Writing Linear Equations MODULE 5

ESSENTIAL QUESTION?

Real-World Video

Linear equations can be used to describe many situations related to shopping. If a store advertised four books for $32.00, you could write and solve a linear equation to find the price of each book.my.hrw.com

How can you use linear equations to solve real-world problems?

Writing Linear Equations 5

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MODULE

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LESSON 5.1

Writing Linear Equations from Situations and Graphs

8.F.4

LESSON 5.2

Writing Linear Equations from a Table

8.F.4

LESSON 5.3

Linear Relationships and Bivariate Data

8.SP.1, 8.SP.2,

8.SP.3

COMMON CORE

COMMON CORE

COMMON CORE

You can use linear equations and their graphs to model real-world relationships involving constant rates of change.

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YOUAre Ready?Personal

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Online Assessment and

Interventionmy.hrw.com

Complete these exercises to review skills you will

need for this module.

Write Fractions as DecimalsEXAMPLE 0.5

___ 0.8

= ? 0.5 × 10 _______

0.8 × 10 = 5

_ 8

- 48 _____ 20

-16 _____ 40

- 40 ____ 0

8 ⟌⎯⎯⎯ 5.000

0.625

Multiply the numerator and the denominator by a power of 10 so that the denominator is a whole number.

Write the fraction as a division problem.Write a decimal point and zeros in the dividend.Place a decimal point in the quotient.Divide as with whole numbers.

n is multiplied by 5. To solve the equation, use the inverse operation, division.

7 is added to k.To solve the equation, use the inverse operation, subtraction.

5n = 205n __ 5

= 20 __

5

n = 4

k + 7 = 9

k + 7 - 7 = 9 - 7

k = 2

Write each fraction as a decimal.

1. 3 _ 8

2. 0.3 ___

0.4 3. 0.13

____ 0.2

4. 0.39 ____

0.75

Inverse OperationsEXAMPLE

Solve each equation using the inverse operation.

5. 7p = 28 6. h - 13 = 5

7. y

_ 3

= -6 8. b + 9 = 21

9. c - 8 = -8 10. 3n = -12 11. -16 = m + 7 12. t ___ -5

= -5

0.375 0.75 0.65 0.52

p = 4

c = 0 n = -4

m = -23 t = 25

h = 18

b = 12y = -18

Unit 2124

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Are You Ready?

Assess ReadinessUse the assessment on this page to determine if students need

intensive or strategic intervention for the module’s prerequisite skills.

Interactive Whiteboards

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whiteboard-ready lessons and

activities.

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Intervention

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homework, quizzes, tests,

and intervention activities.

Prepare your students with

updated practice tests aligned

with Common Core.

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resources online—plan,

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and assignments.

ePlanner

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access all your resources online.

Interactive Answers and

Solutions

Customize answer keys to print

or display in the classroom.

Choose to include answers only

or full solutions to all lesson

exercises.

Author Juli Dixon models successful

teaching practices as she explores the

concept of writing linear equations in

an actual eighth-grade classroom.

Intervention Enrichment

Access Are You Ready? assessment online, and receive

instant scoring, feedback, and customized intervention

or enrichment.

Online and Print Resources

Skills Intervention worksheets

• Skill 26 Write Fractions

as Decimals

• Skill 57 Inverse Operations

Differentiated Instruction

• Challenge worksheets

PRE-AP

Extend the Math PRE-AP

Lesson Activities in TE


Reading Start-Up

Active ReadingTri-Fold Before beginning the module, create

a tri-fold to help you learn the concepts and

vocabulary in this module. Fold the paper

into three sections. Label the columns “What

I Know,” “What I Need to Know,” and “What

I Learned.” Complete the first two columns

before you read. After studying the module,

complete the third column.

VocabularyReview Words✔ linear equation (ecuación

lineal) ordered pair (par

ordenado) proportional relationship

(relación proporcional) rate of change (tasa de

cambio)✔ slope (pendiente)✔ slope-intercept form of

an equation (forma de pendiente-intersección)

✔ x-coordinate (coordenada x)

✔ y-coordinate (coordenada y)

✔ y-intercept (intersección con el eje y)

Preview Words bivariate data (datos

bivariados) nonlinear relationship

(relación no lineal)

Visualize VocabularyUse the ✔ words to complete the diagram. You can put

more than one word in each bubble.

Understand VocabularyComplete the sentences using the preview words.

1. A set of data that is made up of two paired variables

is .

2. When the rate of change varies from point to point, the relationship

is a .

y = mx + b

y x

m b

bivariate data

nonlinear relationship

y-coordinate

slope

slope-intercept form of an

equation, linear equation

x-coordinate

y-intercept

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Reading Start-Up

Have students complete the activities on this page by working alone

or with others.

Visualize VocabularyThe concept web helps students review vocabulary associated with

linear relationships. In each bubble, students should write one or more

words associated with the linear equation y = mx + b.

Understand VocabularyUse the following explanations to help students learn the preview words.

Some relationships are nonlinear relationships. Nonlinear means “not in the shape of a line.” In nonlinear relationships,

the rate of change is not constant. Graphs that represent

nonlinear relationships will not be lines.

Many real-world situations have nonlinear relationships.

For example, the growth of a plant is usually not a linear

relationship because plants do not grow the same amount

every day.

Active ReadingIntegrating Language ArtsStudents can use these reading and note-taking strategies to help them

organize and understand new concepts and vocabulary.

Additional ResourcesDifferentiated Instruction

• Reading Strategies ELL

After

Students will connect that:

• there are various forms of linear

equations

• in nonlinear relationships, the rate

of change can vary from point to

point

In this moduleStudents represent and use linear relationships:

• write an equation in the form y = mx + b to model

a linear relationship between two quantities using

verbal, numerical, tabular, and graphical

representations

• contrast bivariate sets of data that suggest a linear

relationship with bivariate sets of data that do not

suggest a linear relationship from a graphical

representation

Before

Students understand proportional

and linear relationships:

• use tables and verbal descriptions

to describe a linear relationship

• write and graph a linear

relationship

COMMONCORE ELA-Literacy.RST.6-8.7 Integrate quantitative or technical information

expressed in words in a text with a version of that information expressed visually (e.g., in a flowchart, diagram, model, graph, or table).

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The graph shows the temperatures

in degrees Celsius inside the earth at

certain depths in kilometers. Use the

graph to write an equation and find

the temperature at a depth of 12 km.

The initial temperature is 20°C.

It increases at a rate of 10°C/km.

The equation is t = 10d + 20.

At a depth of 12 km, the

temperature is 140°C.

Unpacking the StandardsUnderstanding the standards and the vocabulary terms in the

standards will help you know exactly what you are expected to

learn in this module.

What It Means to YouYou will see how to use a linear relationship between sets of data

to make predictions.

What It Means to YouYou will learn how to write an equation based on a situation that

models a linear relationship.

MODULE 5

In 2006 the fare for a taxicab was an initial charge of $2.50 plus

$0.30 per mile. Write an equation in slope-intercept form that can

be used to calculate the total fare.

The constant charge is $2.50.

The rate of change is $0.30 per mile.

The input variable, x, is the number of miles driven.

So 0.3x is the cost for the miles driven.

The equation for the total fare, y, is as follows:

y = 0.3x + 2.5

8.F.4

Construct a function to model a

linear relationship between two

quantities. Determine the rate of

change and initial value of the

function from a description of a

relationship … Interpret the rate

of change and initial value of a

linear function in terms of the

situation it models, and in terms

of its graph or a table of values.

Key Vocabularyrate of change (tasa de cambio)

A ratio that compares the

amount of change in a

dependent variable to the

amount of change in an

independent variable.

COMMON CORE

8.SP.3

Use the equation of a linear

model to solve problems

in the context of bivariate

measurement data, interpreting

the slope and intercept.

Key Vocabularybivariate data (datos bivariados)

A set of data that is made up of

two paired variables.

COMMON CORE

Visit my.hrw.com to see all the Common Core Standards unpacked.

UNPACKING EXAMPLE 8.F.4

UNPACKING EXAMPLE 8.SP.3

Unit 2126

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Use the examples on this page to help students know exactly what

they are expected to learn in this module.

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Unpacking the Standards

Go online to see a complete unpacking of the Common Core Standards.

