Lesson Plan: Slope As Rate of Change

Lesson Summary

The lesson "Slope as Rate of Change" aims to deepen students' understanding of slope by interpreting it as a rate of change between two quantities. Through real-world examples and interactive activities, students will learn to:

Calculate and interpret rates of change.
Distinguish between rates and ratios.
Apply these concepts to solve practical problems.

Lesson Objectives

By the end of this lesson, students will be able to:

Understand the concept of slope as a rate of change.
Interpret slope in real-world situations involving rates of change.
Calculate and interpret rates of change from given data or graphs.
Solve problems involving rates of change.
Distinguish between rates and ratios.

Common Core Standards

8.EE.B.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph.
8.F.B.4: Construct a function to model a linear relationship between two quantities.
HSF-IF.B.6: Calculate and interpret the average rate of change of a function over a specified interval.

Prerequisite Skills

Understanding of slope as the ratio of the vertical change to the horizontal change.
Familiarity with the slope formula: Slope = $\frac{y_2 - y_1}{x_2 - x_1} $
Ability to plot points on a coordinate plane and graph linear equations.
Basic algebraic skills for solving equations.
Understanding of ratios and proportions.

Key Vocabulary

Slope: The measure of the steepness or incline of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line: \[ \frac{\text{rise}}{\text{run}} \]
Rate of Change: A ratio that describes how one quantity changes in relation to another; in linear relationships, it is represented by the slope of the line.
Ratio: A comparison of two quantities by division, often expressed in the form $ a:b $ or as a fraction $ \frac{a}{b} $.
Unit Rate: A rate in which the second quantity in the comparison is one unit; for example, 60 miles per hour indicates 60 miles per 1 hour.
Proportional Relationship: A relationship between two quantities in which the ratio remains constant; graphically, it is represented by a straight line passing through the origin.
Linear Function: A function that creates a straight line when graphed; it has a constant rate of change 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 line crosses the y-axis on a graph, indicating the value of $ y $ when $ x $ is zero.
Dependent Variable: The variable in a function that depends on the value of another variable; typically represented as $ y $.
Independent Variable: The variable in a function whose value determines the value of the dependent variable; typically represented as $ x $.
Square Root: The number that, when multiplied by itself, gives the original number. Example: $ \sqrt{16} = 4 $.
Desmos: An advanced graphing calculator implemented as a web application, used for creating interactive mathematical graphs and visualizations.

Multimedia Resources

A collection of definitions on the topic of slope: https://www.media4math.com/Definitions--slope
Share this slide show with students to review definitions on the topic of slope: https://www.media4math.com/library/slideshow/student-tutorial-slope-concepts-definitions

Warm-Up Activities

Activity 1: Discussion of Rates and Ratios

Begin by asking students to define a ratio and provide examples. Explain that a ratio is a comparison of two quantities by division, often expressed as a fraction or a:b. Next, introduce the concept of a rate. Explain that a rate is a special type of ratio that compares two different types of quantities, such as distance and time, or cost and number of items.

Show these math clip art images to demonstrate the differences between ratios and rates: https://www.media4math.com/library/75393/asset-preview

Provide additional examples of rates, such as:

Speed (distance traveled per unit of time)
Fuel efficiency (distance traveled per unit of fuel consumed)
Population growth rate (change in population per unit of time)
Inflation rate (change in prices per unit of time)

Emphasize that rates involve different units for the numerator and denominator, while ratios often involve the same units. Display a graph showing the distance traveled by a car over time. Ask students to describe what the graph represents and what information they can gather from it. Guide them to understand that the slope of the graph represents the rate of change of distance with respect to time, which is the speed of the car.

Activity 2: Calculating Rates

Using a calculator, have students compute various rates to reinforce the concept of unit rate and proportional relationships. Provide real-world scenarios such as:

The cost per pound of meat when a 5-pound package costs \$20.
The cost per gallon of gas when a 12-gallon fill-up costs \$42.
The speed of a car traveling 180 miles in 3 hours.

Encourage students to express their answers in unit rate format, such as $ \frac{\text{cost}}{\text{pound}} $ or $ \frac{\text{miles}}{\text{hour}} $.

Activity 3: Skateboarder on a Ramp

Display an image or video of a skateboarder on a ramp. Ask students to observe how the skateboarder’s speed increases as they move down the ramp.

Discuss what happens to the skateboarder’s speed as they descend.
Introduce the concept of acceleration as a rate of change in speed.
Ask students how they would calculate the rate of speed increase over time.

Speed Data Table

Use the following table to analyze the skateboarder’s changing speed over time:

Time (seconds)	Speed (feet per second)
0	0
1	2
2	5
3	9
4	14
5	20

Have students calculate the rate of change (slope) between different time intervals using the formula:

\[ \text{slope} = \frac{\text{change in speed}}{\text{change in time}} \]

Discuss how the increasing speed suggests acceleration and how it connects to slope as a rate of change.

