Letting the Students Come Up with the Questions: AP Calculus Day 1

The homework assignment I gave my AP Calculus class on the first day of school this year was inspired by Michael Pershan’s well-worth-a-read post I don’t focus my classroom on solving problems. (This was his contribution to Sam Shah‘s fabulous Virtual Conference on Mathematical Flavors).

In developing this assignment, I took an old GeoGebra activity I’d created and framed it in an entirely new way. And, wow, was it ever better as a result of this re-framing! If you follow the link to my original activity, you’ll see that in it, I fell into the dangerous teacher trap of telling the students what questions to explore in a situation where there’s no need to do that. In this year’s version, I got rid of all those questions and gave them this instead:

And here’s a sampling of some responses:

As a means of helping students to focus on what their classmates had observed and asked, I began the following class by giving copies of this sampling of responses to small groups of students and asking them to peruse this list and look for something that they had included in their homework that wasn’t included in this sampling and add it to the list. I then asked students to name themes they saw running through the observations and they came up with things like function, position, distance, time displacement, speed, velocity, acceleration, . . . and they began asking and answering questions regarding differences among these terms, which then led us to a discussion of terms like magnitude, slope, and rate of change.

At one point we meandered into a consideration of why we often use s for position and why it’s fine to put x on the vertical axis. We looked at one student’s articulation of their insight around this idea:

Then we took a look at this meme (thanks to Caitlyn Gironda for this idea from her Engage-ify your AP Calculus class presentation):

I pointed out that there were an awful lot of “it”s (or similar vagueness) in the excellent observations they had made, and that these “it”s sometimes meant the particle, sometimes meant time, sometimes meant position, etc, so the next activity was for each group to pick any observation from the list and improve its precision and write both the original statement and the improved one on the board. We then looked at these as a class and suggestions were offered for further improvements or expansions, leading–as it turned out–to additional clarification of some of the themes we had identified earlier.

Finally we took a look at one of the questions that was asked in different forms by lots of people: 

We considered one student’s progression of questions which moved higher and higher up the ladder of abstraction in pursuit of an answer to this question. Along the way the student ended up sketching a graph and noting the following:

and the student finished with the question

Another student picked up where this one left off: 

And all of this from an assignment which didn’t ask students to find or solve anything in particular, but asked them to observe, question, and look for some answers to their own questions!