Topics
4. Proportional Relationships & Lines
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United States of America
Grade 8

4.03 Compare proportional relationships

Lesson
Practice

Introduction

Recall that a  proportional relationship  can be represented in a table, in an equation, verbally, or graphically. A  constant of proportionality  is the constant value of the ratio between two proportional quantities.

In this lesson, we will compare two different proportional relationships represented by graphs, equations, and tables in order to determine which has a greater rate of change.

Compare proportional relationships

In order to compare proportional relationships represented in different ways, we start by identifying the unit rate (or slope) from the graphs, tables, or equation.

Given an equation of proportional relationship:

\displaystyle y=mx
\bm{m}
is the unit rate or slope of the line

Examples

Example 1

Which describes a greater unit rate of change of y with respect to x?

  • Relationship A:

    y=5x

  • Relationship B:

    x0123
    y04.5913.5
Worked Solution
Create a strategy

Find the slope of the two and compare them. A higher slope means a greater rate of change in y.

Apply the idea

For Relationship A, y=5x is in the form y=mx.

Therefore the slope, or unit rate of change, is 5.

For Relationship B, use points \left(1,\,4.5\right) and \left(2,9 \right) to find the slope (any two points can be used):

\displaystyle m\displaystyle =\displaystyle \dfrac{y_2-y_1}{x_2-x_1}Formula for slope
\displaystyle =\displaystyle \dfrac{9-\,4.5}{2-1}Substitute the coordinates
\displaystyle =\displaystyle 4.5Evaluate

The slope of Relationship A is greater than the slope of Relationship B which means that Relationship A has a greater unit rate of change.

Example 2

Which describes a greater unit rate of change of y with respect to x?

  • Relationship A:

    -5
    -4
    -3
    -2
    -1
    1
    2
    3
    4
    5
    x
    -30
    -20
    -10
    10
    20
    30
    y

  • Relationship B:

    x2468
    y12243648
Worked Solution
Create a strategy

Find the slope of the two and compare them. A higher slope means a greater rate of change in y.

Apply the idea

For Relationship A, identify two points on the line and use them to find the slope:

-5
-4
-3
-2
-1
1
2
3
4
5
x
-30
-20
-10
10
20
30
y
\displaystyle m\displaystyle =\displaystyle \dfrac{y_2-y_1}{x_2-x_1}Formula for slope
\displaystyle =\displaystyle \dfrac{20-0}{3-0}Substitute the coordinates
\displaystyle =\displaystyle \dfrac{20}{3}Evaluate

For Relationship B, use points (2, 12) and (4, 24) to find the slope:

\displaystyle m\displaystyle =\displaystyle \dfrac{y_2-y_1}{x_2-x_1}Formula for slope
\displaystyle =\displaystyle \dfrac{24-12}{4-2}Substitute the coordinates
\displaystyle =\displaystyle 6Evaluate
\displaystyle \dfrac{20}{3}\displaystyle >\displaystyle 6Compare slopes

The slope of Relationship A is greater than the slope of Relationship B which means that it has a greater unit rate of change.

Idea summary

When comparing proportional relationships, we need to compare the unit rate of change represented by the slope of the line. Equations written in the form y=mx are handy because they are the easiest form to identify the slope of the line from.

\displaystyle y=mx
\bm{m}
is the slope or the unit rate of change

If we don't have the equation of the line we can use the following to find the slope from the coordinates of two identified points:

\displaystyle m=\dfrac{y_2-y_1}{x_2-x_1}
\bm{m}
is the slope or the unit rate of change
\bm{y_2-y_1}
is the change in y
\bm{x_2-x_1}
is the change in x

Outcomes

8.EE.B.5

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

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