Math H Practice by lighthousecurriculum

Level H

Practice Book

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LEVEL H

Lighthouse Math Practice books

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Lighthouse Math Level H Practice Books • ISBN 978-1-955773-88-1

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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.

Tina is sewing a curtain to cover a square window with an area of 16 square feet. What is the length of one edge of the curtain?

Johnny is creating a sign for an ice cream truck. It must fit in a square space with an area of 25 square feet. What will be the length of one side of the sign?

A square mirror has an area of 100 square inches. The mirror has a frame around its edges.

A dumpster with a square bottom needs to be placed along a 1 meter wall near a construction site. The square bottom has an area of 4 square meters. Will the dumpster fit in this space? Explain your answer.

a. What is the length of one edge?

b. Find the total length of the frame.

A fidget toy in the shape of a cube has a volume of 125 cubic inches.

What is the length of one side of the toy?

Bob is painting a tree house in the shape of a cube with a volume of 64 cubic feet.

What is the length of one side of the tree house?

A square swimming pool with a depth equal to its length and width can be filled to hold 729 cubic feet of water. What is the length, width and depth of the pool? Hint: they are all the same.

A rectangular storage container can hold a volume of 432 cubic meters. When the storage container is divided in half, each part is a cube with a volume of exactly half the original volume. What are the dimensions of one part of the divided container?

Lesson instructions 3

Write repeating decimals as fractions.

Choose the best answer.

1. Which one will give a repeating decimal?

Write 0.7 as a fraction:

2. Which one will NOT give a repeating decimal?

Use division to change the following fractions into decimals. Use bar notation to show if there are any repeating digits.

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

Alex says that 0.4 is the fraction 4 10 because the 4 is in the tenths place. Explain his error.

Alex from problem 1 asks you to explain how he can change 0.4 to a fraction.

Explain how to find the equivalent repeating decimal for 4 11 if you know that 2 11 is equal to 0.18.

Explain why 1 12 and 5 12 equal repeating decimals but 3 12 and 6 12 do not. Hint: put all fractions in simplest form.

Put numbers into categories.

Rational numbers - any number that can be written as a fraction

Integers - positive and negative numbers with no fractional parts

Whole numbers - positive integers and zero

Irrational numbers - numbers with an infinite amount of decimal places with no predictable pattern

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

Write the term that most specifically describes the given number: whole number, integer, rational number or irrational number.

Draw a line to all terms that describe

Jack says that the number 4.2 is an irrational number because it has an infinite amount of decimal places. Is he correct? Explain.

David says that the number 7 is a rational number. Explain why he is correct and what other categories this number falls into.

Parkland School has a square sandbox on its playground. It has an area of 20 square feet. Is the length of the sandbox a rational number?

Parkland School also has a cube shaped storage container on its playground. It can hold a volume of 27 cubic feet. Is the height of the container a rational number?

Estimate to create approximations of irrational numbers.

√30 is between √25 and √36

√30 is between 5 and 6.

√30 is about halfway between 5 and 6.

√30 is about 5.5.

Use perfect squares to approximate the value of the irrational numbers to one decimal place.

Plot the numbers on the number line below.

Answer the question, then write if the number is rational or irrational.

Max has a square garden with an area of 25 square meters. What is the length of his garden?

Jessica bakes a pie with a diameter of 10 inches. What is the circumference of the pie? Hint: C = πd

A landscaper is designing a square garden with an area of 80 square feet.

Approximate the side length of the garden (to one decimal place).

An interior designer chooses a square rug with an area of 55 square feet.

Approximate the side length of the rug (to one decimal place).

Lesson instructions

Compare and order rational and irrational numbers.

Place the numbers on the given number line. Write the correct symbol:

Write true or false, then explain.

We can find the exact location of 49 on the number line.

We can find the exact location of 56 on the number line.

Rosie uses a circular cake pan with a height of 4 inches and a diameter of of 8 inches to bake a cake. She can find the exact volume of her cake.

Peter builds a shed. The rectangular door has a height of 2.5 meters and a width of 1 meter. He can find the exact area of the door.

Simplify.

Simplify square roots and cube roots

4 x 4 = 16 √16 = 4

4 x 4 x 4 = 64 3 √64 = 4

Change the fraction to a decimal.

Change the decimal to a fraction.

Convert repeating decimals to fractions

Place the numbers on the number line.

Use perfect squares to approximate square roots

Jack has a storage box in the shape of a cube. Find the length of the box if the total volume is 125 cubic feet.

Paul is building a fence around his square garden. If the area of his garden is 81 square inches, how much fencing will he need to put up around the entire perimeter of his garden?

A tire has a diameter of 20 inches. Approximate the circumference of the tire.

Fanny wants to bake a pie in a circular pan with a diameter of 8 inches. Approximate the circumference of the pie.

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.

Lesson instructions 1

Simplify the expressions.

Product of Powers Property am × an = am+n

52 x 53 x 36 x 39 = 55 x 315

Match the expression to its simplified version.

1. 62 × 68

2. 610 × 66

3. 59 × 54

4. 520 × 516

5. 37 × 34

6. 38 × 320

7. 25 × 27 × 2 3

8. 29 × 24 × 2

Simplify the expression.

9. m5 × m6 = 11. x3 × x8 =

13. n2 × n2 = 15. y9 × y4 =

17. r6 × r =

19. q2 × q2 × q2 =

21. b7 × b3 × b5 =

23. f 9 × f × f 2 =

25. j2 × j5 × j7 =

27. t4 × t3 × t =

29. y4 × x6 × x5 =

31. h10 × f 4 × h11 =

A. 513

B. 616

C. 328

D. 536

E. 215

F. 610

G. 214

H. 311

10. a9 × b7 × a3 = 12. p5 × d9 × d3 × p2 = 14. 63 × a2 × a6 = 16. 72 × d3 × 75 =

18. 47 × r10 × 4 3 × r3 = 20. 83 × 84 × t3 × t2 =

22. q3 × 62 × 615 × 64 = 24. w7 × 55 × 53 × w7 =

26. 92 × z3 × g8 × z3 × 93 =

28. f10 × q7 × q2 × 103 × f 4 =

30. a6 × 39 × a10 × a14 × b3 =

32. 710 × n5 × m6 × n9 × m14 =

Fred simplifies the expression y6 × y2:

“Since 6 × 2 = 12, then y6 × y2 = y12.”

Is Fred correct? Explain why or why not.

Leo simplifies x4 × x4. He says the missing exponent is 16 because 4 × 4 = 16. Is Leo correct? Why or why not?

Danny simplifies the expression 42 × 53:

“Since 4 × 5 = 20 and 2 + 3 = 5, the answer must be 205.”

Is Danny correct? Explain why or why not.

Kayla is given the expression a6 × a? = a15

She says that the missing exponent is 9 because 6 + 9 = 15. Is Kayla correct? How do you know?

expressions.

Match the expression to its simplified version.

A. x3 B. x6

C. x8

311

315

b? b5 = b3

Dan says the missing exponent is 8 because 8 − 5 = 3. Is Dan correct? How do you know?

Kelly simplifies the expression 86 23 : “Since 8 ÷ 2 = 4 and 6 - 3 = 3, the expression simplifies to 4 3." Is Kelly correct? Explain why or why not.

Chad simplifies the expression w3 w2 : "Since 3 - 2 = 1, then w3 w2 = w." Is Chad correct? Explain why or why not.

Bob simplifies the expression z8 z2 : “Since 8 divided by 2 is 4, then z8 z2 = z4.” Is Bob correct? Explain why or why not.

Determine whether each statement is true or false.

1. x0 = 0

3. Any number raised to the power of 0 is 1.

5. (5 + 3)0 = 1

7. 2x0 = 1

9. 5 -2 = -25

2. Negative exponents create a reciprocal. 4. x-1 × x = 1 6. 20 × 3 -1 = 20 3-1

a0b0 = 1

Rewrite each expression with positive exponents.

b -14 =

Simplify each expression and write the answer with positive exponents.

