Honing My Skills Instruction

Skills instruction was something that I was not good at when I first started teaching. I found it boring compared to big ideas and didn’t really understand why kids didn’t get things from me doing problems on the board and then them doing practice problems with each other or individually or for homework. For me, it was like “What is there to understand that simple practice wont solve?” Well, I have grown a lot since then. This year, I have implemented a lot of new tools (many whiteboard based) that have really helped out with my skills instruction. I feel like I really had a great sequence this year when doing the skills part of integrating with my regular class – I wanted to share and reflect, especially so it’s written down somewhere for me to use in the future, because some of the ideas will be useful when I teach other topics.


Just so you know how I lead up to this seminal topic in Calculus… First, I spent a considerable time (about five 45 minute class periods) exploring everything to do with Riemann Sums, both in terms of pure area and what area means in applied situations. I think that feels like a lot of time, but we tackled the conceptual side of integration very throughly and used that to motivate the idea of an antiderivative/integral. Once we motivated the integral, we focused on learning how to find antiderivatives, which is the part I want to talk about.

1. Guess and Check With a Partner

With inspiration from a great worksheet from Sam, I wanted students to rely on their intuition at first to find andiderivatives, instead of relying on formulae. I’ve tried things like this previously, but it really helped this time to explicitly explain that this is what we were doing – that maybe eventually we can rely on a rule, but we are going to discover the math first. I paired them up with whiteboards and set them out with the list of functions from Sam’s worksheet. Their goal: find the antiderivative of all the functions. The method: each person had a marker. One person would write down a guess for an antiderivative, and the other person would simply take the derivative of this to see if it went back to the original function. They would keep doing this until they got something correct, then write that answer down on the sheet. Then, after one person has been the “guesser’ four or five times, they switch. Example:

For the kids that actually did what I asked (others just kind of started solving them on their own, which is okay I guess), it was a really nice exercise. They worked together really well, and were so excited to tell me the rule that they had made up for integrating power functions. I had them even doing simple substitution, per great suggestion from Sam. They got good at just getting themselves to try something, and getting in the habit of checking all of their answers. One kid at the end of class told me “My brain hurts from thinking so much.” Then, after the students were done, the next class we started by collecting rules they had noticed, and it made a nice little automatic cheat sheet for them. –> SHEET WITH FUNCTIONS HERE

2. Power Rule Folding Game

Next was to tackle more complicated functions with which we could use our rules, mainly negative and fractional powers. I did this same exercise in the fall when learning how to differentiate these functions to much success, and then tried it again with differentiating power functions to much confusion (so I guess the activity has a specific niche). The idea is that everyone starts with a problem, does one step and folds over the sheet so that only their work is visible. Then everyone rotates their problems around. The next person does the next step, and then folds the paper so only their work is visible. The group keeps rotating the papers until they are all done, then they open them up and look for mistakes (if there are any). Example:

This was good for helping them drill some algebraic manipulation and develop the skill of checking their own work for mistakes… all while working very closely collaboratively. –> FOLDABLES HERE

4. Flip-Up Answers for Initial Conditions

After learning basic integration skills, we began to talk about how functions have more than one antiderivative, and how sometimes it is useful to find a specific one. After only one or two examples together, we immediately just started practicing this idea with an activity that I stole from Mimi where I placed problems around the room with the answers on the back, the idea being that students would go solve whatever problems they felt like they needed to. Example:

I enjoyed this for many of the same reasons that Mimi cited in her original post. Students could work at their own pace without feeling like they were falling behind, could pick their own problems, and could move around the room to interact with many different people (which are all huge advantages over just doing a worksheet). –> FLIP PROBLEMS HERE (though the formatting is a bit screwy)

5. Mistake Game

After two days of a little bit more traditional style instruction just to make the connection between the definite integral and area (a lesson that I need to make more discovery based next year), we then did the Mistake Game, an idea from Kelly, which I have described a few times now. Basically students work out problems on whiteboards and hide a mistake in their solution. They then present their work like as if they didn’t make a mistake and the other students have a discussion to try to find their error. The problems I chose for the mistake game where all functions for which you had to do some sort of simplifying before integrating (like distributing or dividing), which ended up being a great way of pushing them a little bit forward while giving them plenty of opportunity to really go in depth discussing this new mechanical process of a definite integral.