Common Core Standards

Content Areas

COMMONCORE Functions—8.F

Use functions to model relationships between quantities.

COMMONCORE Statistics and Probability—8.SP

Investigate patterns of association in bivariate data.

Common Core StandardsLesson

5.1

Lesson

5.2

Lesson

5.3

8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value … .

COMMONCORE

COMMONCORE

8.SP.1 Construct and interpret scatter plots for bivariate measure-ment data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

COMMONCORE

8.SP.2 … For scatter plots that suggest a linear association, informally fit a straight line, … .

COMMONCORE

8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

COMMONCORE


5.1L E S SON Writing Linear Equations from Situations

and Graphs

Interactive Whiteboard Interactive example available online

ADDITIONAL EXAMPLE 1A DJ charges a setup fee plus an hourly

fee to provide music for a dance party.

Use the graph to write an equation in

slope-intercept form to represent the

amount spent, y, on x hours of music.

y = 75x + 100

Engage

ESSENTIAL QUESTIONHow do you write an equation to model a linear relationship given a graph or a description?Use pairs of values for input and output to determine the values for the slope m and the y-intercept b in the equation y = mx + b.

Motivate the LessonAsk: How can you figure out the total cost for different plans for your cell phone when the

fees and charges are not the same? What choices have you made that involved comparing

memberships or rentals? The Explore Activity shows how to write an equation to help you

evaluate and compare costs.

Explore

EXPLORE ACTIVITYConnect Vocabulary ELL Explain the terms potter’s wheel (a rotating disk used for shaping clay) and kiln (an oven for

baking or drying clay or pottery). Ask if any students have worked with clay. Any such

students may be able to explain these terms to the class.

Explain

EXAMPLE 1Questioning Strategies Mathematical Practices • Why do you think that the points (0, 8) and (8, 18) were used to find the slope? Sample

answer: These two points clearly lie at the intersection of vertical and horizontal grid lines.

You can’t be sure of the values of y for the other points on the line.

• Describe the new graph if the membership fee were changed to $10. The y-intercept

moves up to (0, 10); the slope stays the same.

Engage with the WhiteboardDraw arrows showing the rise and run between the two points (0, 8) and (8, 18) on

the graph. Ask students what the y-intercept, rise, and run represent in terms of the

membership fee and rental fee.

YOUR TURNAvoid Common ErrorsStudents may be confused about which axis is which and about the meaning of the

y-intercept. Verify that they understand that the x-value is zero at the y-intercept, and that

the y-intercept is the y-value of the point where the graph crosses the y-axis.

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The student is expected to:

Functions—8.F.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Mathematical Practices

MP.2 Reasoning

COMMONCORE

COMMONCORE

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400

600

500

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Amount on Gift Card

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Video Club Costs

Writing an Equation from a GraphYou can use information presented in a graph to write an equation in slope-intercept form.

A video club charges a one-time membership fee

plus a rental fee for each DVD borrowed. Use the

graph to write an equation in slope-intercept form

to represent the amount spent, y, on x DVD rentals.

Choose two points on the graph, (x1, y1)

and (x2, y2), to find the slope.

m = y2 - y1 ______ x2 - x1

m = 18 - 8 ______ 8 - 0

m = 10 ___ 8

= 1.25

Read the y-intercept from the graph.

The y-intercept is 8.

Use your slope and y-intercept values to write an equation in

slope-intercept form.

y = mx + b

y = 1.25x + 8

Reflect 1. What does the value of the slope represent in this context?

2. Describe the meaning of the y-intercept.

EXAMPLE 1

STEP 1

STEP 2

STEP 3

3. The cash register subtracts $2.50 from a $25

Coffee Café gift card for every medium coffee

the customer buys. Use the graph to write an

equation in slope-intercept form to represent

this situation.

YOUR TURN

If the graph of an equation is a line that goes through the

origin, what is the value of the y-intercept?

Math TalkMathematical Practices

8.F.4COMMONCORE

Find the change in y-values over the change in x-values.

Substitute (0, 8) for (x1, y1) and (8,18) for (x2, y2).

Simplify.

Slope-intercept form

Substitute 1.25 for m and 8 for y.

The value of the y-intercept is zero.

cost per DVD rental

amount spent for 0 rentals, or the membership fee of $8

y = -2.5x + 25

Unit 2128

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? ESSENTIAL QUESTION

Writing an Equation in Slope-Intercept FormGreta makes clay mugs and bowls as gifts at the Crafty Studio.

She pays a membership fee of $15 a month and an equipment

fee of $3.00 an hour to use the potter’s wheel, table, and kiln.

Write an equation in the form y = mx + b that Greta can use to

calculate her monthly costs.

What is the input variable, x, for this situation?

What is the output variable, y, for this situation?

During April, Greta does not use the equipment at all.

What will be her number of hours (x) for April?

What will be her cost (y) for April?

What will be the y-intercept, b, in the equation?

Greta spends 8 hours in May for a cost of $15 + 8($3) = .

In June, she spends 11 hours for a cost of .

From May to June, the change in x-values is .

From May to June, the change in y-values is .

What will be the slope, m, in the equation?

Use the values for m and b to write an equation

for Greta’s costs in the form y = mx + b:

A

B

C

D

How do you write an equation to model a linear relationship given a graph or a description?

L E S S ON

5.1Writing Linear Equations from Situations and Graphs

EXPLORE ACTIVITY

What change could the studio make that would make a difference to the

y-intercept of the equation?


8.F.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value… . Interpret the rate of change and initial value… .

COMMONCORE

8.F.4COMMONCORE

the number of hours Greta uses the studio

the money Greta pays the studio each month

0

$15

$39

$48

+3

+9

3

y = 3x + 15

15

Changing the membership fee changes the y-intercept.

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PROFESSIONAL DEVELOPMENT

Math BackgroundA constant rate of change can be shown by a

linear graph. The rate of change is the slope of

the line. Lines with positive slopes rise from left

to right; lines with negative slopes go down.

The magnitude of the slope describes the

steepness. The line y = x makes a 45° angle with

the x-axis and has a slope of 1. A line with a slope

whose absolute value is between 0 and 1 is less

steep than a 45° line. A line whose absolute value

of the slope is greater than 1 is steeper than a

45° line.

CC Integrate Mathematical Practices MP.2

This lesson provides an opportunity to address

this Mathematical Practices standard. It calls for

students to represent a situation symbolically.

Students read values from a graph and create a

new representation of the linear relationship in

the form of an equation.

Writing Linear Equations from Situations and Graphs 128


ADDITIONAL EXAMPLE 2The cost for 25 square yards of carpet is

$650 including delivery and installa-

tion. The cost for 40 square yards of

installed carpet is $950. Write an

equation in slope-intercept form for

the cost of the installed carpet.

y = 20x + 150

EXAMPLE 2Questioning Strategies Mathematical Practices • What does the rise over the run, or the slope, represent in the problem situation? change

in rent over change in square feet, or rent per square foot

• If the equation for the rent were y = 4 __ 3 x + 25, how would this change the rent? The rent

would be $75 more for any number of square feet.

Focus on Critical ThinkingIn Example 2, one student wrote this equation for the slope: m = 1150 - 750

________ 600 - 900

. Explain what

error was made in this equation. The coordinates for the points must be in the same order

in the numerator and in the denominator.

YOUR TURNFocus on Modeling Mathematical PracticesMake sure that students understand that the number of chores Hari chooses to do is the

independent, or input, variable, and the allowance he receives is the dependent, or output,

variable.

Integrating Language Arts ELL

Encourage English learners to take notes on new terms or concepts and to write them in

familiar language.

Elaborate

Talk About ItSummarize the Lesson

Ask: In a linear relationship represented by y = mx + b, how do you find m, the

value for the constant rate of change, and the y-intercept, b? Use two sets of

x- and y-values to find m, the change in y over the change in x. Then substitute m, x, and y

in y = mx + b and solve to find b.

GUIDED PRACTICEEngage with the Whiteboard

For Exercise 2, have two students label the graph with the coordinates of the two

different points they choose to find the slope. Emphasize that using any two points

on the line will result in the same slope.

Avoid Common ErrorsExercise 2 Ask students to predict whether the slope will be positive or negative before

they do any calculations, and to explain their answer before finding the slope. Students may

enter y1 and y2 in a different order than x1 and x2 and get the opposite slope.