7-8 × 70 × 7-9 =

106 × 10 -2 × x4 =

9-4 × 98

John simplifies the expression 50x-1.

“Since 50 = 0 and x-1 = 1 x , then 0 × 1 x = 0.”

Is John correct? Explain why or why not.

Moses simplifies 1 8-3

“The reciprocal of a negative exponent just makes the result negative, so the answer is 2-3 = -8.”

Is Moses correct? Explain why or why not.

Donna simplifies the expression 3 -1 × 90.

“I can add the exponents -1 + 0 = -1 so the answer is 1 3 .”

Is Donna correct? Explain why or why not.

Jake simplifies x-2 x-5

“Dividing means subtracting exponents so -2 - -5 = 7. The answer is x7.”

Is Jake correct? Explain why or why not.

Lesson instructions 4

Simplify the expressions.

Fraction bars and roots act as grouping symbols.

Choose the first step to solve.

1. 16 + 4 x 3

Take the square root of 16.

Add 16 and 4.

Multiply 4 by 3.

Take the square root of 4.

Simplify the expression.

3. 3 × 2 - 1 + 3 9 - (6 × 2 - 5)

Take the square root of 6.

Add 7 and 5.

Square 5.

Square 7.

5. 8 - 3(5 - 32) 7 - 2 × 6

2. (7 + 5)2 - 6

4. 3(-4) + (-5)(-2) 23 - 2 - (-6)

6. 16 + 3(15 - 3 × 4) (8 - 3)2

A library has 28 boxes with 30 books in each box plus an additional 120 books not in boxes. They want to place the books on the shelves in a room with 8 bookcases. Each book case has 6 shelves. Write an expression and simplify to find out how many books will need to go on each shelf.

Each chicken on a farm lays 6 eggs per week. There are 120 chickens in the North coop and 150 chickens in the south coop. The egg cartons each hold 12 eggs. Write an expression and simplify to show how many cartons are needed per week on the farm.

A square poster is created by taping together four pieces of paper: a rectangular paper which is 5 inches by 4 inches, two square papers with a side length of 4 and another rectangular paper which is 3 inches by 4 inches. Write an expression and simplify to find the length of the square poster.

Two molds shaped as cubes are filled with plaster. One has a side length of 2 inches and another has a side length of 6 inches. How many times greater is the volume of the second mold than the volume of the first? Hint: Divide to find how many times greater.

Lesson instructions 5

Convert to Scientific Notation.

50,000 = 5 × 10,000

Rewrite the number as a multiplication problem.

50,000 = 5 × 10 × 10 × 10 × 10

Break down the power of ten into repeated multiplication.

50,000 = 5 x 104

Express the repeated multiplication as an exponent.

Determine if the number is written in the correct scientific notation. Fix any incorrect values.

Rewrite each number in scientific notation.

1. 9,000 = 2. 0.00012 = 3. 5,000 = 4. 0.53 = 5. 0.007 = 6. 800,000 =

0.0000081 =

2,300 =

0.006 =

54,000 =

0.00000097 =

1,000 =

Rewrite each number in standard form.

× 10 -4 =

0.055 = 14. 360 = 15. 87,000,000 = 16. 0.0023 = 17. 120,000 = 18. 0.0035 = 19. 5 × 102 = 20. 6.5 × 106 = 21. 1.2 × 103 = 22. 4.7 × 10 -2 = 23. 8 × 10 -5 =

5.01 × 10 -2 =

1.5 × 106 =

2.6 × 10 -3 =

7 × 104 =

8.2 × 10 -2 =

1.9 × 10 -5 =

5 × 102 =

6.8 × 103 =

A particle of moon dust has a diameter of about 0.000000082 meters. Write this number in scientific notation.

Earth has about 1.386 × 109 cubic kilometers of water. Write this number in standard form.

A virus has a diameter of about 1.2 × 10 -7 meters. What is the diameter in standard form?

A city's population is estimated at 3,400,000 people. Write this number in scientific notation.

Lesson instructions 6

Put numbers in the same form to compare and order them.

Compare each value using <, >, or =.

1. 5.2 × 104 5.1 × 104

2. 3.7 × 10 -5 3.7 × 10 -6

3. 6.02 × 1023 6.02 × 1022

4. 1.01 × 10 -3 1.1 × 10 -3

5. 8.0 × 106 8.0 × 107

Put the numbers in order: 2,050, 2 x 103, 2.5 x 102, 25,000

2.5 x 102 < 2 x 103 < 2.05 x 103 < 2.5 x 104

250 < 2,000 < 2,050 < 25,000 2.5 x 102 < 2 x 103 < 2,050 < 25,000

6.0 × 108 6.00 × 108

Convert to standard form and then order the numbers from least to greatest.

11. 0.00011, 1.0 × 10 -4, 0.000105, 9.8 × 10 -5

12. 8,900, 8.1 × 10³, 9.0 × 10³, 8.5 × 10³

13. 0.0042, 4.1 × 10 -3, 4.0 × 10 -3, 0.0045

14. 6.0 × 106, 5,500,000, 6,100,000, 5.9 × 106

15. 2.7 × 10 -1, 0.3, 0.25, 2.5 × 10 -2

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

16. 5,500, 5.8 × 103, 5.6 × 103, 5.4 × 103

17. 0.00094, 9.0 × 10 -4, 9.1 × 10 -4, 0.00092

18. 910, 9.0 × 102, 8.7 × 102, 9.2 × 102

19. 0.00012, 1.4 × 10 -4, 1.3 × 10 -4, 0.00011

20. 100,000, 1.1 × 105, 9.9 × 104, 95,000

The depths of three ocean locations are recorded as:

• Site A: -10,912 meters

• Site B: -1.1 × 104 meters

• Site C: -1.09 × 104 meters

Convert all values to scientific notation and order the sites from shallowest to deepest.

Three dust particles are measured under a microscope:

• Particle A: 4.1 × 10 -6 meters

• Particle B: 0.0000039 meters

• Particle C: 4.0 × 10 -6 meters

Convert all measurements to scientific notation and arrange them from smallest to largest.

Three types of lightbulbs last for the following durations:

• Bulb A: 8.5 × 10³ hours

• Bulb B: 8,700 hours

• Bulb C: 8.45 × 10³ hours

Convert each to scientific notation and determine which bulb lasts the longest.

A printing company produces the following number of flyers in a week:

• Monday: 4.2 × 104 flyers

• Tuesday: 42,500 flyers

• Wednesday: 4.15 × 104 flyers

Convert all amounts to scientific notation and determine which day had the highest production.

Lesson instructions 7

Multiply or divide numbers in scientific notation by grouping and following exponent rules.

(4 × 102) • (3 × 105) (4 × 3) • (102× 105) → 12 × 107→ 1.2 × 108 (2.4 × 106) ÷ (4 × 103)

Multiply.

1. (3 × 105) • (2 × 108)

3. (4 × 103) • (5 × 106)

5. (6 × 104) • (3 × 102)

7. (8 × 107) • (2 × 103)

9. (1.2 × 103) • (4 × 105)

11. (5 × 102) • (9 × 104)

13. (7 × 106) • (3 × 102)

15. (6.5 × 105) • (2 × 104) 2. (9 × 103) • (4 × 105)

Divide.

17. (8 × 106) ÷ (4 × 103)

19. (2.7 × 10−4) ÷ (3 × 10−7)

21. (7.5 × 108) ÷ (2.5 × 104)

23. (4.2 × 10−6) ÷ (6 × 10−3)

25. (3.6 × 107) ÷ (4 × 102)

27. (6.4 × 10−2) ÷ (8 × 10−5)

29. (1.6 × 109) ÷ (4 × 103)

31. (9.1 × 10−3) ÷ (7 × 10−6)

(2.2 × 106) • (3 × 102)

(1.5 × 104) • (2 × 105) 8. (3.3 × 103) • (4 × 103) 10. (4.5 × 106) • (5 × 103)

12. (7.2 × 104) • (6 × 102)

14. (5.5 × 105) • (2.5 × 102)

16. (1.6 × 106) • (1.5 × 103)

18. (2.5 × 106) ÷ (5 × 102)

20. (4.5 × 10−5) ÷ (9 × 10−8)

22. (3.3 × 108) ÷ (1.1 × 104)

24. (5.6 × 10−2) ÷ (7 × 10−5)

26. (1.2 × 10−5) ÷ (6 × 10−3)

28. (6.3 × 10−6) ÷ (7 × 10−3)

30. (1.8 × 107) ÷ (6 × 103)

32. (2.7 × 10−4) ÷ (3 × 10−6)

A spacecraft used 4.8 × 106 liters of fuel over 6 × 102 seconds. What was the rate of fuel consumption per second?

