How do you introduce integrals?

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 y=\sin(x) and told them to find the shaded area. Those were my only instructions.

sine

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.

thoughtprocess

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 \int x^2 dx=\frac{x^3}{3}+C 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.

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18 comments

  1. I have also tried a bunch of different methods. We usually end differentiation with “applications” and I sometimes use the physics of displacement/velocity/acceleration to introduce antiderivatives.

    Another method I’ve taken is that I always try to emphasize that one of the most key ideas to the entirety of calculus is estimating smooth, difficult things by estimating it with LOTS of easier things. So, the “slope of the tangent line” comes from doing a LOT of secant lines. And here, I reinforce that idea and then ask them to find the area of a curve like yours (although I usually start out with y = x^2 from 0 to 1). I have thought about putting them in groups with the same function graphed on different kinds of grid paper that have different scales. I try to get them to see that our estimations can get better if we cut the shape into LOTS of little shapes….thus harkening back to limits and moving on to Riemann sums.

    I like (and have used successfully) your idea for introducing antiderivatives. Make them struggle through one or two like x^2, then x^3, then 5x. Then they usually start to get the pattern, so give them a few free ones in the same manner. Then I mix it up with a x^(-2) to see if their pattern still works (“remember to check your answer!”). You can throw in the x^(-1) just to make sure they don’t get TOO hung up on their pattern, but assure them it works for other things like x^(1/2).

    That works well for the analytic students. Some are more graphical. When we do derivatives, I will give them a graph on an unscaled set of axes and ask them to sketch the derivative based on estimations of the tangent slopes (steep and positive, shallow and negative, etc.). You could have them think about doing that process backwards. On the projector (or overhead) you can have the (unlabeled) graphs of x^2 and (1/3)x^3 and ask them to see if that makes sense that it could start that way.

    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.

  2. I’m currently taking AP Calc AB – I thought it might be interesting to present a student POV. In my calculus class, we go straight out of the book as it is presented in the book. So far it’s been Pre-Calc Review, Limits, Differentiate, Applications of Derivative, and now we’re working on Integration.

    My first exposure to integration was actually not in my Calculus class but in my AP Physics C: Mechanics class. At my school, they allow students to take AP Calc & AP Physics concurrently. When it came to Work-Energy, or even the kinetic equations (velocity, acceleration, etc). Pretty much, my Physics teacher tried to explain as best she could but she just said that it’s to “reverse an equation” or add up all the tiny parts.

    My actual introduction to Calculus was first through Anti-Derivative. My teacher has us do a quick exploring activity on how to reverse a derivative. For example, “What gives 2x?” – “x^2”. Then we moved into general sums. Then more specifically to Riemann Sums and now we just covered the two Fundamental Theorems of Calculus. To be honest, I still don’t fully grasp the concept of Riemann sums and I wish he had devoted more time to it rather than just skim over it.

    Essentially, I’ve been taught via the Physics way over the sum or even area way. Learn the formulas, know that it’s the reverse of the derivative, and just use it. I wish we could go into more detail on some aspects but the AP test dictates everything unfortunately.

  3. It is neat (it will be neat) that area under the curve matches the integral… but letting it come after I think is nicer.

    Finding the power rule initially through a problem solving-type approach makes sense.

    Finally, a suggestion. One of the commenters mentioned AP physics and AP calc…

    After you’ve done some antidifferentiation, you could propose finding distance traveled from the velocity function. v(t) = k is boring, but v(t) = at forces some math, “how far in this” (dtime) etc will both lead to (a/2)^2 and to a bunch of skinny rectangles, and it matches the anti-derivative…. and it’s the area of that triangle… There are other choices, but that problem provides a neat link.

  4. I like the activities you are using. I usually teach anti-derivatives as part of my derivatives unit. Your activities are better versions of what I do on that topic. 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.

  5. @David P: Great minds think alike! Um… i LOVE the idea of putting the function on the same graph paper but giving it different scales. Find the area under the curve from 0 to 1. WITH DIFFERENT SCALES. OMG. GENIUS!

    @Daniel: Ah, learning it first in Physics… you don’t really get exposed to the undergirdings of the concept. Blah. But thank you so much for your perspective. Last year I only spent a little time on Riemann Sums and how they relate to integration (because they’re so tedious), but I will take a goodly amt of time thanks to your suggestion.

