Lesson Plan: Representing Proportional Relationships

Lesson Summary

In this lesson, students will explore different ways to represent proportional relationships using tables, graphs, and equations. They will analyze proportional relationships in real-world contexts and learn how to determine the constant of proportionality. The lesson emphasizes how proportional relationships appear as linear functions that pass through the origin when graphed.

Students will practice:

• Using tables to identify proportional relationships.
• Graphing proportional relationships and understanding their characteristics.
• Writing equations in the form $y = kx$, where $k$ is the constant of proportionality.
• Interpreting real-world problems involving proportions.

Lesson Objectives

This lesson can be completed in one 50-minute class period. However, additional practice may be beneficial and could extend into a second class period if time allows.

• Represent proportional relationships using equations
• Graph proportional relationships
• Interpret points on graphs of proportional relationships, particularly (0, 0) and (1, r)
• Explain the meaning of points on graphs in real-world contexts
• Solve proportions using equations and ratio tables

Common Core Standards

• 7.RP.A.2c Represent proportional relationships by equations.
• 7.RP.A.2d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

Prerequisite Skills

• Understanding of proportional relationships
• Basic algebraic concepts
• Familiarity with coordinate plane
• Ability to interpret ratios and unit rates

Key Vocabulary

• Proportional Relationship: A relationship between two quantities where the ratio remains constant.
  • Multimedia Resource: https://www.media4math.com/library/43387/asset-preview
• Constant of Proportionality: The constant ratio between two proportional quantities, represented as $k$ in the equation $y = kx$.
  • Multimedia Resource: https://www.media4math.com/library/43411/asset-preview
• Unit Rate: A rate in which the denominator is 1, such as $\frac{50}{1}$ miles per hour.
  • Multimedia Resource: https://www.media4math.com/library/43395/asset-preview
• Ratio: A comparison of two numbers, often written as a fraction like $\frac{3}{5}$.
  • Multimedia Resource: https://www.media4math.com/library/22157/asset-preview
• Linear Relationship: A relationship that forms a straight line when graphed.

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

Students will complete an equivalent ratio table. This activity will be conducted using an interactive whiteboard or handouts.

Example: A recipe calls for 2 cups of flour for every 3 cups of milk. Complete the equivalent ratio table:

Solution:

Explanation: To fill in the missing values, we use the relationship between flour and milk. For every 2 cups of flour, we need 3 cups of milk. So, we can multiply both the flour and milk amounts by the same factor to find equivalent ratios.

For the second column: 2 × 2 = 4 cups of flour, 3 × 2 = 6 cups of milk
For the third column: 2 × 3 = 6 cups of flour, 3 × 3 = 9 cups of milk
For the fourth column: 2 × 4 = 8 cups of flour, 3 × 4 = 12 cups of milk

Resource: https://www.media4math.com/library/card/equivalent-ratio-tables

Activity 2: Review of Proportions

Start the lesson with a review of proportions by solving an equation algebraically. A proportion is an equation stating that two ratios are equal. To solve for an unknown value, use cross-multiplication.

Example: Solve for $x$ in the proportion:

$$\frac{3}{4} = \frac{x}{8}$$

Solution:

1. Use cross-multiplication: $$3 \times 8 = 4 \times x$$
2. Simplify: $$24 = 4x$$
3. Divide both sides by 4: $$x = \frac{24}{4} = 6$$

Thus, $x = 6$.

Activity 3: Review of Slope and Linear Functions

Next, review the concept of slope and linear functions, focusing on proportional relationships. A proportional relationship can be written in the form:

$$y = kx$$

where $k$ is the constant of proportionality, which represents the slope of the line.

Example: Suppose a car travels at a constant speed of 50 miles per hour. The equation for the distance traveled is:

$$y = 50x$$

where:

• $y$ represents the total distance traveled.
• $x$ represents the number of hours driven.
• $k = 50$, meaning the car travels 50 miles per hour.

Graphing the Equation:

• The graph of $y = 50x$ is a straight line that passes through the origin (0,0).
• The slope of the line is $k = 50$, which means for every increase of 1 in $x$, $y$ increases by 50.

Teach

Introduction

Proportional relationships appear in many real-world scenarios, such as currency exchange rates, speed calculations, and recipe adjustments. In this lesson, students will learn how to represent proportional relationships using tables, graphs, and equations.

Proportional relationships always maintain a constant ratio between two quantities, which is represented by the constant of proportionality. When graphed, they always form a straight line that passes through the origin.

Key Vocabulary

• Proportional Relationship: A relationship between two quantities where the ratio remains constant.
• Constant of Proportionality: The fixed number $k$ in the equation $y = kx$, which represents the ratio of $y$ to $x$.
• Ratio: A comparison of two numbers, often expressed as a fraction $\frac{a}{b}$.
• Unit Rate: A rate in which the denominator is 1, such as $\frac{60}{1}$ miles per hour.
• Linear Relationship: A relationship where the graph forms a straight line.
• Graph of a Proportional Relationship: A straight line passing through the origin (0,0) with a slope equal to the constant of proportionality.

By understanding these key terms, students will be better equipped to identify and analyze proportional relationships using different mathematical representations.

Instruction

This slide show provides an overview of proportions:

https://www.media4math.com/library/slideshow/overview-proportions

 This slide show provides multiple examples of solving proportions algebraically. Review several of these examples before moving on to the more detailed examples below:

https://www.media4math.com/library/slideshow/math-examples-solving-proportions-algebraically

Example 1: Distance and Time

Problem: A car travels at a constant speed of 60 miles per hour. Express the relationship between distance traveled (y) and time (x) as an equation.

