## Two Runners

In this activity, students use a software simulation of two runners along a track. Students control the speed and starting point of the runners, watch the race, examine the graphs, and analyze the time-versus-distance relationships. 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.

Scenario 3Two students face each other from opposite walls and they travel to opposing points with equal strides and pace.Each student takes one step per second.During the activity, have students indicate the point where they pass each other (intersect) using a piece of masking tape on the floor. Discuss the results.

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

Runners, Take Your Mark! (Two Runners) Activity Sheet

Students should visit 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.

Note: 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 student sets both runners at zero by dragging their icons along the tracks and click once on their icons so they are facing the same direction.
- The students should take out their Runners, Take Your Mark! (Two Runners) Activity Sheet, record the step size of "1" for both runners, 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) records results on the graph.

- The students then select the
**Play Button**to run the simulation. After the runners are completely done, the stop button resets the simulation.

- Next, the students set both of the runners at the zero position and choose
**different**step sizes for each runner. - The students then select the
**Slow Run Button**and with each "click" (at least 10 times) records results on the graph. - The students then select the
**Play Button**to run the simulation. After the runners are completely done, the stop button resets the simulation.

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** tab. 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! (Two Runners) Activity Sheet
- Computers with internet access

**Assessment Option**

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.

**Extensions**

- 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.
- The teacher will need to discuss and help students interpret graphs that show two variables (the boy and girl runners) which may be unfamiliar to many students in this age group.

**Questions for Students**

- What do you think the race will look like?
- Who will go farther in 100 "seconds"? (Note: It's convenient to call the units of time "seconds" for discussion purposes, although the simulation runs much faster.)
- 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?

### Single Runner

### Learning Objectives

Students will:

- Identify and describe situations with constant rates of change and compare them (for two runners).

### 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.

### Common Core State Standards – Mathematics

Grade 5, Algebraic Thinking

- CCSS.Math.Content.5.OA.B.3

Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule ''Add 3'' and the starting number 0, and given the rule ''Add 6'' and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.