Lesson Plan: Slope and Rate of Change 

Lesson Summary

This lesson focuses on understanding the concept of slope as a measure of the rate of change in linear functions. Students will learn to calculate slope from various representations, including graphs, tables, and real-world scenarios. The lesson emphasizes interpreting slope in context, enabling students to connect mathematical concepts to everyday situations.

Lesson Objectives

• Understand the concept of rate of change and its relationship to slope.
• Calculate the rate of change from real-life situations.
• Interpret the meaning of slope and rate of change in various contexts.

Common Core Standards

• F.LE.1: Distinguish between situations that can be modeled with linear functions and with exponential functions.
• F.LE.5: Interpret the parameters in a linear or exponential function in terms of a context.

Prerequisite Skills

• Understanding of linear functions and equations.
• Ability to calculate slope from two points.

Key Vocabulary

• Slope: A measure of the steepness of a line, representing the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
• Rate of Change: The ratio that describes how one quantity changes in relation to another; in linear functions, it is equivalent to the slope.
• Linear Function: A function that graphs to a straight line and can be expressed in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
• Y-Intercept: The point where a graph crosses the y-axis, indicating the value of \( y \) when \( x \) is zero.
• Rise: The vertical change between two points on a graph.
• Run: The horizontal change between two points on a graph.
• Positive Slope: A slope that rises from left to right, indicating a positive rate of change.
• Negative Slope: A slope that falls from left to right, indicating a negative rate of change.
• Zero Slope: A horizontal line indicating no change in \( y \) as \( x \) changes; the rate of change is zero.
• Undefined Slope: A vertical line where the change in \( x \) is zero, making the slope undefined.

Multimedia Resources

• Math definitions of terms related to linear equations and functions: https://www.media4math.com/Definitions--LinearFunctions
• Video definitions on the topic of linear equations and functions: https://www.media4math.com/MathVideoCollection--LinearFunctionsDefinitions
• Math definitions of terms related to functions and relations: https://www.media4math.com/Definitions--FunctionsRelations

Warm Up Activities

Choose from one or more activities

Activity 1: Review of Ratios

For students who need a brief review of ratios and rates, show these math clip art images to demonstrate the differences between ratios and rates: 

https://www.media4math.com/library/75393/asset-preview

Activity 2: Desmos

Show distance-vs.-time graphs for constant speed and non-linear speed. Use this Desmos activity, which shows a data set for constant speed and non-linear speed. There are corresponding equations for the two graphs. Point out that this lesson will focus on linear graphs.

https://www.desmos.com/calculator/kdqqdrxef5

Activity 3: Identifying Rates in Real-World Scenarios

Students will analyze different situations where rates are used and determine how they relate to unit rates.

1. Provide the following examples and ask students to identify the rate:
   • A car travels 300 miles in 5 hours.
   • A grocery store sells 4 apples for \$2.
   • A water tank fills at a rate of 15 gallons per 3 minutes.
2. Have students determine the unit rate for each scenario by dividing the quantity by the time or amount.
   • \( \frac{300 \text{ miles}}{5 \text{ hours}} = 60 \) miles per hour.
   • \( \frac{2 \text{ dollars}}{4 \text{ apples}} = 0.50 \) dollars per apple.
   • \( \frac{15 \text{ gallons}}{3 \text{ minutes}} = 5 \) gallons per minute.
3. Discuss how these unit rates represent a constant rate of change, similar to how slope is used in linear functions.

Activity 4: Graphing Unit Rates

Students will use the Desmos Graphing Calculator to visualize unit rates as linear functions.

1. Open the Desmos Graphing Calculator.
2. Have students input equations based on the unit rates found earlier:
   • \( y = 60x \) (miles per hour)
   • \( y = 0.5x \) (cost per apple)
   • \( y = 5x \) (gallons per minute)
3. Students should analyze the slope of each graph and relate it to the real-world context.

Teach

Introduction to Slope and Rate of Change

Review of Slope of a Linear Function

The slope of a line measures its steepness and direction. It is calculated using the formula:

\[ m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} \]

where \( (x_1, y_1) \) and \( (x_2, y_2) \) are two points on the line.

Connecting Slope to Rate of Change

In real-world scenarios, slope represents the rate of change, describing how one quantity changes relative to another. The units of the slope indicate what is changing and at what rate.

