## 5.2.1 Distance, Speed, and Time Relationships Using Simulation Software

This example includes a software simulation of two runners along a track. Students can control the speeds and starting points of the runners, watch the race, and examine a graph of the time-versus-distance relationship. The computer simulation uses a context familiar to students, and the technology allows them to analyze the relationships more deeply because of the ease of manipulating the environment and observing the changes that occur. Activities like this can help students in the upper elementary grades understand ideas about functions and about representing change over time, as described in the NCTM Algebra Standard.

### Follow-Up Questions and Tasks

Think about and discuss the following: What does the graph show? Did what happened match your prediction? If it did, how does the graph show what you predicted? If not, why do you think what happened was different from what you expected?

Click on Get Ready to position the boy and the girl to start a new race. Make a change in one of your settings (e.g., the length of the girl's stride or the boy's starting position). How will this change affect the graph? Run the simulation again and see what happens. Continue making changes and predicting the result. After each run of the simulation, think about what the graph shows and think about what happened and why.

### Discussion

This example illustrates computer software that engages students in the upper elementary grades in ideas about functions and about representing change over time. The software and examples in this activity are based on the Trips software (Clements, Nemirovsky, and Sarama 1996). This software allows students to analyze change by setting the starting positions and length of stride (speed) for two runners. Students then observe the simulated races as they happen and relate the changing positions of the two runners to dynamic representations that change as the events occur. Students can predict the effects on the graph of changing the starting position or the length of the stride of either runner. They can observe and analyze how a change in one variable, such as length of stride, relates to a change in speed. This computer simulation uses a familiar context that students understand from daily life, and the technology allows them to analyze the relationships in this context deeply because of the ease of manipulating the environment and observing the changes that occur.

In this activity, students are working with functional relationships. As students work with this example, they need to be encouraged by the teacher to analyze how a change in the starting position or the length of the stride will affect the time needed to reach the finish lines. Acting out different stories about the "trips" can help students visualize the effect of, for example, increasing the length of the stride or having one runner start in a position ahead of the other runner. As students become familiar with the simulation, they can analyze each situation numerically by building a table showing the relationship between time and distance. By inspecting the track, the graph, and the table, students can become more precise in reasoning quantitatively about the relationships ("The length of the boy's stride is 2, so you know his distance by multiplying the time by 2"). Older elementary school students can relate the boy's and girl's trips proportionally ("The girl goes twice as far as the boy in the same amount of time"). Students can begin to describe rate of change informally by inspecting the slope of the line ("The girl's line is steeper because she is moving faster"). Interpreting two-variable graphs will be unfamiliar to many students in this age group. Part of the teacher's role is to help them connect what is happening on the graph to what is happening on the track: How long does it take for the boy to go the same distance as the girl has traveled in fifty "seconds"? How can you see this demonstrated on the track? On the graph? Where on the track does the girl catch up to the boy? Where is this point on the graph?

Additional Tasks and Questions

Set the starting position and length of stride for both runners. Run the simulation. Now write a story that describes the trip. For example, "The girl is going really fast. She catches up to and passes the boy, who is going slow," or "The girl started way behind the boy, who was already halfway to the tree by the time she got going. She went really fast and caught up to him more and more. Finally, at 75 she passed him and kept going really fast and got to the tree first."

Three motion stories are told below. Before the students use the simulation, have them physically simulate the motion stories (with their bodies). Then develop specific instructions (starting position and length of stride for each runner) to produce the action in the stories. Try out the instructions using the computer simulation above.

*Motion Story 1*. The boy and girl start from the same position. The girl gets to the tree ahead of the boy.*Motion Story 2.*The boy starts behind the girl. The boy gets to the tree before the girl.*Motion Story 3.*The boy starts at the tree and the girl starts at the house. The boy gets to the house before the girl gets to the tree.Look at the two graphs below, which show the results of different motion stories. Develop a set of instructions to produce each trip.

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**Take Time to Reflect**

Do you think students would enjoy using this computer activity? Why or why not? What are they likely to focus on?

How can teachers help students become comfortable moving among various techniques for organizing and representing ideas about relationships and functions?

What important ideas about functions and representing change over time can students learn while working on this activity?

**Acknowledgement**

This activity and applet were adapted with permission from the Trips software, Clements, Nemirovsky, and Sarama (1996). The activity was adapted with permission from Tierney et al. (1998).

**Reference**

- Clements, Douglas H., Nemirovsky, Ricardo, & Sarama, Julie. Trips (computer program). Palo Alto, Calif.: Dale Seymour Publications, 1996.
- Tierney, Cornelia, Ricardo Nemirovsky, Tracy Noble, and Doug Clements.
*Investigations in Number, Data, and Space: Patterns of Change*. Palo Alto, Calif.: Dale Seymour Publications, 1998.