9-12
1

In this lesson, students use remote-controlled cars to create a system of equations. The solution of the system corresponds to the cars crashing. Multiple representations are woven together throughout the lesson, using graphs, scatter plots, equations, tables, and technological tools. Students calculate the time and place of the crash mathematically, and then test the results by crashing the cars into each other.

### Pre-Activity (optional)

The Collision Activity Sheet reviews concepts of systems of equations using cars colliding as an example. It provides a model that students can use as they complete the main activity. If students are already familiar with solving systems of equations, this activity sheet can be omitted to allow more time for the rest of the activity. It is suggested that you lead this pre-activity and students provide input.

When students create the slope-intercept equation, they may struggle with the negative sign when the car is moving back from position 100. Relate a negative sign to a different direction. This is the same concept used for describing subtraction on a number line. Note that the rate of change is actually velocity (speed with direction). For simplicity, the concept of velocity can be omitted.

As students calculate when the cars will crash, show this visually using a graphing calculator to relate the algebraic solution to a graphical representation. It may take several minutes of discussion to ensure students understand this relationship. Make sure students can also determine the solution using algebra. This system of equations lends itself to being solved by substitution.

As students calculate answers, they will likely round their answer to the nearest integer. This may lead to slightly different answers. Discuss with the class what rounding error means in the context of the problem. Since the main lesson uses real data collected by the students, it is reasonable to round the answers since human error will likely cause variation similar to rounding error.

### Preparation

This lesson is designed to use remote-controlled cars, allowing students to relate mathematics to a physical activity. It is strongly recommended that you have enough cars to provide one car for every four students. If a class set of cars cannot be obtained, ask students to bring in battery-operated, remote-controlled cars, or use two remote-controlled cars and large groups for the data collection process. Although larger groups can be used, it is best to divide the class into groups of four so that each student has one role. If larger groups are used, student roles can be rotated. If no cars can be obtained, the activity can be modified to use the sample data provided in the Road Rage Answer Key.

Before beginning the activity in class, find a location appropriate for students to use the remote-controlled cars. A hallway approximately 125 feet long with natural divisions, such as tiles, allows for easy measurement and data collection. If the hallway does not have divisions, use colored masking tape to mark equal intervals, and have students estimate the distance between the units. For example, in 3 seconds, a car can travel the length of 10 cement blocks. In this case, 1 cement block represents 1 interval of distance. If each cement block is measured, then the number of blocks can be converted to length in inches. Counting units simplifies the measurement because the cars move quickly and it is very time consuming to physically measure the distance. Alternately, you could use a football field for this activity, provided you do not use miniature remote-controlled cars, which are only about 2 inches long.

Divide students into groups of 4. Each student should have a role, as outlined on the Road Rage Activity Sheet. You can assign these roles or allow students to choose roles. Students could keep their roles for the duration of the lesson, or rotate so each student assumes multiple roles. Provide a stopwatch and a randomly selected remote-controlled car to each group. Discuss an overview of the steps students will complete in the activity:

1. Collect data by racing the car.
2. Graph your data and find the line of best fit to determine the speed of the car.
3. Determine the equation to find the position of the car, starting from either end of the hallway.
4. Partner with another group and determine when and where the cars will crash by solving a system of equations.
5. Perform a crash test and compare the results to the mathematical solution.
6. Answer and discuss analysis questions.

### Data Collection

Have students practice driving and timing their cars in the classroom prior to going into the long hallway or football field to collect the data. Ensure students are comfortable controlling the cars and collecting accurate measurements. Sample data is provided on the Road Rage Answer Key. As students work through the various parts of the activity, use the answer key to check for reasonable answers and appropriate calculations.

Depending on the size of the space available, several groups can collect data at the same time. Separate the space with clear boundaries for each group. This step should be primarily student-run, with you making sure students are stay focused on collecting all of the necessary data. As students run the trials, they should record the results in the table in Question 1 of the activity sheet. Groups who complete the trials quickly should be encouraged to collect data for longer timings, which will provide a more accurate estimate of speed in the steps that follow.

### Line of Best Fit

Groups then graph their data and estimate a line of fit. Students may need some guidance on how to create a line of fit. Discuss the definitions of key terms, which are provided on the activity sheet. Note: Make sure students are focused on drawing a line between the data points rather than on connecting the points. It is also important to point out that a line of fit does not have to include any of the measured points. This is an excellent opportunity to talk about outliers and why they may have occurred. Data may vary based on who was driving the car.

For this part of the activity, it is only important to determine the speed of the car. Students will calculate this based on the line of fit. Stress to each group can have a different speed depending on which car is used. Also, this is a manual fit based on personal judgment of a "best fit" line, which may cause natural variation. To find the line of best fit in Question 5, have students use either a graphing calculator or an online applet, such as Line of Best Fit. Students should compare the values for speed found by both methods of calculating the line of fit.

### System of Equations

At this point, remind students to use the Collision Activity Sheet to help them create the equations and calculate the crash time and position. Validate that each group has created correct equations starting from both ends of the hallway before pairing the groups. For classroom management reasons, you may want to randomly assign starting locations to the groups. One group starts at 0, and the other group starts at the ending location, position 100.

