Constant of Proportionality Worksheet for 12-Year-Olds | Math

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Instructions

  1. A proportional relationship can be written as y = kx. The number k is the constant of proportionality.
  2. Complete each section in order. Show your work when you calculate.
  3. Remember that k = y ÷ x, as long as x is not 0.
  4. In a proportional relationship, the graph is a straight line that passes through (0, 0).
  5. Use the hints when needed. Complete the challenge questions if you are ready.

Section 1: Spot the Proportional Graph

A graph represents a proportional relationship when it is a straight line passing through the origin, (0, 0).

For each description, write proportional or not proportional. Explain your choice briefly.

  1. A straight line passes through (0, 0), (1, 4), and (2, 8).

    Answer: ____ Explanation: __

  2. A straight line passes through (0, 3), (1, 5), and (2, 7).

    Answer: ____ Explanation: __

  3. A curved line passes through (0, 0), (1, 2), and (2, 8).

    Answer: ____ Explanation: __

  4. A straight line passes through (0, 0), (3, 2), and (6, 4).

    Answer: ____ Explanation: __

  5. A straight line passes through (0, 0), (2, 10), and (4, 20).

    Answer: ____ Explanation: __

Hint: Check both features: Is the graph a straight line? Does it pass through (0, 0)?

Section 2: Find the Constant from a Graph

Read each set of points from a proportional graph. Use one point to find the constant of proportionality.

Formula: k = y ÷ x

  1. A line passes through (0, 0), (1, 3), and (4, 12).

    k = ____

  2. A line passes through (0, 0), (2, 8), and (5, 20).

    k = ____

  3. A line passes through (0, 0), (3, 6), and (7, 14).

    k = ____

  4. A line passes through (0, 0), (4, 10), and (8, 20).

    k = ____

  5. A line passes through (0, 0), (5, 15), and (10, 30).

    k = ____

For each problem above, write the equation of the relationship in the form y = kx.

  1. Equation: ____

  2. Equation: ____

  3. Equation: ____

  4. Equation: ____

  5. Equation: ____

Section 3: Constant of Proportionality from Tables

Find k = y ÷ x for the example row. Then complete each table. The relationship in each table is proportional.

Example

x y k = y ÷ x Equation
2 10 5 y = 5x
3
4
5
6
7
  1. Complete the table and write the equation.
x y k = y ÷ x Equation
4 12 3 y = 3x
1
2
5
7
10
  1. Complete the table and write the equation.
x y k = y ÷ x Equation
3 1.5 0.5 y = 0.5x
2
4
6
8
10
  1. Complete the table and write the equation.
x y k = y ÷ x Equation
5 35 7 y = 7x
1
2
3
8
12

Section 4: Decide Whether a Table Is Proportional

Calculate y ÷ x for at least two rows. If the quotient is always the same, the table is proportional.

1.

x y k = y ÷ x
1 4 4
2
3
5
7
10

The table is proportional: ____

2.

x y k = y ÷ x
1 5 5
2 11
3
4
5
6

The table is proportional: ____

3.

x y k = y ÷ x
2 6 3
4 12
6
8
10
12

The table is proportional: ____

Section 5: Interpret the Constant in Real Life

The constant of proportionality tells how much y changes for every 1 unit of x. Include units in your answers.

  1. A cyclist travels 18 miles in 2 hours at a constant speed.

    a. Find the constant of proportionality.

    k = ____

    b. Write an equation for distance d and time t.

    c. What does the constant mean in this situation?

  2. A store sells 4 notebooks for $12.

    a. Find the constant of proportionality for cost C per notebook n.

    k = ____

    b. Write an equation.

    c. What would 7 notebooks cost?

  3. A recipe uses 3 cups of flour to make 12 muffins.

    a. Find the number of cups of flour needed for each muffin.

    k = ____

    b. Write an equation for cups of flour f and muffins m.

    c. How much flour is needed for 20 muffins?

  4. A car uses 6 gallons of gasoline to travel 180 miles.

    a. Find the number of miles traveled per gallon.

    k = ____

    b. Write an equation for miles m and gallons g.

    c. How far can the car travel using 10 gallons?

Section 6: Show What You Know

For each situation, identify the constant, write an equation, and explain what the constant means.

  1. A dog walker earns $45 for 3 hours of work.

    Constant: ____

    Equation: ____

    Meaning of the constant: __

  2. A printer produces 240 pages in 8 minutes.

    Constant: ____

    Equation: ____

    Meaning of the constant: __

  3. A runner completes 5 kilometers in 25 minutes.

    Constant: ____

    Equation: ____

    Meaning of the constant: __

Optional Challenge

  1. A proportional graph passes through the point (6, 42). What other point on the graph has an x-coordinate of 10?

    Show your work: __

    Answer: ____

  2. Two proportional relationships are shown below.

    • Relationship A: y = 4x
    • Relationship B: A table includes the point (5, 15)

    Which relationship has the greater constant of proportionality? How much greater is it?

  3. Create your own real-world proportional relationship. Include a table with one example row and at least four additional rows, the constant of proportionality, an equation, and a sentence explaining the constant.

x y k = y ÷ x
Example: 2 Example: 8 Example: 4

Equation: ____

Meaning of the constant: __

Answer Key

Section 1

  1. Proportional; it is a straight line and passes through (0, 0).
  2. Not proportional; the line does not pass through (0, 0).
  3. Not proportional; the graph is curved, not a straight line.
  4. Proportional; it is a straight line and passes through (0, 0).
  5. Proportional; it is a straight line and passes through (0, 0).

Section 2

  1. k = 3; equation: y = 3x
  2. k = 4; equation: y = 4x
  3. k = 2; equation: y = 2x
  4. k = 2.5; equation: y = 2.5x
  5. k = 3; equation: y = 3x

Section 3

  1. k = 3; equation: y = 3x. The missing y-values are 3, 6, 15, 21, and 30.
  2. k = 0.5; equation: y = 0.5x. The missing y-values are 1, 2, 3, 4, and 5.
  3. k = 7; equation: y = 7x. The missing y-values are 7, 14, 21, 56, and 84.

Section 4

  1. The completed y-values are 8, 12, 20, 28, and 40. The table is proportional because k = 4 for every row.
  2. The table is not proportional because the first two rows give different constants: 5 ÷ 1 = 5 and 11 ÷ 2 = 5.5.
  3. The completed y-values are 18, 24, 30, and 36. The table is proportional because k = 3 for every row.

Section 5

  1. a. k = 9 miles per hour b. d = 9t c. The cyclist travels 9 miles for every 1 hour.

  2. a. k = $3 per notebook b. C = 3n c. 7 notebooks cost $21.

  3. a. k = 0.25 cup per muffin b. f = 0.25m c. 20 muffins need 5 cups of flour.

  4. a. k = 30 miles per gallon b. m = 30g c. The car can travel 300 miles using 10 gallons.

Section 6

  1. Constant: $15 per hour. Equation: E = 15h. The dog walker earns $15 for every hour worked.
  2. Constant: 30 pages per minute. Equation: p = 30m. The printer produces 30 pages every minute.
  3. Constant: 0.2 kilometers per minute. Equation: d = 0.2t. The runner completes 0.2 kilometers every minute.

Optional Challenge

  1. k = 42 ÷ 6 = 7. When x = 10, y = 7 × 10 = 70. The point is (10, 70).
  2. Relationship A has k = 4. Relationship B has k = 15 ÷ 5 = 3. Relationship A is greater by 1.
  3. Answers will vary. The table must have a constant y ÷ x, and the equation and explanation must match the chosen relationship.
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