Princeton Review

 Lesson Plan: Slope and Rate of Change 

Lesson Objectives

  • Understand slope as a rate of change
  • Calculate slope using different methods
  • Interpret slope in real-world contexts
  • Determine and interpret average rate of change in context

Florida BEST Standards

  • MA.912.F.1.3: Calculate and interpret average rate of change in various representations.
  • MA.912.AR.2.4: Graph linear functions and determine key features.
  • MA.912.AR.2.5: Solve and graph problems modeled with linear functions.
  • MA.912.F.1.1: Classify functions as linear, quadratic, cubic, exponential, logarithmic, rational, or absolute value.
  • MA.912.F.1.2: Determine key features of function graphs including extrema, end behavior, intercepts, domain, and range.
  • MA.912.AR.2.6: Determine and interpret the average rate of change for a linear function in context.

Prerequisite Skills

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

Key Vocabulary

  • Rate of change
  • Slope
  • Linear model
  • Constant rate of change

Warm-up Activity (10 minutes)

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

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

Teach (20 minutes)

Conceptual Framework

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.

Review (10 minutes)

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

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

Assess (10 minutes)

Administer a 10-question quiz to assess students' understanding of the rate of change and slope.

Quiz

  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.

 

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