A memory card transfers data at a rate of 2.4 × 106 bytes per second. If a total of 7.2 × 108 bytes must be sent, how many seconds will it take?

Equation:

Equation: Solution:

A power plant produces 4 × 105 kilowatthours of electricity each day. How much energy does it produce over 2 × 103 days?

A cargo ship carries 7.5 × 104 containers per trip. If it makes 6 × 102 trips in a year, how many containers are delivered?

Equation: Equation:

Lesson instructions 8 Add or subtract numbers in scientific notation. Add.

1. (3.5 × 105) + (2.1 × 103)

3. (6.2 × 106) + (7.5 × 104)

5. (8.0 × 103) + (6.0 × 102)

7. (4.7 × 107) + (9.3 × 105)

9. (2.5 × 10−2) + (1.5 × 10−4)

11. (5.9 × 104) + (3.2 × 102)

13. (7.1 × 10−3) + (2.4 × 10−5)

(4.2 x 10³) + (2.1 x 10²)

1. Match the exponents. (4.2 x 103) + (0.21 x 103)

2. Add the coefficients. (4.2 + 0.21) x 103 = 4.41 x 103

3. Adjust if needed. 4.41 x 10³

10. (6.6 × 105) + (4.4 × 104)

12. (5.4 × 10−3) + (6.0 × 10−5)

14. (7.0 × 106) + (2.0 × 104)

15. (9.6 × 106) + (1.4 × 103) 2. (3.3 × 105) + (6.7 × 102) 4. (4.2 × 10−4) + (8.5 × 10−6) 6. (1.1 × 107) + (9 × 105) 8. (2.8 × 103) + (4.2 × 101)

16. (3.2 × 102) + (1.8 × 10−2)

Subtract.

17. (5.0 × 106) − (2.5 × 104)

19. (9.1 × 103) − (3.1 × 101)

21. (2.6 × 10−2) − (1.0 × 10−4)

23. (7.2 × 104) − (4.2 × 102)

25. (3.5 × 105) − (5.0 × 103)

27. (6.8 × 107) − (8.0 × 105)

29. (4.4 × 106) − (1.0 × 104)

31. (8.6 × 10−3 − (2.1 × 10−4)

18. (1.0 × 108) − (5.5 × 106)

20. (5.7 × 105) − (2.0 × 104)

22. (3.0 × 10−2) − (4.5 × 10−3)

24. (9.9 × 104) − (1.1 × 103)

26. (6.2 × 103) − (2.2 × 102)

28. (7.7 × 106) − (4.0 × 104)

30. (2.8 × 10−1) − (8.0 × 10−3)

32. (1.6 × 106) − (6.0 × 103)

An astronaut’s oxygen tank contains 6.3 × 104 liters of oxygen. A backup tank contains 7.2 × 103 liters. How much oxygen is available altogether?

Equation:

A laser sends 5.5 × 106 pulses in one operation and 1.45 × 105 pulses in a second. What is the total number of pulses?

Equation: Solution: Solution:

An antenna receives 4 × 104 radio signals per hour. If background noise accounts for 6.5 × 103 signals, how many meaningful signals remain?

A battery starts with 7.2 × 105 joules of energy. After running a system, it drops to 1.8 × 105 joules. How much energy was used?

Equation:

Equation: Solution: Solution:

Chapter 2

Lesson Name Date

Lesson instructions 9 Add or subtract numbers in scientific notation.

Operation Steps to Solve Multiplication Multiply Coefficients → Add the Exponents → Adjust Scientific Notation Division Divide the Coefficients → Subtract the Exponents → Adjust Scientific Notation Addition Match the Exponents → Add the Coefficients → Simplify. Subtraction Match the Exponents → Subtract the Coefficients → Simplify.

Circle the larger quantity.

1. A telescope collects data from three missions: Mission A: 3.5 × 1012 bytes, Mission B: 2.9 × 1011 bytes, Mission C: 6.1 × 1010 bytes. Which mission produced the most data?

Mission A Mission B Mission C

2. Three types of cells have these approximate widths: Cell A: 5.2 × 10 -6 meters, Cell B: 3.1 × 10 -8 meters, Cell C: 9 × 10 -7 meters. Which cell is the widest?

3. Three satellites produce the following power per day: Satellite A: 2 × 107 watts, Satellite B: 6.5 × 106 watts, Satellite C: 9.3 × 108 watts. Which satellite produces the most power?

Satellite A Satellite B Satellite C

4. Astronomers measured the distances to three galaxies: Galaxy A is 2.1 × 107 light-years away, Galaxy B is 5.9 × 108 light-years away, Galaxy C is 4.3 × 106 light-years away. Which galaxy is the farthest from Earth?

Galaxy A Galaxy B Galaxy C

Cell A

Cell B

Cell C

A meteor travels 3.6 × 105 km in 1.2 × 102 seconds. Then its speed doubles due to gravity. What is the new speed?

A solar farm generates 4.5 × 108 joules/ hour. After 3 hours, it loses 3 × 108 joules to inefficiency. How much energy is left?

Each satellite launch costs 2 × 109 dollars. A company launches 3 satellites and adds 5 × 108 dollars in research costs. What is the total budget?

A culture of microbes doubles every hour. It starts with 6.0 × 103 microbes and grows for an hour. Then 2.4 × 103 microbes are sampled and removed. How many microbes remain?

Circle the larger quantity.

Product Rule

am × an = am+n

34 x 32 = 34+2 = 36

Quotient Rule

Simplify the expressions. am an = am-n

Simplify Zero and Negative Exponents

a0 = 1 a-m = 1 am 7⁰ = 1 9-7 = 1 97

Convert from scientific notation to standard form. Convert from standard form to scientific notation.

Convert between Standard and Scientific Notation

56 - 53 = 56-3=53 50,000

5 × 10,000

5 × 10 × 10 × 10 × 10 5 x 104

Simplify Expressions

Operation Steps to Solve

Multiplication Multiply Coefficients → Add the Exponents → Adjust Scientific Notation

Division Divide the Coefficients → Subtract the Exponents → Adjust Scientific Notation

Addition Match the Exponents → Add the Coefficients → Simplify.

Subtraction Match the Exponents → Subtract the Coefficients → Simplify.

(9.6 × 106 ÷ (4 × 103)

A telescope uses 2.4 × 103 watts of power per second for observation and an additional 1.8 × 103 watts per second for communication. If it runs for 2.0 × 102 seconds, how much total energy (in joules) is used?

A satellite has a mass of 5.6 × 104 kg. It loses of 1.9 × 103 kg when a module is ejected. The remaining mass is then split evenly between two fuel tanks. What is the mass in each tank?

An underwater vehicle collects 9.2 × 106 bytes of data on its first mission and 7.8 × 106 bytes on the second. It transmits all the data over 10 identical sessions. How much data is transmitted per session?

A sample starts with 6.0 × 105 bacteria. After a chemical is added, the population decreases by 2.5 × 104. The resulting population then doubles. What is the final count?

Chapter 3

In Chapter 3, we will expand on

Equations

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 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.

Lesson instructions 1

Simplify. Then, solve the equation.

1. Complete the distributive property.

2. Combine like terms.

5(6x + 9) – 20x = 105 30x + 45 – 20x = 105 10x +

3. Add/subtract on both sides to isolate the variable.

4. Multiply/divide on both sides to isolate the variable.

Write a simplified equation, then solve.

