Topics
3. Ratios & Proportions
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United States of America
Grade 7

3.03 Proportional relationships in graphs

Lesson
Practice

Introduction

Proportional relationships can be represented in a table, in an equation, verbally, or graphically. Before we discuss graphing proportional relationships, it might be a good idea to do a quick review of the  coordinate plane  .

Proportional relationships in graphs

We know that a ratio compares the relationship between two values; it compares how much there is of one thing compared to another. We can therefore plot pairs of ratios on a coordinate plane.

When plotting ratios on the coordinate plane:

  • The first number in a ratio becomes the x-value on the coordinate plane.
  • The second number in that ratio becomes the y-value on a coordinate plane.

For example, the ratio 2:5 can be represented by the ordered pair \left(2,5\right).

Like we discovered in  proportional relationships in tables,  when quantities are proportional, we can use a ratio table to determine unknown values. We can graph each of these quantities as coordinates.

In a proportional relationship, the coordinates will fall on a straight line that goes through the origin, located at \left(0,0\right). Similarly, any points on the same line are proportional to each other.

Examples

Example 1

Plot 1:3 on the coordinate plane below.

1
2
3
4
x
1
2
3
4
y
Worked Solution
Create a strategy

A ratio of the form x:y, means that the x represents the horizontal position and y the vertical position of the point.

Apply the idea

The horizontal position is 1, and the vertical position is 3.

1
2
3
4
x
1
2
3
4
y

Example 2

Consider the following graph:

5
10
15
\text{green}
5
10
15
\text{red}
a

What ratio has been plotted?

Worked Solution
Create a strategy

Use the graph to find the ratio, x : y.

Apply the idea

The point (1, 2) lies on the line. At this point x=1 and y=2. So the ratio is 1 : 2.

b

Which of the following could be being represented by this graph and ratio?

A
For every 1 green mix, there are 2 red sweets.
B
For every 2 green mix, there are 1 red sweets.
Worked Solution
Create a strategy

Use the axes labels.

Apply the idea

The x-axis is labeled "green", and the y-axis is labeled "red". So the ratio x:y=1:2 is read as 1\text{ green}:2 \text{ reds}.

So the answer is option A.

Example 3

State whether each of the following graphs represents a proportional relationship:

a
5
10
15
20
x
5
10
15
20
y
Worked Solution
Create a strategy

In a proportional relationship, the coordinates will fall on a straight line that goes through the origin.

Apply the idea

The graph shows a straight line that goes through the origin. Therefore, this graph represents a proportional relationship.

b
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
Worked Solution
Create a strategy

In a proportional relationship, the coordinates will fall on a straight line which goes through the origin.

Apply the idea

The graph shows a straight line that does not go through the origin. Therefore, this graph does not represent a proportional relationship.

Idea summary

When plotting ratios on the coordinate plane:

  • The first number in a ratio becomes the x-value on the coordinate plane.
  • The second number in that ratio becomes the y-value on a coordinate plane.

When proportional relationships are graphed, they are straight lines that go through the origin.

Outcomes

7.RP.A.2

Recognize and represent proportional relationships between quantities.

7.RP.A.2.A

Decide whether two quantities are in a proportional relationship, e.g. By testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

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