__Obstacle Course Activity__

## Main Concepts

Calculus

Limits, One-sided limits

Velocity

## Duration

80 minutes in class + 200 min homework or classwork

## Summary of Activity

In order to display the concept of limit in a real-life setting, students film themselves running a short obstacle course (see video). Their video is then the central object of their mathematical scrutiny. The first question they have to answer is: “what was your instantaneous velocity at the instant you crossed the finish line?”. In order to do that, students are required to illustrate limits by calculating multiple average velocities over decreasing intervals. The second is similar to the first one: “what was your instantaneous velocity at the instant you reached the first cone?”. This time, we require students to find the limit “from the left” and the limit “from the right”. As usual, students need to write a reflection and analysis of results to wrap up the project where students formulate and formalize their understanding.

## Notes and Insights

This activity is introduced verbally to students. There is no hand-out. Key elements to explain are:

- pair up with a friend and film each other’s race.
- at the end of the race, accelerate as much as you can for better results.
- there are two locations where you have to approximate the instantaneous velocity: the finish line (one-sided limit) & the mark between the hoola hoops and the slalom (two one-sided limits.
- you can indicate the minimum number of intervals you want. We ask for seven at the end, and three from each side at the first mark.
- you should insist that the last interval be as small as possible (less than 1 meter).
- insist on the importance of using the correct wording between “average velocity” and “instantaneous velocity” in their write-up.

In the video below, you can notice that we added some colored tape at intervals of 1 meter against the blue wall to facilitate distance measurements. It is recommended to put tape at narrower intervals at the two locations where they approximate their instantaneous velocity.

We are usually flexible on the strategies students use to find the distance and time. The changing angle of the camera is obviously an issue to evaluate the distance they run. Time can be more precisely figured out. What we are not flexible on is the exposition of the concept of limits in their write-up. The intervals have to be in decreasing order, narrowing down closely to the expected location. Limits are a very technical and abstract concept that students struggle to grasp. This activity is excellent at bringing out some misconceptions. Students often choose successive intervals that aren’t appropriate. They may also neglect the importance of displaying them in decreasing order to illustrate the concept of limit. Address those misconceptions along the way as needed.

For a toned down version of this activity, teachers may decide to skip the obstacle course part and show the video of Usain Bolt’s stunning performance in Berlin in 2009 (for example here). Then the teacher should ask that simple question: “What is his speed at the exact moment when Usain Bolt crosses the finish line?”. However, it isn’t as valuable. Firstly, the obstacle course allows for a lot of variation in the velocity (it is even negative when we do the loop), whereas Usain Bolt’s velocity near the end is almost constant. Secondly, each student is working on their own race which makes the experience more unique.