Issuu

Level H

Standard Edition

This book belongs to Name

Address

Telephone

School

Teacher

Grade

LEVEL H

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

Lighthouse C URRI C ULUM

Credits Curriculum Writers

Review Team

Layout & Design

Yehudis Leitner Curriculum Coordinator

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Lighthouse Math Level H • ISBN 978-1-955773-87-4 ©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

A better way to teach Dear Educator, Welcome to the Lighthouse Math Curriculum! What makes our curriculum so unique? Lighthouse Math uses a scaffolded approach to learning and mastering math skills. When provided with a solid foundation, students can retain more information and prepare for the next level of skills. Instead of separate workbooks and textbooks, students have everything they need built into one place: a softcover book containing 11 chapters, with each lesson containing review, new skills, and practice. All lessons include step-by-step instructions for clarity, giving all teachers - new as well as seasoned - the tools for success. The books are custom illustrated, providing a vibrant learning experience. They 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. We at Lighthouse CurriculumTM are committed to providing support and guidance to our educators. We look forward to hearing from you and are available to answer any questions you may have.

Sincerely,

Lighthouse Curriculum Team

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

4 5

1 6

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

Lesson 1

the square root

Lighthouse Math

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 �� Table of Contents

Level H

22 24 26 28 30 32 34 36 38 40 42

Lighthouse 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

Lighthouse Math

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 |

Level H

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 �� Table of Contents

Level H

142 144 146 148 150 152 154 156

Lighthouse 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

Glossary

Lighthouse Math

Level H

188 190 192 194 196 198 200 202 204 206 208

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.

Chapter 1 | Skill Checklist

Skill 1: Operations with Rational Numbers Multiplication

Solve.

Same Sign: Positive

−

-5 × 8 =

-3 + 7 =

−

14 ÷ -2 =

-8 − 4 =

-0.5 + -1.9 =

-2.3 × -0.2 =

−

-2.5 − (-1.2) =

-8.1 ÷ -9 =

Different Sign: Negative

−

Division Same Sign: Positive

−

Different Sign: Negative

−

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

out 8 correct

Skill 2: Exponents

Base

Exponent

Match each exponent to the correct expression. 1.

Two to the sixth power

4×4×4×4×4

2×2×2×2×2×2

5×5×5

3×3×3×3×3

5×5×5×5

Solve.

2. 34 =

3. 83 =

4. 25 =

I can solve exponents.

out 4 correct

Skill 3: Converting Between Fractions and Decimals

0.75 =

75 ÷ 25 3 = 100 ÷ 25 4

3 = 3 ÷ 8 = 0.375 8

Convert the decimals to fractions in the simplest form. 1. 2.03 =

1 = 5

3 = 16

8 = 25

I can convert decimals to fractions and fractions to decimals.

out 6 correct

Level H

3. 20.015 =

Convert the fractions to decimals. 4.

2. 0.46 =

Chapter 1

Skill Checklist

Lighthouse Math

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 = ?

7×?= 7× = 63 7× = 70

170 ÷ 3 = ?

481 ÷ 5 = ?

64 is closer to

, so 64 ÷ 7 ,

will be a little greater than 64 ÷ 7 = about

× ? = 170 × = 150 × = 180

170 is closer to

, so 170 ÷ 3

will be a little

than

170 ÷ 3 = about

×?= × = × =

is closer to

than

will be a little ÷

, so 481 ÷ 5 .

= about

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+1=b+1 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

a=b

Skill 6: Solving One-Step Equations Addition

Subtraction

x + 9 = -4 −9 −9 x = -13

x−3=6 +3 +3 x=9

Multiplication

Division

5x = 25 ÷5 ÷5 x=5

3 × x = -2 × 3 3 x = -6 out 4 correct

Lighthouse Math

Solve. 1.

m+4=7

9x = 81

y =5 2

p − 3 = 17

I can solve one-step equations.

Level H

Chapter 1

Skill Checklist

1-1 | Square Roots

PREREQUISITE SKILLS

Solve.

DAI LY REV I E W

SPIRAL REVIEW

3×

= 12

7×

102 =

= 49

9×

122 =

= 81

4×

82 =

= 16

Squaring numbers. 1.

62 =

L E A RN A ND C O NNECT

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. 1. Which expression can you use to solve 49?

7×7

8×8

2. Which expression can you use to solve 4?

4×4

3×3

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

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

2. What number multiplied by itself equals 121?

3. Which expression is NOT equal to the others?

2×2

6×6

Which expression is NOT equal to the others?

Simplify. 5. 25 =

6. - 16 =

7. 1=

8. 81 =

9. - 36 =

10. - 4 =

11. 100 =

12. - 64 =

13. 400 =

14. - 9 =

15. 49 =

16. - 121 =

Fill in the missing numbers. 17. 9 =

18.

21. 1,600 =

22. (-8)2 =

= 25

19. -

= -8

20. 62 =

23.

24.

= 100

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?

B. What is the perimeter of the garden?

CH A L L E N G E 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?

Lighthouse Math

Level H

Chapter 1

Exercise 1

A. What is the length of one wall?

1-2 | Cube Roots

PREREQUISITE SKILLS

Solve.

DAI LY REV I E W

SPIRAL REVIEW

43 =

23 =

49 =

53 =

81 =

Solve. 1.

100 =

L E A RN A ND C O NNECT 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 Choose the correct answer. 3

6×6

6×6×6

6×3

7×7×7

14 × 7

7×7

7×3

1. Which can be used to find 216?

2. 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

Exercise | 1-2 Name Circle the correct answer. 3

10 × 10 × 10

10 × 3

100 × 10

5×5×5

25 × 5

10 × 5

1. Which expression CANNOT be used to find 1,000? 2. Which expression CANNOT be used to find 125? Simplify. 3

4. 8 =

10.

3. -512 = 6. 1 = 9. 512 =

5. - 729 =

7. -64 =

8. 27 =

11. 216 =

125 =

Fill in the missing numbers. 3

12. -27 = 16.

1,000 =

13.

14.

= 729

17. (-4)3 =

18.

= -8

15. 63 =

19.

Use cube roots to solve. 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?

CH A L L E N G E 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? B. What is the side length of the second box? Hint: Use a perfect cube you know to estimate the length. 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.

Lighthouse Math

Level H

Chapter 1

Exercise 2

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.

1-3 | Decimal Patterns

PREREQUISITE SKILLS

Write the place value of the underlined digit.

DAI LY REV I E W

1. 0.245

SPIRAL REVIEW

2. 23.6

3. 0.896

4. 1.275

5. 812.59

Find the cube root. 3

1. 1

2. 64

- 27

L E A RN A ND C O NNECT Some fractions convert to repeating decimals. Repeating decimals can be converted back to a fraction. 1 can be written as a decimal using 3 long division: 0.3333 3 1.0000 − 9 10 − 9 10 − 9 10 − 9 1

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 × 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. x=

6 2 or 9 3

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

x = 0.6

2 . 3

A P P LY Tell whether the fraction is a terminating decimal (T) or a repeating decimal (R). 2 1. 5 T

1 2. 11 R

19 3. 4 R

21 4. 10 R

5 5. 12 R

7 6. 8 R

1 7. 6 R

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

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

4 2. 5

2 3. 3

1 4. 6

5 5. 9

6 6. 12

Change the following decimals into fractions. Write in the simplest form. 7. 0.2

8. 0.5

9. 1.4

10. 2.1

11. 0.8

12. 3.3

13. 0.6

14. 3.7

Use the given information to find patterns. 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?

16. Nick says that since 31 = 0.3, therefore 32 must be 0.6. Do you agree or disagree? Why?

17. Complete the table, then describe any patterns you see. 2 11

3 11

0.09

0.18

0.27

4 11

5 11

9 11 0.54

0.63

10 11

1 11

0.72

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

Lighthouse Math

B. 0.17

C. 0.24

Level H

Chapter 1

Exercise 3

1-4 | Rational and Irrational Numbers

PREREQUISITE SKILLS

Find the square root.

DAI LY REV I E W

SPIRAL REVIEW

- 25 =

49 =

100 =

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

1 = 2

1 = 4

1 = 9

0.4 =

1.7 =

L E A RN A ND C O NNECT Rational numbers can be written in fraction form. This includes all terminating and repeating decimals.

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

Integers

Irrational Numbers

-2, 0, 1, 25, -47

2, π, 47

Whole Numbers

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

Put each number below in the box that describes it most specifically. 1. 4. 7. 10.

0.5 1 3 4 5

-6

-2.7

Rational numbers

Integers

11. π

12.

13. 3.6

14. -1

15. -4

Irrational numbers

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

Exercise | 1-4 Name Write the term that most specifically describes the given number: whole number, integer, rational number or irrational number. 1. 5

2. 0

3. -3

4. -7

5. 10

6. 4.5

7. 7.25

8. 49

9. 2

π 10. 2

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

12. - 16

13. 16 15. -8.5 17. -12

11 14. 3

rational number

16. 3.1416

irrational number

3 19. 4

18. -2.75 20. 3

whole number

21. 25

22. -6

integer

23. 2.3

24. 5

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.

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. A. The circumference of the fountain is 8π. B. The area of the square garden is 62.

CH A L L E N G E 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 =

Level H

Day 3:

Chapter 1

Exercise 4

×e=

Answer the questions about the rational and irrational numbers.

1-5 | Understanding Irrational Numbers

PREREQUISITE SKILLS

Place the following numbers in the correct spot on the numberline

DAI LY REV I E W

SPIRAL REVIEW

-1

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

4.2

1 3

2.75

-7.4

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

64 =

π=

100 =

25 =

L E A RN A ND C O NNECT 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.

A P P LY Use perfect squares to approximate the value of each irrational number. Then place them on the number line. 1. 20

2. 80

20 is between 16 and 25 .

80 is between

20 is between 4 and

80 is between

20 is about

-10

-8

-9

3. - 33 and and

80 is about

-7

-6

-5

-4

-3

-2

-1

- 33 is between -

- 33 is between

- 33 is about

and -

and

. .

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

Exercise | 1-5 Name Use perfect squares to approximate the value of the irrational numbers to one decimal place. 1. 10

2. - 75

3. 118

4. 17

5. - 140

6. 40

7. 62

8. - 53

Plot the numbers on the number line below. 9. 2π

-10

10. -3π

-9

-8

-7

11. π

-6

-5

-4

12. -2π

-3

-2

-1

π 14. 2

13. -π

15. π2

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?

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

and

πd 2

B. Approximate the length of the balloon arch.

C. A pproximate the length of the pool.

A. T he 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.

CH A L L E N G E 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 Given that side a = 5 and side b = 7, approximate the length of side c.

Lighthouse Math

Level H

Chapter 1

Exercise 5

1-6 | Comparing and Ordering Rational and Irrational Numbers

DAI LY REV I E W

PREREQUISITE SKILLS

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

SPIRAL REVIEW

7.5

7.45

-0.346

1.02

1.002

1 3

-0.345 1 4

6.03

6.3

1,720

1,270

Approximate the value of the irrational numbers to one decimal place. 1. 28

2. 79

4. 3π

3. 15

L E A RN A ND C O NNECT A number line can be used to compare and order rational and irrational numbers. 6.23, 6 31 ,

36,

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

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

6 31

6.3

48 6.4

6.5

1 7 10 2

6.6

6.7

6.8

6.9

3.3

100

A P P LY

Compare the numbers with >, <, or =. 1.

1 6

3.14159

- 5

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.

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

1 12. is an irrational number. 3

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

Exercise | 1-6 Name Place the numbers on the given number line. 1. 5, 4, 8, π,

10 3 , 27 3

2.5

3.5

3 2. 48, 2π, 36, 6 , 35, 5.3 4

5.5

6.5

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

8.5

9.5

Write the correct symbol: >, <, or =. 3

4. - 2

-1

5. 8

-1

6. 24

7. 3π

-5.4

-5.3

11.

12. 2π

-3.18

15. -0.8

5.2

18.

9 10

0.99

10. - 16 1 13. 3 17 16. 10

4 0.3

14. -3.81

1.75

17. 5

1 5

4.1

-22.22

-222.2 4 3 4

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.

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?

CH A L L E N G E 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?

Lighthouse Math

Level H

Chapter 1

Exercise 6

1-7 | Review

L E A RN A ND C O NNECT Square Root

Cube Root

a2 = 100 a × a = 100 100 = a

b3 = 64 b × b × b = 64 3 64 = b

a = 10

Types of Numbers

Rational Numbers Can be written as a fraction

- 51 , 1.3, 9 21 , .6

b=4

Integers

Fractions as Repeating Decimals

Do not have fractional parts

Repeating Decimals as Fractions

-4, 0, 5, 31, -17

0.1

2 3

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

Whole Numbers

x = 0.1

Are only positive or 0

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

0, 1, 2, 3, 4

1 9 1 0.1 = 9

Irrational Numbers

= 0.6

Go on forever with no pattern

3, π, 45

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

Write if the fraction represents a terminating or repeating decimal.

1. 6.23

2 2. 6 3

1 7. 8 9

1 8. 4 3

3. 6

4. 6.3

1 9. 2 5

3 10. 2

5. 6π

6. 6.0003

3 11. 1,000

100 12. 6

Level H

Chapter 1

Lesson 7

Lighthouse Math

Exercise | 1-7 Name Simplify. 3

5. - 100

6. 144

11. 27

12. 25

1. 64

2. - 16

3. 1

4. 125

7. 49

8. 4

9. - 36

10. 216

Change the fraction to a decimal. 13.

5 3

14.

7 2

15.

1 6

16.

1 5

17.

4 12

18.

24 5

19. 2

3 10

20.

2 9

21.

4 11

22.

1 4

Change the decimal to a fraction. 23. 3.2

24. 0.7

25. 2.3

26. 9.5

27. 1.6

28. 1.4

29. 2.1

30. 0.8

Put the rational and irrational numbers on the number line. 18

2.5

2.4

4.45

3.74

3.5

4.5

CH A L L E N G E 32. Bobby wants to cover a square countertop with contact paper. He knows the countertop has an area of 900 square centimeters. 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.

Lighthouse Math

Level H

Chapter 1

Exercise 7

31.

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.

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

constant

variable

operation

7y − 3

8a − 9b + 6

Term

Coefficient

Variable

Operation

Constant

Circle the like terms. 3.

4g 7h -h 4r

3a2

8a -a2 3b 3a3

I can correctly label the parts of expressions.

out 4 correct

Skill 2: Place Value Write the place of the underlined digit.

M Hth TTh Th

tth hth m

5 7 8 3 2 . 9 4 © Lighthouse Curriculum. Copying strictly prohibited.

5,007,895,230

104.00753

0.520893

7.058328

Millionths

Hundred Thousandths

Thousandths

Ten Thousandths

Tenths t

Hundredths

Decimal Point

Tens T

Ones

Hundreds

Thousands

Ten Thousands

Millions

Hundred Thousands

Place Value

Write the value of the underlined digit. 4.

56.2069

985,540,000.9

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

Solve. 1.

24 × 1,000 =

8.596 × 100 =

32.007 × 10 =

4,830 ÷ 10 =

7,820 ÷ 1,000 =

9.26 ÷ 100 =

I can multiply and divide by multiples of ten.

out 6 correct

Level H

Chapter 2

Skill Checklist

Lighthouse Math

Name

Skill 4: Rounding Numbers Round to the nearest hundredth.

1.382

2.458

1.382

2.458

1.38

2.46

Round each number to the given place value. 1.

Round 8,652,258.25 to the nearest hundred thousand:

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:

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

1 4

2 3

I can write the reciprocal of a number.

out 7 correct

Skill 6: Order of Operations P

Parentheses

Exponents

M/D

Multiplication/Division

A/S

Addition/Subtraction

out 4 correct

Lighthouse Math

Use PEMDAS to solve. 1.

3+2×7=

7 + 82 × 5 =

5 × 4 − (3 + 1) ÷ 2 =

33 + 3(9 − 3) =

I can solve using the order of operations.

Level H

Chapter 2

Skill Checklist

2-1 | Multiplication with Exponents

DAI LY REV I E W

PREREQUISITE SKILLS

Solve.

SPIRAL REVIEW

(7 − 4) + (-5)2 × 2 =

(4 − -3)2 + -14 ÷ 7 =

25 ÷ -5 + -4 × 32 =

(9 × 2) ÷ 6 − -10 + 23 =

Compare the numbers with >,<, or =. 1.

1 3 7 5 8 3

π 24

L E A RN A ND C O NNECT 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

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.

(3 × 3) × (3 × 3 × 3 × 3)

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

A P P LY

For each expression, circle the terms with the same base. 1.

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 =

10. 83 × 84 × 87 =

11. 63 × 23 × 25 =

12. 57 × 58 =

13. 35 × 33 × 32 =

14. 97 × 32 × 94 =

15. 46 × 411 =

16. 62 × 6 × 69 =

17. 52 × 58 × 65 =

18. 25 × 2 =

19. 710 × 74 × 75 =

20. 109 × 105 × 104 =

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

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

2. p3 × p2 × p5 =

3. x5 × y2 × x3 =

a8 × a6 =

g15 × g8 × g7 =

a6 × b7 × b2 =

b10 × b =

w4 × w2 × w15 =

t5 × t3 × z7 =

10. c9 × c8 =

11. r8 × r10 × r15 =

12. r12 × s3 × s10 × r8 =

13. 22 × x3 × x3 =

14. 83 × 87 × s6 × s10 =

15. 55 × v10 × y11 × 56 × v8 =

16. 48 × y2 × 45 =

17. 93 × x2 × 910 × x14 =

18. a3 × b6 × 23 × 27 × a5 =

19. 65 × a9 × a8 =

20. 45 × d7 × d × d2 =

21. h7 × 74 × g6 × g10 × g2 =

22. 103 × w5 × w10 =

23. f8 × 312 × f11 × f4 =

24. 28 × p3 × s5 × p4 × s10 =

25. 56 × 514 = 5

26. 94 × x3 ×

27. 57 × d

× 59 × d10 = 516 × d18

28. 4

29. 58 × 5

30. 94 ×

Find the missing value.

× 44 = 410

31. 79 × 7 34. 3

= 94 × x9

× a8 = 514 × a8

32. 77 × 74 × b10 = 7

= 715

35. 8

× 38 = 313

× b10

× 85 × c10 = 817 × c10

× 93 × p2 = 97 × p14

33. 68 × a10 × a6 × a 36. 43 × y

= 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. © Lighthouse Curriculum. Copying strictly prohibited.

Is Sara correct? Explain why or why not.

CH A L L E N G E 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?

am • an = ax

Lighthouse Math

Level H

Chapter 2

Exercise 1

2-2 | Division with Exponents

PREREQUISITE SKILLS

Simplify.

DAI LY REV I E W

1. 5.

SPIRAL REVIEW

12 = 16 9 = 18

2. 6.

18 = 27 24 = 36

3. 7.

21 = 35 42 = 60

50 = 65 35 = 56

4. 8.

Simplify the expression. 1.

34 × 36 × 35 =

25 × 53 × 26 =

73 × b5 × b4 =

52 × 5 × 58 =

a7 × 64 × 62 × a2 =

910 × 45 × 46 × 94 =

L E A RN A ND C O NNECT 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. 55 52

Same base

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.

55 52

5×5×5×5×5 5×5

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

A P P LY Simplify the expression. 1.

37 =3 33

109 = 106

45 = 4

610 =6 68

911 = 97

710 = 75

65 × 38 =6 62 × 33

57 × 313 =5 32

95 × 77 × 810 =9 92 × 74 × 83

10.

89 × 714 = 75

11.

35 × 27 = 32 × 25

12.

57 × 38 × 23 = 56 × 35 × 2

×3

×7

×8

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

Exercise | 2-2 Name Simplify the expression. x8 1. 5 = x a11 4. 4 = a b10 7. = b 3x10 10. 5 = x 22a8 13. 2 = a b8 16. 3 5 = 5b

w12 2. 11 = w y5 5. 3 = y a5 × b4 8. 2 = a 4d9 11. 4 = d 65y7 14. 2 3 = 6 y 45 × a6 17. 3 = a

t5 × s4 3. 4 2 = t ×s f11 × g14 6. 11 = g 10 a × b3 × c7 9. 9 = a × b2 × c3 58 × t8 12. 3 4 = 5 ×t 616 × g4 15. 10 = 6 45 × a9 × b11 18. 3 6 = 4 × a × b3

52d6 20. 2 = 52d d 7 y9 23. 2 3 = 7y6 7 y 815 × a8 26. = 815 × a2 a

104 × t12 21. 2 5 = 102 × t 10 × t 54 × p5 = 5 × p5 24. 5 99 × a10 × b 27. = 93 × a6 × b3 9 × a4 × b2

Find the missing value. x12 19. = x2 x 3 22. 4 = 34 3 a3b 25. 5 = a3b6 b

Solve.

