Lesson Plan: Applying Ratios and Proportional Reasoning

Lesson Summary

In this lesson, students will apply their understanding of ratios and proportional reasoning to solve complex, real-world problems. They will utilize various representations, such as tables, graphs, and equations, to model and analyze situations involving proportional relationships. Through engaging activities and practical examples, students will enhance their problem-solving skills and ability to communicate mathematical reasoning effectively.

Lesson Objectives

• Apply ratio and proportional reasoning to solve complex real-world problems
• Use multiple representations to model and solve problems

Common Core Standards

• 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.

Prerequisite Skills

• Understanding of ratios, rates, and proportional relationships
• Problem-solving skills

Key Vocabulary

• Proportion: An equation stating that two ratios are equivalent. For example, if $\frac{a}{b} = \frac{c}{d}$, then $a$, $b$, $c$, and $d$ are in proportion.
  • Multimedia Resource: Definition
• Constant of Proportionality: The constant value ($k$) that relates two proportional quantities, expressed as $y = kx$. It represents the rate at which one quantity changes relative to another.
• Linear Relationship: A relationship between two variables where the rate of change is constant, resulting in a straight-line graph. In the context of proportional relationships, the line passes through the origin.

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: Review of Equivalent Ratios

Present the following multi-step real-world problem involving ratios: 

"A recipe for 12 muffins requires 2 cups of flour and 3/4 cup of sugar. If you want to make 30 muffins, how much flour and sugar will you need?"

Solution using a table of equivalent ratios

1. Create a table with columns for muffins, flour, and sugar.
2. Fill in the given information for 12 muffins.
3. Extend the table to show equivalent ratios for 24 and 36 muffins.
4. Use the table to solve for 30 muffins.

Calculation for 30 muffins

1. Observe that 30 is between 24 and 36 muffins.
2. Find the scale factor: 30 ÷ 12 = 2.5
3. Multiply the original amounts by 2.5: 
   Flour: 2 cups × 2.5 = 5 cups 
   Sugar: 3/4 cup × 2.5 = 1 7/8 cups

Answer: For 30 muffins, you need 5 cups of flour and 1 7/8 cups of sugar.

Activity 2: Review of Tape Diagrams

Objective: Help students visualize ratios and proportional relationships using tape diagrams.

Instructions:

1. Draw a tape diagram on the board to represent the ratio 2:3. Explain that each segment represents one part of the ratio.
2. Provide a real-world scenario: "For every 2 cups of flour, 3 cups of sugar are needed in a recipe. If we use 4 cups of flour, how much sugar is needed?"
3. Have students extend the tape diagram to show an equivalent ratio of 4:6.
4. Encourage students to use tape diagrams to model other proportional relationships, such as the ratio of boys to girls in a classroom.

Activity 3: Review of Double Number Lines

Objective: Reinforce proportional reasoning by using double number lines with whole-number alignments.

Instructions:

1. Draw a double number line with two corresponding scales: one for the number of pencils in a pack and one for the total cost.
2. Use the example: "A store sells packs of pencils at a rate of 3 pencils for $2."
3. Label the number of pencils on the top line (3, 6, 9, 12, 15) and the cost on the bottom line ($2, $4, $6, $8, $10).
4. Ask students: "If a customer buys 12 pencils, how much will they pay?"
5. Discuss how the double number line helps visualize proportional relationships and find missing values.

This activity provides a clear and practical example of using double number lines to solve proportional problems while keeping numbers whole and easy to work with.

Teach

Demonstrate how to approach complex problems using various representations and strategies, such as tables, graphs, and equations. Emphasize the importance of clear communication in problem-solving, encouraging students to explain their reasoning.

Present the following real-world problems and their solutions:

1. Art Application: Color Mixing

Problem: An artist mixes yellow and blue paint in a 3:2 ratio to create green. If they need 25 ounces of green paint, how much yellow and blue paint should they use?

