teaching

How I Teach Calculus: A Comedy (The Chain Rule)

March 12, 2010 Shawn Cornally

This post is a part a larger series documenting the changes I am making to my calculus course. My goals are to implement standards-based grading and to introduce genuine applications of the concepts being taught. I’m not suffering any delusions that any of this is all that ground-breaking, I just want to log the comedy that ensues:

Arguably the most important skill in all of calculus, the chain rule naturally follows the product and quotient rules. I did not discuss the quotient rule here (of course I did with the kids), because I want to show how redundant it is after they gain control of the chain rule. The kids know the quotient rule, but only via a sibling-like relationship to the product rule.

The chain rule requires some serious discussion. It underpins their understanding of calculus, both differential and integral. For any of you who have gone on to apply calculus via physics/chemistry or have taken vector calculus and beyond, it is almost painful how important an understanding of the chain rule is. If you’re reading this, I’ll assume I’m preaching to the choir.

The great question arises again: What physical handle can I give my students to hold on to that will enter them into the zany fun-house that is the chain rule? This time I actually pulled inspiration for this one from the textbook (!):

Invitation – Gears:

I use Larson et al. (7th ed.) as my textbook, and they put this one right there on page 127. It’s the first page of the section, which is generally common practice to skip. If I had students read through an example about gears and then try to talk about it, I’d probably get a lot of smoke and mirrors from them. So, I looked for some real gears, and Playskool helped me out.

I bet a lot of you that teach calculus already use the gear analogy, so you’re probably not too impressed right now, but hey, I don’t write this so you’ll think I’m awesome. I write this stuff so you can see how it worked, which is really all I have to offer: a look in to another set of kids and their reactions.

I left the gears out on the table, and my freshman study hall couldn’t keep their hands off of them. I couldn’t help but wonder how my upperclassmen calculons would respond. They were equally as engaged, and it was fun to say, “Hey, don’t touch the materials until we start!” in a math room.

I had enough cogs to break the kids into two groups each with equal materials. I let them play for a bit (5 minutes) and they built all sorts of interconnected parts and thoroughly enjoyed spinning them. However, I knew that we’d never just magically stumble upon the chain rule. So I pointed them to the configuration as pictured above. I asked: “How does spinning the crank spin the gear on the inside?” What would happen if the read gear were bigger/smaller?

I could feel a sort of tentative understanding brewing. Students described the red gear as:

• “inbetween”
• “controlling”
• “dividing/multiplying”
• I threw out the term, “intermediary.”

This lesson started to drag on. It was really hard to jump from these goofy, colored gears to a genuine understanding of compositions of functions and their rates. So, I decided to break this lesson up, and teach the material, knowing that we would return to the gears to tie everything up. A failure? Perhaps, but only if you expect perfection everyday, which is ludicrous.

My discussion of the chain rule started. We went back over what compositions of functions are and how to make them. I put the standard chain rule formula up on the board and told them it would be best to memorize it at some point:

About now the kids are getting used to this kind of cryptic combination of f’s and g’s. The more conservative among you are considering berating me in the comments about not showing them how this all precipitates from the limit definition. I wish as much as you do that kids cared about that kind of stuff, but mine don’t. I’ve been there. I’ve wanted to show kids the most beautiful proofs, and have! Only to be met with crickets. I’m not saying kids should be shielded from derivations and proofs, I’m just saying that you probably want to do them for more self-indulgent reasons that you want to admit. (I did)

I then do a couple of examples; you know, things with trig functions and coefficients, things with roots, things that would be a bear to expand before differentiating. The kids get the mechanics, but now the gears come back!

Where does that magical formula come from? I remember following my math teachers as if they were shamans, but I don’t want to be one. Here’s a video of the operation:

Larson explains it in the book pretty well, but our setup is not identical to his hypothetical gears. Ours do not have perfect integer multiple radii, nor do they have common axes like his do. So the kids begin with a standard math approach: label stuff. Here’s what they labeled:

They gave the axes names (just like Larson did), and they numbered the gears. The questions then started to fly. There were a lot of kids gone that day, so we really only settled on two:

1. What does the red gear do to x and y?
2. Does the inverted gear (3) behave differently than a normal gear?

The kids began measuring things. Mostly how many rotations of y results in one rotation of u, and then how many of u created one rotation of x. I helped them put this understanding together (I think this is that scaffolding thing?). I did not anticipate how well this would get and Leibniz notion. Here’s what they experimentally determined:

• y has to turn 2.5 times to turn u once
• x has to turn 3 times to turn u once

This was perplexing for a while in the room. Most of us felt like energy wasn’t being conserved or something. How could the green gear be so close to the blue gear? The kids drew all sorts of things to explain the inside of the red gear. Eventually we landed on the concept of gear ratio: a totally unexpected sidebar!

So, to end the gear madness I pushed a bit. What’s the change in y for the change in u? “2.5:1,” they said:

What’s the change in u for the change in x? “1:3,” they said:

How is gear y related to gear x then? This took a bit, but eventually one of my more sheepish students came up with it¹. (I’m buying them pizza):

So, gear y rotates 5 times for every 6 rotations of x. This is the chain rule. We then worked an example problem and connected each piece of the process to its appropriate gear. The extensions of this are endless. The essential connections are:

We then, of course, went through all of the other essential bits of the chain rule.

I feel like these gears could be used in many other places other than calculus. What do you algebra folks think?

Next post will be a derivation, just to show you all that I’m not soft math.

1. I know, I know. Treating Leibniz notation as fractions is a bit off. Hush. I’m a physicist, remember?

Larson et al. Calculus of a Single Variable 7th Ed. Houghton Mifflin Company, 2002. p 127.