Teach

Introduction to Slope as Rate of Change

Show students this video about rates: https://www.media4math.com/library/1796/asset-preview

Explain that slope can be interpreted as the rate of change between two quantities. The rate of change is the ratio of the change in one quantity to the change in another quantity over a specific interval.

Provide real-world examples of rates of change, such as:

Speed (distance traveled per unit of time)
Fuel efficiency (distance traveled per unit of fuel consumed)
Population growth rate (change in population per unit of time)
Inflation rate (change in prices per unit of time)

Demonstrate how to calculate and interpret rates of change from given data or graphs.

Desmos Activity

Show students this Desmos activity about rates: https://www.media4math.com/library/75398/asset-preview

Explain what the graph represents:

The slope of this line is the rate (cost per pound) of fruit.
Changing the value of the slider m changes the rate.
The slider varies from 0 to 5.
The table shows the input values for x (the pounds of fruit) and the output values f(x), which represent the cost for that amount of fruit.

Have students explore this activity to answer the following questions:

What happens to the slope as the rate increases?
What happens to the cost of fruit as the slope increases?

Problem-Solving

Guide students through solving problems involving rates of change. Encourage them to identify the quantities involved, determine the appropriate units, and interpret the meaning of the calculated rate of change in the context of the problem.

Use this collection of math clip art images to show examples of such rates: https://www.media4math.com/library/slideshow/slope-rate-change

Example 1: Finding the Slope as a Rate of Change

Problem: A car travels between two cities. At time 2 hours, the car has traveled 60 miles. At time 5 hours, the car has traveled 150 miles. What is the car’s speed?

Solution:

Identify the given points: $ (2, 60) $ and $ (5, 150) $, where x represents time (hours) and y represents distance (miles).
Use the slope formula: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting values: $ \frac{150 - 60}{5 - 2} = \frac{90}{3} = 30 $

Interpretation: The slope represents the rate of change of distance with respect to time, meaning the car is traveling at 30 miles per hour.

Example 2: Slope as Rate of Change from a Data Table

Problem: A water tank is being filled at a constant rate. The table below shows the amount of water in the tank over time. What is the rate of change?

Time (minutes)	Water in Tank (gallons)
0	50
2	70
4	90
6	110

Solution:

Choose any two points from the table, such as $ (2, 70) $ and $ (6, 110) $.
Use the slope formula: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting values: $ \frac{110 - 70}{6 - 2} = \frac{40}{4} = 10 $

Interpretation: The slope represents the rate at which the tank fills, which is 10 gallons per minute.

Example 3: Identifying Slope as Rate of Change from an Equation

Problem: A taxi company charges a base fare of \$3 plus \$2 per mile traveled. What is the rate of change?

Solution:

The cost equation follows the form $ y = mx + b $, where $ y $ is the total cost and $ x $ is the number of miles.
Given equation: $ y = 2x + 3 $.
Identify the slope $ m = 2 $.

Interpretation: The slope represents the rate of change of cost per mile, meaning the taxi charges \$2 per mile.

Example 4: Real-World Scenario – Slope as Speed on a Hiking Trail

Problem: A hiker climbs a trail that gains 800 feet of elevation over a horizontal distance of 2,000 feet. What is the rate of elevation gain?

Solution:

Use the slope formula: \[ \text{slope} = \frac{\text{rise}}{\text{run}} \]
Substituting values: $ \frac{800}{2000} $
Simplify: \[ \frac{800}{2000} = \frac{4}{10} = \frac{2}{5} \]

Interpretation: The slope represents the rate of elevation gain, meaning the hiker ascends at 2 feet for every 5 feet of horizontal distance.

Example 5: Real-World Scenario – Price Increase Over Time

Problem: The price of gasoline increased from \$3.50 per gallon to \$4.10 per gallon over 6 months. What is the rate of price change per month?

Solution:

Use the slope formula: \[ \text{slope} = \frac{\text{change in price}}{\text{change in time}} \]
Calculate the price change: $ 4.10 - 3.50 = 0.60 $.
Calculate the time change: 6 months.
Compute the rate of change: \[ \frac{0.60}{6} = 0.10 \]

Interpretation: The slope represents the rate of price increase per month, meaning gasoline prices increase by \$0.10 per month.

Review

Summarize the key points of the lesson:

Slope represents the rate of change between two quantities.
The rate of change is calculated by dividing the change in one quantity by the change in another quantity over a specific interval.
Rates of change can be interpreted in various real-world contexts, such as speed, fuel efficiency, population growth, and inflation.
Rates involve different units for the numerator and denominator, while ratios often involve the same units.

Encourage students to ask questions and clarify any remaining doubts.

