Calculus

"Someone told me…"

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.

Honing intuition is hard work (but worth it)

I thought I’d kill birds with stones and (1) try using LaTeX equations while (2) explaining how I honed my calculus classes intuition. Granted, the idea is simple and I think most teachers teach it this way, but I didn’t and my students got confused. So on Day 2 of the chain rule, I had them come in and do the following right away:

Find the derivatives:

A1:
B1:
C1:

A2:
B2:
C2:

A3:
B3:
C3:

Bonus Problems:

D1: Find the derivative of .

D2: Find the derivative of .

And in fact, they were very successful. Instead of showing them a big equation and how to break it down, I instead started small and built it up. And showed them each part of what was going on. And by the end, my students were pretty capable chain-rule-appliers.

There are three things to say about this:

1. It is important to first introduce as because it makes the composition of functions easier to see. (Plus, really surprisingly and disappointingly, some of my students didn’t realize they were the same thing!)

2. There is something really exhilerating about writing really long answers to problems. They love it. I love it. It makes us feel important and like we’re doing something extraordinarily complex. Which we are (to some extent).

3. I think it’s important to polish off this whole chain rule business with something that shows the students that all these things that they’re doing works. Physically worsk. And are just like what we’ve been doing. So what I did after we worked on this worksheet is we went back to a derivative we had initially solved with the chain rule: . I asked them what the equation of the tangent line to this function was at a particular value of . And then I showed them that what we found for the tangent line worked graphically. They saw the tangent line hit the curve in front of them. At this point, there weren’t many gasps (they knew it was going to work), but I think it drove home that (a) hell yeah, it works! and i’m. not. lying. to. you. and (b) this is no different than everything else we’ve been doing. (We’ve been finding tangent lines to tons of functions. These functions are just longer and more gross looking.)

A life of whining

I know, I know, teachers are always complaining.

But what’s even more terrible is that we teachers don’t tend to share our success with each other. We do have successes, I swear– even though they tend to come in small numbers and sporadically. Still, we keep them to ourselves.

So in this post, I’m going to brag.

In my calculus class yesterday, my students struggled hardcore with the chain rule. They didn’t quite get substitution for the more involved problems, and I’ve been trying to hone their intuition explicitly — saying I don’t want them to do problems formally. I want them to practice “seeing” the solutions.

Let’s level here: getting an assortment of calculus students to “see” anything is hard. They like rules, procedures, things they know will always work. Telling them to “hone their intuition” frightens them. There’s less certainty. But the reward of finally getting it is so much greater — because you can suddenly attack incredibly complicated problems.

So I waltzed into my classroom filled with students fretting about not “seeing” the solutions, and said: “Screw the homework for now. We’re going to get this!” And I gave them this sheet with 9 problems on it — I worked hard to come up with the idea the night before — specifically designed to “hone their intuition.” And all I can say is that: it worked fabulously. They were doing really complicated chain rule problems in their heads!

The rest of the class was spent capitalizing on this new understanding. At the end of class, one of my students said that calculus class is the first time she’s enjoyed math since 7th grade when she first learned to solve for x. Which means I must be doing something right.

I left glowing.