Author:
Subject:
Algebra
Material Type:
Lesson Plan
Level:
Middle School
6
Provider:
Pearson
Tags:
6th Grade Mathematics, Distance, Rate, Time
Language:
English
Media Formats:
Interactive, Text/HTML

# Using Graphs As A Visual Representation Of Rate Situations ## Overview

In this lesson, students first watch three racers racing against each other. The race is shown on a track and represented on a graph. Students then change the speed, distance, and time to create a race with different results. They graph the new race and compare their graph to the original race graph.

# Key Concepts

A rate situation can be represented by a graph. Each point on a graph represents a pair of values. In today's situation, each point represents an amount of time and the distance a racer traveled in that amount of time. Time is usually plotted on the horizontal axis. The farther right a point is from the origin, the more time has passed from the start. Distance is usually plotted on the vertical axis. The higher up a point is from the origin, the farther the snail has traveled from the start. A graph of a constant speed is a straight line. Steeper lines show faster speeds.

# Goals and Learning Objectives

• Understand that a graph can be a visual representation of an actual rate situation.
• Plot pairs of related values on a graph.
• Use graphs to develop an understanding of rates.

# Lesson Guide

If possible, project the interactive as students walk into class. Have students follow along and set up their own interactive as you demonstrate how to set up the race. If a projector is not available, give students a few minutes to set up their own interactive; circulate around the classroom to check for accurate setups. Before anyone starts the race, have a short discussion with the class about which car they think will win the race and why. Then allow  students to start their races and observe the race and the corresponding graphs.

Discuss the students’ observations. Try to elicit the following key concepts:

• The vertical axis of the graph is labeled “Distance,” and it is exactly as long as the racetrack.
• The horizontal axis is labeled “Time,” and it shows how long the race took to complete.

ELL: When showing the interactive, be sure that ELLs can follow the explanations. Allow ELLs time to process the information if the pace of the interactive is too fast for them. Ask students if they need to watch it a second time. Consider asking some questions to check for understanding before moving on to the discussion. If during the discussion it becomes apparent that ELLs have failed to understand important parts of the interactive, consider showing it one more time.

# Racing

Use the Race and Graph interactive to set up the following race and see which racer wins.

Choose the street racetrack. Set the track distance to 500 m. Set up the cars with the following speeds:

Orange car: 70 m/s

Yellow car: 85 m/s

Green car: 100 m/s

• What do you notice about the race and its graph? Discuss your observations with a partner.
• Which racer won the race?

INTERACTIVE: Race and Graph

# Lesson Guide

Students use the interactive to explore rate. Discuss with the class that a rate situation can be represented by a graph.

Some possible discussion points are:

• What does a point on the graph represent?
• How are the vertical and horizontal axes labeled?
• What do you think the slope of the line represents?

# Who Is Faster?

The green car traveled faster than the yellow car in the race from Task 1.

• Where can you see this information on the racetrack?
• Where can you see this information on the graph?

INTERACTIVE: Race and Graph

## Hint:

• The green car traveled the fastest. It reached the end of the track first, and its graph is the steepest line.
• The orange car traveled the slowest. It reached the end of the track last, and its graph is not as steep as the other two lines.
• The point at which each racer's graph intersects the 1-second line shows that racer's speed per second.

# Lesson Guide

Discuss the Math Mission. Students will analyze a graph of the race to compare the speeds of the three racers.

## Opening

Analyze a graph to compare the speeds of three racers.

# Lesson Guide

Students use the interactive to set up a graph.

# Mathematical Practices

Mathematical Practice 1: Make sense of problems and persevere in solving them.

• As they work, watch for students who can make sense of the race and model the relationship between distance and time for each racer on the graph.

Mathematical Practice 6: Attend to precision.

• Identify several graphs that vary with regard to level of precision. In particular, focus on what scales students use for the x- and y-axes and on how accurately they plot each race.

# Interventions

Students don't understand the visual representation.

• Where do you see each racer's race in your graph?
• Look at the visual representation of the race and your graph.
• How are they the same?
• How are they different?
• How do you know each racer traveled at a constant speed by looking at your graph?
• How can you tell which racer won by looking at your graph?

Student does not correctly represent units on the graph.

• What does the vertical axis of your graph show? What is the scale? What size intervals did you use?
• What does the horizontal axis of your graph show? What is the scale? What size intervals did you use?

Student has a solution and needs to delve deeper.

• Why did you approach the problem in the way you did?
• What does a point on your graph represent?
• If the [color] racer raced faster, what would happen to its graph?
• If the [color] racer raced slower, what would happen to its graph?

• Prediction graphs will vary, but should show graph lines for all 3 racers.
• Comparisons will vary.
• Explanations will vary.

# Twice as Fast

Now create a new race by changing the speed, distance, and time.

Decide on the track you want and the distance of the track. Then set up the race so that Racer 1 goes twice as fast as Racer 2 and Racer 2 goes twice as fast as Racer 3.