Multimedia Resources

Explain the concept of rate of change and how it relates to slope. Use this video:

https://www.media4math.com/library/44958/asset-preview

Use this slide show to review rate of change:

https://www.media4math.com/library/slideshow/slope-rate-change

 Use this video to demonstrate rates and slope:

https://www.media4math.com/library/1802/asset-preview

 Real-World Application

Present a detailed example of a linear function with a non-zero y-intercept.

• Scenario: Consider a taxi fare system where the fare starts at \$3 (base fare) and increases by \$2 per mile traveled.
• Graph: Plot the fare (y) against the distance traveled (x).
• Equation: The linear function representing this scenario is shown below. In this equation 2 is the slope (rate of change) and 3 is the y-intercept (base fare).

y = 2x + 3

• Calculation: Calculate the fare for different distances:
  • For 0 miles: y = 2(0) + 3 = 3
  • For 1 mile: y = 2(1) + 3 = 5
  • For 2 miles: y = 2(2) + 3 = 7
  • For 3 miles: y = 2(3) + 3 = 9
• Interpretation: The slope (2) indicates that the fare increases by \$2 for every mile traveled, and the y-intercept (3) represents the initial fare when no distance is traveled.
• Use this Desmos activity to explore the graph of the data and the graph of the linear function:

https://www.desmos.com/calculator/fhrxy2ipvt

• Discuss how to interpret the slope and rate of change in different contexts, such as economics, physics, and everyday life.

Real-World Examples of Slope as a Rate of Change

Example 1: Car Speed

Problem: A car travels the following distances over time. Determine the rate of speed.

1. Graph the points on a coordinate plane.

2. Calculate the slope using any two points:

   \[ m = \frac{100 - 50}{2 - 1} = \frac{50}{1} = 50 \]

3. Interpretation: The car is traveling at a rate of 50 miles per hour.
4. The equation of the line is \[ y = 50x \]

Example 2: Water Tank Filling

Problem: A water tank is being filled at a constant rate. The water level (in gallons) over time is recorded as follows:

1. Graph the points on a coordinate plane.

2. Calculate the slope:

   \[ m = \frac{40 - 10}{5 - 0} = \frac{30}{5} = 6 \]

3. Interpretation: The water level is increasing by 6 gallons per minute.
4. The equation of the line is  \[ y = 6x + 10 \]

Example 3: Battery Life Drain

Problem: A phone battery starts at 100% charge and drains at a constant rate while being used. The battery percentage over time is recorded below:

1. Graph the points on a coordinate plane.

2. Calculate the slope:

   \[ m = \frac{70 - 90}{3 - 1} = \frac{-20}{2} = -10 \]

3. Interpretation: The battery percentage decreases by 10% every hour.
4. The equation of the line is  \[ y = -10x + 100 \]

Example 4: Car Depreciation

Problem: A new car is worth \$25,000 and loses \$2,000 in value each year. Write the equation and determine the rate of depreciation.

1. Write the equation (where y is the value of the car and x is the number of years):

   \[ y = 25000 - 2000x \]

2. Convert to slope-intercept form:

   \[ y = -2000x + 25000 \]

3. Graph the line starting at \( (0, 25000) \) with a slope of \( -2000 \).
4. Interpretation: The car loses value at a rate of \$2,000 per year. This is the rate of change.

Example 5: Gym Membership Plan

Problem: A gym charges a one-time registration fee and a monthly membership cost. After 2 months, the total amount paid is \$180, and after 5 months, the total amount paid is \$330. Write the equation representing the cost and determine the monthly membership fee.

1. Let \( x \) be the number of months and \( y \) be the total cost. Identify two points: \( (2, 180) \) and \( (5, 330) \).

2. Find the slope (\( m \)) using the slope formula:

   \[ m = \frac{330 - 180}{5 - 2} = \frac{150}{3} = 50 \]

3. Use the slope-intercept form \( y = mx + b \). Substitute \( (2, 180) \) to find \( b \): [ 180 = 50(2) + b \]\[ b = 80 \]

4. The equation is:

   \[ y = 50x + 80 \]

5. Graph the equation starting at \( (0, 80) \) with a slope of \( 50 \).
6. Interpretation: The monthly membership fee is revealed as the slope, which is \$50 per month, and the registration fee is \$80.