6. Substitution Marker

Then the last skills activity I did with integration was a few days later when we started doing substitution. I had them first try a bunch of substitution problems intuitively, and then showed them how to use a u-substitution. Then, we pulled out the whiteboards and I gave them all a sheet of problems and two markers each. They were to do all u-related work in red, and all original-integral related work in blue. What I wanted them to get comfortable with was envisioning the transition between the variables and helping see how the skeleton of the integral becomes the “outside function” of the backwards chain rule. Example: (actual student work)

This was, again, one of those activities where a bunch of students totally ignored my directions and just solved the problems (and again, not the worst thing), but I think some of the students that did it like this really benefited from using the different colors.

So why did I just ramble about all those activities? I guess what I loved about this whole sequence is how ridiculously much of the instruction for a good week and a half or so was collaborative and engaging, and forced them to think about what could have been routine material in different ways instead of just plowing through worksheets and drills. I feel like I never would have been able to pull something like this off even last year, so I am so grateful (especially to the online community) that I now have a toolbox full of sweet teaching methods. My goal is to try to mix these types of activities more often into various units, since most have skills based components. I would love any other modes of instruction that you use in your classroom to add to my toolbox!!


Side note for the Calculus people: There are a few antiderivative/integral related traps that my students fall into… any ideas on how I can stop these problems before they happen?

  1. I always start with the word “antiderivative” to emphasis that it’s the opposite process of a derivative, and then try to transition to “integral” as soon as possible, but it’s really tough for them to keep the vocabulary straight. I always correct them in class (mostly just trying to replace antiderivative with integral). How do you approach that vocabulary? I even had a hard time writing this post with the correct vocabulary.
  2. Many of my students had a strange barrier this year (that I have never seen before) when finding the area under a curve because they kept thinking of the function you integrate as “the derivative” and the function that you get out as “the original function.” So when we had a function they wanted to find the area underneath, they would take its derivative and then integrate, or some other strange thing like that. How do you introduce the integral as being the opposite of the derivative without getting that misconception (or rather, what did I do in my sequence to imply that)?
  3. I always, always, always have so much trouble convincing some students that u-substitution is only used for specific functions that are “backward chain rules.” But after we learn how to integrate normally, we spend a ton of time on u-substitution, and then some students try to solve EVERYTHING with u-substitutions (like 1/x^6 for example). I spent a lot of time doing activities where we pick out the functions that can be integrated with substitution and those that can’t, but for a lot of students, this obviously did not sink in. Any tips?
  4. I cannot for the life of me get students to remember to add a “dx” when differentiating a u to find a du. So if u = 2x, then du=2xdx. Granted we didn’t do differentials, but I still don’t understand why this was so difficult! I need some sort of conceptual trigger so they can understand why it’s so important…

Posted on May 8, 2012, in Antiderivatives, Calculus and tagged , , , , , , . Bookmark the permalink. 17 Comments.

  1. Maybe I can help with #4… I make mine circle what they’re about to plug in for. Maybe to integrate (pun intended) with your colors, you could have them circle in red everything that is about to be replaced with u and box in red everything about to be replaced with du. Then everything that’s not circled/boxed should stay the same. Especially as you get into definite integrals (where you have to change the bounds to be u-values rather than x-values) this can be a big deal.

    You could also begin the idea before you get to u-sub. Try giving them \int a^2 x^3 dx vs. \int a^2 x^3 da. Show that d_ is an essential part and should match your variable.

    As for #3, my only tip is this. I give mine a “flow chart” of integration techniques (at least up to what we learn in BC) that helps them keep track of the tricks we’ve learned and their order of easiness.
    1. Something we’ve “memorized” (power rule, basic trig, etc.)
    2. Something you can get from geometry (eg \int \sqrt{9-x^2} dx)
    3. u-sub
    4. By Parts
    5. Partial fractions
    6. Trig sub

    If, at any time you change something, start back at the top of the list.