Solution using equivalent ratios:
Set up the proportion: 3 + 2 = 5 parts total
25 ÷ 5 = 5 ounces per part
Yellow: 3 × 5 = 15 ounces
Blue: 2 × 5 = 10 ounces

Answer: The artist needs 15 ounces of yellow paint and 10 ounces of blue paint.

2. Business Application: Sales Commission

Problem: A salesperson earns a 5% commission on all sales. Last month, their total sales were \$45,000. This month, they aim to earn \$3,000 in commission. What should their total sales be?

Solution using an equation

Set up a percent equation:

Sales • 5% = Commission

We know the commission and need to calculate the sales. Sales is the unknown:

x • 5% = 3000

Solve for x:

x = 3000 ÷ 0.05

x = 60,000

Answer: The salesperson needs \$60,000 in total sales to earn \$3,000 in commission.

3. Science Application: Chemical Dilution

Problem: A chemist needs to dilute a solution that is 90% acid. The diluted solution should be 40% acid and have a volume of 225 mL. How much of the original solution and water should be used?

This slide show provides a detailed solution to this problem:

https://www.media4math.com/library/slideshow/applications-ratios-and-proportional-reasoning-mixtures

Emphasize how these different solution techniques (tape diagrams, equivalence tables, and equivalent ratios) can be applied to various real-world problems. Encourage students to choose the method that works best for them in each situation.

Additional Examples

Example 1: Using Equivalent Ratios

Problem: A bakery uses 4 cups of flour to make 6 loaves of bread. How many cups of flour are needed to make 18 loaves?

Solution:

1. Set up the given ratio: $$\frac{4 \text{ cups}}{6 \text{ loaves}}$$
2. Find an equivalent ratio with 18 loaves by scaling up:
   • Multiply both terms by 3: $$\frac{4 \times 3}{6 \times 3} = \frac{12}{18}$$
3. Final Answer: The bakery needs 12 cups of flour to make 18 loaves.

Example 2: Solving with Proportions

Problem: A school has 24 teachers for 480 students. If the school expands to 600 students, how many teachers should it have to maintain the same ratio?

Solution:

1. Set up a proportion: $$\frac{24}{480} = \frac{x}{600}$$
2. Cross multiply: $$24 \times 600 = 480 \times x$$ $$14400 = 480x$$
3. Divide by 480: $$x = \frac{14400}{480} = 30$$
4. Final Answer: The school needs 30 teachers for 600 students.

Example 3: Solving with Tape Diagrams

Problem: A fruit stand sells 3 apples for every 2 oranges. If the stand has 12 apples, how many oranges should they have?

Solution:

1. Draw a tape diagram with two sections: one for apples and one for oranges.
2. Since the ratio is 3 apples to 2 oranges, divide the apple section into 3 equal parts and the orange section into 2 equal parts.
3. Label the tape diagram for the apples as 3, 6, 9, and 12 (in increments of 3).
4. Label the corresponding orange sections as 2, 4, 6, and 8.
5. Since 12 apples correspond to 8 oranges in the diagram, the fruit stand should have 8 oranges.

Example 4: Solving with Double Number Lines

Problem: A grocery store sells 2 pounds of bananas for $3. How much would 8 pounds of bananas cost?

Solution:

1. Draw a double number line with two scales:
   • Top scale: Pounds of bananas (2, 4, 6, 8).
   • Bottom scale: Price ($3, $6, $9, $12).
2. Identify the cost of 8 pounds by following the pattern.
3. Since 8 pounds aligns with $12, the total cost is $12.

Review

Lesson Summary

In this lesson, students applied their understanding of ratios and proportions to solve real-world problems using different mathematical representations. They explored:

• Equivalent Ratios: Finding missing values in a proportional relationship by scaling up or down.
• Proportions: Setting up and solving equations that express two equal ratios.
• Tape Diagrams: Visual models that represent ratios and help in solving problems step by step.
• Double Number Lines: A structured way to visualize proportional relationships and find missing values.

By practicing these methods, students developed flexibility in problem-solving and strengthened their ability to analyze proportional relationships in various contexts.