-10y + 7 – 2y = 55

6t − 9 + 3t = 63

-18 + 3a – 12 – 5a = -16

7(x – 5) = 21

-8(-4d – 6) = 112

-15(-2f + 3) = -135

-6x – 3(3x + 4) = 18

-4(3y – 2) + 8y = −8

Complete the error analysis problem.

-10 – 3(4z + 6) = 20

9(2j + 11) + 5j = -39

13. Carly solved the equation below. Do you agree with her solution? Why or why not?

-4(2r – 13) + 10r = 16

-8r + 52 + 10r = 16

-18r + 52 = 16

-18r + 52 = 16 -18r = -36 r = 2

Lily and Tom are collecting donations for a charity. Lily collected 4 times as much as Tom, and then they received an additional 8 donations from a surprise contributor. If the total amount of donations they now have is 93, how many donations did Tom originally collect?

Cory, Kyle, and Ben are saving money for a trip. Kyle saved 3 times as much as Ben, Cory saved 5 times as much as Kyle, and then they found an extra $12 in a forgotten envelope. If their total savings is now $107, how much did Ben originally save?

A rectangular swimming pool has a length that is twice its width. If the total perimeter of the pool is 54 feet, what is the width of the pool?

A rectangular playground has a width of 5 ft and length that is x times as long as the width. Find the value of x if the area of the playground is 400 square feet.

Lesson instructions 2

Solve the equation.

1. Undo addition/subtraction if applicable.

2. Undo multiplication/division if applicable.

3. Undo exponents/roots if applicable.

4. Remove parentheses when they are alone on one side of the equation.

Write and solve an equation to answer the question.

Lena is saving money to buy a new bike. She earns $12 per hour babysitting but has already spent $18 on snacks. If she wants to have $90 left after buying snacks, how many hours does she need to work?

A teacher is splitting some markers evenly among 18 students. After giving out the markers, each student has 9 markers. How many markers did the teacher start out with?

A square-shaped mural has an area of x square feet. If one side of the mural is 15 feet, what is the total area?

A ball is tossed into the air, and its height h in feet follows the equation h2 = 64. How high did the ball go?

Lesson instructions 3 Get the variable on one side to solve.

1. Get the variable on one side using inverse operations.

2. Use inverse operations to solve.

7. 0.4p – 2.4 = -7.3p + 5.3

16x2 + 140 = 18x2 + 12

Write and solve an equation to answer the question.

Sarah and Josephine are collecting beads. Sarah already has 46 beads and she collects 6 more each day. Josephine has 18 beads and she collects 8 each day. After how many days will they have the same amount of beads?

Alex is doing a walk-a-thon to raise money for charity. One person donates $20 to his team and pledges to give another $0.50 per mile walked. Another donates $5 and pledges to give $3 for each mile walked. How many miles will Alex need to walk for each donor give the same amount of money?

A square with a side length of 3x has the same perimeter as the rectangle shown below. What is the value of x?

Brielle earns $12 per hour babysitting. She also gets a $9 bonus each time she babysits. Her friend Connie earns $15 per hour babysitting with no bonus. After how many hours of work will they earn the same amount? 4x m

3 m

Lesson instructions 4

Simplify. Then, solve the equation.

1. Complete the distributive property and combine like terms.

2. Get the variable to one side of the equation.

3. Use inverse operations to isolate the variable.

Write a simplified equation, then solve.

1. 1.3(2x + 5) + 5.7 = 5(x + 3) + 2

4(3x + 9) = 4(7x – 11) + 16

12x + 36 = 28x -28 -16x + 36 = -28 x = 4

2. 4(3z − 2) + 7 = 2(5z + 1) + 5

4. 2(4t + 3) + 6 = 3(2t + 5) + 4

3. 5(2v + 1) + 3 = 3(3v + 4) + 2

5. 5(3x − 2) + 8 = 2(6x + 4) + 5

6. 2.5(3n − 4) + 1.2 = 1.5(4n + 2) − 1.3

7. 3.2(2x + 1) − 4 = 2(3x − 5) + 1.6

8. 5(m2 − 7) + 18 = 2m2 − (2m2 − 3)

9. 1 2 (6t + 4) + 3 = 2 3 (5t + 6) + 1

10. 3 4 (4b + 8) + 7= 1 2 (10b + 2) + 4

11. 2.5(4x + 2) − 3 = 3(2x − 1) + 2.0

12. 3(-r2 + 7) + 12 = -2r2 − 3

Write and solve an equation to answer the question.

A crafting magazine charges an initial fee of $15 and $10 per month to maintain the subscription. A different crafting magazine charges $25 per month without an initial fee. How many months would you need to subscribe for the cost to be the same for both magazines?

A cleaning company charges a $18 deposit plus $20 per hour of work. Another company charges $28 plus $15 per hour. How many hours will each cleaner have to work for the cost to be the same?

A manufacturer sells a robot toy to two stores for x dollars. Store A charges 9 dollars more than 2 times the manufacturer's price. Store B charges 3 times the manufacturer's price. A customer buys 4 robots at Store A and 3 robots at store B and spends the same amount of money. How much is the manufacturer's price?

There are x notebooks in a package. Ronnie buys 5 packages and each package costs $10 less than 2 times the number of books in each pack. George buys 3 packages and each package costs $6 more than twice the number of books in each package. If they spent the same amount of money, how many books are in each package?

Lesson instructions 5 State whether each equation has one solution, no solution, or infinitely many solutions.

one solution no solution infinitely many solutions The isolated variable equals one number. The remaining values don’t equal one another. The remaining values equal one another.

Write if each equation has one solution, no solution, or infinitely many solutions.

3(2x + 5) = 6x + 18

2(4y − 3) + 8 = 3(y + 4) + 10

4(2a + 3) = 5a + 12

3(3c + 4) = 9c + 12

4(3x + 5) = 2(6x + 10)

5(2y − 4) = 3(3y + 2)

30(2x + 4) = 5(6x + 8) + 20

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

4(3x + 2) + 1 = 12x + 9

1 2 (8x + 6) = 3x + 4

1.5(2x − 1) = 3x + 2

3 4 (2y+5) = 2y + 9

1 3 (9x + 3) = 2x + 7

1 2 (8x + 4) = 4x + 2

0.6(5x + 7) = 2x + 4

2(15x − 10) = 3x + 2 19. 0.4(10x + 5) = 2x + 6

1 3 (24x − 15) = 2 3 (12x + 12)

8.2(3x + 9) = 3(8.2x + 24.6)

Sarah is buying tickets to the zoo. One ticket vendor sells tickets for $12 each with a $5 processing fee, while another vendor sells tickets for $10 each with a $9 processing fee. Is there an amount of tickets Sarah can buy at which the cost is the same from both vendors? If so, how many?

Lily is buying snacks for a party. One store charges $6 per snack with a $5 shipping fee, and another store charges $6 per snack with a $7 shipping fee. Is there an amount of snacks Lily can buy at which the cost is the same from both stores? If so, how many?

John is looking for a gym membership. Gym A charges $20 per month with a $10 registration fee, while Gym B charges $18 per month with a $15 registration fee. Is there an amount of months John can pay for at each gym at which the cost is the same from both gyms? If so, how many?

Debbie buys office supplies from two different stores. The first store charges her $3 per pen and $2 for an ink refill. The second store charges her $3 per pen and $2 for a box of erasers. Is there an amount of pens Debbie can buy at which the cost is the same from both store? If so, how many?

Lesson instructions 6

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

A photographer charges a $100 session fee, plus $20 per photo. Another photographer charges a $120 session fee, plus $15 per photo. For how many photos will the total cost for both photographers be the same?

Write an equation:

100 + 20p = 120 + 15p

Solve: p = 4

Answer: The cost is the same when buying 4 photos.

Write and solve an equation for each word problem. Then, answer the question.

1. Lily is buying notebooks. One store charges $5 per notebook plus a $2 shipping fee, while another store charges $4 per notebook plus a $4 shipping fee. How many notebooks must she buy for both stores to charge the same amount?

