Grade 7 Workbook — Chapter 7: Proportional Relationships: Tables, Graphs, and $y = mx$

SOL 7.PFA.1 · Companion to Textbook Chapter 7

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Item numbers match the textbook exactly and run continuously from 1 to 118.

PAGE 1 — Chapter opener

Chapter 7 · Proportional Relationships: Tables, Graphs, and $y = mx$

Standard 7.PFA.1

In this chapter you will:

• Find the slope $m$ as the rate of change from a table, a graph, or a situation
• Write the equation $y = mx$ for a proportional relationship
• Identify a positive, negative, or zero slope from a graph
• Graph a line from a point and a slope, or from an equation
• Connect a situation, a table, an equation, and a graph of the same relationship

Words to know: rate of change · constant rate of change · slope · rise · run · slope triangle · origin · proportional relationship · direct variation

Graphing note: keep a straightedge handy. Every line in this chapter is straight and passes through the origin.

PAGE 2 — Slope is rise over run

7.1 Rate of Change as Slope

FIGURE: fig1-slope-triangle.png (full width)

Fill in the blanks.

Slope is the ratio of ________ to ________.

$$m = \frac{\phantom{\text{rise}}}{\phantom{\text{run}}} = \frac{\text{change in } y}{\text{change in } x}$$

In a proportional relationship the rate of change is ______________ — it never varies.

The graph of a proportional relationship is a straight line through the ______________, the point ( ____ , ____ ).

Read the slope triangle in the figure above.

Rise = ______ Run = ______ $m =$ ______

PAGE 3 — Through the origin, or not

The Origin Test

FIGURE: fig7-proportional-or-not.png (full width)

Circle one for each graph.

FIGURE: fig6-two-slope-triangles.png (full width)

Both triangles sit on the same line.

Small triangle: rise ______ over run ______ = ______

Large triangle: rise ______ over run ______ = ______

What does this show about slope? _______________________________________________

PAGE 4 — Practice 7.1

Finding Slope

Guided practice

1. A line passes through the origin and $(4, 8)$. $m =$ ______

2. A line passes through the origin and $(5, 15)$. $m =$ ______

3. Rise 6, run 4. $m =$ ______

4. Rise $-3$, run 1. $m =$ ______

5. A faucet fills 12 gallons in 4 minutes. Rate of change = ______ gallons per minute

Independent practice

6. Find each slope.

7. Origin and $(6, 9)$. $m =$ ______

8. Origin and $(2, -10)$. $m =$ ______

9. $(0, 0)$ and $(8, 6)$. $m =$ ______

10. Explain why the slope of a proportional relationship is the same number as its unit rate.

11. Application. A printing press prints 1,500 pages in 6 minutes at a steady rate.

    Rate of change: ______ pages per minute Pages in 10 minutes: ______

12. Reasoning. Jae says the slope of the line through the origin and $(4, 10)$ is $\tfrac{4}{10} = \tfrac{2}{5}$. Find the mistake and give the correct slope.

    Mistake: _______________________________________________

    Correct slope: ______

PAGE 5 — Exit ticket 7.1

Exit Ticket · Lesson 7.1

Name: ________________________ Date: ____________

13. A line passes through the origin and $(3, 12)$. $m =$ ______

14. Rise $-6$, run 2. $m =$ ______

15. A hose delivers 45 liters in 5 minutes. Rate of change = ______ liters per minute

16. What does rate of change mean, and why does it stay constant in a proportional relationship?

PAGE 6 — The quotient test

7.2 Finding Slope from a Table

FIGURE: fig3-table-to-graph.png (full width)

Divide $y$ by $x$ in EVERY row. Same quotient every time = proportional, and that quotient is $m$.

Run the test on the table in the figure.

Proportional? ______ $m =$ ______

Watch out. The table $x$: 1, 2, 3 with $y$: 5, 9, 13 climbs by a steady 4 each time. Is it proportional? ______

Why or why not? _______________________________________________

PAGE 7 — Guided practice 7.2

Reading Slope from Tables

17. $x$	1	2	3	4
    $y$	9	18	27	36

    $m =$ ______

18. $x$	2	4	6
    $y$	1	2	3

    $m =$ ______

19. $x$	1	2	3
    $y$	$-5$	$-10$	$-15$

    $m =$ ______

20. $x$	3	6	9
    $y$	12	24	36

    $m =$ ______

21. Is this proportional? Show your quotients.

Quotients: ______, ______, ______   Proportional? ______

PAGE 8 — Independent practice 7.2

Tables, Slopes, and Missing Values

22. Each table is proportional. Find each slope.

23. $x$	0	1	2	3
    $y$	0	2.5	5	7.5

    $m =$ ______

24. Proportional. Find the slope, then fill in the missing value.

$m =$ ______   Missing value: ______

25. One table is proportional and one is not.

    Table 1 — $x$: 1, 2, 3 · $y$: 3, 6, 9 Table 2 — $x$: 1, 2, 3 · $y$: 3, 5, 7

    Proportional table: ______ How you decided: _______________________________________________