Example 6: Standard Form to Slope-Intercept Form (Car Rental Pricing)

Problem: A car rental company charges a flat fee plus a daily rental rate. The total cost, \( y \), for renting a car for \( x \) days follows the equation:

\[ 40x - y = -200 \]

1. Rearrange into slope-intercept form:
2. Identify the slope and y-intercept:\[ -y = -40x - 200 \]\[ y = 40x + 200\]
   • Slope: \( m = 40 \), meaning the daily rental rate is $40 per day.
   • Y-Intercept: \( b = 200 \), meaning there is an initial flat fee of $200.
3. Interpretation: The rental cost increases by $40 for each additional day the car is rented.

Review

Summary of Key Concepts

In this lesson, students learned that the slope of a line represents the rate of change in a linear function. Slope is calculated using the formula:

\[ m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} \]

Students also explored:

• How to calculate and interpret slope from tables, graphs, and equations.
• The four types of slope:
  • Positive slope: The function increases as \( x \) increases.
  • Negative slope: The function decreases as \( x \) increases.
  • Zero slope: A horizontal line where \( y \) remains constant.
  • Undefined slope: A vertical line where \( x \) remains constant.
• Converting standard form equations to slope-intercept form to analyze the rate of change.

Key Vocabulary

• Slope: The measure of the steepness of a line, defined as the ratio of the vertical change (rise) to the horizontal change (run).
• Rate of Change: The amount one quantity changes in relation to another; in linear functions, this is the slope.
• Y-Intercept: The point where a function crosses the y-axis, representing the initial value when \( x = 0 \).
• Positive Slope: A line that rises from left to right, indicating a positive rate of change.
• Negative Slope: A line that falls from left to right, indicating a negative rate of change.
• Zero Slope: A horizontal line where there is no change in \( y \).
• Standard Form: A linear equation written as \( Ax + By = C \), which can be converted to slope-intercept form.
• Linear Function: A function that graphs as a straight line and follows the equation \( y = mx + b \).

Multimedia Resources

Review rate of change in the context of converting units of measurement. Use this video:

https://www.media4math.com/library/1803/asset-preview

Example 1: Positive Slope (Delivery Fee)

Problem: A food delivery service charges a base fee plus an additional cost per mile driven. The total cost of deliveries is recorded below:

1. Calculate the slope:

   \[ m = \frac{12 - 8}{3 - 1} = \frac{4}{2} = 2 \]

2. Find the equation using \( y = mx + b \):

   \[ y = 2x + 6 \]

3. Interpretation: The delivery service charges $6 as a base fee and $2 per mile driven.

Example 2: Negative Slope (Cooling Liquid)

Problem: A hot cup metal cools over time. The recorded temperatures are:

1. Calculate the slope:

   \[ m = \frac{160 - 180}{5 - 0} = \frac{-20}{5} = -4 \]

2. Find the equation using \( y = mx + b \):

   \[ y = -4x + 180 \]

3. Interpretation: The metal cools at a rate of 4°F per minute.

Example 3: Zero Slope (Flat Road)

Problem: A highway has a section that remains at the same elevation for several miles. The elevation at different distances is:

1. Calculate the slope:

   \[ m = \frac{500 - 500}{4 - 2} = \frac{0}{2} = 0 \]

2. Find the equation:

   \[ y = 500 \]

3. Interpretation: Since the slope is 0, the highway remains at a constant elevation.

Quiz

Answer the following questions.

1. Define the rate of change.
   
    
2. Calculate the rate of change for the points (2, 3) and (5, 11).
   
    
3. What does a slope of 0 indicate about a line?
   
    
4. Interpret the slope of a line that represents the cost of apples over time.
   
    
5. Calculate the slope of the line passing through (1, 2) and (4, 8).
   
    
6. Explain the difference between a positive and negative slope.
   
    
7. Given a table of values, determine the rate of change.
   
    
8. How does the rate of change relate to the steepness of a line?
   
    
9. What is the slope of a horizontal line?
   
    
10. Describe a real-life situation where understanding the rate of change is important.

Answer Key

1. The rate of change is the ratio of the change in the dependent variable to the change in the independent variable.
2. (11-3)/(5-2) = 8/3
3. A slope of 0 indicates a horizontal line.
4. The slope represents the rate at which the cost of apples changes over time.
5. (8-2)/(4-1) = 6/3 = 2
6. A positive slope indicates an increasing relationship, while a negative slope indicates a decreasing relationship.
7. Calculate the differences in y-values and x-values and divide.
8. The greater the rate of change, the steeper the line.
9. The slope of a horizontal line is 0.
10. Understanding the rate of change is important in situations like calculating speed, growth rates, and financial trends.