• Sketch a graph to predict what the graph of the race will look like, then start the race.
• Compare your graph with the race graph in the interactive.
• Explain your thinking in setting up the race and your prediction as compared with the results.

INTERACTIVE: Race and Graph

## Hint:

What scale will you use for the axes?

# Preparing for Ways of Thinking

Look for the following types of responses about the student graphs to share during Ways of Thinking:

• Time is on the horizontal axis and distance is on the vertical axis—this is the standard way to graph the relationship.
• Distance is on the horizontal axis and time is on the vertical axis. Speed is usually graphed with time on the horizontal axis and distance on the vertical axis. However, some students may put distance on the horizontal axis and time on the vertical axis.
• Scaling for the axes differs from the scaling shown in the race graph or it is not in equal increments.

SWD: Students with disabilities may need help understanding the academic vocabulary in this lesson. If possible, provide support (such as images, bold text, or highlights) for the graphing vocabulary and/or prompt students to refer to their vocabulary resources as they complete the tasks in this lesson.

ELL: Make sure you point to the graphs to highlight the parts of the graphs that are similar and different. Remember that beginning ELLs may rely heavily on these visual representations.

# Challenge Problem

• Check to see that students' graphs reflect the instructions. Students are to set up a graph that shows the following situation:
• The graph of the orange line is the steepest line.
• The graph of the green line is not as steep as the other two lines.
• The graph of the blue line is exactly in between the other two lines.

# Prepare a Presentation

• Explain how the speed and the graph of the racers are related. Use your work to support your explanation.

# Challenge Problem

Set up a race that shows the following situation.

• The graph of the orange racer is the steepest line.
• The graph of the green racer is not as steep as the other two lines.
• The graph of the blue racer is exactly in between the other two lines.

INTERACTIVE: Race and Graph

# Mathematics

• If you have students who graph distance on the horizontal axis, display their work next to a student graph that has time on the horizontal axis. Discuss which graph shows more clearly that the green racer moved twice as fast as the orange racer. Students will be analyzing different mathematical models and justifying their conclusions.
• Is it easier to see the relationships between the various speeds when time is tracked on the horizontal axis? Why or why not?
• Also discuss the idea of the independent variable and the dependent variable. Because the time varies and the distance is calculated, time is the independent variable. The independent variable is usually graphed on the x-axis.
• Display a graph that has correctly scaled axes and a graph that does not (for example, a graph with unevenly spaced intervals on the distance axis). Discuss what it means to scale an axis and why irregular intervals result in graphs that convey inaccurate information. This discussion will help students understand why it is important to be precise in mathematics and allow students to critique the reasoning of others.
• What do you notice about the scales of the horizontal axes?
• Why do you think it is important to use evenly spaced intervals?
• How could you fix the scale on this graph [point to a graph with unevenly spaced intervals]?
• Now show several graphs that represent the race accurately. Ask students to explain the relationship between the race and its representation as a graph.

SWD: You may want to highlight the x- and y-axes using different colors and the coordinates in matching colors to help students grasp the concept visually.

ELL: Monitor that ELLs understand the meaning of each of these questions. If there are concerns, explain them one by one to allow them to fully participate in the activity.

# Ways of Thinking: Make Connections

Take notes as your classmates make presentations about the relationship between the speed and the graph of their racers.

## Hint:

• What does a point on the graph represent?
• Explain how this line on the graph represents this racer’s race.
• Which racer won the race? How is this shown on the graph?
• What is the independent variable in this situation?
• How did you choose the scale for your axis?

# Lesson Guide

Discuss the summary with students. Make sure the following topics are touched on in the discussion:

• A situation can be represented by a graph.
• Each point on a graph represents a pair of values. In the race graph, each point represents the time and the distance that the racer traveled in that amount of time.
• Time is usually plotted on the horizontal axis. The farther right a point is from the origin, the more time has passed from the start.
• Distance is usually plotted on the vertical axis. The higher up a point is from the origin, the farther the racer has traveled from the start.
• A graph of a constant speed is a straight line.
• Steeper lines show faster speeds.

ELL: You may want to write these key points on a poster so that students can refer back to them throughout the unit. When working with ELLs, it is important to provide supplementary materials, such as graphic organizers, to illustrate new concepts and vocabulary necessary for mathematical learning.

# Summary of the Math: Understanding Graphs

• A situation can be represented by a graph.
• Each point on a graph represents a pair of values. In the race graph, each point represents the time and the distance that the racer traveled in that amount of time.
• Time is usually plotted on the horizontal axis. The farther right a point is from the origin, the more time has passed from the start.
• Distance is usually plotted on the vertical axis. The higher up a point is from the origin, the farther the racer has traveled from the start.
• A graph of a constant speed is a straight line.
• Steeper lines show faster speeds.

## Hint:

• What does a point on the graph represent?
• What does a straight line on the graph mean?
• What does the steepness of a line mean?