Off in the BC side of things, some weeks ago we were talking about curve sketching. In my local parlance, I refer to f, f', and f'' as a stack. We need to learn where we are in the stack, and what information can be used to take us "up" or "down" that stack.

We started with sketching a polynomial and identifying regions where the behavior was increasing or decreasing. We translate that into positive or negative values for the function "down" one level in the stack. Similarly, though they didn't know algebraic integration at the time, we talked about how to make connections "up" between the values of a graph and the behavior those values were connected to.

Normally it's a long process in AB, but with the size of my BC class we knocked it out in a couple days. Throughout the sketching we added the ability to justify minimums, maximums, and points of inflection as well as identify regions that were concave up or down. For their design project, they used Desmos to create a polynomial, use prime notation to plot its first and second derivative, and then split it by its critical points. At the end they had to justify all the important features of their original, or "top" curve.

I've been using this class to integrate use of notation better. Desmos support for derivatives works great here, because although they could expand their initial polynomial and manually determine the first and second derivative, that was not the point of the exercise. I wanted them to work on how to make a mathematical justification for various points of interest on a function.

It's likely the AB students will follow up with a version of this project. It may not be as dense, but the "create your own stack of functions" aspects will play a key part.