Today in Calculus, I was waxing euphoric about why what we’re about to embark upon is amazing — how we’re eventually going to be able to find volumes and surface areas of strange figures. Not your standard spheres or cylinders or cones, but strange, exotic figures.

So I decide to open WinPlot and produce a surface created by revolving around the x-axis.

As I pressed “Enter” to generate the graph, I immediately recognized that we were going to have a problem. But it was too late.

I could have picked any number of other functions, but I decided to pick . Great.

We all had a laugh. Ah, high schoolers.

Moral: Be careful what you plot in class.

Recently, I posted a bit asking people how they introduce integrals. And I got a ton of different responses, which was wonderful. I am going to copy a few bits of comments here, but I really recommend that if you teach calculus, you take a moment to read them all in their entirety.

David P.: I sometimes use the physics of displacement/velocity/acceleration to introduce antiderivatives. […] I am only in my 3rd year teaching, so I’ve not found a “best” way yet. I really like the “surprise” of the FTC that areas and slopes seem like they should have no connection whatsoever, but that they’re almost as related at + and -. So, sometimes I even just say, “ok, we’re done with that section, let’s move on to something else” and then try to surprise them when we get to the connection.

Andy: I usually teach anti-derivatives as part of my derivatives unit. […] I also show how it applies to the position, velocity, acceleration problems. Then I transition to my integration unit. I don’t even tell them about integration and anti-differentiation being related. I just talk about area and then when we get the the fundamental theorem, I am able to drop the crazy idea on them that integration and differentiation are related. I enjoy their reactions to that.

Nick H.: Generally, I like the ask questions first teach skills later approach.

TwoPi: I usually start off with velocity examples, and in each case link the displacement calculation (or approximation) to the geometry of computing the area under the graph of velocity versus time. So start with constant velocity, then linear velocity (and sometimes nice quick applications involving stopping distances for cars at various initial velocities).

This year I focused on anti-derivatives. On the first day, I just said: the derivative of is , so the antiderivative of is . That’s all. The rest of the class had students struggle through finding simple antiderivatives (PDF and PDF). On the second day, I gave students a method to solving antiderivatives, a method which builds their intuition (PDF). And on the third day, I had students just practice, practice, practice.

Then I gave them a quiz — 17 questions asking for the antiderivatives of functions from to to . Moreover, I didn’t give partial credit. If a negative sign was missing, or a constant was incorrect, I took off full credit for the problem.

The average grade for both sections was an A-.

So I have to say that my approach this year worked. I’ll deal with -substitution and all that nonsense later. But the fact is, my students will be able to soon integrate some pretty hard stuff without resorting to -substitution.

Many of the comments talked about working with position, velocity, and acceleration graphs to start out. I think after I teach the area under curves and Riemann sums, I will go into this topic. Honestly, I was hesitant to start integration with position/velocity/acceleration because anything physics related tends to make my students convulse. They are scared of physics. I wanted to make sure that they didn’t shut down completely before we even start.

(However, I am excited to derive from first principles. I hope to hear lots of oohs and aahs.)

I’m putting a call out to calculus teachers and calculus aficionados out there. I want to know how you transition to teaching integration, and why you cho0se to do it that way. And if you have any activities, investigations, etc., that you can send me, I’d love to have them (and post them here for other calculus teachers).

I’m not super pleased with, but I don’t hate, what I’m going to be doing tomorrow.

Here’s the deal. I just gave my last test on differentiation today, and tomorrow I’m transitioning to teach integration. I teach a regular (non AP) calculus class, so we can take our time. At the moment, I’m grappling with two things: (1) whether to teach anti-differentiation first and the notion of “area under the curve” second, or vice versa, and (2) how to make integration intuitive.

Last year, I transitioned by giving students a graph of and told them to find the shaded area. Those were my only instructions.

Some students made triangles, some students guestimated, some students made rectangles. I don’t remember all the different approaches. But then we had a discussion about how they estimated their areas, which then led to me transitioning to Riemann sums and a general introduction to the whole new unit. The thing I emphasized: “In all your previous math classes, you only learned how to find areas and volumes of silly little figures, like squares and cubes and maybe you remember a nonagon or cone. But what about crazy, strange, weird areas? Volumes of crazy, strange, weird figures? Did you ever wonder where the formula for the volume of a sphere come from? Calculus not only can answer questions about position, velocity, and acceleration, and how to maximize and minimize quantities, but it can do all this other stuff too.”

This year I’m not going to talk about areas under curves (yet). I’m going to start with two days of practicing antidifferentiation. I’m not going to say much to transition to this new material except to say that derivatives were the first part of the course and antiderivatives will be the second. And that we’ll soon be able to do a lot with them, like we found out we could with differentiation… Then I’m going to introduce the idea of the “opposite of differentiation” and spend the entire period having students build their intuition.

First, they’re going to do a matching game in pairs (PDF). We’ll then quickly debrief, but not really go into depth about any question.

Second, they’re going to work in a different set of pairs on just playing around with finding the antiderivative, by intuition and guess and check. I want them to learn to think through a problem. So I typed up what goes through my head when I try to do an antiderivative.

And then I’m letting them loose on a set of problems which should hopefully introduce them to some basic integration rules (PDF). I think it’ll take the whole period. And we’ll spend the next day debriefing. I want them to struggle through integration now. I want them to see why instead of memorize the power rule. I anticipate it to be kind of hellish for them; they — like most students — want formulaic ways to do calculus.

But just as I struggled to hone my students intuition (see my previous blog post) for differention, I wanted to make something similar for integration.

We’ll see what happens tomorrow.