2. Anna is hosting a party. She rents a hall for $150 and pays $20 per hour for the rental. Her friend Lisa rents a hall for $100 and pays $30 per hour. How many hours must they rent the halls for the total cost to be the same?

3. The cost of renting a bike for a day is $5 plus $3 per hour. Another company charges $7 per hour with no base fee. For how many hours of riding will the cost be the same?

4. A bookstore sells books for $10 each with a $5 shipping fee. Another bookstore sells the same books for $8 each with a $27 shipping fee. How many books must be bought from each store to spend the same amount of money?

Ethan and Eli are planning to buy the same new art set. Ethan has $100 saved up, and he earns $10 for every lawn he mows. Eli has $120 saved up, and he earns $5 for every lawn he mows. How many lawns will they each have to mow to have the same amount of money?

Two businesses are selling the same product at different prices. Business A sells a chair for $30 each with a $10 shipping fee. Business B sells the same chair for $27 each with a $19 shipping fee. How many chairs must be bought for the total cost be the same from both businesses?

A car rental company charges $45 per day with a one-time insurance fee of $25, while another rental company charges $35 per day with a one-time insurance fee of $55. How many days must you rent a car for the cost to be the same at both companies?

A student and a teacher are collecting books for a classroom library. The student has already collected 50 books and plans to collect 10 more books each month. The teacher has already collected 80 books and plans to collect 4 more books each month. How many months will they have to collect books so that they collected the same amount of books?

Solve each equation.

2(4x + 1) = 3x + 2(2x – 3)

1. Complete the distributive property and combine like terms: 8x + 2 = 7x – 6

2. Get the variable to one side: x + 2 = -6

3. Use inverse operations to solve: x=-8

Write if each equation has one solution, no

or infinitely many solutions.

Write and solve an equation. Then, answer the question.

A babysitter charges $12 per hour plus a $15 transportation fee. Another babysitter charges $10 per hour plus a $25 transportation fee. For how many hours of babysitting will their total amount be the same?

A car wash business charges $10 for a basic wash for members. They charge $4 to become a member. Another car wash business charges $7 for a basic wash for members. They charge $10 to become a member. For how many washes will the cost be the same at both businesses?

One catering service charges $500 for the event plus $30 per guest. Another catering service charges $220 for the event plus $40 per guest. For how many guests will the total cost be the same?

A construction company charges $250 per hour plus a $500 equipment fee. Another construction company charges $300 per hour plus a $300 equipment fee. For how many hours of work will the total charges be the same?

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.

Use the rule to fill in the table and find the ordered pairs. Then, plot them on the graph.

Use the rule to fill in the input/output table. Then, list the ordered pairs and plot them on graph below.

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

Each input (x) in a function has exactly one unique output (y).

Circle the functions.

(2, 8)(-1, 3)(-2,7) (2, -4)(5, 3) b. x = 2

Circle the sets of ordered pairs that are functions.

2. (−2,5),(0,3),(2,1),(4,−1)

(2,10),(4,20),(6,30),(8,40)

Circle the tables that are functions.

Circle the equations that are functions.

Circle the scenarios that are functions.

Each person has a unique phone number.

Each book in the library has a specific shelf.

Each student is assigned one locker.

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

Apply the vertical line test. Then, circle the graphs that are functions. 1. (1,2),(2,3),(3,4),(4,5) 5. (−1,0),(0,1),(−1,2)

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

(−2,6),(0,4),(−2,8),(3,2)

(5,2),(5,3),(6,4)

Each person has multiple email addresses. The words bat, bark and jam have multiple meanings.

Explain why the relationship is not a function.

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

Explain why the relationship is not a function.

Lesson instructions

Graphs of Functions

The graph shows the runner’s distance from home over time. He runs to the park at steady pace for 2 minutes. He sits at the park from minutes 2 to 7, then runs towards home at slower pace.

Read the scenario and circle the graph that matches.

Sammy walks to a restaurant at a steady pace for 10 minutes. He stays at the restaurant and eats for 30 minutes. Then he walks home at the same pace.

Ava rides her bike to the library. It takes her 7 minutes to get there at a steady pace. She spends 20 minutes inside reading. Then she bikes home in 5 minutes.

Match the graph with the scenario.

3. Walking to school, stopping to tie a shoe, then continuing to school.

4. Riding a roller coaster at the state fair. 5. Going for a jog, taking a short break, jogging back.

Use the graph to choose the best answer.

Use the graph to fill in the blanks.

a. From minutes 0-10 the distance

b. From minutes 10-30 the distance

c. From minutes 30-40 the distance increases increases increases decreases decreases decreases stays the same stays the same stays the same Samantha started hiking from the base of a hill. She hiked uphill for minutes, rested at an elevation of feet for minutes, and then hiked back down to the base in minutes.

Use the graph to answer to the questions.

a. How much water is in the bottle to start?

b. After 20 minutes there is a sharp drop in the amount of water in the bottle. Why might that have occurred?

a. Between which months is the amount of money decreasing?

b. Between which months does the amount of money stay the same?

c. Between which months is the amount of money increasing?

Linear functions increase or decrease at a steady rate and form a straight line when graphed.

Circle linear or nonlinear.

Circle the linear equations. Cross out the nonlinear equations.

Plot the points on the graph. Then circle linear or nonlinear.

Use the rule to fill in the table and graph. Then, circle linear or nonlinear.

y = 10x y = 2x - 3 y = x2 y = 3x + 2

Find the slope and y-intercept of the line.

Slope = 1 2 y-intercept = 3

Slope: y-intercept:

Slope: y-intercept: Slope: y-intercept:

Write the slope and y-intercept. Is the graph increasing or decreasing?

Slope: y-intercept: Slope: y-intercept: Slope: y-intercept:

Circle the graph that has a slope of 2. Circle the graph that has a slope of 3

Use the graph to answer the questions.

How tall is the tree to start?

Hint: The start is at month 0.

Is the tree increasing or decreasing in size?

Choose the scenario that could be represented by the graph.

What is the slope? What is the y-intercept? Is the tank gaining fuel or losing fuel?

Choose the scenario that could be represented by the graph.

a) The height of a burning wax candle over time.

b) The number of children enrolled in school.

c) The height of mountain climber over time.

d) The amount of water in a leaky tank over time.

a) The amount of water in a tank as you fill it up.

b) The amount of water in a tank as you empty it.

c) The elevation of a kite.

d) The number of people at an event before it begins.

Lesson instructions 6

Find the slope and y-intercept of the line.

Point One: (0,5)

Point Two: (6,13)

When x is 0, y is 5. The y-intercept is 5.

Find the slope and the y-intercept from the table.

Find the slope of the line passing through the points.

Use the slope formula to find the missing value.

4. (2,3) and (6,7)

7. Slope: 1 2 Point 1: (4,10) Point 2: (6, y) 8. Slope: 4 Point 1: (-3, y) Point 2: (1, 18)

(–3,4) and (1,–4)

(0,6) and (5,6)

Find the slope. Then tell what it represents.

Slope = Maya walks at a rate of miles per hour.

Slope = The balloon is rising at a rate of meters per minute.

Slope = The car is driving at a rate of miles per hour.

Slope = The liquid in the container is emptying at a rate of milliliters per second.

Similar, or proportionate, triangles can be drawn along straight lines. Set up a proportion to find a missing side.

Circle all the triangles with the same slope as the grey triangle.

Find the missing side of the triangle.

Find the missing coordinate.

Write the equation of a line in slope intercept form.

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

Write the equation of the line that has a y-intercept of 6 and a slope of -2.

y = -2x + 6

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

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

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

Write the equation of the line that goes through origin and has a slope of 5.

Write the equation of the line with a slope of -2 and a y-intercept of 10.

Write the equation of the line that passes through the origin and has a slope of -1.

Write the equation of the line with a slope of 8 and a y-intercept of -3.

Use the y-intercept and the slope to graph a linear equation.

Graph: y = 1 2 x - 1

1. Plot the y-intercept at (0, -1). 2. Use the slope to find more points on the line by going up 1 and right 2. 3. Connect the points with a ruler.

