I told my calculus class, in the last 15 minutes of class on Monday, that:
“Somebody told me something, and I don’t know if it’s true. They said that if there’s a cubic equation that hits the x-axis three times, then there’s a point of inflection, and it will be the average of these three x-intercepts. I don’t know if I believe them. It sounds plausible, but I’m skeptical. Because why would the zeros be related to the inflection point? And why would they be related to it in such an elegant way? Crazy talk, I said.”
Of course, they called me on it, saying that of course I knew if it was true or not. So I chuckled, and said that they got me, and of course I know if it’s true or not. But I wanted them to figure it out. So I asked them to figure out what the problem is, and guess how we would solve it.
By the end of the class, we together (but letting them lead the discussion) determined that our general equation for the cubic would be f(x)=k(x-a)(x-b)(x-c), where the x-intercepts are at a, b, and c.
Then for homework I let them loose on figuring out if what I was told was true.
The next day (today), I asked them how far they got. One person solved it, and a few had the right idea, but got frustrated with the algebra. No one “checked” to see if the inflection point truly was an inflection point (if at that point, the function switched curvature from being concave up to concave down or vice versa). But going over the solution together was awesome because:
1. I got to reinforce that a, b, and c in this equation are constants, not variables (a few were confused about that)
2. I got to show them a quick way to “foil” out (x-a)(x-b)(x-c)
3. I got to remind them how to prove something is an inflection point
4. I got to show them what a formal proof looks like
and most importantly,
I got a few of them to see how cool it was. I basically told them why I loved problems like this… because even though the algebra can get hairy, even though you might make a wrong turn somewhere along the line, we were able to show something that is totally not intuitive. To use the words I used in class, that “the payoff is worth so much more than the work.” And even though only one person solved it on their own, I think a few of my students felt that ownership as we solved it together in class.
In theory and practice it was 30 minutes of class well spent. I should do more of these sorts of problems. Hard things we do together in class.