“Since 6 divided by 2 is 3,

46 : 42

29. Lisa simplifies the expression

46 = 43.” 42

58 : 53

“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.

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

Lighthouse Math

Level H

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.

Chapter 2

Exercise 2

28. Danny simplifies the expression

2-3 | Negative and Zero Exponents

PREREQUISITE SKILLS

Find the reciprocal.

DAI LY REV I E W

1. 5. SPIRAL REVIEW

2 = 5 10 = 11

1 = 8 5 = 6

2. 6.

4 = 3 1 = 9

3. 7.

4. 8.

6 = 7 2 = 13

Simplify the expression. 1.

48 = 45

95 = 94

x11 = x6

a14 = a6

L E A RN A ND C O NNECT 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

5 = 25

x =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.

3-3

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

1 33

1 x-4

Negative Power Property

x 1

a-m =

1 am

Some expressions require multiple steps to simplify.

43 • 45 • x-4 48 • x2

48 • x-4 48 • x2

40 • x-6

1 x6

Multiply.

Divide.

Apply zero and negative power properties.

A P P LY Circle the correct answer. 1. What is the value of 70? 0

5. Simplify (x2y3)0. x2y3

2. Which expression equals 1? 01

101

6. Simplify 2x0. 2

3. Evaluate 4 • 30. 4

7. What is x0 + x0? 0

4. Which expression does not equal 1? 1000

5a0

8. 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

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

1 2. -4 = 2

63 3. 5 = 6

4. 10-11 =

1 5. -7 = 4

52 6. 9 = 5

7. 5-8 =

1 8. -8 = 6

94 9. 10 = 9

10. 7-4 =

1 11. -15 = 10

47 12. 10 = 4

13. 9-11 =

14.

1 = 9-6

7 15. 3 = 7

16. a-5 =

1 17. -3 = x

x 18. 8 = x

19. y-8 =

1 20. -4 = b

y6 21. 11 = y

22. c-5 =

1 23. -2 = d

c5 24. 12 = c

25. a-10 =

1 26. -20 = a

a2 27. 5 = a

28. b-14 =

29. a-2 × a0 × a-3 =

30. 6-5 × y4 × 6-7 × y0 =

31. 40 × 5-2 × b-9 × b4 =

a-1 × a-2 32. 0 = a

8-10 × 86 33. -4 0 = 8 ×8

8-3 × d8 34. 5 = 8 × d-7

35. 50 × b-2 × b3 =

36. 93 × 9-7 × 98 × b-2 =

37. q0 × q-1 × 53 × 5-5 × 5-4 =

3-1 × 36 38. 4 -6 = z ×z

2-4 × b5 39. 7 = 2 × b9

40.

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

Solve. 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.

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

CH A L L E N G E 43. Simplify. Show your work.

Lighthouse Math

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

Level H

Chapter 2

Exercise 3

2-4 | Order of Operations with Exponents and Roots

DAI LY REV I E W

PREREQUISITE SKILLS

Solve.

SPIRAL REVIEW

(10 − 4) + (-1)2 × 2 =

(4 − 8)2 + -14 ÷ 7 =

45 ÷ -5 + -3 × 32 =

(6 × 2) ÷ 3 − -6 =

b-6 =

c-4 =

Rewrite with positive exponents. 1.

4-3 =

8-5 =

1 = b-2 1 = a-8

1 = 8-3 1 = 6-5

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

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. 9 + 8(5) − 6

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

5 + 30 8− 13 − 6

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

9 + 40 − 6 49 − 6 7−6=1

8−

35 7

8−5=3

A P P LY Add parentheses around the parts of the expression that should be grouped. Then simplify. 24 + 9 = 1. 10 − (-1)

2. 8(3) + 12 =

62 − 17 = 3. 3(3)

-14 + -6 = 5. 2(5)

6. 2(4) − 7 =

7 + 16 − (-2) =

= =

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

Exercise | 2-4 Name Simplify the expression. 12 + 8(2) 1. 5 − (-2)

18 − 30 2. 2 − 1 4

6(10) +5 3. 8 − 12

121 4. × 52 − 3 11

5. -1(9 − 25) + 3

6. 10 − (-39) − 2.5

7. 5(15) + 2(12.5)

8. 64 + (32 − 2)

52 + 19 − 22 9. 4−9

101 − 72 + 12 10. -7 − 6

20 − 4 × (4 + 6) 11. 2 3 − 10

12. (1 − 4)² + (6 − 2)²

Write an expression and simplify. 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?

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.

CH A L L E N G E Simplify. 15.

16.

(12 − 8)2 + (45 − 48)2 (5 − 4(8))³

Lighthouse Math

Level H

72 + 9 + 1 14 − (-13)

Chapter 2

Exercise 4

2-5 | Converting Numbers to Scientific Notation

DAI LY REV I E W

PREREQUISITE SKILLS

Multiply.

SPIRAL REVIEW

0.61 × 100 =

1.5 × 1000 =

0.003 × 100 =

18 × 100 =

0.2 × 10,000 =

6 × 100 =

Simplify. 1.

4 + 12(2) − 3 + 2

1 + 52 −3 3 − 16

8(4) − 2 3−8

- (-4) − (-8) + 10

L E A RN A ND C O NNECT Scientific notation is a way to rewrite very large or very small numbers using powers of ten. Converting from Standard Form to Scientific Notation Rewrite the number as a multiplication problem.

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

2 × 104

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

85,000

8.5 × 104

0.0025

25 × 10-4

100,000

1 × 105

0.00009

9 × 105

Correct?

Fix Incorrect Values

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

Level H

Chapter 2

Lesson 5

Lighthouse Math

Exercise | 2-5 Name Rewrite each number in scientific notation. 1. 3,200 =

2. 420 =

3. 35,000 =

4. 0.005 =

5. 0.0000003 =

6. 0.000045 =

7. 78,000 =

8. 9,100 =

9. 0.003 =

10. 0.00041 =

11. 0.71 =

12. 8,900,000 =

13. 6,000,000 =

14. 1,000 =

15. 0.082 =

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

17. 1.2 × 10 -4 =

18. 9.8 × 105 =

19. 7.1 × 104 =

20. 5 × 106 =

21. 1.5 × 10 -3 =

22. 6 × 10 -2 =

23. 4.3 × 10 -1 =

24. 2.4 × 104 =

25. 2.5 × 105 =

26. 3.9 × 10 -5 =

27. 7.75 × 10 -6 =

28. 9 × 10 -3 =

29. 6.7 × 103 =

30. 8 × 104 =

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

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.

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

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

Lighthouse Math

Level H

36. Sara writes the number 0.00042 into scientific notation: 4.2 × 10-3 Is Sara correct? Explain why or why not.

Chapter 2

Exercise 5

2-6 | Comparing Numbers in Scientific Notation

PREREQUISITE SKILLS

DAI LY REV I E W

Write the numbers in order from least to greatest.

SPIRAL REVIEW

0.001, 0.12, 0.0012

1.801, 1.081, 1.81

0.45, 0.405, 0.054

0.002, 0.00022, 0.02

0.62, 0.6, 0.602

0.541, 0.451, 0.154

Convert to scientific notation. 1.

0.00015

40,600

0.099

8,000,000,000

0.034

5,800

L E A RN A ND C O NNECT 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. 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

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

The numbers ordered from least to greatest: 2.5 × 10 -2, 2 × 103, 2,050, 25,000

A P P LY Compare each value using <, >, or =. 1.

4.6 × 105

3.2 × 106

9 × 104

8.1 × 104

6.3 × 10 -3

6.3 × 10-2

1.5 × 108

7.2 × 10 -5

9.1 × 10-6

2.4 × 106

2.3 × 107

5.01 × 10 -1

5.01 × 10-1

3.9 × 102

3.9 × 103

4.2 × 10 -4

4.21 × 10-4

10. 8 × 107

Level H

Chapter 2

Lesson 6

7.99 × 107

Lighthouse Math

Exercise | 2-6 Name Compare each value using <, >, or =. 1. 5.2 × 103

5,000

2. 7 × 102

700

3. 3.4 × 104

34,000

4. 8.1 × 102

8,000

5. 2 × 106

200,000

6. 4.5 × 101

7. 9.9 × 105

999,000

8. 6 × 102

1,000

9. 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 × 105

601,000

14. 2.5 × 103

25,000

15. 9.1 × 102

910

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

18. 0.0003; 3.2 × 10 -4; 2.9 × 10-4

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

20. 6.5 × 103; 5,900; 7,100

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

22. 3,100; 2.9 × 103; 3.3 × 103

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

24. 0.006; 5.8 × 10 -3; 0.005

25. 7000; 6.9 × 103; 6,500; 7.1 × 103

26. 920; 9.1 × 102; 9.3 × 102

27. 0.34; 3.3 × 10 -1; 3.5 × 10-1; 0.36

CH A L L E N G E 28. A student says: "5.2 × 105 is less than 6.3 × 104 because 5.2 is less than 6.3." Is the student correct? Explain why or why not using place value and powers of ten.

Lighthouse Math

Level H

29. Two students write the following expressions: Student A: 3.4 × 106 Student B: 34 × 105 Are they equivalent? Explain your answer with a conversion or simplification.

Chapter 2

Exercise 6

Convert to scientific notation and then order the numbers from least to greatest.

2-7 | Multiplication and Division with Scientific Notation

DAI LY REV I E W

PREREQUISITE SKILLS

Solve.

SPIRAL REVIEW

45 ÷ 100 =

960 ÷ 10,000 =

0.5 × 1,000 =

94 × 10 =

89 ÷ 1,000 =

2 ÷ 100 =

1.2 × 104

1,200

4.8 × 10 -2

0.048

9.2 × 108

92,000,000

6 × 104

60,000

Compare using >, <, or =. 1.

8 × 10 -5

7.05 × 106

0.008 7,500,000

L E A RN A ND C O NNECT 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. Multiplication:

(4 × 102) × (3 × 105) (4 × 3) × (102 × 105)

12 × 107

1.2 × 108

Separate values.

Simplify.

Adjust.

Division:

(2.4 × 106) ÷ (4 × 103)

2.4 × 106 4 × 103

2.4 106 × 3 4 10

0.6 × 103

6 × 102

Separate values.

Simplify.

Adjust.

A P P LY Adjust each number to be in the correct scientific notation. 1.

0.45 × 107 =

11 × 103 =

128 × 10 -4 =

0.04 × 108 =

54 × 103 =

280 × 105 =

0.03 × 109 =

0.1 × 10 -5 =

Level H

Chapter 2

Lesson 7

Lighthouse Math

Exercise | 2-7 Name Multiply. 1.

(3.5 × 103) • (7 × 104)

(1.3 × 10 -5) • (4 × 10-3)

(8.4 × 10-8) • (5 × 103)

(2.1 × 107) • (2 × 103)

(9.1 × 102) • (3 × 106)

(6.7 × 104) • (3 × 103)

(6.3 × 104) • (2 × 105)

(5.9 × 106) • (4 × 10-2)

(2.8 × 107) • (3 × 102)

10. (4.1 × 103) • (7 × 104)

11. (4.6 × 10 -3) • (4 × 10-4)

12. (3.8 × 105) • (2.5 × 103)

13. (7.9 × 105) • (2 × 103)

14. (7.6 × 10 -2) • (1.2 × 1010)

Divide. 15. (6 × 108) ÷ (2 × 104)

16. (7.2 × 105) ÷ ( 8 × 102)

17. (9 × 10−3) ÷ (3 × 10−5)

18. (6 × 104) ÷ (2 × 10−6)

19. (2.4 × 107) ÷ (6 × 103)

20. (4.5 × 107) ÷ (1.5 × 103)

21. (8.1 × 10−6) ÷ (9 × 103)

22. (2.4 × 10−2) ÷ (6 × 10−5)

23. (5 × 106) ÷ (2.5 × 103)

24. (2.7 × 107) ÷ (3 × 10−4)

25. (1.5 × 10−2) ÷ (3 × 10−6)

26. (5 × 105) ÷ (1.25 × 10-3)

27. (3.6 × 109) ÷ (1.2 × 104)

28. (4.2 × 10−3) ÷ (2.1 × 10−6)

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: (9 × 105) ÷ (6 × 107)

Equation:

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.

CH A L L E N G E 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?

Lighthouse Math

Level H

Chapter 2

Exercise 7

2-8 | Addition and Subtraction with Scientific Notation

DAI LY REV I E W

PREREQUISITE SKILLS

Solve.

SPIRAL REVIEW

0.25 + 0.065 =

2.04 + 0.81 =

9.8 − 0.02 =

1.6 − 0.8 =

0.012 + 0.05 =

1.901 + 0.25 =

Solve. 1.

4 × 1012 • 5 × 10-7 =

6 × 10 -3 • 8 × 10-8 =

1.8 × 107 ÷ 9 × 105 =

3.6 × 10 -5 ÷ 4 × 10-3 =

5 × 1010 • 3.2 × 10 -4 =

2.8 × 10 -2 • 6 × 103 =

L E A RN A ND C O NNECT When adding or subtracting numbers in scientific notation, both numbers must have the same power of 10. 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. Next, add or subtract the coefficients. Keep the common power of 10. The final answer should be adjusted to proper scientific notation. Addition:

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

(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 × 10-2) − (0.2 × 10-2)

(1.4 − 0.2) × 10-2

Change to the same power of ten.

Add the numbers.

1.2 × 10-2 Simplify.

A P P LY Solve. 1.

(5.2 × 104) + (3.1 × 104)

(7 × 106) − (2.5 × 106)

(1.6 × 10 -3) + (2.9 × 10-3)

(9.3 × 107) − (1.8 × 107)

(6.1 × 105) + (3.8 × 105)

(4.2 × 10 -6) − (1.2 × 10-6)

(8 × 108) + (1.5 × 108)

(3.5 × 103) − (2 × 103)

Level H

Chapter 2

Lesson 8

Lighthouse Math

Exercise | 2-8 Name Add. 1.

(6.3 × 106) + (2.4 × 104)

(6.6 × 107) + (4 × 105)

(8.1 × 105) + (9 × 103)

(2.2 × 106) + (5.5 × 103)

(3.5 × 10 -2) + (1.2 × 10-4)

(5.1 × 104) + (9 × 102)

(2 × 107) + (5 × 105)

(3.3 × 10 -4) + (8.7 × 10-6)

(4.4 × 103) + (3.1 × 102)

10. (9.7 × 105) + (2 × 103)

11. (7.8 × 106) + (6 × 104)

12. (4.2 × 10 -5) + (1.8 × 10-3)

Subtract. 13. (7.5 × 105) − (2.5 × 103)

14. (7 × 105) − (2 × 102)

15. (3.2 × 106) − (1 × 104)

16. (4.2 × 10−2) − (2.2 × 10 -4)

17. (6 × 10 -3) − (2 × 10-4)

18. (9 × 106) − (1 × 104)

19. (9.1 × 104) − (1.1 × 102)

20. (2.3 × 104) − (1.3 × 102)

21. (8.3 × 105) − (4 × 103)

22. (1.6 × 105) − (5 × 103)

23. (1 × 108) − (7.5 × 105)

24. (6.2 × 10 -3) − (4.2 × 10-5)

Write an equation. Then, solve. 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?

Equation:

Solution:

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?

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

• Ship B receives 6.75 × 106 liters.

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?

Lighthouse Math

Level H

Chapter 2

Exercise 8

2-9 | Applications of Scientific Notation

DAI LY REV I E W

PREREQUISITE SKILLS

SPIRAL REVIEW

Solve. 1.

(7 × 104) • (2 × 10-9) =

(9 × 10 -6) • (3 × 102) =

(4.5 × 10 -2) ÷ (5 × 101) =

(6 × 103) • (4 × 10-7) =

Solve. 1.

(7 × 106) + (3 × 105) =

(2.5 × 10 -4) + (4.5 × 10-6) =

(9.8 × 10 -7) − (3 × 10-8) =

(5.2 × 103) + (3.1 × 102) =

L E A RN A ND C O NNECT 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. How much total fuel did the spaceship use during the entire mission? Step 1: M ultiply 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 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

2. 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

The mass of a mosquito

California’s economy

Vermont’s economy

3. The mass of an ant is about 4 × 10-3 grams, while the mass of a mosquito is about 2.5 × 10-2 grams. 4. California’s economy produces 3.9 × 1012 dollars per year. Vermont’s economy produces 4.3 × 1010 dollars per year.

Level H

Chapter 2

Lesson 9

Lighthouse Math

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

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:

Equation:

Solution:

3. 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?

4. 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:

Solution:

5. The human body makes 2 × 106 new blood cells per second. How many red blood cells does the body make in a week? (1 week ≈ 6 × 105 seconds)

6. A satellite sends 6 × 108 bytes of data. Then, it sends an extra 2 × 106 bytes in a burst. How much data did it send?

7. 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?

8. 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?

CH A L L E N G E 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?

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?

Lighthouse Math

Level H

Chapter 2

Exercise 9

Solve.

2-10 | Review

L E A RN A ND C O NNECT

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

x7 x2

x15

Zero Exponent Rule

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

1 x4

x-4

Multiplying and Dividing in Scientific Notation

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 × 105

x7 − 2 x5

(4 × 6)(10² × 10³)

(3 × 10²) + (2 × 10³)

2.4 × 106

(0.3 × 10³) + (2 × 10³)

(0.3 + 2) × 10³

2.3 × 103

A P P LY Simplify each expression. 1.

34 × 38

95 × 92 × b4 × b8

84 × a7 83 × a4

76 × 74

y3 × y × 27 × 25

109 × y9 102 × y8

x4 × x2

94 9

48 × 43 × x10 45 × x3 × x4

10. 48 × 43 × 45

11.

z10 z5

12.

28 × a9 × y4 22 × a3 × y

Level H

Chapter 2

Lesson 10

Lighthouse Math

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

43 × 4-5

b-6 × b3 × c-8 × c8

a-2 × a4

28 × 3 × 3-8 × 3-4

y0 × y-7

1 z0

9-8 × t3 93 × t a-3 × b5 a-5 × b3 9-3 × r × s5 94 × r7 × s2

Convert to scientific notation. 10. 4,500

11. 0.00000044

12. 980,000

13. 0.00081

14. 6,000,000,000

15. 0.007

16. 72,000,000

17. 0.00000000032

18. 5,600

19. 5 × 108

20. 2.4 × 10 -2

21. 9.81 × 107

22. 3.2 × 104

23. 8.9 × 102

24. 1 × 102

25. 7.1 × 10 -3

26. 4.7 × 10 -6

27. 4.5 × 10 -8

28. (3.2 × 104) + (4.5 × 104)

29. (7.2 × 10 -2) ÷ (8 × 102)

30. (9.4 × 10 -4) − (5.1 × 10-4)

31. (7.1 × 10 -3) − (2.4 × 10-3)

32. (9 × 105) − (3.2 × 105)

33. (1.2 × 103) × (4 × 102)

Convert to standard form.

Simplify each expression.

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?

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?

CH A L L E N G E 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.

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?

Level H

Chapter 2

Exercise 10

Solve.

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.

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

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.

10. 4(6 + 4)

5(3 + 2) 5(

) + 5(

)

(

)

Identity Property

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

Use the identity property to fill in the blanks. 11. 5 + 0 =

12. 32 ×

= 32

13. 873 +

= 873

I understand and can use properties of operations.

out 13 correct

Skill 2: Inverse Operations © Lighthouse Curriculum. Copying strictly prohibited.

Match equations that show inverse operations. Operation

Inverse

2×3=6

2+3=5

8÷4=2

2×4=8

2+4=6

6÷3=2

5−3=2

6−4=2

Use inverse operations to solve. 5.

Level H

89 × 0.367 ÷ 0.367 =

I can identify and use inverse operations.

out 6 correct

7 − 15 + 15 =

Chapter 3

Skill Checklist

Lighthouse Math

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

Apply the distributive property to each expression. 3.

