Sharing a intro calculus idea with my younger son inspired by Steven Strogatz’s Infinite Series appendix

Last week Steven Strogatz released two previously unpublished appendicies for his book Infinite Powers:

I had originally included two appendices in Infinite Powers, but they felt too mathy and got cut. Still, some of you might find them interesting. Here's a rough draft of appendix 1, about a clever way that Fermat integrated x^n with his bare hands [PDF]: https://t.co/gutuj0R38A

— Steven Strogatz (@stevenstrogatz) November 19, 2019

My older son and I did a fun project with Fermat’s idea. He’d taken calculus last year and the ideas Strogatz shared made for a really nice calculus review:

Sharing Appendix 1 to Steven Strogatz’s Infinite powers with my son

My younger son is in 8th grade and has not taken calculus. I thought some of the ideas about finding areas under simple curves would be interesting, so I tried sharing some of those ideas this morning.

We started by taking a look at the first page of Strogatz’s appendix and then talked about finding the area under for small values of

Now we moved on to the case . He had the really neat idea of thinking that this piece of the parabola might be a quarter circle. That idea made for a great little exploration:

I asked for another idea had he decided to chop the parabola up into rectangles. This isn’t an idea that came out of the blue because we have talked about some intro calculus ideas before. I was still happy to have this idea jump to the front of his mind, though:

Finally, I shared the full Riemann sum calculation with him so that he could see how to arrive at the exact answer of 1/3. This part was not as much an exploration for him as it was just me showing him now to do the sum. I was ok with this approach as there is plenty of time after 8th grade to dive into the details of Riemann sums:

I’m very happy that Strogatz shared these unpublished appendixes. They are yet another great way for kids to see some introductory ideas from Calculus.