’

S’’

R’’

Rotate PQRS 90º clockwise; translate PQRS up 1, right 2

Lighthouse Math

P’’

-4

Exercise 6

Have students create their own transformation path (without saying which transformations took place) and switch with a partner to solve.

CHALLENGE AND EXPLORE For problem 10, have students use a piece of tracing paper to help them map the new figure. Once the figures are plotted, have students observe the two figures and rule out any transformation that will not make sense based on preservation rules.

155

Have students work in pairs to identify which transformations were made to the shapes in the problems in the Apply section. Discuss the answers and students’ reasoning as a class. Next, complete problem 1 in the Exercise section as a class to ensure student comprehension. Then, have students complete problems 2-6 independently. Have students check their work with a partner and then review the answers as a class, addressing any problems that gave students difficulty. Next, complete problem 7 together as a class to model this type of problem. Then, have students complete problems 8 and 9 independently.

ACTIVITY Transformation Telephone: The purpose of this activity is to have students review combining transformations in a collaborative and engaging way. Divide students into groups of four and give each group four sheets of graph paper with a simple shape on each. The first student writes a two-step transformation sequence for the shape and passes it to the second student, who performs the transformations and draws the new image. The third student receives only the final image and must describe what sequence of transformations they think produced it. The fourth student then compares the original instructions, the drawing, and the guessed sequence for accuracy. If correct, student 4 becomes student 1, student 1 becomes student 2, and so on. The group picks a new problem from the sheets and repeats this process. If incorrect, student 4 must correct before picking a new problem.

COMMON ERRORS Students may confuse the x- and y-axes. Students may confuse a rotation and a reflection. Students may confuse clockwise and counterclockwise.

ASSESS Check problems 2, 6, 8, and 9 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 8-7 8-7 | Review

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will review concepts introduced in Chapter 8.

Translations

Materials

y Vocabulary cards y Task cards

A’

-5

B’

C’

B’

-5

A(-4, 4) → A’(4, 4) B(-4, 1) → B’(4, 1) C(-1, 1) → C’(1, 1)

-5

C’

-5

Dilations

Rotating a figure involves turning it 90º, 180º, or 270º clockwise or counterclockwise.

A’

Dilating a figure involves making it larger or smaller by a given scale factor.

-5

A(-4, 4) → A’(4, 4) B(-4, 1) → B’(1, 4) C(-1, 1) → C’(1, 1)

-5

Reflecting a figure involves mirroring it across the x-axis, y-axis, or another given line.

A’

Rotations

B’

-5

A(0, 2) → A’(0, 4) B(-2, -2) → B’(-4, -4) C(2, -2) → C’(4, -4)

C’

Guiding Questions: 1. What key words in this definition help you know which transformation it describes? [Answers will vary.] 2. Does this transformation change the orientation, the size, or both? [Answers will vary.] 3. Can someone restate this definition in their own words? [Answers will vary.] 4. Does anyone agree or disagree? Why or why not? [Answers will vary.]

A(-4, 4) → A’(2, -2) B(-4, 1) → B’(2, –5) C(-1, 1) → C’(5, -5)

C’

PRE-LESSON WARM-UP Start with a review of all the vocabulary terms learned in this chapter. Prepare a pile of all the vocabulary words from this chapter and a pile of all the definitions. Be sure to include words like prime marks and the preservation rules as definitions. Place all the vocabulary terms on the board face up and all the definitions on a desk face down. Have a student come up and choose a definition card, read it aloud, and then find the match on the board. Students sitting can help the student at the board at their request.

Translating a figure involves moving it up, down, left, or right a given number of units.

-5

Reflections 5

A’

X’

-10

C’ C

B’ B

Y’ -10

156

W’ -10

R’ S’

Q’ 10

Z’ R

-10

translation

rotation

translation

rotation

translation

rotation

reflection

dilation

reflection

dilation

reflection

dilation

Level H

Chapter 8

Lesson 7

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect)

Begin by writing the four transformations on the board. Beneath each transformation, list the words “shape,” “size,” and “orientation.” Then, have students come up and circle what is preserved and cross out what changes for each transformation. Hold a discussion about which aspect is preserved in all transformations and why. [Shape. In a translation, reflection, and rotation, the figure is only moving to a different place on the graph. In a dilation, even though the figure changes size, the figure gets bigger or smaller in proportion to the original figure, so the shape will be preserved.] Next, have students come up to add the definition for each transformation on the board. Have students look at the chart in the Learn and Connect section. Going box by box, ask students to explain each transformation that occurred to the blue triangle and how the image and the coordinates show this. [Translation: to the right 6 and down 6; Reflection: over the y-axis; Rotation: 90° clockwise; Dilation: by a factor of 2]

APPLY AND DEVELOP SKILLS (Practice) Have students pair up to complete the problems in the Apply section. Review the answers and strategies used as a class. Next, have students complete all of the problems in the Exercise section independently, checking with a partner as they go. Review the answers together to ensure understanding. Lighthouse MATH Level H | Teacher's Guide

Exercise | 8-7 Name Translate each figure. 1.

Move down 2, right 5 B

A D A’ -10

Move up 6, left 8

B’ C’

M’

-10

D’

N’

L’

H N

O’M

H’ L

Have students use an anchor chart with a table of the different transformations and their properties. Students can also use tracing paper when needed.

F’

-10

STRUGGLING LEARNERS

Move down 10, left 4

G’

-10

Reflect each figure. 4.

Reflect across x-axis T’

Reflect across y-axis

E’

Q’ R’ -10

Q T

X’

-10

R S

Y’

-10

W’

Z’

Z -10

EARLY FINISHERS

Reflect across x = -2

S’

X 10

D’ J’ I’

I J

G’ H’ F’

Have students choose any topic from the chapter and create an error analysis problem. Have them exchange with a partner to solve.

E F

-10

Rotate each figure. Write the ordered pair for each new vertex. 8.

Rotate 180º

-10

V’

W’

U’

T’ 10

-10

J’

-10

P’10 O’ -10

Q’ R’

-10

L’

J’(2, -3), K’(1,-10), L’(9,-10)

O'(2,4), P'(2,9), Q'(6,8), R'(6,3)

Dilate each figure. Write the ordered pair for each new vertex. 10. Scale factor = 2

11. Scale factor =

A’ A

-10

D D’

C -10

W’

T’

V’

U’

L M O 10 Q P O’

-10

U Q’

C’

A’(-8, 8), B’(8, 8), C’(8, -8), D’(-8, -8)

Lighthouse Math

-10

12. Scale factor = 3 L’ M’

S’

W 10

1 3 10

B’

-10

Chapter 8

For an added challenge, have students return to problem 5 in the Exercise section, draw in the line y = x, and reflect the original image over that line. If students need assistance, tell them the rule that reflecting an image over the line y = x will change its coordinates from (x,y) to (y,x).

P’ -10

S’(0, 3), T’(3, 1), U’(3, -1), V’(-3, -1), W’(-3, 1)

Level H

CHALLENGE AND EXPLORE

Rotate 90º counterclockwise

K’

T’(9, 4), U’(2, 4), V’(2, 8), W’(9, 8)

Rotate 90º clockwise

L’(-3, 6), M’(0, 6), N’(0, -3), O’(6, -3), P(6, -6), Q(-3, -6)

Exercise 7

157

Transformation Showdown: The purpose of this activity is to have students review all transformations and their properties learned in Chapter 8. Split the class into groups. Prepare a pile of task cards for each team with transformation tasks from each lesson in this chapter. On the count of 3, each team opens the top card and has two minutes to work together and solve. When you say, “3, 2, 1 Showdown!”, students lift their task card with the answer. The groups that answered correctly get a point, and the task card is quickly reviewed on the board. Repeat with all task cards. The team with the most points wins.

COMMON ERRORS Students may confuse the x- and y-axes. Students may reflect an image but forget to flip orientation. Students may confuse clockwise and counterclockwise Students may forget to keep the image in proportion with the preimage.

ASSESS Check one problem from each section in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

ACTIVITY

Chapter 9

158

In Chapter 9, we will expand upon

Volume Volume is a measure of how much 3D space a shape takes up. • We will find the volume of cylinders. • We will find the volume of cones. • We will find the volume of spheres. • We will solve real-world problems using formulas for volume.

159

Level H | 9-0 Chapter 9 | Skill Checklist

Skill 1: Naming 3D Shapes Cylinder

Cone

Write the name of the 3D shape. 1.

Sphere

cylinder

cone

sphere

I can identify 3D shapes.

out 4 correct

Skill 2: Radius and Diameter

Objective and Learning Goals

radius

Students will review skills needed for Chapter 9:

diameter

Write radius or diameter. 1.

radius

r = 8 cm

r = 12 in d = 24 in

diameter

d = 16 cm

r = 7.5 ft d = 15 ft

I can find the radius and the diameter.

out 6 correct

Find the missing radius or diameter.

The diameter is two times the radius. The radius is half the diameter.

y Naming 3D shapes y Radius and diameter y Area of a circle y Prisms y Multiplying decimals

Skill 3: Area of a Circle Find the area of the circle.

A = πr2 5m

Use 3.14 for π

78.5 cm2

4. 15 ft

Level H

10 cm

50.24 m

A = (3.14)(52) A = (3.14)(25) A = 78.5 m2

160

4m 2

out 4 correct

cylinder

Chapter 9

176.63 ft2

9 cm

254.34 cm2

I can find the area of a circle.

Skill Checklist

Lighthouse Math

Skill 1: Naming 3D Shapes

Skill 2: Radius and Diameter

Skill 3: Area of a Circle

y Review the names of the 3D shapes in the gray box. y Ask students for some identifying features of each shape. y Have students complete problems 1-4 and review the answers and strategies as a class.

y Remind students that the radius of a circle is the distance from the center of the circle to the edge of the circle. y Remind students that the diameter is the distance from one edge to the other edge passing through the center point of the circle. y Review the relationship between radius and diameter. y Have students complete problems 1-6 and review the answers and strategies as a class.

y Remind students of the formula for finding the area of a circle. y Using the example in the gray box, evaluate the area of the given circle. y Have students complete problems 1-4 and review the answers and strategies as a class.

Lighthouse MATH Level H | Teacher's Guide

Name

Skill 4: Prisms A prism is a 3D shape with two parallel bases connected by rectangular faces.

Write B on the bases and H next to the height of the prism. 1.

The height (H) of a prism stretches from one base to the other.

To find the volume of any prism, multiply the area of its base by its height.

Find the volume of the prism. 4.

2 cm

3 ft 10 cm

3 ft

8 in

3 in

8 ft

4 in

6 cm

120 cm3

48 in3

72 ft3

h b

I can find bases, heights, and volumes of prisms.

out 6 correct

Skill 5: Multiplying Decimals

out 3 correct

Lighthouse Math

Skill 4: Prisms y Remind students that a prism is a 3D shape with two bases and some rectangular faces. y Use the gray box to show students that the distance between the two bases is called the height. y Remind students that to find the volume of a prism, they first need to find the area of the base and then multiply it by the height of the prism. y Have students complete problems 1-6 and review the answers and strategies as a class.

12.35 9.8

121.03

3.41 0.08

0.2728

6.51 1.35

8.7885

I can multiply decimals.

Level H

Chapter 9

Skill Checklist

161

Evaluate.

14.3 1.25 715 2860 + 14300 17.875 ×

Skill 5: Multiplying Decimals y Remind students that when we multiply decimals, we don’t need to line up the decimal points. y Work through the example in the gray box with student input and then review the final decimal placement. y Remind students that the number of decimal places in the question is the number of decimal places in the answer. y Have students complete problems 1-3 and review the answers and strategies as a class.

Lighthouse MATH Level H | Teacher's Guide

Level H | 9-1 9-1 | Volume of Cylinders

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to use the formula for the volume of a cylinder to solve problems.

Find the area (A = πr2) of each circle. Round your answers to the nearest hundredth.

SPIRAL REVIEW

2. 7.5 cm

area = 78.5 in

Translation

Rotation

Reflection

Dilation

A figure reflects

A figure becomes bigger or

down, left, or right.

around a point.

across a line.

smaller by a scale factor.

V = πr2h V = π(2)2(10) V = π(4)(10) V = (3.14)(40) V ≈ 125.6 m3

10 m

A cylinder is a 3D shape that has two circular bases. To find the volume of a cylinder, use the formula V = πr2h, where r represents the radius of the circular bases and h represents the height of the cylinder.

If given the diameter, divide it by two to find the radius.

A P P LY Circle the formula that could be used to find the volume of each cylinder. 1.

7 in 18 in

V = π(18)2(7)

V = π(7)2(18)

12 cm 8 cm

V = π(12)2(8)

V = π(12)2(4)

C. V = π(14)2(18)

V = π(8)2(12)

D. V = π(18)2(14)

V = π(4)2(12)

Fill in the blanks to determine the volume of each cylinder. Use 3.14 for pi. Round your answers to the nearest tenth. 3.

V = πr2h V = π( 1 )2( 9 ) V = π( 1 )( 9 )

6 in

14 in

9 π V ≈ 28.3 m3

V = πr2h

V = π( 3 )2( 14 ) V = π( 9 )( 14 )

5 cm

8 cm

V = πr2h V = π( 8 )2( 5 ) V = π( 64 )( 5 )

V = 126 π

V = 320 π

V ≈ 395.6 in3

V ≈ 1,004.8 cm3

Vocabulary

Volume - the amount of space found within a 3D shape Cylinder - a 3D shape that has two circular bases

A figure spins

Volume is the amount of space found within a three-dimensional (3D) shape. It is measured in cubic units.

area = 754.39 m

L E A R N A ND C O NNE C T

PRE-LESSON WARM-UP

10 m

31 m

A figure moves up,

To find the volume of any shape, find the area of the shape’s base and multiply that value by the shape’s height.

Ask students to tell you the difference between area and volume. [Area is a measure of how much space a 2D/flat object takes up. Volume is a measure of how much space a 3D/solid object takes up.] Ask students to tell you how to find the volume of a prism. [V = area of the base x height of the prism] Ask them to use this formula to find the area of the triangular prism below. [V = 60 m3]

area = 176.63 cm

Explain what happens during each transformation. 1.

Vocabulary y Volume - the amount of space found within a 3D shape y Cylinder - a 3D shape that has two circular bases

10 in

Guiding Questions: 1. What is a prism? [A 3D shape with two parallel bases connected by rectangular faces whose length is the height of the prism] 2. What do we need to know to find the volume of a prism? [The shape of the base, how to find its area, and the height of the prism] 3. What is important to remember about bases and heights? [They are always perpendicular (at a 90° angle) to each other]

162

Level H

Chapter 9

Lesson 1

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Segueing from the pre-lesson warm up about prisms and their volumes, introduce the cylinder, a prism with a circular base. Draw a cylinder on the board. Tell students that the same volume formula applies to the cylinder because it is a prism. Ask students what information we need to know to be able to find its volume. [The area of its base and its height] Ask students what dimension, besides height, needs to be filled into the picture. [the radius of the circular base] Ask students to help you write out the formula for the volume of a cylinder. [V = πr2h] Next, draw a radius on one of the bases and write 2 m next to it. Then, label the height of the cylinder as 10 m. Ask students to calculate the volume of the cylinder using 3.14 for pi. [V = 125.6 m3] Tell students that to follow the volume formula, you can multiply the radius by itself, then multiply by the height, and then multiply by 3.14. Ask students: What would we need to do if we were given the diameter of the circular base and not the radius? [Divide the diameter by 2 to find the radius before putting it in the formula for volume.]

APPLY AND DEVELOP SKILLS (Practice) Have students work on problems 1 and 2 in the Apply section independently, then discuss the answers as a class. Next, work together to fill in the blanks for problems 3 and 4 and then have students find the final answers on their own Lighthouse MATH Level H | Teacher's Guide

Exercise | 9-1 Name Find the volume of each cylinder. Use 3.14 for pi. Round each answer to the nearest hundredth. 2.

4 cm

20 in

15 ft

11 cm

STRUGGLING LEARNERS

10 cm

Have students create a formula sheet that will be added to throughout the chapter. The first formula to be added will be V = πr2h. Have students use highlighters to find the two important dimensions. Tell them that when highlighting the radius, if they encounter a diameter, they should immediately divide it by two and write it in the diagram.

13 m 7 in 6 ft

552.64 cm3

423.9 ft3

3,077.2 in3

1,020.5 cm3

14 m

2.2 in

18 m 2m

9 cm

17 m

5 in 7 cm

226.08 m3

346.19 cm3

19.00 in3

3,176.11 m3

EARLY FINISHERS

Solve each word problem. Use 3.14 for pi. Round each answer to the nearest hundredth. 10. Harper’s Soup produces cylindrical cans of soup that have a diameter of 10 centimeters and a height of 12 centimeters. How much soup can each can hold?

Sparkle Water is designing a new CO2 gas tank in the shape of a cylinder. The tank has a radius of 4 inches and a height of 10 inches. How much CO2 gas can the tank hold when it is full? 502.4 in3

942 cm3

11. A cylindrical box at Bob’s Packaging has a radius of 3 feet and a height of 18 feet. How much could this box hold?

12. Jerry’s Juice is creating a new cylindrical container for its fruit punch. The container has a diameter of 9 inches and a height of 12 inches. What is the volume of fruit punch the container can hold when filled to the top?

508.68 ft3

763.02 in3

CH AL L ENGE 13. Find the missing height of the cylinder given the volume is 3,384.92 ft3. Use 3.14 for pi.

14 ft

V = 3,384.92 ft3

22 ft

Level H

Chapter 9

Exercise 1

Have students find cylinders around the classroom. Then, have them measure the radius (or diameter if it is easier) and the height of the cylinder. Then, they should sketch a picture of their cylinder and calculate its volume. If they are unable to find a true cylinder, have them measure a shape that is similar to a cylinder, find its volume, and estimate how the true measurement would differ based on how the shape differs from a true cylinder.

CHALLENGE AND EXPLORE

h ft

Lighthouse Math

163

and complete problem 5. Once students have a good understanding, have them work on problems 1-8 in the Exercise section independently, checking their answers with a partner. Review the answers as a class, then work together to set up the formulas for problems 9-12. Have students complete the problems on their own. Review the answers as a class, ensuring that students use the correct cubic unit.

ACTIVITY Height of the Impact: The purpose of this activity is for students to see that the volume of a cylinder is more affected by the radius of the base than by the height of the cylinder. Divide the class into groups of four. Each student in the group will need to calculate the volume of a different cylinder. Then, students will compare the volumes of each cylinder. Finally, hold a class discussion about what affects the volume of a cylinder more: changing its radius or its height. Use the following dimensions for the four cylinders: Cylinder One: radius 2, height 2 [V = 25.12] Cylinder Two: radius 6, height 2 [V = 226.08] Cylinder Three: radius 2, height 6 [V = 75.36] Cylinder Four: radius 2 height 18 [V = 226.08] Students should notice that Cylinder Two has a radius that is triple the radius of Cylinder One, but its volume is nine times as large as Cylinder One and that Cylinder Four has a height that is three times the height of Cylinder Three but its volume is only three times the volume of Cylinder Three. These results tell us that changing the radius has more effect on the volume than changing the height.

For problem 13, students will need to use the formula for the volume of a cylinder to find the missing height. Remind students that we can work backwards using inverse operations to solve for h. Also remind them that they are given a diameter, not a radius.

COMMON ERRORS Students may use the diameter instead of the radius. Students may forget to square the radius. Students may forget to multiply by the height. Students may not use a cubic unit in their final answer.

ASSESS Check problems 4, 5, and 10 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

Level H | 9-2 9-2 | Volume of Cones

PREREQUISITE SKILLS

Objective and Learning Goals

82 + 7 71

DAI LY REVI EW

y Students will be able to use the formula for the volume of a cone to solve problems.

Simplify each expression.

SPIRAL REVIEW

(52)(3) 75

72 – 16 33

102 ÷ 4 25

83 – 92

Find the volume of each cylinder. Round your answers to the nearest tenth. 1.

3 in

141.3 in3

5 in

75.4 m3

10 ft

8 ft

4. 628 ft3

12 m

1,017.4 m3

Vocabulary y Cone - a 3D shape with one circular base that extends to a single point

L E A R N A ND C O NNE C T V = 31 πr2h

A cone is a 3D shape with one circular base that extends to a single point.

Materials

10 m

To find the volume of a cone, use the formula V = 31 πr2h, where r represents the radius of the circular bases and h represents the height of the cone.

y Notecards

Then draw a cone with the same dimensions.

V ≈ 41.87 m3

22 ft

8 ft

[V = 31.4 in3]

V = 31 (3.14)(40)

The height is perpendicular to the base, not one of the slanted sides.

Circle the formula that could be used to find the volume of each cone.

Draw a cylinder on the board with a radius of 1 inch and a height of 10 inches and have students find the volume of the cylinder.

