Starting Calculus with Area Functions

So I decided to try a new beginning to (non-AP) calculus this year. Instead of doing an algebra bootcamp and diving into limits, I decided to teach kids a new kind of function transformation. I’d say this is something that makes my classroom uniquely mine (this is my contribution to Mission 1 of Explore the MTBoS). I don’t think anyone else I know does something like this.

You see, I was talking with a fellow calculus teacher, and we had a big realization. Yes, calculus is hard for kids because of all the algebra. But also, calculus involves something that students have never seen before.

It involves transformations that morph one graph into another graph. And not just standard up, down, left, right, stretch, shrink, reflect transformations. Although they do transform functions, they don’t make them look too different from the original. Given a function and a basic up, left, reflect, shrink transformation of it, you’d be able to pair them up and say they were related… But in calculus, students start grappling with seriously weird and abstract transformations. For example: if you hold an f(x) graph and an f'(x) graph next to each other — they don’t look alike at all. You would never pair them up and say “oh, these are related.”

So I wanted to start out with a unit on abstract and weird function transformations. Turns out, even though the other teacher and I had brainstormed 5 different abstract function transformations, I got so much mileage out of one of them that I didn’t have to do anything else. You see: I introduced my kids to integrals, without ever saying the word integrals. Well, to be fair, I introduced them to something called the area transformation and the only difference between this and integrals is that we can’t have negative area. [1]

You can look at this geogebra page to see what I mean by area functions.

Here’s the packet I created (.docx)

That packer was just the bare backbones of what we did. There was a lot of groupwork in class, a lot of conceptual questions posed to them, and more supplemental documents that were created as I started to realize this was going to morph into a much larger unit because I was getting so much out of it. (I personally was finding so much richness in it! A perfect blend of the concrete and the abstract!)

Here are other supplemental documents:

2013-09-16 Abstract Functions 1.5

2013-09-17 Abstract Functions 1.75

2013-09-20 Area Function Concept Questions

2013-09-23 Abstract Functions 1.9375

The benefits I’m already reaping:

  • It’s conceptual, so those kids who aren’t strong with the algebraic stuff gain confidence at the start of the year
  • Kids start to understand the idea of integration as accumulation (though they don’t know that’s what they are doing!)
  • Kids understand that something can be increasing at a decreasing rate, increasing at a constant rate, or increasing at an increasing rate. They discovered those terms, and realized what that looks like graphically.
  • Kids already know why the integral of a constant function is a linear function, and why the integral of a linear function is a quadratic function.
  • Kids are talking about steepness and flatness of a function, and giving the steepness and flatness meaning… They are making statements like “because the original graph is close to the x-axis near x=2, not much area is being added as we inch forward on the original graph, so the area function will remain pretty flat, slightly increasing… but over near x=4, since the original function is far from the x-axis, a lot of area is being added as we inch forward on the original graph, so the area function shoots up, thus it is pretty steep”
  • Once we finish investigating the concept of “instantaneous rate of change” (which is soon), kids will have encountered and explored the conceptual side of both major ideas of calculus: derivatives and integrals. All without me having used the terms. I’m being a sneaky teacher… having kids do secret learning.

I mean… I worked these kids hard. Here is a copy of my assessment so you can see what was expected of them.

I love it.

Love. It.



I’m going to put a picture gallery below of some things from my smartboards.

This slideshow requires JavaScript.

[1] To be super technical, I am having kids relate f(x) and \int_{0}^{x} |f(t)|dt



  1. As an 8th grade math teacher, I rarely get the opportunity to think about calculus. I love being able to read blogs about teaching methods at this level— it gets my brain working!

    This is a really cool activity that makes students think about the math in a different way than they are used to thinking.

  2. First of all, thanks for being a part of the MTBoS leadership that planned this blogging event. I like to blog, but this is the exact time of the year when it gets hard to do so admist other responsibilities. The nudge is appreciated.

    I teach 8th grade, so I won’t be able to incorporate this awesome idea into my classroom. I do have a ton of former students come back to visit who are mightily struggling in calculus and I wish they could be in your class. You’re clearly meeting your students where they are and making calc “totes” fun.

  3. I love how this gets students thinking about the concepts of integrals without getting them tied up in all the formal algebra. I definitely never would have thought about doing it at the beginning of the year, but I can see how it will help students later as they tie the ideas together with the calculus more formally. It sounds like it got them thinking about some awesome topics!

  4. After this unit ended, I have been doing a lot with slope. Well, average rate of change, and problematizing the idea of instantaneous rate of change.

    1. Sorry, I guess my question wasn’t clear. Most calculus courses begin with slope. I’m curious about what made you decide to start with area instead. I am intrigued, and want to think about this.

      (To me, it seems so much easier to start with slopes and rate of change. I am allegedly supposed to use the same textbook the rest of my department uses. I provide suggested homework from the book, but I skip around wildly, because I don’t want to do limits at the beginning of the course – and many other choices I’ve made that don’t go with the book. If I tried to do area first, I can’t imagine how I’d use the textbook. But, if it seems like a good enough idea, maybe I can find a way.)

