## Single Runner

In this activity, students use a software simulation of one runner along a track. Students control the speed and starting point of the runner, watch the race, examine a graph, and analyze the time-versus-distance relationship. This activity helps students understand, describe, and compare situations involving constant rates of change.

To introduce this activity, ask two student volunteers to stand in front of the classroom to physically demonstrate and discuss the results of each of the following scenarios:

Scenario 1. Two students start from the same position at one end of the classroom. One student takes giant-steps while the other takes baby-steps.Each student takes one step per second.Who gets to the other end of the classroom first? How many steps are taken? Discuss the results.

Scenario 2.One student starts behind the other at the same end of the classroom, both walking with equal stride and pace.Each student takes one step per second.Who gets to the other end of the classroom first? How many steps does each student take? Discuss the results. Ask students to predict the effect of changing the length of stride.

Place students into teams of two and distribute a Runners, Take Your Mark! (Single Runner) Activity Sheet to each group.

Runners, Take Your Mark! (Single Runner) Activity Sheet

Students should open the Runner Simulation Tool.

Working together, partners share the responsibility of "Mouse Driver" and "Reader/Recorder". The "Reader/Recorder" will read the directions from the activity sheet and record observations while guiding the activity. The "Mouse Driver" controls the action of the mouse and movement on the computer screen. Partners should switch roles until all have moved the runner.

- Be sure to tell students about two key assumptions used in this activity.

(a) The runner always takes**one step per second**(no matter how big the step size is).

(b) We will**measure time in seconds**, even though the actual movement in the simulation will probably be much faster. - To begin, the students select either the male or female runner. To do this the student "clicks" upon the male or female icon in the box next to the graph of the runner they DON’T want to use. This will cause that runner to temporarily vanish from the running line and graph. Next the students set the runner to zero by dragging the icon along the track and clicking until the runner is facing the direction of running from left to right.
- The students should take out their Runners, Take Your Mark! (Single Runner) Activity Sheet, record the step size of "1", and set the step size on the interactive
applet to "1".
- The students then select the
*Slow Run Button*and with each "click" (at least 10 times), results are recorded on the graph. - The students then select the
*Play Button*to run the simulation. After the runner is completely done, the stop button resets the simulation. - Next, the students set the runner’s step size to 2, select the
*Slow Run Button*and record the results on the graph. - Repeat this with the step sizes of 4 and 5 and record the results. Students may vary the runner’s step size all the way up to 15.

### Teacher Note

In this race simulation software, the finish time is rounded up to the nearest whole number. Thus, for example, if a runner starts at 0 with step size 3, the finish time shown will be 34, rather than 33 1/3. Students may notice this and comment that 34 × 3 does not equal 100. They may notice that with step size of 3, and one step per second, the finish time should be 33 1/3 seconds. Please be aware of this limitation of the software as you teach the lesson.

The closing should be structured so that students can review and
pull together what they have learned. Include questions or tasks that
encourage students to reflect on their work. For example, have students
consider the
**Questions for Students** (below). In so doing they will
consolidate what they have learned. Furthermore, this will provide an
opportunity for you and the students to assess what they have learned
and what they still want or need to understand. This will give you
ideas for further instruction.

- Runners, Take Your Mark! (One Runner) Activity Sheet
- Computers with internet access

**Assessment Options**

Review and interpret the results shown on the graphs:

- Describe what you observe with the runner’s step size and time.
- Explain the relationship between starting point and time.
- With increasing step sizes, predict how many steps to the hundred line.
- What would you have to do to change the slope to be at a greater or lesser angle?

**Extensions**

- Suppose the length of the runner's stride (step size) is 2. You know that in this simulation the runner always takes one step per second. Thus, you can find the distance traveled by the runner by multiplying the time (in seconds) by 2.
- Students can begin thinking about proportional change. For example, see what happens when the the step size is doubled.
- Students can analyze the situation and rate of change based on the slope of the line. For example, steeper slopes mean faster speed, or parallel lines mean same speed.

**Questions for Students**

1. If the runner is using a longer stride (larger step size), how does this affect the number of steps it takes to get to the finish line?

[Longer strides (larger step sizes) result in less steps needed to get to the finish line.]

2. Compare and contrast the differences and similarities between 2 steps, 4 steps and 5 steps.

3. How can you see the differences in steps sizes demonstrated on the track?

4. What happens to the graph when you start the runner further down the track?

### Two Runners

### Learning Objectives

Students will:

- Identify and describe situations with constant rates of change and compare them.
- Make and test predictions about step sizes and finish times.

### NCTM Standards and Expectations

- Represent and analyze patterns and functions, using words, tables, and graphs.

- Investigate how a change in one variable relates to a change in a second variable.