26. A proportional table contains $(6, 8)$ and $(9, 12)$. Find $m$ two ways.

    By dividing $y$ by $x$: ______ By change in $y$ over change in $x$: ______

27. Application. Complete the check, then answer.

Quotients: ______, ______, ______   Proportional? ______   $m =$ ______ dollars per hour

Earnings for 12 hours: ______

28. Reasoning. How can you tell from a table alone whether a relationship is proportional? Why is checking one row not enough?

PAGE 9 — Exit ticket 7.2

Exit Ticket · Lesson 7.2

Name: ________________________ Date: ____________

29. $x$	1	2	3	4
    $y$	11	22	33	44

    $m =$ ______

30. $x$	4	8	12
    $y$	$-2$	$-4$	$-6$

    $m =$ ______

31. Is this proportional? Explain.

32. Why is dividing $y$ by $x$ in every row a valid test for proportionality?

PAGE 10 — The equation $y = mx$

7.3 Writing $y = mx$

$$y = mx$$

Fill in the blanks.

$m$ stands for the ______________, the constant rate of change.

Substituting $x = 0$ gives $y =$ ______, so the graph always passes through the ______________.

A relationship such as $y = 3x + 5$ is ______ proportional, because at $x = 0$ it gives $y =$ ______.

Three ways in, one equation out.

PAGE 11 — Practice 7.3

Writing Equations

Guided practice

33. $x$	1	2	3
    $y$	4	8	12

    $y =$ ____________

34. Origin and $(3, 21)$. $y =$ ____________

35. A worker earns $14 per hour. $y =$ ____________ where $x =$ ____________ and $y =$ ____________

36. Origin and $(4, -12)$. $y =$ ____________

37. For $y = 6x$, find $y$ when $x = 9$. $y =$ ______

Independent practice

38. Write $y = mx$ for each proportional table.

39. Origin and $(6, 4)$. $y =$ ____________

40. Origin and $(1, -9)$. $y =$ ____________

41. For $y = 2.5x$: when $x = 14$, $y =$ ______ when $y = 60$, $x =$ ______

PAGE 12 — Applications 7.3

Equations in the Real World

42. Application. A bus travels at a steady 55 miles per hour.

    Equation: $y =$ ____________ where $x =$ ____________ and $y =$ ____________

    Distance in 4.5 hours: ______ miles

    Time to travel 302.5 miles: ______ hours

43. Application. Almonds cost $6.40 per pound.

    Equation: $y =$ ____________

    Cost of 3.5 pounds: $______

    Pounds you can buy with $40: ______

44. Reasoning. A phone plan charges $20 per month plus $5 per gigabyte of data.

    Cost for 0 gigabytes: $______

    Why can this not be written as $y = mx$?

PAGE 13 — Exit ticket 7.3

Exit Ticket · Lesson 7.3

Name: ________________________ Date: ____________

45. $x$	1	2	3
    $y$	15	30	45

    $y =$ ____________

46. Origin and $(8, 2)$. $y =$ ____________

47. A copier makes 40 copies per minute. $y =$ ____________ Copies in 7 minutes: ______

48. How do you find $m$ from a graph when you have no table?

PAGE 14 — Three directions

7.4 Positive, Negative, and Zero Slope

FIGURE: fig2-three-slopes.png (full width)

Complete the table. Always read a graph left to right.

Zero slope, precisely.

$y = 0x$ simplifies to $y =$ ______. Every point on it has a $y$-coordinate of ______.

The graph is a ______________ line lying along the ______-axis.

Zero slope is horizontal, NOT vertical. A vertical line has a run of zero, and dividing by zero is undefined, so a vertical line has no slope at all.

PAGE 15 — Practice 7.4

Naming the Sign

Guided practice

49. $y = 5x$ → ____________

50. $y = -8x$ → ____________

51. $y = 0x$ → ____________

52. Origin and $(2, -6)$. $m =$ ______ Sign: ____________

53. Origin and $(7, 0)$. $m =$ ______

Independent practice

54. Classify each slope.

55. Order least steep to steepest: $y = 2x$, $y = 6x$, $y = \tfrac{1}{2}x$

    ---

56. A line through the origin falls from left to right. What can you say about $m$? ____________

57. Describe the graph of $y = 0x$ in one sentence, then name three points on it.

_______________________________________________ Points: ______, ______, ______

58. Origin and $(-2, -10)$. $m =$ ______ Sign: ____________ (Be careful.)

59. Application. Snow falls at a steady 2 inches per hour. Sign of the slope: ____________

    Why are the slopes of contextual graphs in this chapter positive?