10 in

V = 31 π(4)(10)

A P P LY

PRE-LESSON WARM-UP

1 in

V = 31 π(2)2(10)

V = 31 π(6)2(2)

V = 31 π(2.5)2(4)

V = 31 π(2)2(6)

V = 31 π(5)2(4)

V = 31 π(22)2(4) V = 31 π(22)2(8)

V = 31 π(1)2(6)

V = 31 π(4)2(2.5)

V = 31 π(4)2(22)

V = 31 π(6)2(1)

V = 31 π(4)2(5)

V = 31 π(8)2(22)

Fill in the blanks to determine the volume of each cone. Use 3.14 for pi. Round your answers to the nearest tenth. V = 31 πr2h V = 1 π( 4 )2( 10 )

4. 10 ft 4 ft

3 V = 31 π( 16 )( 10 ) V = 31 160 π

V = 31 πr2h V = 1 π( 3 )2( 8 )

5. 8 in 6 in

V ≈ 167.5 ft3

3 V = 31 π( 9 )( 8 ) V = 31 72 π

V ≈ 75.4 in3

V = 31 πr2h V = 1 π( 2.5 )2( 5 )

V = 31 π(6.25)( 5 ) V = 1 31.25 π 3

V ≈ 32.7 m3

Vocabulary Cone - a 3D shape with one circular base that extends to a single point

10 in

1 in

Tell them that the volume of a cone with the same dimensions (a radius of 1 inch and a height of 10 inches) has a volume of 10.47 in3. Have students discuss first with a partner and then as a class what they think is the relationship between a cylinder’s volume and a cone’s volume and what they think the formula for the volume of a cone could be. Tell students to round the volumes and use the Guiding Questions below to help them with this process. Guiding Questions: 1. Which shape has a bigger volume? [cylinder] 2. About how many times greater is the volume of the cylinder than the volume of the cone? [3 times] 3. Why is it important that we compared shapes that have the same dimensions? [So that the only difference between the two is their shape] Lighthouse MATH Level H | Teacher's Guide

164

Level H

Chapter 9

Lesson 2

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Using the images from the pre-lesson warm-up, hold a discussion with students about how a cylinder and a cone are the same and how they are different. [Both have a circular base, but the cone only has one, while the cylinder has two. They both have a height, but the cylinder’s height goes from one circular base to the other while the cone’s height goes from the circular base to a tip. The sides of a cone come in towards each other and meet at a point.] Emphasize that the cylinder and the cone are very similar because of their circular bases, and therefore, the formula for the volume of a cone is similar to the formula for the volume of a cylinder. Write the formula for the volume of a cylinder on the board (V = πr2h) and remind students that we discovered in the warm-up that the cylinder was three times bigger than the cone with the same dimensions. Ask them what we should add to the formula for the volume of a cone to show this. [Divide by 3, or multiply by 31 ] Write the formula for the volume of a cone on the board and have students practice finding the volume using a radius of 2 m and a height of 10 m. Tell students that to follow the volume formula, you can multiply the radius by itself, then multiply by the height, then multiply by 3.14, and then divide by 3. Remind students that the height is perpendicular to the base and not one of the slanted sides.

Exercise | 9-2 Name Find the volume of each cone. Use 3.14 for pi. Round each answer to the nearest hundredth. 1.

7 in

6 ft

Remind students that multiplying by 31 is the same as dividing by 3. Have students add V = 31 πr2h to the formula sheet for the volume of a cone.

10 yd 18 yd

12 cm

3 in

STRUGGLING LEARNERS

5 cm

4 ft

65.94 in3

37.68 ft3

314 cm3

847.8 yd3

8. 5 in

EARLY FINISHERS

Have students return to problems 1-8 in the Exercise section and sketch a cylinder with the same dimensions as each cone (using the given figures to help them). Have them find the volume of the cylinders they sketched. Ask them if sketching the cylinders helped them to see why the volume of a cone is 31 the volume of a cylinder and what strategy they used to find the new volumes.

2.1 in

14 m

58.61 m3

3,014.4 cm3

23.08 in3

14.13 m3

Solve each word problem. Use 3.14 for pi. Round each answer to the nearest hundredth. 10. Planet Popcorn is selling popcorn in coneshaped paper holders. Each cone has a diameter of 8 inches and a height of 9 inches. What is the volume of popcorn one cone can hold?

Frosty’s Ice Cream is designing a new cone for their sundaes. The cone has a radius of 3 inches and a height of 7 inches. How much ice cream can the cone hold if it is filled to the top?

150.72 in3

65.94 in3

11. A science lab uses small cone-shaped containers to measure out powder. One container has a radius of 1.5 cm and a height of 5 cm. What is the volume of powder it can hold?

12. The Art Spot is designing cone-shaped molds for clay sculptures. Each mold has a diameter of 13 inches and a height of 8 inches. What is the maximum volume of clay the mold can hold?

11.78 cm3

353.77 in3

CH AL L ENGE 13. Find the missing radius of the cone if the volume is 301.44 ft3. Use 3.14 for pi. Round your answer to the nearest hundredth.

18 ft

CHALLENGE AND EXPLORE For problem 13, students will need to find the missing radius of the cone with the given height and volume. Remind students that they can work backwards using inverse operations to solve for the missing piece.

x ft

x = 4 ft

Lighthouse Math

V = 301.44 ft3

Level H

Chapter 9

Exercise 2

165

Have students work on problems 1-3 in the Apply section independently, then discuss the answers as a class. Next, work together to fill in the blanks for problems 4 and 5, have students find the final answers on their own, and complete problem 6. Once students have a good understanding, have them work on problems 1-8 in the Exercise section independently, checking their answers with a partner. Review the answers as a class, then work together to set up the formulas for problems 9-12. Have students complete the problems on their own. Review the answers as a class, ensuring that students use the correct cubic unit.

ACTIVITY Cylinder and Cone Competition: The purpose of this activity is for students to practice finding the volumes of cones and cylinders and to remember the difference between their formulas. Divide students into teams. Give each team a set of four note cards with a radius (or a diameter) and a height. When the game begins, each team will need to calculate the volume of the cylinder with that radius and height and the volume of the cone with the same radius and height. The team to finish first with all correct answers wins. At the end, hold a discussion about what strategies students used to work efficiently.

COMMON ERRORS Students may forget to multiply by 1 3 or divide by 3 when finding the volume of a cone.

ASSESS Check problems 3, 4, and 10 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 9-3 9-3 | Volume of Spheres

PREREQUISITE SKILLS

Objective and Learning Goals DAI LY REVI EW

y Students will be able to use the formula for the volume of a sphere to solve problems.

Find the circumference of each circle. Use 3.14 for pi. Round your answers to the nearest tenth. 1.

SPIRAL REVIEW

31.4 m

10 m

37.7 in

6 in

3 yd

18.8 yd

Find the volume of each cone. Round your answers to the nearest tenth. 1.

84.8 in3

9 in

Vocabulary

29.3 m3

6 in

y Sphere - a round 3D shape where every point is the same distance away from the center

47.1 ft

15 ft

314 ft3

12 ft

184.2 in3

11 in

5 ft

8 in

L E A R N A ND C O NNE C T V = 43 πr3

A sphere is a round 3D shape where every point is the same distance away from the center.

Materials y Modeling clay

Remember to cube the radius, not square it.

V = 43 π(2)3

V = 43 (3.14)(8) V = 33.49 m3

To find the volume of a sphere, use the formula V = 43 πr3.

A P P LY

PRE-LESSON WARM-UP

3. 18 m

11 m

15 m

V = 43 π(11)3

V = 43 π(18)3

V = 43 π(22)3

V = 43 π(9)2

V = 43 π(7.5)2 V = 43 π(15)3

V = 43 π(11)2

V = 43 π(18)2

V = 43 π(7.5)3

V = 43 π(22)2

V = 43 π(9)3

V = 43 π(15)2

Fill in the blanks to determine the volume of each cone. Use 3.14 for pi. Round your answers to the nearest tenth. V = 43 πr3

4. 3m

V = 43 πr3

V = 43 π( 3 )3 V = 4 π( 27 )

10 in

V = 43 π( 5 )3 V = 4 π( 125 ) 3

V ≈ 113.1 m3

V = 43 πr3

6. 12 ft

Guiding Questions: 1. How does a sphere differ from a cylinder and cone? [It’s all round; it doesn’t have a base or a height, it doesn’t have a flat top or bottom, it doesn’t have a point at the top.] 2. What dimension does a sphere share with a cylinder and cone? [radius/diameter] 3. What are some spheres you have seen in the real world? [ball, globe, pearls]

Ask students to define or describe a sphere. Have students give examples spheres from the real world and talk about what makes them the same or different from the shapes we have already studied.

Circle the formula that could be used to find the volume of each cone.

V ≈ 523.3 in3

V = 43 π( 6 )3 V = 4 π( 216 ) 3

V ≈ 904.3 ft3

Vocabulary Sphere - a round 3D shape where every point is the same distance away from the center

166

Level H

Chapter 9

Lesson 3

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect)

Tell students that a sphere is another 3D shape that is related to a circle because it has a radius/diameter but unlike a cylinder and a cone, which both have heights, the points on the edge of a sphere are all equidistant, or the same distance, from the center of the shape. Tell students that to find the volume of a sphere, we use the formula V = 43 πr3h. Point out to students that this formula differs from the formula for a cone because it has an improper fraction ( 43 instead of 31 ), and the radius is cubed rather than squared. Tell students that to follow the volume formula, you can multiply the radius by itself and then multiply it by itself again. Then, multiply by 3.14. Then, multiply by 4 and divide by 3. Have students practice using this formula to find the volume of a sphere with a radius of 2 meters.

APPLY AND DEVELOP SKILLS (Practice) Have students work on problems 1-3 in the Apply section independently, then discuss the answers as a class. Next, work together to fill in the blanks for problems 4 and 5, have students find the final answers on their own, and complete problem 6. Once students have a good understanding, have them work on problems 1-8 in the Exercise section independently, checking their answers with a partner. Review the answers as a class, then work together to set up the formulas for problems 9-12. Have students complete the problems on their own. Review the answers as a class, ensuring that students use the correct cubic unit. Lighthouse MATH Level H | Teacher's Guide

Exercise | 9-3 Name Find the volume of each sphere. Use 3.14 for pi. Round each answer to the nearest hundredth. 1.

STRUGGLING LEARNERS

1 in 7m

6 ft

113.04 ft3

4.19 in3

1,436.03 m3

33.49 yd3

8. 10 in

5 cm

Have students add the formula for the volume of a sphere to their anchor chart. Allow them to work with a list of the perfect cubes to help with calculations. Remind students that multiplying by 43 is the same as multiplying by 4 and dividing by 3.

4 yd

2.2 cm

18 m

523.33 cm3

3,052.08 m3

4,186.67 in3

EARLY FINISHERS

5.57 cm3

Have students return to problems 1-3 in the Apply section and calculate the volumes of the given spheres.

Solve each word problem. Use 3.14 for pi. Round each answer to the nearest hundredth. 10. Amazing Aquarium Supplies sells glass fishbowls shaped like spheres. One bowl has a diameter of 20 inches. How much water can it hold if filled to the top?

Snowy Days Company builds snow globes with a spherical glass dome. If the radius of the glass dome is 4 inches, what is the volume of the snow globe's dome?

267.95 in3

11. Cheers Chocolates creates truffles in the shape of spheres. Each truffle has a radius of 2 centimeters. What is the volume of one truffle?

12. Tom sells decorative balloons shaped like spheres. One balloon has a diameter of 14 inches. What is the balloon's volume when blown up?

33.49 cm3

1,436.03 in3

CH AL L ENGE 13. Find the radius of the sphere if the volume is 2,143.57 ft3.

14. Find the diameter of the sphere if the volume is 904.32 yd3.

8 ft

Lighthouse Math

CHALLENGE AND EXPLORE

4,186.67 in3

For problems 13 and 14, students will need to work backwards to find the radius or diameter of a sphere with a given volume. Allow students to reference a list of perfect cubes to aid in their calculations if needed.

12 yd

Level H

Chapter 9

Exercise 3

167

Model It: The purpose of this activity is for students to compare the three 3D shapes related to circles and how their volumes are affected by their radius. Give students three slices of different colored modeling clay that are exactly the same size. Have them shape one slice into a sphere, one slice into a cylinder, and one slice into a cone. Ask them to compare the diameters of each shape. They will need to cut the sphere down the center in order measure its diameter. Hold a discussion about which diameter is the biggest and which is the smallest and why they think that is. [The sphere spreads out its volume evenly, so it’s more “efficient.” It can hold the most volume for the least diameter. The cone must have a wider diameter to hold the same amount of volume due to all of the missing space above its base where it comes to a point. The cylinder is in between the two extremes.]

COMMON ERRORS Students may square the radius instead of cubing it. Students may multiply by 31 instead of 43 .

ASSESS Check problems 2, 3, and 11 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

ACTIVITY

Level H | 9-4 9-4 | Solving Real-World Problems with Volume

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will be able to solve real-world problems involving the volumes of cylinders, cones, and spheres.

Solve each word problem.

SPIRAL REVIEW

A shipping box is 12 inches long, 8 inches wide, and 6 inches tall. 3 What is the volume of the box? 576 in

A brick mold has dimensions of 10 inches by 4 inches by 3 inches. What is the volume of one brick?

120 in3

Find the volume of each shape. 8m

1. 15 m

2. 753.6 m3

11 in

103.62 in3

4. 33.49 m3

10 ft

3 in

523.33 ft3

Materials y Dice

L E A R N A ND C O NNE C T When solving real-world volume problems, look for shape-related words to determine the volume formula you need to use. After this, look for important information like radius, diameter, and height.

PRE-LESSON WARM-UP

Real-World Problem

Create a table on the board with the names of each shape from the chapter in its own column. Have students help you create a list of real-life items of each shape.

y Cone: ice cream cone, traffic cone, party hat, funnel, teepee, megaphone, sharpened pencil tip, mountain y Cylinder: cans of food, cup or glass, paper towel tube, battery, candle, pipe, rolling pin, chalk, column, glue stick

Guiding Questions: 1. Are all the items we listed true spheres, cones, or cylinders? [Answers may vary. Have students justify their answers.] 2. Why might we want to find the volume of any of the items we listed? [To figure out how much material needs to be used to make the item; to figure out much food, drink, or other substance the item can hold]

y Sphere: ball, globe, pearl, marble, bubbles, beads, planets, oranges

Formula

A company is designing a new water tank in the shape of a cylinder. The tank has a radius of 4 feet and a height of 10 feet. How much water can the tank hold when it is full?

Volume of a cylinder

An ice cream shop is designing a new cone for their sundaes. The cone has a radius of 3 inches and a height of 7 inches. How much ice cream can the cone hold if it is filled to the top?

Volume of a cone

A ball has a diameter of 10 inches. How much air is needed to completely fill the ball?

Volume of a sphere

Solution

V = π(4)2(10) V = π(16)(10)

V = πr2h

V ≈ 502.4 ft3 V = 31 π(3)2(7) V = 31 π(9)(7)

1 2 πr h 3

V ≈ 65.94 in3 V = 43 π(5)3

V = 43 π(125)

4 3 πr 3

V ≈ 523.3 in3

A P P LY Match each problem with the equation used to solve it.

168

Sugar Unlimited is making a giant candy in the shape of a sphere with a diameter of 6 inches. How much candy mixture is needed to create one candy?

V = πr2h

Happy Camper Company is designing a metal canister to boil water over a fire. The canister is in the shape of a cylinder with a radius of 5 inches and a height of 8 inches. How much water can the canister hold?

V = 31 πr2h

At a carnival, a new game involves tossing rings into a funnel-shaped target. The funnel is a cone with a radius of 6 inches and a height of 9 inches. What is the volume of space inside the funnel?

V = 43 πr3

Level H

Chapter 9

Lesson 4

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Tell students that as we discussed in the pre-lesson warm-up, it can be helpful to know the volume of a 3D shape in order to figure out how much material needs to be used to make it or to figure out how much a substance the item can hold. Tell students that when solving volume word problems, they will need to use key words from the chapter to figure out which formula to use. Ask students to brainstorm what those key words could be. [cylinder, cone, sphere, radius, diameter, height] Read each word problem from the Learn and Connect section aloud for students and ask them what shape the problem is about, which formula we need to use, and what is the radius and (if applicable) height. Have students work in pairs to fill in the formula for each problem and solve it. Then, go through the process together. Conclude by emphasizing that each answer must have the proper unit for volume (a cubed unit).

APPLY AND DEVELOP SKILLS (Practice) Have students work with a partner to solve problems 1-3 in the Apply section. Review the answers as a class, asking students to name key words that helped them. Next, work together as a class to set up problems 1 and 2 in the Exercise section. Have students complete the problem on their own to find the answers, and then continue to work independently on problems 3-10. Review the answers and strategies used as a class to ensure understanding. Lighthouse MATH Level H | Teacher's Guide

Exercise | 9-4 Name Solve each word problem. Round your answers to the nearest hundredth. 2.

Sammy’s Soups is manufacturing soup cans shaped like cylinders. Each can has a radius of 3 inches and a height of 6 inches. How much soup can one can hold?

A fuel company is making metal drums to store oil. Each drum is in the shape of a cylinder with a diameter of 8 feet and a height of 9 feet. How much oil can one drum hold?

Cosmic Creations makes globes shaped like spheres. One of their best sellers has a diameter of 12 centimeters. What is the volume of that globe?

452.16 ft3

904.32 cm3

A ball is shaped like a sphere with a diameter of 9 inches. How much air is needed to completely fill the ball?

Smoothie World uses cups shaped like cylinders. One size has a radius of 2 inches and a height of 10 inches. How much smoothie does this cup hold when full?

381.51 in3

EARLY FINISHERS

125.6 in3

Pete’s Pizza Shop uses a cone-shaped container to hold their pizza sauce. The container has a radius of 5 inches and a height of 7 inches. How much sauce fits in the container?

A small ball has a diameter of 2.7 inches. How much air is needed to completely fill the ball?

183.17 in3

Have students create their own word problems, one for each shape, for a partner to solve.

10.3 in3

CHALLENGE AND EXPLORE

10. A hardware store sells paint in cans shaped like cylinders. One can has a radius of 5 inches and a height of 12 inches. How much paint can the can hold when full?

A bakery uses cone-shaped pastry molds with a radius of 1.5 inches and a height of 5 inches. How much filling fits in one pastry mold?

11.78 in

942 in

CH AL L ENGE 11. A party hat shaped like a cone has a radius of 4 inches and a volume of 100 cubic inches. What is the height of the hat? Round your answer to the nearest whole number.

12. A ball has a volume of 113.04 cubic inches. What is the diameter of the ball?

6 in.

Lighthouse Math

Students should use the formula sheet they have compiled throughout the chapter as they work through each problem. Have students highlight key words, such as the shape name, as well as radius, diameter, and height. Encourage students to immediately divide diameters by 2 so as not to forget to use the radius when finding the volume.

32.71 in

169.56 in

STRUGGLING LEARNERS

Slush Rush sells snow cones in paper cups shaped like cones. Each cup has a radius of 2.5 inches and a height of 5 inches. What is the volume of ice needed to fill one snow cone cup to the top?

For problems 11 and 12, students will need to use their volume formulas to work backwards and find a missing height and diameter. Remind students that the formulas for volume use the length of the radius, not the diameter.

6 in.

Level H

Chapter 9

Exercise 4

169

Shape Calculation Game: The purpose of this activity is to have students practice solving problems for all the different shapes from the chapter together with their peers. Divide students into pairs and give each a six-sided die. One student will roll the die three times. The first roll will tell the student the shape, the second roll will tell them the radius, and the third roll will tell them the height (if applicable). Rolling 1 or 2 on the first roll means cylinder, 3 or 4 means cone, and 5 or 6 means sphere. The number rolled on the second roll is the length of the radius. The number rolled on the third roll is the height. Students will not roll a third time if they roll 5 or 6 on the first roll. The second student will write down the formula needed to calculate the volume of the shape, sketch a picture of the shape, and fill in the radius and height. Then, the students will calculate the volume of the shape together. When they finish, they will switch rolls.

COMMON ERRORS Students may confuse the formulas for the volume of a cylinder, cone, and sphere. Students may use a diameter instead of a radius. Students may forget to multiply by the height.

ASSESS Check problems 5, 6, and 7 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

ACTIVITY

Level H | 9-5 9-5 | Review

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will review the concepts covered in Chapter 9.

Cylinder

Cone

PRE-LESSON WARM-UP

Sphere

h h

Sketch the picture of the ice cream cone below on the board and ask students to find the total amount of ice cream (in cm3), assuming that the cone is filled completely, that the shape of the ice cream above the cone is a perfect hemisphere (half-sphere), and that both shapes have a diameter of 8 centimeters.

V = πr2h

1 2 πr h 3

4 3 πr 3

A P P LY Match each problem with its solution.

6.5 cm

[V = 242.83 cm3]

Guiding Questions: 1. How do we find the volume of a hemisphere? [Use the formula for the volume of a sphere, then divide by 2.] 2. How do we find the total volume in both shapes? [Add them together.]

8 cm

170

Al’s Party Supplies sells large helium balloons that are spherical with a diameter of 14 inches. How much helium is needed to fill one balloon completely?

282.6 in3

Fifi’s Frozen Yogurt serves yogurt in cone-shaped cups with a radius of 2 inches and a height of 6 inches. What is the volume of yogurt that fits in one cup?

14.13 in3

Josie’s Juice Bar packages juice in metal cans that are shaped like cylinders. Each can has a radius of 3 inches and a height of 10 inches. How much juice does one can hold when full?

48,833.28 in3

A gardener collects rainwater in a barrel shaped like a cylinder. The barrel has a radius of 18 inches and a height of 48 inches. How much water can the barrel hold when full?

1,436.03 in3

A baker fills a piping bag shaped like a cone with frosting. The bag has a radius of 2 inches and a height of 9 inches. How much frosting can the bag hold when it is full?

37.68 in3

A single scoop of ice cream is shaped like a sphere with a diameter of 3 inches. What is the volume of one scoop of ice cream?

25.12 in3

Level H

Chapter 9

Lesson 5

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect)

Have the students work in groups to create a poster displaying the three shapes and their volume formulas that were studied in this chapter. Students should draw a cylinder, a cone, and a sphere, and after writing their formulas next to them, they should embellish their drawings to make them look like a real-life cylinder, cone, and sphere. Next, they should work together to think of a real-world problem that could apply to each shape. They should write their problems under each shape and then give their poster to another group to solve. When all the groups finish, they should get their original posters back, check the work done by the other group, and present their posters to class.

APPLY AND DEVELOP SKILLS (Practice) Have students work with a partner to solve problems 1-6 in the Apply section, then review the answers as a class and address any questions that came up along the way. Next, have students work independently on problems 1-12 in the Exercise section and go over the answers and strategies used to ensure understanding. Finally, have students work with a partner on problems 13-18, where students analyze the three shapes they have studied, each with the same sized radius, to see which has the largest volume when their other dimensions are held constant. They will see that the cylinder has the largest volume, then the sphere, and then the cone. Have a class discussion as to why this may be. Lighthouse MATH Level H | Teacher's Guide

Exercise | 9-5 Name Find the volume of each cylinder. Use 3.14 for pi. Round each answer to the nearest hundredth. 1.

7 cm

13 cm

4. 5m

12 ft

STRUGGLING LEARNERS

17 cm 22 cm

Students should use their anchor charts throughout the lesson as well as a perfect squares and perfect cubes chart to assist with their calculations.