Exercise 3 Students may use the wrong independent variable. Have them consider

whether temperature is dependent upon chirps or chirps dependent upon temperature.

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129 Lesson 5.1

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5. Hari’s weekly allowance varies depending on the number of

chores he does. He received $16 in allowance the week he did

12 chores, and $14 in allowance the week he did 8 chores.

Write an equation for his allowance in slope-intercept form.

YOUR TURN

Guided Practice

1. Li is making beaded necklaces. For each necklace, she uses 27 spacers,

plus 5 beads per inch of necklace length. Write an equation to find how

many beads Li needs for each necklace. (Explore Activity)

a. input variable:

b. output variable:

c. equation:

2. Kate is planning a trip to the beach. She estimates her average speed

to graph her expected progress on the trip. Write an equation in

slope-intercept form that represents the situation. (Example 1)

Choose two points on the graph to find the slope.

m = y2 - y1

_____ x2 - x1 =

Read the y-intercept from the graph: b =

Use your slope and y-intercept values to

write an equation in slope-intercept form.

3. At 59 °F, crickets chirp at a rate of 76 times per minute, and at 65 °F, they

chirp 100 times per minute. Write an equation in slope-intercept form that

represents the situation. (Example 2)

4. Explain what m and b in the equation y = mx + b tell you about the graph

of the line with that equation.

ESSENTIAL QUESTION CHECK-IN??

Input variable: Output variable:

m = y2 - y1

_____ x2 - x1 = Use the slope and one of the ordered

pairs in y = mx + b to find b. = · + b; = b

Write an equation in slope-intercept form.

y = 0.5x + 10

the length of the necklace in inches

the total number of beads in the necklace

y = 5x + 27

y = -60x + 300

y = 4x - 160

The slope of the graphed line is m, and the y-intercept is b.

temperature

100 4 65 -160

chirps per minute

0 - 300 ______

5 - 0 = -300

_____ 5

= -60

100 - 76 _______

65 - 59 = 24

__ 6

= 4

300

Unit 2130

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My Notes

600 square feet for $750900 square feet for $1150

West Main St. Office Rentals

Offices for rent at convenient locations.

Monthly Rates:


Writing an Equation from a DescriptionYou can use information from a description of a linear relationship to find the

slope and y-intercept and to write an equation.

The rent charged for space in an office building is a

linear relationship related to the size of the space

rented. Write an equation in slope-intercept form

for the rent at West Main Street Office Rentals.

Identify the input and output

variables.

The input variable is the square

footage of floor space.

The output variable is the monthly rent.

Write the information given in the problem as ordered pairs.

The rent for 600 square feet of floor space is $750: (600, 750)

The rent for 900 square feet of floor space is $1150: (900, 1150)

Find the slope.

Find the y-intercept. Use the slope and one of the ordered pairs.

y = mx + b

750 = 4 _ 3

· 600 + b

750 = 800 + b

-50 = b

Substitute the slope and y-intercept.

y = mx + b

y = 4 _ 3

x - 50

Reflect 4. Without graphing, tell whether the graph of this equation rises or falls

from left to right. What does the sign of the slope mean in this context?

EXAMPLEXAMPLE 2

STEP 1

STEP 2

STEP 3

m = y2 - y1 ______ x2 - x1

= 1150 - 750 __________ 900 - 600

= 400 ____ 300

= 4 __ 3

STEP 4

STEP 5

8.F.4COMMONCORE


Substitute for y, m, and x.

Multiply.

Subtract 800 from both sides.


Substitute 4 __ 3

for m and -50 for b.

Slope is positive, so the graph rises from left to right.

This means that the rent increases as the square

footage increases.

129Lesson 5.1

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Multiple RepresentationsHave students discuss the different ways they

know to describe the slope of a line. Have them

write a list, such as this: rise over run, change in

y over change in x, y2 - y1

_____ x2 - x1

, slope, constant rate of

change. Have them sketch a line that has a

positive slope, another with a negative slope,

and a line with a slope of zero (horizontal), and a

line with an undefined slope (vertical).

Visual CuesDraw a linear graph and mark two points on it in

red. Then, on the same graph, mark two different

points on it in blue. Ask students to discuss

whether the slope of the line will be greater

between the two red points or between the two

blue points. Lead them to see that the slope of a

given linear graph is always the same between

any two points on the line.

Additional ResourcesDifferentiated Instruction includes:

• Reading Strategies

• Success for English Learners ELL

• Reteach

• Challenge PRE-AP

DIFFERENTIATE INSTRUCTION


Math Trainer


Personal

Online homework

assignment available

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Evaluate

GUIDED AND INDEPENDENT PRACTICE

Concepts & Skills Practice

Explore Activity

Writing an Equation in Slope-Intercept Form

Exercise 1

Example 1

Writing an Equation from a Graph

Exercises 2, 7–9, 12–15

Example 2

Writing an Equation from a Description

Exercises 3, 5–6, 10–11


• Leveled Practice worksheets

Lesson Quiz available online

5.1 LESSON QUIZ

1. Lee charges $3 for a basket and $2.50 for each pound of fruit picked at the orchard. Write an equation in y = mx + b form for the total cost of x pounds of fruit from the orchard.

2. A camp charges families a fee of $625 per month for one child and a certain amount more per month for each additional child. Use the graph to write an equation in slope-intercept form to represent the amount a family with x addi-tional children would pay.

3. Identify the y-intercept in question 2 above. Tell what the y-intercept means in this context.

4. A driving range charges $4 to rent a golf club plus $2.75 for every bucket of golf balls you hit. Write an equation that shows the total cost c of hitting b buckets of golf balls.

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Exercise Depth of Knowledge (D.O.K.) Mathematical Practices

5–6 2 Skills/Concepts MP.4 Modeling

7–9 2 Skills/Concepts MP.6 Precision


12–15 2 Skills/Concepts MP.2 Reasoning

16–18 3 Strategic Thinking MP.3 Logic

COMMONCORE

8.F.4COMMONCORE

COMMONCORE 8.F.4

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Answers1. y = 2.50x + 3

2. y = 225x + 625

3. b = 625, the fixed fee for one child

4. c = 2.75b + 4

131 Lesson 5.1

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The graph shows the activity in a savings account.

12. What was the amount of the initial deposit that started this

savings account?

13. Find the slope and y-intercept of the graphed line.

14. Write an equation in slope-intercept form for the activity in this savings

account.

15. Explain the meaning of the slope in this graph.

16. Communicate Mathematical Ideas Explain how you decide which part

of a problem will be represented by the variable x, and which part will be

represented by the variable y in a graph of the situation.

17. Represent Real-World Problems Describe what would be true about

the rate of change in a situation that could not be represented by a

graphed line and an equation in the form y = mx + b.

18. Draw Conclusions Must m, in the equation y = mx + b, always be a

positive number? Explain.

FOCUS ON HIGHER ORDER THINKING

$1000

m = 500; b = 1000

y = 500x + 1000

The amount of money in the savings account increases

by $500 each month.

Examine the problem and decide what quantity you

start with, or the input, and what quantity you are trying

to find, or the output. Use the input quantity for x and

the output quantity for y.

The rate of change would not be constant. Using

different pairs of points in the slope formula would

give you different results.

No. A negative number for m means the dependent

variable is decreasing as the independent variable

increases, so the graph falls from left to right.

Unit 2132

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Independent Practice5.1

5. A dragonfly can beat its wings 30 times per second. Write an equation in

slope-intercept form that shows the relationship between flying time in

seconds and the number of times the dragonfly beats its wings.

6. A balloon is released from the top of a platform that is 50 meters tall.

The balloon rises at the rate of 4 meters per second. Write an equation in

slope-intercept form that tells the height of the balloon above the ground

after a given number of seconds.

The graph shows a scuba diver’s ascent over time.

7. Use the graph to find the slope of the line. Tell what the

slope means in this context.

8. Identify the y-intercept. Tell what the y-intercept means in

this context.

9. Write an equation in slope-intercept form that represents the diver’s

depth over time.