    @jd: After what I saw yesterday, I think I’m never going back.

    @Andy: Interesting… you show them how to do position/vel/acc problems BEFORE showing them integration. I never thought of that, because I’m always like “displacement/position is the area under the curve!” I like the graphical relationship when showing that, but I’ll have to think this through.

    Thanks everyone for your extensive and thoughtful comments. They are really helpful!

  6. Hi Sam, as a side note I liked the brain 1-2 conversation in the way that it points out a bit of self-awareness. I’d think kids in ap calc would generally be observant of things like noticing the errant 3 from the initial guess at x^2, but it’s a nice way to get them watching for errors and figuring out what you do with coefficients.

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

    How are you going to make the leap to the areas under the curve? Seems like they’ll be good at doing the formula stuff, do you just do last year’s lesson after the new strategy this year?

  7. 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).

    At some point, the generalization to accumulating the rate-of-change of a quantity over time gets introduced, which then suggests that this Riemann sum process is an inverse of the finding a rate of change process that they’ve already mastered. Voila, a plausible handwaving of the Fundamental Theorem.

  8. This semester, I tried a completely different way to transiition into integration. I took the point of view that the most important application of the antiderivative is not solving area problems, but rather helping to solve differential equations. I introduced a little of the basic terminology of differential equations and then showed them some standard examples. In the course of trying to solve these, the students were naturally led to asking the question, “what function has my function as its derivative?” The students quickly discovered the power rule and, with some effort, substitution. I used some artificial differential equations to motivate the other rules. Only after the students knew how to solve the integrals and how they are used to solve differential equations did I begin mentioning areas. When I brought those out, the students were sold on their usefulness.

  9. When I was a student, I remember finding derivatives confusing, but integration very natural, it fit with what I remembered from geometry about the method of finding the area of a circle by smaller and smaller arcs (Archemedes?) and what my boyfriend (now my husband) had been explaining his college engineering classes. Sometimes what I visualized was a crocheted pattern, I was making sweaters and lace at the time.

    Of course, encouraging your students to fall in love with engineering students and talk about math is not a real practical suggestion, so how did your introduction go over?

    -Christine

  10. @Adam: I like the idea of doing simple differential equations. I think I’ll add that next year.

    @Christine: I thought it went fabulously. You remind me that I should make a followup post. I’ll try to get to that this week.

  11. I’ve only taught this content a few times, but the way I liked best was along the lines of what @Andy suggests above. I teach antidifferentiation for a day or so. I indicate that it’s worthwhile by bringing up the dist/velo/accel relationship. We’ve always gone down that chain, but it’s really not unreasonable to go back up, and antidifferentiation will do that. I haven’t used the graphicl approach that @DavidP suggests (given f’, sketch f), but I do love that exercise when doing derivatives (who doesn’t?), so expect that I’d also appreciate it for the antiderivative.

    After just that I say, let’s start something new: I’m tired of only being able to compute areas of circles polygons (really only triangles and rectangles, about the limit of my memory). Let’s, given a function f, and bounds a and b, try to find the area ‘under’ f. Then talk about computing this via Riemann sums, but point out how difficult that is, and wish there was a better way. Then, I say, hey, just for grins, let’s let F(x)=int_a^x f(x) dx, and think about it’s derivative. Walking through the proof of FTC, we find F’=f, and I have them remind me that this is called antidifferentiation.

  12. I liked the idea of introducing integrals with the concept of differential equations.its really good

  13. I introduce antiderivatives with a full-fledged game of of Jeopardy. I suppose this could be done digitally, but I have derivatives printed out and taped to the whiteboard, with the dollar value taped on top. I let the students work in teams for this. There isn’t any application built into this game, so the usefulness aspect isn’t emphasized.

  14. I play Jeopardy. Most kids are familiar with this, and I introduce the concept as a game of Jeopardy where I give you the answers (the derivatives) and the students have to give the question (the original function) It makes integrals very intuitive at least for the power rule, and all the trig they memorized for derivatives. Some students it even helps with the chain rule and u-substitution as these are more challenging (I save those for higher point values and daily doubles)

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