20 ft

2,000.18 cm3

251.2 m3

2,260.8 ft3

4,991.03 cm3

Find the volume of each cone. Use 3.14 for pi. Round each answer to the nearest hundredth. 5.

1.5 ft

16 m

6 in

2 in

EARLY FINISHERS

15 in

3 ft

25.12 in

Have students create a word problem to match one problem from each of the first three sections on the Exercise page. Then, have them trade problems with a friend and solve.

5 in

602.88 m

7.07 ft

98.13 in3

Find the volume of each sphere. Use 3.14 for pi. Round each answer to the nearest hundredth. 9.

10.

11.

12.

14 ft

267.95 m3

10 m

1 in

523.33 m3

4.19 in3

CHALLENGE AND EXPLORE

1,436.03 ft3

Find the volume of each figure. Use 3.14 for pi and round your answers to the nearest tenth. Notice that the shapes have the same radiuses and heights. Then, fill in the blanks to compare the shapes using their volumes. 14.

15.

13.

6m 6m 3m

V = 169.56 m3

V = 56.52 m3

V = 113.04 m3

16. The

cone

can hold the least volume. It can hold 31 the volume of the

cylinder .

17. The

sphere

can hold the second least volume. It can hold more than the

cone

the 18. The

but less than

cylinder . cylinder

Lighthouse Math

can hold the most volume. It can hold three times the volume of the

Level H

Chapter 9

Exercise 5

cone

Problems 13-18 in the Exercise section can be compared to the Activity done in Chapter 9-3 (Model It). Discuss with students how, when the radius and height were held constant in problems 13-15, the cylinder held the most volume, then the sphere, then the cone. But in Model It, the volume was held constant, and we saw that the cone had the largest diameter, the cylinder was in the middle, and the sphere had the smallest diameter. Discuss why this is, how diameter (or radius) is related to the volume of each shape, and how this affects each formula.

171

Shape Calculation Game Take Two: For extra practice, have students do the Shape Calculation Game from Chapter 9-4. This activity can be modified for students who are ready for a challenge by having them roll more than once to create a compound shape. Then, they should sketch the shape they have created and find its volume. Students can play around with their compound shape to make it reflect something in the real world. For example, if a sphere and a cylinder are rolled, they can make them look like a ball balancing on an upturned bucket.

COMMON ERRORS Students may square the radius instead of cubing it when finding the volume of a sphere. Students may forget to divide by three when finding the volume of a cone. Students may not use a cubic unit in their answer.

ASSESS Check the odd-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

ACTIVITY

Chapter 10

172

In Chapter 10, we will study

Scatter Plots, Association, and Probability Organizing data can help us to understand and interpret that data and use it to make predictions. • We will draw scatter plots and look for associations between variables. • We will draw lines of best fit and use them to make predictions. • We will put information into two-way tables. • We will interpret two-way tables to look for associations.

173

Level H | 10-0 Chapter 10 | Skill Checklist

Skill 1: Plotting Points from a Table x

Plot the points from the table on the graph. 10

14 12

6 4 2 0

2 0

I can plot points from a table on a graph.

out 5 correct

Skill 2: Writing Equations in Slope-Intercept Form

Objective and Learning Goals

y = mx + b m = slope b = y-intercept

Write the equation.

Students will review skills needed for Chapter 10:

-5 -10

3. 5

-5

y = 2x - 4

-5

y = 4x + 4

y = -3x +2

y = -2x + 6 I can write an equation from a graph.

out 3 correct

Skill 3: Positive and Negative Slopes Positive slope

As the x-values increase, so do the y-values.

Negative slope As the x-values increase, the y-values decrease.

Write if the slope is positive or negative. 0

174

positive

out 3 correct

-5

y Plotting points from a table y Writing equations in slope-intercept form y Positive and negative slopes y Organizing a sample space in a table y Percentages y Probability

5 -5

-10

Level H

Chapter 10

negative

positive

I can identify positive and negative slopes.

Skill Checklist

Lighthouse Math

Skill 1: Plotting Points From a Table

Skill 2: Writing Equations in Slope-Intercept Form

Skill 3: Positive and Negative Slopes

y Remind students that the values from an x/y table can be changed into coordinates that can be plotted on a graph. y Ask students which value comes first in a coordinate, the x or the y. [x first, then y] y Review the table and graph in the gray box. y Have students complete problems 1-5 and review the answers and strategies as a class.

y Have students tell you what each part of the formula y = mx + b stands form. [m is the slope, b is the y-intercept] y Remind students that we can find the slope from a graph by looking at the rise and the run of the line. The slope is the rise divided by the run. y Remind students that the y-intercept is the y-value at the point where the line crosses the y-axis. y Have students complete problems 1-3 and review the answers and strategies as a class.

y Remind students that lines with positive slopes increase from left to right because as the x-values increase, the y-values also increase. y Remind students that lines with negative slopes decrease from left to right because as the x-values increase, the y-values decrease. y Have students say one example for each kind of slope, using the graphs in the gray box as a reference. y Have students complete problems 1-3 and review the answers and strategies as a class.

Lighthouse MATH Level H | Teacher's Guide

Name

Skill 4: Organizing a Sample Space in a Table Jenny flips a coin with heads (H) and tails (T) and then rolls a six-sided die. She lists the possible outcome combinations in the table. 1

H4 H5

Create a table for each sample space. 1.

12 possible outcomes

An ice cream shop sells vanilla, chocolate, strawberry, and pecan ice cream. Customers can choose from a cup, a sugar cone, or a waffle cone.

CV CC CS CP

ScV ScC ScS ScP

Wc WcV WcC WcS WcP

A cafe gives you a choice of wheat bread or rye bread and four different toppings: cheese, hummus, avocado, and egg.

W WC WH WA WE R

RC RH RA RE

I can create a table for a sample space.

out 2 correct

Skill 5: Percentages Find the percent. Round to the nearest hundredth.

8 out of 54 students 8 54 0.1481 14.81%

14 out of 20 apples are red.

25 out of 80 students wear glasses. 31.25%

100 out of 120 beans are spotted. 83.33%

12 out of 94 books are missing. 12.77%

70%

I can find percents to the nearest hundredth.

out 4 correct

As likely as not

Likely

1 4

1 2

3 4

0.25

0.5

0.75

25%

50%

75%

Circle the likelihood of the event happening. 1.

It has rained 60% of the days in April. It is unlikely likely that it will rain during the rest of April.

He makes 80% of the shots he takes. He is unlikely likely to make the shot.

35% of people are wearing white. It is unlikely that a person wearing white will be picked.

5% of tickets are winners. It is unlikely that the next ticket will be a winner.

likely

I can find the likelihood of an event happening.

out 4 correct

Lighthouse Math

likely

Level H

Chapter 10

Skill Checklist

175

Skill 4: Organizing Sample Space in a Table

Skill 5: Percentages

Skill 6: Probability

y Tell students that in a situation with more than one variable, we can show and organize possible outcomes using a table. y Use the situation in the gray box in which a coin is flipped and a six-sided die is rolled as an example for this concept. y Ask students what the possible outcomes are for flipping a coin [heads or tails] and for rolling a six-sided die [1, 2, 3, 4, 5, or 6], and show them how to write these outcomes down the left side and across the top of the table. y Show students that each box in the table combines two outcomes together to show a possible compound outcome. y Have students complete problems 1-2 and review the answers and strategies as a class.

y Remind students that a percentage is a ratio of a number compared to 100. y Use the example ratio in the gray box to change from a ratio to a fraction to a decimal (using division) to a percent (by multiplying by 100). y Have students complete problems 1-4 and review the answers and strategies as a class.

y Remind students that probability is the theoretical chance that an event will occur, and it can be written as a percentage, a fraction or a decimal. y Use the graphic in the gray box to remind students of which probabilities show that an event is likely to occur and which show that an event is unlikely to occur. y Have students complete problems 1-4 and review the answers and strategies as a class.

Lighthouse MATH Level H | Teacher's Guide

Unlikely

Skill 6: Probability

Level H | 10-1 10-1 | Understanding Scatter Plots 5

PREREQUISITE SKILLS

y Students will be able to construct and interpret scatter plots to describe patterns of association between two quantities.

DAI LY REVI EW

Objective and Learning Goals

Graph each point from the table on the coordinate plane. 1.

SPIRAL REVIEW

Input

-2

Output

-2

-5

5 -5

Find the volume of each shape. Use 3.14 for pi. Round your answer to the nearest tenth. 6m

8 in

197.8 m3

134 in3

9 ft

3,052.1 ft3

4 in

Vocabulary L E A R N A ND C O NNE C T

y Scatter plot - a graph that shows the relationship between two sets of data y Association - the relationship between two sets of data y Positive association - as x increases, y increases. y Negative association - as x increases, y decreases. y No association - no obvious pattern

A scatter plot is a graph that shows the relationship between two sets of data. It shows one dot for each piece of data. When you look at a scatter plot, you can see the association. This is the pattern you see (or don’t see) between two sets of numbers. Positive Association As x increases, y increases.

Guiding Questions: 1. How did you choose to match up your pairs? [Each outcome is related to its scenario.] 2. Why might we want to take data on one of the scenarios listed? [To see how the two variables are connected and to help us make predications or prepare for the future. For example, tracking the number of pages read and the time spent reading can tell us the reading speed. Finding the number of days of rainfall and the number of inches of water collected can tell us the average amount of rainfall over time, etc.]

Lighthouse MATH Level H | Teacher's Guide

Create two columns on the board. Column 1 is titled Scenario and should have the following information: Number of pages read, number of days with rainfall, time riding a bike, minutes of exercise, number of tickets sold, players’ jersey number. Column 2 is titled Outcome and should have the following information: Calories burned, inches of water collected, player height, time spent reading, distance traveled, money earned. Have students work in pairs to match up each scenario with its outcome. Then, discuss the Guiding Questions below.

Test Score

Test Score Week

Week

As the weeks increase, the test scores increase.

y Graph paper y Index cards

No Association No obvious pattern

Test Score

Materials

PRE-LESSON WARM-UP

Negative Association As x increases, y decreases.

Week

As the weeks increase, the test scores decrease.

As the weeks increase, the test scores are random.

A P P LY Label the association that describes the data graphed in each scatter plot. 1.

positive association

negative association

no association

Vocabulary Scatter plot - a graph that shows the relationship between two sets of data Association - the relationship between two sets of data

176

Level H

Chapter 10

Lesson 1

Positive association - as x increases, y increases. Negative association - as x increases, y decreases. No association - no obvious pattern

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Begin the lesson by drawing a simple table with x- and y-values on the board. Review with students that when plotting points on a graph, the x-value represents the input, and the y-value represents the output. Explain that when we want to represent a scenario visually and track how the variables change together, we can use a scatter plot. Explain that a scatterplot is a visualization of the relationship between two numerical sets of data. Have students look at the three different scatter plots in the Learn and Connect section. Explain that each one shows a different relationship between the x-coordinate (number of weeks) and the y-coordinate (test scores). Ordered pairs were collected and then plotted as dots on a graph. Ask students what they notice about the collection of dots on each graph. [The dots on the graph on the left are grouped together forming an upward slope, the dots on the graph in the middle are grouped together forming a downward slope, and the dots on the graph on the right are not grouped together but are all over the place.] Tell students that sometimes on a scatter plot, a pattern, or association, can begin to appear as more data is added. Review the three types of associations, discussing what each means in their context: positive (as the weeks go on, the test scores increase), negative (as the weeks go on, the test scores decrease), and no association (no clear connection between the number of weeks and the test scores). For positive and negative associations, prompt students to connect the patterns they see to positive and negative slopes, reinforcing how scatter plots build on prior graphing knowledge.

Exercise | 10-1 Name Graph each point to create a scatter plot for each table. Then, fill in the blanks to describe the association. 100

Time Studied (hours)

Test Score (out of 100)

STRUGGLING LEARNERS

The scatter plot has a

Test Score

positive association.

Have students draw an anchor chart with what positive, negative, and no association graphs will look like for use during this chapter.

As the time spent

40 20 2

studying increases , the test score increases .

Time Studied (hours)

$30,000

Year

2025

2026

2027

2028

2029

Car’s $24,500 $21,000 $19,500 $17,500 $15,000 Value

The scatter plot has a negative association.

$24,000

Car’s Value ($)

$18,000

EARLY FINISHERS

As the time in years increases , the car’s

$12,000 $6,000 '25 '26 '27 '28 '29

Have students come up with their own scenario and ordered pairs to plot, and have them trade with a partner who will then plot and name the association the graph shows.

value decreases .

Year

Heights of 8th graders (inches)

# of Friends

The scatter plot has

association.

As the height increases ,

the number of friends

2 60

has

CHALLENGE AND EXPLORE

pattern.

Height (inches)

4.0

GPA

3.9

3.8

3.6

3.4

3.3

3.1

2.6

The scatter plot has a negative association.

3.2 2.4

As the number of absences increases ,

1.6 0.8 4

# of absences

GPA

the GPA decreases .

# of absences

CH AL L ENGE 5.

Describe the association between the temperature outside and the number of ice creams sold.

Describe two variables that have no association.

Positive association; as the temperature goes up,

Answers will vary. Sample answer: The time a

more ice creams are sold.

person wakes up in the morning and their height

Lighthouse Math

Level H

Chapter 10

Exercise 1

For problem 5, prompt students to find the association by asking, “As the temperature increases, do you think that more or less people will buy ice cream?” For problem 6, have students ensure that the two variables they suggest are not related to one another.

177

Play a quick game of Showdown using the problems in the Apply section. Have students observe the scatter plot, then write which association it is on a piece of paper. When “3,2,1, showdown” is called, students raise their answer. Scan the room to see which students may need clarification. Work on problem 1 in the Exercise section together. Next, have students work in pairs to complete problems 2-4. Then pull the class back together to review the answers and strategies used.

ACTIVITY Build the Scatter Plot: The purpose of this activity is for students to create a scatter plot and determine its association in a hands-on way. Prepare sets of index cards with one ordered pair on each card. Divide the class into groups. Give each group graph paper and a set of index cards. Have students work together to plot all the points. Then, have students label their graph with the type of association: positive, negative, or no association. Next, ask students to think of a scenario that depicts their scatter plot. Have them label the x- and y-axes and give their graph a title. Finally, have groups share their scatter plot and scenario with the class.

COMMON ERRORS Students may confuse no association and negative association. Students may confuse the input and output when plotting data on a graph.

ASSESS Check problem 4 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 10-2 10-2 | Modeling Linear Associations

PREREQUISITE SKILLS

y Students will be able to describe patterns in scatter plots and draw a line of best fit.

DAI LY REVI EW

Objective and Learning Goals

Write a linear equation in slope-intercept form given each piece of information.

SPIRAL REVIEW

slope = 21 y-intercept = -3

slope = 6 y-intercept = 4

y = 2x − 3 y = 6x + 4

slope = -5 y-intercept = 0

y = -5x

slope = 43 y-intercept = 9

y = 4x + 9

State whether each scatter plot displays a positive, negative, or no association. 2.

Vocabulary

no association

y Line of best fit - a line drawn on a scatter plot that is close to or passes through most of the points

negative association

positive association

A line of best fit is a line you can draw on a scatter plot that is close to or passes through most of the points. Lines of best fit can help you make predictions.

y Task cards y Ruler

You can write an equation for a line of best fit.

Monty is renovating his backyard. The scatter plot shows how his available money decreases over several months as he spends on the project.

Slope =

14 − 12 2 = = -2 3−4 -1

Money (thousands)

PRE-LESSON WARM-UP

y = -2x + 20

(3, 14) line of best fit

(4, 12)

y-intercept = 20

The slope of -2 represents the amount of money Monty spends monthly. The y-intercept of 20 shows that he started with $20,000.

Use the equation to make predictions.

y = -2(9) + 20 4

y = -18 + 20 2

After 9 months, Monty might only have about $2,000 left.

y=2

Guiding Questions: 1. How do we find the slope of a line? [change in y divided by change in x] 2. How do we find the y-intercept of a line? [where it crosses the y-axis, what the y-coordinate is when x = 0.] 3. What is slope-intercept form? [y = mx + b, where m is the slope, and b is the y-intercept]

positive association

L E A R N A ND C O NNE C T

Materials

Have students write the equation of a line that passes through the points (0, 5) and (2, 1). If needed, write the formula for finding slope on the board as a reminder for students. [y = -2x + 5] Tell students that today, we will be using this skill to analyze data on scatterplots and make predictions.

A P P LY Circle the scatter plot with the correct line of best fit. 1.

102

100

102

100

102 100

94 0

94 5

Vocabulary Line of best fit - a line drawn on a scatter plot that is close to or passes through most of the points

178

Level H

Chapter 10

Lesson 2

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect)

Have students observe the graph in the Learn and Connect section. Explain that the graph shows the amount of money that Monty has in his bank account over several months, during which time he is doing renovations on his backyard. Ask students: What is happening to the amount of money he has over time, and why do you think that is? [It is decreasing because he is spending it on his renovation project.] Tell students that we can use this data to predict how much money Monty will have left after nine months. We can show it visually by creating a line of best fit. Explain that a line of best fit is a straight line that we can draw through the center of the data points on graphs with a positive or negative association. Say: Use a ruler to draw a line of best fit, making sure that the amount of points above and below the line are about the same. The line does not have to pass through all the points, but it needs to pass through the middle of the data. Next, tell students that we can write an equation of the line of best fit in slope-intercept form. Have students use the points (3,14) and (4,12) to find the slope of this line. Tell students that to find the slope, they can use any two points on the line that are easiest to read, including the y-intercept. Ask: What does a slope represent? [how much the y-value changes when x is increased by one] Ask: What does the slope -2 tell us? [Monty spends $2,000 a month.] Next, have students find the y-intercept. Ask: What does the y-intercept represent on a graph? [the starting value] What does the y-intercept represent in this problem? [How much money Monty started out with] Tell students that we can input any number of weeks in the place of x to predict the amount of money (y) that Monty will have at that time. Ask students to find how much money Monty might have after 9 months. [$2,000] Be sure to explain that Lighthouse MATH Level H | Teacher's Guide

Exercise | 10-2 Name Draw the line of best fit. The number of hours students played games instead of studying and test scores

Test Score

100 98 96 94 10

The number of salespeople at a car dealership and the total money earned Dollars (hundred thousands)

102

Have students write a checklist for themselves: Identify the pattern/ association, find the line of best fit, find the slope, find the y-intercept, write in y = mx + b form, and predict. Have students circle two points on the line of best fit as well as the y-intercept.

8 6 4 2

Number of Salespeople

Hours Spent Playing

STRUGGLING LEARNERS

The number of workers on construction sites and the number of houses built Number of Houses

Number of Workers

Write an equation for each line of best fit. Then, answer the questions. 8th-grade Test Scores

4. 100

EARLY FINISHERS

A. What does the slope mean? The test scores increase by 5 points for every 1 hour of studying.

Test Score

y = 5x + 50

Equation:

Have students go back to problems 4 and 5 in the Exercise section and make two new predictions given the data.

B. What does the y-intercept mean?

The score a student received with 0 hours of studying

20 2

Hours of Study

C. Predict the score a student would get with 10 hours of studying. 100%

# of Healthy Flowers

Number of Flowers After Days with No Water

Equation:

CHALLENGE AND EXPLORE

y = -2x + 40

A. What does the slope mean? The number of healthy flowers decreases by 2 each day without water.

50 40

B. What does the y-intercept mean?

There were 40 healthy flowers at the start, before any days passed.

20 10 2

# of Days

C. Predict the number of healthy flowers after 20 days without watering. 0 healthy flowers

CH AL L ENGE 6.

Would a scatterplot showing the relationship between the price of lemonade and the amount of cups sold likely have a positive or negative slope? Explain.

For problem 6, hold a discussion about what happens when prices go up and what happens when prices go down. Do consumers buy more or less? Then have students apply this information to the problem to come to the conclusion that the more expensive the lemonade is, the less cups will be sold.

Negative. As the price would go up, people would buy less because they would not want to spend more. Since this is a negative association, it would have a negative slope.

Lighthouse Math

Level H

Chapter 10

Exercise 2

179

a prediction is not an answer for sure. It is just an educated guess, so we write our answer as “about $2,000.”

Have students complete the problem in the Apply section independently, and discuss the answer as a class. Next, have students work independently on problems 1-3 in the Exercise section. Review the answers as a class to ensure understanding. For problems 4 and 5, have students work in pairs and review the answers and strategies used to solve.

ACTIVITY Beat the Clock: The purpose of this activity is for students to practice drawing lines of best fit. Give each student a card with a scatterplot with a positive or negative association on it. Have students place the end of their ruler where they think the y-intercept should be. Then, have them adjust their ruler so that it goes through the center of the data. They should count how many points fall above the top edge of the ruler. Then keeping the top edge of the ruler where it is, they can flip the ruler upside down to count the number of points that fall below this edge. If the amount of points are about the same, and the line seems to go through the center of the data, instruct students to draw in their line of best fit. Next, have them find the equation of their line using the y-intercept and the slope formula. They should use two points that they can find on their line of best fit to find the slope. Students should trade their card with a partner to check their work.

COMMON ERRORS Students may believe that the line of best fit should go through all the data points. Students may think that the line of best fit should go through the first and last data points on the scatter plot.

ASSESS Check problems 1 and 4 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 10-3 10-3 | Constructing Two-Way Tables

PREREQUISITE SKILLS

Objective and Learning Goals

1. 2. 3.

DAI LY REVI EW

y Students will be able to construct two-way tables. Students will be able to read a two-way table.

The dot plot below shows the number of books each student in Ms. Harper’s class read. Answer the related questions.

SPIRAL REVIEW

Guiding questions: 1. What information will the school need to collect to answer their question? [How each student gets to school, what time they arrive at school, whether this time is late or not] 2. How would you organize this information so the relationship is clear and easy to understand? [In a table or a graph]

Number of Books Read

Line of Best Fit: y = 3x + 20 How much money (y) is earned after working 10 hours (x)? $50

Line of Best Fit: y = -4x + 120 How many gallons of gas are left in the tank (y) after driving 15 miles (x)? 60

L E A R N A ND C O NNE C T A group of students were asked if they are righties or lefties and which writing utensil they prefer to use. The results are shown:

Student

Hand

right

left

right

left

right

left

pen pencil pencil pen

pen

other pencil pen

other

pen

Writing Utensil

We make data like this easier to understand and interpret by putting it in a two-way table. The top row and left column show the categories. The last column and row show the totals. The numbers in the middle show the frequency of each response.