2(3x + 4)

9 + 5y + y − 2

4(5x + 9)

-3(8j − 6)

-5(3n + 5) + 20

Simplify each expression.

6x + 8

6(7p + 2) − 10p

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

out 6 correct

Skill 4: Translating Expressions • Total • Plus • Sum • More than • Increased by • Difference • Less than • Minus • Subtracted from • Decreased by

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

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

• Product • Times • Of • Twice • Double

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

degrees colder than 3 half the temperature today

pool has 5,000 gallons of water and A each minute another 9 gallons are added © Lighthouse Curriculum. Copying strictly prohibited.

+ × ÷

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

out 4 correct

Lighthouse Math

Solve. 1.

3m + 4 = 7

-9x − 9 = 81

y −3=5 2

5p + 2 = 17

I can solve two step equations.

Level H

Chapter 3

Skill Checklist

3-1 | Simplifying Equations

PREREQUISITE SKILLS

Solve. x 2. = 7 7

DAI LY REV I E W

1. 15y = 225

SPIRAL REVIEW

3. 0.2g = 64

4. -11.7 + v = -18.2

Simplify. 1.

6.3 × 107 ÷ 9 × 104 =

3.6 × 10 -3 ÷ 6 × 103 =

L E A RN A ND C O NNECT An equation is a math sentence with an equal sign. Some equations need certain parts simplified before they can be solved. 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.

5(4x – 2) – 19x = 20 Simplify:

Solve:

5(4x – 2) – 19x = 20

x – 10 = 20 + 10 + 10

20x – 10 – 19x = 20

x = 30

x – 10 = 20

Once the like terms are combined, solve the equation.

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

-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

Follow the steps to solve each equation.

2. -31 = -23x + 32 + 24x 3. 36 = 2(2g – 6) 4. -27v – 35 + 53v = 251

Simplify

Solve

Combine the like terms:

Subtract 32 from both sides:

Apply the distributive property:

Add 12 to both sides: Divide by 4:

Combine the like terms:

Add 35 to both sides: 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

Exercise | 3-1 Name Write a simplified equation. Then solve. 1. -3x + 5 – 2x = 20

2. 7y − 4 + 2y = 41

-10 + 4a – 3 – 2a = 17

4. -5(6x – 12) = 0

5. 7(2s + 3) = -35

-2.5(4r – 6) = 15

7. -8x – 4(2x + 3) = 28

8. -3(5y − 4) + 10y = -3

-6p – 2(7p + 1) = 18

Equation:

Solution:

Equation:

Solution:

Equation:

Solution:

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?

4x + 3 ft

CH A L L E N G E 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. B. Write an equation if the total donation is $120. C. Solve the equation to find out the total number of tickets sold.

Lighthouse Math

Level H

Chapter 3

Lesson 1

3-2 | Solving Multi-Step Equations

PREREQUISITE SKILLS

Simplify.

DAI LY REV I E W

SPIRAL REVIEW

3. 81

2. 82

1. 93

4. 36

Solve for x. 1. 4x + 3x = 49

2. 4(x − 2) = 4

3. 2.5(6x + 1) + 1.5 = 40

4. -5(3x − 4) − 5x = 85

L E A RN A ND C O NNECT 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.

Inverse Operations

−

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. 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.

There will be two possible answers, one positive and one negative.

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

x +4=5

8−x + 7 = 20 6

x−6 =8

x−8 = 12 10

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

Exercise | 3-2 Name Solve. 1. x – 3.5 = 12.6

g 2. = -16 -8

3. v2 = 144

4. s = 2.5

5. 76 + x = 29

6. -5x – 2 = 13

7. 3y + 7 = 22

8. x2 + 9 = 25

9. 3( c) = 12

10. w2 – 5 = 20

z = 23 6

k 12. – 7 = -5 15

13. -2d – 9.5 = 12.5

14. p + 8 = 10

15. 3(j2) = 27

n–2 = -8 17. 10

18. (p + 2)2 = 100

19. -2c = 10

20. 5x2 + 3 = 128

11. 18 +

16. x + 3 = 4

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

A. How are the equations different?

B. In which equation would you first subtract 8?

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

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?

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?

CH A L L E N G E 25. A farmer is building a rectangular pen. Its dimensions are shown to the right.

x2 + 5 ft

A. If the perimeter is equal to 106 ft, what is the value of x? B. Using the value of x from part a, find the area of the pen.

2x2 ft

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

Lighthouse Math

Level H

Chapter 3

Lesson 2

Write and solve an equation for each word problem.

3-3 | Equations with Variables on Both Sides

DAI LY REV I E W

PREREQUISITE SKILLS

SPIRAL REVIEW

Evaluate. 1.

-72.2 + 61.9 =

625.5 ÷ 4.5 =

-4 21 × 2 41 =

-8 32 × 4 45 =

Solve. 2. -13 +

1. -8x + 4 = 20

2 x=3 3

3. x2 – 22 = 27

4. 5x + 11 = 6

L E A RN A ND C O NNECT Some equations have variables on both sides of the equal sign. When you have a problem like this, follow the steps below: 1. U se inverse operations to get the variable to one side of the equation and combine like terms. 2. I solate the variable using inverse operations and following the reverse order of operations - SADMEP.

-4x2 – 82 = -7x2 + 26 + 7x2 + 7x2 3x2 – 82 = + 82 3x2 ÷3 x2 x

26 + 82 108 ÷3

= 36 = 6 or -6

A P P LY Complete each step to solve the equations.

12m + 6 = 10m – 18

4 3 1 1 z– = z+ 5 4 5 4

Subtract to combine the variables.

Subtract to isolate the variable.

Add to isolate the variable.

Divide to isolate the variable.

Multiply by the reciprocal to isolate the variable.

8x + 12 = 16 – 3x

2b + 20 = 9b – 22

Add to combine the variables.

Subtract to combine the variables.

Subtract to isolate the variable.

Add to isolate the variable.

Divide to isolate the variable.

Level H

Chapter 3

Lesson 3

Lighthouse Math

Exercise | 3-3 Name Solve each equation. 1. 5x + 7 = 3x – 7

2. 4b + 9 = 3b + 1

3. -6y – 5 = -3y + 10

4. -10d – 21 = 5d + 9

5. 11g – 8 = 7g + 4

6. 4.5t + 1.2 = 2.5t – 2.8

7. 3.2v – 1.6 = 1.2v + 2.4

3 1 8. m + 5 = m + 9 4 4

9. 2x2 – 36 = x2

10. 6y2 + 4 = 5y2 + 20

11. 5s2 – 54 = 2s2 + 138

12. 7y – 3 = 9y

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

27x + 56 = 13x + 42 − 13x − 13x 14x = 42 ÷ 14x ÷ 14x x=3

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?

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?

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?

CH A L L E N G E 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?

Lighthouse Math

Level H

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

Chapter 3

Lesson 3

Write an equation for each word problem and solve.

3-4 | More Equations with Variables on Both Sides

DAI LY REV I E W

PREREQUISITE SKILLS

SPIRAL REVIEW

Simplify each expression using the distributive property. 1.

-5(3z +

2 ) 5

18(-4b + 6)

6.7(2h – 6)

3x(x – 12)

Solve each equation. 1. 5x + 7 = -2x + 56

2. t – 17 =

1 t – 11 3

3. 2.1y – 4.1 = 18.3 – 1.1y

k + 13 =k–2 4. 4

L E A RN A ND C O NNECT Some equations with variables on both sides of the equal sign need to be simplified before they can be solved. When you have a problem like this, follow the steps below: 1. Complete the distributive property.

4x + 12 = 8x – 6x + 12 4x + 12 = 2x + 12 – 2x – 2x 2x + 12 = 12 – 12 – 12

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

2x = 0 ÷2 ÷2

4. Use inverse operations to isolate the variable.

4(x + 3) = 8x – 3(2x – 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)

Complete the distributive property.

Combine like terms.

Subtract.

Add.

Subtract.

Divide.

Level H

Chapter 3

Lesson 4

Lighthouse Math

Exercise | 3-4 Name Solve each equation. 1. 2(3x + 4) + 5 = 4(x + 5) + 3

2. 4(2t + 3) + 2t = 3(3t + 5) − 1

3. 3(x + 6) = 2(x + 2) + 10

4. 50g – 3(7g – 9) = 61 + 5(5g – 10)

5. 25k − 3(5k − 8) = 40 + 2(6k − 7)

7. 2.5(3x - 4) + 1.2 = 1.5(4x + 2) – 1.3

8. 5(g2 – 7) + 18 = 2g2 – (2g2 – 3)

40x − 12x + 10 = 32 + 20x − 24

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?

9. 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.

CH A L L E N G E 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? B. What is the value of the length and width?

4x + 8 ft

C. What is the area of the space?

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

Level H

Chapter 3

Lesson 4

3-5 | Identifying the Number of Solutions

DAI LY REV I E W

PREREQUISITE SKILLS

Simplify.

SPIRAL REVIEW

15m – 13 + 16m – 9

-59 – 36g – 62 + 24g

5.7x + 19.6 – 2.3x – 4.2

2 4 2 3 t– + t– 3 5 3 5

Solve each equation. 1. 2(8x + 3) = 3(-3x – 8)

2. 6x + 18 – 3x = 4(x – 3)

3. 5(2x + 3) = -3(4x – 2) + 10

L E A RN A ND C O NNECT

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 – 4p – 4p 2p + 22 = -8 – 22 – 22 2p = -30 ÷2 ÷2 p = -15

12m + 18 = 6(2m + 3) 12m + 18 = 12m + 12 – 12m – 12m 18 = 12

10q + 4 = 2(5q + 2) 10q + 4 = 10q + 4 – 10q – 10q 4=4

A P P LY Choose if each equation has one solution, no solution, or infinitely many solutions. 1.

3(x + 2) + 5 = 3(x – 1) + 8

one solution

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

no solution

5(6x + 4) – 11 = 7(4x + 7)

infinitely many solutions

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

Exercise | 3-5 Name Write if each equation has one solution, no solution, or infinitely many solutions. 1. 12a + 18 = 6(2a + 3)

2. 4(x + 2) = 2(3x + 1) + 6

3. 6(3x + 2) = 18x + 12

4. 9(x − 2) = 3(3x + 4) + 3

5. 10(y − 3) = 2(5y − 6) + 4

6. 6x − 4 = 2(3x + 4) − 12

7. 4(2x − 5) = 8x − 8 – 12

8. 5(2x − 3) + 10 = 3(4x − 2)

9. 6(8x − 3) + 7 = 4(12x + 2)

10. 3(12x + 9) = 7(8x + 6) + 21

11. 125(2x + 10) = 250x + 1,250

12. 9(4x − 8) – 12x = 3(8x + 10) + 20

13. 1.2(4x + 5) = 5x + 16.8

1 3 14. (24x − 20) = (8x + 12) 4 4

2 15. (10x + 10) = 4x + 4 5

Solve each word problem. 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?

CH A L L E N G E 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

Lighthouse Math

No Solution

Level H

Chapter 3

Infinitely Many Solutions

Lesson 5

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.

3-6 | Equation Applications

PREREQUISITE SKILLS

Translate each verbal model into an equation. 2. The quotient of a number and 5, plus four 4. The product of eight and a number, minus six

DAI LY REV I E W

1. Eleven more than three times a number 3. Eighteen less than a number squared

SPIRAL REVIEW

Solve each equation. 1. 5(2x – 3) = 7x + 3 + 3x

2. 5x + 4 = 8(x + 9)

3. 4x – 15 + 2x = 3(2x – 15)

L E A RN A ND C O NNECT 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? 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. 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. + $ saved already

w= $ saved per week

+ $ saved already

$ saved per week

Explain the solution in the context of the problem:

Level H

Chapter 3

Lesson 6

Lighthouse Math

Exercise | 3-6 Name

1. 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?

2. 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?

3. 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?

4. 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?

5. 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?

6. 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?

7. 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 A L L E N G E 8. 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.

Lighthouse Math

Level H

Chapter 3

Lesson 6

Write and solve an equation for each word problem. Explain the solution in context of the problem.

3-7 | Review

L E A RN A ND C O NNECT 5(4y2 − 3) = 7y2 + 49 − 3y2 Apply the distributive property.

Combine like terms.

5(4y2 − 3) = 7y2 + 49 − 3y2

20y2 − 15 = 7y2 + 49 − 3y2

20y2 − 15 = 4y2 + 49

Get the variable to one side of the equation.

20y2 − 15 = 4y2 + 49 − 4y2 − 4y2 16y2 − 15 =

Follow SADMEP to solve.

Addition/Subtraction

Multiplication/Division

Exponents/Roots

16y2 − 15 = 49 + 15 + 15 16y2 = 64

16y2 = 64 ÷ 16 ÷ 16 y2 = 4

y2 = 4 y = 2 or -2

A P P LY Choose if the equation has one solution, no solution, or infinitely many solutions. 1.

21x + 59 = 7(3x + 9) – 10

one solution

-60 + 16z = 4(4z – 15)

no solution

4x + 6 = 2x + 14

infinitely many solutions

Match each equation with its correct solution. 4.

-3x = 6

x = -36

5x2 + 13 = 138

x = -6

-30 = -26 +

x 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

x = 5 or x = -5

11. 2(2x2 + 8) + 80 = 4(3x2 + 6)

Level H

Chapter 3

Lesson 7

Lighthouse Math

Exercise | 3-7 Name Write if each equation has one solution, no solution, or infinitely many solutions. 1. 14t – 22 = 2(7t + 9)

2. 6(12x + 5) = 9(5x + 8) + 12

3. 12(2g + 6) = 24g + 72

4. -7m + 3(4m + 5) = 50

5. -36 = -6(9x + 1) – 3

6. 5(2x2 – 12) = 580

7. 3d + 6 = d + 26

8. (11y + 24)² = 49y2

2 1 1 3 9. v – 2 = v – 3 3 4 3 4

10. 8(4x + 8) = 5x + 7(4x – 10)

11. 6(3x + 5) = 5(3x + 3) – 6

12. 10(2x + 3) – 4x = 4(2x + 4)

Solve each equation.

Write and solve an equation for each word problem. 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.

CH A L L E N G E 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? 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

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.

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.

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

Dependent Variable The thing that is affected by the change, output

Choose the correct equation for each scenario.

The y variable in y = 3x

A teacher places 3 pencils (p) in each pencil box (b). 3b = p

3p = b

3+p=b

Ariel (a) eats two more cookies than Dana (d). 2+a=d

d+2=a

2d = a

2a = d

I can identify independent and dependent variables.

out 4 correct

3+b=p

Skill 2: Input/Output Tables Complete the tables using the equations.

c = 1.53f # of flowers

Cost

y = 3x

$1.53

$3.06

$4.59

$6.12

2. y

n 3 n

16 24

k 1

12 9

10 81

I can use an equation to complete an input/output table.

out 3 correct

Level H

d=c+5

Chapter 4

Skill Checklist

Lighthouse Math

Name

Skill 3: The Coordinate Plane Label the parts on the coordinate plane. 1. y-axis 4

(-2, 3)

Quadrant II

Quadrant I

2 1

-4

-3

-2

-1

x-axis

(1, 1) 1

-axis x y-axis origin Quadrant I Quadrant II Quadrant III Quadrant IV

-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. a

3. c (2, 4)

4. b

5. d (6, 9)

b a 0

I can find and plot points on the coordinate plane.

out 5 correct

Skill 4: Equivalent Ratios Brand B

$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

100 = 1,000

Brand A

I can identify and find equivalent ratios.

out 5 correct

Skill 5: Absolute Value

|-3| = 3

|3| = 3

out 3 correct

Lighthouse Math

Solve. 1.

|5| =

|-3| =

|-10| =

I can find the absolute value of a number.

Level H

Chapter 4

Skill Checklist

4-1 | Introduction to Graphing

DAI LY REV I E W

PREREQUISITE SKILLS

Find the letter for the ordered pairs.

(-4, -3)

(-3, 3)

(3, 2)

(3, -3)

-5

5 D

C -5

SPIRAL REVIEW

Solve. 1.

x + 7 = -5

2. y2 = 81

3. -2a + 4a − 10 = 20

4. 2x + 6 = 4x − 8

L E A RN A ND C O NNECT 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.

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. 1. Rule: y = 5x x

2. Rule: y = 2x + 3 y

-1

3. Rule: y = x + 1 y

-2

3 4

Vocabulary 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

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

2. Rule: y = 2x

3. Rule: y = -x

Ordered pairs

-2

-1

Ordered pairs

-5

Use the graph to fill in the input/output table. 4.

-5

6. A bike rental shop charges a $2.00 fee plus $2 per hour to rent a bike. C. Graph the ordered pairs. Then, connect them with a line.

CH A L L E N G E Draw a line to connect the dots. Then, write the rule. 7.

(3, 7) 5

(4, 9)

(2, 5)

(3, 16)

Rule: 10

(1, 3)

(-1, 8)

Rule:

(2, 11)

(-2, 11)

6 3

(1, 6)

(0, 1) 0

(4, 21)

-3

Lighthouse Math

Level H

Chapter 4

Exercise 1

Rule:

(0, 5) (1, 2) 3 (2, -1)

A. Write an equation to model the cost (y) of renting a bike for x hours. B. Create an input/output table.

4-2 | Functional Relationships

DAI LY REV I E W

PREREQUISITE SKILLS

Solve for y for each given x.

SPIRAL REVIEW

y = 2x + 3 x = 1, y =

y=x−1

x = -1, y =

y = 5x

y = 3x − 1

x = 0, y =

Write four ordered pairs from each table. 1.

x -1 0 1 2

y 0 1 2 3

-3

-2

-1

L E A RN A ND C O NNECT 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. Function

Not a Function

Word form

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)

(1, 2), (1, 3), (2, 4)

x y

Table Equation

0 2

1 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

2 6

-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)

x y

8.5 2.4

8.5 2.5

8.6 2.6

2 -2

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

Exercise | 4-2 Name Tell if the ordered pairs represent a function. Write YES for function or NO for not a function. 1. (1, 2), (2, 3), (3, 4), (4, 5)

(1, 5), (2, 5), (3, 5), (4, 5)

(1, 2), (1, 3), (2, 4)

(5, 1), (6, 2), (5, 3)

(0, 0), (1, 1), (2, 2), (3, 3)

(7, 10), (8, 11), (9, 12), (10, 13)

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

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.

-5

13.

-5

5 -5

-5

Use the equation x = | y | to complete the table and answer the questions. A. I s the equation a function? How do you know?

x y

-1

B. Find another example of an input that gives more than one output.

CH A L L E N G E 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

Lighthouse Math

Level H

Chapter 4

Code Entered Snack Received

Exercise 2

14.

4-3 | Graphs of Functions

DAI LY REV I E W

PREREQUISITE SKILLS

SPIRAL REVIEW

Write ordered pairs using the table. 1.

x -1 0 1 2 3

y -2 0 2 4 6

Write yes or no to show if each of the following are a function. 1.

y = 2x + 3

y=7

L E A RN A ND C O NNECT 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)

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. 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)

Circle the graph that matches each scenario.

2 1 0

10 15 20 Time (minutes)

Distance from Home (miles)

John's Jog

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

2. The recipe calls for 2 cups of flour for every cup of water.

John's Jog Distance from Home (miles)

1. 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.

Cups of Flour

A P P LY

16 12 8 4 0

4 6 Cups of Water

Lighthouse Math

Exercise | 4-3 Name Look at the graphs. Choose the real-life scenario that matches. 1.

Marks's Money 40

Number of Bunnies in the City 35,000 30,000 25,000

Bunnies

Dollars

30 20

20,000 15,000 10,000

5,000 2

4 6 Hours Tutoring

1,000

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. T here are 1,000 bunnies in the city, and every year, there are 90 more bunnies than the year before.

Use the graphs to answer the questions. 3. 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)

4. A bucket is used to catch a drip from the ceiling. A. What most likely happened after 5 hours?

B. After 9 hours was the ceiling still dripping? How do you know? 0

Hours

CH A L L E N G E Mike's Distance from Home

Meters

5. Describe the scenario displayed by the graph.

28 24 20 16 12 8 4 0

Minutes

Lighthouse Math

Level H

Chapter 4

Exercise 3

Quarts of Water

Water in the Bucket 12 10 8 6 4 2

4-4 | Linear Functions

PREREQUISITE SKILLS

Is this relation a function? Circle Yes or No.