10. The formula for converting Celsius temperatures to Fahrenheit

temperatures is a linear equation. Water freezes at 0 °C, or 32 °F, and it

boils at 100 °C, or 212 °F. Find the slope and y-intercept for a graph that

gives degrees Celsius on the horizontal axis and degrees Fahrenheit on

the vertical axis. Then write an equation in slope-intercept form that

converts degrees Celsius into degrees Fahrenheit.

11. The cost of renting a sailboat at a lake is $20 per hour plus $12 for

lifejackets. Write an equation in slope-intercept form that can be used to

calculate the total amount you would pay for using this sailboat.

8.F.4COMMONCORE

y = 20x + 12

y = 4x + 50

y = 0.125x - 10

y = 30x

m = 9 _ 5

; b = 32; y = 9 _ 5

x + 32 where y = °F and x = °C

m = 0.125; the diver ascends at a rate of

0.125 m/s

-10; the diver starts 10 meters below the water’s surface.

131Lesson 5.1

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Activity available online my.hrw.comEXTEND THE MATH PRE-AP

Activity Give each pair of students a sheet of graph paper marked with the x- and

y-axes, and pencils or pieces of wire or spaghetti to use for the lines to make quick

graphs. One student calls out a slope (for example, m = 0 or 1, or 0.5, or 5, or -1) and

the other places the line on the graph in the approximate position going through the

origin, (for example: horizontal, bisecting QI, less steep than y = x, steeper than y = x,

or bisecting QII).

Then have one student call out a slope and a y-intercept and the other place the line.

If students have trouble, have them place a line with the same slope through the

origin and then move it to go through the y-intercept.


5.2L E S S O N

Writing Linear Equations from a Table


ADDITIONAL EXAMPLE 1The Dailey family uses maple sap to make syrup. The table shows the temperature of the sap as it heats. Graph the data, and find the slope and y-intercept from the graph. Then write the equation for the graph in slope-intercept form.

Time (h) 0 1 2 3 4

Temp (°F) 38 83 128 173 218

°F = 45h + 38

EngageESSENTIAL QUESTION

How do you write an equation to model a linear relationship given a table? Use pairs of input and output values to determine the slope m and the y-intercept b in the equation y = mx + b.

Motivate the LessonAsk: Have you ever recorded measurements in a table? Tables are often used in science labs to record temperatures, weights, and other measures. Graphing the information in a table provides another way to look at the data.

ExploreHave a student provide a number between 10 and 20. Then have students make a table with integer x-values from 1 through 10. Fill in the y-values so that x and y for each column add to the selected number. Ask students to write an equation for the relationship.

ExplainEXAMPLE 1

Questioning Strategies Mathematical Practices • Describe informally what is happening in this experiment. The temperature of a fish tank is measured at the beginning as 82 degrees Fahrenheit. This temperature gradually falls over 5 hours at a steady rate until it reaches 72 degrees Fahrenheit.

• Could you find the equation from the table without drawing a graph? Explain. Yes; choose two points from the table to find the slope, and use (0, 82) to find the y-intercept.

Engage with the WhiteboardHave students draw lines to extend the x-axis and the graphed line. Where will the line meet the x-axis if the rate of cooling stays the same? Students should check their

prediction by substituting 0 for y in the equation and solving for x (41 hours).

YOUR TURNAvoid Common ErrorsRemind students to refer to the slope formula on the previous page when calculating the slope. They should write out the formula and substitute the coordinates of two points.

EXAMPLE 2Questioning Strategies Mathematical Practices • How can you find the y-intercept when it is not given as one of the points in the table? Substitute m and any point in the table as (x, y) to solve for b in y = mx + b.

• What does the y-intercept represent in this situation? the base price without minutes

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Common Core StandardsThe student is expected to:

Functions—8.F.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.


MP.4 Modeling

COMMONCORE

COMMONCORE

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ESSENTIAL QUESTION

Graphing from a Table to Write an EquationYou can use information from a table to draw a graph of a linear relationship

and to write an equation for the graphed line.

The table shows the temperature of a fish tank during an experiment.

Graph the data, and find the slope and y-intercept from the graph. Then

write the equation for the graph in slope-intercept form.

Time (h) 0 1 2 3 4 5

Temperature (°F) 82 80 78 76 74 72

Graph the ordered pairs from

the table (time, temperature).

Draw a line through the points.

Choose two points on the graph to

find the slope: for example, choose

(0, 82) and (1, 80).

m = y2 - y1 ______ x2 - x1

m = 80 - 82 _______ 1 - 0

m = -2 ___ 1

= -2

Read the y-intercept from the graph.

b = 82

Use these slope and y-intercept values to write an equation

in slope-intercept form.

y = mx + by = -2x + 82

EXAMPLEXAMPLE 1

STEP 1

STEP 2

STEP 3

STEP 4

STEP 5

How do you write an equation to model a linear relationship given a table?

L E S S ON

5.2Writing Linear Equations from a Table

Which variable in the equation y = mx + b shows

the initial temperature of the fish tank at the beginning

of the experiment?


8.F.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value... . Interpret the rate of change and initial value… .

COMMONCORE

8.F.4COMMONCORE

Use the slope formula.

Substitute (0, 82) for (x1, y1) and (1, 80) for (x2, y2).

Simplify.

The variable b, or the y-intercept, shows the initial temperature.

133Lesson 5.2

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5 10 15 20

Time (s)

100,000

200,000

300,000

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m3)

Water Releasedfrom Hoover Dam

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Animated Math

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1. The table shows the volume of water released by Hoover Dam over

a certain period of time. Graph the data, and find the slope and

y-intercept from the graph. Then write the equation for the graph

in slope-intercept form.

Water Released from Hoover Dam

Time (s) Volume of water (m3)

5 75,000

10 150,000

15 225,000

20 300,000

YOUR TURN

Writing an Equation from a TableThe information from a table can also help you to write the equation that

represents a given situation without drawing the graph.

Elizabeth’s cell phone plan lets her choose how many minutes are included

each month. The table shows the plan’s monthly cost y for a given number

of included minutes x. Write an equation in slope-intercept form to

represent the situation.

Minutes included, x 100 200 300 400 500

Cost of plan ($), y 14 20 26 32 38

Notice that the change in cost is the same for each increase of

100 minutes. So, the relationship is linear. Choose any two ordered

pairs from the table to find the slope.

m = y2 - y1 ______ x2 - x1

= (20 - 14)

__________ (200 - 100)

= 6 ____ 100

= 0.06

Find the y-intercept. Use the slope and any point from the table.

y = mx + b 14 = 0.06 ⋅ 100 + b 14 = 6 + b 8 = b

Substitute the slope and y-intercept.

y = mx + b y = 0.06x + 8

EXAMPLE 2

STEP 1

STEP 2

STEP 3

Which variable in the equation y = mx + b tells you the volume of water

released every second from Hoover Dam?


8.F.4COMMONCORE


Substitute for y, m, and x.

Multiply.

Subtract 6 from both sides.


Substitute 0.06 for m and 8 for b.

m = 15,000; b = 0; y = 15,000xthe slope, or m

Unit 2134

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Math BackgroundA table of values generated from measurements

such as temperature will often not have a

constant rate of change and cannot be precisely

described by a linear relationship. In many cases,

however, the ordered pairs can be plotted and a

line of best fit can be drawn through them. The

slope and y-intercept of the line of best fit can be

determined and used to write an equation that

describes the relationship.


This lesson provides an opportunity to address

this Mathematical Practices standard. It calls for

students to apply mathematics to problems

arising in everyday life, society, and the work-

place. Students apply what they know about

linear relationships to problems arising from an

experiment measuring changes in temperature,

measuring the flow of water, and examining the

cost of a cell-phone plan. They relate details of

everyday relationships to the formal summary of

a linear equation in mathematical terms.

Writing Linear Equations from a Table 134

Focus on ReasoningHow can you tell, without drawing the graph, where the line of the graph would start and

whether it rises or falls from left to right? Would the line be more or less steep than the

graph for y = x? Explain. The y-intercept tells where the line starts (0, 8); the positive slope

of 0.06 tells that the line rises. Since the slope is less than one, the graph is less steep than

y = x.