PRE-LESSON WARM-UP

All possible answers to one question The number of students who are righties and prefer a pencil

Righty

Lefty

Total 5

Pen

Pencil

Other

Total

All possible answers to the other question

Total number of students surveyed

Ask students: Why might a school want to know the relationship between how students get to school and the number of latenesses? Have students think, then discuss with a pair, and finally, share with the class. [to see if specific modes of transportation cause students to come to school later, to see if bus schedules or routes need to be changed] Then, discuss the Guiding Questions together as a class.

Make predictions given the line of best fit equation and x value.

Vocabulary y Two-way table - table used to organize two separate pieces of data y Categories - topics that people were surveyed about listed at the top and left y Frequency - the number of times people responded in a specific way

How many students read 1 book? 3 students 7 books How many books did no student read? How many students read 4 books? 2 students

The principal surveyed students about their favorite subjects and whether they prefer group or individual work.

A teacher surveyed students to find out whether they prefer reading fiction or nonfiction and if they like to read at home or at school.

Prefers Group Work

Prefers Individual Work

Total

Fiction

Nonfiction

Total

Math

At Home

Science

At School

Total

A. How many students prefer group work? B. How many students like math and prefer individual work? 8

A. How many students were surveyed? B. How many students prefer nonfiction and read at school? 17

Vocabulary Two-way table - used to organize two separate pieces of data Categories - topics that people were surveyed about listed at the top and left

180

Level H

Chapter 10

Lesson 3

Frequency - the number of times people responded a specific way

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect)

Read the opening paragraph of the Learn and Connect section and have students observe the table. Ask: What variables are being looked at in this scenario? [whether students are right-handed or left-handed and their choices of writing utensils] Say: We can create a table that will make it easier for us to understand and interpret this data. The colorful table below the gray table does that. It is called a two-way frequency table. Explain that this table is used to show how often (the frequency) something occurs when we have more than one category for each variable. Go through the categories and where they are placed in the table. Then, explain to students that each square in the center of the table is the intersection of two variables and shows one possible combination, for example, the combination of a student who is a righty and chooses a pen. Then, show students how the end of each row and column shows the total number for the category in that row and column and that the total number of students surveyed is written in the right column and bottom row (the intersection of all the totals). Have students confirm that the totals in each row and column and the grand total of students surveyed match up.

Lighthouse MATH Level H | Teacher's Guide

Exercise | 10-3 Name Construct the two-way tables given the information. Then, answer each question. 1.

Student

Read

yes

Detention

yes

Read

Doesn’t Read

Total

B. How many students don’t read and haven't had detention?

Have students use a different colored highlighter for each possible combination of data before counting and inputting the frequencies into their two-way tables.

A. How many students said they had detention?

Detention No Detention Total

STRUGGLING LEARNERS

A teacher asked students whether they read on the weekend and whether they’ve ever gone to detention.

5 6

C. How many students read and have had detention?

EARLY FINISHERS

Anna asked customers at the grocery store whether they eat vegetables and if they’ve been to the doctor in the last year. Customer

Vegetables

yes

Doctor

yes

Doctor

No Doctor

Total

Vegetables

No Vegetables

Total

Have students return to problem 3 in the Exercise section. Ask them to find the percentage of people who are for playing sudoku, against playing sudoku, and those that have no opinion. Then, have them find the percentage of people that fall into each age range.

A. Have more customers seen a doctor or not seen a doctor? More have seen a doctor.

B. Of the customers who haven’t seen a doctor, do more eat vegetables or don’t eat vegetables? More don’t eat vegetables.

CHALLENGE AND EXPLORE

Use the totals to fill in the missing information. A survey asked people their age and whether they were for or against playing Sudoku puzzles.

For

Against

No Opinion

Total

Ages 21-40

Ages 41-60

Over 60

Total

105

200

CH AL L ENGE 4.

A school surveyed 80 students about whether they eat breakfast daily and whether they feel alert during first period. The data is summarized below: A. What percent of students surveyed feel alert but did not eat breakfast? 15% B. What percent of students surveyed do not feel alert, but did eat breakfast? 12.5%

Lighthouse Math

Level H

Chapter 10

Alert

Not Alert

Total

Breakfast

No Breakfast

Total

Exercise 3

For problem 4, remind students that a percentage is found by dividing the part by the whole and multiplying by 100. Tell students they will have to find the correct box for each percentage they need to calculate. Ask them to think about what is the part and what is the whole for each [the whole for both is the grand total of 80.]

181

Work as a whole class to complete the problems in the Apply section. Then, have students complete problems 1-3 in the Exercise section independently. Review the problems and discuss the answers as a class. Ask students: How did you determine the numbers for each category in the table? What must be true about the sum of the frequencies in a table?

ACTIVITY Interactive Class Poll: The purpose of this activity is to have the class gather data on two categories of variables and to compile their data in a two-way table. Ask all students the same compound question: How many siblings do you have, and do you have glasses or not? Write each student’s answer on the board as a number and yes or no (for example, a student with four siblings and no glasses would be “4, No”). Then, have students work with a partner to compile the data collected into a two-way table. Have students compare their tables with another pair when finished.

COMMON ERRORS Students may put a frequency into the wrong cell. Students may miscount items. Students may not calculate totals for rows, columns, or the grand total.

ASSESS Check problem 3 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 10-4 10-4 | Interpreting Two-Way Tables

PREREQUISITE SKILLS

Objective and Learning Goals

DAI LY REVI EW

y Students will calculate percentages within rows or columns in a two-way table (relative frequencies). Students will use percentages to describe if there is an association between two variables.

Answer each percent problem.

3. SPIRAL REVIEW

Vocabulary

18 is what percent of 40? 45% What number is 40% of 90?

15 & Older

14 & Younger

Total

How many people prefer painting?

Painting

How many people are 15 and older?

Drawing

How many total people were surveyed?

Total

Guiding Questions: 1. How did you complete the above problems? [Divided the part by the whole and multiplied it by 100] 2. If each percentage represents the portion of students wearing a white shirt in two different classes, which class is more likely to have a student wearing a white shirt? [the second class with a higher percent]

Lighthouse MATH Level H | Teacher's Guide

They asked two questions: Of students who live in an apartment, what percentage bus? Of students who live in a house, what percentage bus?

Apartment

House

Total

Bus

Car

Total

Students who bus and live in an apartment 12 = = 0.6 = 60% Students who live in an apartment 20 Students who bus and live in a house 12 = = 0.3 = 30% Students who live in a house 40 If the percentages were close to each other, that would mean that it would be equally likely for students who live in houses or apartments to take the bus. Since the percentages are not close, we can say that there is an association between the type of homes students live in and the way they get to school. Students who live in apartments are more likely to bus. © Lighthouse Curriculum. Copying strictly prohibited.

66 is what percent of 88? [75%]

22 is what percent of 88? 25%

Use the two-way table to answer each question.

y Dice y Paper

18 is what percent of 40? [45%]

6 is 12% of what number?

The 8th graders wanted to see if there is an association between the type of homes students live in and the way they get to school. They conducted a survey and recorded the answers in the two-way table.

Materials

Write the following questions on the board and instruct students to solve:

L E A R N A ND C O NNE C T

y Two-way table - table used to organize two separate pieces of data y Association- the relationship between two sets of data

PRE-LESSON WARM-UP

A P P LY Use the two-way table to answer each percentage question. 1.

Gabby surveyed customers in a book shop to find out whether they like hardcover or paperback books and if they read more than 5 or less than 5 books a month.

More Than 5 Less Than 5 Total

Hardcover

Paperback

Total

100

A. Of the customers who prefer hardcover books, what percentage reads more than 5 books a month? B. Of all customers, what percentage prefer paperback books and read less than 5 books a month?

20 = 0. 67 = 67 % 30 50 = 0. 5 = 50 % 100

Vocabulary Two-way table - used to organize two separate pieces of data

182

Level H

Chapter 10

Association - the relationship between two sets of data

Lesson 4

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Tell the class that today, we will use percentages to find out if there is an association between the variables in a table. Read the opening paragraph of the Learn and Connect section and have students observe the two-way table. Read the two questions that are being posed. Work through how to find these percentages, being sure to review the formula for finding a percentage (part/whole x 100). Explain that the total for the first percentage comes from the bottom of the first column because we are looking for the percent of all people living in an apartment who bus. Similarly, the total for the second percentage comes from the bottom of the second column because we are looking for the percent of all people living in a house who bus. Ask students to explain when we would use the totals at the end of each row. [If we were looking for the percent of people who bus who live in an apartment or a house] Finally, explain that we can compare the percentages we found to see if there is an association between the variables. Since these percentages are not close together, we can reasonably conclude that there is an association between whether someone takes the bus and whether they live in a house or an apartment. Ask students what conclusion we can draw from this data. [People who live in an apartment are more likely to take the bus.] Tell students that if the percentages were close together, we would not be able to reasonably drawn conclusions like this. Work as a class to find the percentage of people who live in an apartment and take a bus and the percentage of people who live in an apartment and take a car. Have students draw a conclusion based on these percentages. [60%; 40%; People who live in an apartment are more likely to take the bus than a car.]

Exercise | 10-4 Name Find two percentages rounded to the nearest whole number. Then, circle the word that best completes each sentence to describe the association. 1.

STRUGGLING LEARNERS

Mrs. Monroe asked students if they use a calculator on their math homework and if they like or dislike math. Calculator

No Calculator

Total

Like Math

Dislike Math

Total

Before starting to calculate a percentage, have students underline and circle the part and whole, respectively.

A. Of the students who use a calculator, what percentage likes math? 54% B. Of the students who don’t use a calculator, what percentage likes math? 55%

There is an no association between using a calculator and liking math. A student less likely to like math compared to a student who who uses a calculator is equally more doesn’t use a calculator. 2.

EARLY FINISHERS

A dentist surveyed patients to determine whether they floss daily and whether they’ve had a cavity in the last year.

Cavity

Floss

Don't Floss

Total

No Cavity

Total

Have students return to problem 2 in the Exercise section. Ask: If 15 more patients were added and of those, 11 said they flossed and had cavities, while four said they did not floss and did not have cavities, what are the new percentages and how does that change the association?

A. Of the patients who have no cavities, what percentage floss? 67% B. Of the patients who have cavities, what percentage floss? 29%

There is an no association between flossing and cavities. A patient who had no cavities was equally more less likely to have flossed compared to a patient who had cavities. An ice cream store recorded the types of ice creams ordered throughout the day. Cone

Cup

Total

Chocolate

149

210

Vanilla

105

115

220

Strawberry

130

Total

293

267

560

A. Of all the chocolate ice creams, what percentage were served in a cone? 71%

CHALLENGE AND EXPLORE

B. Of all the vanilla ice creams, what percentage were served in a cone? 48% C. Of all the strawberry ice creams, what percentage were served in a cone? 30% © Lighthouse Curriculum. Copying strictly prohibited.

There is an no association between the flavor of ice cream and if it was served in a cone or a cup. Chocolate Vanilla Strawberry ice cream is most likely to be served in a cone. Chocolate Vanilla Strawberry ice cream is least likely to be served in a cone.

CH AL L ENGE 4.

Are students who like pineapple on their pizza more likely to eat pizza weekly? Explain.

Eat Pizza Weekly

Don’t Eat Pizza Weekly

Total

Like Pineapple

Yes. 75% of students who like pineapple eat pizza weekly, while

Don’t Like Pineapple

only 50% of students who don’t like pineapple eat pizza weekly.

Total

100

Lighthouse Math

Level H

Chapter 10

Exercise 4

For problem 4, students will need to compare the percentages of students who eat pizza weekly from each category, those who like pineapple on pizza and those who don’t like pineapple on pizza.

183

Work together as a class to complete the problem in the Apply section, being sure to point out the different totals being used in the calculations and where they come from. Have students work with a partner on problems 1-3 in the Exercise section, then review the answers and strategies used as a class.

ACTIVITY Roll and Tally: Divide students into pairs and give each two dice and a sheet of paper. Have students draw a 7×7 table. They should designate and label the top row for player A with the numbers 1, 2, 3, 4, 5, and 6, and do the same for the leftmost column for player B. Each player takes one die and starts to roll. Students should record the results of each roll in a list outside of the grid. For example, if player A rolls a 3, and player B rolls a 5, they will record 3,5. Repeat the process of roll and record until the students hit 50 rolls. Then, have students put their data into their two-way table by marking off the number of times each roll was rolled. For example, if 6,6 appeared on their list a total of five times, write a 5 in the bottom right cell of the table. Then, have students work together to answer the following questions: 1) What percentage of times did Player A roll a 3? 2) Of all the times Player B rolled a 5, what percentage of those rolls matched with Player A rolling a 2? 3) Which roll combination happened most often and what percentage of the total rolls does that represent?

COMMON ERRORS Students may use the wrong totals when finding percentages.

ASSESS Check problem 3 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 10-5 10-5 | Review

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will review concepts introduced in Chapter 10.

Use scatter plots to represent the relationship between independent and dependent variables. Create a line of best fit to further interpret the data and make predictions. Positive Association As x increases, y increases.

y Index cards

Lighthouse MATH Level H | Teacher's Guide

Hours Studying

Number of Germs

line of best fit

Hours Cleaning

As the hours increase, the test scores increase.

Divide students into groups. Give each group a list of the vocabulary words from the chapter and the corresponding Guiding Question to discuss with their groups. Students should define each word and answer the related Guiding Question. Words to include are: scatter plot, association, positive association, negative association, no association, line of best fit, two-way table, and frequency.

No Association No obvious pattern

Test Score

PRE-LESSON WARM-UP

Number of Pets

As the hours increase, the germs decrease.

Use two-way tables to represent multiple pieces of information. This information can be used to answer questions about percentages.

As the pets increase, the income is random.

On time

Not On Time

Eats Breakfast

Total 7

Doesn’t Eat Breakfast

Total

A P P LY Construct each scatter plot. Fill in the blanks with positive, negative, increases, or decreases to describe the association. Number of attempts

Number correct

The scatter plot

shows a positive

association. As

the number of

attempts increases , 10

Number of attempts

184

Heating Cost ($)

Number correct

Guiding Questions: 1. On a scatter plot, what are the three types of associations we may see? [positive, negative, no association] 2. Why is it helpful to look for associations between variables? [So we can make predictions about the data] 3. When there is a positive association, how do the data points change as you move from right to left on the graph? [They get higher.] 4. When there is a negative association, how do the data points change as you move from right to left on the graph? [They get lower.] 5. What does no association tell us about the two variables on a graph? [There is no connection between them; neither one influences the other.] 6. How can I use the line of best fit to help make predictions? [Use the slope and y-intercept to make an equation for the line. Then, input a variable for x to get a possible outcome.] 7. Give an example of categories you would like to observe on a two-way table. [Answers will vary.] 8. How can we use the frequencies in a two-way table to predict if there is an association between the variables? [We can calculate percentages and compare them. If the percentages are close, there is no association. If they differ greatly, there is an association.]

Negative Association As x increases, y decreases.

Income

Materials

the number

Chapter 10

Heating Cost ($)

100

The scatter plot

shows a negative

association. As

the temperature

increases , the 20

80 100

Temperature (ºF)

correct increases .

Level H

Temperature (ºF)

Lesson 5

heating cost decreases .

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Review the three associations on the chart in the Learn and Connect section with the class. For each graph, ask students to identify the two variables and describe how they are related within that context. [As the hours of studying increase, test scores increase; as the hours of cleaning increase, the number of germs decreases. There is no relation between the number of pets someone has and their income.] Ask students how we can use the line of best fit. [Find the equation of the line using the slope and y-intercept, and plug in an x-value to make a prediction.] Ask students how to create a line of best fit. [Use a ruler to draw a line that follows the trend of the data with the same number of data points above and below the line.] Ask: Why can no line of best fit be drawn for the graph with no association? [because there is no trend] Move to the twoway table, and read the sentence on the left. Have students observe the table, and ask: What are the categories that are being observed? [number of students who come to school on time and not on time, number of students who eat breakfast and do not eat breakfast] Ask: What does each cell of the table tell? [The number of students who satisfy a specific combination of variables] What does the total represent? [The total number of students for each column and row. The grand total is the total number of students being observed altogether.] Finally, ask students to guide you to find out the percentage of students who eat breakfast and come on time to school. [16%]

Exercise | 10-5 Name Draw a line of best fit. Then, answer the questions about the scatter plot. 1.

A travel agency tracked how many days people vacationed and how much money they spent.

800 600 400 200 4

Write common formulas on the board, for example, the formula for calculating percentages and the slope formula. Have students highlight key words in problems with two-way tables to determine how to set up the fraction in order to find the percentage. Allow students to use calculators as needed.

Number of Mistakes on Test

Money Spent ($)

1,000

STRUGGLING LEARNERS

The scatter plot shows the relationship between the number of hours of sleep students got and the number of mistakes they made on a test.

8 6 4 2

Number of Days

Hours of Sleep

A. Write an equation for the line of best fit.

y = 50x

y = -x + 9

B. Describe the association.

There is a positive association because as the number

There is a negative association because as the number of

of days increases, the money spent increases.

hours of sleep increases, the number of mistakes decreases.

C. Predict the money spent if someone vacationed for 15 days. $750

EARLY FINISHERS

C. Predict the number of mistakes made after 8 hours of sleep. 1 mistake

Have students create an error analysis problem using a scatter plot or a twoway table. Then, have them switch with a partner to solve.

Use the two-way table to answer the questions. A school nurse recorded whether students washed their hands before lunch and whether they had been sick in the past month.

Sick

Washed Hands

Didn’t Wash Hands

Total

Not Sick

Total

140

A. Of the students who washed their hands what percentage was sick? Round your answer to the nearest whole percent. 22%

CHALLENGE AND EXPLORE

B. Of the students who didn't wash their hands, what percentage was sick? Round your answer to the nearest whole percent. 80%

There is an no association between washing hands and getting sick. A student who washed their hands was equally more less likely to have been sick compared to a student who didn’t wash their hands.

CH AL L ENGE 4.

Is it more or less likely that a student who sleeps 7 or more hours feels more rested? Explain. More likely. 18 out of 25 (72%) of students who sleep 7+ hours feel well rested. Only 4 out of 25 (16%) of students who sleep less than 7 hours feel well rested. This shows a strong association between more sleep and feeling better in the morning.

Lighthouse Math

Feels Well Rested

Doesn’t Feel Well Rested

Total

Sleeps ≥ 7 Hours

Sleeps <7 Hours

Total

Level H

Chapter 10

Exercise 5

For problem 4, students will need to look for and compare the percentage of students who sleep more than seven hours and feel rested and the percentage of students who sleep less than seven hours and feel rested. Since these percentages differ greatly, there is an association between the amount of sleep students get and whether or not they feel well-rested.

185

Have students work in partners to complete problems 1 and 2 in the Apply section. Review the answers and strategies used as a class, asking students to explain why the association is true. For example, “When the temperature rises, people tend to shut off their heat. Therefore, heat is not being used as much, so the bill for heat will go down.” Have students complete problems 1-3 in the Exercise section independently, checking with a partner. Discuss the answers and strategies used as a class when complete.

ACTIVITY Problem Round Up: The purpose of this activity is to have students create a two-way table of their own. Divide students into groups and have them select two variables with at least two variations to collect data on, for example, whether or not a student wears glasses and the number of siblings a student has. They should collect data from students in the class and then create a two-way table with the results. They should calculate the percentages for each category and decide if there is an association between their two variables. When complete, have groups present their findings to the class and discuss the significance of the results.

COMMON ERRORS Students may believe the line of best fit should connect all the data points. Students may put a frequency into the wrong cell. Students may use the wrong totals when finding percentages.

ASSESS Check problems 2 and 3 in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Chapter 11

186

In Chapter 11, we will conclude with

Review This chapter includes all of the concepts covered in the book. • We will review the real number system as well as square roots, cube roots, repeating decimals, and irrational numbers. • We will review operations with exponents as well as scientific notation and operations in scientific notation. • We will review how to solve linear equations. • We will review linear functions and graphing. • We will review proportional relationships. • We will review how to solve systems of equations. • We will review angles, triangles, and the Pythagorean theorem. • We will review congruence and transformations. • We will review finding volume. • We will review scatter plots, associations, and probability.

187

Level H | 11-1 11-1 | Real Numbers Review

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will recall all concepts having to do with rational numbers and irrational numbers, including classifying numbers, ordering and comparing numbers, square roots and cube roots, changing fractions to decimals, and changing repeating decimals to fractions.

Square Root

Cube Root

Types of Numbers

a2 = 100 a × a = 100 100 = a

b3 = 64 b × b × b = 64 3 64 = b

Can be written as a fraction

a = 10

b=4

Fractions as Repeating Decimals

2 3

PRE-LESSON WARM-UP

0.6666 3 2.0000 − 18 20 − 18 20 − 18 20 − 18 2

Present the following riddle to students to review the different vocabulary terms from the chapter:

Guiding Questions: 1. If a number can be written as a fraction, which category does it fall into? Which category is it excluded from? [It must be a rational number. It cannot be an irrational number.] 2. What perfect square between 50 and 100 is also a perfect cube? [64]

Do not have fractional parts

Repeating Decimals as Fractions

-4, 0, 5, 31, -17

0.1

y Notecards y Clips

Have students work in groups to figure it out and write their answer on a piece of paper. Discuss as a class, being sure to touch on the vocabulary words from the chapter.

- 51 , 1.3, 9 21 , .6 Integers

Materials

I am a number that can be written as a whole number or a fraction. I am less than 100 but more than 50. You can take my square root and my cube root and get an integer for both. What number am I? [64]

Rational Numbers

Whole Numbers Are only positive or 0

x = 0.1

= 0.6

0, 1, 2, 3, 4

× 10 × 10 10x = 1.1 − x (or 0.1) 9x = 1 ÷9 ÷9

−x

Irrational Numbers

1 9 1 0.1 = 9 x=

Go on forever with no pattern

3, π, 45

A P P LY Write the numbers in the correct box based on their type. 1.