DAI LY REV I E W

1. (-1,1)(0,0), (1,1), (2,4),(3,9) Yes

200

2. y=x+3

Yes

100

Yes 100

SPIRAL REVIEW

200

Label and sketch the graph described. 1. 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.

L E A RN A ND C O NNECT 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

Nonlinear Function

(3, 11)

(2, 9) (1, 7)

5 (0, 5)

-5

-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 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

Exercise | 4-4 Name Graph the points to determine if the function is linear or nonlinear. 1.

-1 0 1

-1

-3

-5

Linear or Nonlinear

-10

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 x

4. y

y = x2 x

-2

-1

2 0

Linear or Nonlinear

-5

Write an equation, complete the table, and graph the points to determine if the function is linear or nonlinear. x

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 A L L E N G E 10. Write a linear equation and a nonlinear equation. Linear equation:

Lighthouse Math

Nonlinear equation:

Level H

Chapter 4

Exercise 4

5. 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.

4-5 | Slope and y-intercept

PREREQUISITE SKILLS

Write each fraction in simplest form.

DAI LY REV I E W

1. SPIRAL REVIEW

25 5

6 12

5 20

45 9

Circle the graph that shows a linear function. 60

20 0

-5

L E A RN A ND C O NNECT 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

Positive Slope As the x-values increase, so do the y-values.

Negative slope As the x-values increase, the y-values decrease.

A P P LY Find the slope and y- intercept. 1.

7 6 Rise 2 5 4 3 2 1

Run 5

0 1 2 3 4 5

slope = y-intercept =

7 6 5 4 Run 1 3 Rise 1 2 1 0 1 2 3 4 5

7 Run 2 6 5 Rise 6 4 3 2 1 0 1 2 3 4 5

slope = y-intercept =

-5-4-3 -2 -1

Rise -2

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

Run 3

slope = y-intercept =

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

Exercise | 4-5 Name Choose any two points on the graph. Find the rise and the run. Then, find the slope.

Rise:

6 4

Run:

2 0

4 6 8 Attempts

Point 1:

Money Earned This Month 100

Point 2:

8 Points

Point 1:

Points Scored

Money Earned

Point 2:

Rise:

60 40

Run:

20 0

Slope:

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

slope =

240

y-intercept =

increasing

2 Time (h)

Distance (miles)

Distance Traveled Based on Fuel Used

slope =

200

y-intercept =

160 120

increasing

80 40

decreasing

10 Liters

decreasing

Circle the correct graph. 5. Which graph has a slope of -2? Graph A

Graph B

-5

Graph B

-5

6. Which graph has a slope of 4?

-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 A L L E N G E 8. What happens to a graph if we change the y-intercept?

Lighthouse Math

9. If a line goes through the origin, what is the y-intercept?

Level H

Chapter 4

Exercise 5

4-6 | Find Slope and y-intercept from a Set of Coordinates

DAI LY REV I E W

PREREQUISITE SKILLS

Solve.

SPIRAL REVIEW

5 + (-3) =

-4 + (-6) =

-7 + 9 =

6 − (-2) =

-3 − 4 =

-10 − (-5) =

Find the slope and y-intercept. Is the line increasing or decreasing? 5

slope =

increasing

y-intercept =

decreasing

L E A RN A ND C O NNECT 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.

y2 − y1 x2 − x1

Slope formula:

(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

When x is 0, y is 1. The y-intercept is 1.

A P P LY Find y-intercept from the table. 1.

-2

-1

-2

y-intercept:

-1

y-intercept:

Fill in the blanks to find the slope. 4. (x1, y1) → (1, 3) (x2, y2) → (5, 15)

y2 − y1 15 − = x2 − x1 −1

5. (x1, y1) → (3, 14) (x2, y2) → (4, 18)

y2 − y1 = x2 − x1

− −

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

Exercise | 4-6 Name Find the slope of a line passing through two points. 2. (-2, 4) and (2, -4)

1. (3, 7) and (5, 17) Label: x1 =

y1 =

x2 =

y2 =

Plug into formula: − −

Label: x1 =

y1 =

x2 =

y2 =

Plug into formula: m=

− −

Find the slope of the line that passes through the points. 3. (1, 3) (2, 7)

4. (-2, -1) (-8, 2)

6. (-7, 3) (-2, 5)

7. (-4, 0) (0, -3)

8. (6, 8) (-3, 7)

(0,0) (-4,-1)

Use the slope formula to find the missing y-value. 9. slope = -3 Point 1: (2, y) Point 2: (6, 5)

10. slope =

1 2

Point 1: (4, y) Point 2: (8, 6)

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

(4, 6)

6 4

(2, 3)

50 1

Hours

Slope = Water is emptying at a rate of per hour.

Days

Slope = The plant is growing at a rate of per day.

gallons

inches

CH A L L E N G E 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)

Level H

14. Slope =

2 1

Chapter 4

Exercise 6

Points: (3, 5) and (x, 9)

Height (in inches)

Gallons of Water

350 300

4-7 | Slope and Similar Triangles

PREREQUISITE SKILLS

Solve the proportions.

DAI LY REV I E W

SPIRAL REVIEW

3 x = 4 8

x 3 = 4 8

6 3 = x 4

6 4 = x 3

Find the slope given the coordinates. 1. (3, 2) and (4, 1)

2. (12, 5) and (8, 6)

Slope =

3. (1, 3) and (4, 9)

Slope =

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

Slope =

L E A RN A ND C O NNECT 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.

C (4, 12)

These triangles are similar triangles. They are proportionate, meaning that they are the same shape but not the same size.

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. Ratio of sides:

Orange Triangle

B (2, 6)

Blue Triangle

3 4

Vertical Side Length Horizontal Side Length

3 1

6 2

A (1, 3)

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.

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

Exercise | 4-7 Name Set up a proportion to find the missing side. 1.

x 1

3 2

1 = 3

1 = x

x 4

5 2

4 = 5

Find the missing side of the triangle. 4.

6. y

3 x

2 1

Find the missing coordinate. 7.

(4, y)

(x, 10)

(-3, 0)

6 15

3 (-1, -4)

2 (0, -2) (3, y)

(-5, -5)

7.5

(-20, -12.5)

(-4, -16)

CH A L L E N G E 10. Use the map to find the total distance of the red trail.

Whiterock Park Trail Map

North

18 m West

East South

ow much longer are the blue and red trail H together than the black trail?

Black Trail: 30 meters Blue Trail: 10 meters 4m

Lighthouse Math

Level H

Chapter 4

Red Trail: ? meters

Exercise 7

4-8 | Slope-Intercept Form

PREREQUISITE SKILLS

Find the slope and y-intercept from the graph.

DAI LY REV I E W

Slope:

y-intercept = -5

SPIRAL REVIEW

-5

y-intercept =

Circle the triangles with the same slope as the gray triangle. 8

6 4

3 12

L E A RN A ND C O NNECT 10

All linear equations can be written in slope-intercept form. y = mx + b m is the slope.

b is the y-intercept. -5

-10

The line on the graph has a slope of The equation of the line is y =

1 and a y-intercept of -2. 3

3 -5

1 x − 2. 3 -10

A P P LY

Write the slope (m) and the y-intercept (b) of each equation. 1. y = 5x − 3 m=

2. y = -2x + 10 b=

4. y = -x + 4 m=

5. y=

7. y=x+1 m=

b= 1 x 2 b=

8. y = 3x − 15 b=

6. y = -4x − 5

3. y = 7x + 8

m= 9.

1 y = - x + 2 8 m=

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

Exercise | 4-8 Name Write the equation given the slope (m) and the y-intercept (b). 1 b=1 6

1. m = 2 b = -1

2. m = -

6. m = 3 b = 4

7. m = -1 b = 1

3. m = 3 b = 0

4. m = -2 b = 5

5. m = 9 b = -1

1 b=2 3

9. m = -9 b = 2

10. m =

8. m =

1 b = -10 4

Use the graph to write the equation in slope-intercept form. 5

11.

-5

12.

-5

17.

-5

14.

-5

16.

-5

15.

13.

-10

-5

18.

-5

-10

-5

19.

y = 3x − 1

y = 21 x − 0

y = 4x

y = 8x

Circle the lines that pass through the origin. y = 43 x + 5

CH A L L E N G E 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?

Lighthouse Math

Level H

21. A line passes through the points (0, 7) and (3, 12). What is the equation of the line in slope-intercept form?

Chapter 4

Exercise 8

4-9 | Graphing Lines from Slope-Intercept Form

PREREQUISITE SKILLS

Use the slope formula to find the slope (m) between the given points. 2. (9, 7) and (15, 4)

DAI LY REV I E W

1. (3, 4) and (5, 9)

m= SPIRAL REVIEW

3. (-1, -2) and (-6, -3)

4. (6, -1) and (4,-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

5. m = -1 b = 5

L E A RN A ND C O NNECT An equation written in the slope-intercept form gives all the information needed to graph a line and find coordinates on that line. y = 2x + 1 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)

1 2

-5

y = 2x + 1 y = 2(-2) + 1 y = -4 + 1 y = -3 The point (-2, -3) is on the line.

To find another point on the line, plug in any x-value and solve to find the y-value.

Here’s how to graph a line using this equation:

A P P LY Follow the directions to find a point on the given line. 1. y = 3x + 5

2. y=

1 x−3 2

Plug in 2 for x: y = 3(

)+5

Plug in 6 for x: y = 21 (

Solve to find y: y = y=

Write the coordinates: (2,

)

3. y = -4x + 1 )−3

Plug in 1 for x: y = -4(

−3

Write the coordinates: (6,

Solve to find y: y = y= )

)+1 +1

Write the coordinates: (

)

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

Exercise | 4-9 Name Graph the linear equations on the coordinate plane. 1 1. y=- x+2 2

2. y = 3x − 1

3. y = -x − 2

-5

1 5. y=- x+2 3

4. y = 2x − 1

2 x 3

6. y=

-5

Find two sets of coordinates (x, y) on each line.

(

)(

8. y = -4x + 1

)

(

)(

9. y=

)

(

1 x−4 3

)(

10. y = -2x + 7

)

(

)(

11. y=

)

(

1 x − 10 5

)(

7. y = 8x − 2

)

CH A L L E N G E 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 from the starting line (y) over time (x).

B. Graph each equation on the coordinate plane.

Lighthouse Math

Level H

Chapter 4

Exercise 9

4-10 | Modeling and Interpreting Linear Functions

PREREQUISITE SKILLS

Write the slope (m) and the y-intercept (b) of each equation.

DAI LY REV I E W

1. y=

1 x−5 2

m= SPIRAL REVIEW

2. y=x+2 b=

3. y = -3x b=

2 x−4 3

4. y=

Find the point on each line when x = 4. Give your answer as an ordered pair. 1 1 1. y=x+6 2. y = -2x + 1 3. y= x+3 4. y=- x−2 4 2 (4, ) (4, ) (4, (4, ) )

5. y = 5x − 1 (4,

)

20 (0, 20)

L E A RN A ND C O NNECT 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.

Dollars

To write a linear equation that models a real-life scenario, you need to know four things:

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

A P P LY Fill in the table. Scenario

Output (y)

Input (x)

squirrel has 10 acorns. Each day, he A collects 4 more.

Number of acorns

Time (days)

The water meter reads 36 L. Each hour, it decreases by 2 L.

A tire receives 1 psi of air with each pump. It begins with 12 psi.

Slope (m)

Y-intercept (b)

Amount of water in liters

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

Exercise | 4-10 Name Read the scenario and then answer the questions. 1. Karen has $15 in her bank account. She receives $10 each week for babysitting. A. Write an equation B. What does x represent? C. What does y represent? D. Graph the equation. Label the axes. E. How long will it take Karen to save $45? F. Put a dot on the graph that represents the answer to E.

2. An oven cools down at a rate of 50 degrees per hour. The oven starts at a temperature of 350°F. A. Write an equation B. What does x represent? C. What does y represent? D. Graph the equation. Label the axes. E. What temperature will the oven be after 5 hours? F. Put a dot on the graph that represents the answer to E.

500

250

Use the table to graph the line. Then, write the equation and answer the questions. Money Earned (y)

120

A. What are the coordinates of the y-intercept? B. Explain what the y-intercept means.

200

C. How much money is earned for each item?

100

Equation:

Number of Items Sold

D. How many items will need to be sold to earn $180? E. If 5 items are sold, how much money is earned?

CH A L L E N G E 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 B. What is the y-intercept of the line?

. The x variable represents

C. What is the slope of the line?

D. What does the slope of the line mean given the context?

Lighthouse Math

Level H

Chapter 4

Exercise 10

Number of Items Sold (x)

Money Earned ($)

4-11 | Comparing Linear Functions 5

PREREQUISITE SKILLS

Find the slope (m) and y-intercept (b) for each.

DAI LY REV I E W

1 1. y = ( )x − 8 3

SPIRAL REVIEW

-1

3. -5

m= b=

-5

Write an equation. 1. Ann has 12 marbles. She collects 3 marbles each day.

2. There are 14 gallons of gas in a tank. 1.5 gallons are used per day.

3. Mike reads 20 pages of his book each day. He begins on page 5.

L E A RN A ND C O NNECT We can compare functions by looking at their slopes and y-intercepts. There are two buckets collecting water from two different leaky sinks.

Bucket A x hours

y i nches

Slope = 2

y-intercept = 0

of water

Bucket A fills at a rate of 2 inches per hour. Bucket B fills at a rate of 21 inch per hour.

Bucket B

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.

Inches of water

Bucket A is filling faster than Bucket B. 5

Slope = 21

Hours

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

2. Function B

-10

-1

3. Function C

y = 5x

m =

b =

Chapter 4

5. Least Slope: 6. Greatest y-intercept:

m =

Level H

4. Greatest Slope:

Lesson 11

7. Least y-intercept:

Lighthouse Math

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

Function C

-2

-1

-6

-3

-10

y = 4x − 2

-10

b =

The slope of Function A is

than the slope of Function B.

5. The slope of Function B is

than the slope of Function C.

6. The y-intercept of Function B is

b =

than the y-intercept of Function A.

Three different functions show the drop in temperature over time. Compare them, then answer the questions.

Temperature Change in Boston x

-1

-6

-11

-16

8 6

Temperature Change in Albany

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 A L L E N G E 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.

Lighthouse Math

Level H

Chapter 4

Exercise 11

4-12 | Review

L E A RN A ND C O NNECT 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.

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)

Slope: the rate the water is leaking

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.

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.

m= b=

8− = −0 y=

-2

m= x+

Level H

Chapter 4

4− = − -2 y=

-4

m= x+

Lesson 12

0− = −0 y=

Lighthouse Math

Exercise | 4-12 Name Determine if the relationship is a function. 1.

-1

Function

Not a Function

(0, 1) (1, 2) (1, 3) (4, 5)

Function

Not a Function

Function

Not a Function

Determine if the function is linear. 4.

5. x

-1

Linear

Non-linear

Linear

y = 6x − 14

Non-linear

Linear

Non-linear

Write the equation of the line. 10

8. -10

-4

9. Danielle has 34 stickers. Each week, she receives 5 more stickers to add to her collection.

-10

Find the slope (m), y-intercept (b), and equation of each function. Then, answer the questions.

Degrees

b= 10

Hours

Equation:

b= Hours

Equation:

12. A. What was the starting temperature on Monday? B. What was the starting temperature on Tuesday? C. On which day did the temperature increase faster?

11. Temperature Change on Tuesday Degrees

10. Temperature Change on Monday

CH A L L E N G E 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. -10

-10

Lighthouse Math

Level H

Chapter 4

Exercise 12

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.

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. 5 stars 4 smiles

Write the rate. Make sure to label the units. 2.

Tina makes 20 sandwiches in 2 hours.

12 donuts cost $14.50. I can write ratios and rates.

out 3 correct

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

1 8

2 16

18 24

8 12

5 15

15 45

For each fraction, write two equivalent fractions. 4.

4 = 6

5 = 7

100 = 1,000

Find the missing number that would make the fractions equivalent. 6.

Level H

2 = 3 9

6 3 = 44

I can identify and find equivalent fractions and ratios.

out 8 correct

1 5 = 2

Chapter 5

Skill Checklist

Lighthouse Math

Name

Skill 3: Setting up a Proportion Match each scenario to its proportion. 1. Eric paints 4 walls in 15 minutes. How many minutes will it take him to paint 10 walls?

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? 7 tbsp

2. Jake mows 15 lawns in 4 hours. How many lawns can he mow in 10 hours? 3. Moses hikes 4 meters in 10 minutes. How long will it take him to hike 15 meters? 4. Lisa can write 10 sentences in 4 minutes. How many sentences can she write in 15 minutes?

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

(6, 5)

(4, 4)

Line A =

Line B =

Line C =

6 4

1 0

Slope =

9 10

Line D =

-10 -8

-6

-4

-2

rise y − y1 = 2 run x2 − x1

-2

1 5−4 = 2 6−4

-6

Slope =

-4

-8 -10

out 4 correct

Lighthouse Math

I can find the slope from a graph.

Level H

Chapter 5

Skill Checklist

Find the slope of each line on the graph.

5-1 | Proportions

DAI LY REV I E W

PREREQUISITE SKILLS

Solve.

SPIRAL REVIEW

2x − 3 = 7

(x + 7) = -2.5 4

4x + 5 = 1

6(x + 2) = -12

x + 10 = 8 2

3(x − 6) = 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 =

L E A RN A ND C O NNECT 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

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. 5 miles x miles = 1. 2 hours 6 hours

x pages 30 pages = 2. 4 minutes 6 minutes

18 pencils 42 pencils = 3. 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

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?

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

Pages

Bottles

135

270

405

540

Minutes

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

4. 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?

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

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

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

9. 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?

CH A L L E N G 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?

Lighthouse Math

Level H

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

Solve by writing a proportion.

5-2 | Graphs of Proportional Relationships

DAI LY REV I E W

PREREQUISITE SKILLS

Find the unit rate. 1.

A car travels 210 miles in 3 hours.

A pack of 6 markers costs $4.20.

3. A recipe uses 12 eggs to make 4 cakes. SPIRAL REVIEW

Determine the better buy. 1. A 32-oz bottle of juice costs $4.80, or a 20-oz bottle costs $3.00.

2. A 10-pack of pencils is $2.90, or a 15-pack is $4.20.

L E A RN A ND C O NNECT 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. 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.

4-hour rental → $48 7-hour rental → $84

Unit rate:

Slope:

$24 $12 = 2 hours 1 hour

Rise: 24 = 12 Run: 2

(7, 84)

Cost ($)

2-hour rental → $24

100

Run = 2

Slope 12 (4, 48)

Rise = 24 20

(2, 24)

Hours Rented

(0, 0)

A P P LY

1. A smoothie shop charges $8 for 2 smoothies and $10.50 for 3 smoothies.

10 5 0

Number of Smoothies

2. A student earns $30 for 2 hours and $45 for 3 hours.

Earnings ($)

Graph the two given points and determine if the relationship is proportional.

Cost ($)

Renting a boat costs $12 per hour.

50 25 0

Hours Worked

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

Level H

Chapter 5

Lesson 2

Lighthouse Math

Exercise | 5-2 Name Find the unit rate based on the graph.

Total Cost ($)

550

9.0 7.2 5.4 3.6

(4, 3.60)

1.8

(2, 1.80)

440 330

Time (mins)

(9, 495)

(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 1 2 3 4 5 6 7 8 9 10

Temperature (°C)

-2 -4 -6 -8 -10

Graph each proportional relationship. Determine the slope. 5. A recipe uses 3 cups of flour for every 2 batches of cookies.

Cost of Notebooks

Zoo Ticket Pricing

Flour Needed for Cookie Batches

100

20 15 10 5 0

3 4 5

12 9 6 3 0

9 10

Total Cost ($)

Cups of Flour

Total Cost ($)

6. A zoo sells 4 tickets for $36.

Number of Notebooks

3 4 5

60 40 20

9 10

3 4 5

Batches of Cookies

Number of Tickets

Slope =

7. The temperature starts at 0 degrees and drops 8 degrees per minute.

8. A delivery truck drives 180 miles in 3 hours.

Time (Minutes) 3 4 5

-20 -40 -60

9 10

600

480 360 240 120 0 0

-80

Scuba Diver's Depth Over Time

Depth (Feet)

Temperature (OF)

9. As a scuba diver begins at sea level and descends at a rate of 36 feet every 2 minutes.

Distance Traveled by Delivery Truck Distance (Miles)

Temperature Dropping

3 4 5

9 10

-10 -20 -30 -40

9 10

-50 -60

Time (Hours)

-100

Slope =

CH A L L E N G E 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.