Talk About ItCheck for Understanding

Ask: If the base rate increased from $8 to $12 but the per minute rate stayed the

same at $0.06 per minute, does the slope change? The y-intercept? Write a new

equation for a base fee of $12. no; yes; y = 0.06x + 12

YOUR TURNConnect to Daily Life

Students may not understand how sales commission works. Give one or more examples,

using points from the table to make this clear, explain: The salesperson is paid $250 per

week regardless of how many computers are sold. If 10 computers are sold, the total pay is

$250 plus $75 for each of the 10 computers.

Elaborate

Talk About ItSummarize the Lesson

Ask: How can you use values given in a table of a linear relationship to draw a

graph and write an equation in y = mx + b form? Use values from the table to

draw a graph. Use two points to find the slope. To find the y-intercept, either read it from

the graph or substitute m and any point in the table as (x, y) to solve for b in y = mx + b.


For Exercise 1, have students mark on the graph several points that are not in the

table, such as the points that tell how much is left on the pass after 3 rides and after

14 rides. Label the points with their coordinates.

Focus on CommunicationHave students informally describe the information that is shown in the table for

Exercises 2–5, and what it might mean for someone who is climbing mountains. It gets

colder the higher you climb.

Avoid Common ErrorsExercise 5 To avoid calculating an incorrect value, remind students that they can use the

table to predict, or check, the reasonableness of the temperature. They should see that the

temperature must be between 35 °F and 43 °F.


ADDITIONAL EXAMPLE 2Zara made an initial deposit to a bank

account and then added a fixed

amount every week. The table shows

the money in her account. Write an

equation in slope-intercept form to

represent the situation.

Number of

weeks, x1 2 3 4 5

Balance ($), y 140 160 180 200 220

y = 20x + 120

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Animated MathWrite Equations for

Lines

Students explore changing the slope

and y-intercept of a line in a dynamic

graphing tool.

135 Lesson 5.2

4

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4 8 12 16 20Number of rides

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1. Jaime purchased a $20 bus pass. Each time he rides the bus, a certain amount is deducted from the pass. The table shows the amount, y, left on his pass after x rides. Graph the data, and find the slope and y-intercept from the graph or from the table. Then write the equation for the graph in slope-intercept form. (Example 1)

Number of rides, x 0 4 8 12 16

Amount left on pass ($), y 20 15 10 5 0

The table shows the temperature (y) at different altitudes (x). This is a linear relationship. (Example 2)

Altitude (ft), x 0 2,000 4,000 6,000 8,000 10,000 12,000

Temperature (°F), y 59 51 43 35 27 19 11

2. Find the slope for this relationship. 3. Find the y-intercept for this relationship.

4. Write an equation in slope-intercept form that represents this relationship.

5. Use your equation to determine the temperature at an altitude of 5000 feet.

6. Describe how you can use the information in a table showing a linear relationship to find the slope and y-intercept for the equation.

ESSENTIAL QUESTION CHECK-IN???

Guided Practice

m = -1.25; b = 20; y = -1.25x + 20

m = 51 - 59 _______ 2000 - 0 = -8 ____ 2000 = -0.004 b = 59

y = -0.004x + 59

Use two data points from the table to find the slope, and

then locate the point on the table where x is 0 to identify the

y-intercept.

y = -0.004(5000) + 59 = 39 °F

Unit 2136

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Math TrainerOnline Assessment

and Intervention

Personal

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Reflect2. What is the base price for the cell phone plan, regardless of how many

minutes are included? What is the cost per minute? Explain.

3. What If? Elizabeth’s cell phone company changes the cost of her plan as shown below. Write an equation in slope-intercept form to represent the situation. How did the plan change?

Minutes included, x 100 200 300 400 500

Cost of plan ($), y 30 35 40 45 50

4. A salesperson receives a weekly salary plus a commission for each computer sold. The table shows the total pay, p, and the number of computers sold, n. Write an equation in slope-intercept form to represent this situation.

Number of computers sold, n 4 6 8 10 12

Total pay ($), p 550 700 850 1000 1150

5. To rent a van, a moving company charges $40.00 plus $0.50 per mile. The table shows the total cost, c, and the number of miles driven, d. Write an equation in slope-intercept form to represent this situation.

Number of miles driven, d 10 20 30 40 50

Total cost ($), c 45 50 55 60 65

YOUR TURN

Explain the meaning of the slope and y-intercept of

the equation.


$8; $0.06; in the equation for the table, y = 0.06x + 8, the

y-intercept, 8, is an initial amount that does not depend

on the rate of change. The slope, 0.06, represents the

rate of change, which is the per-minute cost.

y = 0.05x + 25; The fee was increased by $17, and the

per-minute rate was decreased by 1 cent.

p = 75n + 250

c = 0.50d + 40

The slope, or m, is the commission of $75 for each computer sold and the y-intercept, or b, is the base salary of $250.

135Lesson 5.2

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8_MFLESE056718_U2M05L2.indd 135 13/11/13 1:57 PM

TechnologyStudents can see the effect of increasing the absolute value of m, the slope, by graphing these equations on the same screen of a graphing calculator: y = 0.5x, y = 1x, y = 1.5x, y = 2x, y = 5x. They can also see the effect of increasing the value of b, the y-intercept, by graphing these equations on the same screen of a graphing calculator: y = x - 2, y = x + 1, y = x + 3, y = x + 5. All of these equations have a positive slope. Ask students to conjecture how the graph would change if they had a negative slope. They can verify their conjectures by graphing y = x + 3 and y = -x + 3.

Communicating MathHelp students be precise in their use of math-ematics language by discussing the difference between “What is the y-intercept?” and “At what point on the graph does the line cross the y-axis?” The y-intercept is a single value, b, the y-coordinate of the point, which is (0, b).

Students often ask why m is used to represent the slope. Mathematicians do not know a definite answer to this, although m has been used for slope for centuries. Some students like to remember the meaning of m by relating it to mountain slope.

Additional ResourcesDifferentiated Instruction includes: • Reading Strategies • Success for English Learners ELL

• Reteach • Challenge PRE-AP




Math Trainer


Personal

Online homework


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Evaluate



Example 1

Graphing from a Table to Write an Equation

Exercises 1, 7–8

Example 2

Writing an Equation from a Table

Exercises 2–5, 9–13




5.2 LESSON QUIZ

The table shows Eli’s distance from

home as he rides his bike at a steady

rate after meeting a friend.

Time (h) 1 2 3 4 5

Distance (mi) 14 26 38 50 62

1. Graph the data.

2. Find the slope and y-intercept.

3. How fast does Eli ride his bike, in miles per hour?

4. Write an equation in y = mx + b form that represents the miles, y, that Eli goes in x hours.

5. If each distance were halved, what would the equation be? What would it mean in terms of the problem? Explain.

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8.F.4COMMONCORE

COMMONCORE 8.F.4




12 2 Skills/Concepts MP.4 Modeling

13 2 Skills/Concepts MP.3 Logic


COMMONCORE

Exercise 9 combines concepts from the Common Core cluster “Use

functions to model relationships between quantities.”

Answers1.

2. m = 12; b = 2

3. 12 mi/h

4. y = 12x + 2

5. y = 6x + 1; Eli starts 1 mile from home and goes 6 mi/h.

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137 Lesson 5.2

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12. Finance Desiree starts a savings account with $125.00. Every month, she

deposits $53.50.

a. Complete the table to model the situation.

Month, x

Amount in

Savings ($), y

b. Write an equation in slope-intercept form that shows how much

money Desiree has in her savings account after x months.

c. Use the equation to find how much money Desiree will have in

savings after 11 months.

13. Monty documented the amount of rain his farm received on a monthly

basis, as shown in the table.

Month, x 1 2 3 4 5

Rainfall (in.), y 5 3 4.5 1 7

a. Is the relationship linear? Why or why not?

b. Can an equation be written to describe the amount of rain? Explain.

14. Analyze Relationships If you have a table that shows a linear

relationship, when can you read the value for b, in y = mx + b, directly

from the table without drawing a graph or doing any calculations? Explain.

15. What If? Jaíme graphed linear data given in the form (cost, number).

The y-intercept was 0. Jayla graphed the same data given in the form

(number, cost). What was the y-intercept of her graph? Explain.


y = 53.50x + 125.00

0

125.00

1

178.50

2

232.00

3

285.50 339.00

No, the change between weeks is constant, but the

change in the amount of rain is not constant.

No; there is no apparent pattern in the table.

