Goal: Creating the headache for which finding the slope of the curve at a point is the aspirin.
Process: Students will see an odometer and speedometer, but the speedometer is hidden. They will attempt to figure out the speed of the car over time.
ACT I: Watch the video. What do you notice? What do you wonder?
The main question I'm aiming at: Can we find the speed of the car at each moment in time? The answer is clearly no -- but can we use the data to approximate the speed of the car over time. Let's do that.
What happened: I did this first minute of the first AP Calculus class of the year. Since students had never done something quite like this and the summer cobwebs were still in need of shaking off, the questions were less mathy than I hoped and expected. But we did get there.
Examples:
Why is the video so shaky?
Why are we watching this?
Why is there a picture of a pie?
But we did get there: How fast was the car going?
Pretty quickly the group established that it wasn't going to be possible to answer that question for every moment. They concluded that they could find the average speed of the car over the interval, but given the information they had, they could only figure out the average speed for every tenth-mile.
A student pointed out that if we had more digits on the odometer, we could figure out more frequent average speeds.
Which led to the big, happy moment of the day when a student said:
Wait -- this means we can never know how fast the car is going, because we always need two points, which means time has passed and it's just another average speed.
Headache! The aspirin: Calculus (Part I), The Derivative.
ACT II: Watch the video again, gathering data as you go.
Graph the data (on Desmos or by hand) and explain what your graph shows. How can you use the graph to approximate the speed of the car at different times in the interval? Make a table and a graph of the your approximate speeds.
What happened: The data was pretty straightforward to gather, so the graphs were fairly good. What was nice was that the time vs. distance graph looked reasonably linear -- looking at it, one might assume constant speed. Once students looked at the intermediate speeds, though, it became clear that this wasn't the case, and the time vs. speed graph illuminated that fact.
ACT III: The video unmasked
Watch the unmasked video.
How do your estimates of the speed compare to the actual speed? What caused problems?
Critical concluding question: If you had a graph of the actual distance traveled at each moment in time, how could you use the graph to find the speed of the car at any given moment?
What happened: The general behavior of the students' graphs matched the behavior of the speedometer pretty well. The actual maximum speed differed from the estimated actual speed, and we discussed why that was.
It wasn't hard for the students to see that if the time interval could be smaller (meaning more accurate distance information), the speed calculation would be more accurate. The smaller we make that interval, the more accurate we are. If we could get that interval to zero then...... but wait, that's not possible.
Or is it?
My conclusions:
Overall, I was pleased with how this went. It put students in the position where they couldn't help but ask the first big question of calculus. I think, too, it will set them up well for understanding the difference between average rate of change and instantaneous rate of change.
Some things I could do to improve the lesson:
1) Add a story -- nothing wrong with adding a little fun.
2) I could add a clock to the dashboard. It would make it a bit easier to gather the data. The video player timer disappears every so often. This might also make it possible to have a somewhat longer video, as fast-forwarding would be easier.
3) Embedding the video in a Desmos lesson is definitely something I'll do. I did this for lesson #2 and it worked really well.
Happy to take any suggestions!
