Equivalent Ratios and Proportional Relationships 

Lesson Summary

In this lesson, students delve into the concept of equivalent ratios, building upon their foundational understanding of ratios. They will learn to identify, generate, and represent equivalent ratios using various methods, including ratio tables and graphs. Through real-world applications, students will explore proportional relationships and develop strategies to solve problems involving ratios and proportions. This lesson aligns with Common Core Standards 6.RP.A.1 and 6.RP.A.3, ensuring a comprehensive grasp of equivalent ratios and their practical significance.

Lesson Objectives

• Understand the concept of equivalent ratios
• Generate equivalent ratios
• Use ratio tables to represent equivalent ratios
• Identify proportional relationships in tables and graphs
• Solve real-world problems involving ratios and proportions

Common Core Standards

• 6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
• 6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
• 6.RP.A.3a Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
• 6.RP.A.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

Prerequisite Skills

• Understanding of ratios
• Multiplication and division skills

Key Vocabulary

• Equivalent Ratios: Ratios that express the same relationship between quantities, even though the numbers may differ. For example, 2:3 and 4:6 are equivalent because both represent the same proportional relationship.
  • Multimedia Resource: https://www.media4math.com/library/43384/asset-preview
• Ratio Table: A structured arrangement of equivalent ratios displayed in rows and columns, used to identify patterns and solve problems related to proportional relationships.
• Scale Factor: A number by which all components of a ratio are multiplied or divided to produce an equivalent ratio. For instance, multiplying both terms of the ratio 3:4 by 2 yields the equivalent ratio 6:8.
  • Multimedia Resource: https://www.media4math.com/library/43390/asset-preview
• Proportional Relationship: A consistent relationship between two quantities where their ratio remains constant. This means that as one quantity changes, the other changes in a way that the ratio between them stays the same.
  • Multimedia Resource: https://www.media4math.com/library/22139/asset-preview
• Proportion: An equation that asserts two ratios are equal. For example, the statement 1/2 = 2/4 indicates that the two ratios form a proportion.
  • Multimedia Resource: https://www.media4math.com/library/43387/asset-preview

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 of these activities.

Activity 1: Review of Ratios

Review the following definitions:

• Ratio: https://www.media4math.com/library/22157/asset-preview
• Visualizing ratios: https://www.media4math.com/library/43385/asset-preview

Show two sets of objects with the same ratio (e.g., 2 apples to 3 oranges, and 4 apples to 6 oranges). Ask students to describe the relationship between the two sets. Encourage them to use ratio language and explain why these ratios are equivalent. 

Activity 2: Brief Review of Equivalent Fractions

Before introducing equivalent ratios, students will review equivalent fractions to reinforce the concept of proportional relationships.

Instructions:

1. Write the following fraction pairs on the board and ask students if they are equivalent:
   • $\frac{1}{2}$ and $\frac{2}{4}$
   • $\frac{3}{5}$ and $\frac{6}{10}$
   • $\frac{4}{7}$ and $\frac{8}{14}$
2. Have students simplify the second fraction in each pair by dividing both the numerator and denominator by their greatest common factor (GCF).
3. Discuss how multiplying or dividing both terms of a fraction by the same number does not change its value, laying the foundation for understanding equivalent ratios.
4. Introduce the idea that just like equivalent fractions, ratios can be scaled up or down while maintaining the same relationship.

Activity 3: Using Desmos to Simplify Fractions

Students will use the Desmos Scientific Calculator to practice simplifying fractions, reinforcing the concept of proportional relationships.

Instructions:

1. Open the Desmos Scientific Calculator.
2. Have students enter the fractions from Activity 1 into the calculator using division:
   • Type "1 ÷ 2" and "2 ÷ 4" to compare their decimal values.
   • Repeat with "3 ÷ 5" and "6 ÷ 10" to verify equivalence.
3. Guide students to notice that equivalent fractions have the same decimal representation.
4. Explain that this property also applies to ratios, and later in the lesson, they will use similar methods to verify equivalent ratios.
5. Also make a note that equivalent fractions can be simplified using the calculator.

Teach

Definitions

Introduce the following definitions:

• Equivalent ratios: https://www.media4math.com/library/43384/asset-preview
• Visualizing equivalent ratios: https://www.media4math.com/library/43386/asset-preview
• Proportion: https://www.media4math.com/library/43387/asset-preview
• Proportional relationship: https://www.media4math.com/library/22139/asset-preview
• Scale factor: https://www.media4math.com/library/43390/asset-preview

Define equivalent ratios as ratios that represent the same relationship between two quantities. 

Connect equivalent ratios to equivalent fractions. For example, the ratio 2:3 can be written as the fraction 2/3. Multiplying both the numerator and the denominator by the same number (e.g., 2) gives 4/6, which is equivalent to 2/3. 

Therefore, 2:3 and 4:6 are equivalent ratios.