If there is a point (0, y) in the table, then y = b because

at the y-intercept the value of x is zero.

0; Jaíme’s graph contained (0, 0). Since Jayla’s data were

the same, but with x and y switched, her graph also

contained (0, 0).

$713.50

4

Unit 2138

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Name Class Date

Independent Practice

7. The table shows the costs of a large cheese pizza with toppings

at a local pizzeria. Graph the data, and find the slope and

y-intercept from the graph. Then write the equation for the

graph in slope-intercept form.

8. The table shows how much an air-conditioning repair company

charges for different numbers of hours of work. Graph the data,

and find the slope and y-intercept from the graph. Then write the

equation for the graph in slope-intercept form.

Number of hours (h), t 0 1 2 3 4 5

Amount charged ($), A 50 100 150 200 250 300

9. A friend gave Ms. Morris a gift card for a

local car wash. The table shows the linear

relationship of how the value left on the

card relates to the number of car washes.

a. Write an equation that shows the number of dollars left on the card.

b. Explain the meaning of the negative slope in this situation.

c. What is the maximum value of x that makes sense in this context? Explain.

The tables show linear relationships between x and y. Write an equation in

slope-intercept form for each relationship.

10.

11.

5.2

Number of toppings, t 0 1 2 3 4 5

Total cost ($), C 8 10 12 14 16 18

Number of car washes, x 0 8 12

Amount left on card ($), y 30 18 12

x -2 -1 0 2

y -1 0 1 3

x -4 1 0 6

y 14 4 6 -6

8.F.4COMMONCORE

m = 2; b = 8; C = 2t + 8

y = -1.5x + 30

y = x + 1 y = -2x + 6

The amount of dollars left decreases as the number

of car washes increases.

20; after 20 washes there is no money left on the card

m = 50; b = 50; A = 50t + 50

137Lesson 5.2

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Activity To extend the relationships between values in a table, points on a graphed

line, and ordered pairs that satisfy a linear equation, have students draw graphs that

use the other three quadrants. For example, ask them to create a table of ordered

pairs for the equations y = x + 3 and y = -x - 2 using both negative and positive

values of x. Have them graph the points in the table and draw a line through the

points. Discuss situations that might be represented by such equations, for example,

charging purchases, debt, temperatures below zero, diving below sea level.


5.3L E S SON

Linear Relationships and Bivariate Data

Engage

ESSENTIAL QUESTIONHow can you contrast linear and nonlinear sets of bivariate data? The graph of a linear set of data is a single line; the graph of a nonlinear set of data is not a single line.

Motivate the LessonAsk: Does a table of paired values always graph as a single line?

ExploreIf you made a table showing the side length of a square and its area and then graphed

those points, would the relationship be linear?

Explain

EXAMPLE 1Questioning Strategies Mathematical Practices • According to the line on the graph, what is the height of the handrail from the floor where

the stairway starts? Does that height satisfy the equation? Explain. 3 ft; yes, (0, 3) satisfies

the equation because 3 = 0.8(0) + 3.

• Would you get a different value for the rate of change, or slope, if you chose two other

points? Explain. No; the slope is constant for a linear relationship.

Engage with the Whiteboard On the graph in Step 1, have students plot the point where the graph crosses the

y-axis and label the point with its coordinates.

YOUR TURNAvoid Common Errors In Exercise 1, students may misread the second point on the graph as (10, 50). Remind

students to determine unlabeled values on an axis by looking at both the closest lesser axis

label and the closest greater axis label.

Talk About It Check for Understanding

Ask: In Exercise 1, why does the line not go through the origin? There is an initial

fee at the beginning, regardless of the number of minutes.

EXAMPLE 2Questioning Strategies Mathematical Practices • Why does the graph of the line not go through the origin? there is an initial cost of $3

• How would you describe in words the cost of taxi rides shown in the graph? The cost is $3

plus $2 for every mile traveled.


ADDITIONAL EXAMPLE 1The charge for a cheese pizza changes

as the number of toppings changes.

Show that the relationship is linear, and

then find the equation for the

relationship.

A line passes through all the graphed

points so the relationship is linear.

y = 1.5x + 8

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The student is expected to:

Statistics and Probability—8.SP.1

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.


Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.


Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.


MP.6 Precision

COMMONCORE

COMMONCORE

COMMONCORE

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Find the equation of each linear relationship.

1. 2.

YOUR TURN

Hours (x) Number of units (y)

2 480

15 3,600

24 5,760

30 7,200

48 11,520

55 13,200

Making PredictionsYou can use an equation of a linear relationship to predict a value between data points that you already know.

The graph shows the cost for taxi rides of different distances. Predict the cost of a taxi ride that covers a distance of 6.5 miles.

Write the equation of the linear relationship.

(2, 7) and (6, 15)

y = mx + b

15 = 2(6) + b

15 = 12 + b

3 = b

The equation of the linear relationship is y = 2x + 3.

You can check your equation using another point on the graph. Try (8, 19). Substituting gives 19 = 2(8) + 3. The right side simplifies to 19, so 19 = 19. ✓

EXAMPLE 2

STEP 1

m = 15 - 7 ______ 6 - 2

= 8 __ 4

= 2

8.SP.3COMMONCORE

Select two points.

Fill in values for x, y, and m.

Simplify.

Solve for b.

Simplify.

Calculate the rate of change.

y = 4x + 20 y = 240x

Unit 2140

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How can you contrast linear and nonlinear sets of bivariate data?

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Finding the Equation of a Linear RelationshipYou can use the points on a graph of a linear relationship to write an equation for the relationship. The equation of a linear relationship is y = mx + b, where m is the rate of change, or slope, and b is the value of y when x is 0.

A handrail runs alongside a stairway. As the horizontal distance from the bottom of the stairway changes, the height of the handrail changes. Show that the relationship is linear, and then find the equation for the relationship.

Show that the relationship is linear.

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ft)

(5, 7)(10, 11)

(15, 15)

(25, 23)(20, 19)

O

Write the equation of the linear relationship.

Choose two points Choose a point and use the slope to

to find the slope. substitute values for x, y, and m.

(5, 7) and (25, 23) y = mx + b

7 = 0.8(5) + b

7 = 4 + b

3 = b

The equation of the linear relationship is y = 0.8x + 3.

EXAMPLEXAMPLE 1

STEP 1

STEP 2

m = 23 - 7 ______ 25 - 5

= 16 ___ 20

= 0.8

L E S S O N

5.3 Linear Relationships and Bivariate Data

EssEntial QuEstion

What does the slope of the equation represent in this situation? What does

the y-intercept represent?


8.SP.1

Construct and interpret scatter plots for bivariate measurement data... . Describe patterns such as...linear association, and nonlinear association. Also 8.SP.2, 8.SP.3

COMMONCORE

8.SP.2COMMONCORE

All of the points (5, 7), (10, 11), (15, 15), (20, 19), and (25, 23) lie on the same line, so the relationship is linear.

Sample answer: The slope represents the increase in height of the handrail per foot of horizontal distance. The y-intercept represents the height of the handrail at the bottom of the stairway.

139Lesson 5.3

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Math BackgroundSimple interest is paid on the amount of an investment or loan. The amount of interest does not change if the interest earned is added to the original amount or if interest charges increase the amount of a debt. With compound interest, interest is paid on the accumulated interest as well as the original amount. In the case of an investment, that means the interest is also earning interest (unless it is withdrawn from the investment), and in the case of a loan, the debtor pays interest on the interest unless loan pay-ments are large enough to offset the interest charges.


This lesson provides an opportunity to address this Mathematical Practices standard. It calls for students to precisely communicate mathemati-cal ideas and reasoning. Students use tables and graphs and equations to represent linear and nonlinear relationships. Students use these multiple representations to compare and contrast linear and nonlinear relationships and to communicate their understanding.

Linear Relationships and Bivariate Data 140

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ADDITIONAL EXAMPLE 2The graph shows the distance of a train

from a landmark as it travels at a

constant speed. Use the graph to

predict what the distance will be after

7.5 hours. 400 miles

Engage with the Whiteboard Have students extend the graph and predict the cost of various rides greater

than 8 miles.

Focus on Modeling Mathematical PracticesHave students discuss why extending the line to the left by graphing this relationship in

other quadrants, such as II or III, does not make sense in the context of the real-world

problem.

