It’s probably unconventional to talk about the limit definition of the derivative before learning limits, but hey, that’s what I did this year. We tried to connect the activity we did yesterday trying to find the instantaneous velocity of a rolly chair with graphical and numerical depictions of the same concept (calculating average rates of change for a function at smaller and smaller intervals, and looking at what happens to the accuracy of secant lines when the points become closer and closer together). I even went so far to write up in algebraic notation what we were doing, throwing the limit sign up there right at the end (which they had never seen before). From this, they guessed at what the idea of a limit is. Though they weren’t quite right, I think the activity we did yesterday really gave them an intuitive feel for what a limit is. I’m hoping that with some exposure to the limit definition of the derivative a few times over the course of the next week, with a great conceptual underpinning supporting the idea, the task of understanding one of the most fundamental parts of Calculus will be easier.
On another note, this was one of the weakest classes I had all year – it was a lot of me talking, and not much variety of activity. I’m try to avoid this, but with some topics it almost seems inevitable!