# 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 |

Ask students to:

- Determine if this is a proportional relationship
- Identify the constant of proportionality (pace in minutes per mile)
- Predict the time for a 13-mile run (half marathon)

Solutions:

- Yes, this is a proportional relationship. The ratio of time to distance is constant (8:1).
- The constant of proportionality is 8 minutes per mile.
- 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

Is the relationship between x and y proportional?

x y 2 6 4 12 6 18 8 24 - What is the constant of proportionality in the relationship from question 1?

- Does the graph of y = 2x + 1 represent a proportional relationship?

- If a car travels 240 miles in 4 hours at a constant speed, what is the constant of proportionality?

- In the equation y = kx, what does k represent?

- Is the origin always included in the graph of a proportional relationship?

- If 3 shirts cost $24, how much would 5 shirts cost in this proportional relationship?

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

Does the table represent a proportional relationship?

x y 0 3 2 5 4 7 6 9 - If y is proportional to x and y = 15 when x = 3, what is the constant of proportionality?

## Answer Key

- Yes
- 3
- No
- 60 miles per hour
- The constant of proportionality
- Yes
- $40
- 1.25 cubic inches per ounce
- No
- 5

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