This year I am using mistakes to prompt discussion in class and make homework dissection more engaging. In my AP Calculus AB class, I use Active Calculus as the text, and many of its activities and exercises are perfect for an Activity Builder conversion. Such was the case when I assigned this activity to work on section 1.3 Ex 1 in Active Calculus.
The very first slide has students drawing some secant and tangent lines on a function. Most students in all three sections did pretty well with it, but I saw a few common errors I wanted to address. So channeling my Talk Less course from TMC16, I made up these two slides.
Each table had a minute to respond to the prompt. The best discussion happened with the second slide when I asked students to conjecture on what the student was thinking when he drew the green line (which was supposed to be tangent). When I started asking this question in the first of 3 sections, I didn’t really have a good idea myself.
Students in each class proposed that the student thought tangent means it can only intersect the function ONCE (like with a circle). The student may have thought this was the only way to accomplish such a definition. I’d never considered that. It led to a great wrap up and clarification.
Here’s another slide I used from that activity.
There was one more unexpected observation from that activity. After the exercise from the text was done, I built in a few slides at the end where students would connect elements from the limit definition of the derivative to its traditional graphical counterpart. It started with an animated version (slide 8) and then asked questions about specific terms and where they link up. On slide 9, I asked where the two points for the AROC were. Most answered in the way I would have expected, by selecting the 2 points in their one spot. But then I got this.
Exactly one student in each section had a response like this. And again, students were able to put themselves in that one student’s shoes and realize that in fact, that second point is not static and can be in many different places. I would have never done that on my own.