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ESSENTIAL QUESTION

EXPLORE ACTIVITY

L E S SON

7.1Linear Relationships in the Form y = mx + b

How do you use tables and verbal descriptions to describe a linear relationship?

Discovering Linear RelationshipsMany real-world situations can be described by linear relationships.

Jodie pays $5 per ticket for a play and a one-time $2 convenience fee.

The table shows the total cost for different numbers of tickets.

Number of tickets 1 2 3 4 5

Total cost ($) 7 12 17 22 27

Describe a pattern for the row showing the number of tickets bought.

Describe the pattern for the row showing total cost.

Out of the total cost paid, how much does the actual ticket account for?

Reflect1. How much more than $5 does Jodie pay for one ticket? What

if she buys 5 tickets? Explain.

2. Analyze Relationships Describe the total amount paid in dollars

based on the number of tickets.

A

B

C

Expressions, equations, and relationships—7.7 The student applies mathematical process standards to represent linear relationships using multiple representations....

7.7

225Lesson 7.1

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Math On the Spotmy.hrw.com

Representing Linear Relationships Using a TableIn a linear relationship between two quantities,

as one quantity changes by a constant amount, the

other quantity also changes by a constant amount.

Proportional relationships are a special kind of linear

relationship.

A man’s shoe size is approximately 3 times his foot length in inches

minus 22. Use a table to represent the relationship between foot length

and shoe size.

Make a table. Label the top row Foot length (in.) and the bottom

row Shoe size.

Enter some foot lengths in inches. Since it is impossible to have a

negative shoe size, pick a foot length that when multiplied by 3

will be greater than 22.

Think:

3 × 7 = 21

3 × 8 = 24

Make a table relating foot length to show size.

Foot length (in.) 8 9 10 11 12

Shoe size 2 5 8 11 14

Reflect3. Analyze Relationships If someone had a foot length of 13 inches, how

can you use the table to determine his shoe size?

4. Critical Thinking Foot lengths do not have to be whole numbers. Give

an example of a non-whole number foot length you could have chosen

when filling in the table and find the approximate shoe size. What

should a person do if their foot length does not correspond to a whole

or half shoe size? Explain.

EXAMPLE 1

STEP 1

STEP 2

STEP 3

Math TalkMathematical Processes

7.7

Why is foot length on the top and shoe size on

the bottom? 21 - 22 = -1; this cannot be a man’s shoe size.

24 - 22 = 2; start the table at 8 inches.

Remember that a man’s shoe size is 3 times his foot length in inches minus 22.

Unit 4226

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My Notes

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5. Lea’s house is 350 meters from her friend’s house. Lea walks to her

friend’s house at a constant rate of 50 meters per minute. Use a table to

represent the relationship between time and the distance Lea has left to

walk to her friend’s house.

YOUR TURN

Luis will participate in a walkathon for

charity. He received a pledge from his

aunt, and the table shows the relationship

between the miles walked by Luis and the

amount his aunt pledged.

Use the table to give a verbal description

of the relationship between miles walked

and amount pledged.

Miles walked 1 2 3 4 5

Amount pledged ($) 31.50 33 34.50 36 37.50

Look for patterns in the different values for miles walked and

amount pledged.

33 - 31.50 = 1.50

2 - 1 = 1

1.50 ____

1 = 1.50

In the table, each value for the number of miles walked is

1 greater than the previous one, and each amount pledged is

$1.50 greater than the previous one.

EXAMPLEXAMPLE 2

STEP 1

Representing Linear Relationships Using a Verbal DescriptionJust as you can create a table given a verbal description of a linear relationship,

you can also create a verbal description given a table. To do so, look for

patterns so that you can determine how a change in one quantity affects

another. Then put the patterns into words by making a general statement

about the relationship.

7.7

Find the difference in the number of miles walked.

Find the difference in the amounts pledged.

Find the rate that represents the amount pledged per mile walked.

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The relationship between the cost of an online advertisement for a movie

and the number of times it is clicked on is shown in the table.

8. Use the table to give a verbal description of the relationship.

Number of clicks 10 20 30 40 50

Cost ($) 150.50 151 151.50 152 152.50

9. What is the cost for the advertisement if it is clicked 1000 times?

10. Is there a lower limit for the number of clicks? Is there an upper limit?

Explain.

YOUR TURN

Determine how much more

money than $1.50 Luis’s aunt is

pledging for 1 mile walked.

$31.50 - $1.50 = $30

Give a verbal description for the relationship between the miles

walked by Luis and amount of money pledged by his aunt.

Luis’s aunt pledged $30 plus an additional $1.50 for each mile

he walks.

