How I Teach Calculus: A Comedy (u-substitution)
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:
I wanted to introduce integrating more complicated functions in a way that was more meaningful than just saying, “Hey, I bet you can’t get this one!”
My students and I have officially gotten on the calculus shuttle at this point. They have a firm grasp of differential and integral calculus conceptually, and have seen that, just like most other operators, the derivative has an inverse. (the integral!) They like this, and we build a sort of family tree of math where [ + ] and [ - ] live together, and [ * ] and [ / ] live together, and ^ with log, finally d/dx with the integral.
The time now comes to flesh out our ability to do integrals. It is obvious to the kids that they can really only handle polynomials. I wanted to introduce integrating more complicated functions in a way that was more meaningful than just saying, “Hey, I bet you can’t get this one!” That method works pretty well, but it was the first day of the year where temperatures were going to break 70 degrees, so I pretty much had to go outside. (It’s a law in Iowa that you must celebrate with song and victuals the first day of the year that doesn’t require a coat.)
Here’s what I (always) think to myself, “What will necessitate the usage of more complicated functions than polynomials, that will also be outside?” Answer: I have no idea! I check out a class set of digital cameras, and away we went. Their instructions: Take pictures of things you think are beautiful.
Cornally gambles. This could most definitely blow up in my face. What if their pictures don’t give any inspiration back to calculus? What if I can’t figure out how to relate anything? What if they don’t take this seriously, and as soon as we’re out the door they scramble away into the hills never to seen from again except for when they scavenge a few sheep from a local farmer in the wee hours of the morn? These are the things teachers think about. The kids just said, “OK” and they participated just fine.
We walked around our campus and the surrounding area, which is fairly rural, for about 20 minutes. We talked about all sorts of things. When do frogs come back from under the mud? Why do streams have ripples on the bottom? What makes a good photo? We talked about Gestalt principles a bit, framing, and composition flow. In the end they took pictures of things like trees, streams, parking lot cracks, or whatever else they wanted. Being outside was refreshing, too (and legally mandated by NCLB…)
We went back to the room and they downloaded their images. I asked the question, “Do you see any math?” A few stares, but when they realized I was serious, they started looking. Many of them found sine/cosine waves in things, some saw parabolas and circles. One that struck gold was this:
Super simple. This student’s goal with the photo, as he said, “I wanted to show the crack as the rising action from left to right, it looks kind of like root [square root function] but also looks like a letter y.”
“How minimalist of you,” I replied. “What root function is that?” So we busted out Grapher (or whatever graphing tool) and they started trying to fit a function to this simple crack in the sidewalk.
A student bubbled, “What if they wanted to fix the crack, Mr. C? What would they do?”
I said, “I’m not sure, I bet they break up the whole square and replace it.”
“Couldn’t they just cut out the crack and poor new concrete and have it join to the old concrete?”
I said, “I’m not sure if wet cement bonds to cured cement.”
Another student pipes up, as they’re working in Grapher, “I read this article about that for my chemistry project. They made a self-healing concrete. Part of it stays dry after it sets.”
“Sweet, How much of that do we need? I bet it’s expensive,” says the original photographer.
I don’t answer. They start thinking. I didn’t plan this, I have no idea what’s happening. I know your pancreas is probably knotting at this interchange, but this actually happened. I guess I can put those original teacher neuroses to bed a bit; if you let them, kids will come up with some awesome stuff. The story continues:
Student A: Hey, if I found the area from the crack on down that’s the area of the chunk we’d need.
Student B: If we drew a line just under the crack we could save that good concrete and just fill a little gap.
Cornally: How can you find the area of a little swatch like that?
Student A: Just find the bigger area and minus the littler area.
Student B: Yeah.
Something like this:
…and another lesson is in the books. Area between two curves. I love it when they do my job for me. So, the students continued on trying to fit functions to this picture. They ended up going outside again to measure the crack so they could get the numbers right (This was my idea, admittedly). Here are the screen shots from Grapher of a couple things they tried:
The ellipse part blew my mind. They went online to look up ellipses, because they weren’t happy with how the root functions got closer together. My next step in this lesson would have been to fit these functions using a statistical model, but we ran out of time. We went with what we had.
So, the students set up an integral to get the areas under the root functions and realized pretty quick that they didn’t know how to find the antiderivatives. Some students bumbled ahead doing the reverse-power rule, but a quick differentiation check let those students know that this method was obviously not applicable (I really believe in this attempt at Piaget-style accommodation. You may know this in science as a “discrepant event.”)
Now the traditional lesson on u-substitution (change of variables, reverse-chain rule, whatever) begins. How can we make this integral doable? What is flummoxing us? The students think, and deftly say, “The thing in the root.”
I pondered just letting them try to figure this out on their own, but experience has shown me that this specific kind of freedom rarely fruits. So I Pied-Pipered a bit. The total problem we decided to set up and solve was:
We attacked the first integral. (I’m going to solve this problem for those of you who are students that happen to arrive here for help. If you’re a teacher, you can probably skip to the end.). The integrand is a composition of functions. You needs to sub-out the inner composed function, like:
The problem most students have arises here and for some unknown reason is poorly explained. You absolutely cannot do this integral because it is mixing variables. It is asking you to perform an integral with respect to the variable x, but you have the variable u. This makes no mathematical sense. You must create a du. The only way we know how to do this is by finding the derivative of u, good thing that we defined u earlier!
You might be asking yourself what about our limits of integration (a and b). They most certainly are left over from the world of x. We can leave them until later, when we rid ourselves of the integral and have plugged the x‘s back in. Notice that the goal of substitution is to create an integral that is doable. We can most certainly do the integral of root u with respect to u.
We then finished our initial area problem for the concrete. We also took into account that the concrete is probably a few inches thick to find volume. They looked up how much concrete was, and we solved some of their questions.
After this introduction, we moved through the standard treatment of u-substitution, and all of the formalities therein. I make an special point to discuss problems where the substitution for dx doesn’t cancel anything, and you have to use your definition of u again. This blog is for the things that I’m doing that I think you all might want to hear about. It is not exhaustive of my total classroom behaviors.
All in all, this lesson was probably the biggest gamble of anything I’ve done this year. I asked for a lot of maturity and really had no idea where it was headed. I almost didn’t write about it for fear that you’d all find this ridiculous or unhelpful, but I’d really like to add evidence to the lessons-don’t-need-to-be-totally-figured-out position. Sometimes fluidity is freeing. Also, 70 degree days in Iowa are fleeting.