# Lesson Plan: Slope and Rate of Change

## 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

- 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* = 2*x* + 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

- Define the rate of change.

- Calculate the rate of change for the points (2, 3) and (5, 11).

- What does a slope of 0 indicate about a line?

- Interpret the slope of a line that represents the cost of apples over time.

- Calculate the slope of the line passing through (1, 2) and (4, 8).

- Explain the difference between a positive and negative slope.

- Given a table of values, determine the rate of change.

- How does the rate of change relate to the steepness of a line?

- What is the slope of a horizontal line?

- Describe a real-life situation where understanding the rate of change is important.

## Answer Key

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

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