Lighthouse Math

Level H

Chapter 5

11. Can you think of three ways to find the unit rate from a graph of a proportional relationship?

Exercise 2

4. Each notebook costs $2.50.

5-3 | Comparing Proportional Relationships

PREREQUISITE SKILLS

DAI LY REV I E W

Find the unit rate. 1.

A cyclist rides 60 miles in 4 hours.

A grocery store sells 5 pounds of apples for $7.50.

3. A machine produces 300 parts in 12 minutes. 4. A student earns $84 after working 7 hours. SPIRAL REVIEW

Find the unit rate. 1.

-5

5 -5

-5

L E A RN A ND C O NNECT 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. 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 Problems (y)

Unit Rate:

14 = 1.4 10

Equation: y = 1.4x

Problems

Minutes (x)

Time Mannie Spends on Math

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.

Write an equation for each scenario.

y = 4x

A lemonade stand earns $3 for every cup sold.

1 y = ( )x 3

A car travels at a constant speed of 60 miles per hour.

y = 4.25x

Each zoo ticket costs $9.

Level H

Chapter 5

Lesson 3

Lighthouse Math

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

Photographer's Rates 250

Total Cost ($)

16 12

(6, 12)

8 (3, 6)

4 0

Lawns Mowed

200 150

(6, 150)

100 (3, 75)

50 0

Number of Pretzels

Photo Sessions

Write the equation based on the table. 4.

5. Hours (x)

Tickets (x)

Miles (y)

120

180

270

Cost $ (y)

21.75

36.25

43.50

65.25

Packs of Chips (x)

Boxes (x)

Cost $ (y)

5.00

12.50

17.50

25.00

Snacks (y)

Solve.

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?

9. 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

Cost ($)

11. Printer A prints 90 booklets in 6 minutes. Printer B is represented by the graph. Which printer prints faster?

30 25 20 15 10 5 0

CH A L L E N G E 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?

Lighthouse Math

Level H

Chapter 5

Exercise 3

8. 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?

5-4 | Review

L E A RN A ND C O NNECT 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: a c = b d

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

From a table

Use k in the equation y = kx.

y = 3x → 3 is the unit rate.

Divide y by x in any column.

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

A P P LY Write a proportion. Then solve. Question

Proportion

A recipe uses 3 cups of sugar to make 24 cookies. How many cups of sugar are needed to make 40 cookies?

A car travels 180 miles in 4 hours. How far can it travel in 6 hours at the same speed?

A printer produces 120 pages in 8 minutes. How many pages will it print in 15 minutes?

It costs $7.50 for 3 pounds of apples. How much will 5 pounds of apples cost?

100

Level H

Chapter 5

Lesson 4

Solution

Lighthouse Math

Exercise | 5-4 Name Find the unit rate to determine the better buy. Option A

Option B

Better Buy

12-oz bottle of shampoo for $4.80

18-oz bottle of shampoo for $6.75

3 notebooks for $6.60

5 notebooks for $10.50

5-pound bag of rice costs $6.25

2-pound bag of rice costs $2.80

Graph the proportional relationship. Then write an equation for the graph. 4. $30 is earned in 2 hours.

5. 3 notebooks cost $7.50. 10

Cost ($)

Money Earned ($)

Time (hours)

Number of Notebooks

Solve. Explain your answer. 7. Rachel types at a speed of 42 words per minute. Donna’s typing speed is shown in the table below. Who types faster?

8. 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?

Minutes (x)

Words Typed (y)

138

230

368

460

9. 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

8 Cost ($)

Distance (Miles)

36 18 0

Time (Hours)

6 4 2 0

5 Pounds

CH A L L E N G E 10. Can you tell if a graph is in a proportional relationship if you know just one point?

Lighthouse Math

Level H

Chapter 5

Exercise 4

101

6. 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?

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

Chapter 6 | Skill Checklist

Skill 1: Graphing in Slope-Intercept Form y = mx + b

Graph the lines. 10

m = slope b = y-intercept

y = 2x − 5

y =

1 x+4 2

-10

-5

m=2 b = -5

-10

10 5

2 2. y=- x−7 3

-10

-5 -5 -10

-6

3. 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 The Distributive Property

Apply the distributive property. 1. 3(4x + 7)

2(3x + 4) 2(3x) + 2(4) 6x + 8

2. 5(6x − 8)

3. 8(8x + 9)

Factoring

45x + 35 ÷5 ÷5 5(9x + 7)

Factor each expression. 4. 18x + 10

Level H

6. 100x − 36

I can apply the distributive property and factor expressions.

out 6 correct

104

5. 56x + 8

Chapter 6

Skill Checklist

Lighthouse Math

Name

Skill 3: Evaluating Expressions Using Substitution x=4

5(3x + 8)

Evaluate.

5(3(4) + 8) 5(12 + 8) 5(20)

1. 2 − 4x when x = 6

100

3. 14x + 3 when x =

out 4 correct

2. 9.2 + 5.3x when x = 1.1

1 2

x 4. + 5 when x = -9 3

I can evaluate expressions using substitution.

Skill 4: Solving Equations

out 4 correct

Solve the equations using inverse operations. 1. 3x − 7 = 32

x 2. + 8 = 15 2

3. -5x + 4 = 29

x 4. − 12 = -15 4

I can solve equations using inverse operations. © Lighthouse Curriculum. Copying strictly prohibited.

4x + 6 = -14 −6 −6 4x = -20 ÷4 ÷4 x = -5

Skill 5: Zero Pairs 6 + -6 = 0

Fill in the blanks to make the statements true.

x + -x = 0

1. 3+

2. -15 +

3. 4x +

1 4. - y+ 2

out 4 correct

Lighthouse Math

I can apply the additive inverse property.

Level H

Chapter 6

Skill Checklist

105

6- 1 | Introduction to Systems of Linear Equations

PREREQUISITE SKILLS

Graph each line given the slope and the y-intercept. 5

DAI LY REV I E W

1 1. m= 2

2. m = -3 -5

b=2

b=1

-5

1 3. m=4 -5

b=0

-5

-5 5

SPIRAL REVIEW

Write an equation for each proportional relationship. 1.

2. 4 cookies cost $12

-5

5 -5

L E A RN A ND C O NNECT 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 x+4 3 2 y= x−2 3

The lines have different slopes.

5 -5

The lines intersect at every point.

y = 3x + 2

y = -3x + 6

Infinite solutions

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

5 -5

2. -5

3. 5

-5

5 -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

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

4. -5

-5

For each system of equations, write if the slope and y-intercept are the same or different. y = -2x + 1

y = 6x + 4

y = -2x + 1

y = 21 (18x – 14)

y = -6x – 4

y=x

y = -x

Determine how many solutions exist for each system of equations. 5

9. -5

-5

13.

10.

y = 13x – 6 y = -5x + 4

11. 5

-5

14.

12. -5

-5

y = 12x – 9

15.

y = 3x

-5

16.

y = 1.5(2x + 6)

y = 3(4x – 2)

y = 32 (15x – 9) y=

8x – 6

Determine how many solutions exist for each system of equations. Explain why. 18. y = -5x + 3 and y = -5x + 7

CH A L L E N G E 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: Sam: Will Alex and Sam ever spend the same amount of money on their trips? If so, after how many miles? Explain.

Lighthouse Math

Level H

Chapter 6

Exercise 1

107

17. y = -x + 3 and y = x + 3

6-2 | Solutions of Systems of Equations

PREREQUISITE SKILLS

Evaluate each expression. 2. -8x – 3 if x = 5

DAI LY REV I E W

1. 5x + 2 if x = -4

SPIRAL REVIEW

3. 18x + 11 if x = 2

4. 9x + 5x if x = 3

State whether each system has one solution, no solution, or infinite solutions. 1.

y = -5x – 4

y = 21 x + 8

y = 11x – 7

y = -5x + 3

y = 3x + 8

y = 31 x + 2 y = -3x + 2

y = 3x + 8

L E A RN A ND C O NNECT 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. To determine if an ordered pair is a solution to a system, substitute the x and y values into both equations.

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. On a graph, this is the point where both lines meet.

3x + 4y = 5 has no solution 3x + 4y = 9 because 3x + 4y cannot equal both 5 and 9. The system

A P P LY Match each system of equations with its solution. 1.

y = 3x + 1

y= x +7

y = -2x + 5

y = 4x − 7

y = 6x + 3 A. (2, 1)

y = -3x − 3

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= x

y = -3x + 6

y = -2x + 8

y=x+6

y = 2x + 4

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

Exercise | 6-2 Name Determine if the given ordered pair is a solution to the system. 1.

(-2, 8)

(5, 5)

(4, -2)

(2, 3)

(2, 1)

y = -4x

y=x

y = -x + 6

y = 3x − 3

y = 2x − 3

y = x + 10

y = 3x − 10

y = 2x − 6

y = -x + 7

y = -x + 5

(0, 0)

(3, -8)

(4, 6)

10. (-1, 1)

(6, -4)

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. 11. (10,

12. (

)

1 13. (- , ) 3 9x + 6y = 3

, 4)

-x + y = 5

2x + 2y = 12

y = 2x − 5

y = 2x

14. (

-9x + 2y = 5

15. (

, 0)

, 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. x+y=3

16.

y=x−1 Solution: (

)

x+y=3

y=x-1

x + y = 15

17.

y = 2x

Solution: (

)

x + y = 15

y = 2x

18. Explain why the following system has no solution.

-2x + 7y = 11 -2x + 7y = 2

CH A L L E N G E 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. Simplify both equations. Does it have one solution, no solution, or infinitely many solutions? Explain how you know.

Lighthouse Math

Level H

Chapter 6

Exercise 2

109

Answer the question.

6-3 | Solving Systems of Equations by Graphing

PREREQUISITE SKILLS

Graph each linear equation. y=

DAI LY REV I E W

1 x–1 3

-5

y = -3

-5

y = -3x

-5

SPIRAL REVIEW

-5

Determine if each ordered pair is a solution of the system. 1.

(1, 3)

y = 2x + 1

y = -3x + 6

(2, 0)

y = -x + 4

(2, 4)

y= x –2

y= x +2 y = -2x + 5

L E A RN A ND C O NNECT 5

You can find the solution to a system of equations by graphing. 1. Graph each line.

2. Find where the two points intersect. Write the solution as an ordered pair.

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

A P P LY Graph each system of equations if necessary. Then choose the correct solution. y = -x

y = - 32 x + 4

y=3

(3, -3)

(-3, 3)

(3, 3)

y = - 41 x − 1

-5

5 -5

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

Exercise | 6-3 Name Write the solution to each system of equations. 5

-5

3. 5

-5

4. -5

-5

Solve each system of equations by graphing. 5.

y = 2x

y = -x – 3

y = -2x + 1

y = - 21 x + 3

y = -2x + 4 -5

-5

y= x -5

-5

Solve each word problem by graphing. 8. 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:

9. 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: y = -10x + 80 y = 30x + 40

y = 10x + 60 y = 30x + 20

When will Flora and Lena have the same amount of money?

100

160 120 80 40 0

80 60 40 20 0

Number of Weeks

200

Money ($)

Money Saved ($)

When will both Liam and Manuel have earned the same amount of money? How much will they have earned?

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 B. Graph the equations. How many of each type of cookie did they sell?

Lighthouse Math

Level H

Money Raised

CH A L L E N G E 20 16 12 8 4 0

# of Cookies

Chapter 6

Exercise 3

111

6-4 | Isolating Variables

DAI LY REV I E W

PREREQUISITE SKILLS

Solve each equation for x.

SPIRAL REVIEW

7x + 14 = 35

2x + 9 = x – 5

3 x – 10 = -2 4 1 2 x – 6 = 2x + 3 2

Determine if each ordered pair is a solution of the system. 1.

(1, 5)

y = 2x + 3

(3, 1)

y = -x + 6

y = 3x – 4

(4, 3)

y = - 21 x + 5 y = 3x – 9

y = -2x + 7

L E A RN A ND C O NNECT Look at the following equation:

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: A dd 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: I f 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+

1 y 3

y = 3 − 3x

A P P LY

Follow the steps to solve for x and y.

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

4x = 12 – 3y

3y = 12 – 4x

Simplify.

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

Exercise | 6-4 Name What does x equal? Isolate the x variable to find out. 1. y = 2x + 8 x= 6. 3x – 18y = 9 x=

2. y = -3x – 15

3. 4x – y = 24

x= 7. 2y = 6x – 4

4. -4x + 5y = 12

3 1 8. 4x+ 2 y=6

5. 5x + 2y = 10

9. y = 21 x – 3

x= 10. y = - 25 x + 4

What does y equal? Isolate the y variable to find out. 11. 2x + y = 7 y= 16. 4x + 6y = -8 y=

12. 4y – x = 12 y= 17. 31 x + 21 y = 6

14. 21 x + 8y = 5

13. 6x – 2y = 10 y=

18. -18 = -3y + 4x

15. -5x + 3y = 27

19. 2x – 14 = 7y

y= 20. 35 x + 65 y = 4

Read the problems. Isolate a variable to solve. 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.

21. You are planning a rectangular garden using 50 feet of fencing. The perimeter formula is 2L + 2W = P. 2L + 2W = 50

1.5b + 0.9c = 150

Rewrite this equation so that it will you tell you the width you need to use for any given length.

Rewrite this equation so that it will tell the number of loaves of bread they can make with any given number of cakes.

CH A L L E N G E 23. Use the following equation to perform each task. A. Isolate the x variable.

2 4 1 2 x + y – 7 = x + y – 18 3 5 2 3 B. Isolate the y variable.

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.) D. What do you notice about the x-intercept and y-intercept in relation to the isolated equations?

Lighthouse Math

Level H

Chapter 6

Exercise 4

113

6-5 | Solving Systems of Equations by Substitution

DAI LY REV I E W

PREREQUISITE SKILLS

Evaluate each expression if a = -4, b = 7, and c = 31 . 1. -9c + 5b

SPIRAL REVIEW

2. 5a – 18c

3. -ab

4. 12c – b

3. 20 = 3x – 5y

4 1 4. x + y = -4 5 2

Solve each equation for y. 2. -8x – 4y = 36

1. 2x + y = 17

L E A RN A ND C O NNECT 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: I solate a variable. Check to make sure at least one equation has an isolated variable. If not, isolate a variable in one of the equations. 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 – 2x = 3 4x + 4y = 24 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).

y – 2x = 3 + 2x + 2x y = 2x + 3

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)

A P P LY Fill in the blanks to solve the system by substitution. 1.

y=x+2 5x + y = -4 A. W e know that y is equal to

B. We can substitute for y in the equation 5x + y = -4. 5x + x=

C. Now we know that x= .

D. We can substitute for x in the equation y = x + 2. y= y =

= -4 x + 2 = -4 x=

E. Now we know that y= . Solution: ( , )

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

Exercise | 6-5 Name Solve each system of equations using substitution. 1.

y = 2x + 1

3x + y = 16

x=y–4

2x + y = 10

y = 5x

x + y = 18

x = 3y

2x – y = 10

y = 3x + 6 2x + y = -4

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

4x – 2y = 12

8x + 16y = 24

-x + 5y = 30 6x + 10y = -20

10.

1 3 2 x– 4 y=4 -2x + 45 y = -16

First isolate the y variable, then solve each system of equations using substitution. 11.

3x + y = 10 2x – y = 0

12.

5x – 2y = 16 -3x + y = -9

13.

6x – y = 7

14.

2x + 3y = -1

18x – 6y = -12 7x – 2y = -9

15.

3 4 x + 2y = 7

x – 3y = -2

Use substitution to find the solution to each system of equations. Answer the questions. 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?

y=x+3

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?

120x + 45y = 4,350 x + y = 60

200y + 100x = 2,100

CH A L L E N G E 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

Chapter 6

Exercise 5

115

6-6 | Solving Systems of Equations by Elimination

DAI LY REV I E W

PREREQUISITE SKILLS

Use the distributive property to simplify. 1.

SPIRAL REVIEW

2. -9(7x + 3)

5(3x – 8)

3. -7(8x – 2y + 4)

Solve each system of equations using substitution. 1.

y=x+3

5x – y = 12

x = -2y – 2

3x + 2y = 6

x + 3y = -11

-3x – 3y = 9

-9x – 3y = 3 4x + 2y = 2

L E A RN A ND C O NNECT 2x – 4y = 10

Look at the two equations:

What happens when you add them?

4x + 4y = 8

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. 1) See if a variable can be canceled out. If not, multiply each equation so that the coefficients of one variable are the same.

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. W e know that and can cancel if you the equations -4x + y = 8 2x – y = 6 =

B. Now we can solve for . -2x = 14 x= C. Now we know that x equals .

D. We can substitute for x in either equation to find y. -4(

)+y=8 +y=8 y=

E. Now we know that y equals . Solution: ( , )

Vocabulary Elimination - the process of removing a variable using addition or subtraction

116

Level H

Chapter 6

Lesson 6

Lighthouse Math

Exercise | 6-6 Name For system of equations, determine what number each equation needs to be multiplied by, so the variables can cancel out. 1.

3x + 4y = 3 6x – y = -3

Multiply by cancel x

-4x – 3y = -7 x – y =4

Multiply by to cancel

8x – 6y = -20

Solve each system of equations using elimination. 3.

6x – 5y = -8

3x + 5y = 26

3x – 2y = 4 5x + 4y = 14

x + 5y = -19 3x + 5y = -27

6x + y = 60

-16x + 7y = 30

2x + 5y = -11

-2x + 3y = -4 5x – y = -3

3x + 2y = 0

7x + 7y = 105

10.

14x + 3y = 85 10x – 6y = 20

Write a system of linear equations to represent the word problem. Then use elimination to solve the problem. 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?

x + y = 73

x + y = 200

2x + 21 y = 68

4x + 2y = 520

CH A L L E N G E 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. B. Write a system of equations to represent this situation. Think: How many equations will you need in order to solve this system? C. Find out how many projects came from each grade level.

Level H

Chapter 6

Exercise 6

117

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?

6-7 | Applications of Systems of Equations

PREREQUISITE SKILLS

Determine if the ordered pair is a solution to the system. 2. (2, 8) 9x + y = 26 x + y = 10

DAI LY REV I E W

1. (1, 14) 15x + 21y = 8 x = 14y SPIRAL REVIEW

3. (0.1, 1.6) 2x + 3y = 5 y = 6x + 1

Solve each system using elimination. 1.

x + y = 45

x–y=3

x + y = 34

7x + 3y = 71 4x + 2y = 44

2x + 3y = 89

L E A RN A ND C O NNECT 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. A ssign variables to each unknown value. x = # of adult tickets y = # of child tickets

B. Write one equation to represent the number of tickets. x + y = 100

Step 2: Solve using any method.

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.

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. Two bags together weigh 45 pounds. The heavier bag weighs 3 pounds more than the lighter one. How much does each bag weigh?

118

2. 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

7x + 3y = 75

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

Lighthouse Math

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

x + y = 20 2x + 4y = 54

A. Why is there a 2x in the second equation and only x in the first equation?

B. What do x and y represent? How do you know?

C. How many of each type of animal does the farmer have?

2. 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?

3. 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?

4. 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?

5. 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?

CH A L L E N G E 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.

B. How many of each type of school supplies did Adam buy?

Lighthouse Math

Level H

Chapter 6

Exercise 7

119

Write a system of equations for each problem. Solve using any method.

6-8 | Review

L E A RN A ND C O NNECT 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? Write a system of equations: x + y = 45 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

# of Tulip Pots

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

A P P LY Determine how many solutions exist for each system of equations. 5

1. -5

2. 5

-5

3. -5

-5

4. 5

-5

5 -5

one solution

no solution

infinite solutions

120

Level H

Chapter 6

Lesson 8

Lighthouse Math

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?

