We’ve been doing u-substitution for the last couple of days in Calculus. Too soon to tell, but I think it’s going ok.

Friday, we started. We had already talked about how antiderivative rules are just derivative rules, but backwards. So, we reviewed the chain rule. Then I wrote on the board, and we talked about what each part meant. Next up, examples: . What’s g? g’? f? Ok, now we find F, plug in g, and add C. You try some! (I like starting with square roots because it’s really obvious what the “inside” function is. Exponents are also good. I think in the past, I’ve had them integrate powers of 2x-1 until the powers were so high that expansion became impractical. I skipped that this year, but I may go back to it next year.)

I’m pretty sure this is how I was taught u-substitution at first. It’s how my textbook lays it out: do pattern matching, then get into actual substitution. I’ve never taught it that way, but it seemed like it was time. It seemed to help.

So, after they do a problem or two, I show them a trick: u-substitution. We rework the original example, and then they rework their previous examples. For the rest of the day, I throw up a bunch of problems on the board, and they work through them. That was Friday.

I find that a lot of students, in all areas of math, don’t know when they’re done. When have they answered the question? With u-substitution, once they’ve found the anti derivative, they could be done. It’s really just convention that we switch everything back to x. It is, however, the convention, so they have to do it. So, once we’ve actually found the antiderivative, I find these words coming out of my mouth:

“Ok, so we found the antiderivative. But, see, u-substitution is like a bad rebound. I got rid of my x and found u. I don’t want my x anymore; that’s all behind me – I want u! And things are going great, but suddenly I realize, I don’t actually want u. I want my x back. So I’m going to ditch u and go back to my x.”

It never gets more than a few chuckles, but they do remember.

I didn’t see them again until Wednesday, when I had them for two hours. There were a few questions on the homework, but most of them were about places where I’d messed up on the key. That’s always a good sign – they understand what’s going on, and are confused by mistakes. A better sign is when they realize that those are mistakes and email me about it, but this is still good.

We wrapped our our tale on Wednesday: dealing with situations where the integrand needs some massaging, like , and then dealing with definite integrals. I was pretty happy that the class figured out what to do and I didn’t have to tell them. I was even happier about what happened a few minutes later when they were trying a few problems themselves.

A tardy student walked in. Everything we’d done was still on the board. He pulled out his notes, and I walked over and explained what was going on on the board. He nodded like it all made sense, so I left him to try it on his own. Later in the hour, I walked past his table, and asked his neighbor how it was going. She said something like, “I’ve got the second and third, but I’m having trouble with the first.” I was about to open my mouth and start talking when I heard him say, “Oh, I’ve got the first. Here, do you want to see?” I walk off quietly and have a happy dance on the inside. I’ll take peer teaching over mine just about any day of the week; the fact that he got there without even being around for my original overview made it even sweeter.

Because this was a Wednesday, I had two hours, which was probably about 30 minutes more than I used. That wasn’t bad – they started their homework, which meant they could work on it and ask me questions while I’m still there. But I feel like the lesson could have gone deeper. We got into some questions about what we’re actually doing when we find the new bounds with a definite integral, and I found myself talking about how when we do a u-substitution, we’re transforming the horizontal axis, and we’re finding the new coordinates on the horizontal axis. That’s true, and that makes sense to me, but I know a lot more math than these kids do. Complex analysis made me pretty comfortable with transformations of a plane, but that wasn’t until grad school, so I doubt they’re really happy with it. I’d like to find something – on Desmos maybe? – that gives a visual of what’s happening to the graph when we do u-substitution, but I’m not sure how to put that together. Any ideas, MTBoS?

What do you mean by “integrate powers of (2x-1) until the powers were so high that expansion became impractical”? I too enjoy square roots and fractional powers in integrals. My favorite method to solve is by finding some series representation, but there isn’t always a simple solution that results; sometimes one must leave it as a series.

Perhaps you might also ask them to think about what they could do to make integrating those powers more reasonable; if I recall correctly, that was something Feynman used to like asking his students, to allow them to think outside the box. I’m not a fan of a rigid curriculum. It helps at first because it’s easy to get organized, but then nothing is unexpected, everything is set in stone. I made a point of stating that in my high school calculus class.

You sound like a great teacher. I’m sure your students are lucky to have you.

All best.

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Without u-substitution, if I ask students to integrate (2x-1), then , then , they’ll probably turn this into 2x-1, , and , then find the antiderivative. But if I ask about $(2x-1)^{10}$, they usually see expanding the binomial as too much work. It gives u-substitution some motivation, but there are certainly other ways to motivate it – I didn’t use this particular example this year. Does that make more sense?

What do you mean by “rigid curriculum”? Do you mean you don’t have a fixed lesson plan and examples you do every year? Or that you don’t have a definite list of topics you cover? Or something in between?

I like the Feynman question. He was always good at getting people to think.

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