Comparing Rates: Equations vs. Graphs

Algebra 1 Grades High School 1:25 Video
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Lesson Description

Learn how to compare the rate of change in linear equations and graphs to determine which represents a faster or slower relationship between variables.

Video Resource

Comparing rates | Linear equations and functions | 8th grade | Khan Academy

Khan Academy

Duration: 1:25
Watch on YouTube

Key Concepts

  • Unit Rate
  • Slope as Rate of Change
  • Comparing Linear Relationships

Learning Objectives

  • Students will be able to determine the unit rate of a linear equation.
  • Students will be able to determine the unit rate from a graph.
  • Students will be able to compare the unit rates of linear equations and graphs.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of unit rate and its connection to slope. Discuss how unit rate represents the change in 'y' for every unit change in 'x'.
  • Video Presentation (5 mins)
    Play the Khan Academy video 'Comparing rates | Linear equations and functions | 8th grade | Khan Academy'. Ensure students pay attention to how the unit rate is identified in both the equation and the graph.
  • Guided Practice (10 mins)
    Work through examples similar to the video, one with an equation (e.g., y = 4x) and another with a graph. Guide students in finding the unit rate (slope) in each case. For the graph, emphasize identifying two clear points and calculating rise over run.
  • Independent Practice (10 mins)
    Provide students with worksheets containing equations and graphs. Ask them to determine and compare the unit rates. Increase complexity by varying the scale of the graph.
  • Wrap-up and Discussion (5 mins)
    Recap the key concepts and address any questions. Lead a short discussion on real-world scenarios where comparing rates is useful (e.g., comparing the speed of two cars, the cost of two products per unit).

Interactive Exercises

  • Rate Race
    Divide students into groups and present them with pairs of equations and graphs. The first group to correctly identify and compare the unit rates wins a point. This can be done using whiteboards or online collaboration tools.
  • Graphing Challenge
    Give students an equation and ask them to create a graph representing the same relationship. Then, give them another equation with a different rate and ask them to graph it on the same coordinate plane and compare the rates.

Discussion Questions

  • How is the unit rate related to the slope of a line?
  • In what real-world situations might you need to compare rates represented by equations and graphs?
  • What are some strategies for finding the unit rate from a graph, especially when the points are not perfectly on the grid lines?

Skills Developed

  • Calculating Slope
  • Interpreting Graphs
  • Comparing Rates of Change

Multiple Choice Questions

Question 1:

What does the unit rate represent in a linear equation?

Correct Answer: The slope

Question 2:

Which equation has a smaller unit rate: y = 3x or y = 5x?

Correct Answer: y = 3x

Question 3:

On a graph, how do you find the unit rate?

Correct Answer: Calculate the rise over run

Question 4:

A line on a graph increases 2 units on the y-axis for every 1 unit on the x-axis. What is the unit rate?

Correct Answer: 2

Question 5:

Which representation (equation or graph) is more convenient for visually comparing rates?

Correct Answer: Graph

Question 6:

If one car travels 60 miles in one hour, and another car travels according to the equation y = 55x (where y is distance and x is time), which car is traveling faster?

Correct Answer: The first car

Question 7:

What is another name for unit rate when discussing linear equations?

Correct Answer: Slope

Question 8:

Which of the following equations has the largest rate of change? y = 2x, y = (1/2)x, y = 5x, y = x

Correct Answer: y = 5x

Question 9:

What is the significance of a steeper line when representing data on a graph?

Correct Answer: A larger rate of change

Question 10:

The equation y = 7x is being compared to a line on a graph. If the line's slope is 6, which has a greater rate of change?

Correct Answer: The equation

Fill in the Blank Questions

Question 1:

The unit rate of an equation is also known as the ____.

Correct Answer: slope

Question 2:

To find the unit rate on a graph, you calculate the ____.

Correct Answer: slope

Question 3:

The equation y = 8x has a unit rate of ____.

Correct Answer: 8

Question 4:

A steeper line on a graph indicates a ______ unit rate.

Correct Answer: larger

Question 5:

When comparing rates, a rate that changes more quickly will have a ______ slope.

Correct Answer: steeper

Question 6:

The fraction used to calculate the slope of a line from a graph is rise over _____.

Correct Answer: run

Question 7:

In the equation y = mx + b, 'm' represents the ____ rate.

Correct Answer: unit

Question 8:

A line that is perfectly horizontal has a slope, and therefore a unit rate, of _____.

Correct Answer: zero

Question 9:

If a line rises 4 units for every 2 units it runs, its rate of change is _____.

Correct Answer: 2

Question 10:

Comparing rates helps us decide which relationship changes more _____ per unit of x.

Correct Answer: quickly

Educational Standards

A1.A.4.1:Analyze, use and apply mathematical models and other data sets (e.g., graphs, equations, two points, a set of data points) to calculate and interpret slope and the x- and y-intercepts of a line. A1.F.3.1:Identify and generate equivalent representations of linear functions, graphs, tables, and real-world situations. A1.F.2.1:Distinguish between linear and nonlinear (including exponential) functions. Understand that linear functions grow by equal intervals (arithmetic) and that exponential functions grow by equal factors over equal intervals (geometric).

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