3.1

2.4

2π

3 4

-7.2525

-10

10. -9

188

Rational Numbers

3.1, 2.4, 3 , -7.2525 4

Integers

Irrational Numbers

-10, -9

12, 2π

Whole Numbers

49, 8

Level H

Chapter 11

Lesson 1

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Use the table in the Learn and Connect section to review the concepts from this chapter. First, discuss how taking the square root is the inverse of squaring and how we can use perfect squares we know to find square roots. Ask students what kind of number we simplify to when taking the square root of a perfect square [integer/whole number] and what we get when taking the square root of a non-perfect square [decimal (usually an irrational number)]. Next, repeat this process with the box on cube roots. Then move on to converting fractions to decimals using long division. Go over the algebraic process for converting repeating decimals to fractions, and ask students if they recall any of the patterns they found when doing this process [Fractions with a denominator of nine will always be a repeating decimal with the numerator of the fraction repeating.]. Finally, categorize all these numbers using the Types of Numbers chart. Be sure to go over the differences between whole numbers and integers and rational numbers and irrational numbers.

APPLY AND DEVELOP SKILLS (Practice) Complete the chart in the Apply section together, being sure to address any misconceptions along the way. Next, model problems 1, 4, 5, 11, and 21 in the Exercise section. Have students work independently on the rest of the problems on the page. Review the answers and strategies used to solve. Lighthouse MATH Level H | Teacher's Guide

Exercise | 11-1 Name Simplify. 1.

100

- 49

-7

- 36

-6

1 0.125 8

13.

3 5

0.6

14.

7 10

1 9

18.

40 13.3 3

19.

- 16

-4

10.

-3

0.7

15.

1 0.83 12

5 0.45 11

20.

3 0.75 4

8 125

-27

STRUGGLING LEARNERS Provide struggling learners with anchor charts to help them remember the steps for long division and changing repeating decimals to fractions. Also give them a chart outlining the different classifications for the numbers.

Change the fraction to a decimal. 11.

1 3

0.3

12.

16.

11 3

3.6

17. 2

2.1

EARLY FINISHERS Early finishers can come up with their own riddles and trade with a partner to guess the number.

Change the decimal to a fraction. 9

82 3

25. 8.6

22. 0.6

2 3

26. 0.4

4 9

23. 5.3

24. 6.5

24 9

27. 2.4

28. 0.7

65 9

CHALLENGE AND EXPLORE

7 9

21. 1.2

Put the numbers on the number line. 29.

14 3

2.35

9 2.5

9 2

π π

3.94

9 2

4.5

3.5

CH AL L ENGE

For problems 30-32, ask students what they think the four in the root symbol means [fourth root]. Tell students that the fourth root of 625 tells us what number multiplied by itself four times equals 625. Ask students how they could find the fourth root [take the square root, then take the square root again].

Solve. 30.

625 =

Lighthouse Math

31.

16 =

Level H

32.

Chapter 11

Exercise 1

81 =

189

Numbers on the Line: The purpose of this activity is to order numbers on a number line in an interactive way. Fold several notecards in half like a tent and write different types of numbers on one flap. Make sure to include integers, decimals, fractions, repeating decimals, square roots, cube roots, and irrational numbers between -10 and 10. Fasten a string from one side the room to the other and drape a notecard with the number 0 in the middle of the line. Drape a notecard with the number -10 on the left end of the line and another with the number 10 on the right end. Have students work in groups to drape their numbers in the correct place on the number line, encouraging discussion. Be sure to remind them about spacing, and if any numbers are equivalent, provide clips so that they can be placed in the same spot.

COMMON ERRORS Students may forget the steps for changing a repeating decimal to a fraction.

ASSESS Check the odd-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

ACTIVITY

Level H | 11-2 11-2 | Exponents and Scientiﬁc Notation

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will work with exponents and scientific notation by applying the properties of exponents, including the product of powers, the quotient of powers, zero, and negative exponents, to simplify expressions. They will express, compare, and order numbers in both scientific notation and standard form, perform arithmetic operations with numbers in scientific notation, and solve real-world problems involving scientific notation to interpret and compare magnitudes effectively.

Product of Powers Property

Quotient of Powers Property

When multiplying, if the base is the same, add the exponents.

When dividing, if the base is the same, subtract the exponents.

x6 · x9

Negative Exponents

Any number or variable with a zero exponent is equal to 1.

Negative exponents indicate a reciprocal. They can be written as a positive exponent by flipping the numerator to the denominator.

Guiding Questions: 1. How do we multiply numbers with the same base and different exponents? [Keep the base the same and add the exponents.] 2. How do we divide numbers with the same base and different exponents? [Keep the base the same and subtract the exponents.]

Lighthouse MATH Level H | Teacher's Guide

Adding and Subtracting in Scientific Notation

Separate the coefficients and the powers of 10. Simplify. Put in proper scientific notation.

Change to the same power. Add the coefficients. Keep the power of 10. Simplify.

(4 × 10²)(6 × 10³) 24 × 10

1 x4

x-4

Multiplying and Dividing in Scientific Notation

(4 × 6)(10² × 10³)

(3 × 10²) + (2 × 10³)

(0.3 × 10³) + (2 × 10³)

(0.3 + 2) × 10³

2.4 × 10

2.3 × 103

A P P LY Match the expressions. 1.

x4 • x-2 • x0

x-3 x-2

x5 •

A. x9

x2 • x3 x-6

B. x5

1 x-5

x3 •

C. x4

D. x2

x7 0 •x x A

E. x11

Simplify the expression and rewrite with positive exponents only. 6. 10.

Next, have students tell you why we might want to write a number in scientific notation. [If it is a very big or very small number with many digits or zeros, it won’t take up as much space.] Write the number 4.1 × 108 on the board and ask: Is this a very big or a very small number? [Big] What is the coefficient? [4.1] What is always true about the coefficient in a number written in scientific notation? [It is between 1 and 10, not including 10.] Ask students to give you an example of a very small number written in scientific notation [Any number with a coefficient between 1 and 10 and a power of 10 with a negative exponent]

x7 − 2 x5

Zero Exponent Rule

y Whiteboards y Dry-erase markers

Tell students that in math, everything has an opposite. Ask them to tell you the opposite of addition. [subtraction] Ask them to tell you the opposite of multiplication. [division] Tell them that today, we will use these ideas to review rules for multiplying and dividing exponents.

x7 x2

x15

x0 1

Materials

PRE-LESSON WARM-UP

x6 + 9

190

1 x2

x3 • x-5 70 7-4

5x-2y3 10x4y-1 x3y-1 11. x-2y0 7.

Level H

1y4 2x6 x y

58 • 57

12.

y-2 y7

Chapter 11

Lesson 2

515 1 y9

3p-6 35p-3 40z-3 13. -1 2 z •z 9.

1 34p3 1 z4

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Write the following expression on the board: x5 • x3. Ask students how we can simplify this expression using a shortcut. [Add the exponents together and rewrite using the same base; x8.] Remind students that this rule works only when the bases are the same. Next, write x9 ÷ x12 on the board. Ask students to simplify this one using a similar rule. [Subtract the exponents, x-3.] Ask students how we can rewrite this answer using only positive exponents. [A negative exponent can be written as a reciprocal with a positive exponent; x13 .] Next, write x0 on the board and ask students to tell you its value. [1] Transition to scientific notation. Write the example multiplication problem from the Learn and Connect section on the board. Ask students to walk you through the steps to solve. [Separate and sort the coefficients and the powers of 10, multiply each set, and adjust so that it is written in proper scientific notation.] Next, rewrite the same example but as a division problem instead. Ask students how they will solve this differently. [Divide the coefficients 4 and 6 to get 0.67, subtract the exponents on the powers of 10 to get 10-1, and rewrite in proper scientific notation, 6.7 × 10-2.] Finally, write the example addition problem from the Learn and Connect section on the board. Ask students what the first step is to help them simplify this expression. [Change one of the numbers so that they have the same power of 10.] Remind students that it is helpful to change the number with the smaller power to match the number with the larger power. [3 × 102 becomes 0.3 × 103.] Have students finish the process. [Add the coefficients to get 2.3 and keep the power of 10 the same. Write as 2.3 × 103.] Remind students to always check that their answers are in proper scientific notation.

Exercise | 11-2 Name Fill-in the missing exponents. m5 • m4 = m 9

(q3 • s5) 1 = 1 (q 4 • s 5 ) q

x5 • 3 3 • 3-7 • x 8 =

(v-7 • u -8 ) 1 6. = 8 10 v3 uv (z11 • z-6) 10. =z1 z4

x13 34

t8 • t -7 = t

47 • 4 9 • 4-7 = 49

97 1 7. = 2 99 9 (w10 • w 6 ) 11. = w7 w9

STRUGGLING LEARNERS

y10 • y7 = y 17

p7 8. =1 p7 r7 12. = 62r2 7 (6 • r 5 • 6 -9 )

Create a reference sheet that outlines the steps for simplifying each type of exponent and scientific notation problem. Include simple examples for each rule and space for students to add their own notes. As students work through practice problems, have them first identify which rule they are using before solving.

Convert to scientific notation or standard form. 13. 5,600 5.6 × 10

14. 89,000,000 8.9 × 10

17. 7.45 × 10

-4 0.000745

18. 5.0 × 10

21. 190 1.9 × 10

-6 0.000005

22. 9.01 × 10 -2 0.0901

15. 0.00000082 8.2 × 10

16. 6.2 × 103 6,200

-4 19. 0.00047 4.7 × 10

20. 0.0061 6.1 × 10

23. 42,500 4.25 × 10

24. 2.09 × 102

−7

-3

209

Solve. 25. (2.3 × 104) + (5.7 × 104) = 8 × 10

26. (3 × 102) • (4 × 105) = 1.2 × 10

28. (5.6 × 106) ÷ (8 × 102) = 7 × 10

29. (8 × 103) + (2.5 × 103) =1.05 × 10

30. (3.2 × 104) ÷ (4 × 102) = 80

-2 31. (4.9 × 10 ) + (3.8 × 10 ) = 8.7 × 10

10 32. (6.5 × 10 ) • (2 × 10 ) = 1.3 × 10

33. (7.2 × 106) − (3.2 × 105) = 6.88 × 10

-2

27. (6.2 × 105) − (1.1 × 105) = 5.1 × 10

EARLY FINISHERS

Solve. 34. The city of New Andora has a population of 2.3 × 106 people. The population is expected to grow by 1.2 × 105 people next year. What will the population of New Andora be next year?

35. A single human cell contains about 2.0 × 102 centimeters of DNA when stretched out. A person has approximately 3.7 × 1013 cells. What is the total length of DNA in one human body? 37. The Lyra-7 telescope can spot objects as small as 2.1 × 10−9 meters in diameter. A space rock passing by the telescope has a diameter of 1.05 × 10-5 meters. How many times larger is the space rock compared to the smallest object the telescope can see?

3.312 × 1013 pages

5,000 times

CH AL L ENGE 38. Why is scientific notation considered easier to use when working with very large or very small numbers? Give an example from science or technology.

39. Why is it important to write numbers with the same exponent when adding or subtracting in scientific notation? It makes sure the place values line up correctly and that the final numbers are accurate.

Scientific notation helps shorten very large or small numbers and makes calculations easier. For example, scientists use 3.0 × 108 meters per second to represent the speed of light instead of writing 300,000,000.

Lighthouse Math

Level H

Chapter 11

Exercise 2

191

APPLY AND DEVELOP SKILLS (Practice) Have students work independently to solve problems 1-13 in the Apply section, then have them check and discuss their answers with a partner. Review the answers as a class and address any questions or problems that students found challenging. Have students work independently on problems 1-37 in the Exercise section, checking with a partner as they go. Conclude with a full-class review, selecting a few representative problems from each section to work through on the board, and reinforce key concepts.

ACTIVITY Solving Group Showdown: The goal of this activity is for students to practice and apply key concepts from the unit by solving exponent and scientific notation problems collaboratively. Begin by dividing the class into five groups and giving each group a whiteboard and a dry-erase marker. Write one problem on the board and have groups work together to solve it on their whiteboards. When a group finishes, they hold up their board to display their answer. The first group with the correct answer earns five points, the second group earns four points, the third group earns three points, and so on. After each round, reveal the correct answer and briefly explain which rule applies (for example, multiplying exponents, dividing powers, or performing operations in scientific notation). Continue for several rounds, using a mix of problem types. The group with the most points at the end wins.

7.4 × 1015 cm

2.42 × 106

36. A printing company prints 9.2 × 109 pages every second using high-speed machines. If the machines run non-stop for 3.6 × 103 seconds, how many pages will the company print in total?

Have students select one problem from the lesson that they got wrong and write a sentence detailing where they went wrong. If they got them all right or have not checked their answers yet, have them choose a problem type they found challenging and write an explanation of how to solve it step-bystep.

CHALLENGE AND EXPLORE Have students work in pairs to solve problems 38 and 39. Afterward, review the problems as a class, discussing how the skills from this lesson connect to real-world applications such as scientific measurement, scaling, or growth patterns. Emphasize the importance of understanding these concepts beyond the classroom and how they are used in practical contexts.

COMMON ERRORS Students may multiply or divide exponents instead of adding and subtracting. Students may think exponents raised to the power of zero equal zero. Students may write their final answers in incorrect scientific notation.

ASSESS Check the odd-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

Level H | 11-3 11-3 | Linear Equations

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will solve linear equations by applying strategies such as simplifying expressions, using the distributive property, and managing variables on both sides. They will classify equations based on their solutions as having one solution, no solution, or infinitely many solutions, and apply these skills to create and solve equations in real-world contexts.

-4(3x2 + 4) = 8x2 – 33 – 3x2 Apply the distributive property.

Combine like terms.

Get the variable to one side of the equation.

-4(3x + 4) = 8x – 33 – 3x

-12x – 16 = 8x – 33 – 3x

-12x2 – 16 = 8x2 – 33 – 3x2

-12x2 – 16 = 5x2 – 33

-12x² – 16 = 5x² – 33 – 5x² – 5x² -17x² – 16 = - 33

Guiding Questions: 1. How can we check our answers and be sure they are correct? [By plugging our answers back into x in the original equation. If the result is an equation where both sides are the same number/equal, then our answer is correct.] 2. How will we know if our answer is wrong? [If, after plugging in our answer to check it, the equation is not balanced - the two sides of the equal sign are different numbers our answer is incorrect.]

Addition/Subtraction

Multiplication/Division

Exponents/Roots

-17x – 16 = -33 + 16 + 16 -17x2 = - 17

-17x = - 17 ÷ -17 ÷ -17 x2 = 1

x2 = 1

x = 1 or -1

A P P LY Choose if the equation has one solution, no solution, or infinitely many solutions.

PRE-LESSON WARM-UP

21x + 59 = 7(3x + 9) – 10

-60 + 16z = 4(4z – 15)

no solution

4x + 6 = 2x + 14

infinitely many solutions

one solution

Tell students that you found two answers to this problem: 3 and -3. Ask them to work with a partner to prove that your answers are correct. Ask two students to write their work on the board, one for each answer, and hold a discussion about looking for equality when checking answers to equations.

y Poster board y Markers

12(x2 − 2) = 3x2 + 57

Follow SADMEP to solve.

Materials

Write the following problem on the board:

192

-4x = 6

x=1

4x2 + 2 = 18

x = -9

-5 = -12 +

x 3

x = 2 or x = -2

3(6x – 1) + 7 = 22

x = 9 or x = -9

2.3x + 1.9 = 5.6x – 11.3

x = 11

3x – 21 = 2x + 60

x = -1

10.

1 1 (200x + 28) + 21 = (150x – 60) + 5 3 2

x=4

11. 3(2x + 5) + 80 = 4(3x + 7) + 1

x = 21

Level H

Chapter 11

Lesson 3

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Write out the multi-step equation example -4(3x² + 4) = 8x² + 33 − 3x² on the board, leaving space to show each stage of the solution. Ask students what the first step to solve should be. [Use the distributive property.] Have a student complete this step on the board. [-12x² − 16 = 8x² + 33 − 3x²] Ask a different student to describe and complete the next step. [combining like terms on the right side to get -12x² − 16 = 5x² − 33] Ask why combining like terms is a helpful step. [It makes the equation easier to manage.] Ask students for the next step. [Subtract 5x² from both sides so that we only have an x on one side of the equation; -17x² − 16 = −33] Ask another student for the next step. [Follow SADMEP (the reverse of PEMDAS) to isolate the variable; -17x² = -17; x² = 1; x = 1 or -1] Discuss how using a approach like SADMEP prevents skipped steps. Conclude by reviewing the complete process as a class, highlighting how distribution, combining like terms, and inverse operations all work together to simplify multi-step equations and reach the correct solution.

APPLY AND DEVELOP SKILLS (Practice) Work through problems 1-5 in the Apply section as a class. For problems 1-3, remind students how to determine whether an equation has one solution, no solution, or infinitely many solutions, and discuss how to identify each by examining the simplified form. For problems 4 and 5, work through each Lighthouse MATH Level H | Teacher's Guide

Exercise | 11-3 Name Solve each equation. 1.

2(y + 5) + 3y = 40

-3(2a – 5) + 4a – 6 = 25

y=6

25(t – 2) + 3(2t + 1) – 4 = 73

3(p2 – 4) + 2(p2 + 1) = 35

t=4

-3x + 8 = 2x – 12

a = -8

p = 3 or -3

5q + 9 = -q + 3

x=4

q = -1

STRUGGLING LEARNERS

(m – 4) + 3(m + 1) = 22 2 m=6 (2r – 3) + 2(r + 5) = 17 3 r=3 z – 7 = (3z) – 1 2 4 z = -24

Provide students with a reference sheet and have them write out the acronym SADMEP, labeling each part to represent the reverse order of operations (Subtraction, Addition, Division, Multiplication, Exponents, and Parentheses). As they solve each equation, have them label every step according to this process and write out their work in full sentences or clear mathematical statements.

Write if each equation has one solution, no solution, or infinitely many solutions. 10. 3(x + 4) – 2x = 10 One solution

13. 7(a – 2) + 3 = 7a – 11

11. 5(2y – 3) = 10y – 15 Infinitely many solutions

14. 6m – 9 = 3(2m + 4)

Infinitely many solutions

12. (4z – 8) = 2z – 5 2 No solution 15. 2(3t – 5) + 4 = 6t – 6

No solution

Infinitely many solutions

Write and solve an equation for each word problem. Explain the solution in context of the problem.

200 + 12b = 620 → b = 35 books; They can print 35 yearbooks with their budget.

150 + 3c = 450 → c = 100 bars; They need to sell 100 candy bars to reach their goal.

EARLY FINISHERS

18. The student council wants custom shirts for a fundraiser. Bright Threads charges a $50 setup fee plus $9 per shirt. ColorWave Designs charges a $20 setup fee plus $12 per shirt. For how many shirts will the total cost be equal?

19. Lakeview Middle School is comparing two catering companies. Flavor Fest charges $100 to deliver and $7 per student. Party Perfect charges $50 to deliver and $9 per student. For how many students will the total cost be the same?

50 + 9t = 20 + 12t → 10 shirts; If they order 10 shirts, both companies cost the same.

100 + 7n = 50 + 9n → n = 25 students; For 25 students, both caterers charge the same total cost.

CH AL L ENGE 20. Sara’s homework time is given by S = 55 + 10m, where m is the number of math problems. Deborah’s homework time is D = 25 + 15m. Sara thinks she will always take longer because 55 is greater than 25. Is she correct? Explain your reasoning and find when their times are equal. Sara will not always take longer than Deborah. They will take the same amount of time to complete 6 problems. Sara will take longer to complete less than six problems. If they are completing more than 6 problems, Deborah will take longer.

Lighthouse Math

Level H

Chapter 11

Exercise 3

193

step together on the board, pausing after every calculation to reinforce the use of SADMEP and to ensure students understand the reasoning behind each operation. After completing these as a class, have students work on problems 6-11 with a partner, then review the answers together, clarifying any misconceptions. Next, have students work on problems 1-25 in the Exercise section in small groups. Review the answers as a class. Have students help you solve the word problems on the board, being sure to discuss the key words that tell us how to write the equation for each.

ACTIVITY Create a Scenario Poster: Students will work collaboratively to create a realworld scenario that involves solving a multi-step equation with one solution. After dividing the class into small groups, provide each group with a poster board and markers. Each group will brainstorm and write a creative, realistic scenario that can be represented by a multi-step equation, such as comparing costs, tracking progress, or modeling change over time. On the poster, groups must include their written scenario, the equation that represents the situation, the step-by-step process used to solve it, the final solution, and a clear explanation of what the solution means in the context of their story. Once the posters are complete, students will share their work with the class, explaining how their equation connects to the scenario and how they verified their answer.

Ask students who finish early to identify one concept or skill they found challenging earlier in the unit. They should return to Chapter 3 and practice additional problems related to that topic.

CHALLENGE AND EXPLORE Have students work in pairs to solve problem 20 and then review the solution as a class. Discuss what the numbers 10 and 15 represent in the equation [the rate at which they each solve problems] and what the constants (55 and 25) might represent. [how many math problems each already solved] Have the pairs test different numerical values to see how each affects the outcome.

COMMON ERRORS Students may not follow SADMEP when solving. Students may confuse equations that have no solution with equations that have infinite solutions.

ASSESS Check the odd-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

17. The 8th graders already made $150 for their trip and plan to sell candy bars for $3 each. Their goal is to raise $450. How many candy bars must they sell?

16. The student council pays a $200 setup fee to print the yearbook and $12 per book printed. Their total printing budget is $620. How many yearbooks can they print without going over budget?

Level H | 11-4 11-4 | Linear Relationships and Functions Review

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will understand linear functions and their features and will recognize them and analyze them in a table, graph, equation, and word form.

Materials

Table

A water tank is leaking water at a rate of 2 liters an hour. When it was full, the tank had 14 liters of water.

y Notecards

Graph

Amount of Water Left in the Tank

y = -2x + 14 x represents time in hours

y represents liters of water 0

Slope: the rate the water is leaking

Read the following example of a linear function aloud and have students create a table of values, a graph, and an equation to model the function:

y2 - y1 x2 - x1

Slope: rise over run

Slope: m

12 - 10 2 = =-2 1-2 -1

Rise: -2 = -2 Run: 1

-2

y-intercept: the amount of water the tank started with

y-intercept: the point where the x value is 0

y-intercept: the point where the line crosses the y-axis

y-intercept: b

It started with 14 liters.