Key Vocabulary

• Ratio: A comparison between two quantities, often written as a fraction, using a colon (3:4), or in words ("3 to 4").
• Equivalent Ratios: Ratios that express the same relationship but are scaled versions of one another (e.g., 2:3 is equivalent to 4:6).
• Proportion: An equation that shows two ratios are equal, such as $\frac{a}{b} = \frac{c}{d}$.
• Constant of Proportionality: The fixed value $k$ that relates two variables in a proportional relationship, where $y = kx$.
• Tape Diagram: A visual model that represents ratios using bar segments.
• Double Number Line: A pair of aligned number lines used to compare proportional relationships.

Activities

Students work in groups to solve complex ratio and proportion problems, presenting their solutions to the class. Encourage the use of tables of equivalent ratios and other representations. Use problems from https://www.media4math.com/LessonPlans/RatiosRatesGr6 as a starting point, adapting them to be more challenging.

Additional Worked-Out Examples

Example 1: Using Equivalent Ratios

Problem: A recipe calls for 2 cups of sugar for every 5 cups of flour. How much sugar is needed for a batch that uses 20 cups of flour?

Solution:

1. Set up the equivalent ratio: $$\frac{2}{5} = \frac{x}{20}$$
2. Find the scale factor: $20 \div 5 = 4$.
3. Multiply the sugar by the same factor: $2 \times 4 = 8$.
4. Final Answer: 8 cups of sugar are needed.

Example 2: Solving with Proportions

Problem: A car travels 180 miles in 3 hours. At the same speed, how far can it travel in 5 hours?

Solution:

1. Set up a proportion: $$\frac{180}{3} = \frac{x}{5}$$
2. Cross multiply: $$180 \times 5 = 3x$$ $$900 = 3x$$
3. Divide by 3: $$x = 300$$
4. Final Answer: The car can travel 300 miles in 5 hours.

Example 3: Using a Tape Diagram

Problem: At a pet store, the ratio of cats to dogs is 5:2. If there are 15 cats, how many dogs are there?

Solution:

1. Draw a tape diagram with 5 equal sections for cats and 2 equal sections for dogs.
2. Label the sections for cats as 5, 10, and 15.
3. Since each section represents 3, the corresponding sections for dogs are 2, 4, and 6.
4. Final Answer: The pet store has 6 dogs.

Example 4: Using a Double Number Line

Problem: A juice company sells 3 bottles of juice for $5. How much would 12 bottles cost?

Solution:

1. Draw a double number line with:
   • Top scale: Number of bottles (3, 6, 9, 12).
   • Bottom scale: Price ($5, $10, $15, $20).
2. Identify the cost of 12 bottles from the pattern.
3. Final Answer: The total cost is $20.

Quiz

Answer the following questions.

1. A car travels 210 miles on 7 gallons of gas. How many miles can it travel on 12 gallons?
   
    
2. In a school, the ratio of boys to girls is 5:6. If there are 132 students in total, how many boys are there?
   
    
3. A recipe calls for 2 1/4 cups of flour for 9 servings. How much flour is needed for 15 servings?
   
    
4. If you earn \$720 for working 40 hours, what is your hourly wage? 
   
    
5. A map has a scale of 1 inch : 25 miles. If two cities are 7.5 inches apart on the map, what is their actual distance?
   
    
6. In a bag of marbles, the ratio of red to blue marbles is 3:5. If there are 24 red marbles, how many blue marbles are there?
   
    
7. A store offers a 15% discount on all items. If an item originally cost $80, what is its discounted price?
   
    
8. If 3 pounds of apples cost $4.50, how much would 7 pounds cost?
   
    
9. A rectangle has a length-to-width ratio of 4:3. If its perimeter is 70 inches, what are its dimensions?
   
    
10. In a survey, 3 out of every 8 people preferred brand A. If 600 people were surveyed, how many preferred brand A?
    
     

Answer Key

1. 360 miles
2. 60 boys
3. 3 3/4 cups
4. \$18/hr
5. 187.5 miles
6. 40 blue marbles
7. $68
8. $10.50
9. Length: 20 inches, Width: 15 inches
10. 225 people

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