- Why does the sum of the first n squares turn out to be n(n+1)(2n+1)/6? (I tried to think about it with drawings, but got nowhere. I looked up "sum of first n squares visual" and found this discussion at a blog called Understanding. This pdf was linked to
~~, and it's the best thing I found~~. But Jason Dyer pointed me to this much better visual.*Now I get it*.) - What if we wanted that same shape, but could afford a little extra weight, and wanted to find the area out to a variable right edge, labeled R? That was a way to rehash what we'd done the week before, with a little bit more generality.
- A car part that's under some crazy function.

We haven't done many derivatives yet, so he didn't notice that R^3/3 was the anti-derivative of x^2 (at x=R). I asked him to find the area under a triangle with angled side y=x and right side at x=r. That was easy. I wondered what the area under y=x^3 would be, and he saw the pattern, and guessed it. We haven't proved it yet.

We tried to find the area under y=2^x, but we'd never done derivatives of exponential functions yet, so we weren't able to finish that one.

We have one more session before I go on vacation. Maybe we'll tie up some loose ends, or maybe we'll play with something less strenuous. I'll let him decide.

re: sum of first n squares, I like this visual proof:

ReplyDeletehttp://mathoverflow.net/questions/8846/proofs-without-words/8851#8851

That is

ReplyDeletewaybetter than the one I linked to! I totally see it now. Lovely...And it starts to give me more of a feel for why cones and pyramids have the 1/3 in their volume formulas. If what I'm thinking is right, then a 4D pyramid or cone (whatever that is) would have a 1/4 in its volume formula! Wow! Thanks, Jason! (No, I'm not excited, I always jump up and down like this in the morning.)