When x is 0, y is 14.

(0,14)

Slope:

It is losing water at a rate of 2 liters an hour.

It takes a car two hours to drive 100 miles. At this speed, how far will it travel in three hours?

Guiding Questions: 1. What types of x values make sense in this scenario? [Positive x values] 2. Is the graph increasing or decreasing? [Increasing] 3. What is the slope, and what does it represent? [50 mph - the speed of the car] 4. What is the y-intercept, and what does it represent? [0 miles, the starting distance]

7 Time (hours)

PRE-LESSON WARM-UP

Be sure that students define their variables [x represents time in hours, y represents distances in miles] and choose x values that make sense [1, 2, 3, 4]. If needed, help them to calculate matching y-values [50, 100, 150, 200]. Remind them that their table lists coordinates that can be graphed and connected with a line to show the relationship. When creating the equation, remind them of the formula y = mx + b with m being the slope and b being the y-intercept. [y = 50x]

Slope-Intercept Form

Liters of water

Word Form

A P P LY Find the slope. 1.

194

-2

-1

(3, 17) (8, 7)

1 x+5 4

-10

Level H

-2

Chapter 11

1 4

Lesson 4

-3 2

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Use the scenario described in the table in the Learn and Connect section to review with students how to find the slope and y-intercept (the defining characteristics of a linear function) using word form, a table, a graph, and an equation. Remind students that the slope represents the rate of the function, and it tells us about the steepness of the line on the graph of the function. Remind them as well that the y-intercept of the function is the starting point of the function, or the y-value when x = 0. It is where the line crosses the y-axis. When talking about slope from a graph, we use the terms rise and run. Tell students that sometimes a rise can be a drop, and we use this term to talk about vertical change, while we use the term run to talk about horizontal change. Finally, remind students that in order to be a function, each input (x-value) must have exactly one output (y-value), and to be a linear function, the graph must be a straight line (the rate of change stays the same always).

APPLY AND DEVELOP SKILLS (Practice) Work together as a class to find the slope of all the linear functions in the Apply section. Then have students work independently on all the problems in the Exercise section. Review the problems together as a class to ensure understanding, and correct any misconceptions.

Lighthouse MATH Level H | Teacher's Guide

Exercise | 11-4 Name Determine if the relationship is a function. 1.

-1

-2

Function

Not a Function

Function

STRUGGLING LEARNERS

(0, 1) (1, 1) (2, 1) (3, 1)

Not a Function

Function

Struggling learners should use the chart in the Learn and Connect section to help them with understanding and solving problems in different forms.

Not a Function

Determine if the function is linear. 10

y = -2x + 9

6. -10

-2

-8

EARLY FINISHERS

-10

Linear

Non-Linear

Linear

Non-Linear

Linear

Non-Linear

Early finishers should choose any problem from 1-8 in the Exercise section. Their task is to create a scenario that matches the representation by defining the variables and interpreting the slope and y-intercept. They should also represent the function in one other form. Repeat as time allows.

Write the equation of the line. 10

7. -10

8. 10

-1

-7

-3

-10

A pot of water is set to boil. It begins at a temperature of 68°. The temperature increases by 1 21 °F every minute. y = 3 x + 68

y = 4x − 3

y = 3x + 4

Find the slope (m), y-intercept (b), and equation of each function. Then answer the questions. 11. Water Usage in Cedarwood

m = -10

b = 50

40 Liters

Liters

12.

-5 2

B. Which city started off with more water? Elmwood

b = 40

20 10

Hours

Equation: y = -10x + 50

A. How much water was in Cedarwood’s tank to 40 liters start?

C. Which city used water at a faster rate? Elmwood

Equation: y = -2.5x + 40

CH AL L ENGE 10

13. A line has a y-intercept of 1 and passes through the point (2, -6). Find the slope, write an equation, and graph the line. -10

Slope = - 7

Lighthouse Math

y=-7x+1 2

Level H

-10

Chapter 11

Exercise 4

10. Water Usage in Elmwood

For problem 13, help students understand that when we are given the y-intercept, we are given a coordinate (0, b). We can then use this and the other coordinate to find the slope using the slope formula. Alternatively, students can plot the two points and connect them with a straight line and then use the rise/run to find the slope and create an equation.

195

Match the Model: The purpose of this activity is to help students become more fluent at changing between word form, tables/sets of coordinates, graphs, and equations of the same function. Provide students with notecards of different functions represented in the different ways. Have them walk around the room, teaming up with other students who have cards showing the same function represented in a different way. They should define the slope and y-intercept once all four students have joined together, and then they should share with another group and check their work.

COMMON ERRORS Students may confuse slope and y-intercept when writing equations of lines in the form y = mx + b.

ASSESS Check the even-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

ACTIVITY

Level H | 11-5 11-5 | Proportional Relationships

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will work with proportional relationships by finding unit rates, solving proportions, graphing relationships, writing equations, and comparing different representations to solve real-world problems.

a×d=c×b

3 12 = x 18

Example:

3 × 18 = 12 × x

4.5 = x

54 = 12x

How to Find the Unit Rate in a Proportional Relationship

PRE-LESSON WARM-UP

From an equation

Use k in the equation y = kx.

From a table

Divide y by x in any column.

y = 3x → 3 is the unit rate.

Write the following questions on the board:

Miles (y)

120

180

Hour (x)

60 ÷ 2 = 30 miles per hour

Guiding Questions: 1. What does your rate tell you about your situation? [It shows how much happens for every one unit of the other quantity, like words per minute or miles per hour.] 2. Which representation makes it easiest to compare your rate with your partner’s? [Answers vary; e.g., a graph or equation shows who has the larger rate because the line is steeper or the constant (k) is greater]

Lighthouse MATH Level H | Teacher's Guide

Find the slope.

1,000 800 600 400 200

(12, 600) (4, 200)

0 2 4 6 8 10 12 14 16 18 20

Number of Hours

(4, 200) → 200 ÷ 4 = $50 per hour

Have students write down their responses individually. Then, have them pair up with a partner and compare their situations. Ask them to find a way to represent one of the comparisons using a table, a graph, or an equation. Have students share with the class and then briefly discuss as a class how these representations help compare two quantities. Tell students that today, we will be reviewing proportional relationships represented in different ways.

From a graph

Total Cost ($)

Cost of a Venue Rental

y How many math problems can you solve in 10 minutes? y How many pages of a book could you read in one hour? y How many laps can you run in five minutes?

Match the scenario to its solution. D

196

Sarah ran 3 miles in 24 minutes. At the same pace, how long would it take her to run 5 miles?

Lisa types 360 words in 12 minutes. How long will it take her to type 600 words?

It takes 18 minutes to walk 0.6 miles. How long will it take to walk 1.2 miles at the same pace?

Level H

Chapter 11

A train travels 90 miles in 60 minutes. How long will it take to go 135 miles at that rate?

A machine prints 40 pages in 15 minutes. How long does it take to print 120 pages?

A hose fills 10 gallons in 5 minutes. How long will it take to fill 22 gallons at the same rate?

Lesson 5

20 minutes

90 minutes

45 minutes

40 minutes

11 minutes

36 minutes

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Begin by reminding students about what a proportional relationship is using the opening text from the Learn and Connect section. Then, go through the example, which uses a proportion to find a missing quantity. Next, have a student define unit rate [the rate per one unit] and use the table to review how to find a unit rate in different ways. Explain that this review focuses on finding the unit rate from equations, tables, and graphs. Start with the equation example, y = 3x. Ask students what the 3 could represent and review that the coefficient in front of x (the k in y = kx) is the unit rate. Prompt students to explain that this shows how y changes for each unit of x. Next, review the table with Miles (y) and Hours (x). Have students recall how to find the rate by dividing y by x in any column. Confirm that the ratio stays constant across all pairs, showing a proportional relationship. Then, review the graph of “Cost of a Venue Rental.” Ask students how to find the unit rate from the graph and guide them to identify it as the slope. Walk through the example given, being sure that students divide the change in y by the change in x. Emphasize that proportional graphs always form a straight line through the origin. Conclude by connecting all three methods. Review that the unit rate, slope, and constant of proportionality all represent the same value in different forms.

Exercise | 11-5 Name Determine the better buy. 1.

Which is the better buy for orange juice?

Which pack of pencils offers the best deal?

STRUGGLING LEARNERS

Which is the better buy for cereal?

32 oz for $4.48

10 pencils for $2.90

12 oz for $3.60

48 oz for $5.76

20 pencils for $5.60

18 oz for $5.22

64 oz for $8.80

15 pencils for $3.90

24 oz for $7.92

Create a reference sheet for the topics in the lesson that students can use when solving problems. Have them write out the steps for each problem as they solve to reinforce learning.

Graph the proportional relationship and find the slope. 4.

A car can travel 70 miles in 2 hours.

100

Slope:

60 20 0

A factory produces 120 toys in 4 hours.

Car Travel Distance

Toy Production

140

Number of Toys

Miles

140

100

Slope:

60 20

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Hours

Have students select one topic from the lesson that they found challenging and create two to three example problems related to it. Ask them to write out the steps to solve each one and explain their reasoning in words.

Slope:

A printer can print 90 pages in 6 minutes. Number of Pages

Number of Muffins

Muffins Made

4 0

Hours

A recipe uses 3 cups of flour to make 12 muffins. 16

EARLY FINISHERS

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Pages Printed

120 90

Slope:

30 1

Cups of Flour

CHALLENGE AND EXPLORE

Minutes

Solve.

Hours (x)

Income (y)

$62.50

$100

$125

$150

Who earns more per hour?

The equation y = 6x represents the number of math problems Hannah can solve in x minutes. The graph below shows how many math problems Rose can solve per minute. Who can Math Problems Solved Over Time solve math 50 problems faster?

William

Minutes

The amount of money Eric makes per hour is shown in the table below. William’s income is given by the equation y = 14x.

Problems Solved

Hannah

CH AL L ENGE

To solve problem 10, first have students define a linear equation and a proportional relationship. Then, discuss which parts overlap and which do not. Have students provide examples of linear equations that are proportional relationships and linear equations that are not proporational relationships.

10. Why is every proportional relationship also a linear equation, but not every linear equation represents a proportional relationship? Proportional relationships are always linear because they are straight lines. However, a linear equation does not have to pass through the origin, while the graph of a proportional relationship does.

Lighthouse Math

Level H

Chapter 11

Exercise 5

197

Have students work in pairs to complete problems 1-6 in the Apply section. Review as a class to reinforce understanding of how to find the unit rate. Then, have students work in groups to solve problems 1-9 in the Exercise section. Review as a class and go over any problems students found challenging.

ACTIVITY Unit Rate Stations: Students will review how to find and interpret unit rates by working with equations, tables, and graphs in a hands-on, group-based activity. Set up three stations around the classroom, each focused on one representation of proportional relationships. At the equation station, provide examples such as y = 2x or y = 0.5x, and have students identify the unit rate and give an example of what it represents in context. At the table station, give students tables showing pairs of values, such as distance and time or cost and quantity, and have them find the unit rate by dividing y by x in each row to confirm the ratio remains constant. At the graph station, display proportional graphs and have students determine the unit rate by finding the slope between two points on the line. Divide students into small groups and have them rotate through each station every few minutes. After all groups rotate through the stations, review as a class and discuss how the unit rate connects across all three forms, emphasizing that the unit rate, slope, and constant of proportionality all represent the same relationship in different ways.

COMMON ERRORS Students may divide x by y when finding a unit rate. Students may think that graphs for proportional relationships don’t have to go through (0,0).

ASSESS Check the odd-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 11-6 11-6 | Systems of Equations

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will understand and solve systems of linear equations by analyzing their solutions as one solution, no solution, or infinitely many solutions. They will apply methods such as graphing, substitution, and elimination to find solutions and interpret points of intersection. Additionally, students will use these techniques to model and solve real-world problems involving systems of linear equations.

A food stand sold 60 items in one night. They sold x sandwiches that each cost $12, and y bottles of soda that each cost $6. At the end of the night, they made $528 altogether. How many of each item did they sell? Solve by Graphing

Solve by Substitution

Solve by Elimination

x + y = 60 → y = -x + 60

x + y = 60 → y = 60 – x

x + y = 60 → 6(x + y = 60)

12x + 6y = 528 → y = -2x + 88

12x + 6y = 528

100

12x + 6(60 – x) = 528 12x + 360 – 6x = 528 6x + 360 = 528 6x = 168 x = 28

# of Puzzles

80 60 40

(28,32)

# of Building Sets

28 + y = 60 y = 32 (28, 32)

28 Sandwiches 32 Bottles of Soda

Materials y Notecards

80 100

6x + 6y = 360 – 12x + 6y = 528 -6x = -168 x = 28 28 + y = 60 y = 32 (28, 32) 28 Sandwiches 32 Bottles of Soda

A P P LY Match the graph to the number of solutions. 5

PRE-LESSON WARM-UP Write the following systems on the board and ask students to work with a partner to predict, without solving, which system has one solution, which one has infinitely many solutions, and which one has no solution.

3x + 2y = 14 x−y=1 [one solution] 4x − y = 3 4x − y = 7 [no solution] 2x + 4y = 10 x + 2y = 5 [infinitely many solutions] Review and discuss students’ predictions as a class and ask students to explain the reasoning behind their guesses. Then, solve each system together to check the accuracy of their predictions. Guiding Questions: 1. What does it mean when a system has one solution? [It means the two lines intersect at exactly one point, and there is a single pair of values that satisfies both equations.] 2. What does it mean when a system has infinite solutions? [It means the two lines are the same line, and every point on the line satisfies both equations.] 3. How can you tell when a system has no solutions? [The equations form parallel lines with the same slope but different intercepts, so they never intersect.]

Lighthouse MATH Level H | Teacher's Guide

-5

2. 5

-5

3. 5

-5

one solution

no solution

infinitely many solutions

2x + 3y = 21 4x − y = 7

-5

Determine the solution to each system of equations. 4.

198

x + y = 18 3x + 2y = 46

x + 2y = 19 3x − y = 8

5x + 4y = 38 x − 2y = 2

(10,8)

(4,8)

(4,1)

(8,10)

(5,7)

(6,1)

(3,5)

(12,6)

(7,6)

(4,3)

(4,4)

(6,12)

(9,5)

(6,2)

(5,3)

Level H

Chapter 11

Lesson 6

(2,6)

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Begin by reading aloud the scenario in the Learn and Connect section and asking students to help you create two equations from the given information. [x + y = 60 and 2x + 6y = 528]. As you create the equations, have students explain their reasoning. Emphasize that the first equation represents the number of items sold, while the second equation represents the total amount of money made. Ask students: What are the steps for solving this system using a graph? [Put each equation in slope-intercept form, graph both, and look for the coordinates at the point of intersection.] Ask students what the point of intersection means in the context of the problem. [They sold 28 sandwiches and 32 bottles of soda.] Next, ask students for the first step for solving the system using substitution. [Isolate one of the variables in one equation.] Ask them which equation seems easier to do this for. [The first one] Have students work with a partner to solve the system using substitution, and then have one pair put their work on the board for the class to look over. Remind students that we are solving for both x and y. Point out that the solution when solving with substitution matches the coordinates we found when graphing. Finally, have students tell you the first step to solve the system using elimination. [Multiply the first equation by six so the coefficients in front of the ys match.] Ask students whether we will need to add or subtract these equations to eliminate a variable. [Subtract] Ask: Which variable will be eliminated, and which will we solve for? [y will be eliminated, and we will solve for x.] Ask: How will we solve for y afterward? [By substituting the solution for x in either equation] Have students work with a partner to solve. Emphasize that all three approaches lead to the same solution, and ask the class which method felt most efficient and why.

Exercise | 11-6 Name Solve each system of equations by graphing. 1.

y=5–x y = 2x – 1

2x + 3y = 6 x + 2y = 4

-5

Create a reference sheet that outlines each method for solving systems (graphing, substitution, elimination) with clear, step-by-step instructions. Have students take the first problem from each set in the Exercise section and solve it using all three methods for extra practice and comparison.

-5

Solution: (0, 2)

Solution: (2, 3)

y = - 31 x + 3

-5

STRUGGLING LEARNERS

y = - 31 x + 1

y=4–x y = 2x + 1

-5

Solution: (1, 3)

Solution: no solution

Solve each system of equations using substitution. 5.

y = 2x x + 2y = 5

y=x–1 x + y = -5

y = 7 – 3x 2x + y = 5

(-2, -3)

(1, 2)

(2, 1)

x = 2y 3x – y = -5

EARLY FINISHERS

(-2, -1)

Solve each system of equations using elimination. 9.

x+y=3 2x + y = 7

10.

x + 2y = -5 3x + 2y = -7

(4, -1)

11.

12.

4x + 5y = 14 6x – 7y = -8 (1, 2)

(-1, -2)

Ask students to choose one set of questions from the Exercise section and solve it using a different method than they used the first time to confirm their answers and compare strategies.

3x + 4y = 17 2x + 5y = 16 (3, 2)

Solve. 14. A school ordered a total of 48 notebooks in two different colors. The number of blue notebooks was 6 more than the number of green notebooks. How many blue and how many green notebooks were ordered?

x + y = 84 x = 3y (63, 21)

13. Two groups of students collected a total of 84 cans for a food drive. Group A collected 3 times as many cans as Group B. How many cans did each group collect?

x + y = 48 x=y+6 (27, 21)

CH AL L ENGE 15. Match each system to the best way to solve and explain your reasoning. A

y = 2x + 1 y = -x − 1

Reasoning:

easy to graph

Lighthouse Math

graphing

x+y=6 x=6−y equations easy to plug in Reasoning: B.

elimination

Level H

Chapter 11

3x + 4y = 18 6x + 4y = 22 coefficients match Reasoning: C.

Have students work in pairs to discuss problem 15 and explain their reasoning for each part. Then review as a class, highlighting how choosing an efficient method helps them solve systems more effectively.

substitution

Exercise 6

199

Work through problems 1-7 in the Apply section as a class. For problems 1-3, emphasize how the graph helps determine the number of solutions. For problems 4-7, demonstrate how to plug in each answer choice to identify the one that satisfies both equations. After the review, have students work in groups to solve problems 1-14 in the Exercise section. Conclude by reviewing the answers as a class and addressing any problems students found challenging to clear up remaining misconceptions.

ACTIVITY Systems of Equations Shuffle: Students will practice solving different types of systems using multiple methods. Place students in groups and give each group ten notecards. Each group creates ten different systems of equations and writes each system on the front of a card. Once all groups have finished, have them rotate their entire set of cards to another group and receive a new set from a different group. Students then solve each system using one of three methods and show all steps on the back of the card. After solving, each group selects one example solved by graphing, one by elimination, and one by substitution to share with the class, explaining why the chosen method worked well for that particular system.

COMMON ERRORS Students may only solve for one variable and forget to solve for the other. Students may add equations instead of subtracting or vice versa.

ASSESS Check the even-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 11-7 11-7 | Angles, Triangles, and the Pythagorean Theorem Review

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will be able to recognize the different relationships between angles formed by intersecting lines as well as parallel lines crossed by a transversal and the relationships between angles inside and outside a triangle. Finally, students will be able to use the Pythagorean theorem to solve for an unknown side in a right triangle.

Angle Relationships Complementary Angles

Supplementary Angles

Adjacent Angles

Vertical Angles

Angles that add up to 90°

Angles that add up to 180°

Angles that share a vertex and a side

Formed by intersecting lines, these angles are directly across from one another and equal.

50°

70°

90° – 50° = a a = 40°

180° – 70° = d d = 110°

e is adjacent to f

67°

113°

g = 67° i = 113°

Angle Relationships Formed by Parallel Lines and a Transversal Alternate Interior

Alternate Exterior

Corresponding

Same Side Interior

Same Side Exterior

Equal

Add to 180°

Materials y Table of angle relationships

Angle Relationships in a Triangle

Right Triangles The Pythagorean Theorem

hy p

PRE-LESSON WARM-UP

e g

h i

Guiding Questions: 1. What do supplementary angles add up to? [180°] 2. What do complementary angles add up to? [90°] 3. What kinds of lines form alternate exterior angles? [Parallel lines crossed by a transversal] 4. What is the sum of all the angles in a triangle? [180°]

Lighthouse MATH Level H | Teacher's Guide

a+m m

(x1, y1)

(c)

(x2, y2)

b+m

d=m

leg (b)

c = 180°

a+m

a2 + b2 = c2

distance = √(y2 - y1)2 + (x2 − x1)2

A P P LY Draw a line to match the name of the relationship between angle a and angle b. 1. a

200

[Examples of answers: y a and e are supplementary and adjacent y a and d are vertical angles y b and c are complementary and adjacent y b and f are vertical angles y c and f are supplementary and adjacent y d and e are supplementary and adjacent y e and h add up to f (the exterior angle of the triangle) y e and g are alternate interior angles y d and h are alternate interior angles y h and i add up to a because they are corresponding angles]

c m

Draw the following on the board and challenge students to recall how the marked angles are related and if they can come up with more than one angle relationship that describes them.

ote

leg (a)

The Distance Formula

3. a

5. a

b a

adjacent

vertical

corresponding

alternate exterior

complementary

angles

Level H

Chapter 11

Lesson 7

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Use the chart in the Learn and Connect section to review the different angle relationships. Remind students that some angles are related because they are equal, and other angles are related because they add up to 90° (complementary angles) or 180° (supplementary angles, same side interior or exterior angles in parallel lines, interior angles in a triangle). Some angles, such as adjacent angles, are just related by their position to each other, not by their angle measures. Finally, review the Pythagorean theorem with students and how to use to it to solve for a missing side in a triangle.

APPLY AND DEVELOP SKILLS (Practice) Have students work in pairs to match the angles with their relationship in the Apply section. Then, go over the answers to ensure students understand and to clear up any misconceptions. Next, complete problems 1, 2, and 12 in the Exercise section together as a class to model setting up equations for different types of figures. Have students complete the rest of the problems independently, checking with a partner as they go. Review the answers as a class to ensure understanding.

Exercise | 11-7 Name Write an equation. Then, solve for x. 1.

