# Lesson Plan: Introduction to Ratios and Unit Rates

## Lesson Objectives

- Compute unit rates associated with ratios of fractions
- Apply unit rates to solve real-world problems involving different quantities
- Use ratio reasoning to convert measurement units
- Graph linear functions representing unit rates

This lesson plan is designed for one 50-minute class period but could be extended to two periods for more in-depth practice and discussion.

## Common Core Standards

7.RP.A.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

## Prerequisite Skills

- Basic fraction knowledge
- Understanding of division
- Graphing linear functions

## Key Vocabulary

- Ratio
- Unit rate
- Equivalent ratios
- Conversion factor
- Linear function
- Slope

## Warm-up Activity (10 minutes)

Present students with the following scenario:

A car travels 180 miles in 3 hours.

Ask students to:

- Calculate the speed of the car in miles per hour
- Determine how far the car would travel in 5 hours at this rate
- Convert the speed to kilometers per hour (use 1 mile ≈ 1.60934 km)

Discuss how these calculations relate to unit rates and conversions between different units.

## Teach (30 minutes)

### Definitions

**Ratio:**A comparison between two quantities, often expressed as a:b, a to b, or a/b.**Unit rate:**A ratio where the second term is 1, used to compare quantities per unit.**Equivalent ratios:**Ratios that represent the same relationship between quantities.**Conversion factor:**A ratio used to convert one unit of measurement to another.**Linear function:**A function that produces a straight line when graphed, often in the form y = mx + b.**Slope:**The steepness of a line, calculated as the change in y divided by the change in x.

Use this slide show to review these and related definitions that are used in this lesson:

__https://www.media4math.com/library/slideshow/ratio-and-rate-definitions__

### Instruction

Use this slide show to introduce rates and unit rates:

__https://www.media4math.com/library/slideshow/introduction-rates__

As time allows show the following three examples of rates.

### Example 1: Physics - Speed Calculation (Finding the Unit Rate)

A cyclist travels 36 miles in 2.4 hours.

**Question:** What is the cyclist's speed in miles per hour?

**Solution:**

Unit rate (speed) = Distance ÷ Time = 36 miles ÷ 2.4 hours = 15 miles per hour

The cyclist's speed is 15 miles per hour.

### Example 2: Business - Production Planning (Calculating and Using Unit Rate)

A factory produces 540 widgets in a 12-hour shift.

**Questions:**

- What is the production rate in widgets per hour?
- How many widgets can the factory produce in an 8-hour shift at this rate?

**Solution:**

a) Unit rate = Total widgets ÷ Total hours = 540 widgets ÷ 12 hours = 45 widgets per hour b) Production in 8 hours = 45 widgets/hour × 8 hours = 360 widgets

The factory produces 45 widgets per hour and can produce 360 widgets in an 8-hour shift.

### Example 3: Engineering - Fuel Efficiency (Graphing and Calculating Unit Rate)

A car's fuel consumption can be represented by the following data points. For each gallon of gas used, the car travels a certain number of miles.

Gallons | Miles |

2 | 60 |

5 | 150 |

8 | 240 |

**Questions:**

- Graph these points on a coordinate plane with gallons on the x-axis and miles on the y-axis.
- Draw a line through these points.
- Calculate the slope of this line.
- What does the slope represent, and what are its units?

**Solution:**

Slope = Change in y ÷ Change in x = (240 - 60) ÷ (8 - 2) = 180 ÷ 6 = 30 The slope represents the fuel efficiency of the car. Its units are miles per gallon (mpg).

The car's fuel efficiency is 30 miles per gallon.

## Review (5 minutes)

Use this video to review unit rates:

__https://www.media4math.com/library/1797/asset-preview__

If time allows, work through another detailed example with the class:

A coffee shop sells 180 cups of coffee in 3 hours.

**Questions:**

- What is the unit rate of coffee sales per hour?
- How many cups of coffee can they expect to sell in an 8-hour day at this rate?
- Graph the relationship between hours and cups of coffee sold from 0 to 10 hours.
- Using the graph, estimate how long it would take to sell 300 cups of coffee.

Guide students through each step, encouraging participation and discussion. If necessary, refer to these grade 6 lessons on ratios for additional review:

## Assess (5 minutes)

Administer the 10-question quiz to gauge understanding.

## Quiz

- If a car travels 240 miles in 4 hours, what is its speed in miles per hour?

- Convert 65 miles per hour to kilometers per hour. (1 mile ≈ 1.60934 km)

- A recipe calls for 2/3 cup of flour for every 1/4 cup of sugar. How much flour is needed for 1 cup of sugar?

- If 18 widgets are produced in 1.5 hours, what is the production rate in widgets per hour?

- A delivery truck uses 12 gallons of gas to travel 180 miles. What is its fuel efficiency in miles per gallon?

- Convert 30 miles per gallon to kilometers per liter. (1 gallon ≈ 3.78541 liters, 1 mile ≈ 1.60934 km)

- If 5/8 kg of rice costs \$2.40, what is the price per kg?

- A printer prints 80 pages in 2 minutes. How many pages can it print in 7 minutes?

- If 3/4 cup of flour weighs 90 grams, how many grams does 1 cup of flour weigh?

- A train travels 450 km in 3 hours. What is its speed in meters per second?

## Answer Key

- 60 mph
- 104.6 km/h
- 2 2/3 cups of flour
- 12 widgets per hour
- 15 mpg
- 12.75 km/L
- \$3.84 per kg
- 280 pages
- 120 grams
- 41.67 m/s

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