2. Is (-2, 6) a solution to this system of equations?

2x + y = 11

x + y = 10

x – 2y = -2

2x – y = 1

Solve each system of equations by graphing. 3.

y = 3x

y = 21 x + 1

y = -x + 4 5

-5

y = 21 x – 3

-5

y = - 31 x – 1

y = - 32 x – 4

y = -5x – 5

-5

y=0

-5

5 -5

Solve each system of equations using substitution or elimination. 7.

y = 3x + 1 2x + y = 11

y=x+4

-2x + y = 0

y = 3x x + y = 12

10.

x – 5y = 0 x + y = 18

11.

-x + y = 2 3x + 2y = 24

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?

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?

CH A L L E N G E 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?

Lighthouse Math

Level H

Chapter 6

Exercise 8

121

Write and solve a system of equations for each word problem.

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

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

I can identify points, lines, and angles.

out 6 correct

Skill 2: Types of Angles Circle the type of angle shown. a right angle is 90°

an acute angle is less than 90°

a straight angle is 180°

right

acute

straight

obtuse

an obtuse angle is more than 90° but less than 180°

124

right

acute

straight

obtuse

out 4 correct

I can classify angles.

Level H

Skill Checklist

Chapter 7

Lighthouse Math

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. Scalene Triangle No sides equal

Isosceles Triangle Two sides equal

Equilateral Triangle All sides equal

Acute Triangle All angles acute (<90°)

Right Triangle One right angle (90°) © Lighthouse Curriculum. Copying strictly prohibited.

Obtuse Triangle One obtuse angle (>90°)

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 out 9 correct

Lighthouse Math

1. 82 =

2. 602 =

3. 2.52 =

4. 0.42 =

5. 9=

6. 400 =

7. 16 =

8. 6,400 =

9. 8,100 =

I can evaluate squares and square roots.

Level H

Chapter 7

Skill Checklist

125

7-1 | Understanding Angle Relationships

PREREQUISITE SKILLS

Write the name of the highlighted angle using three letters with the letter for the vertex in the middle.

DAI LY REV I E W

W Z

SPIRAL REVIEW

Solve for x. 2. 12 + 5x = 47

1. 3x − 16 = 14

3. 144 = 4x + 20

4. 32 = 8x − 6

L E A RN A ND C O NNECT Angles are formed by lines that meet at a vertex. Angle relationships can be used to find a missing value. Complementary Angles

Supplementary Angles

Angles that add up to 90°

Angles that add up to 180°

b = 90°

d = 180°

m B

h i

Angles that share a vertex and side

e is adjacent to

AEB is supplementary to AEC. When added together, they equal 180°.

x°

Adjacent Angles

Formed by intersecting lines, these angles are directly across from one another and equal j

Vertical Angles

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

126

Level H

Chapter 7

Lesson 1

Lighthouse Math

Exercise | 7-1 Name Write whether the angles are complementary or supplementary. Then find the missing angle. 1.

x° 80° x°

52°

x°

68°

x°

124°

Write whether the angles are adjacent or vertical. Then find the missing angle. 5.

x°

8. x°

121°

40° x°

x° 25°

75°

34°

76°

Write an equation. Then solve to find the value of x. 10.

11. x°

(x + 22)°

12.

112°

(3x - 6)°

(3x)° x°

120°

x°

CH A L L E N G E 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.

2x°

m ACD =

Lighthouse Math

m ACE =

m DCE =

Level H

m ECB =

Chapter 7

Exercise 1

(x + 30)° A

x° C

127

7-2 | Parallel Lines and Transversals

PREREQUISITE SKILLS

Find the missing angle. 1.

DAI LY REV I E W

89°

SPIRAL REVIEW

3. b

42°

122°

165°

Find the value of x. 1.

(3x + 1)° 115°

3. (15x - 11)° 41°

105° x° 81°

x° x°

L E A RN A ND C O NNECT 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

Sketch

Example from diagram

Relationship

Alternate Interior

equal

AGH and

GHD

Alternate Exterior

equal

AGE and

DHF

Corresponding Angles

equal

GHD and

EGB

Same Side Interior

add to 180°

BGH and

GHD

Same Side Exterior

add to 180°

EGB and

DHF

> symbol on lines shows that they are parallel

What is another example from the diagram?

A P P LY Name the angle relationship. 1.

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

Exercise | 7-2 Name Refer to the diagram. Name all the angle pairs from each category.

1. Alternate Interior

2 4

2. A lternate Exterior

3. Corresponding

4. Same Side 5. Same Side Interior Exterior

3 5

6 8

Name the angle relationship, then find the value of x. 6.

x°

35°

105°

x°

150°

x° x°

20°

Use what you know about angle relationships to write an equation. Then solve. 10.

(3x)°

11.

12.

(4x + 1)° 25°

(8x + 5)° (x + 4)°

102°

(10x + 7)°

13.

(5 - 2x)°

y°

(6x + 5)°

A. J onathan wrote the equation 6x + 5 = 2x + 3 to solve for x in the diagram. Do you agree with him? Why or why not?

B. Solve for y. Explain how you found your answer.

(2x + 3)°

CH A L L E N G E 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.

2 4

3 5

6 8

Lighthouse Math

Level H

Chapter 7

Exercise 2

129

14. Refer to the diagram below to answer the questions.

7-3 | Angle Relationships in Triangles

DAI LY REV I E W

PREREQUISITE SKILLS

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.

L E A RN A ND C O NNECT b

We can use rules we already know to find another rule.

Rule: All the angles in a triangle add up to 180°.

m a + m b + m c = 180°

Rule: Supplementary angles add up to 180°.

m c + m d = 180°

Since both (m a + m b) and m d equal 180° when added to m c, we can say: New rule m a+m b=m d

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°

67° 75°

38°

x°

63°

40°

x°

21°

101°

x°

45°

84°

Find the measure of the missing angle. 6.

24°

31°

42° x°

x° 71°

33°

57°

x°

82°

10.

91° 25°

x°

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

Level H

Chapter 7

Lesson 3

Lighthouse Math

Exercise | 7-3 Name Write an equation and then solve for x. 1.

24° x + 6°

120° x°

3x°

2x° (7x + 5)°

50°

(x - 20)° (x + 8)°

52°

x°

3x°

x°

Use the shapes and rules you know to answer the questions. 6.

A. F ind the value of x.

x°

(2x)°

7. The hexagon below is composed of 6 identical equilateral triangles. A. What is the measure of each angle in an equilateral triangle?

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? C. What is the sum of all the angles in a hexagon?

8. 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

C. What is true about angles that are opposite each other in a parallelogram?

CH A L L E N G E 9. Find the value of w, x, y and z. A

74°

w° x°

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°

y= y° B

Lighthouse Math

z° D

Level H

Chapter 7

Exercise 3

131

B. Given what you know about angles in a triangle, what is the sum of the angles in a parallelogram?

7-4 | Introduction to the Pythagorean Theorem

DAI LY REV I E W

PREREQUISITE SKILLS

Solve for x.

SPIRAL REVIEW

1. x2 = 36

2. x2 = 16

3. 10 + x2 = 91

4. x2 + 28 = 37

5. x2 − 8 = 41

Find the value of x. 1.

20°

60°

x°

73°

115°

40°

3x°

x°

L E A RN A ND C O NNECT 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.

a +b =c 2

ote

leg (a)

We can find a missing side of a right triangle using a formula called the Pythagorean theorem. 2

hy p

(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

0.8 meters

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

Exercise | 7-4 Name Use the Pythagorean theorem to find the hypotenuse of the right triangle. 1.

1.6

2.4

1.8

4 1.2

Use the Pythagorean theorem to find the missing leg of the right triangle. 5.

1.25

13 10

0.75

b b

Use the Pythagorean theorem to solve for x. 9.

10.

11.

x 4.8

x= 12

3.6

12. Look at the triangle and Trevor’s work. Did he use the Pythagorean theorem correctly?

32 + 72 = x2 9 + 49 = x2 58 = x2 7.6 = x

CH A L L E N G E 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? B. The ladder can extend 2 more meters. About how high up the wall can it reach when extended?

Lighthouse Math

Level H

Chapter 7

Exercise 4

133

7-5 | Distance on the Coordinate Plane

DAI LY REV I E W

PREREQUISITE SKILLS

Solve. 2. 82 + 18

1. 4 − 32

SPIRAL REVIEW

3. 2(42 + 9)

4. (22 + 12)

5. 45 − 52

Use the Pythagorean theorem to find the missing side. 1.

3 4

L E A RN A ND C O NNECT The Pythagorean theorem can also be used to find the length of a line segment on the coordinate plane. Step 1: D raw 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

(8, 5)

The length of line segment AB is 5 units. 0

All of this can be done in one formula called the distance formula:

c = (y2 − y1)2 + (x2 − x1)2

A P P LY Complete the steps to find the distance between the two points.

1. Draw two lines to make a right triangle. 2. Use the coordinates to find the length of each leg of the triangle.

a= b= 3. Use the Pythagorean theorem to find the length of c. 4. Explain why c is the distance between the two points.

-32

-16

(24, 16)

-16

(-24, -20) -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

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 − (20, 5)

-50

)2 + (

c= (

)2 + (

c = (y2 − y1)2 + (x2 − x1)2 10

− 20)2 )2 -10

(-4, -2)

-50

c = (6 −

(2, 6)

)2 + (

c= (

)2 + (

− 4)2 )2

-10

Find the distance between the two points on the coordinate plane. 3.

-10

120

-120

Length =

-120

Find the distance between the two points.

Length =

6. (3.2, 1.3) and (5.6, 4.5)

7 (-3, -4) and (1.8, 2.4)

Length =

5. (0, 4) and (-8,-2)

Length =

CH A L L E N G E 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?

400 300 200 100 100

Lighthouse Math

Level H

Chapter 7

Exercise 5

200 300 400

135

7-6 | Applying the Pythagorean Theorem

PREREQUISITE SKILLS

Circle H if the bolded side is a hypotenuse. Circle L if it is a leg.

DAI LY REV I E W

2. H

SPIRAL REVIEW

3. H

4. H

Find the distance between the two points. 1. (-6,-2) and (0, 6)

3. (3,8) and (-2, 9)

2. (2,1) and (5, 5)

L E A RN A ND C O NNECT 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.

30 yards

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.

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

2. 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

Exercise | 7-6 Name Circle the equation that will help solve the problem. Then find the missing piece.

1. 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? c2 = 82 + 62

60 + 80 = c2

The diagonal is

62 + b2 = 82 long.

2. 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

122 + b2 = c2 long.

Use the pictures and the Pythagorean theorem to answer the questions. 1.6 km coastline

4. 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?

1.2 km

3. 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?

5 1.2

ile

0.75 mile

Draw a diagram. Then answer the question. 6. 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?

CH A L L E N G E 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 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?

Lighthouse Math

Level H

Chapter 7

Exercise 6

137

5. 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?

7-7 | Review

L E A RN A ND C O NNECT 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

g 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

(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

138

Level H

Chapter 7

Lesson 7

Lighthouse Math

Exercise | 7-7 Name Find the value of x. 1.

4. (x + 50)°

(x + 60)°

x°

(3x)°

(2x + 14)°

x° 2x°

x°

(20 + 2x)°

(7x)°

x°

(x + 80)°

(4x)°

x° (2.4x)°

42° 72°

(x - 20)°

Use the Pythagorean theorem or distance formula to find the missing length. 9.

10. 25 in

11. (6, 4)

x (0, -4)

15 in

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?

CH A L L E N G E 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 Avenue C

Chapter 7

Exercise 7

70°

Avenue D

139

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

Chapter 8 | Skill Checklist

Skill 1: Drawing Lines, Angles, and Shapes A C

E F

EFG or

H J

ABC

LMN

HIJ

I can understand line, angle, and shape notations.

out 4 correct

Skill 2: Lines of Symmetry 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. 1.

I can draw lines of symmetry.

out 4 correct

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 more 3

less

12 × 6

12 ×

5 × 83 will be less than 5.

out 4 correct

142

Level H

Chapter 8

than 9. 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.

Skill Checklist

Lighthouse Math

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. 3 units to the right, 4 units up , ) (

-10

2. 7 units to the right, 2 units down ( , )

-10

Point O is the origin: (0,0)

3. 3 units to the left, 8 units up ( , )

Point A is 5 units to the left of the origin and 4 units below it at (-5, -4).

4. 6 units to the left, 9 units down ( , )

-10

out 4 correct

-10

I can navigate the coordinate plane fluently.

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 =.

1. A: 10

2. B: 3. C: A

4. D:

y=6

-10 -10

Draw the lines on the graph.

B C

5. y = -1

x = -4

D -10

6. x=8 -10

out 8 correct

Lighthouse Math

I can draw vertical and horizontal lines from an equation.

Level H

Chapter 8

Skill Checklist

143

8-1 | Congruent and Similar Figures

PREREQUISITE SKILLS

Solve each proportion. 3 x = 5 20

DAI LY REV I E W

SPIRAL REVIEW

9 72 = 10 x

7 14 = x 16

x 15 = 20 25

Find the missing side using the Pythagorean theorem. 1.

2. x in

3 in

4.2 m

17 ft

5.6 m

4 in

x ft

10 in

15 ft

6 in

x in

L E A RN A ND C O NNECT 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

ABC

JKL

is congruent to

70° 13 in

A and B and C and

Corresponding angles are equivalent.

50°

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.

8 in T

12 in 110°

110°

70°

M and N and O and P and

Corresponding angles are equivalent.

x = 20

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

similar

12 55° U

65°

Y 60°

55°

10 m

congruent

NO OP = RS ST 4 10 = 8 x 4 10 8 = 10 × =x 8 x 4

M 6 in N 110° 110° 4 in 4 in 70° 70° P O 10 in

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

Exercise | 8-1 Name Identify the corresponding angle or side. M

65°

18 ft

50° 50°

11 ft

M and

NO and

B and

L F

18 ft

122° C

58°

65°

65° N

T 122°

11 ft

10 in

58° U

CD and

M and

KL and

Determine the unknown measurements of the congruent figures. 4.

18 m K 115°

65°

26 m

24 m M

VU =

60° V

UV =

44 ft

42° Q 50 ft

60°

78°

34 ft

32 m

10 in

42°

28 in

9 in

XY =

Determine the unknown measurements of the similar figures. F

D 64 m C

76°

65° 39° E 80 m

114°

66°

76° G

10 in

x in

66° D

12.

55°

80°

x= 96 m

12 m

6 in

20 in

C x in

11.

8 in

6 in

5 in

x= 40 in

8 in

7 in

x= 10.

16 m H

45°

64 m

CH A L L E N G E 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.

Lighthouse Math

Level H

Chapter 8

Exercise 1

145

8-2 | Translations

DAI LY REV I E W

PREREQUISITE SKILLS

Find the distance between the ordered pairs using the distance formula.

SPIRAL REVIEW

(0, 0) and (6, 8)

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

(2, -2) and (3.8, 0.4)

(7, 15) and (10.6, 19.8)

State whether each set of shapes is congruent or similar. 1.

18 cm

6 cm

2 cm

5 in

11 in

6 cm

20 in

5 in

16 in

4 in 12 in

5 in

3 in

L E A RN A ND C O NNECT 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.

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. 10

2. Q

X’

T -10

-5

Q’

S 5

R’

-10

Y’

-5

T’

Z’

-5

S’ Y

-10

Move up/down

units.

Move up/down

units.

Move right/left

units.

Move right/left

units.

Circle one

Circle one Circle one

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

Exercise | 8-2 Name Write a rule to describe the translation of the black shape to the purple shape. A’

10 L’

M’

A O’

B’ -6

C’

-10 L

N’ 10

-10

-6

G H 10

G’ H’

J’ I’

-10

Translate each shape using the given rule. Label each vertex. 4. Move 5 units left and 9 units up.

5. Move 5 units right and 3 units down.

6. Move 7 units right and 13 units up. 10

L M P Q N O S

D -10

-10

U F

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. 8

-6

A. Write the new coordinates 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? 6

CH A L L E N G E 9

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? C. P ’Q’R’S’ was then translated and point Q’’ was plotted at (1,-1). What are the directions for the translation?

Lighthouse Math

Level H

Chapter 8

-6

Exercise 2

147

8-3 | Reflections

DAI LY REV I E W

PREREQUISITE SKILLS

Draw the line(s) of symmetry in each shape. 1.

SPIRAL REVIEW

Describe the translation that occurs from one ordered pair to the other. 2. (-4, 0) → (-7, 5)

1. (3, 2) → (5, -1)

3. (5, -9) → (-3, -2)

4. (2, 4) → (6.5, -2.5)

L E A RN A ND C O NNECT 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. You can think about the figure being folded over the line. If the two shapes would completely align, the original figure was reflected correctly. 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

C’

-5

A’

B’

A’

Reflected across the y-axis

C C’

-5 B’

-5

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

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’

W’ X’ B C

-3

-5

J K

J’ K’

-5

-2

M’

W X

Z’ Y’

-5

L’

G’

F’

-5

E’

Graph each of the given reflections. Label each vertex. 5. Reflection across y-axis 5

6. Reflection across y=2

R S

I K

-5

7. Reflection across x=4

8. Reflection across x-axis 5

-5 -1

-5

-1

X 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 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?

-5 -2

CH A L L E N G E 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. 5

A. Graph both quadrilaterals. B. What are the coordinate points of the transformed quadrilateral? 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’’.

Lighthouse Math

Level H

-5

Chapter 8

Exercise 3

149

C. Use the slope formula to calculate the slope of side AB and A’B’. Do they have the same slope?

8-4 | Rotations

DAI LY REV I E W

PREREQUISITE SKILLS

Find the slope between each set of points.

SPIRAL REVIEW

(3, 6) and (5, 10)

(-1, -2) and (-8, -4)

(-5, 4) and (-2, 5)

(0, 3) and (10, -7) 10

Use the coordinate plane to reflect each point. Write the new ordered pair. 1. (5, 6) reflected across the x-axis

2. (-2, -4) reflected across the y-axis

3. (-1, -8) reflected across the x-axis

-10

L E A RN A ND C O NNECT 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 -5

B’

C A’

B C’

A’

B’

-5

A’

C’

-5

90° clockwise or 270º counterclockwise: (x, y) → (y, -x)

C’ A

-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’

A’ D’ 5

V’

T’ S’

B’ C’

-5

3. L’

K’

-5

J’

K L -5

-5

G’ H’

-5

F’

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

Exercise | 8-4 Name Rewrite each set of points given the degree of rotation around the origin. 90° clockwise/ 270º counterclockwise (y, -x)

(-4, 7)

(3, 8)

(-2, -3)

(6, -1)

180° clockwise/ counterclockwise (-x, -y)

270° clockwise/ 90º counterclockwise (-y, x)

Rotate each shape around the origin using the given angle of rotation. Label each vertex. 6. 90º clockwise rotation

5. 180º rotation 5

O N

-5

G H M

-5

7. 90º counterclockwise rotation

270º clockwise

5F C

I 5

-5

X -5

-5

D 5

V W

-5

Complete each step. 10

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 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?

-10

C. Use the slope formula to calculate the slopes of side AB and A’B’. Do they have the same slope? -10

CH A L L E N G E 5

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. -5

A. Graph both triangles. B. What are the coordinate points of the transformed triangle?

Lighthouse Math

-5

Level H

Chapter 8

Exercise 4

151

8-5 | Dilations

Find the missing value in each set of similar shapes. 1.

8 m 42° 10 m

80° 6m

SPIRAL REVIEW

58°

3 in

28 cm

32 in

x in

80° xm

58°

12 in

42° 16 m

DAI LY REV I E W

PREREQUISITE SKILLS

49 cm

x cm 7 cm

Write the new ordered pair given the rotation. 2. (-5, -8) rotated 180º

1. (3, 1) rotated 90º counterclockwise

3. (-4, 2) rotated 90º clockwise

4. (6, 6) rotated 270º counterclockwise

L E A RN A ND C O NNECT 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. 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(-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)

-5

1 2 A(-4, 4) → (-4 × 21 , 2 × 21 ) → A’(-2, 2)

A’ A

C’

Scale Factor =

B(-4, 2) → (-4 × 21 , 2 × 21 ) → B’(-2, 1)

C(0, 2) → (0 × 21 , 2 × 21 ) → C’(0, 1)

-5

A’

B’

C’

A P P LY Circle whether the dilation of the black figure is an enlargement or a reduction. 1.