60. Reasoning. A classmate says a zero-slope line is vertical. Explain the error.

PAGE 16 — Exit ticket 7.4

Exit Ticket · Lesson 7.4

Name: ________________________ Date: ____________

61. $y = -7x$ → ____________

62. Origin and $(3, 12)$. $m =$ ______ Sign: ____________

63. What does the graph of $y = 0x$ look like?

64. How can you decide the sign of a slope from a graph without any arithmetic?

PAGE 17 — Graphing $y = mx$

7.5 Graphing a Proportional Relationship

FIGURE: fig5-graph-from-equation.png (half width, left)

FIGURE: fig4-graph-from-point-and-slope.png (half width, right)

The five steps, from an equation.

1. Plot the ______________, $(0, 0)$.
2. Write $m$ as $\dfrac{\text{rise}}{\text{run}}$. A whole number $m$ becomes $\dfrac{\ \ \ }{1}$.
3. Move right by the ______ and up or down by the ______.
4. Plot a third point as a ______________.
5. Draw a straight line with arrows on both ends.

From a point and a slope. Start at the given point, apply the slope forward and backward. Going backward should land you on ______________, which confirms the relationship is proportional.

PAGE 18 — Guided practice 7.5

Graph It

Use one grid per item.

BLANK COORDINATE GRID: x from -6 to 6, y from -8 to 8, unit gridlines, axes labeled x and y (repeat 5 times, small)

65. Graph $y = 3x$. Two points besides the origin: ( ____ , ____ ) and ( ____ , ____ )

66. Graph $y = -2x$. Two points besides the origin: ( ____ , ____ ) and ( ____ , ____ )

67. Graph the proportional line through the origin with slope $\tfrac{1}{2}$. Two whole-number points: ( ____ , ____ ) and ( ____ , ____ )

68. A proportional line passes through $(3, 6)$ with slope 2. Graph it.

    Does it pass through the origin? ______ Equation: $y =$ ____________

69. Graph $y = \tfrac{2}{5}x$. Two whole-number points: ( ____ , ____ ) and ( ____ , ____ )

PAGE 19 — Independent practice 7.5

More Graphing

BLANK COORDINATE GRID: x from -10 to 10, y from -10 to 10, unit gridlines (repeat 3 times, small)

70. Graph each and name two points other than the origin.

71. A proportional line passes through $(4, 10)$.

    $m =$ ______ $y =$ ____________ One more point: ( ____ , ____ )

72. A proportional line passes through $(-2, 6)$ with slope $-3$.

    Check: $-3 \times (-2) =$ ______ Equation: $y =$ ____________ One more point: ( ____ , ____ )

73. Graph $y = \tfrac{5}{2}x$. When $x = 6$, $y =$ ______

74. A line through the origin has slope 0. Describe its graph, then name a point other than the origin.

_______________________________________________ Point: ( ____ , ____ )

75. Application. Cookies cost $1.50 each.

    Equation: $y =$ ____________ 2 cookies: ( ____ , ____ ) 6 cookies: ( ____ , ____ )

    Why is only the first quadrant used? _______________________________________________

76. Reasoning. Why is knowing one point besides the origin enough to graph the whole line?

PAGE 20 — Exit ticket 7.5

Exit Ticket · Lesson 7.5

Name: ________________________ Date: ____________

BLANK COORDINATE GRID: x from -6 to 6, y from -10 to 10, unit gridlines (repeat 2 times, small)

77. Graph $y = 5x$. Two points: ( ____ , ____ ) and ( ____ , ____ )

78. Graph the proportional line through the origin with slope $-4$. Two points: ( ____ , ____ ) and ( ____ , ____ )

79. A proportional line passes through $(6, 4)$. $m =$ ______ $y =$ ____________

80. How does the origin give you a free second point when graphing $y = mx$?

PAGE 21 — Four ways to say the same thing

7.6 Connecting All Four Representations

FIGURE: fig8-four-representations.png (full width)

Where does the slope hide? Complete the table.

Say the point out loud. On a graph of gallons against miles for a car getting 32 miles per gallon, the point $(5, 160)$ means:

PAGE 22 — Guided practice 7.6

Moving Between Representations

81. A printer prints 5 pages per minute. $y =$ ____________

82. For $y = 7x$, complete the table.

83. A line passes through the origin and $(2, 10)$. $y =$ ____________

84. $x$: 3, 6, 9 and $y$: 21, 42, 63.