Graph the linear equations on the coordinate plane.

y = -2x

y = 3x - 2

y = x - 1

Find the coordinates on the line

y = 1 5 x + 8 when x = 5.

Hint: plug in 5 for x and solve for y. Give your answer as an ordered pair.

Find the coordinates on the line y = 4x - 3 when y = 17.

Hint: plug in 17 for y and solve for x. Give your answer as an ordered pair.

Find the coordinates on the line y = -2x + 1 when x = -3.

Find the coordinates on the line y = - 1 2 x - 4 when y = -5.

Define the input, output, slope and y-intercept of a scenario to write an linear equation.

A bird takes off from a tree that is 5 feet in the air. He ascends at a rate of 1 foot per second. Write an equation to show his height (y) after x seconds.

y = x + 5

Input: seconds

Output: height (ft)

Slope: rate of 1 ft/sec y-intercept: 5 ft high to start

Define the input, output, slope and y-intercept of each scenario. Then, write an equation.

1. Emma is on page 65 of her book. Each day, she reads 20 pages.

2. There are 120 liters of water in the tank. It is leaking at a rate of 5 liters per minute.

3. Sam has $12. He makes $2 for each lawn he mows.

4. Chad has $3,000 in the bank. Each week he takes $50 out of his account.

5. A farmer has planted 8 rows of seeds so far. He plants at a rate of 3 rows per hour.

6. Kayla has crocheted 15 rows of a scarf. It takes her a minute to crochet a row.

7. Abby fills up an empty tank of water at a rate of 1.4 quarts per minute.

Number of pages read Time (days)

Time (minutes)

Answer the questions about the linear equations.

Joseph has saved $100 so far. He saves $50 more each week.

a. Write an equation to model the scenario.

b. Graph the equation. Label the axes and give the graph a title.

A water tank has 300 gallons of water in it. A leak causes it to lose water at a rate of 20 gallons per hour.

a. Write an equation to model the scenario.

b. Graph the equation. Label the axes and give the graph a title.

Use the graph to answer the questions.

a. How many minutes does it take to heat the mixture up to 120°?

b. What temperature is the mixture after 4 minutes?

a. How much energy does the battery have after 15 minutes?

b. How long until the battery has no charge?

Chapter 4

Lesson instructions 11

Function A

Compare the slopes and y-intercepts of graphs, tables and equations.

Function B Function C

y = 1 2 x + 4

slope: 2 y-intercept: -1

slope: -2 y-intercept: -6

slope: 1 2 y-intercept: 4

Function A has the greatest slope.

Function C has the greatest y-intercept.

Find the slope and y-intercept of each function. Then, circle the function with the greatest slope.

1. Function A

Function B

Function C

slope: y-intercept:

2. Function D

slope: y-intercept:

Function E

slope: y-intercept: slope: y-intercept:

Use Functions A-F to answer the questions below.

3. Which function has the greatest y-intercept?

4. Which functions have negative y-intercepts?

Function F

slope: y-intercept: slope: y-intercept:

5. Which function has the least slope?

6. Which function has the least y-intercept?

Compare the functions. Then, answer the questions.

a. Which tank starts off with more water?

b. Which tank is leaking faster?

y = 10x + 25 y = 80x + 10 Tank A Pool B

a. Which plant starts off the tallest?

b. Which plant grows the fastest?

a. Which pool is filling the fastest?

b. Which pool starts off empty?

a. Which car starts off at a further distance?

b. Which car is driving at a faster speed?

Chapter 4 Lesson 12

Determine if the relationship is a function.

Each input in a function has exactly one output.

Is this a function? (3,4) (3,5) no

Determine if the function is linear.

A linear function has a slope that doesn’t change and its graph is a straight line.

Is this linear? y = 3x + 6 yes

Find the slope and the y-intercept of the function. Then, write the equation.

The slope (m) is the rise the run. m = y2 - y1 x2 - x1

The y-intercept (b) is the y-value when x = 0 and is found at point (0,b) on the graph.

Slope-intercept form: y = mx + b

Equation:

Chapter 4

Level H Word Problems

Lesson 12

Graph the line.

y = x - 6

Graph the line.

y =1 3 x + 2

Circle the function with the higher slope.

Function A: y = 6x + 10

Function B:

Circle the function with the higher y-intercept.

Function A: y = -2x + 5

Function B:

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.

Solve the proportion.

3 miles 4 minutes = x miles 12 minutes

x students 20 tables = 12 students 15 tables

Use the table to find the unit rate.

A cyclist rides at a constant speed. Write his speed as a unit rate.

5. A water tank fills at a steady rate. Write the unit rate of gallons per minute.

6. A taxi company charges a consistent rate per mile. Write the unit rate.

Determine the better buy.

A 20-oz bottle of ketchup for $3.60 or a 32-oz bottle for $5.12?

A 10-pack of socks for $15.50 or a 6-pack for $9.60?

A 4-pack of light bulbs for $7.00 or a 2-pack for $3.80?

A 15-oz jar of honey for $6.00 or a 25-oz jar for $9.50? 11. A 1.2-lb bag of frozen vegetables for $2.04 or a 2-lb bag for $3.40?

A bakery uses 6 eggs to make 24 muffins. How many eggs are needed to make 60 muffins?

A factory produces 150 toys in 5 hours. How many toys can it produce in 8 hours at the same rate?

A student reads 45 pages in 30 minutes. How many pages will the student read in 1 hour?

A store sells 4 pounds of apples for $6.20. What would 7 pounds of apples cost at the same rate?

Determine if the relationship is proportional.

This graph represents a proportional relationship because:

→ It passes through the origin (0,0)

→ It has a constant unit rate

Determine if the scenario represents a proportional relationship. If it does, find the unit rate.

Scenario Proportional? Unit Rate

1. A bakery sells 4 cookies for $6 and 8 cookies for $12.

2. A taxi charges $10 for 2 miles and $15.50 for 3 miles.

3. A tutor earns $60 for 4 hours and $90 for 6 hours.

4. A bike rental costs $12 for 2 hours and $18 for 3 hours.

5. A shirt costs $20, two shirts cost $40, three shirts cost $63.

6. A 2-liter bottle of juice is $2.80, and a 3-liter bottle is $4.50.

7. A train travels 120 miles in 2 hours and 180 miles in 3 hours.

Graph each proportional relationship. Then write the unit rate.

8. A music teacher charges $60 for 3 lessons.

9. Each pair of socks costs $2.75.

10. A car drives 150 miles using 5 gallons of gas.

instructions 3 Write equations for proportional relationships to compare them.

Cereal A costs $0.25 per ounce at the grocery store. Cereal B’s cost is shown in the graph. Which is the better buy?

Write the equation based on each scenario given.

Scenario Unit Rate Equation

1. A photographer charges $25 per session.

2. A printer prints 90 pages in 3 minutes.

3. A smoothie shop sells each drink for $4.50.

4. A person walks 9 miles in 3 hours.

5. A student types 180 words in 9 minutes.

Write the equation based on the graph.

Write the equation based on the table.

Painting Company A charges $300 for 5 rooms. Painting Company B’s price is shown in the table. Who charges less per room?

Rooms 3 6 9 Cost ($) 210 420 630

Two friends are selling handmade bracelets to raise money. Hannah’s sales are shown on a graph with the point (4, 48). Emma’s sales are shown in the table below. Who earns more per bracelet?

Earnings ($) 30 50 70

Bracelets Sold 3 5 7

Chef A uses 3 eggs for every 2 servings. Chef B’s egg usage is plotted on the graph below. Who uses more eggs per serving?

Egg Usage for Chef B

Judy earns $72 for 6 hours of tutoring. Kayla’s hourly rate is shown in the equation: y = 13.50x. Who earns more per hour?

Solve the proportion.

b = c d

× d = b × c

Determine the better buy.

Step One: Find the Unit Rate

$3.50 for 5 = 3.5/5 = $0.70 each

$5.60 for 7 = 5.6/7 = $0.80 each

Step Two: Compare the unit rates.

$3.50 for 5 is the better buy.

Write an equation.

1. A recipe uses 2 tablespoons of oil to make 10 muffins. How many tablespoons of oil are needed to make 25 muffins?

2. It costs $12 to buy 3 notebooks. How much will it cost to buy 7 notebooks?

Find the unit rate.

y = kx, k → unit rate

From an equation… The coefficient of x is the unit rate.

From a table… Divide y by x in any column.

From a graph… Choose a point and divide y by x.

3. A 16-oz bottle of juice costs $4.80. A 24-oz bottle of the same juice costs $6.00. Which is the better buy?

4. A 10-pack of pencils costs $3.90, and a 15-pack costs $5.25. Which is the better buy?

7. Find the unit rate (miles per hour).

8. A graph shows the point (5, 40) on a line that passes through the origin. What is the unit rate?

Abe finishes typing a 420 word document in 6 minutes. Ben finishes typing a 330 word document in 5 minutes. Ben says that he is faster because he finished first. Is he correct? Use unit rates to prove your answer.

A bakery uses 4 cups of sugar to bake 48 muffins. One morning, the baker needs to prepare a larger batch of 72 muffins for a school event. How many cups of sugar will the baker need to keep the recipe proportional?

Graph the proportional relationship of a car that travels 180 miles in 3 hours. Then write the equation.

A water tank is being filled at a constant rate. After 3 minutes, it has 24 liters of water. Graph the proportional relationship then, write the equation.

Equation:

A Car's Travels Water in a Tank

Chapter 6

In Chapter 6, we will discover

Linear Systems

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.

Determine how many solutions exist for each system.

Determine how many solutions exist for each system of equations.

9. y = 18x – 11 y = 18x + 9

13. y = 1 4 (16x – 24) y = 4x – 6 10. y = -7x + 4 y = 2.5x – 3 14. y = 4 5 (50x – 35) y = 40x + 20 11. y = -8(x + 2) y = -8x – 16 15. y = 32x + 12 y = 3 4 (32x + 12) 12. y = 5(-3x + 4) y = -15x – 4 16. y = 6 y = 9

Explain why the system of equation has one solution, no solution, or infinitely many solutions.

17. y = -5x y = 5x 18. y = 2x + 1 y = 2x – 1

Two caterers offer different packages for a taco party. Tito’s Tacos charges a $30 flat fee and $2 per taco. This is represented by the equation y = 2x + 3. Taco Tuesdays charges a $50 flat fee and $2 per taco. This is represented by the equation y = 2x + 5. Will the two caterers ever charge the same total amount for the same number of tacos? Explain your reasoning.

Two theaters are comparing their ticket revenue. Nothing But Movies sells tickets for $10 and had $200 in pre-sales, which is represented by the equation y = 10x + 200. Megaplex also sells tickets for $10 and had $200 in pre-sales, which is represented by the equation y = 10x + 200. How many tickets would they need to sell to earn the same revenue? Explain your answer.

Two hikers are walking toward the same destination from different starting points. Leslie starts at mile 0 and walks at 3 miles per hour, which is represented by the equation y = 3x. Bonnie starts at mile 2 and walks at 2 miles per hour, which is represented by the equation y = 2x + 2.

Will the hikers ever be at the same location at the same time? Explain.

A train and a car leave the same city traveling at different speeds. The train starts 1 hour earlier and travels at 60 mph, which is represented by the equation y = 60x. The car starts later and goes 70 mph, which is represented by the equation y = 70(x – 1)

Will the car catch up to the train? Explain.

Lesson instructions 2

An ordered pair is a solution to the system if it works in BOTH equations.

Is (-1,2) a solution to this system of equations?

y = -2x

y = 3x – 2

1. Substitute the x and y values.

2 = -2(-1) → 2 = 2

2 = 3(-1) – 2 → 2 = -5 No, it is not a solution.

Determine if the given ordered pair is a solution to system.

1. Is (1,3) a solution to this system of equations?

y = 2x + 1

y = -x + 4

3. Is (2,5) a solution to this system of equations?

y = 5

y = 2x + 1

5. Is (2,4) a solution to this system of equations?

y = 6 – x y = 2x

7. Is (0,0) a solution to this system of equations? y = 9x y = 4x + 7

2. Is (1,-1) a solution to this system of equations?

y = -3x + 2

y = x – 5

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

y = -3x y = x – 8

6. Is (-3,0) a solution to this system of equations? y = 0

y = -2x – 2

8. Is (2,8) a solution to this system of equations?

y = 3x + 2

y = -3x + 2

9. Is (-4,4) a solution to this system of equations? 5x + 2y = -12 -3x – 3y = 0

10. Is (11, 2) a solution to this system of equations? 5x – 10y = 35 x + y = -13

Stacy stated that the ordered pair (4, 2) is a solution to the following system of equations.

y = 2x – 6 y = 5x

Do you agree with her statement? Why or why not?

Fill in the tables to find the solution to the system.

5x + 2y = 26 2x + y = 10

Solution: ( , )

Cory stated that the ordered pair (-5, -6) is a solution to the following system of equations.

9x – 5y = -15 -2x + 7y = -32

Do you agree with his statement? Why or why not?

Fill in the tables to find the solution to the system. y = 7x y = 5x + 20

( , )

Lesson instructions 3

Solve a system of equations by finding the point of intersection on the graph.

y = x – 2

y = - 1 3 x + 2

1. Graph both equations.

2. The solution is where the lines intersect: (3, 1)

Write the solution to each system of equations.

Solve each system of equations by graphing.

Read the problem, graph the equations, and find and interpret the solution to the system.

In a science experiment, Container A starts at 80°F and cools at a rate of 5°F per hour. Container B starts at 40°F and warms at a rate of 5°F per hour. This can be represented by the following system of equations:

y = −5x + 80

y = 5x + 40

Ella reads 10 pages per day. Katie starts ahead with 20 pages already read and reads 5 pages per day. This can be represented by the following system of equations:

y = 10x

y = 5x + 20

The Climbing Crew is 60 feet up a wall and climbing at 10 feet per minute. Team Awesome is only 20 feet up but climbs at 15 feet per minute. This can be represented by the following system of equations:

y = 10x + 60

y = 15x + 20

Billy has $40 already saved and puts away $15 every week. Noah starts with $10 but saves $25 each week. This can be represented by the following system of equations:

y = 15x + 40

y = 25x + 10

Lesson instructions 4

Use inverse operations to isolate a variable in an equation. 2x + 4y = 16

Solve for x:

2x + 4y = 16 - 4y -4y

2x = 16 - 4y ÷2 ÷2 x = 8 - 2y

Solve for y:

Isolate the x variable in each equation.

Isolate the y variable in each equation. 1. y = -10x – 60

3x – y = 6

A group has $136 to spend on tickets to a show. Adult tickets (y) cost $16 each and a child's ticket (x) is $8. The equation 16y + 8x = 136 represents the possible number of each type of ticket that can be purchased. Rewrite this equation to show the number of adult tickets (y) that can be purchased given any number of child's tickets (x).

Mark has $40 to spend at a carnival. Soda is $2 for a cup (c) and games (g) are $4 each. The equation 2c + 4g = 40 represents how many cups of soda he can buy and how many games he can play. Rewrite this equation so that it will show the number of cups of soda (c) he can buy for any given number of games (g) purchased.

Lena has 72 feet of fencing to put up around her garden. If the length of her garden is L and the width of her garden is W, the equation 2W + 2L = 72 represents the possible lengths and widths her garden can be with this fencing. Rewrite the equation to show the width of the garden (W) for any given length.

Jack needs to reach a point 5 miles away. He moves at a speed of 2 miles per hour when it is windy, but at a speed of 4 miles per hour when it is not. This scenario can be represented by the equation 2x + 4y = 5. Rewrite the equation to find the amount of time spent traveling without wind (y) given any amount of time with wind (x).

Lesson instructions 5 Use substitution to solve a system of equations.

x – y = 3 2x + 3y = 11

1. Solve for x or y in one of the equations. x = y + 3

2. Substitute this expression into the other equation and solve.

2(y + 3) + 3y = 11 → 5y + 6 = 11 → 5y = 5 → y = 1

3. Substitute again to solve for the other variable.

x – 1 = 3 → x = 4

Solution = (4, 1)

Solve each system using substitution.