-10

A D

A’ D’

B’ C’

X’

Y F’

G’

I Z’

Y’ -10

-10

enlargement or reduction

I’

H’

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

Exercise | 8-5 Name Identify the scale factor of each dilation from the black shape to the purple shape. L 10

-10 U X U’

M -10

L’

V’

W Y

O’

M’

N’

X’

R’

W’ T

Z’

N -10

Y’

S T’

-10

S’

Graph the dilated image using the given scale factor. Label each vertex. 4. Scale factor = 3

5. Scale factor = 2 10

6. Scale factor =

-10

C D

1 4 J

-10

10 M

-10

Complete each step. 7. 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.

A. Write the coordinates of X’, Y’, and Z’ after the dilation. B. Find the distance of XY and X'Y'. C. Show that the side length of XY is proportional to X’Y’. 10 5

CH A L L E N G E 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 .

-10

A. Graph both HIJK and H’I’J’K’. B. List the ordered pairs for H’I’J’K.

Lighthouse Math

-10

Level H

Chapter 8

Exercise 5

153

8-6 | Combining Transformations 5

DAI LY REV I E W

PREREQUISITE SKILLS

Graph each ordered pair on the coordinate plane. 1.

J(3, 5)

F(-1, -3)

R(0, 0)

C(4, -2)

-5

5 -3

SPIRAL REVIEW

Write the new ordered pair given the scale factor. 2. (-3, -3) dilated by a 3. (10, -2) dilated by scale factor of 31 a scale factor of 5

1. (4, -2) dilated by a scale factor of 2

4. (0, 6) dilated by a scale factor of 41

L E A RN A ND C O NNECT 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’’

-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

-10

C’

D’

A’

B’

-10

translation reflection rotation dilation

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

Exercise | 8-6 Name Complete each transformation. Label each vertex. 1. Translate down 2, left 4 1 Dilate by a scale factor of 2

2. Rotate 270º clockwise Reflect across the x-axis

W -10

D 10

-10

4. Reflect across x = -3 Rotate 270º counterclockwise

-10

5. Translate down 13, right 6 Rotate 90º counterclockwise G

-10

3. Translate up 7, right 10 Reflect across y-axis

1 6. Dilate by a scale factor of 5 Reflect across y = 4 K

H I

-10

N -10

-10

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 Translation:

-10

D’’

C’’ 10

A’’

B’’

Rotation: 180º rotation Translation:

Dilation: scale factor of 2 Reflection:

P’’

O’’

Q 10

-10

R’’ S’’

M’’

N’’

-10

M N

Q’’ -10

-10

CH A L L E N G E 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.

-3

-4

Lighthouse Math

Level H

Chapter 8

Exercise 6

155

8-7 | Review

L E A RN A ND C O NNECT Translations 5

Translating a figure involves moving it up, down, left, or right a given number of units.

-5

A’

-5

B’

Reflections 5

C’

B’

-5

A(-4, 4) → A’(2, -2) B(-4, 1) → B’(2, –5) C(-1, 1) → C’(5, -5)

C’

A(-4, 4) → A’(4, 4) B(-4, 1) → B’(4, 1) C(-1, 1) → C’(1, 1)

-5

Rotations 5

B’

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’

B’

-5

A(0, 2) → A’(0, 4) B(-2, -2) → B’(-4, -4) C(2, -2) → C’(4, -4)

C’

A P P LY

Circle all of the transformations that took place. 1.

A’

X’

-10

C’ C

B’ B

Y’ Z

-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

Exercise | 8-7 Name Translate each figure. 1. Move down 2, right 5 B

3. Move down 10, left 4

2. Move up 6, left 8

D -10

-10

M L

-10

O -10

-10

Reflect each figure. 4. Reflect across x-axis

6. Reflect across x = -2

5. Reflect across y-axis

I J

HG -10

Q T

-10

W Z

S -10

-10

X 10

E F

-10

Rotate each figure. Write the ordered pair for each new vertex.

-10

K -10

-10

Rotate 90º counterclockwise

J -10

8. Rotate 90º clockwise

-10

7. Rotate 180º

Dilate each figure. Write the ordered pair for each new vertex. 10. Scale factor = 2

11. Scale factor =

S B

-10

12. Scale factor = 3

-10

Lighthouse Math

1 3

L M -10

U -10

Level H

Chapter 8

O Q P

-10

Exercise 7

157

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

Chapter 9 | Skill Checklist

Skill 1: Naming 3D Shapes Cylinder

Cone

Write the name of the 3D shape.

Sphere

I can identify 3D shapes.

out 4 correct

Skill 2: Radius and Diameter radius

diameter

Write radius or diameter. 1.

Find the missing radius or diameter.

The diameter is two times the radius. The radius is half the diameter.

4. r = 12 in d=

5. r=

6. r=

d = 16 cm

d = 15 ft

I can find the radius and the diameter.

out 6 correct

Skill 3: Area of a Circle Find the area of the circle.

A = πr2 5m

Use 3.14 for π

A = (3.14)(52) A = (3.14)(25) A = 78.5 m2

10 cm

9 cm

15 ft

out 4 correct

160

Level H

Chapter 9

I can find the area of a circle.

Skill Checklist

Lighthouse Math

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.

3 ft

2 cm

10 cm

3 ft

8 in

3 in

8 ft

4 in

6 cm

h b

I can find bases, heights, and volumes of prisms.

out 6 correct

Skill 5: Multiplying Decimals

out 3 correct

Lighthouse Math

12.35 × 9.8

3.41 × 0.08

Evaluate.

14.3 × 1.25 715 2860 + 14300 17.875

6.51 × 1.35

I can multiply decimals.

Level H

Chapter 9

Skill Checklist

161

9-1 | Volume of Cylinders

PREREQUISITE SKILLS

Find the area (A = πr2) of each circle. Round your answers to the nearest hundredth.

DAI LY REV I E W

SPIRAL REVIEW

10 in

2. 7.5 cm

area =

31 m

area =

Explain what happens during each transformation. 2. Rotation

1. Translation

3. Reflection

4. Dilation

L E A RN A ND C O NNECT 2m

Volume is the amount of space found within a three-dimensional (3D) shape. It is measured in cubic units.

V = πr2h V = π(2)2(10) V = π(4)(10) V = (3.14)(40) V ≈ 125.6 m3

10 m

To find the volume of any shape, find the area of the shape’s base and multiply that value by the shape’s height. 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.

7 in 18 in

V = π(18)2(7)

V = π(7)2(18)

12 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)

8 cm

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 = π( 9m

V = πr2h )2(

)

V = π(

)(

)

V≈

6 in

V = π( 14 in

V = πr2h )2(

)

V = π(

)(

)

V≈

in3

5 cm

8 cm

V = πr2h V = π(

)2(

)

V = π(

)(

)

V≈

cm3

Vocabulary Volume - the amount of space found within a 3D shape Cylinder - a 3D shape that has two circular bases

162

Level H

Chapter 9

Lesson 1

Lighthouse Math

Exercise | 9-1 Name Find the volume of each cylinder. Use 3.14 for pi. Round each answer to the nearest hundredth. 1.

4 cm

20 in

15 ft

11 cm

10 cm

13 m 7 in 6 ft

14 m

2.2 in

18 m 2m

9 cm

17 m

5 in 7 cm

9. 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?

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?

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?

CH A L L E N G E 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 h ft

Lighthouse Math

Level H

Chapter 9

Exercise 1

163

Solve each word problem. Use 3.14 for pi. Round each answer to the nearest hundredth.

9-2 | Volume of Cones

PREREQUISITE SKILLS

Simplify each expression.

DAI LY REV I E W

1. SPIRAL REVIEW

82 + 7

(52)(3)

72 – 16

102 ÷ 4

83 – 92

Find the volume of each cylinder. Round your answers to the nearest tenth. 1.

3 in

5 in

10 ft

8 ft

12 m

L E A RN A ND C O NNECT V = 31 πr2h

A cone is a 3D shape with one circular base that extends to a single point.

V = 31 π(2)2(10)

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.

V = 31 π(4)(10)

V = 31 (3.14)(40)

The height is perpendicular to the base, not one of the slanted sides.

V ≈ 41.87 m3

A P P LY Circle the formula that could be used to find the volume of each cone. 1.

2. 6m

8 ft

22 ft

V = 31 π(6)2(2)

V = 31 π(2.5)2(4)

V = 31 π(22)2(4)

V = 31 π(2)2(6)

V = 31 π(5)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 = 31 π(

10 ft 4 ft

V = 31 π( V = 31

V≈

V = 31 πr2h

5. )2( )(

) )

V = 31 π(

8 in 6 in

π ft3

V = 31 π( V = 31

V≈

V = 31 πr2h

6. )2( )( π in3

) )

V = 31 π(

)2(

)

)(

)

V≈

V = 31 π( V = 31

Vocabulary Cone - a 3D shape with one circular base that extends to a single point

164

Level H

Chapter 9

Lesson 2

Lighthouse Math

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

5 cm

6 ft

10 yd 18 yd

12 cm

3 in

4 ft

8. 5 in

2.1 in

14 m

9. 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?

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?

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?

CH A L L E N G E 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

V = 301.44 ft3

x ft

Lighthouse Math

Level H

Chapter 9

Exercise 2

165

Solve each word problem. Use 3.14 for pi. Round each answer to the nearest hundredth.

9-3 | Volume of Spheres

PREREQUISITE SKILLS

Find the circumference of each circle. Use 3.14 for pi. Round your answers to the nearest tenth.

DAI LY REV I E W

SPIRAL REVIEW

10 m

15 ft

6 in

3 yd

Find the volume of each cone. Round your answers to the nearest tenth. 1.

3. 7m

9 in 6 in

4. 12 ft

11 in

5 ft

8 in

L E A RN A ND C O NNECT V = 43 πr3

A sphere is a round 3D shape where every point is the same distance away from the center.

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 Circle the formula that could be used to find the volume of each cone. 1.

3. 18 m 15 m

V = 43 π(11)3

V = 43 π(18)3

V = 43 π(7.5)2

V = 43 π(22)3

V = 43 π(9)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 π(

V≈

V = 43 π(

V = 43 πr3

5. 10 in

) 3

V = 43 π(

V≈

V = 43 π(

V = 43 πr3

6. 12 ft

11 m

) 3

V = 43 π(

V = 43 π( V≈

)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

Exercise | 9-3 Name Find the volume of each sphere. Use 3.14 for pi. Round each answer to the nearest hundredth. 1.

1 in 7m

6 ft

4 yd

8. 10 in

5 cm

2.2 cm

18 m

9. 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?

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?

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?

CH A L L E N G E 13. Find the radius of the sphere if the volume is 2,143.57 ft3.

Lighthouse Math

Level H

14. Find the diameter of the sphere if the volume is 904.32 yd3.

Chapter 9

Exercise 3

167

Solve each word problem. Use 3.14 for pi. Round each answer to the nearest hundredth.

9-4 | Solving Real-World Problems with Volume

PREREQUISITE SKILLS

Solve each word problem.

DAI LY REV I E W

1. A shipping box is 12 inches long, 8 inches wide, and 6 inches tall. What is the volume of the box? SPIRAL REVIEW

2. A brick mold has dimensions of 10 inches by 4 inches by 3 inches. What is the volume of one brick?

Find the volume of each shape. 1.

15 m

11 in

4. 2m 10 ft

3 in

L E A RN A ND C O NNECT

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. Real-World Problem

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

V = 31 π(3)2(7)

1 V = πr2h 3

V ≈ 65.94 in3

A ball has a diameter of 10 inches. How much air is needed to completely fill the ball?

Volume of a sphere

V = πr2h

Solution

V = π(4)2(10) V = π(16)(10) V ≈ 502.4 ft3

V = 31 π(9)(7) V = 43 π(5)3

V = 43 π(125)

4 V = πr3 3

V ≈ 523.3 in3

A P P LY Match each problem with the equation used to solve it.

168

1. 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

2. 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

3. 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

Exercise | 9-4 Name

1. 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?

2. 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?

3. 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?

4. 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?

5. A ball is shaped like a sphere with a diameter of 9 inches. How much air is needed to completely fill the ball?

6. 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?

7. 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?

8. A small ball has a diameter of 2.7 inches. How much air is needed to completely fill the ball?

9. 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?

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?

CH A L L E N G E 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.

Lighthouse Math

Level H

12. A ball has a volume of 113.04 cubic inches. What is the diameter of the ball?

Chapter 9

Exercise 4

169

Solve each word problem. Round your answers to the nearest hundredth.

9-5 | Review

L E A RN A ND C O NNECT

Cylinder

Cone

Sphere

h h

V = πr2h

1 2 πr h 3

4 3 πr 3

A P P LY

Match each problem with its solution.

170

1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

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

17 cm 22 cm

20 ft

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

15 in

3 ft 2 in

5 in

Find the volume of each sphere. Use 3.14 for pi. Round each answer to the nearest hundredth. 9.

10.

11.

12.

14 ft 10 m

1 in

13.

14.

15. 6m

6m 3m

16. The

can hold the least volume. It can hold 31 the volume of the

17. The

can hold the second least volume. It can hold more than the

the 18. The

Lighthouse Math

m3 . but less than

. can hold the most volume. It can hold three times the volume of the

Level H

Chapter 9

Exercise 5

171

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.

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

Chapter 10 | Skill Checklist

Skill 1: Plotting Points from a Table x

Plot the points from the table on the graph. x

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 y = mx + b m = slope b = y-intercept

Write the equation. 5

-5 -10

2. 5

-5

3. 5

-5

5 -5

-10

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

out 3 correct

174

Level H

Chapter 10

I can identify positive and negative slopes.

Skill Checklist

Lighthouse Math

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

12 possible outcomes

out 2 correct

Create a table for each sample space. 1. 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.

C Sc Wc

2. A cafe gives you a choice of wheat bread or rye bread and four different toppings: cheese, hummus, avocado, and egg. I can create a table for a sample space.

Skill 5: Percentages Find the percent. Round to the nearest hundredth.

8 out of 54 students 8 54 0.1481 14.81% out 4 correct

1. 14 out of 20 apples are red.

2. 25 out of 80 students wear glasses.

3. 100 out of 120 beans are spotted.

4. 12 out of 94 books are missing.

I can find percents to the nearest hundredth.

Unlikely

As likely as not

Likely

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.

1 4

1 2

3 4

0.25

0.5

0.75

25%

50%

75%

2. He makes 80% of the shots he takes. He is unlikely likely to make the shot. 3. 35% of people are wearing white. It is unlikely that a person wearing white will be picked. 4. 5% of tickets are winners. It is unlikely that the next ticket will be a winner.

out 4 correct

Lighthouse Math

likely

I can find the likelihood of an event happening.

Level H

Chapter 10

Skill Checklist

175

Skill 6: Probability

10-1 | Understanding Scatter Plots 5

DAI LY REV I E W

PREREQUISITE SKILLS

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

9 ft

4 in

L E A RN A ND C O NNECT 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.

Negative Association As x increases, y decreases.

Week

Test Score Week

As the weeks increase, the test scores increase.

Test Score

No Association No obvious pattern

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

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)

The scatter plot has a

Test Score

association.

As the time spent

40 20 2

studying

, the

test score

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

$24,000

Car’s Value ($)

association.

$18,000

As the time in years

$12,000

, the car’s

$6,000 '25 '26 '27 '28 '29

value

Year

Heights of 8th graders (inches)

# of Friends

The scatter plot has

association.

6 4

As the height

the number of friends 62

has

, 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

3.2

association.

2.4

As the number of

1.6

absences

0.8 4

the GPA

# of absences

GPA

# of absences

CH A L L E N G E 5. Describe the association between the temperature outside and the number of ice creams sold.

Lighthouse Math

Level H

6. Describe two variables that have no association.

Chapter 10

Exercise 1

177

10-2 | Modeling Linear Associations

DAI LY REV I E W

PREREQUISITE SKILLS

Write a linear equation in slope-intercept form given each piece of information.

SPIRAL REVIEW

1. slope = 21 y-intercept = -3

2. slope = -5 y-intercept = 0

3. slope = 6 y-intercept = 4

4. slope = 43 y-intercept = 9

State whether each scatter plot displays a positive, negative, or no association. 2.

L E A RN A ND C O NNECT 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. 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)

20 16

y = -2x + 20

(3, 14) line of best fit

(4, 12)

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

y-intercept = 20

A P P LY Circle the scatter plot with the correct line of best fit. 1.

102

100

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

Exercise | 10-2 Name Draw the line of best fit.

Dollars (hundred thousands)

102 100

Test Score

2. The number of salespeople at a car dealership and the total money earned

98 96 94 10

3. The number of workers on construction sites and the number of houses built

Number of Houses

1. The number of hours students played games instead of studying and test scores

8 6 4 2 2

6 4 2 2

Number of Salespeople

Hours Spent Playing

Number of Workers

Write an equation for each line of best fit. Then, answer the questions. 8th-grade Test Scores

Equation:

100

A. W hat does the slope mean?

Test Score

80 60

B. W hat does the y-intercept mean?

40 20 2

Hours of Study

Number of Flowers After Days with No Water

Equation:

A. W hat does the slope mean?

B. W hat does the y-intercept mean?

30 20 10 2

C. P redict the number of healthy flowers after 20 days without watering.

# of Days

CH A L L E N G E 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.

Lighthouse Math

Level H

Chapter 10

Exercise 2

179

# of Healthy Flowers

C. P redict the score a student would get with 10 hours of studying.

10-3 | Constructing Two-Way Tables

PREREQUISITE SKILLS

The dot plot below shows the number of books each student in Ms. Harper’s class read. Answer the related questions.

DAI LY REV I E W

1. 2. 3. SPIRAL REVIEW

How many students read 1 book? How many books did no student read? How many students read 4 books?

Number of Books Read

Make predictions given the line of best fit equation and x value. 1. Line of Best Fit: y = 3x + 20 How much money (y) is earned after working 10 hours (x)?

2. Line of Best Fit: y = -4x + 120 How many gallons of gas are left in the tank (y) after driving 15 miles (x)?

L E A RN A ND C O NNECT 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.

All possible answers to one question The number of students who are righties and prefer a pencil

Righty

Lefty

Total

Pen

Pencil

Other

Total

All possible answers to the other question

Total number of students surveyed

A P P LY

Answer the questions using the two-way table. 1.

he principal surveyed students about their T favorite subjects and whether they prefer group or individual work. Prefers Group Work

Prefers Individual Work

Total

Math

Science

Total

teacher surveyed students to find out whether A they prefer reading fiction or nonfiction and if they like to read at home or at school. Fiction

Nonfiction

Total

At Home

At School

Total

A. How many students prefer group work? B. How many students like math and prefer individual work?

A. How many students were surveyed? B. How many students prefer nonfiction and read at school?

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

Exercise | 10-3 Name Construct the two-way tables given the information. Then, answer each question. 1. A teacher asked students whether they read on the weekend and whether they’ve ever gone to detention. Student

Read

yes

Detention

yes

A. How many students said they had detention?

Detention No Detention Total Read

B. How many students don’t read and haven't had detention?

Doesn’t Read Total

C. How many students read and have had detention?

2. 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

A. Have more customers seen a doctor or not seen a doctor?

Vegetables No Vegetables

B. Of the customers who haven’t seen a doctor, do more eat vegetables or don’t eat vegetables?

Total

3. A survey asked people their age and whether they were for or against playing Sudoku puzzles.

For Ages 21-40 Ages 41-60

Over 60

Total

Against

No Opinion

Total

15 20

200

CH A L L E N G E 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? B. What percent of students surveyed do not feel alert, but did eat breakfast?

Lighthouse Math

Level H

Chapter 10

Alert

Not Alert

Total

Breakfast

No Breakfast

Total

Exercise 3

181

Use the totals to fill in the missing information.

10-4 | Interpreting Two-Way Tables

DAI LY REV I E W

PREREQUISITE SKILLS

Answer each percent problem.

SPIRAL REVIEW

18 is what percent of 40?

6 is 12% of what number?

What number is 40% of 90?

22 is what percent of 88?

Use the two-way table to answer each question.

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

L E A RN A ND C O NNECT 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. 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.

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?

30 100

= 0.

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

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. 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

A. Of the students who use a calculator, what percentage likes math? B. Of the students who don’t use a calculator, what percentage likes math?