      1. A couple reasons:

        1) I really want kids to realize that calculus — the big ideas anyway — aren’t “hard.” So I figure by starting the year with something like “area functions” and getting them to think about some of the ideas without all the gross mathematical language or symbols which can be scary was a pretty neat idea. And then when we get to integrals, I’m hoping that the initial foray into area functions will stick with them. And the language they used to describe things “you’re adding area at an increasing rate” or “you’re adding area at a decreasing rate” (and thus, the area graph looks like BLAH or BLAH).

        2) I really wanted to start my kids off this year thinking about “strange functions” that are based off of original functions. So eventually they will learn the derivative fxn. But the original fxn and the derivative fxn look very different — almost like they are not related. But there are deep connections between the two that make them intimately related. I was thinking if I could use a different “strange function” at the beginning of the year to get them thinking like that… and I realized area functions were perfect. (I had a few others in mind, but once I saw the good thinking the kids were doing, I just focused on that).

    2. (I’m afraid my comments are going to be all out of order, but I don’t see a reply button on your latest comment.)

      I just re-read your post in more detail. And now I think I understand. You’re not doing the whole area thing now, just area as one example of an abstract function transformation. Is that right?

      I’d love to see the other 4 abstract function transformations you brainstormed.

      This might be fun to do at the beginning of my course, without thinking of it as a unit. Just an activity to broaden their thinking.

      1. Yes! You got it! I was going to make mine an activity, and it turned into a week and a half exploration because they were saying good things. But that’s about it. Then we went straight into “normal” calculus (slopes).

        But totally just something quick to get them thinking.

  5. “But in calculus, students start grappling with seriously weird and abstract transformations. For example: if you hold an f(x) graph and an f'(x) graph next to each other — they don’t look alike at all. You would never pair them up and say “oh, these are related.””

    Well, it depends on what you mean by “related.” If related means “similar” in the usual geometric sense, and if that’s all you’ve got, then no. Obviously, rigid transformations don’t change the shape of the graph, so all rigid transformations, even in combination, produce “similar” (even congruent, if lifting into 3-space is permissible) graphs from whatever you start with. Adding dilation to the mix makes it a little harder, but you still have similarity to go with, and while you might mistakenly see to graphs as similar, you probably aren’t going to go WAY wrong.

    But the relationship between the graphs of f(x) and f'(x) have a relationship, of course. It’s simply not one that kids are used to seeing in math class (and while they may have seen it in science classrooms, they probably weren’t thinking too heavily about the relationships among the position, velocity, and acceleration graphs for some object in motion, nor dreaming of how the functions can be derived from each other through differentiation or integration. How could they?

    What they certainly can learn to do is see that the graph of f'(x) shows how the slope of the graph of f{x} is changing. And there is software, for one thing, to show that relationship. There are hands-on exercises and other physical ways to model and even experience the changes and the relationship.

    I say none of this to take away from what you describe doing, which is fine. But I’m not sure there is anything weird about the transformation of the function graph into the graph of its derivative. It does take some visualization skills that might not be easy for everyone, but I suspect that traditional calculus instruction has made use of far fewer “tools” and models than I would expect from a contemporary, creative, progressive teacher, particularly a teacher who recognizes that left to his/her own devices, it is not very likely that a beginning calculus student who doesn’t know that there is a relationship between the derivative of a function and the slope of that function at a given value of x is going to leap to the conclusion that the graph of f'(x) shows the change in slope of the graph of f(x) as x changes.

    1. I think we’re saying the exact same thing! Kids aren’t exposed to transformations that do anything but preserve the same general shape. That’s all that I meant by “weird” — it’s weird and strange to them on first learning.

      I don’t (obviously) deny that there is a relationship between f(x) and f'(x). I just started this year off trying to highlight that there can be relationships between graphs that aren’t intuitive on first glance, but there are deep connections between the two graphs!

      1. Sam, I’m sure we’re basically on the same page here, and what you’re doing is interesting. What really worries me is how weak kids are (haven’t been asked to think about, haven’t be led to understand) the relationship between graphs and what they might be describing in “the real world.” So asking kids to put stories to graphs or match graphs with appropriate stories catches many of them completely off-guard.

        It’s worth noting, however, that there was a study done at the University of Michigan about a dozen years ago in which they interviewed successful calculus students (I think that means kids who finished at least one calculus course with an A, but I heard Hy Bass talk about this long enough ago that some of the details have slipped into the ether). They were each asked to explain in their own words what the derivative of a function meant.

