Lesson Plan: Unit Rates and Complex Fractions

Lesson Objectives

This lesson can be completed in one 50-minute class period. You can also refer to this grade 6 lesson:

In this lesson students will be shown how to:

Calculate unit rates
Solve problems involving unit rates
Work with complex fractions in ratios

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 understanding of ratios
Division of fractions

Key Vocabulary

Unit rate
Complex fraction
Denominator
Numerator
Tape diagram

Warm-up Activity (10 minutes)

Find a ratio involving fractions

Example: In a recipe, 3/4 cup of flour is used for every 1/3 cup of oil. What is the ratio of flour to oil?
Solution: The ratio of flour to oil is 3/4 : 1/3, which can be simplified to 9:4 by multiplying both fractions by 12 (the least common multiple of 4 and 3).
Have students practice with similar examples, such as ratios of ingredients in recipes or measurements in construction projects.

Teach (20 minutes)

Definitions

Unit rate: A ratio that compares quantities of different items, where the second quantity is one unit.
Complex fraction: A fraction that contains fractions in the numerator, denominator, or both.
Denominator: The bottom number in a fraction, representing the number of equal parts the whole is divided into.
Numerator: The top number in a fraction, representing the number of parts being considered.
Tape diagram: A visual model that uses rectangles to represent ratios and proportions.

Use this slide show to review these and other definitions:

https://www.media4math.com/library/slideshow/definitions-ratios-rates-and-complex-fractions

Instruction

Introduce this video, which covers ratios with fractions. Have students develop the technique of transforming these ratios into those with whole numbers:

https://www.media4math.com/library/1792/asset-preview

Use this slide show to demonstrate examples of ratios with fractions:

https://www.media4math.com/library/slideshow/math-examples-simplifying-ratios-fractions

Explain the concept of unit rates and demonstrate how to calculate them with complex fractions using the following examples:

Example 1. Recipe Scaling

A recipe for making 20 cookies calls for 3/4 cup of sugar and 2/3 cup of flour. How much flour and sugar is needed to make 35 cookies?

First, find the unit rate for each ingredient per cookie:
Sugar: (3/4) ÷ 20 = 3/80 cup per cookie
Flour: (2/3) ÷ 20 = 1/30 cup per cookie
Then, multiply each unit rate by 35 to find the amounts needed for 35 cookies:
Sugar: (3/80) * 35 = 105/80 = 1 25/80 cups = 1 5/16 cups
Flour: (1/30) * 35 = 35/30 = 1 5/30 cups = 1 1/6 cups
The unit rates are 3/80 cup of sugar per cookie and 1/30 cup of flour per cookie.

Example 2. Garden Plot Fencing

A rectangular garden plot is 3 1/2 feet wide and 4 2/3 feet long. Find the ratio of the width to the length. What is the minimum amount of fencing to purchase to have double fencing around the garden?

Ratio: 3 1/2 : 4 2/3 = 21/6 : 28/6 = 21:28 (simplified to 3:4)
Calculate the perimeter of the garden:
2 * (3 1/2 + 4 2/3) = 2 * (3 1/2 + 4 4/6) = 2 * (3 1/2 + 4 2/3) = 2 * 8 1/6 = 16 1/3 feet
Double the perimeter for double fencing:
2 * 16 1/3 = 32 2/3 feet
The minimum amount of fencing to purchase is 32 2/3 feet.

Example 3. Scale Model Ratio

In a scale model, 3/8 inch represents 2 1/4 feet of actual size. What is the ratio of model size to actual size? If the scale model is 6 inches tall, how tall is the actual structure?

Ratio: 3/8 : 2 1/4 = 3/8 : 9/4 = 1:6
To find the actual height, use equivalent ratios:
1 unit : 6 units
6 inches : x feet
Set up the proportion: 1 : 6 = 6 : x
Cross multiply: 1 * x = 6 * 6
Solve for x: x = 36 feet
The actual structure is 36 feet tall.

Review (10 minutes)

Practice calculating unit rates in various contexts and simplifying complex fractions

Example 1 (Recipe)

In a cake recipe, the ratio of flour to sugar is 2 3/4 : 1 1/2. Express this as a complex fraction and simplify it.

Solution: (2 3/4) ÷ (1 1/2) = 11/4 ÷ 3/2 = (11/4) * (2/3) = 11/6 ≈ 1.83
This means there are about 1.83 parts flour for every 1 part sugar.

Example 2 (Scale Model)

A model car is built at a scale of 1:24. The actual car's length is 15 3/4 feet. What is the length of the model car?

Solution: Set up the ratio: 1 : 24 = x : (15 3/4)
Cross multiply: 24x = 15 3/4
Solve for x: x = (15 3/4) ÷ 24 = 63/4 ÷ 24 = 63/96 = 21/32 ≈ 0.66 feet or about 7 7/8 inches

Assess (10 minutes)

10-question quiz

Quiz

A recipe for 12 muffins calls for 1/3 cup of oil and 1/2 cup of sugar. What is the ratio of oil to sugar expressed as whole numbers?
In a scale model, 1/2 inch represents 3 feet. If the model is 4 inches long, how long is the actual object?
A garden is 2 3/4 feet wide and 3 1/2 feet long. What is the ratio of width to length in simplest form?
If 2/3 of a cake serves 8 people, how many people will a whole cake serve?
A runner completes 4 km in 1/4 hour. What is the runner's speed?
In a recipe, the ratio of flour to sugar is 1 3/4 : 1 1/4. Express this as ratio with whole numbers.
A model train is built at a scale of 1:87. If the actual train is 52 1/2 feet long, how long is the model in inches?
If 5/8 of a gallon of paint covers 100 square feet, how many square feet will 1 gallon cover?
A car travels 45 3/4 miles in 3/4 hour. What is its speed in miles per hour?
In a fruit salad, the ratio of apples to oranges is 2 1/3 : 1 3/4. How many cups of oranges are needed if 4 cups of apples are used?

Answer Key

2:3
24 feet
11:14
12 people
16 km per hour
7/5
7.24 inches (approximately)
160 square feet
61 miles per hour
3 cups of oranges

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