Princeton Review

 Lesson Plan: Unit Rates and Complex Fractions

Lesson Objectives

This lesson can be completed in one 50-minute class period. 

In this lesson students will be shown how to:

  • Review unit rates
  • Introduce complex fractions
  • Simplify complex fractions
  • Solve real-world problems involving unit rates and complex fractions

Florida BEST Standards

MA.7.AR.3.1: Apply previous understanding of percentages and ratios to solve multi-step real-world problems.

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

  1. 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?

     
  2. In a scale model, 1/2 inch represents 3 feet. If the model is 4 inches long, how long is the actual object?

     
  3. 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?

     
  4. If 2/3 of a cake serves 8 people, how many people will a whole cake serve?

     
  5. A runner completes 4 km in 1/4 hour. What is the runner's speed?

     
  6. In a recipe, the ratio of flour to sugar is 1 3/4 : 1 1/4. Express this as ratio with whole numbers.

     
  7. 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?

     
  8. If 5/8 of a gallon of paint covers 100 square feet, how many square feet will 1 gallon cover?

     
  9. A car travels 45 3/4 miles in 3/4 hour. What is its speed in miles per hour?

     
  10. 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

  1. 2:3
  2. 24 feet
  3. 11:14
  4. 12 people
  5. 16 km per hour
  6. 7/5 
  7. 7.24 inches (approximately)
  8. 160 square feet
  9. 61 miles per hour
  10. 3 cups of oranges

 

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