1 ( 3 x + 66)°

(x - 10)°

(2x)° (x + 24)°

x°

STRUGGLING LEARNERS

4. (x - 2)°

Allow students to use the chart in the Learn and Connect section to assist them in remembering the angle relationships and the Pythagorean theorem.

(4x)°

(5x)°

x − 10 = 1 x + 66

6x = 90 15

2x + 22 = 180 79 x=

(0.3x + 1)°

2x°

6x = 180 30

2x°

x°

(0.4x - 3)°

82°

88°

2x + 82 = 180 49 x=

0.3x + 1 = 0.4x − 3 40 x=

(x - 13)°

145°

3x + 75 = 180 35 x=

EARLY FINISHERS

(1.5x)°

2.5x = 145 58 x=

Early finishers should add in angle measures to the missing angles in all the problems in the Exercise section, keeping in mind that many angles may be equal.

Use the Pythagorean theorem or distance formula to find the missing length. 9.

10.

11. (6,4)

26 in

x 3m

x (0,-4)

24 in

10 in

CHALLENGE AND EXPLORE 10

12. A 1.3 meter ladder leans against a wall. The bottom of the ladder is placed half a meter away from the wall. How high up the wall does the ladder reach? 1.2 meters

CH AL L ENGE Find the value of the x and y. 13. y° 20°

Lighthouse Math

x° 80°

160

14.

Level H

Chapter 11

x°

140°

130

50°

y°

Exercise 7

For problems 13 and 14, encourage students to sketch parts of each figure off to the side to help them search for angle relationships without the distraction of other lines.

201

Real Angles Challenge: The purpose of this activity is to have students apply what they have learned to the real world. Students will look around the classroom and outside the window for angle relationships in real life. Provide them with a table of the different angle relationships and challenge them to find one of each. Have them sketch what they saw and say if the angles are equal. If not, they should describe how they are related (e.g., add up to 180°).

COMMON ERRORS Students may think that all the angles in a relationship are equal. Students may forget to square or take the square root when solving for an unknown using the Pythagorean theorem.

ASSESS Check the even-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

ACTIVITY

Level H | 11-8 11-8 | Transformations and Congruence

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will recognize and perform geometric transformations by describing and performing translations, reflections, rotations, and dilations on the coordinate plane, understanding their properties and effects on figures. They will describe sequences of transformations that map one figure onto another, identify similar figures, and use transformations, including dilations, to demonstrate similarity.

Translations

Have students compare their transformed figures with a partner, checking that each step was applied correctly and discussing any differences they notice. Review the transformations as a class by choosing a few volunteers to share their original coordinates and walk through the entire sequence on the board. Guiding Questions: 1. How did each transformation change your figure? Name the transformation done in each step. [The first one changed its position (translation), the second one flipped it (reflection), the third one turned it (rotation).] 2. Which transformations preserve the shape and size, and which ones change them? [Translations, reflections, and rotations keep the size; dilations change it.]

Lighthouse MATH Level H | Teacher's Guide

B’

C’

Translating a figure involves moving it up, down, left, or right a given number of units.

A’

C’

B’

-5

A(-4, 4) → A’(2, -2) B(-4, 1) → B’(2, –5) C(-1, 1) → C’(5, -5)

B’

A’

C’

-5

Dilations

Rotating a figure involves turning it 90º, 180º, or 270º clockwise or counterclockwise.

A’

Dilating a figure involves making it larger or smaller by a given scale factor.

-5

A(-4, 4) → A’(4, 4) B(-4, 1) → B’(1, 4) C(-1, 1) → C’(1, 1)

-5

Reflecting a figure involves mirroring it across the x-axis, y-axis, or another given line. A(-4, 4) → A’(4, 4) B(-4, 1) → B’(4, 1) C(-1, 1) → C’(1, 1)

-5

Rotations

A(0, 2) → A’(0, 4) B(-2, -2) → B’(-4, -4) C(2, -2) → C’(4, -4)

B’

C’

-5

A P P LY Write the numbers in the correct box based on their type. D

1. A’

B’

A B -10

C’

-10

A’

-10

reflection

C’

Level H

Chapter 11

Lesson 8

rotation

10 B’

D’

-10

A. translation

202

C’ -10 C

D C D’

Have students follow each of the following steps, each time, tracing the new image. y Move the shape 3 units right and 2 units down. y Reflect it across the y-axis. y Rotate it 90 degrees counterclockwise about the origin.

A’

-5

y Notecards y Graph paper y Polygon blocks

Provide students with graph paper and a block in the shape of a polygon. (A notecard could work also.) Tell students to start by drawing a coordinate plane on the graph paper. Then, tell students to place the shape anywhere they want on the coordinate plane and trace it.

-5

Materials

PRE-LESSON WARM-UP

Reflections

-10

B' 10

B A

-10

dilation

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Review the chart in the Learn and Connect section with students. As you go through it, remind students that the figures are changing in a predictable way based on set rules. Ask students to notice what stays the same and what changes in each example. For translations, remind students that the figure slides or shifts its position without flipping or turning. Have students figure out the directions for the translation done in the top left box of the chart, noting how point A (-4, 4) becomes A’ (2, -2). [The x-value increased by 6, and the y-value decreased by 6, so it was a translation right 6, down 6.] For reflections, point out that reflecting across the y-axis changes the sign of the x-coordinate, and reflecting across the x-axis changes the sign of the y-coordinate. Remind students that any line can be a line of reflection and that the new figure should be the same distance away from this line as the old figure. For rotations, review the rules for coordinate changes for each rotation. A 90° clockwise turn: (x,y) → (y,-x); A 90° counterclockwise turn: (x,y) → (-y,x); a 180° turn (x,y) → (-x, -y). Have students figure out which rotation was done in the bottom left box of the chart. [90° clockwise] Remind students that dilations scale a figure so that they become larger or smaller proportionally. Ask students to figure out how the figure in the bottom right box was dilated. [by a factor of 2] Ask: What happens to a figure when each coordinate is multiplied by a scale factor that is a fraction? [It would make the figure smaller.] Close by highlighting that translations shift, reflections flip, rotations turn, and dilations resize, all while maintaining the original shape of the figure. Recognizing these patterns helps students predict and check their transformations accurately.

Exercise | 11-8 Name Draw the transformed figure and label the coordinates of the new vertices. 2.

Dilate by a scale factor of 2 10

A’: ( 0 , 0 ) B’: ( -6 , 8 )

-10

Y Z’

Rotate 90º clockwise 10

L’

-10

H’: ( -2 , -8 )

F’

Translate 2 units left and reflect across the x-axis R

Q’: ( -4 , -3 )

-10

R’

x = 68 y= 2

z= 7

42° 15

70°

8. y

83° 83°

97°

z = 18

83°

97°

97° z 10

Alex states that ABCD and A’B’C’D’ are similar because ABCD was dilated by a factor of 2 and then translated 10 left and 2 up. Justin states that ABCD and A’B’C’D’ are similar because all the side lengths are proportional. Who is correct? Explain your reasoning.

-10

Both are correct, they are just using different definitions to show similarity.

Lighthouse Math

-10

x = 97 y= 4

7 5

C’ B’ A

x°

1 2

-10

CH AL L ENGE 9.

Have students create a new figure and apply any transformation to their figure. They should then give their figure to a partner to check and have their partner complete a different transformation on it. They can continue to pass the figure back and forth, transforming it differently each time.

A’

D’: ( -5 , 2 )

42° z

G’

D’

C’: ( -4 , -3 )

-10

68° 10

EARLY FINISHERS

Fill in the missing information for the congruent figures. 7.

Rotate 180º and dilate by a factor of

B’: ( -2 , -3 )

Q’

P’

A’: ( -1 , 2 )

R’: ( -3 , -7 )

H’ -10

-10

P’: ( -7 , -6 )

Provide a reference sheet that outlines each transformation and the coordinate rules for each. Have students write the original and new coordinates for each problem to help them organize their work.

-10

M’

J’

M’: ( 3 , -2 )

Z Y’

F’: ( -2 , -3 )

X W’

G’: ( 6 , -4 )

K’

-10

L’: ( 2 , -6 )

K’: ( -2 , -2 )

W X’

Z’: ( -2 , 2 )

C’

Translate 5 units right and 3 units down J’: ( -3 , -6 )

-10

Y’: ( 5 , 2 )

B’

X’: ( -2 , 6 )

A A’

C’: ( 6 , 8 )

W’: ( 5 , 6 )

-10

STRUGGLING LEARNERS

Reflect across the y-axis

Level H

Chapter 11

Exercise 8

A’

B B’ -10

D 10 D’

CHALLENGE AND EXPLORE Have students work with a partner to solve problem 9. Review the solution as a class and discuss what it means for shapes to be similar, confirming proportionality by comparing corresponding side ratios.

C C’

203

APPLY AND DEVELOP SKILLS (Practice)

ACTIVITY Transformation Relay: Students will practice transforming figures on the coordinate plane while working collaboratively. Divide the class into groups and prepare two sets of notecards, one set with the coordinates of various shapes and another set with single-step and two-step transformations. One member from each group draws one card from the coordinates stack and one from the transformations stack. Students will work together to find the new coordinates and draw the transformed figure. The first group to solve correctly earns four points, the second earns three, the third earns two, and the last earns one. Continue rounds until the target number of points is reached, rotating through group members each time.

COMMON ERRORS Students may confuse the x- and y-axes. Students may reflect an image but forget to flip its orientation. Students may confuse clockwise and counterclockwise. Students may forget to keep the image in proportion with the preimage. Students may not count the proper distance for both sides of the line of reflection.

ASSESS Check the odd-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

Work through problems 1-4 in the Apply section as a class, reviewing the key characteristics of each transformation as they appear. Next, assign students to small groups to work through problems 1-8 in the Exercise section. Finish by reviewing the solutions as a class, calling on volunteers to demonstrate selected problems on the board.

Level H | 11-9 11-9 | Volume

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will solve problems involving the volumes of cylinders, cones, and spheres by applying their respective formulas. They will compare the volumes of these shapes with identical dimensions and use their understanding to tackle real-world applications involving these three-dimensional figures.

Cylinder

Cone

Sphere

h h

Materials

V = πr2h

y Highlighters y Notecards y Dice

1 2 πr h 3

4 3 πr 3

A P P LY Match each scenario to the corresponding volume. Use 3.14 for π. Round each answer to the nearest hundredth.

Have students take out a piece of paper and sketch a cone, a cylinder, and a sphere. Ask them to label the radius, diameter, and height on each sketch. Have students check their drawings with a partner to confirm correct placement and understanding.

Then lead a brief discussion about how volume is determined for each shape, emphasizing how the radius, height, or diameter affects the final volume. Guiding Questions 1. How would changing the radius affect the volume of each shape? Which shape’s volume would change the most? [Increasing the radius increases the volume more than increasing the height by that same amount. The sphere’s volume changes the most.] 2. How are the volume formulas for cylinders, cones, and spheres the same, and how are they different? [They all use the radius of the shape, but a sphere’s radius is cubed. A cylinder and a cone have a height that is multiplied, but a sphere does not.]

Lighthouse MATH Level H | Teacher's Guide

PRE-LESSON WARM-UP

204

A candle is made in a cylindrical mold with a radius of 4 cm and a height of 10 cm.

A paper cup is shaped like a cone with a radius of 3 cm and a height of 9 cm.

A rubber ball has a radius of 6 cm.

A glass marble has a radius of 2.8 cm.

A pencil holder is a cylinder with a radius of 2.5 cm and a height of 8 cm.

A small funnel is shaped like a cone with a radius of 4 cm and a height of 12 cm.

84.78 cm3

91.91 cm3

157 cm3

904.32 cm3

502.4 cm3

200.96 cm3

Level H

Chapter 11

Lesson 9

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Use the chart in the Learn and Connect section to review the names of the shapes that were studied and their volume formulas. Ask: What do r and h represent? [Radius and height] Ask: How are the formulas for the cylinder and cone the same? How are they different? [They both use r2h, but for the cone, you divide by 3.] How is the formula for the sphere different? [It uses r3, and it is multiplied by 43 .] Have students find the volume of a cylinder and a cone, each with a radius of 2 inches and a height of 5 inches. [63.8 in3; 20.9 in3] Then, have them find the volume of a sphere with a diameter of 4 inches. [33.5 in3] When modeling a calculation, ask, “After plugging the radius and height into the equation, what operation comes first?” [Squaring or cubing the radius] Remind students that when they are given the diameter, they need to remember to divide it by 2 to find the radius before squaring or cubing it.

Exercise | 11-9 Name Find the volume of each cylinder. Use 3.14 for pi. Round each answer to the nearest hundredth. 1.

4 cm

5 cm

10 ft

Provide a reference sheet that lists all three volume formulas. Have students highlight the given information in each problem and color-code the parts of the formula that match what they are solving for.

3.5 m

6 ft

471 ft3

251.2 cm3

STRUGGLING LEARNERS

8 cm

11.25 cm

18 m

692.37 m3

565.2 cm3

Find the volume of each cone. Use 3.14 for pi. Round each answer to the nearest hundredth. 5.

6 ft

12 m

10.8 in

10 in

3 ft 5 in

282.6 in

EARLY FINISHERS

6 in

150.72 m

113.04 ft

94.2 in3

Ask students to create their own comparison problems involving cones, cylinders, and spheres. They should write one problem comparing a cone and cylinder, one comparing a cone and sphere, and one comparing a cylinder and sphere.

Find the volume of each sphere. Use 3.14 for pi. Round each answer to the nearest hundredth. 9.

10.

11.

12. 13 ft

904.32 m3

6 in

5 in

113.04 in3

523.33 in3

1149.76 ft3

Find the volume of each figure. Use 3.14 for pi and round your answers to the nearest tenth. Notice that the shapes have the same radius and height. Then, fill in the blanks to compare the shapes using their volumes. 14.

15.

13.

10 m 10 m 5m

785

261.7 m3

V = 523.3 m3

1 the volume of the 3

16. The

cone

can hold the least volume. It can hold

17. The

sphere

can hold the second least volume. It can hold more than the

the 18. The

cylinder . cone

but less than

cylinder . cylinder

Lighthouse Math

can hold the most volume. It can hold three times the volume of the

Level H

Chapter 11

Exercise 9

cone

CHALLENGE AND EXPLORE

Have students work with a partner to solve problems 13-15. Afterwards, discuss as a class that we can use these examples to compare the volumes of cylinders, cones, and spheres because they have the same radius and height. After completing problems 16-18, ask students how they can use the formulas and the pictures of the shapes to come to the same conclusions.

205

Work through problems 1 and 2 in the Apply section as a class to review the steps for solving volume problems. Have students work independently on problems 3-6. Review the answers as a class, walking through any problems students found challenging. Then have students work with a partner on problems 1-12 in the Exercise section. Review the answers as a class to ensure understanding.

ACTIVITY Roll for Volume: Students will work in pairs to practice calculating volume. Give each pair three notecards and two dice. Have students draw one shape on each of the notecards (cylinder, cone, and sphere), turn them upside down, and shuffle them. Students draw a card to determine the shape and roll the dice to generate dimensions. For cylinders and cones, one die represents the radius and the other represents the height. For spheres, add the numbers rolled to determine the radius. Students should solve at least two volume problems for each shape. Review as a class and invite pairs to share their results.

COMMON ERRORS Students may forget to square or cube the radius. Students may forget to multiply by 3.14 or the height. Students may forget to divide by 3 or multiply by 43 .

ASSESS Check the even-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Level H | 11-10 11-10 | Scatter Plots, Association, and Probability

Objective and Learning Goals

L E A R N A ND C O NNE C T

y Students will be able to analyze and interpret data by constructing and interpreting scatter plots and two-way tables, identifying patterns of association between variables, and using linear models to make predictions and solve problems.

Negative Association As x increases, y decreases.

Materials

Week

y Poster board y Stopwatches y Rulers

Test Score

Week

As the weeks increase, the test scores increase.

As the weeks increase, the test scores decrease.

This information can be used to answer questions about percentages.

Guiding Questions: 1. Do the points form any noticeable pattern, or are they scattered randomly? [Answers will vary based on the data.] 2. Does the trend appear to be positive, negative, or neither? How can you tell? [Answers will vary based on the data. A positive trend will have points getting progressively higher on the graph further to the right, and a negative trend will have points starting off high and getting lower moving to the right. No trend will have points scattered all over the graph.]

Lighthouse MATH Level H | Teacher's Guide

As the weeks increase, the test scores are random.

Completed Homework

Did Not Complete Homework

Total

Sixth Grader

Seventh Grader

Total

15 68.2% of sixth graders = 22 completed homework.

A P P LY Fill in the scatterplot based on the table and then circle the pattern of association. Study Time (min.)

Mistakes

What is the pattern of association?

12 10

8 6

4 2 10

positive association negative association

What is the pattern of association?

8 6 4

positive association

negative association

no association

Study Time (min.)

206

Pages Read New Words Learned

New Words Learned

Mistakes

Create a coordinate grid on a piece of poster board. Label the y-axis “Number of Hours of Sleep Last Night” and the x-axis “Number of Siblings.” As a class, collect both pieces of information from each student. Then, plot each student’s data point on the grid. Once all points are added, examine the scatterplot to see whether any patterns or trends appear. Discuss whether the graph shows a relationship and what that relationship might mean.

Week

Use two-way tables to represent multiple pieces of information.

PRE-LESSON WARM-UP

No Association No obvious pattern

Pages Read

Level H

Chapter 11

Lesson 10

Lighthouse Math

INTRODUCE THE LESSON (Learn and Connect) Use the chart in the Learn and Connect section to review what positive, negative, and no association look like. Highlight how the trend of the points helps determine the type of relationship. As students observe the examples, ask: “Do the points move upward, downward, or show no clear pattern as you move from left to right on the graph?” and “What does the line of best fit help you notice about how the variables change together?” Emphasize that the line of best fit is not an exact match to the data but shows the overall pattern. Discuss how the slope and y-intercept of the line of best fit help us understand the trend of the data. Ask: If this line continued, what prediction could you make? Guide students to recognize that graphs with points that are further from the line of best fit show a weaker association, while closely clustered points suggest a stronger one. Next, have students examine the two-way table and review how it organizes categorical information. Point out how rows, columns, and totals work together to tell a bigger story about the data. As students examine the table, ask: What do you notice when you compare the rows? [More students in both grades completed homework than not.] How do the totals help you answer questions about the group as a whole? [You can find percentages.] Have students find the percentage of sixth graders who did not complete their homework [31.8%] and compare that to the percentage of seventh graders who did not complete their homework. [35.7%; a slightly higher percentage of seventh graders did not complete their homework.] Ask students if they think, based on these percentages, that grade level influences homework completion. [No, the percentages are too close to show an association.]

Exercise | 11-10 Name Draw a line of best fit. Then, answer the questions about the scatter plot. 2.

A batch of cookies was left to cool after baking. The scatterplot shows the temperature of the cookies as they cooled.

STRUGGLING LEARNERS

A store tracked the number of bikes sold after different price discounts.

500

400

Number Sold

Temperature (OF)

300 200 100

Have students use a transparent ruler to draw lines of best fit so that they can clearly see the number of points above and below the line they will draw.

6 4 2

100

A. Write an equation for the line of best fit.

Cooking Time (minutes)

EARLY FINISHERS

$10

Price Discount

A. Write an equation for the line of best fit.

y = -2.5x + 450

Ask students to create lines of best fit for problems 1-2 in the Apply section and write a short statement explaining the meaning of both the slope and the y-intercept.

y=x+1

B. Describe the association. There is a negative association because as the cooling time increases, the temperature decreases.

B. Describe the association. There is a positive association because as the price discount increases, the number sold increases.

C. Predict the temperature after 120 minutes. 150 degrees

C. Predict the number of bikes sold 13 bikes after a $12 price discount.

CHALLENGE AND EXPLORE

Fill in the two-way table and answer the questions that follow. Round answers to the nearest percent. Art Club

Total

Orchestra Member

Total

Of the students in the band, what percentage are in the music club?

Of the students in the orchestra, what percentage are in the music club?

46%

50%

There is an no association between being in the in the music club and being in the band or orchestra. A student who is in the music club is equally likely to be in the band more less than the orchestra.

CH AL L ENGE 6.

A scatterplot shows a strong positive trend between hours practicing a skill and success rate, except for one point that is far below the trend. How can this single point affect the accuracy of the line of best fit?

That point can pull the line of best fit downward and make predictions less accurate.

Lighthouse Math

Level H

A line of best fit predicts that when study time is zero minutes, the quiz score will be negative. Why is this prediction not realistic?

Music Club Band Member

For problems 6 and 7, have students work in groups to discuss possible solutions. Review the results as a class and invite groups to share the reasoning behind their explanations. For problem 7, ask students why a line of best fit may give a non-realistic answer such as the one in this scenario.

A negative score is impossible, so the prediction isn’t realistic.

Chapter 11

Exercise 10

207

Have students complete problems 1 and 2 in the Apply section on their own. Discuss as a class how to determine the type of association in each graph and what visual clues support that decision. Work together as a class on problem 1 in the Exercise section, and then have students complete problems 2-5 independently. Conclude by reviewing the problems as a class, addressing any questions students found challenging and clarifying common misconceptions.

ACTIVITY Scatter Plot Posters: The purpose of this activity is for students to collect real data, create a scatter plot, and interpret the relationship between two variables. Students will work in groups of at least five and receive a poster board and a stopwatch. Each group selects one simple timed activity for all members to perform, such as jumping jacks, push-ups, counting aloud, or tossing a ball. Students should complete the activity for as long as they are able and work together to record how long they did it for and how many of the action they were able to do in that time. On the poster board, groups should label both axes, create an appropriate scale, and plot their data points to form a scatter plot. They should determine if there is an association between their variables and draw a line of best fit if applicable. To finish the activity, each group will present their poster to the class with an explanation. Follow up with a brief whole-class discussion comparing trends and observations across groups.

COMMON ERRORS Students may believe the line of best fit should connect all the data points. Students may think that a line of best fit represents actual values.