There is an no association between using a calculator and liking math. A student who uses a calculator is equally more less likely to like math compared to a student who doesn’t use a calculator. 2. A dentist surveyed patients to determine whether they floss daily and whether they’ve had a cavity in the last year. Floss

Don't Floss

Total

Cavity

No Cavity

Total

A. Of the patients who have no cavities, what percentage floss? B. Of the patients who have cavities, what percentage floss?

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. 3. An ice cream store recorded the types of ice creams ordered throughout the day. 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? B. Of all the vanilla ice creams, what percentage were served in a cone? C. Of all the strawberry ice creams, what percentage were served in a cone?

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 A L L E N G E 4. Are students who like pineapple on their pizza more likely to eat pizza weekly? Explain.

Lighthouse Math

Eat Pizza Weekly

Don’t Eat Pizza Weekly

Total

Like Pineapple

Don’t Like Pineapple

Total

100

Level H

Chapter 10

Exercise 4

183

Cone

10-5 | Review

L E A RN A ND C O NNECT 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.

Negative Association As x increases, y decreases.

line of best fit

Income

Number of Germs

Test Score

No Association No obvious pattern

Hours Studying

Hours Cleaning

As the hours increase, the test scores increase.

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.

Number of Pets

As the pets increase, the income is random.

On time

Not On Time

Total

Eats Breakfast

Doesn’t Eat Breakfast

Total

A P P LY

Number of attempts

Number correct

The scatter plot

shows a

association. As

the number of

attempts 10

Number of attempts

184

Heating Cost ($)

Number correct

Construct each scatter plot. Fill in the blanks with positive, negative, increases, or decreases to describe the association.

the number correct

Level H

Temperature (ºF)

Heating Cost ($)

100

The scatter plot

shows a

association. As

the temperature

, the 20

Chapter 10

80 100

Temperature (ºF)

Lesson 5

heating cost .

Lighthouse Math

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.

2. 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. 10

Number of Mistakes on Test

Money Spent ($)

1,000 800 600 400 200 4

8 6 4 2

Number of Days

Hours of Sleep

A. Write an equation for the line of best fit.

B. Describe the association.

C. Predict the money spent if someone vacationed for 15 days.

C. Predict the number of mistakes made after 8 hours of sleep.

Use the two-way table to answer the questions. 3. A school nurse recorded whether students washed their hands before lunch and whether they had been sick in the past month. Didn’t Wash Hands

Total

Sick

Not Sick

Total

140

A. Of the students who washed their hands what percentage was sick? Round your answer to the nearest whole percent. B. Of the students who didn't wash their hands, what percentage was sick? Round your answer to the nearest whole percent.

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 A L L E N G E 4. Is it more or less likely that a student who sleeps 7 or more hours feels more rested? Explain.

Lighthouse Math

Feels Well Rested

Doesn’t Feel Well Rested

Total

Sleeps ≥ 7 Hours

Sleeps <7 Hours

Total

Level H

Chapter 10

Exercise 5

185

Washed Hands

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

11-1 | Real Numbers Review

L E A RN A ND C O NNECT Square Root

Cube Root

a2 = 100 a × a = 100 100 = a

b3 = 64 b × b × b = 64 3 64 = b

a = 10

Types of Numbers

Rational Numbers Can be written as a fraction

- 51 , 1.3, 9 21 , .6

b=4

Integers

Fractions as Repeating Decimals

Do not have fractional parts

Repeating Decimals as Fractions

-4, 0, 5, 31, -17

0.1

2 3

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

= 0.6

Whole Numbers

x = 0.1

Are only positive or 0

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

0, 1, 2, 3, 4

1 9 1 0.1 = 9

Irrational Numbers

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. 8

3. 2.4

4. 2π

3 5. 4

6. -7.2525

7. -10

8. 12

9. 49

10. -9

188

Rational Numbers

Integers

Irrational Numbers

Whole Numbers

Level H

Chapter 11

Lesson 1

Lighthouse Math

Exercise | 11-1 Name Simplify. 1.

100

- 49

- 36

1 8

13.

3 5

1 9

18.

40 3

- 16

125

10.

14.

7 10

15.

1 12

19.

5 11

20.

3 4

-27

Change the fraction to a decimal. 11.

1 3

12.

16.

11 3

17. 2

21. 1.2

22. 0.6

23. 5.3

24. 6.5

25. 8.6

26. 0.4

27. 2.4

28. 0.7

Change the decimal to a fraction.

Put the numbers on the number line. 29.

2.35

2.5

9 2

3.5

3.94

4.5

CH A L L E N G E Solve. 30.

625 =

Lighthouse Math

31.

32.

16 =

Level H

Chapter 11

Exercise 1

81 =

189

11-2 | Exponents and Scientific Notation

L E A RN A ND C O NNECT

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

x7 x2

x15

Zero Exponent Rule

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

1 x4

x-4

Multiplying and Dividing in Scientific Notation

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 × 105

x7 − 2 x5

(4 × 6)(10² × 10³)

(3 × 10²) + (2 × 10³)

2.4 × 106

(0.3 × 10³) + (2 × 10³)

(0.3 + 2) × 10³

2.3 × 103

A P P LY Match the expressions. 1. x4 • x-2 • x0

2. x5 •

x-3 x-2

A. x9

x2 3. -6 • x3 x

B. x5

1 4. -5 x

C. x4

D. x2

5. x3 •

x7 0 •x x

E. x11

Simplify the expression and rewrite with positive exponents only. 6. x3 • x-5 70 10. -4 7

190

5x-2y3 7. 4 -1 10x y x3y-1 11. -2 0 x y

Level H

Chapter 11

8. 58 • 57 12.

Lesson 2

y-2 y7

3p-6 9. 5 -3 3p 40z-3 13. -1 2 z •z

Lighthouse Math

Exercise | 11-2 Name Fill-in the missing exponents. 1. m5 • m4 = m

2. t8 • t = t

(q3 • s5) 1 = 1 5. (q • s ) q

(v-7 • u ) 1 6. 3 = 8 v uv (z11 • z-6) 10. =z z4

x5 • 3 • 3-7 • x

x13 34

47 • 4 • 4-7 = 49

4. y10 • y7 = y

97 1 = 99 9 (w10 • w ) 11. = w7 w9

p7 =1 p r7 = 62r2 12. (67 • r • 6 )

Convert to scientific notation or standard form. 13. 5,600

14. 89,000,000

15. 0.00000082

16. 6.2 × 103

17. 7.45 × 10 -4

18. 5.0 × 10 -6

19. 0.00047

20. 0.0061

21. 190

22. 9.01 × 10 -2

23. 42,500

24. 2.09 × 102

Solve. 25. (2.3 × 104) + (5.7 × 104) =

26. (3 × 102) • (4 × 105) =

27. (6.2 × 105) − (1.1 × 105) =

28. (5.6 × 106) ÷ (8 × 102) =

29. (8 × 103) + (2.5 × 103) =

30. (3.2 × 104) ÷ (4 × 102) =

31. (4.9 × 10 -2) + (3.8 × 10-2) =

32. (6.5 × 103) • (2 × 106) =

33. (7.2 × 106) − (3.2 × 105) =

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?

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?

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?

CH A L L E N G E 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.

Lighthouse Math

Level H

39. Why is it important to write numbers with the same exponent when adding or subtracting in scientific notation?

Chapter 11

Exercise 2

191

Solve.

11-3 | Linear Equations

L E A RN A ND C O NNECT -4(3x2 + 4) = 8x2 – 33 – 3x2 Apply the distributive property.

Combine like terms.

Get the variable to one side of the equation.

-4(3x2 + 4) = 8x2 – 33 – 3x2

-12x2 – 16 = 8x2 – 33 – 3x2

-12x2 – 16 = 5x2 – 33

-12x² – 16 = 5x² – 33 – 5x² – 5x² -17x² – 16 = - 33

Follow SADMEP to solve.

Addition/Subtraction

Multiplication/Division

Exponents/Roots

-17x2 – 16 = -33 + 16 + 16 -17x2 = - 17

-17x2 = - 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. 1.

21x + 59 = 7(3x + 9) – 10

one solution

-60 + 16z = 4(4z – 15)

no solution

4x + 6 = 2x + 14

infinitely many solutions

Match each equation with its correct solution. 4. -4x = 6

x=1

5. 4x2 + 2 = 18

x = -9

x 3

x = 2 or x = -2

7. 3(6x – 1) + 7 = 22

x = 9 or x = -9

8. 2.3x + 1.9 = 5.6x – 11.3

x = 11

9. 3x2 – 21 = 2x2 + 60

x = -1

1 1 10. (200x + 28) + 21 = (150x – 60) + 5 3 2

x=4

11. 3(2x + 5) + 80 = 4(3x + 7) + 1

x = 21

6. -5 = -12 +

192

Level H

Chapter 11

Lesson 3

Lighthouse Math

Exercise | 11-3 Name Solve each equation. 1. 2(y + 5) + 3y = 40

2. -3(2a – 5) + 4a – 6 = 25

(m – 4) + 3(m + 1) = 22 3. 2

4. 25(t – 2) + 3(2t + 1) – 4 = 73

5. 3(p2 – 4) + 2(p2 + 1) = 35

(2r – 3) + 2(r + 5) = 17 6. 3

7. -3x + 8 = 2x – 12

8. 5q + 9 = -q + 3

z – 7 = (3z) – 1 9. 2 4

Write if each equation has one solution, no solution, or infinitely many solutions. 10. 3(x + 4) – 2x = 10

11. 5(2y – 3) = 10y – 15

(4z – 8) = 2z – 5 12. 2

13. 7(a – 2) + 3 = 7a – 11

14. 6m – 9 = 3(2m + 4)

15. 2(3t – 5) + 4 = 6t – 6

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?

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?

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?

CH A L L E N G E 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.

Lighthouse Math

Level H

Chapter 11

Exercise 3

193

Write and solve an equation for each word problem. Explain the solution in context of the problem.

11-4 | Linear Relationships and Functions Review

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.

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)

Slope: the rate the water is leaking

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.

y2 - y1 x2 - x1

Slope: rise over run

Slope:

A P P LY Find the slope. 1.

-2

-1

194

2. (3, 17) (8, 7)

3. y=

1 x+5 4

-10

Level H

Chapter 11

Lesson 4

Lighthouse Math

Exercise | 11-4 Name Determine if the relationship is a function. 1.

-1

-2

Function

2. (0, 1) (1, 1) (2, 1) (3, 1)

Not a Function

Function

Not a Function

Function

Not a Function

Determine if the function is linear. 10

4. y = -2x + 9

6. -10

-2

-8

10 -10

Linear

Non-Linear

Linear

Non-Linear

Linear

Non-Linear

Write the equation of the line. 10

7. -10

8. 10

-1

-7

-3

-10

9. A pot of water is set to boil. It begins at a temperature of 68°. The temperature increases by 1 21 °F every minute.

Find the slope (m), y-intercept (b), and equation of each function. Then answer the questions. 11. Water Usage in Cedarwood

20 10

40 Liters

12.

B. Which city started off with more water?

A. How much water was in Cedarwood’s tank to start?

10 Hours

Equation:

Hours

C. Which city used water at a faster rate?

Equation:

CH A L L E N G E 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

-10

Lighthouse Math

Level H

Chapter 11

Exercise 4

195

10. Water Usage in Elmwood

11-5 | Proportional Relationships

a×d=c×b

Example:

3 12 = x 18

3 × 18 = 12 × x

4.5 = x

54 = 12x

How to Find the Unit Rate in a Proportional Relationship From an equation

From a table

Use k in the equation y = kx.

y = 3x → 3 is the unit rate.

Divide y by x in any column.

Miles (y)

120

180

Hour (x)

60 ÷ 2 = 30 miles per hour

From a graph

Total Cost ($)

Cost of a Venue Rental

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

A P P LY Match the scenario to its solution.

196

1. Sarah ran 3 miles in 24 minutes. At the same pace, how long would it take her to run 5 miles?

2. A train travels 90 miles in 60 minutes. How long will it take to go 135 miles at that rate?

3. Lisa types 360 words in 12 minutes. How long will it take her to type 600 words?

4. A machine prints 40 pages in 15 minutes. How long does it take to print 120 pages?

5. It takes 18 minutes to walk 0.6 miles. How long will it take to walk 1.2 miles at the same pace?

6. A hose fills 10 gallons in 5 minutes. How long will it take to fill 22 gallons at the same rate?

Level H

Chapter 11

Lesson 5

20 minutes

90 minutes

45 minutes

40 minutes

11 minutes

36 minutes

Lighthouse Math

Exercise | 11-5 Name Determine the better buy. 1. Which is the better buy for orange juice?

2. Which pack of pencils offers the best deal?

3. 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

Graph the proportional relationship and find the slope. 4. A car can travel 70 miles in 2 hours. Car Travel Distance

100

Number of Toys

Miles

140

5. A factory produces 120 toys in 4 hours.

Slope:

60 20 0

Toy Production

140 100

Slope:

60 20

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Hours

Slope:

7. A printer can print 90 pages in 6 minutes. Number of Pages

Number of Muffins

Muffins Made

4 0

Hours

6. A recipe uses 3 cups of flour to make 12 muffins. 16

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Pages Printed

120 90

Slope:

60 30 1

Cups of Flour

Minutes

Hours (x)

Income (y)

$62.50

$100

$125

$150

9. 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?

Problems Solved

8. The amount of money Eric makes per hour is shown in the table below. William’s income is given by the equation y = 14x.

Who earns more per hour?

Minutes

CH A L L E N G E 10. Why is every proportional relationship also a linear equation, but not every linear equation represents a proportional relationship?

Lighthouse Math

Level H

Chapter 11

Exercise 5

197

Solve.

11-6 | Systems of Equations

L E A RN A ND C O NNECT 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

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

-5

2. 5

-5

3. 5

-5

one solution

no solution

infinitely many solutions

Determine the solution to each system of equations. 4. x + y = 18 3x + 2y = 46

198

5. x + 2y = 19 3x − y = 8

6. 5x + 4y = 38 x − 2y = 2

7. 2x + 3y = 21 4x − y = 7

(10,8)

(4,8)

(4,1)

(2,6)

(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

Lighthouse Math

Exercise | 11-6 Name Solve each system of equations by graphing. 1.

=5–x y y = 2x – 1 5

-5

x + 3y = 6 2 x + 2y = 4 5

-5

Solution:

y = - 31 x + 3

-5

y = - 31 x + 1

y = 4 – x y = 2x + 1

-5

Solution:

-5

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

x = 2y 3x – y = -5

11.

4x + 5y = 14 6x – 7y = -8

12.

3x + 4y = 17 2x + 5y = 16

Solve each system of equations using elimination. 9.

x+y=3 2x + y = 7

10.

x + 2y = -5 3x + 2y = -7

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? © Lighthouse Curriculum. Copying strictly prohibited.

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?

CH A L L E N G E 15. Match each system to the best way to solve and explain your reasoning. y = 2x + 1 y = -x− 1

x+y=6 x=6−y

Reasoning:

Reasoning: A.

Lighthouse Math

3x + 4y = 18 6x + 4y = 22

graphing

Reasoning: elimination

Level H

Chapter 11

substitution

Exercise 6

199

11-7 | Angles, Triangles, and the Pythagorean Theorem Review

L E A RN A ND C O NNECT 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

g 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

(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 Draw a line to match the name of the relationship between angle a and angle b. 1. a

200

3. a

5. a

b a

adjacent

vertical

corresponding

alternate exterior

complementary

angles

Level H

Chapter 11

Lesson 7

Lighthouse Math

Exercise | 11-7 Name Write an equation. Then, solve for x. 1.

1 ( 3 x + 66)°

(x - 10)°

4. (2x)° (x + 24)°

x°

(x - 2)°

(4x)°

(5x)°

x= 7.

(0.3x + 1)°

2x°

x= 8.

2x°

x°

(0.4x - 3)°

82°

88°

(x - 13)°

145°

(1.5x)°

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

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?

CH A L L E N G E Find the value of the x and y. 13. y° 20°

Lighthouse Math

x°

14.

x= 80°

Level H

Chapter 11

x°

140°

50°

y°

Exercise 7

201

11-8 | Transformations and Congruence

L E A RN A ND C O NNECT

Translations

-5

A’

-5

B’

C’

Reflections

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)

A(-4, 4) → A’(4, 4) B(-4, 1) → B’(4, 1) C(-1, 1) → C’(1, 1)

-5

Rotations

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(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. 1.

2. A’

B’

A B -10

-10 C

C’

-10

A’

-10

reflection

Level H

Chapter 11

Lesson 8

10 B’

D’

C’

-10

A. translation

202

A’

C’

D C D’

rotation

-10

B' 10

B C

-10

dilation

Lighthouse Math

Exercise | 11-8 Name Draw the transformed figure and label the coordinates of the new vertices. 1. Dilate by a scale factor of 2 ,

)

B’: (

)

C’: (

A -10

)

10 C

-10

3. Translate 5 units right and 3 units down ,

)

K’: (

)

L’: (

)

M’: (

-10

10 M

)

Q’: (

)

R’: (

)

X’: (

)

Y’: (

)

Z’: (

)

10 W

-10

F’: (

)

G’: (

)

H’: (

)

-10

A’: (

)

B’: (

)

C’: (

)

D’: (

)

10 A

42°

70° 15

70°

x°

83°

8 83°

83°

97°

7 5

83°

97°

97° z 10

CH A L L E N G E 9. 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

A’

B’ -10

Lighthouse Math

-10

x°

-10

8. x=

42° z

1 2

6. Rotate 180º and dilate by a factor of

Fill in the missing information for the congruent figures. 68°

-10

7. x=

-10

5. Translate 2 units left and reflect across the x-axis P’: (

4. Rotate 90º clockwise

J’: (

W’: (

Level H

Chapter 11

Exercise 8

D 10 D’

C C’

203

A’: (

2. Reflect across the y-axis

11-9 | Volume

L E A RN A ND C O NNECT

Cylinder

Cone

Sphere

h h

V = πr2h

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.

1. A candle is made in a cylindrical mold with a radius of 4 cm and a height of 10 cm.

204

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

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

4. 11.25 cm

18 m

10 ft

8 cm

3.5 m

6 ft

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

6 in

Find the volume of each sphere. Use 3.14 for pi. Round each answer to the nearest hundredth. 9.

10.

11.

12. 13 ft

6m 6 in

5 in

13.

14.

15. 10 m

10 m 5m

1 the volume of the 3

16. The

can hold the least volume. It can hold

17. The

can hold the second least volume. It can hold more than the

the 18. The

Lighthouse Math

m3 . but less than

. can hold the most volume. It can hold three times the volume of the

Level H

Chapter 11

Exercise 9

205

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.

11-10 | Scatter Plots, Association, and Probability

Negative Association As x increases, y decreases.

Week

Test Score

No Association No obvious pattern

Week

As the weeks increase, the test scores increase.

Week

As the weeks increase, the test scores decrease.

Use two-way tables to represent multiple pieces of information. This information can be used to answer questions about percentages.

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

Study Time (min.)

Mistakes

What is the pattern of association?

12 10 8

positive association

negative association

6 4 2 10

New Words Learned

What is the pattern of association?

8 6 4

positive association

negative association

no association

Study Time (min.)

206

Pages Read

New Words Learned

Mistakes

Fill in the scatterplot based on the table and then circle the pattern of association.

Pages Read

Level H

Chapter 11

Lesson 10

Lighthouse Math

Exercise | 11-10 Name Draw a line of best fit. Then, answer the questions about the scatter plot. 2. A store tracked the number of bikes sold after different price discounts.

500

400

Number Sold

Temperature (OF)

1. A batch of cookies was left to cool after baking. The scatterplot shows the temperature of the cookies as they cooled.

300 200 100

6 4 2

100

Cooking Time (minutes)

$10

Price Discount

A. W rite an equation for the line of best fit.

A. Write an equation for the line of best fit.

B. D escribe the association.

B. Describe the association.

C. P redict the temperature after 120 minutes.

C. Predict the number of bikes sold after a $12 price discount.

Fill in the two-way table and answer the questions that follow. Round answers to the nearest percent. Art Club

Band Member

Orchestra Member

Total

3. Of the students in the band, what percentage are in the music club?

Total

32 31

4. Of the students in the orchestra, what percentage are in the music club?

5. 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 more less likely to be in the band than the orchestra.

CH A L L E N G E 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?

Lighthouse Math

Level H

7. 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?

Chapter 11

Exercise 10

207

Music Club

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