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 subtitution.
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:
- Collect data by racing the car.
- Graph your data and find the line of best fit to determine the speed of the car.
- Determine the equation to find the position of the car, starting from either end of the hallway.
- Partner with another group and determine when and where the cars will crash by solving a system of equations.
- Perform a crash test and compare the results to the mathematical solution.
- 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 that each group can have a different speed depending on which car is being 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.