Key Vocabulary

Slope: The rate of change between two points on a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run): \[ \text{slope} = \frac{\text{rise}}{\text{run}} \]
Rate of Change: A ratio that describes how one quantity changes in relation to another. In a linear relationship, it is represented by the slope.
Linear Function: A function that creates a straight line when graphed and can be written in the form $ y = mx + b $, where $ m $ is the slope.
Proportional Relationship: A relationship where two quantities increase or decrease at a constant rate, represented by a straight line passing through the origin.
Unit Rate: A rate where the denominator is 1, such as 60 miles per hour or $2 per gallon.
Y-Intercept: The point where a line crosses the y-axis, representing the starting value when $ x = 0 $.
Independent Variable: The variable that represents the input or cause, often labeled as $ x $.
Dependent Variable: The variable that represents the output or effect, often labeled as $ y $.

Example 1: Slope as Rate of Change in Business

Problem: A coffee shop tracks its daily earnings. On Monday, the shop earned \$200, and on Friday, it earned \$500. Assuming a constant rate of increase, what is the daily rate of earnings?

Solution:

Identify the given points: $ (1, 200) $ for Monday and $ (5, 500) $ for Friday.
Use the slope formula: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting values: $ \frac{500 - 200}{5 - 1} = \frac{300}{4} = 75 $.

Interpretation: The coffee shop earns an additional \$75 per day.

Example 2: Slope from a Data Table – Bike Rental Cost

Problem: A bike rental company charges a base fee and an hourly rate. The table below shows the total cost for different rental times. What is the hourly rate?

Time (hours)	Total Cost ($)
1	12
2	18
3	24
4	30

Solution:

Choose two points, such as $ (1, 12) $ and $ (3, 24) $.
Use the slope formula: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting values: $ \frac{24 - 12}{3 - 1} = \frac{12}{2} = 6 $.

Interpretation: The hourly rental rate is \$6 per hour.

Example 3: Slope as Speed in a Marathon

Problem: A runner covers 10 miles in 1.5 hours and 16 miles in 2.5 hours. What is the runner's speed in miles per hour?

Solution:

Identify the given points: $ (1.5, 10) $ and $ (2.5, 16) $.
Use the slope formula: \[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting values: $ \frac{16 - 10}{2.5 - 1.5} = \frac{6}{1} = 6 $.

Interpretation: The runner’s speed is 6 miles per hour.

Final Review Questions

Explain how slope represents a rate of change in different real-world contexts.
Find the slope of a line passing through the points (4, 10) and (8, 22).
Given the equation $ y = 5x + 2 $, identify the rate of change.
A hiker climbs 1,200 feet over 4 miles. What is the slope?
Using the bike rental table, predict the cost for a 6-hour rental.

Quiz

Answer the following questions.

What does the slope of a graph represent in terms of rate of change?
If a car travels 120 miles in 3 hours, what is its average speed (rate of change of distance with respect to time)?
The population of a city increased from 50,000 to 60,000 in 5 years. What is the annual population growth rate?
A company's revenue increased from \$2 million to \$3.5 million over a period of 2 years. What is the rate of change of revenue with respect to time?
The graph below shows the distance traveled by a car over time. What is the car's average speed between t = 2 hours and t = 4 hours?
Interpret the rate of change in the following scenario: The cost of a product increases by \$5 for every additional unit purchased.
A company's profit increased from 100,000 to 150,000 over a period of 2 years. If the rate of change of profit remained constant, what would be the company's profit after 4 years?
The graph below shows the temperature change over time. What is the rate of change of temperature between t = 2 hours and t = 5 hours?
Explain the difference between a positive rate of change and a negative rate of change in the context of population growth.
A car travels 180 miles in 3 hours. If the car maintains the same rate of change, how far will it travel in 6 hours?

Answer Key

Answer: The slope of a graph represents the rate of change between the two quantities plotted on the x and y axes.
Answer: Average speed = Distance / Time = 120 miles / 3 hours = 40 miles per hour.
Answer: Annual population growth rate = (Change in population / Initial population) / Time = (10,000 / 50,000) / 5 years = 0.04 or 4% per year.
Answer: Rate of change of revenue = (Change in revenue / Time) = (\$3.5 million - \$2 million) / 2 years = \$0.75 million per year.
Answer: Average speed = (Change in distance) / (Change in time) = (Distance at t = 4 hours - Distance at t = 2 hours) / (4 hours - 2 hours).
Answer: The rate of change represents the change in cost per additional unit purchased, which is \$5 per unit.
Answer: Rate of change of profit = (\$150,000 - \$100,000) / 2 years = \$25,000 per year. Profit after 4 years = \$100,000 + (4 × \$25,000) = \$200,000.
Answer: Rate of change of temperature = (Change in temperature) / (Change in time) = (Temperature at t = 5 hours - Temperature at t = 2 hours) / (5 hours - 2 hours).
Answer: A positive rate of change indicates that the population is increasing over time, while a negative rate of change indicates that the population is decreasing over time.
Answer: Rate of change (speed) = Distance / Time = 180 miles / 3 hours = 60 miles per hour. Distance traveled in 6 hours = Rate of change × Time = 60 miles per hour × 6 hours = 360 miles.