### Car Crash & Data Analysis

For the car crash trials, have students collect data one at a time. Before beginning this step, mention to students not to run their cars until their turn because the batteries run down, affecting speed and crash results. This step should otherwise be student run. Let pairs of groups decide on an order or have them pick an order from a hat. Let the groups decide who races the car, and let them swap drivers for each of the three trials.

To conclude the activity, students should discuss their results with the class, including details about variation and what occurred. Reasons for variation are discussed in more detail in the Road Rage Answer Key. If students have difficulties controlling the cars for long distances, expect the estimates to be inaccurate, but discuss why this occurred.

### Reference

http://www.mathprojects.com/ — Monster Cars lesson

Assessment Options

1. Ask groups to present their results to the class and compare them. Have the groups discuss the differences among solutions and why they occurred. Students may compare the speed of the cars and the accuracy of the initial data collection. If two large groups were used instead of smaller groups, assign different questions from the activity sheet to each group.
2. Ask each group to present the answer to one of the analysis questions.
3. Students can answer a set of questions to show the impact of different speeds on the graphical and algebraic results. Have students complete the What If? activity sheet. Answers are provided at the end of the activity sheet.
What If? Activity Sheet and Answer Key
Several key points of the lesson can be addressed using this activity sheet. When students are asked what would happen if their car was twice as fast, use the opportunity to discuss why the car that starts as position 100 is closer to the starting position after completing a trial run. Students should realize that since their car has doubled its speed, it will travel farther than it did previously. When students are asked what will happen if both their cars are twice as fast, use this opportunity to discuss how the position is the same but the time is shorter. Make sure students understand that this is not the same solution (x changed although y did not). Question them as to why the solution changed.

Extensions

There are several variations for this lesson. Vary speeds (doubling, halving, etc.) and starting locations. For example, the slower car can have a head start, then students can calculate when the faster car will catch up.

Questions for Students

1. What are some of the factors that may have caused your result to vary from the predicted result?

[If different students raced the car, the results can vary. In addition, the battery power for the car can affect the speed. Students could also suggest other factors.]

2. How could you redesign the data collection and the crash test to provide a closer result?

[Ways to control the factors contributing to error include such things as choosing the same person to race the car throughout the data collection and crash activities, and collecting additional data.]

3. What does it mean to say that your predicted result is a solution of the system of equations?

[Students should understand that a solution to a system of equations is where the lines represented by the equations intersect. For the cars it is the time and location where they will crash because the graph plots time vs. position.]

4. How many solutions are there for the system of equations? Why?

[One. Looking at the graph, 2 lines can intersect at 1 point at most.]

5. How can you tell from the data table, with reasonable certainty, that your algebraic solutions are correct?

[The algebraic solution is the place where the x and y values in one table are equal to the x and y values in the other table.]

6. Why are the values of time and position the same for both cars when they collide?

[Although this may seem redundant, this question reinforces that a single solution satisfies both equations. For the cars to collide, they must be at the same location at the same time.]

7. If your car was twice as fast, how would the graph of the car’s movement over time change?

[Students should relate this to a steeper graph that shows a greater distance traveled in a shorter amount of time. Students should also be able to determine that the coefficient of x (the slope) doubles.]

8. Which method was easiest for determining when the cars would crash (table, graph, or algebraic computation)? Why?

[Expect different opinions on this and take the time for students to discuss their choices. If a student cannot pick a method, it may indicate that you need to review the solution techniques.]

Teacher Reflection
• How did your lesson address auditory, tactile and visual learning styles?
• Did students demonstrate understanding of the materials presented?
• Did students make the connection between a solution of a system of equations and the time and location of the crash?
• How did students communicate that they understand the meaning of a solution?
• What were some of the ways that students illustrated their active engagement in the learning process?
• What issues, if any, arose with classroom management? How did you correct them? If you use this lesson in the future, what could you do to prevent these problems?

### How Did I Move?

6-8, 9-12
A common problem when students learn about the slope-intercept equation y = mx + b is that they mechanically substitute for m and b without understanding their meaning. This lesson is intended to provide students with a method for understanding that m is a rate of change and b is the value when x = 0. This kinesthetic activity allows students to form a physical interpretation of slope and y-intercept by running across a football field. Students will be able to verbalize the meaning of the equation to reinforce understanding and discover that slope (or rate of movement) is the same for all sets of points given a set of data with a linear relationship.

### How Should I Move?

6-8, 9-12
This investigation uses a motion detector to help students understand graphs and equations. Students experience constant and variable rates of change and are challenged to consider graphs where no movements are possible to create them. Multiple representations are used throughout the lesson to allow students to build their fluency with in how graphs, tables, equations, and physical modeling are connected. The lesson also allows students to investigate multiple function types, including linear, exponential, quadratic, and piecewise.

### Learning Objectives

Students will:

• Collect data and graph a scatter plot to determine the speed of a remote-controlled car.
• Create a line of best fit using estimation and technology.
• Use tables, graphs, and algebraic calculation to determine when their cars will crash with another group's car.
• Validate their calculations by crashing the cars into each other.
• Analyze why their time and location estimates for the crash may not be the same as a real-life trial.

### NCTM Standards and Expectations

• Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships.
• Write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases.
• Approximate and interpret rates of change from graphical and numerical data.