2/10/25

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Grade 8 Mathematics

Teacher Facing
February 18, 2025

Unit 13: Proportional and Nonproportional Relationships
Lesson 8: Multiple Representations of Linear
Relationships

Do Now

2

Write the equation of the line shown on the graph.

3

Academic Vocabulary

Linear Equation

y = mx + b

slope

y-intercept

4

Unit Rate

Constant of
proportionality

y
x

Academic Vocabulary

k =

5

LO: SWBAT represent linear proportional and non-
proportional situations with tables, graphs, and equations
in the form of y = kx and y = mx + b, where b ≠ 0.

DOL: Given a set of problems, students represent linear
proportional and non-proportional situations with tables,
graphs, and equations in the form of y = kx and y = mx + b,
where b ≠ 0 in at least 4 of 5 questions.

MATH.8.5B Represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b, b ≠ 0.

MATH.8.5A Represent linear proportional situations with tables, graphs, and equations in the form of y = kx.

6

Jade is refilling her fish tank after cleaning it. The table shows the amount of water in
the tank while she is filling it.

Time

(minutes),

x

Amount of

water

(gallons), y

1

2

16

3

4

5

6

How much water is in the tank after 1 minutes?

8

There is _______ gallons of water in the tank after
1 minute.

Complete the table.

What equation could you write
to represent the amount of
water in the tank after x
minutes?

7

Let's graph the equation.

Time

(minutes), x

Amount of
water
(gallons), y

1

2

16

3

4

5

6

8

24

32

40

48

y = 8x

Time(minutes), x

Amount of water, y

This represents a proportional
relationship because the graph increase
at a constant rate and it goes through
(0,0).

x

y

8

Mr. Sanders fills the family swimming pool with water. After 3 hours, the
water level had risen to 4 feet. Which representation shows a
proportional relationship between the time and height of the water level?
Select TWO correct answers.

A. y = x

A. The water rises

at 1.5 feet per
hour.

C.

D. E.

hours

feet

2

3

3

4.5

4

8

y
y

x

x

Time (hours)

Time (hours)

Height of water (feet)

Height of water (feet)

What equation could you
write to represent the total
cost of buying x computers?

9

A company pays $500 for each computer, plus an $80 shipping fee.

2

3

4

5

x

Number of

computers in

each order

Total Cost
(dollars)

1

$580

$1,080

What does the $500 represent? What does $80 represent?

$500 represents the _________________________, the $80
represents ___________________________.

What is the cost of
purchasing one computer?

Complete the table.

$1,580

$2,080

$2,580

$(500x + 80)

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Let's graph the relationship.

2

3

4

5

Number of

computers in

each order

Total Cost
(dollars)

1

$580

$1,080

y = 500x + 80

$1,580

$2,080

$2,580

Number of Computers, x

Total Cost, y

Initial
Value

(1, 580)

(0, 80)

Where can you see
the initial value on

the graph? the

slope?

11

Jackson sells toolboxes for farming tools. He plans on selling them during a fair this
weekend. Jackson estimates he will sell the boxes for $30 each with an additional
shipping fee of $10, regardless of order size. Which representation models the
relationship between number of boxes sold, x, and total amount earned, y?

x

y

0

30

1

40

2

50

x

y

0

10

1

40

2

70

y = 10x + 30

A.

B.

C.

D.

Number of boxes, x

Amount earned, y

13

The linear graph shown is a model of the length of track remaining in a bobsled run, over time.

Complete the table.

Time

(seconds), x

Track Length

(meters), y

0

4

10

15

The line has a

______________ slope.

Calculate the slope.

Is the slope of the line
positive or negative?

Write an equation to represent
the relationship between time

and track length

Initial Value

Open Ended

Answer the question and explain why or why not.

14

Veronica is an English tutor. She charges $60 for each tutoring
session plus $0.75 per mile that she has to drive to and from the
session.

Part A: Complete the table below to
represent this scenario.

Part B: Write an equation to represent
the relationship between the number of
tutoring sessions, x, and the amount
she earns, y.

Number of
sessions, x

Amount Earned,

y

0

2

4

6

8

15

Jerri paid $0.50 for one orange at the Farmer's Market. Let x represent the
number of oranges and y represent the total amount he spent. Which of the
following could represent this scenario? Select THREE correct answers.

B.

C.

D.

E.

F.

A. y = 0.50+ x y = 0.50x

y

0

0.50

1

1.00

2

1.50

x

y

3

1.50

4

2.00

5

2.50

16

Jeb is reading a book for History class. He reads the same number of
pages each night. The graph shows the number of pages remaining
to read over time. He reads the same number of pages every night.

Part B: Write an equation to represent the
relationship between time, x, and the number
of pages remaining, y.

Part A: How many pages does he read each
night?

17

Which representation shows the relationship described by the
equation y = –2x?

B.

C.

D.

x

y

2

–2

4

–4

6

–6

x

y

2

–1

4

–2

6

–3

18

The graph below shows the balance remaining on Jason’s loan after
he makes each payment.

Part A: How much is each payment?

Part B: Write an equation to represent the
relationship between the number of payments
made, x, and the balance remaining, y.

19

A hot-air balloon is released at ground level, and it rises into the air at a constant rate.
After 5 seconds the height of the balloon is 20 feet. The balloon continues to rise at the
same rate.

Which table shows the relationship between the time in seconds, x, and the height of the
balloon in feet, y?

B.

C.

D.

Time, x

Height, y

10

2.5

20

5.0

30

7.5

40

10.0

50

12.5

Time, x

Height, y

10

40

20

60

30

80

40

100

50

120

Time, x

Height, y

10

25

20

35

30

45

40

55

50

65

Time, x

Height, y

10

40

20

80

30

120

40

160

50

200

20

Mr. Cloud is writing an autobiography. He has already written 32 pages. Each day, he writes
another 5 pages.

Part A: Write an equation to represent the
relationship between the number of days, x, and the
number of pages written, y.

Part B: Graph the relationship.

Number of Days , x

Number of pages, y

Open Ended

Please explain.