Reflect6. Make a Prediction How could you find the amount pledged by Luis’s

aunt if Luis walks 7 miles? What is the amount pledged?

7. What If...? Luis’s mother decides to also pledge $15 plus and additional

$3 per mile. If Luis wants to earn the same amount from his mother and

his aunt, how far must he walk? What is the amount he will earn from

each person?

STEP 2

STEP 3

Luis’s aunt gives an additional $30 more than the $1.50 per mile.

Luis’s aunt only gives the additional $30 one time.

Unit 4228

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Guided Practice

1. The age of a cat 2 years or older can be approximately converted into

human years by multiplying by 4 and adding 16. Use a table to represent

the relationship between cat age and human years. (Example 1)

Label the rows of the table.

Choose numbers to represent the ages of the cat. Choose numbers that are

2 or greater, since the relationship described is only for cats 2 years or older.

Complete the table by calculating the value for Human years based on

the description.

2. The yearly cost of a community college based on the number of credits

taken is shown in the table. Use the table to give a verbal description of

the relationship between credits and cost. (Explore Activity and Example 2)

Credits 3 6 9 12 15

Cost ($) 175 250 325 400 475

Look for patterns in the different values for credits and cost.

Each value for credits is greater than the previous one,

and each value for cost is greater than the previous one.

This means that 1 credit corresponds to in cost.

Determine how many more dollars than that it costs

to take 3 credits.

It costs - = more than to

take 3 credits.

Give a verbal description for the relationship between credits and cost.

The yearly cost of the community college is plus

for each credit taken.

STEP 1

STEP 2

STEP 3

3. When using tables and verbal descriptions to describe a linear relationship,

why is it useful to convert from one to another?

ESSENTIAL QUESTION CHECK-IN??

229Lesson 7.1

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Independent Practice

Name Class Date

7.1

A teacher is making multiple copies of a 1-page homework assignment.

The time it takes her in seconds is 2 times the number of copies she

makes plus 3.

4. What does the 3 represent in this scenario? What does the 2 represent?

5. What is the total number of seconds it takes for the teacher to make

1 copy? 2 copies? 3 copies? By how many seconds does the total time

increase for each copy?

6. Represent Real-World Problems Represent the relationship between

the number of copies made and time in seconds in the table below.

Rosalee parks at a metered space that still has some time left. She adds

some dimes to the meter. The table below represents the number of

minutes left based on the number of dimes inserted into the meter.

Dimes 4 8 12 16 20

Minutes 22 38 54 70 86

7. How many minutes does 1 dime correspond to?

8. Based on your answer to exercise 7, how many minutes should you

receive for inserting 4 dimes?

7.7

Unit 4230

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9. Analyze Relationships Give a verbal description of the relationship

between dimes and the number of minutes left on the meter.

10. Look at your answer for exercise 9. What does each of the numbers in the

answer represent?

The cost in dollars of a loaf of bread in a bakery is equal to 2 minus

0.25 times the number of days since it was baked.

11. What is different about this description compared to most of the other

descriptions you have seen in this lesson?

12. Make a Conjecture Is there a point at which the linear relationship

between days and dollars no longer makes sense?

13. Represent Real-World Problems Represent the relationship between

days and dollars in the table below.

14. Find the number of days it will take the price to reach $0.25.

231Lesson 7.1

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Work Area

The relationship between the number of years since a tree was

transplanted and its height in inches is shown in the table.

Years 2 4 5 8 9

Height (in.) 34 50 58 82 90

15. What is different about this table compared to the other tables you have

seen in this lesson?

16. Analyze Relationships Can you give a description of the relationship

between the years since the tree was transplanted and its height in

inches? If so, what is it?

17. Communicate Mathematical Ideas Suppose you are analyzing the

relationship between time and distance given in a table, and there are

4 values for each quantity. You divide distance 2 minus distance 1 by

time 2 minus time 1. You then divide distance 4 minus distance 3 by

time 4 minus time 3 and get a different answer. What can you say

about the relationship? Explain.

18. Persevere in Problem Solving There is a linear relationship between

a salesperson’s sales and her weekly income. If her sales are $200, her

income is $500, and if her sales are $1,200, her income is $600. What is the

relationship between sales and income?

19. Critique Reasoning Molly orders necklace kits online. The cost of the

necklace kits can be represented by a linear relationship. Molly’s order of

3 kits cost $12.50. Another order of 5 kits cost $17.50. Molly decides that

the kits cost $5 each. Is she correct? Explain.

FOCUS ON HIGHER ORDER THINKING

Unit 4232

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