I learned a lot of tough lessons in my first year teaching Calculus. Around March I started doing myself some favors and found ways to roll a bunch of topics backwards into Pre-Cal. That bet paid off and we're easily a month ahead in Calculus this year.

I also made a note to address another huge Calculus topic in Pre-Cal: the relationship between position and velocity. For whatever reason, kids see this stuff in physics but never see as much more than a formula they're supposed to pull of a sheet. They ought to subtitle the AB curriculum Physics Makes Sense Now with all their emphasis on position, velocity, and acceleration.

We have this stretch early on where we talk about function behavior. It's pretty typical, using words like increasing, decreasing, constant, and undefined to describe things and the domain for which that's valid. It goes hand in hand with Piecewise Functions.

This year we did all that and then added some context.

Graphs increase and decrease sure, but what if that graph represents something physical? What could induce those changes in behavior? That is, the slope of the direction change is the velocity at which the change happens. While introducing this I drew something similar to this graph and had a kid act it out, setting a reference point in the middle of the room and having him/her walk back and forth. We even had a discussion about whether the steepness of the direction change matters. The surprising thing for most of them was the connection between constant position and zero velocity. Often they only remember slope as a positive or negative idea, not so much the zero aspect. Nothing crazy, just a small placeholder for the ones I hope to see next year.

The nice immediate pay off is within a day or so a student of mine was working on a physics worksheet covering this exact same concept, only adding a velocity calculation step.