Generate Equivalent Ratios

Use this video to introduce the concept of equivalent ratios. In the video are three math examples that involve equivalent ratios.

https://www.media4math.com/library/1795/asset-preview

Show that multiplying or dividing both terms of a ratio by the same non-zero number creates an equivalent ratio. 

Example: 

2:3 = (2×2):(3×2) = 4:6. 

Emphasize that this is similar to finding equivalent fractions.

A useful application of equivalent raios is simplifying ratios that include fractions. Use this slide show to demonstrate examples of this technique, but reinforce that this is an application of equivalent fractions:

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

Deepen Understanding of Ratio Tables

Review the concept of ratio tables introduced in Lesson 1. Explain that ratio tables help organize and compare equivalent ratios systematically. 

Use this slide show to explore an application of equivalent fractions and ratio tables in the context of cooking:

https://www.media4math.com/library/slideshow/applications-equivalent-ratios-cooking

Demonstrate how to fill in missing values in a ratio table by scaling up or down. Show how to use the table to find equivalent ratios and solve problems.

Real-World Applications

Show this video, which focuses on ratios with fractions, to continue working with equivalent ratios:

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

Proportions

Explain that proportional relationships show a constant ratio between two quantities. Introduce the concept of a proportion, which is an equation stating that two ratios are equivalent (e.g., 2/3 = 4/6). 

Use this video to introduce proportions:

https://www.media4math.com/library/1798/asset-preview

 Demonstrate how to recognize proportional relationships in tables (constant ratio) and graphs (straight line through the origin).

Example 1: Scaling Up a Recipe

Problem: A cookie recipe calls for 2 cups of flour and 3 cups of sugar to make 12 cookies. How much flour and sugar are needed to make 36 cookies?

Solution:

1. Find the factor to scale up from 12 cookies to 36 cookies. Since $36 \div 12 = 3$, multiply both parts of the ratio by 3.
2. Scale up the flour ratio: $$\frac{2}{12} \to \frac{4}{24} \to \frac{6}{36}$$
3. Scale up the sugar ratio: $$\frac{3}{12} \to \frac{6}{24} \to \frac{9}{36}$$

Final Answer: To make 36 cookies, you need 6 cups of flour and 9 cups of sugar.

Example 2: Scaling Down a Mixture

Problem: A paint company uses a 4:3 ratio of white paint to blue paint to create a specific shade. If a customer only needs a smaller batch with 2 gallons of white paint, how many gallons of blue paint are needed?

Solution:

1. The original ratio is 4:3 (white to blue).
2. To scale down from 4 gallons of white to 2 gallons, determine the factor: $2 \div 4 = \frac{1}{2}$.
3. Multiply the entire ratio by $\frac{1}{2}$: $$\frac{4}{3} \to \frac{2}{1.5}$$

Final Answer: The customer needs 1.5 gallons (or 1 ½ gallons) of blue paint.

Example 3: Using a Map Scale to Find Distance

Problem: A map has a scale of 1 inch representing 50 miles. If the distance between two cities on the map measures 3.5 inches, what is the actual distance between the cities?

Solution:

1. The base ratio is 1 inch : 50 miles.
2. To scale up to 3.5 inches, multiply both numbers by 3.5: $$\frac{1}{50} \to \frac{3.5}{175}$$

Final Answer: The actual distance between the two cities is 175 miles.

Example 4: Estimating Wildlife Population

Problem: Biologists capture, tag, and release 40 deer in a wildlife reserve. A week later, they capture a sample of 100 deer and find that 8 of them are tagged. Estimate the total deer population in the reserve.

Solution:

1. The base ratio is 40 tagged deer in the total population.
2. In a sample of 100 deer, 8 are tagged. To find the factor: $40 \div 8 = 5$.
3. Multiply 100 by 5 to estimate the total population: $$\frac{40}{\text{Total Population}} \to \frac{8}{100} \to \frac{40}{500}$$

Final Answer: The estimated deer population in the reserve is 500.

Example 5: Mixing Ingredients in a Factory

Problem: A factory produces a cleaning solution using a 4:1 ratio of water to disinfectant. If they need to produce a batch with 80 gallons of water, how much disinfectant should be added?

Solution:

1. The base ratio is 4:1 (water to disinfectant).
2. To scale up from 4 gallons of water to 80 gallons, determine the factor: $80 \div 4 = 20$.
3. Multiply the disinfectant amount by 20: $$\frac{4}{1} \to \frac{80}{20}$$

Final Answer: The factory should add 20 gallons of disinfectant.

Example 6: Using a Ratio Table to Scale Up a Recipe

Problem: A recipe requires 3 cups of sugar to make 6 servings of cake. How much sugar is needed to make 24 servings?

Solution:

We can use a ratio table to find the equivalent ratio.

1. The original ratio is 3:6 (sugar to servings).
2. To find the number of cups needed for 24 servings, we scale up by multiplying both numbers by 4.
3. The ratio 3:6 is equivalent to 12:24.

Final Answer: 12 cups of sugar are needed for 24 servings.