YOUR TURNTalk About It

Check for Understanding

Ask: What is the hourly pay graphed in this relationship? $15

EXPLORE ACTIVITYConnect Vocabulary ELL Help all students to understand the term bivariate data by first pronouncing it slowly

so that they hear all the syllables. Then circle the bi- and connect it to a bi-cycle (2 wheels),

and underline vari- and connect it to variable or change. So bivariate data is data in pairs of

2 variables that change in some way. The change is constant (linear) or is not constant

(nonlinear).

Questioning Strategies Mathematical Practices • Look at the table. How are the two options alike? The amount at the beginning is the

same. How do they differ? The rate at which they increase is not the same; Option B grows

faster than Option A.

• Read the problem. How are the two options alike? Both have interest rates of 5%. Which

words tell you the important difference between these two options? simple interest and

compounded annually

Engage with the Whiteboard Have students extend the coordinate plane and the two lines, and make conjectures

about the total amounts after 25 and 30 years. Students should be able to determine

that for Option A, simple interest, there will be $450 after 25 years and $500 after 30 years.

Students will only be able to estimate the values for Option B; the exact values are $677.27

after 25 years and $864.39 after 30 years.

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Contrasting Linear and Nonlinear DataBivariate data is a set of data that is made up of two paired variables. If the

relationship between the variables is linear, then the rate of change (slope) is

constant. If the graph shows a nonlinear relationship, then the rate of change

varies between pairs of points.

Andrew has two options in which to invest $200. Option A earns

simple interest of 5%, while Option B earns interest of 5% compounded

annually. The table shows the amount of the investment for both options

over 20 years. Graph the data and describe the differences between the

two graphs.

Graph the data from the table

for Options A and B on the same

coordinate grid.

Find the rate of change between pairs of

points for Option A and classify the relationship.

Option A Rate of Change

(0, 200) and (5, 250) m = 250 - 200________ 5 - 0

=

(5, 250) and (10, 300)

(10, 300) and (15, 350)

The rate of change between the data values is , so

the graph of Option A shows a relationship.

STEP 1

STEP 2

Option A Option B

Year, x Total ($) Total ($)

0 200.00 200.00

5 250.00 255.26

10 300.00 325.78

15 350.00 415.79

20 400.00 530.66

EXPLORE ACTIVITY 8.SP.1COMMONCORE

10

constant

linear

m = 300 - 250 ________

10 - 5 = 10

m = 350 - 300 ________

15 - 10 = 10

Unit 2142

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Use your equation from Step 1 to predict the cost of a 6.5-mile taxi

ride.

y = 2x + 3

y = 2(6.5) + 3

y = 16

A taxi ride that covers a distance of 6.5 miles will cost $16.

Reflect3. What If? Suppose a regulation changes the cost of the taxi ride to

$1.80 per mile, plus a fee of $4.30. How does the price of the 6.5 mile

ride compare to the original price?

4. How can you use a graph of a linear relationship to predict an unknown

value of y for a given value of x within the region of the graph?

5. How can you use a table of linear data to predict a value?

STEP 2

Paulina’s income from a job that pays her

a fixed amount per hour is shown in the graph.

Use the graph to find the predicted value.

6. Income earned for working 2 hours

7. Income earned for working 3.25 hours

8. Total income earned for working for

five 8-hour days all at the standard rate

YOUR TURN

Substitute x = 6.5.

Solve for y.

Sample answer: Read the y-value of the point on the

graph that corresponds to the given x-value.

It is the same. The new equation is y = 1.8x + 4.3,

so y = 1.8(6.5) + 4.3 = 11.7 + 4.3 = 16.

$30

$48.75

$600

Sample answer: Use the table of values to write

an equation and then use the equation to make

the prediction.

141Lesson 5.3

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ModelingTo help students understand the difference

between amount of change and rate of change,

have them make a table (similar to the one in

the Explore Activity) showing the side of a

square, its perimeter, and its area. Then have

them graph these two sets of points for P and A.

Ask students to compare the resulting graphs.

Help them realize that, beyond a side of 4, the

area increases much faster than the perimeter.

TechnologyHave students make the comparison described

in Modeling by using a graphing calculator to

graph and compare y = 4x and y = x2 (for the

perimeter and area of a square with side x).

Use a window that shows the first quadrant

only and lead students to see that when x has

a value greater than 4, the y-value of the curve

is increasing much faster than that of the

straight line.


• Reading Strategies

• Success for English Learners ELL

• Reteach

• Challenge PRE-AP



Elaborate

Talk About It Summarize the Lesson

Ask: If you were given two graphed lines, one of a linear set of data and the other

of a nonlinear set of data, how could you tell which line goes with which set of

data? The graph of the linear set of data will be a single line.


In Exercises 1–2, have students draw a line through the points and the y-axis. Have

them label the y-intercept. In Exercises 4–5, have students attempt to draw a line

through all the points.

Avoid Common Errors Exercise 2 Some students may attempt to draw a line through all of the points and the

origin and decide that the relationship is not linear. Remind students that the origin is not

necessarily part of the graph unless it is marked as such.

Exercises 4–5 Remind students that the line must contain all of the points and that a line

is always straight.

143 Lesson 5.3

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Guided Practice

Use the following graphs to find the equation of the linear relationship. (Example 1)

1. 2.

3. The graph shows the relationship between

the number of hours a kayak is rented

and the total cost of the rental. Write an

equation of the relationship. Then use the

equation to predict the cost of a rental that

lasts 5.5 hours. (Example 2)

Does each of the following graphs represent a linear relationship?

Why or why not? (Explore Activity)

4. 5.

6. How can you tell if a set of bivariate data shows a linear relationship?

ESSENTIAL QUESTION CHECK-IN??

Yes, because the graph has a

constant rate of change.

Sample answer: Graph the data points. If the points lie

along a straight line, the data is linear.

y = 30x

y = 20x + 30; $140

No, because the graph does not

have a constant rate of change.

y = 2.5x + 2

Unit 2144

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Find the rate of change between pairs of points for Option B and

classify the relationship.

Option B Rate of Change

(0, 200) and

(5, 255.26)m = 252.26 - 200

__________ 5 - 0

≈

(5, 255.26) and

(10, 325.78)

(10, 325.78) and

(15, 415.79)

The rate of change between the data values is ,

so the graph of Option B shows a relationship.

Reflect9. Why are the graphs drawn as lines or curves and not discrete points?

10. Can you determine by viewing the graph if the data have a linear or

nonlinear relationship? Explain.

11. Draw Conclusions Find the differences in the account balances to the

nearest dollar at 5 year intervals for Option B. How does the length of

time that money is in an account affect the advantage that compound

interest has over simple interest?

STEP 3

10.5

not constant

nonlinear

Sample answer: Yes. If all of the data lie on a line, then

the data have a linear relationship. If the graph is a

curve (or not a line), then the data have a nonlinear

relationship.

$55, $71, $90, $115; the longer that the money is in

the account, the greater the relative advantage of it

earning compound interest will be.

Sample answer: You can calculate the value of the

account at any time as the money grows. All points

along the line or curve are reasonable and possible.

m = 325.78 - 252.26 ____________

10 - 5 ≈ 14.7

m = 415.79 - 325.78 ____________

15 - 10 ≈ 18.0

143Lesson 5.3

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Evaluate



Example 1

Finding the Equation of a Linear Relationship

Exercises 1–2, 13

Example 2

Making Predictions

Exercises 3, 13

Explore Activity

Contrasting Linear and Nonlinear Data

Exercises 4–5, 7–13




5.3 LESSON QUIZ

x 1 2 3 4 5

y 5 8 12 14 17

1. Does this table represent a linear relationship? Why or why not?

2. Change one value in this table so that this table does represent a linear relationship.

Explain whether or not each

relationship is linear.

3. The number of inches in a student’s height and the height in feet

4. The radius of a circle and its area

5. An amusement park charges a price for admission and a price for each ride. Joe spent $10.50 and went on 4 rides. Janie spent $14.25 and went on 9 rides. Write a linear equation for the amount spent and predict the amount you would spend to go on 17 rides.

