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 Lesson Plan: Unit Rates and Complex Fractions

 

Lesson Summary

This lesson introduces seventh-grade students to the concepts of unit rates and complex fractions, emphasizing their application in real-world scenarios. Students will learn to:

  • Calculate Unit Rates: Determine the ratio of two quantities, where the second quantity is one unit, to facilitate comparisons.​
  • Simplify Complex Fractions: Simplify fractions that have fractions in the numerator, denominator, or both.​
  • Solve Real-World Problems: Apply knowledge of unit rates and complex fractions to practical situations, such as recipe adjustments and measurements.

Lesson Objectives

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: 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.

Multimedia Resources

 

 

Warm Up Activities

Choose from one or more activities.

Activity 1: Ratio of Ingredients with Fractions

Objective: Reinforce students’ understanding of ratios that involve fractions.

Instructions:

  1. Present the scenario: "A baker uses \( \frac{3}{4} \) cup of sugar and \( \frac{1}{2} \) cup of flour to make a batch of cookies. What is the ratio of sugar to flour?"
  2. Set up the ratio: \[ \frac{\frac{3}{4} \text{ cup sugar}}{\frac{1}{2} \text{ cup flour}} \]
  3. Simplify by dividing the fractions: \[ \frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2} \]
  4. Express the ratio in different forms:
    • Fraction form: \( \frac{3}{2} \)
    • Standard form: 3:2
    • Word form: "3 to 2"
  5. Conclude: "For every 3 parts sugar, there are 2 parts flour."

Activity 2: Ratios with Three Items

Objective: Help students understand ratios involving three different quantities.

Instructions:

  • Present a scenario: "A sports equipment store has 18 basketballs, 24 soccer balls, and 30 baseballs. Write the ratio of basketballs to soccer balls to baseballs."
  • Have students express the ratio in standard form:

18:24:30

  • Ask: "Can this ratio be simplified?" Guide them to find the greatest common factor (GCF), which is 6.
  • Divide each term by 6:

\[ 18 \div 6 = 3 \]

\[ 24 \div 6 = 4 \]

\[ 30 \div 6 = 5 \]

  • Write the simplified ratio: 3:4:5.
  • Conclude: "The simplified ratio of basketballs to soccer balls to baseballs is 3:4:5."

Activity 3: Ratios with Decimals

Objective: Introduce students to ratios that include decimals and how to convert them into whole-number ratios.

Instructions:

  1. Write the ratio: "A store sells 1.5 liters of juice for every 2.5 liters of soda. Express the ratio as whole numbers."
  2. Convert to a fraction: \( 1.5:2.5 = \frac{1.5}{2.5} \).
  3. Multiply both terms by 10 to eliminate decimals: \( 15:25 \).
  4. Simplify by dividing both numbers by 5: \( 3:5 \).
  5. Conclude: "The ratio of juice to soda is 3:5."

 

 

Teach

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: Finding the Ratio of Two Quantities (Standard Form)

Problem: A baker uses \( \frac{5}{6} \) cup of sugar for every \( \frac{2}{3} \) cup of flour. What is the ratio of sugar to flour?

Solution:

  1. Write the ratio in standard form: \[ \frac{5}{6} : \frac{2}{3} \]
  2. Rewrite as a complex fraction: \[ \frac{\frac{5}{6}}{\frac{2}{3}} \]
  3. Divide the fractions by multiplying by the reciprocal: \[ \frac{5}{6} \times \frac{3}{2} = \frac{15}{12} \]
  4. Simplify: \[ \frac{15}{12} = \frac{5}{4} \]
  5. Write in standard ratio form: \[ 5:4 \]
  6. Final Answer: The ratio of sugar to flour is 5:4.

Example 2: Finding the Speed of a Hiker

Problem: A hiker covers \( \frac{7}{8} \) of a mile in \( \frac{1}{4} \) of an hour. What is the hiker's speed in miles per hour?

Solution:

  1. Write the speed as a rate of distance over time: \[ \frac{\text{miles}}{\text{hours}} = \frac{7}{8} \div \frac{1}{4} \]
  2. Rewrite as a multiplication problem using the reciprocal: \[ \frac{7}{8} \times \frac{4}{1} \]
  3. Multiply the fractions: \[ \frac{7 \times 4}{8 \times 1} = \frac{28}{8} \]
  4. Simplify: \[ \frac{28}{8} = \frac{7}{2} \]
  5. Convert to a mixed number: \[ 3 \frac{1}{2} \text{ miles per hour} \]
  6. Final Answer: The hiker's speed is 3 1/2 miles per hour, which is a reasonable pace for hiking.

Example 3: Using Ratios in a Recipe

Problem: A soup recipe calls for \( \frac{2}{3} \) cup of broth for every \( \frac{1}{4} \) cup of rice. What is the ratio of broth to rice?

