Unit 4, Lesson 17
Two Related Quantities, Part 2
Accelerated 6
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In this Warm-up, students reason about the relationship between distance, rate, and time to solve a problem. The purpose is to activate the idea that constant speed can be represented by a set of equivalent ratios associating distance traveled and elapsed time. In the longer activity that follows, students represent the relationship between these two quantities using a table, equations, and graphs.
As students work, monitor for different representations used (particularly tables) as well as for students who calculate each wheelchair’s speed in meters per second or each wheelchair’s pace in seconds per meter.
Arrange students in groups of 2. Give students 3–4 minutes of quiet work time and 1–2 minutes to share their thinking with a partner. Follow with a whole-class discussion.
If needed, remind students of tools or strategies that may be appropriate for solving this problem, including double number line diagrams or tables of equivalent ratios. Consider allowing students to use calculators to ensure inclusive participation in the activity.
A wheelchair user is considering two electric wheelchairs. Wheelchair A can travel 25 meters in 10 seconds. Wheelchair B can travel 13 meters in 5 seconds.
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Invite students to share how they know which wheelchair can travel faster. If one of the following ways of reasoning is not mentioned by students, bring it to students’ attention:
Next, discuss how students found the time it would take each wheelchair to travel 195 meters. Ask students who used different reasoning strategies to share. If no student used a table of equivalent ratios, as shown in the Student Response, display a pair of blank tables and discuss how to use the table to reason about the distance each wheelchair can travel in 1 second and the time it takes to travel 1 meter.
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To Gather
Colored pencils
In this activity, students calculate and compare the unit rates in meters per minute for three robotics teams and consider the graphs and equations that describe the distance–time relationship. As students interpret the data values, graphs, and equations in context, they reason abstractly and quantitatively (MP2).
Arrange students in groups of 2. Give students access to colored pencils and 5–8 minutes of quiet work time, followed by a whole-class discussion. Invite students to share what they know about robotics, including any experience they may have with building and programming robots.
Diego, Elena, and Andre joined different teams for a robotics competition. Each team programmed a robot to travel at a constant rate.
Complete each table to show how far each team’s robot traveled during the competition.
| time (minutes) | distance traveled (meters) |
|---|---|
| 1 | |
| 1 | |
| 2 | 6 |
| 4 | |
| 5 |
| time (minutes) | distance traveled (meters) |
|---|---|
| 1 | |
| 1 | |
| 2 | |
| 4 | 11 |
| 5 |
| time (minutes) | distance traveled (meters) |
|---|---|
| 1 | |
| 1 | |
| 2 | |
| 4 | |
| 5 | 17.5 |
Graph points to show the progress of each robot. Use a different color or symbol for each robot.
Let’s say that
Explain why
Complete the table with equations that can relate distance and time for the other two robots. Make sure that in each equation, the independent and dependent variables are as shown. Be prepared to explain your reasoning.
| equation | independent variable | dependent variable | |
|---|---|---|---|
| Diego's team | |||
| Elena's team | time | distance | |
| Andre's team | distance | time |
The goal of the discussion is to ensure that students understand how the table, graph, and equations represent the situation and make connections across the representations.
Invite students to share their responses to the last two sets of questions and explain their reasoning. In particular, ask them how they identified the independent and dependent variables for the equation for Diego’s robot, and how they figured out how to write equations when the independent and dependent variables are specified.
Point out the standard convention of showing the independent variable on the horizontal axis of a graph.
If time permits, consider asking questions such as:
Equations are very useful for solving problems that involve constant speed. Here is an example.
A boat travels at a constant speed of 25 miles per hour.
We can write equations to help us answer questions like these.
Let's use
When we know the time and want to find the distance, we can write:
In this equation, if
This equation can help us find
When we know the distance and want to find the time, we can write:
This equation can help us find
We can also graph the two equations we wrote to get a picture of the relationship between the two quantities. By convention, the independent variable is represented on the horizontal axis.
A graph of 10 points plotted in the coordinate plane with the origin labeled "O". The horizontal t axis is labeled "time in hours". The numbers 0 through 10, in increments of 2, are indicated, and there are vertical gridlines midway between. The vertical axis is labeled "distance traveled in miles". The numbers 0 through 250, in increments of 25, are indicated, and there are horizontal gridlines midway between. The data are as follows: 1 comma 25. 2 comma 50. 3 comma 75. 4 comma 100. 5 comma 125. 6 comma 150. 7 comma 175. 8 comma 200. 9 comma 225. 10 comma 250.
Problems about constant speed can also be solved using other strategies, such as by making a table of equivalent ratios.
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