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7–8 2 Skills/Concepts MP.3 Logic


13 2 Skills/Concepts MP.4 Modeling

14 3 Strategic Thinking MP.3 Logic

15 3 Strategic Thinking MP.2 Using Structure


COMMONCORE

8.SP.1, 8.SP.2, 8.SP.3COMMONCORE

COMMONCORE 8.SP.1, 8.SP.2, 8.SP.3

Answers1. No, the slope is not the same between

2 and 3 and between 3 and 4.

2. Change y to 11 for x = 3.

3. Yes, the rate of change is a constant 12 inches per foot.

4. No; the rate of change is not constant.

5. a = 0.75r + 7.5; $20.25

145 Lesson 5.3

Work Area

14. Make a Conjecture Zefram analyzed a linear relationship, found that the slope-intercept equation was y = 3.5x + 16, and made a prediction for the value of y for a given value of x. He realized that he made an error calculating the y-intercept and that it was actually 12. Can he just subtract 4 from his prediction if he knows that the slope is correct? Explain.

15. Communicate Mathematical Ideas The table shows a linear relationship. How can you predict the value of y when x = 6 without finding the equation of the relationship?

16. Critique Reasoning Louis says that if the differences between the values of x are constant between all the points on a graph, then the relationship is linear. Do you agree? Explain.

17. Make a Conjecture Suppose you know the slope of a linear relationship and one of the points that its graph passes through. How could you predict another point that falls on the graph of the line?

18. Explain the Error Thomas used (7, 17.5) and (18, 45) from a graph to find the equation of a linear relationship as shown. What was his mistake?

y = 79x + b

45 = 79 · 18 + b

45 = 1422 + b, so b = −1377

The equation is y = 79x − 1377.


m = 45 − 7 _______ 18 − 17.5 = 38 ___ 0.5 = 79

x y

4 38

8 76

12 114

Yes; decreasing the value of b by 4 decreases the value

of mx + b by 4 because the value of mx stays the same.

No; the rate of change must be constant, and the rate

of change is the difference in y-values divided by the

difference in x-values.

Find the equation of the linear relationship using the

slope and given point, and then insert any x-value to

find a y-value on the graph of the line.

He calculated the slope incorrectly.

Sample answer: Because x = 6 lies

halfway between x = 4 and x = 8, the y-value should lie

halfway between the corresponding y-values.

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Name Class Date

Independent Practice5.3

7. 8.

Explain whether or not you think each relationship is linear.

9. the cost of equal-priced DVDs and the number purchased

10. the height of a person and the person’s age

11. the area of a square quilt and its side length

12. the number of miles to the next service station and the number of kilometers

13. Multistep The Mars Rover travels 0.75 feet in 6 seconds. Add the point to the graph. Then determine whether the relationship between distance and time is linear, and if so, predict the distance that the Mars Rover would travel in 1 minute.

Does each of the following tables represent a linear relationship? Why or why not?

Day Height (cm)

5 30

8 76.8

14 235.2

Number of boxes Weight (kg)

3 15

9 45

21 105

8.SP.1, 8.SP.2, 8.SP.3COMMONCORE

Yes, because the rate of change

Linear; the rate of change is the cost of a DVD, which is constant.

Not linear; the rate of growth is less as a person gets older.

Not linear; the rate of change in the area of a square

increases at a greater rate than the side length increases.

Linear; the rate of change between units is

the conversion factor, which is constant.

The relationship is linear; the equation of the

linear relationship is y = 0.125x, so the Mars

Rover would travel 7.5 feet in 60 seconds.

is constant.

No, because the rate of change is

not constant.

145Lesson 5.3

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8_MFLESE056718_U2M05L3.indd 145 13/11/13 1:57 PM


Activity The formula for compound interest is A = P ( 1 + r __ n ) nt , where A is the total

amount of money after n years, including interest, and P is the principal, or initial investment. The annual rate of interest as a decimal is r. If the interest is compounded annually, n = 1, since n is the number of times the interest is compounded per year. The total number of years is t. Use a calculator or an online compound interest calculator to find the amount after 4 years for Option B of the Explore Activity. $243.10



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MODULE QUIZ

5.1 Writing Linear Equations from Situations and GraphsWrite the equation of each line in slope-intercept form.

1. 2.

5.2 Writing Linear Equations from a TableWrite the equation of each linear relationship in slope-intercept form.

3. 4.

5.3 Linear Relationships and Bivariate DataWrite the equation of the line that connects each set of data points.

5. 6.

7. Write a real-world situation that can be represented by a linear

relationship.

ESSENTIAL QUESTION

x 0 100 200 300

y 1.5 36.5 71.5 106.5

x 25 35 45 55

y 94 88 82 76

y = 30x + 20

y = 0.35x + 1.5

y = 2 _ 3

x + 80 __

3 y = −3x + 140

y = −10x + 60

y = −0.6x + 109

Sample answer: A video game rental store charges

$3 per game and a membership fee of $10.

147Module 5

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Intervention Enrichment

Access Ready to Go On? assessment online, and receive

instant scoring, feedback, and customized intervention

or enrichment.

Online and Print Resources


• Reteach worksheets

• Reading Strategies ELL

• Success for English

Learners ELL


• Challenge worksheets

PRE-AP

Extend the Math PRE-AP

Lesson Activities in TE

Additional ResourcesAssessment Resources includes

• Leveled Module Quizzes

Ready to Go On?

Assess MasteryUse the assessment on this page to determine if students have

mastered the concepts and standards covered in this module.


Lesson Exercises Common Core Standards

5.1 1–2 8.F.4

5.2 3–4 8.F.4

5.3 5–6 8.SP.1, 8.SP.2

147 Unit 2 Module 5




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MODULE 5 MIXED REVIEW

Selected Response

1. An hourglass is turned over with the top part

filled with sand. After 3 minutes, there are 855

mL of sand in the top half. After 10 minutes,

there are 750 mL of sand in the top half.

Which equation represents this situation?

A y = 285x

B y = −10.5x + 900

C y = −15x + 900

D y = 75x

2. Which graph shows a linear relationship?

A

B

C

D

3. What are the slope and y-intercept of the

relationship shown in the table?

A slope = 0.05, y-intercept = 1,500

B slope = 0.5, y-intercept = 1,500

C slope = 0.05, y-intercept = 2,000

D slope = 0.5, y-intercept = 2,000

4. Which is the sum of 3.15 × 107 +

9.3 × 106? Write your answer in

scientific notation.

A 4.08 × 107

B 4.08 × 106

C 0.408 × 108

D 40.8 × 106

Mini-Task

5. Franklin’s faucet was leaking, so he put

a bucket underneath to catch the water.

After a while, Franklin started keeping

track of how much water was in the

bucket. His data is in the table below.

a. Is the relationship linear or nonlinear?

b. Write the equation for the relationship.

c. Predict how much water will be in the

bucket after 14 hours if Franklin doesn’t

stop the leak.

x 10,000 20,000 30,000

y 2,500 3,000 3,500

Hours 2 3 4 5

Quarts 5 6.5 8 9.5

Assessment Readiness

CC

BB

CC

AA

linear

23 quarts

y = 1.5x + 2

148 Unit 2

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Assessment Readiness Tip Encourage students to highlight or

underline important information to help them solve problems.

Item 1 Students need to focus on the information they will need

to find the slope and y-intercept of the equation. They should

highlight “3 minutes” and “855 mm of sand” and then “10 minutes”

and “750 mm of sand” to isolate what they need.

Item 5 Students, after reading the entire problem, should realize

that they only need the information in the table. They should

highlight the table and then use that information to answer the

questions.

Avoid Common ErrorsItem 2 Remind students that a linear relationship has every single

point in the graph on a line. Some students may incorrectly think

that a relationship that is mostly linear, as in answer choice C, is

acceptable.

Item 3 Some students might invert the slope since the x-values

are written in the top row of the table and the y-values are in the

bottom row. Remind the students to write out the formula for

finding the slope, and check that they are substituting from the

correct cells of the table.

Additional Resources

Assessment Readiness


Items Grade 8 Standards Mathematical Practices

1 8.F.4 MP.4

2 8.SP.1 MP.4

3 8.F.4 MP.2

4* 8.EE.4

5 8.F.4 MP.4

* Item integrates mixed review concepts from previous modules or a previous course.


Documents

Writing Linear Equations MODULE 5