ASSESS Check the even-numbered problems in the Exercise section. Lighthouse MATH Level H | Teacher's Guide

APPLY AND DEVELOP SKILLS (Practice)

Assessments Lighthouse Math ASSESSMENTS

Level H

Name

Date

Circle rational or irrational. 01. -1.23

a. rational

b. irrational

02.

a. rational

b. irrational

Circle terminating or repeating. 7

03. 3

b. repeating

04. 5

a. terminating

a. 15.6

b. 2.5

06.

25 3

a. 25.3

b. 3.25

c. 2.5

d. 6.15

c. 8.6

d. 8.3

a. 94

4 b. 10

c. 54

d. 44

a. 3

b. 5

c. 7

d. 9

a. 2

b. 4

c. 6

d. 8

a. terminating

b. repeating

Change the fraction to a decimal. 15

05. 6

Change the decimal to a fraction. 07. -4.2

a. - 42

b. -4 52

c. -4 51

4 d. - 10

a. 3

b. 5

c. 7

d. 9

a. 2

b. 4

c. 6

d. 8

08. 0.4

Simplify. 09.

11.

216

10.

12.

Write > or <. 13. 2.2 < 2.2

14.

10 <

15.

1 2

> 0.25

16.

1 3

> 0.3

Choose the best approximation for each number. 17.

208

a. 6.1

b. 6.9

c. 7.1

d. 7.9

Assessments

18. 2

a. 2.21

b. 2.23

c. 3.14

d. 6.28

Lighthouse Math Level H Teacher's Guide

Assessments Lighthouse Math ASSESSMENTS

Level H

Name

Date

Simplify the expression. 01. x3 • x5

03.

05.

z2 • z9 z6 • z0

a4b-5c8 a6b2c0

02. y14 ÷ y8

a. x2

a. y22

b. x15

b. y6

c. x10

c. y2

d. x8

d. y7 04.

a. z7 b. z6

a-4 • a3 a-8 • a1 • a5

a. a0 b. a1

c. z5

c. a2

d. z4

d. a3 06.

a. ac2b7 8

6x9y2z-1 3xy5z-5

8 4 a. 3xy7z

b. a2b3c8

8 4 b. 2xy3z

2 7 c. abc7

9 3 c. 2xz6y

Change the number to scientific notation. 08. 0.000000233

a. 1.4 × 105

a. 2.33 × 10-6

b. 14 × 105

b. 233 × 10-7

c. 1.4 × 106

c. 23.3 × 10-7

d. 14 × 106

d. 2.33 × 10-7

Change the number to standard form. 09. 7.4 × 105

10. 8.1 × 10-5

a. 7,400,000

a. 0.000081

b. 740,000

b. 0.00081

c. 74,000

c. 0.0000081

d. 7,400

d. 810,000

Evaluate. 11.

(3.2 × 104) × (1.5 × 106)

13. (1.3 × 103) + (2.8 × 104)

12. (7.2 × 10-2) ÷ (6 × 10-5)

a. 4.8 × 1010

a. 1.2 × 10-7

b. 4.8 × 10-10

b. 1.2 × 10-3

c. 4.8 × 106

c. 1.2 × 103

d. 4.8 × 1024

d. 1.2 × 107 14. (6.5 × 10-7) − (2.5 × 10-6)

a. 29.3 × 104

a. 1.85 × 10-7

b. 4.1 × 104

b. -1.85 × 10-6

c. 4.1 × 103

c. 4 × 10-7

d. 2.93 × 104

d. 4 × 10-6

Lighthouse Math Level H Teacher's Guide

Assessments

209

07. 1,400,000

Assessments Lighthouse Math ASSESSMENTS

Level H

Name

Date

Solve the equation. 01. 15 + 5x = 5(6x − 7)

a. x = - 54

02. -10 − 4(1 − 6x) = -30

b. x = 2

03. -9.5(2x + 3) = 12.25 − 14x

c. No solution

d. Infinite solutions

a. -8.15

04. 2(-10x − 3) = -20x + 8

c. No solution

a. x = 3

7 b. x = 20

d. Infinite solutions

06. 2(5x2 − 20) = 6(x2 + 10)

a. x = 2.2 or -2.2

b. x = 3 or -3

b. x = 5 or -5

c. No solution

d. Infinite solutions

a. x = -1.7 b. x = 1.7

09. -6x = 52 + 2(-3x + 4)

a. x = -0.35 c. No solution

d. Infinite solutions

07. -8.4x + 14 = -2(4.2x − 7)

b. x = - 32

c. No solution

b. 3.25

05. 4x2 + 6 = x2 + 33

a. x = 32

08. 4 x + 8 = 2 x − 7

a. x = -60 b. x = 60

c. No solution

d. Infinite solutions

a. x = -5

10.

4x = 6

a. x = 3

b. x = 5

b. x = 9

c. No solution

d. Infinite solutions

Choose the equation that will help solve the problem. 11.

Mr. Roberts has 12 more pencils than pens. He has three times as many pencils as pens. How many pens (x) does he have?

210

a. x − 12 = 3x b. 4x = 12 c. x + 12 = 3x d. 3x + 12 = x

Assessments

12. The cost of renting a canoe is $24 an hour plus a $2 fee for life jackets. The cost of renting a bike is $15 an hour plus a $20 fee for helmets. At how many hours (x) will the cost be the same?

a. 24x − 2 = 15x + 20 b. 24x + 2 = 15x − 20 c. 2x + 24 = 20x + 15 d. 24x + 2 = 15x + 20

Lighthouse Math Level H Teacher's Guide

Assessments Lighthouse Math ASSESSMENTS

Level H

Name

Date

Determine if the relationship is a function. 01.

-1

-2

02. (0, 2) (1, 2) (2, 2) (3, 2)

a. Function b. Not a Function

Determine if the relationship is linear. 03.

-1

-5

04. y = 2x2

a. Linear b. Non-Linear

Find the y-intercept. 05.

-1

a. 0 b. 2 c. 4 d. 6

06. y = 4x + 15

a. 4 b. 15 c. 0 d. 19

a. 4 b. 6 c. 8 d. 24

08. (3, 5) and (5, 1)

a. -2 b. 2 c. -4 d. 4

a. y = 6x + 3 b. y = -3x + 6 c. y = 3x + 6 d. y = x + 3

10.

07.

-1

Write the equation of the line. 09.

-1

a. y = - 21 x + 1 b. y = -2x + 1 c. y = -2x - 1 d. y = - 21 x - 1

-5

Answer the question about the graph. 11.

Gallons of Gas in a Tank

12.

How full was the tank to start?

At what rate is the tank filling with water?

Liters of Water in a Tank

Liters of Water

Gallons of Gas

a. 12 gal

b. 40 gal

c. 42 gal

a. 2 liters/min

b. 2 liters/hour c. 5 liters/min d. 5 liters/hour

d. 50 gal 1

Hours

Lighthouse Math Level H Teacher's Guide

Assessments

Minutes

211

Find the slope.

Assessments Lighthouse Math ASSESSMENTS

Level H

Name

Date

Choose the proportion that will help solve for the missing piece. 01. A recipe uses 6 cups of flour to make 2 loaves of bread. How much flour is needed to make 8 loaves of bread?

a. 68 = 2x b. 62 = 8x

02. A car drives 100 miles in two hours on the highway. If the driver wants to go 220 miles, how many hours will he have to drive?

a. 100 = 220 2 x x b. 220 = 100 2

Use a proportion to find the missing piece. 03. Ann buys 2 pounds of cheese for $22.46. How much would she pay for 0.70 pounds of cheese?

a. $11.46 b. $44.92 c. $7.86 d. $5.73

04. A printer prints 105 pages in a minute and a half. How many pages can it print in 5 minutes?

a. 70 pages

06. y = 4x

a. 4

b. 350 pages c. 210 pages d. 525 pages

Find unit rate. 05.

b. 2

b. 8

c. 5

c. 12

d. 10

d. 16

Liters of Water in a Tank Liters of Water

07.

a. 0

a. 1 liter per min

Minutes

08. 2,000 tons of waste is produced in 5 days.

a. 100 tons per day

b. 1.5 liters per min

b. 200 tons per day

c. 2 liters per min

c. 300 tons per day

d. 2.5 liters per min

d. 400 tons per day

Choose the equation of the line. 09.

a. y = 2x

b. y = 21 x -5

10.

b. y = x

c. y = 2x + 2

-5

d. y = 21 x + 2

-5

a. y = 2x

-5

c. y = 3x d. y = 31 x

Compare the proportions. 11.

John runs 5 miles in 45 minutes. David’s miles are shown in the chart below.

Who runs faster?

miles

a. John

minutes

b. David

212

Assessments

12. The cost of eggs in Supermart is given by the equation y = 3.12x where x is the number of cartons of eggs purchased. In Big Market, 3 cartons of eggs cost $10.05.

Which is the better buy? a. Supermart b. Big Market

Lighthouse Math Level H Teacher's Guide

Assessments Lighthouse Math ASSESSMENTS

Level H

Name

Date

Determine if the system has one solution, no solution or infinite solutions. 10

01.

02.

a. One Solution

a. One Solution -10

b. No Solution

-10

c. Infinite Solutions

c. Infinite Solutions -10

03.

b. No Solution

-10

y = 3x − 5 y = 3x + 5

04.

a. One Solution b. No Solution

y = 2x - 4 y = -2x - 4

a. One Solution b. No Solution c. Infinite Solutions

c. Infinite Solutions Solve the system using any method. 3x + y = -21 x + y = -5

07.

b. (1, 3)

3x + 4y = 10 2x + 2y = 6

09.

06.

a. (0, 7)

y = -6x − 3 y = -x + 2

a. (1, 2) b. (-1, 3)

c. (-2, 5)

c. (-1, -2)

d. (-8, 3)

d. (1, 3) 08.

a. (2, 1) b. (2, 2)

4x + 9y = -6 5x + 3y = 9

a. (-2, -3) b. (3, -2)

c. (1, 2)

c. (2, 3)

d. (1, 1)

d. (-3, 2) 10.

a. (-1, 4)

a. (-1, -6) -10

05.

b. (6, 1)

-10

c. (-6, -1)

b. (2, 1) c. (1, -2) d. (-1, 2)

d. (-6, 1) -10

-10

Circle the correct answer to the question. 11.

Tank A begins with 100 liters of water which is leaking at a rate of 10 liters per minute. Tank B begins with 70 liters of water and is leaking at a rate of 5 liters per minute.

A. Circle two equations which can be written.

B. After how many minutes will the two tanks have the same amount of water?

C. How much water will be in each tank when they are equal?

a. y = 10x − 100

a. 5 minutes

a. 10 liters

b. y = -10x + 100

b. 6 minutes

b. 20 liters

c. y = -70x + 5

c. 7 minutes

c. 30 liters

d. y = -5x + 70

d. 8 minutes

d. 40 liters

Lighthouse Math Level H Teacher's Guide

Assessments

213

Assessments Lighthouse Math ASSESSMENTS

Level H

Name

Date

Circle the correct equation. 01.

a. 2x + 20 = 90 50°

(x + 30)° (x − 10)°

02.

a. 4x = 90 b. 4x = 180

b. 2x + 20 = 180 3x°

c. x + 30 = x − 10

x°

c. 3x = x d. 4x = 120

d. 2x + 20 = 50 03.

a. 2x = 90

04.

a. 7x + 45 = 90

b. 2x = 180

6x°

(x + 45)°

c. x + x = 45

x x

b. 7x + 45 = 180 c. 6x = x + 45 d. 7x = 45

d. x + x = 80

Circle the name of the angle relationship shown. 05.

a. Alternate interior a b

06. a

b. Alternate exterior b

c. Complementary

a. Alternate interior b. Alternate exterior c. Corresponding d. Same side interior

d. Same side interior Find the value of x.

07.

a. x = 10

08. x°

b. x = 20

5x° (3x + 40)°

a. x = 10 x cm 6 cm

3x°

56°

10.

b. x = 12

x ft

d. x = 16

d. x = 14

a. x = 11 ft 13 ft

c. x = 14 8 cm

b. x = 7 c. x = 10

c. x = 30 d. x = 40

09.

a. x = 4

b. x = 9 ft c. x = 5 ft

12 ft

d. x = 3 ft

Circle the correct answer to the question. 11.

One end of a 50 inch wooden beam leans against a wall at point that is 40 inches above the ground. The other end rests on the ground.

214

How far from the wall is the other end of the wooden beam? a. 20 inches b. 30 inches

12. A 100-foot rope is tied to a wall. The other end of the rope is staked in the ground 80 feet from the wall.

How high up the wall is the rope tied? a. 10 feet b. 40 feet

c. 40 inches

c. 50 feet

d. 50 inches

d. 60 feet

Assessments

Lighthouse Math Level H Teacher's Guide

Assessments Lighthouse Math ASSESSMENTS

Level H

Name

Date

Circle the name of the transformation shown. 01.

02.

a. translation

b. reflection

b. reflection c. rotation

-10

d. dilation -5

03.

04.

a. translation

-10

a. translation

b. reflection

c. rotation

d. dilation

-10

d. dilation

-10

c. rotation

-10

d. dilation

Circle the coordinates for point B’ after the transformation given. 05. Translate down three and left four.

06. Rotate 90° clockwise.

a. (1, 1) b. (-3, -4)

B -10

b. (4, 3)

c. (-7, 1)

-10

d. (1, -7)

-10

08. Reflect over the y-axis. a. (-9, 12)

b. (3, 4)

c. (-9, -12)

-10

d. (9, 12)

-10

a. (4, -3)

b. (9, -12)

-10

c. (-4, 3) d. (-4, -3)

-10

07. Dilate by a factor of 3.

a. (3, 4)

c. (3, -4) d. (4, 3)

-10

A transformation was applied to the white image to create the gray image. Choose the correct transformation. 09.

B -10

C C’ 10 B’

C’

A’ -10

12.

b. reflection over y-axis

C 10

A’ -5

d. rotation 90° clockwise

B A

-5

Assessments

c. rotation 90° clockwise d. rotation 90° counterclockwise a. Dilation by a factor of 3

B’

c. reflection over y = x

Lighthouse Math Level H Teacher's Guide

b. reflection over y = 1

-10

a. translation down 5 left 2

-10

A’

-10

d. rotation 180°

-10

A B’

B’

c. reflection over x-axis

a. translation left 8, down 2

C’

b. reflection over y-axis

A A’

11.

10.

a. translation down 10

b. Dilation by a factor of 21

C’

c. Dilation by a factor of 2 15

d. Dilation by a factor of -2

215

Assessments Lighthouse Math ASSESSMENTS

Level H

Name

Date

Write the letter of the volume formula next to its corresponding shape. 01.

02.

03.

a b. V = 31 r2h

a. V = r2h

c c. V = 43 r3

Circle the volume of each shape. 05.

a. 25.12 m3 2m

06.

10 cm

a. 2,512 cm3

b. 37.68 m3

b. 628 cm3

8 cm

c. 75.36 m3

c. 251.2 cm3

d. 301.44 m3 07.

a. 87.92 m3 4m

d. 125.6 cm3 08.

b. 43.96 m3

c. 307.72 m3

c. 14.65 m3 7m

d. 29.31 m

a. 113.04 m3 3m

11.

a. 4,186.67 in3 b. 523.33 in3

10 in

c. 28.26 m3

c. 104.67 in3

d. 12.56 m3

d. 41.87 in3 12. 12 ft

b. 26.17 in3 5 in

d. 923.16 m3

10.

b. 37.68 m3

a. 13.08 in3 5 in

b. 131.88 m3

09.

a. 43.96 m3

a. 904.32 ft3 b. 452.16 ft3

c. 32.71 in3

c. 150.72 ft3

d. 98.13 in3

d. 50.24 ft3

Choose the formula that would help to answer the question. 13. What volume of helium would be needed to fill a spherical balloon with a diameter of 8 inches?

a. V = 43 83

b. V = 43 43 c. V = 31 83

14. How much ice cream can an 8-inch tall sugar cone with a radius of 6 in hold when filled to the top?

d. V = 31 43

216

Assessments

a. V = 32 × 8 b. V = 62 × 8 c. V = 31 32 × 8

d. V = 31 62 × 8

Lighthouse Math Level H Teacher's Guide

Assessments Lighthouse Math ASSESSMENTS

Level H

Name

Date

Circle the best description of the association shown. 01.

02. a. Positive

40 30

10 2

Hours

b. Negative

20 10

c. No Association 1

a. Positive

b. Negative

Points

Occurrences

c. No Association 1

03.

Attempts

04. a. Positive

Dollars

Number Correct

b. Negative

20 10

c. No Association 1

Hours

a. Positive

40 30

b. Negative

c. No Association

Time

Circle the correct equation for the line of best fit. 06. 50

a. y = 1.5x + 45

b. y = 2.5x + 10

30 20

c. y = 5x + 10

10 2

Hours

b. y = 5x + 45

30 20

c. y = -5x + 50

d. y = 7x + 10

a. y = 5x + 50

Dollars

Number of Organisms

Days

d. y = -5x + 45

05.

Use the two-way table below to answer questions 7-10. Walks to work

Bikes to work

Drives to work

Takes a bus to work

Total

Lives in a city

Lives in a town

Total

100

07. How many people were surveyed in total?

08. Of the people who live in a city, what percent drive to work?

a. 50 b. 100 c. 33

10. Using your answers to questions 8 and 9, circle if there is an association between living in a city or a town and driving to work.

a. 27% b. 37% c. 33% d. 54%

Lighthouse Math Level H Teacher's Guide

b. 10% c. 20% d. 50%

d. 37 09. Of the people who live in a town, what percent drive to work?

a. 2%

Assessments

a. There is an association b. There is no association

217

Assessments Lighthouse Math ASSESSMENTS

Level H

Name

Date

Circle the best answer. 01. Change 56 to a decimal.

03. Simplify the expression.

a. 0.3

02. Simplify 81

b. 0.56

b. 9

c. 0.6

c. 40.5

d. 0.83

d. 6,561

a. 6x2y

04. Multiply.

b. 4x2y (2.4 × 105) × (3.8 × 104)

d. 4x6y

05. Solve for x.

a. x = -1

06. Solve for x.

a. y = -2x + 6 b. y = -2x − 6

8x + 5 = 7

c. x = 5.5 d. x = 6

08. Find the slope of the line that passes through (6, 7) and (9, -2).

a. 3 b. -3 c. 31

c. y = 6x − 2 -10

a. x = 4.5 b. x = 5

c. x = 91

d. x = -9

07. Find the equation of the line.

c. 6.2 x 105 d. 6.2 x 109

b. x = 1 4(3x − 5) = -25 + 7x

a. 9.12 x 105 b. 9.12 x 109

c. 4y x6

8x-4y3 2x2y2

a. 8

d. - 31

d. y = 6x + 2

-10

09. The number of students in a school is proportionate to the number of teachers. If there are 15 teachers in a school with 300 students, how many teachers would be in a school with 500 students? 11.

Solve the system. -2x − 9y = -25 -4x − 9y = -23

a. 10

10. Choose the statement that is not true about the proportional relationship modeled by y = 5x.

b. 15

a. It passes through the point (0, 0).

c. 20

b. It has a unit rate of 5.

d. 25

c. It passes through the point (3, 10). d. It passes through the point (4, 20).

a. (1, 9)

12. Solve the system. y = 2(3x − 2) x=3−y

b. (-1, 9) c. (1, 3) d. (-1, 3)

218

Assessments

a. (1, 2) b. (-1, 2) c. (2, 1) d. (2, -1)

Lighthouse Math Level H Teacher's Guide

Assessments Lighthouse Math ASSESSMENTS

Level H

Name

Date

Circle the best answer. 13. Find the value of x.

14. Find m ABC. C

a. x = 180 3x − 10 2x + 10

a. 60°

b. 100°

b. x = 100 E

c. x = 52

60°

c. 120° d. 180°

d. x = 36 H

15. How was quadrilateral ABCD transformed?

16. What will be the new coordinates of point A if ABC is dilated by a factor of 21 ?

-5

a. reflection over x = 1

B’

b. reflection over the y-axis

D’

c. rotation 90° counterclockwise

-10

c. (2, 2)

-10

d. translation left 2 up 1 17.

b. (2, 4) A

d. (1, 2)

18. Find the volume.

Find the volume. 8m

a. V = 461.58 in3

a. V = 904.32 m3 6m

b. V = 301.44 m3 c. V = 75.36 m3

7 in

b. V = 153.86 in3

3 in

c. V = 131.88 in3

d. V = 226.08 m

d. V = 197.82 in3

19. Find the volume.

20. What percent of people ages 15-25 like spicy food? a. V = 33.5 ft3 Ages 15-25

b. 30.8%

c. V = 4.2 ft

Ages 26-36

c. 69%

Total

135

d. 58.6%

d. V = 8.4 ft

21. Circle the correct association for the scatterplot.

22. Circle the equation for the line of best fit.

a. positive association

Molecules

b. negative association c. no association d. best association Minutes

Lighthouse Math Level H Teacher's Guide

a. y = 21 x + 1

b. y = 21 x + 21

Molecules

a. 14.8%

b. V = 16.7 ft

2 ft

Likes Dislikes Total spicy spicy

c. y = x + 9 Minutes

Assessments

d. y = x + 1

219

C’

A’ A

a. (1,4)

Glossary Term

Description

Chapter

Term

Description

Adjacent angles

angles that have a common side and share a vertex

7.1

Cone

9.2

Alternate exterior angles

equal angles on opposite sides of a transversal but outside the parallel lines

7.2

a 3D shape with one circular base that extends to a single point

Congruent figures

8.1

Alternate interior angles

equal angles on opposite sides of a transversal but inside the parallel lines

7.2

figures that have the same shape and size with equivalent angle and side measurements

Coordinate plane

4.1

Angle

the space between two lines that cross; measured in degrees

7.1

a grid that helps to locate points using two number lines: x-axis and y-axis; divided into 4 quadrants labeled l, ll, lll, lV

Approximation

a number that is close to another number and is used in its place

1.5

Corresponding

sides or angles in congruent or similar shapes that match up with one another

8.1

Association

the relationship between two sets of data

10.1, 10.4

Corresponding angles

7.2

Base

a number multiplied by itself in a power

2.1

equal angles in the same position on a transversal relative to different parallel lines

Counterclockwise

8.4

the item that gives you more for your money, found by comparing unit rates

5.1

the opposite direction that a clock's hands move

Cube root

the inverse of cubing

1.2