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

• A collection of definitions on the topic of ratios, proportions, and percents: https://www.media4math.com/Definitions--RatiosProportionsPercents
• A student tutorial slide show on definitions on the topic of ratios, proportions, and percents: https://www.media4math.com/library/slideshow/student-tutorial-ratios-proportions-and-percents-definitions

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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