# 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

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:

xy
26
412
618
824
1030

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

### 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)
00
21
42
63
84
• 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)
52
104
156
208
• 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 (\$)
100,0005,000
200,00010,000
300,00015,000
400,00020,000

1. Determine if this is a proportional relationship
2. Identify the constant of proportionality
3. Calculate the commission for a \$350,000 house sale Solutions: 1. Yes, this is a proportional relationship. The ratio of commission to house price is constant (1:20 or 0.05). 2. The constant of proportionality is 0.05 or 5%. 3. 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) 324 540 756 1080 Ask students to: 1. Determine if this is a proportional relationship 2. Identify the constant of proportionality (pace in minutes per mile) 3. Predict the time for a 13-mile run (half marathon) Solutions: 1. Yes, this is a proportional relationship. The ratio of time to distance is constant (8:1). 2. The constant of proportionality is 8 minutes per mile. 3. Time for a 13-mile run: 13 * 8 = 104 minutes or 1 hour and 44 minutes ## Assess (10 minutes) Use this 10-question quiz for assessment. ## Quiz 1. Is the relationship between x and y proportional? xy 26 412 618 824 2. What is the constant of proportionality in the relationship from question 1? 3. Does the graph of y = 2x + 1 represent a proportional relationship? 4. If a car travels 240 miles in 4 hours at a constant speed, what is the constant of proportionality? 5. In the equation y = kx, what does k represent? 6. Is the origin always included in the graph of a proportional relationship? 7. If 3 shirts cost$24, how much would 5 shirts cost in this proportional relationship?

8. What is the constant of proportionality if 8 ounces of a liquid occupy 10 cubic inches?

9. Does the table represent a proportional relationship?

xy
03
25
47
69
10. If y is proportional to x and y = 15 when x = 3, what is the constant of proportionality?

1. Yes
2. 3
3. No
4. 60 miles per hour
5. The constant of proportionality
6. Yes
7. \$40
8. 1.25 cubic inches per ounce
9. No
10. 5