I was very nervous to start Pace Car. When Mrs. F and I originally learned to do this activity, I believe some tears ensued. I did warn my students that tears may fall during this learning exercise, and it might be me (again) or them (maybe again?), but that’s part of the learning struggle for right now. We completed the table using current velocity (v) for Top Car. It was then discovered that Top Car had too little of a delta-x when using current velocity as part of its next-x equation: next-x-top-car(x, v): x + (v * delta-t)
There was disagreement whether or not to use current-v versus v when representing current velocity. However, my students were just lucky to end up with the efficient (lazy?) teacher who didn’t want to write “current” repeatedly. Mrs. F and Mr. R went with the current-v notation. I’m not sure if that brought anymore understanding to the whole Pace Car activity, but it brought upon confusion when our students tried to help each other out. There was some audible debates that went on with which notation to use.
For Middle Car, we used next-x-middle-car(x, v): x + (next-v(v) * delta-t). It was at this point that the students had trouble differentiating between “next-v times v” and “next-v of v.” At the beginning of discussing Middle Car, I found out how intentional I really needed to be with my words. We talked about the function notation and how f(x) looked like next-v(v). From here on out, students started to refer to next-v(v) as “next-v of v” and trying to multiply next-v and velocity decreased by a lot.
When we got to Bottom Car, the students intuitively knew to find the average between current velocity and next velocity. After talking to Mrs. F, I realized they didn’t actually understand WHY and how finding the average worked out. The next class, they learned a little calculus. We talked about how the velocity is different at every point between 3s and 4s. We arbitrarily picked the numbers 3 and 4, but I’m not sure if the students can be flexible enough to shift their thinking for other intervals. Anyway, we talked about how there is the possibility of having an infinite number of points (velocities), and how nobody has time to find the average for an infinite number of points in a time interval. So we decided as a class that we would just pick the beginning of the time interval (current-v) and the end of the time interval (next-velocity).
The code for Bottom Car ended up being- next-x-bottom(x, v): x + (avg-v(v) * delta-t). Some of the students found out really quickly that they’re going to have to use the whole definition of average velocity.
We’re taking our Pace Car Quiz this week, I’m going to see how well we did in presenting Pace Car this way.
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