Solution:
• The unit rate is 60 miles per hour
• Let y represent the distance traveled and x represent the time
• Equation: y = 60x

Example 2: Circle Geometry

Problem: Provide a data table and derive the equation showing that the slope is π. This could also be a hands-on activity in which students are measuring the diameters and circumferences for different circular objects. (Using a string, wrap it around the circular shape and then measure the length of the string to find the circumference.)

Solution:

• The ratio C/d is constant and approximately equal to 3.14 (π)
• This suggests a proportional relationship: C = πd
• The equation is y = πx, where y is the circumference and x is the diameter
• The slope of this line is π, which is the constant of proportionality

Example 3: Hourly Wages

Problem: Start with a data table, derive the equation, and solve a proportion.

Solution:

• Calculate the unit rate: 120 / 8 = $15 per hour
• The equation is y = 15x, where y is wages earned and x is hours worked

Solving a proportion: If someone works for 10 hours, how much will they earn?
• Use the equation: y = 15(10) = $150

Example 4: Buying Fruit by the Pound

Suppose a grocery store sells apples for $2.50 per pound. The relationship between the total cost ($y$) and the number of pounds ($x$) can be expressed using the equation:

$$y = 2.5x$$

where:

• $x$ represents the number of pounds of apples.
• $y$ represents the total cost.
• $k = 2.5$, meaning the cost per pound is $2.50.

Table of Values:

Graphing: When plotted, this equation forms a straight line passing through the origin (0,0), confirming it is a proportional relationship.

Example 5: Road Trip – Driving at a Constant Speed

A car is traveling at a constant speed of 65 miles per hour. The relationship between the distance traveled ($y$) and the time ($x$) in hours is given by:

$$y = 65x$$

where:

• $x$ represents the number of hours driven.
• $y$ represents the total distance traveled.
• $k = 65$, meaning the car travels 65 miles per hour.

Table of Values:

Graphing: This equation forms a straight line through the origin, illustrating a proportional relationship where the slope (constant of proportionality) is 65.

By analyzing these examples, students can see how proportional relationships appear in everyday life and how they can be represented using tables, equations, and graphs.

Review

Lesson Summary

In this lesson, students explored proportional relationships and how they can be represented using tables, graphs, and equations. They learned that proportional relationships maintain a constant ratio, represented by the constant of proportionality $k$, and that their graphs always form a straight line passing through the origin.

Key takeaways from the lesson:

• Proportional relationships can be identified in real-world scenarios such as cost per item, speed calculations, and currency exchange rates.
• The equation for a proportional relationship is written as $y = kx$, where $k$ is the constant of proportionality.
• Tables of proportional relationships maintain a consistent ratio between corresponding values.
• The graphs of proportional relationships are linear and always pass through the origin (0,0).

Key Vocabulary

• Proportional Relationship: A relationship between two quantities where the ratio remains constant.
• Constant of Proportionality: The fixed number $k$ in the equation $y = kx$, which represents the ratio of $y$ to $x$.
• Ratio: A comparison of two numbers, often expressed as a fraction $\frac{a}{b}$.
• Unit Rate: A rate in which the denominator is 1, such as $\frac{60}{1}$ miles per hour.
• Linear Relationship: A relationship where the graph forms a straight line.
• Graph of a Proportional Relationship: A straight line passing through the origin (0,0) with a slope equal to the constant of proportionality.

Students will practice solving real-world problems using ratio tables and proportions.

Example 1

A store sells 5 notebooks for \$7. How many notebooks can be bought for \$42?

Solution using ratio table:

We can see that 30 notebooks can be bought for \$42.

Example 2

A car travels 210 miles in 3 hours. At this rate, how long will it take to travel 350 miles?

Solution using proportion:
• Set up the proportion: 210/3 = 350/x, where x is the time in hours
• Cross multiply: 210x = 3 * 350
• Solve for x: 210x = 1050
x = 1050/210 = 5
• Therefore, it will take 5 hours to travel 350 miles at this rate.

Quiz

Answer the following questions.

1. Write an equation to represent a proportional relationship where y is 3 times x.
   
    
2. Graph the proportional relationship y = 2.5x.
   
   
    
3. What does the point (0,0) represent in a proportional relationship?
   
    
4. In the equation y = kx, what does k represent?
   
    
5. If a graph of a proportional relationship passes through the point (4,10), what is the unit rate?
   
    
6. A car travels 240 miles in 4 hours at a constant speed. Write an equation to represent this relationship.
   
    
7. In a proportional relationship, what does the point (1,r) represent?
   
    
8. Graph the equation y = 0.5x.
   
   
    
9. A recipe calls for 2 cups of flour for every 3 cups of milk. Write an equation to represent the relationship between flour (y) and milk (x).
   
    
10. If 4 oranges cost $2, how much would 10 oranges cost? Set up and solve a proportion.

Answer Key

1. y = 3x
2. ]
3. The starting point or initial value of the relationship
4. The constant of proportionality or unit rate
5. 2.5 (10 ÷ 4 = 2.5)
6. y = 60x, where y is distance and x is time
7. The unit rate of the proportional relationship
8. 
9. y = (2/3)x
10. 4/2 = 10/x, 4x = 20, x = 5. 10 oranges would cost $5.

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