Part 1: Identify the Relationship

Directions: Decide whether each relationship is proportional. Show evidence by finding the constant of proportionality, comparing ratios, or checking the graph points.

Reference: A proportional relationship has the form y = kx, where k is the constant of proportionality. In a table, y ÷ x must be the same for every row. Its graph is a straight line through the origin, (0, 0). The point (1, k) shows the unit rate.

x: 1, 2, 4, 6

y: 3, 6, 12, 18

x: 2, 3, 5, 8

y: 6, 9, 16, 24

x: 1, 3, 5, 7

y: 4, 12, ______, 28

Part 2: Situations and Graph Evidence

Answer Key

1. Proportional. The unit rate is 3 in every row: 3 ÷ 1 = 3, 6 ÷ 2 = 3, 12 ÷ 4 = 3, and 18 ÷ 6 = 3. Thus, y = 3x.

2. Not proportional. For example, 6 ÷ 2 = 3, but 16 ÷ 5 = 3.2. The unit rates are not constant.

3. The missing value is 20. The constant of proportionality is 4 because y = 4x.

4. Yes. It is already in the form y = kx, and k = 5.

5. Not proportional. The equation has a starting value of 2, so its graph would not pass through the origin.

6. Proportional. Both unit rates are $2 per notebook: $6 ÷ 3 = $2 and $10 ÷ 5 = $2.

7. Yes. The points form a straight line through (0, 0), and the unit rate is 3. The point (1, 3) means 1 unit of the input corresponds to 3 units of the output.

8. No. Although the points may form a straight line, the line does not pass through the origin, so the relationship is not proportional.

9. y = 4x. Yes, it is proportional because the cost is a constant $4 per mile and there is no starting fee.

10. Recipe A is proportional because 5 ÷ 2 = 2.5 and 10 ÷ 4 = 2.5. Recipe B is not proportional because 11 ÷ 4 = 2.75.

Challenge 1. Answers may vary. Example: proportional: (1, 4), (2, 8), (3, 12); nonproportional: (1, 4), (2, 8), (3, 13). The first has a constant ratio of 4; the second does not.

Challenge 2. Example: y = 2.5x, with points (0, 0) and (4, 10). The point (1, 2.5) represents a unit rate of 2.5 output units for every 1 input unit.

Exit Ticket: Not proportional. The first two rates are $4 per mile, but $11 ÷ 3 is not $4. If proportional, (1, 4) would mean 1 mile costs $4.