LESSON
7.1
?
Linear Relationships in
the Form y = mx + b
ESSENTIAL QUESTION
Expressions,
equations, and
relationships—7.7 The
student applies
mathematical process
standards to represent linear
relationships using multiple
representations....
How do you use tables and verbal descriptions to describe
a linear relationship?
7.7
EXPLORE ACTIVITY
Discovering Linear Relationships
Many real-world situations can be described by linear relationships.
Number of tickets
1
2
3
4
5
Total cost ($)
7
12
17
22
27
TICKET
ADMIT
ONE
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.
A Describe a pattern for the row showing the number of tickets bought.
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B Describe the pattern for the row showing total cost.
C Out of the total cost paid, how much does the actual ticket account for?
Reflect
1.
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.
Lesson 7.1
225
Representing Linear
Relationships Using a Table
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In 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.
EXAMPLE 1
7.7
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.
Math Talk
STEP 1
Make a table. Label the top row Foot length (in.) and the bottom
row Shoe size.
STEP 2
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.
Mathematical Processes
Why is foot length on the
top and shoe size on
the bottom?
Think:
3 × 7 = 21
21 - 22 = -1; this cannot
be a man’s shoe size.
Remember that a man’s
shoe size is 3 times his
foot length in inches
minus 22.
24 - 22 = 2; start the
table at 8 inches.
Make a table relating foot length to show size.
3 × 8 = 24
STEP 3
Foot length (in.)
8
9
10
11
12
Shoe size
2
5
8
11
14
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Unit 4
3.
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.
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Reflect
YOUR TURN
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.
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Representing Linear Relationships
Using a Verbal Description
Just 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.
EXAMPL 2
EXAMPLE
7.7
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My Notes
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.
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Use the table to give a verbal description
of the relationship between miles walked
and amount pledged.
Miles walked
Amount pledged ($)
STEP 1
1
2
3
4
5
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
____
= 1.50
1
Find the difference in the amounts pledged.
Find the difference in the number of miles walked.
Find the rate that represents the amount
pledged per mile walked.
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.
Lesson 7.1
227
STEP 2
STEP 3
Determine how much more
Luis’s aunt only gives the
money than $1.50 Luis’s aunt is
additional $30 one time.
pledging for 1 mile walked.
Luis’s aunt gives an additional $30 more
$31.50 - $1.50 = $30
than the $1.50 per mile.
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.
Reflect
6.
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
$3 per mile. If Luis wants to earn the same amount from his mother and
his aunt, how far must he walk? What is the amount he will earn from
each person?
YOUR TURN
The relationship between the cost of an online advertisement for a movie
and the number of times it is clicked on is shown in the table.
Number of clicks
Cost ($)
10
20
30
40
50
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.
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Unit 4
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8. Use the table to give a verbal description of the relationship.
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
STEP 1
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
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STEP 2
Determine how many more dollars than
to take 3 credits.
It costs
take 3 credits.
STEP 3
in cost.
=
more than
that it costs
to
Give a verbal description for the relationship between credits and cost.
The yearly cost of the community college is
plus
for each credit taken.
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ESSENTIAL QUESTION CHECK-IN
3. When using tables and verbal descriptions to describe a linear relationship,
why is it useful to convert from one to another?
Lesson 7.1
229
Name
Class
Date
7.1 Independent Practice
7.7
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Intervention
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?
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?
230
Unit 4
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6. Represent Real-World Problems Represent the relationship between
the number of copies made and time in seconds in the table below.
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?
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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.
Lesson 7.1
231
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?
FOCUS ON HIGHER ORDER THINKING
Work Area
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.
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Unit 4
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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.