Lesson Plan: Identifying Proportional Relationships

Lesson Objectives

This lesson can be completed in one 50-minute class period but may require additional time depending on your class.

Identify proportional relationships in various representations
Understand and calculate the constant of proportionality
Apply proportional reasoning to solve real-world problems

TEKS Standards

7.4A: Represent constant rates of change in various forms
7.4C: Determine the constant of proportionality
7.4D: Solve problems involving ratios, rates, and percents

Prerequisite Skills

Understanding of ratios and unit rates
Basic graphing skills

Key Vocabulary

Proportion
Constant of proportionality
Origin
Linear relationship

Additional Resources

As needed, refer to these grade 6 lesson plans:

Warm-up Activity (10 minutes)

Identify equivalent ratios in a table

Example: Given the following table of values, identify pairs of numbers that form equivalent ratios:

x	y
2	6
4	12
6	18
8	24
10	30

Solution: All pairs form equivalent ratios (1:3), as y is always 3 times x. You can use this Desmos activity to graph the coordinates and find the line of best fit:

https://www.desmos.com/calculator/gazl6xwvfg

Teach (20 minutes)

Definitions

Proportion: An equation stating that two ratios are equal
Constant of proportionality: The constant ratio between two proportional quantities
Origin: The point (0,0) on a coordinate plane
Linear relationship: A relationship that forms a straight line when graphed

Use this slide show to review these other related definitions:

https://www.media4math.com/library/slideshow/definitions-proportions-and-proportional-relationships

Instruction

Example 1. Science: Hooke's Law (Force and Spring Extension)

This slide show discusses Hooke's Law in detail.

https://www.media4math.com/library/slideshow/application-linear-functions-hookes-law

Use it as background to frame the following problem solving scenario:

A physics student is investigating the relationship between the force applied to a spring and its extension. She records the following measurements:

Force (N)	Extension (cm)
0	0
2	1
4	2
6	3
8	4

Graph: Plot points and observe direct proportion. Use this Desmos activity:
https://www.desmos.com/calculator/lr5b4efzgq
Constant of proportionality: Extension/Force = 0.5 cm/N
Explain how this demonstrates a direct proportional relationship.
Discuss real-world applications, such as in the design of suspension systems or measuring instruments.

Example 2. Engineering: Gear Ratios

Use this slide show to demonstrate this problem solving scenario with gear ratios:

https://www.media4math.com/library/slideshow/applications-gear-ratios

Here is a summary of the scenario

An engineer is designing a gear system for a new machine. She needs to determine the relationship among the number of teeth for each of the gears:

Gear A has 20 teeth
Gear B has 12 teeth
Gear C has 8 teeth

Determine the number of turns gear C has to make in order for gears A and B complete at least one turn.

The gear ratio, in simplified form is this: 5:3:2
The revolution ratio, in simplified form is this: 2:3:5
Gear C must complete at least 2.5 turns for gears A and B to complete at least one turn

Example 3. Art: Color Mixing

Word problem: An artist is creating a new shade of green by mixing yellow and blue paint. He wants to ensure he can consistently reproduce this color:

Yellow Paint (mL)	Blue Paint (mL)
5	2
10	4
15	6
20	8

Graph: Plot points and observe direct proportion
Constant of proportionality: Blue/Yellow = 0.4
Explain how this is used in creating consistent shades of green
Discuss applications in graphic design, painting, and digital art

Review (10 minutes)

Practice identifying proportional relationships in various representations
Determine the constant of proportionality in different contexts
Have students work in pairs or small groups to analyze given data sets

Example 1 (Business): Sales Commission

A real estate agent earns a 5% commission on each house sale. The following table shows the commission earned for different house prices:

House Price (\$)	Commission (\$)
100,000	5,000
200,000	10,000
300,000	15,000
400,000	20,000

Ask students to:

Determine if this is a proportional relationship
Identify the constant of proportionality
Calculate the commission for a \$350,000 house sale

Solutions:

Yes, this is a proportional relationship. The ratio of commission to house price is constant (1:20 or 0.05).
The constant of proportionality is 0.05 or 5%.
Commission for \$350,000 sale: 350,000 * 0.05 = \$17,500

Example 2 (Sports): Running Pace

A runner is training for a marathon and records her distance and time for several runs:

Distance (miles)	Time (minutes)
3	24
5	40
7	56
10	80