Solution:

  1. Write the ratio in standard form: \[ \frac{2}{3} : \frac{1}{4} \]
  2. Rewrite as a complex fraction: \[ \frac{\frac{2}{3}}{\frac{1}{4}} \]
  3. Divide the fractions by multiplying by the reciprocal: \[ \frac{2}{3} \times \frac{4}{1} = \frac{8}{3} \]
  4. Write in standard ratio form: \[ 8:3 \]
  5. Final Answer: The ratio of broth to rice is 8:3.

Example 4: Ratio of Juice Concentrate to Water

Problem: A juice mix contains \( \frac{3}{5} \) cup of concentrate for every \( \frac{2}{3} \) cup of water. What is the ratio of concentrate to water?

Solution:

  1. Write the ratio in standard form: \[ \frac{3}{5} : \frac{2}{3} \]
  2. Rewrite as a complex fraction: \[ \frac{\frac{3}{5}}{\frac{2}{3}} \]
  3. Divide the fractions by multiplying by the reciprocal: \[ \frac{3}{5} \times \frac{3}{2} = \frac{9}{10} \]
  4. Write in standard ratio form: \[ 9:10 \]
  5. Final Answer: The ratio of juice concentrate to water is 9:10.

 

 

Review

Lesson Summary

In this lesson, students explored unit rates and complex fractions, learning how to simplify ratios that involve fractions and apply them to real-world problems. They practiced:

  • Expressing ratios in standard form (a:b).
  • Using complex fractions to solve unit rate problems.
  • Simplifying ratios by dividing fractions.
  • Applying ratios and unit rates to real-world situations, such as recipes, speed, and measurements.

By mastering these concepts, students can confidently analyze proportional relationships and use mathematical reasoning in everyday applications.

Key Vocabulary

  • Ratio: A comparison of two quantities, often expressed in the form a:b.
  • Unit Rate: A ratio in which the second term is one unit (e.g., miles per hour, price per item).
  • Complex Fraction: A fraction where the numerator, denominator, or both contain fractions.
  • Reciprocal: The inverse of a fraction, used when dividing fractions (e.g., the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \)).
  • Proportional Relationship: A relationship where two quantities maintain a constant ratio.

Additional Worked-Out Examples

Example 1: Finding the Ratio of Ingredients

Problem: A salad dressing recipe requires \( \frac{3}{5} \) cup of olive oil and \( \frac{2}{3} \) cup of vinegar. What is the ratio of olive oil to vinegar?

Solution:

  1. Write the ratio in standard form: \[ \frac{3}{5} : \frac{2}{3} \]
  2. Rewrite as a complex fraction: \[ \frac{\frac{3}{5}}{\frac{2}{3}} \]
  3. Multiply by the reciprocal: \[ \frac{3}{5} \times \frac{3}{2} \]
  4. Multiply: \[ \frac{3 \times 3}{5 \times 2} = \frac{9}{10} \]
  5. Write in standard form: \[ 9:10 \]
  6. Final Answer: The ratio of olive oil to vinegar is 9:10.

Example 2: Finding the Speed of a Cyclist

Problem: A recreational cyclist covers \( \frac{9}{2} \) miles in \( \frac{3}{4} \) of an hour. What is the cyclist's speed in miles per hour?

Solution:

  1. Write the speed as a rate: \[ \frac{\text{miles}}{\text{hours}} = \frac{9}{2} \div \frac{3}{4} \]
  2. Rewrite as a multiplication problem using the reciprocal: \[ \frac{9}{2} \times \frac{4}{3} \]
  3. Multiply: \[ \frac{9 \times 4}{2 \times 3} = \frac{36}{6} \]
  4. Simplify: \[ \frac{36}{6} = 6 \]
  5. Final Answer: The cyclist's speed is 6 miles per hour, which is a reasonable speed for leisurely biking.

Example 3: Finding the Ratio of Paint Colors

Problem: A paint mixture contains \( \frac{2}{5} \) gallon of red paint and \( \frac{3}{10} \) gallon of blue paint. What is the ratio of red to blue paint?

Solution:

  1. Write the ratio in standard form: \[ \frac{2}{5} : \frac{3}{10} \]
  2. Rewrite as a complex fraction: \[ \frac{\frac{2}{5}}{\frac{3}{10}} \]
  3. Multiply by the reciprocal: \[ \frac{2}{5} \times \frac{10}{3} \]
  4. Multiply: \[ \frac{2 \times 10}{5 \times 3} = \frac{20}{15} \]
  5. Simplify: \[ \frac{20}{15} = \frac{4}{3} \]
  6. Write in standard form: \[ 4:3 \]
  7. Final Answer: The ratio of red to blue paint is 4:3.

 

 

Quiz

Answer the following question.

 

  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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