        The striking thing was that virtually none of them could say that the derivative of a function gives the slope of the function f(x) at a given value of x. They could explain how to FIND the derivative of a polynomial (that was the most common response, but the meaning was not accessible, if in fact they even knew it at all. That, to me, is a frightening indictment of K-14 math education. We train kids to pass procedural tests. But we don’t seem inclined to have them think about what they’re doing, what it means, how it connects with other things they know, either in the everyday world, in more technical applications, or with other mathematics they’ve learned/done.

        So it’s not terribly surprising that they aren’t inclined to extend their thinking past whatever transformation relationships they’ve been taught explicitly. Their assumption, I suspect, if they think about it at all, is that “This is all” + “This is all I need know if this isn’t all.”

        I love when someone comes up with an idea s/he hasn’t been taught or led to. My son came into the living room a few years ago, when he was in Algebra II, and asked me if there could be quadratic or higher degree asymptotes. I’m embarrassed to admit that I dismissed the question as silly without giving it any real thought. He came back about 20 minutes later and showed me a quadratic asymptote on his graphing calculator. I’ve never been more pleased to be wrong.

      2. I agree.

        This first unit is EXACTLY getting kids to articulate what you are talking about…”We train kids to pass procedural tests. But we don’t seem inclined to have them think about what they’re doing, what it means, how it connects with other things they know, either in the everyday world, in more technical applications, or with other mathematics they’ve learned/done.”

        If you look at the assessment, it requires kids to explain the reasons undergirding everything. Nothing is taken for granted.

        So as long as your indictments aren’t directed towards this unit, I feel good. Because this unit is actually going against everything you’re saying is wrong with traditional calc education.

  6. The request to “explain in their own words what the derivative of a function meant” is terribly vague. I wouldn’t know whether the questioner was looking for the graphical interpretation that Michael wants, the limit definition, the formula for the specifc derivative, a physical interpretation like velocity, or something else entirely. I’ve always hated that sort of vague question in math classes.

    Note: I don’t see that sot of vagueness in Sam’s handout. Where he asks for an explanation, it is for a specific phenomenon, and the explanation can be made in concrete terms without much ambiguity.

  7. “You see, I was talking with a fellow calculus teacher, and we had a big realization. Yes, calculus is hard for kids because of all the algebra. But also, calculus involves something that students have never seen before.”

    This is such a money statement. Calculus is hard for students for a lot of reasons. Specifically, it’s hard for students who may not be used to having a math class be hard. This is something that I’m seeing so far this year. Some students are struggling for the first time. They’ve never looked at a progress report and seen a “C+” so they are having to explore new emotions and develop new habits.

  8. @gasstation: I attended that talk more than a decade ago. I’m sure the question was better-phrased than my attempt to recall it. I know that it was pretty damned clear what the researchers were NOT asking: how to find a derivative, but that was pretty much what people offered, or vague ideas that danced around the subject from a procedural perspective, not at all in the ballpark of what was asked.

    I don’t find that surprising. And these were not low-level students at a third tier university barely scraping by, but kids who had excelled in the standard undergraduate calculus at U of Michigan. I think if you search on-line, you may find the actual question that was posed, but in any event, I don’t want you to get the impression that it was ill-posed just because I haven’t remembered it accurately.

  9. I love this approach, too. In fact, I used it this past spring when I taught calculus to nurses and biologists. It was a great success, conceptually (algebraically, they were still devastatingly awful). Instead of referring to it as an area function, I called it an accumulation function. This made it more natural to talk about negative quantities (I used a weight function as my primary example, and the students all could relate to gaining and losing weight). It took us about three (50-minute) sessions to get to a point where we could relate the accumulation function to the rate of change function via the fundamental theorem of calculus (stated without any of the usual symbology, of course).

    1. I love that we think alike!!! Did you post anything about this — or do you have any materials you could share? When you did the weight fxn, was your graph’s y-axis the “change in weight” or was it just “weight”?

      I REALLY like the idea of using weight because it’s so concrete. I hope hope HOPE I remember this when I get to integrals. I have a terrible memory…

      1. I’m a bit embarrassed to show you my hastily designed in-class handouts, but here are links to them:

        Click to access s13-math-130-in-class-01-accumulation-rate-of-change.pdf

        Click to access s13-math-130-in-class-02-accumulation-rate-of-change.pdf

        Click to access s13-math-130-in-class-03-fundamental-theorem-of-calculus.pdf

        For weight (which we talked about in class constantly, but which does not appear on the handouts), the y-axis was the change in weight. The beauty was that by day three, when I asked what the accumulation of the change in weight function was, the class almost unanimously agreed that it was the net change in weight. When I gave them the FTOC handout, they asked why something so obvious was a fundamental theorem. Giggle giggle.

    2. Hastily designed, huh? They look pretty nice at first glance. Thanks for sharing! Did you make up the notation yourself?

      1. Thanks, Sue! I did come up with the notation myself. If I had it to do over again, though, I would forgo the notation and let the students come up with their own. The notation did NOT impress them.

  10. I might have to try this. But college students are expected to do outside homework, and our text will support this even less than the way I do it now.

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