1. 5x + 5y = 40 x – y = 2 2. 6x + 3y = -3 x – y = 4 3. 3x – 3y = -6 2x + 2y = 16

-2x – 16y = -8 5x – 3y = -23

4x – 2y = 12 8x + 16y = 24

5. -8x – 9y = -25 2x + 2y = 14 9. 4x + 3y = 18 2x – y = 4 10. 6x – 2y = 12 x + y = 6 11. 5x – 2y = -2 -3x + 4y = -10 12. -3x – 8y = 8 -11x + 6y = -6 13. -2x + 2y = -4 4x – y = -7 6. x – y = 2 3x – 2y = 10

-x + 5y = 30 6x + 10y = -20 14. x + 2y = 10 3x – y = 9 15. 9x – 3y = 6 -6x + 3y = 3 16. y = 7 - 3x 10x + 5y = 15

Use substituion to solve the system of equations and answer the question.

A bakery sells large cupcakes (y) for $4 more than small cupcakes (x). They sold 90 large cupcakes and 110 small cupcakes for a total of $960. How much does each type of cupcake cost?

y = 4 + x 110x + 90y = 960;

A school sells adult-sized (y) and childsized (x) sweatshirts. Adult sweatshirts cost $8 more than child sweatshirts. 70 adult and 130 child sweatshirts are sold for $3,160. How much does each type of sweatshirt cost?

y = x + 8 130x + 70y = 3,160

A student group sold cookies (x) and brownies (y) at a bake sale. Each cookie sold for $2 and each brownie for $3. They sold 80 items in total and earned $200. How many of each item were sold?

x + y = 80

2x + 3y = 200;

Stuffed bears (x) cost $8 and toy cars (y) cost $5. A vendor sold 90 toys in total and earned $600. How many of each toy were sold?

x + y = 90 8x + 5y = 600;

Lesson instructions 6

Add or subtract the equations in a system to solve using elimination.

1. Line up the equations based on their variables.

2. Look for variables whose coefficients match. If there's no match, multiply one or both equations to make a match.

3. Add or subtract to cancel out one variable.

4. Solve for the remaining variable.

5. Substitute this value into either equation to solve for the other variable.

Solve each system of equations with substitution by solving for x.

1. 4x + 3y = 26 4x – 2y = -4

4x + 5y = 18 2x – 5y = -6

-4x + 3y = 14

5. 2x + 3y = -4

+ 3y = 15

2x + y = 21

-6x + 2y = 74

+ y = 3 9. 5x – 7y = -45

-10x – 3y = -87

– 5y = 18

+ 10y = 50 13. -7x – 6y = 92

– 3y = -54

– 4y = -36

– 5y = 13

– 7y = 151

+ 4y = 28

3x + 2y = 14

- y = 8

Use elimination to solve the system of equations and answer the question.

A school is collecting donations for a fundraiser. They received only $5 bills and $10 bills. Altogether, they collected 60 bills worth $450. How many of each type of bill did they collect?

x + y = 60

5x + 10y = 450

A wall is made of 84 bricks. The number of tan bricks is 3 more than twice the number of red bricks. How many of each color brick are there?

x + y = 84

y = 2x + 3

A bakery sold small and large loaves of bread. Small loaves cost $3, and large loaves cost $5. On Monday, they sold a total of 180 loaves and made $720. How many small and large loaves did they sell?

x + y = 180

3x + 5y = 720

A theater sold adult and child tickets. Adult tickets cost $12, and child tickets cost $8. The theater sold 250 tickets and made $2,560. How many of each type of ticket were sold?

x + y = 250

12x + 8y = 2,560

Lesson instructions 7 Write and solve a system for the word problem.

A snack shop sold 120 snacks in one day. Chips cost $2 while sandwiches cost $6. The shop made $400 total. How many of each were sold?

1. Write the system.

x + y = 120

2x + 6y = 400

2. Solve the system using any method.

x = 80 chips; y = 40 sandwiches

Solve each system of equations using any method.

1. At a bake sale, 80 items were sold for a total of $250. Cookies sold for $2 each, and cakes sold for $5 each. This can be represented by the following system of equations where x equals the number of cookies and y equals the number of cakes.

x + y = 80

2x + 5y = 250

How many cookies and cakes were sold?

2. The duration of a show was 4 hours and 30 minutes (270 minutes total). The first act was 30 minutes longer than the second. This can be represented by the following system of equations where x equals the length of the first act and y equals the length of the second act.

x + y = 270

x = y + 30

How long was each act?

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

3. At a book fair, 100 books were sold for a total of $680. Hardcover books sold for $8, and paperback books sold for $4. Write a system where x represents the number of hardcover books sold and y represents the number of paperback books sold. How many hardcover books were sold?

4. Abbie spent 90 minutes on homework in total. She spent twice as much time on math as on science. Write a system where x represents the amount of time Abbie spent on math homework and y represents the amount of time she spent on science homework. How many minutes did she spend on each subject?

Sally and her daughter, Bella, are 47 years old combined. Sally is 8 more than twice as old as Bella. Write a system of equations where x equals Sally’s age and y equals Bella’s age. How old are Sally and Bella?

Ned is a basketball coach. He splits 18 players into groups of 2 and 3. There are 7 groups total. Write a system of equations where x equals the number of pairs and y equals the number of trios Ned created. How many pairs and trios are there?

A cafeteria stocks 30 bottles of fruit juice. Apple juice bottles have 200 mL of juice while orange juice bottles have 300 mL of juice. Altogether the cafeteria has 7,500 mL of juice. Write a system of equations where x equals the number of bottles of apple juice and y equals number of bottles of orange juice. How many bottles of each type do they have?

A total of 72 students went on a field trip. They traveled in vans and buses. Each van holds 6 students. Each bus holds 24 students. There were 6 vehicles in total. Write a system of equations where x equals the number of vans and y equals the number of buses. How many of each vehicle were used?

Determine how many solutions exist for each system of equations.

Determine if the given ordered pair is a solution to system.

(2, 3)

(3, 1)

Is (1,3) a solution to this system? y = 3x x + y = 5

No, it is not a solution.

Solve each system of equations by graphing.

1. Graph both equations.

2. The solution is the ordered pair at the point where the lines intersect: (3, 1)

1. Substitute the x and y values. 3 = 3(1) → 3 = 3 1 + 3 = 5 → 4 = 5 x – y = 3 2x + 3y = 11

y = 1 4 x – 3 y = -x + 2

Solve each system of equations using any method.

1. Isolate: x - y = 3 → x = 3 + y

2. Substitute: 2(3 + y) + 3y = 11 → y = 1

3. Substitute: x - (1) = 3 → x = 4

Solution: (4,1) y = x – 2 y = - 1 3 x + 2

A school sold 120 snack packs during a fundraiser. They sold two types: popcorn bags and cookie boxes. Popcorn bags cost $2 each, and cookie boxes cost $4 each. They made $360 in total. Write a system of equations to represent this scenario. How many of each did they sell?

A total of 150 tickets were sold for a concert. Adult tickets cost $12, and student tickets cost $6. If the total amount collected was $1,200, how many of each ticket were sold? Write a system of equations to solve.

There are 28 students in a class. The teacher forms groups with either 2 students or 4 students in each group. There are a total of 10 groups. Write a system of equations to represent this scenario. How many 2-student groups and how many 4-student groups were formed?

Jill spent $47 at a book fair buying only books and posters. Books cost $8, and posters cost $2. If Jill bought 11 items total, how many of each did she buy? Write a system of equations to solve.

Chapter 7

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.

Lesson instructions 1

Complementary Angles

Angles that add up to 90°

Supplementary Angles

Angles that add up to 180°

Vertical Angles

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

Adjacent Angles

Angles that share a vertex and side

Write whether the angles are complementary or supplementary. Then, find the missing angle.

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

Write an equation, then solve to find the value of x.