    $m =$ ______ $y =$ ____________ A point besides the origin: ( ____ , ____ )

85. For $y = 1.5x$, the ordered pair when $x = 4$ is ( ____ , ____ )

PAGE 23 — Independent practice 7.6

Putting It All Together

86. A car travels 28 miles on each gallon. $y =$ ____________

Describe the graph: _______________________________________________

87. A line passes through the origin and $(5, -15)$. $y =$ ____________

88. $x$: 2, 5, 10 and $y$: 3, 7.5, 15. $m =$ ______ $y =$ ____________ When $x = 20$, $y =$ ______

89. Give the slope of each.

90. Application. A wind turbine produces 15 kilowatt-hours each hour. $y =$ ____________

What does $(6, 90)$ mean? _______________________________________________

91. Application. Rice costs $2.80 per kilogram. $y =$ ____________

Cost of 4.25 kilograms: \$______

92. Reasoning. One student uses a table, another uses a graph. How do you check they describe the same relationship?

PAGE 24 — Exit ticket 7.6

Exit Ticket · Lesson 7.6

Name: ________________________ Date: ____________

93. For $y = 9x$, complete the table.

94. A line passes through the origin and $(4, 10)$. $m =$ ______ $y =$ ____________

95. $x$: 3, 6 and $y$: 4.5, 9. $m =$ ______ $y =$ ____________

96. Which representation would you use to find $y$ when $x = 250$? Why?

PAGE 25 — Chapter 7 review, part 1

Chapter 7 Review

Part A · Slope and writing $y = mx$

97. Origin and $(4, 6)$. $m =$ ______

98. $x$: 2, 4, 6 and $y$: 11, 22, 33. $m =$ ______ $y =$ ____________

99. A machine fills 24 bottles per minute. $y =$ ____________ Bottles in 15 minutes: ______

100. $x$: 1, 2, 3 and $y$: $-8$, $-16$, $-24$. $m =$ ______ $y =$ ____________

101. Origin and $(10, 4)$. $y =$ ____________

102. A cyclist rides 13 miles per hour. $y =$ ____________ Distance in 3.5 hours: ______ miles

Part B · Positive, negative, and zero slope

103. Classify each.

104. Describe the graph of a zero-slope line through the origin, and name two points on it.

_______________________________________________ Points: ( ____ , ____ ) and ( ____ , ____ )

105. A line rises from left to right. Sign of $m$: ____________

106. Steeper: $y = \tfrac{1}{5}x$ or $y = 5x$? ____________ Why? _______________________________

PAGE 26 — Chapter 7 review, part 2

Chapter 7 Review (continued)

BLANK COORDINATE GRID: x from -10 to 10, y from -10 to 10, unit gridlines (repeat 6 times, small — two rows of three)

Part C · Graphing from a point and the slope

107. Graph the proportional line through $(2, 8)$. $m =$ ______ $y =$ ____________

108. Graph the proportional line through $(3, -6)$. $m =$ ______ $y =$ ____________

109. Graph the proportional line through $(8, 6)$. $m =$ ______ $y =$ ____________

Part D · Graphing from an equation

110. Graph $y = 3x$. Two points: ( ____ , ____ ) and ( ____ , ____ )

111. Graph $y = -\tfrac{3}{2}x$. Two whole-number points: ( ____ , ____ ) and ( ____ , ____ )

112. Graph $y = \tfrac{1}{4}x$. Two whole-number points: ( ____ , ____ ) and ( ____ , ____ )

PAGE 27 — Chapter 7 review, part 3

Chapter 7 Review (continued)

Part E · Application and reasoning

113. A landscaper spreads 35 square feet of mulch per minute. $y =$ ____________

 Describe the graph: _______________________________________________

 What does $(10, 350)$ mean? _______________________________________________

114. $x$: 2, 3, 5 and $y$: 9, 13.5, 22.5. $m =$ ______ $y =$ ____________ When $x = 11$, $y =$ ______

115. Why is $x$: 1, 2, 3 with $y$: 6, 11, 16 not proportional?

116. Pump A: $y = 18x$. Pump B fills 95 gallons in 5 minutes.

     Pump B rate: ______ gallons per minute Faster pump: ______

     On one graph, how would you see it? _______________________________________________

117. Why does every proportional graph pass through the origin? Use $y = mx$.

118. Describe the graphs of $y = 2x$, $y = -2x$, and $y = 0x$ on one grid.

 What do all three share? _______________________________________________

Canva production notes

• Page size: 8.5 × 11 in, 0.75 in margins
• Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
• Blank coordinate grids: pages 18, 19, 20, and 26 need student grids. Use light gray unit gridlines, black axes with arrowheads, and label only every other tick so the grid stays readable at reduced size. Six small grids fit a page in two rows of three.
• Figure widths: fig1, fig2, fig3, fig6, fig7, and fig8 are landscape and go full width; fig4 and fig5 are portrait and sit side by side at half width on page 17.
• Tables: leave every answer cell at least 0.4 in tall so students can write a fraction such as $\tfrac{3}{2}$ without cramping.
• Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow.