7.RP.A.2 7th Grade Ratios & Proportional Relationships

Constant of Proportionality

Recognize and represent proportional relationships between quantities, identifying the constant of proportionality and writing equations of the form y = kx.

How to explain it

Students identify the constant of proportionality k from tables, graphs, equations, and verbal descriptions, represent proportional relationships with equations of the form y = kx, and interpret points on the graph of a proportional relationship in terms of the situation, including the meaning of the points (0, 0) and (1, k).

The anchor students hold onto: RATIO: divide y by x for each pair. CONSTANT: that shared value is k. EQUATION: write y = kx. INTERPRET: the graph passes through (0, 0), and the point (1, k) shows the constant.

The constant k is the unit rate for one x. Next you apply proportional reasoning to percent problems in 7.RP.A.3, and in 8th grade this same k becomes the slope of a line.

Worked examples

Example 1 Find k from a Table

Find k: (2, 6), (4, 12), (5, 15)

Step 1Ratio: 6 ÷ 2 = 3, 12 ÷ 4 = 3, 15 ÷ 5 = 3

Step 2Same ratio every pair — k = 3

Step 3Equation: y = 3x

Step 4A: k = 3, y = 3x

Answerk = 3, y = 3x

Example 2 Find k from Words

A plant grows 5 cm every 2 days

Step 1Ratio: k = y ÷ x = 5 ÷ 2 = 2.5

Step 2Constant: 2.5 cm of growth per day

Step 3Equation: y = 2.5x

Step 4A: k = 2.5, y = 2.5x

Answerk = 2.5, y = 2.5x

Common mistakes

What students write Flipping the ratio: k = x ÷ y instead of y ÷ x

The fix k = y ÷ x — divide the y-value by the x-value

Try this A proportional table includes the pair (4, 28). Reza finds the constant by computing 4 ÷ 28 and writes k = 1/7. Find his mistake, then find the correct k and equation.

What students write Adding instead of multiplying: 2 to 6 means add 4

The fix Proportions multiply: y is always k times x

Try this A proportional relationship includes the pair (2, 6). Mia says (3, 7) must also belong to it, because she added 1 to each number. Find her mistake, then find the correct y-value for x = 3.

Teacher tip

Head off the two predictable errors before they happen. First: k = y ÷ x — divide the y-value by the x-value Second: Proportions multiply: y is always k times x