Illuminations: Movement with Functions

# Movement with Functions

 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.

### 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

### Materials

 Stop watches Remote-controlled cars (strongly suggested, but alternatives are described below) Rulers Colored Masking Tape Collision Activity Sheet (optional pre-activity) Road Rage Activity Sheet What If? Activity Sheet (optional) Road Rage Answer Key

### Questions for Students

 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.] 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 shuch things as choosing the same person to race the car throughout the data collection and crash activities, and collecting additional data.] 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.] 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.] 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.] 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.] 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.] 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 indicae that you need to review the solution techniques.]

### Assessment Options

 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. Ask each group to present the answer to one of the analysis questions. 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. 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. Other extensions are provided at Math Projects Journal. Search for the Monster Cars lesson.

### 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?

### NCTM Standards and Expectations

 Algebra 9-12Use symbolic algebra to represent and explain mathematical 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.

### References

 http://www.mathprojects.com/ — Monster Cars lesson
 This lesson was prepared by Jamie Chaikin as part of the Illuminations Summer Institute.

3 periods

### Web Sites

 More and Better Mathematics for All Students
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