Review

Summary of the Lesson

In this lesson, students explored equivalent ratios and how they can be used to solve real-world problems. Key takeaways include:

• Understanding Equivalent Ratios: Ratios that express the same relationship can be scaled up or down using multiplication or division.
• Scaling Up: When increasing the total quantity while maintaining the same ratio, multiply both terms of the ratio by the same factor.
• Scaling Down: When reducing the total quantity while maintaining the same ratio, divide both terms of the ratio by the same factor.
• Using a Map Scale: Map distances can be converted to real-world distances by using equivalent ratios.

Key Vocabulary Review

• Equivalent Ratios: Two or more ratios that represent the same proportional relationship.
• Scaling Up: Multiplying both parts of a ratio by the same factor to increase the total quantity.
• Scaling Down: Dividing both parts of a ratio by the same factor to decrease the total quantity.
• Proportion: An equation showing that two ratios are equivalent.
• Ratio Table: A table that lists pairs of equivalent ratios.
• Map Scale: A ratio that relates map distances to real-world distances.

Multimedia Resources

Use this video to review ratio problems:

https://www.media4math.com/library/1794/asset-preview

Guide students through practice exercises:

1. Create equivalent ratios for 3:4
2. Complete a ratio table for the relationship "for every 2 blue marbles, there are 5 red marbles"
3. Identify whether given relationships are proportional using tables and graphs
4. Solve a problem involving equivalent ratios in a real-world context
5. Convert measurements using ratio reasoning (e.g., convert 12 inches to feet using the ratio 12 inches : 1 foot)

Example 1: Scaling Up a Recipe

Problem: A pancake recipe calls for 2 cups of flour and 3 cups of milk to make 8 pancakes. How much flour and milk are needed to make 24 pancakes?

Solution:

1. The base ratio is 2:3 (flour to milk).
2. To scale up from 8 to 24 pancakes, find the factor: $24 \div 8 = 3$.
3. Multiply both terms of the ratio by 3: $$\frac{2}{8} \to \frac{6}{24}$$ $$\frac{3}{8} \to \frac{9}{24}$$

Final Answer: To make 24 pancakes, you need 6 cups of flour and 9 cups of milk.

Example 2: Scaling Down a Mixture

Problem: A cleaning solution is made using a 5:2 ratio of water to disinfectant. If a smaller batch requires only 2.5 gallons of water, how much disinfectant should be used?

Solution:

1. The original ratio is 5:2 (water to disinfectant).
2. To scale down from 5 gallons to 2.5 gallons of water, determine the factor: $2.5 \div 5 = \frac{1}{2}$.
3. Multiply the entire ratio by $\frac{1}{2}$: $$\frac{5}{2} \to \frac{2.5}{1}$$

Final Answer: The smaller batch should use 1 gallon of disinfectant.

Example 3: Using a Map Scale

Problem: A map has a scale of 1 inch representing 75 miles. If the distance between two cities on the map is 4 inches, what is the actual distance?

Solution:

1. The base ratio is 1 inch : 75 miles.
2. To scale up to 4 inches, multiply both parts of the ratio by 4: $$\frac{1}{75} \to \frac{4}{300}$$

Final Answer: The actual distance between the two cities is 300 miles.

Example 4: Using a Ratio Table to Scale Down a Mixture

Problem: A lemonade recipe calls for 10 cups of water and 4 cups of lemon juice. If a smaller batch requires only 5 cups of water, how much lemon juice is needed?

Solution:

We use a ratio table to find the scaled-down equivalent ratio.

1. The original ratio is 10:4 (water to lemon juice).
2. Since we need only 5 cups of water, we divide both numbers by 2.
3. The ratio 10:4 is equivalent to 5:2.

Quiz

Answer the following questions.

1. What is an equivalent ratio to 2:5?
   
    
2. If 3 pizzas cost $24, how much would 5 pizzas cost in this proportional relationship?
   
    
3. Complete the ratio table: 2:3, 4:?, 6:9, ?:15
   
    
4. Is the relationship between x and y proportional if y = 2x + 1?
   
    
5. What is the constant of proportionality in the ratio 12:3?
   
    
6. If a recipe calls for 2 cups of flour for every 3 cups of milk, how many cups of milk are needed for 8 cups of flour?
   
    
7. Which pair of ratios is equivalent: 4:7 and 8:14, or 3:5 and 9:16?
   
    
8. In a proportional relationship, if x increases by a factor of 3, what happens to y?
   
    
9. In a fruit basket with a 3:2 ratio of apples to oranges, how many oranges are there if there are 15 apples?
   
    
10. If the ratio of white to brown eggs in a carton is 5:3, how many white eggs are there if there are 12 brown eggs?
    
     

Answer Key

1. 4:10 (or any ratio where the second number is 2.5 times the first)
2. $40
3. 2:3, 4:6, 6:9, 10:15
4. No (y-intercept is not 0)
5. 4
6. 12 cups
7. 4:7 and 8:14
8. y also increases by a factor of 3
9. 10 oranges
10. 20 white eggs

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