    • yeah the circle will do the trick… that’s one of those things that is so simple, and seems so obvious when you read it. “ah, why wasn’t i doing that all along!” thanks

  2. Thoughts:
    1. I always emphasize that these are two different things: the indefinite integral is the family of all antiderivatives, and if I’m asking for *an antiderivative* then there better not be a “+C”, although “+5” is just fine. (Of course, the nature of an arbitrary constant takes some getting used to, but that’s to be expected.)
    2. Haven’t had that misconception, so can’t give much useful feedback. Maybe give them several linear examples whose areas they can check with geometry formulas?
    3. I tend to use language like “useful strategy”: we can try to make that substitution, but then we get an integral like \displaystyle \int \frac{1}{u^6}\ dx or \displaystyle \int \frac{1}{u^6}\cdot \frac{1}{6x^5}\ du which just doesn’t get us any closer to being able to integrate, so it’s not a useful strategy. I think expecting them to be able to identify when u-sub is a useful strategy at the outset is less important than instilling a resilience in them that gets them to the point where they can say: “Well, that strategy isn’t helping. What else can I try?” Besides, it can be a subtle difference, when u-sub works, and when it doesn’t: \displaystyle \int \frac{2x+5}{x^2+5x+8}\ dx vs \displaystyle \int \frac{2x+6}{x^2+5x+8}\ dx . Maybe one answer is to introduce other strategies sooner, and just keep coming back to u-sub? I think having students *build* a flow chart of strategies is a great idea, but I would never give them one.
    4. When I have courses that don’t “do” differentials, I always say that the most useful notation for the derivative in this context is \frac{du}{dx} , and we want to substitute in for dx, so we need to solve for dx first.
    u=x^3+1 \\ \frac{du}{dx} =3x^2 \\ du=3x^2dx \\ dx=\frac{1}{3x^2}du
    I think if I had been told it just needed to be stuck on there, I would probably never write it either. Of course, I suppose it might partially depend on how you’re teaching substitution. I always do explicit substitutions, but more and more lately, it seems people (and books) don’t bother with an explicit substitution. I can see how it would be easier (and more tempting) to just fudge the differentials in that case.

    • RE: USUBS – I like the idea of resiliency, that helps. The problem I have had is that students will do crazy stuff to make a u-sub work (like divide by variables on the outside). But I think framing it like that in my discussion might help them frame it like that in their head.
      RE: DIFFERENTIALS – Besides the snarky dig at my use of the word “do” :), this is helpful – That


      how I taught it, with the du/dx (well, the first three steps), but after a few times like that I started skipping the middle step. But I think you and gasstationwithoutpumps have made me realize that instead I think I should keep that in for a while longer, maybe forever. I have preferred explicit subs with my non-AP class, and less so with my AP class (except in situations where you need it like integral (x*sqrt(x-1))dx… sorry I’m not a pro at LaTeX haha).

      Thanks for your thoughtful comments.

  3. Different idea for #4: Have them do u=2x, du/dx=2, so du = 2 * dx. If you have taught derivatives with the standard du/dx notation, this 2-step process should avoid the du=2 problem.

    • yeah that’s how i went about it when i first taught it… they were startled that we were allowed to move the dx, so it became more like math magic without understanding. so i did that a few times and then said you can skip that step and just write blah blah dx. But maybe I shouldn’t have them skip that step and keep the 2-step process like you suggested. and then some kids can skip it if they get used to it. thanks!

  4. I just read an interesting article posted (on twitter I think….) by someone else that talked about teaching differentials and how it’s uncommon now, but maybe shouldn’t be. Anyway, it has me thinking that I might teach differentials next year. It’ll make Euler’s method a bit easier for the BC kids at least. http://www.math.oregonstate.edu/bridge/papers/CMJdifferentials.pdf

    • Whoa, that’s great – I have definitely talked a bit about the concept behind differentials, but we never calculated them, and some of the ways that that paper talks about using them are really clever, especially in the way it gives you flexibility. I think it would be a hard sell conceptually, but might be worth it.

      • I really like the way they talk about introducing it geometrically from the delta x and delta y notation being the change in variables to the dx. They also mention the dx and dy representing a limiting process of the typically algebraic delta x and delta y. If you teach limits earlier you could probably use some kind of argument with a limit as delta x -> 0 it becomes dx or something. Which of course is exactly what the dx is… but anyway. Something I’m strongly considering for next year.

      • yeah great suggestion. thanks!

  5. If you do go the route of the differential, the Silvanus Thompson book _Calculus Made Easy_ is beautiful: https://docs.google.com/viewer?url=http%3A%2F%2Fdjm.cc%2Flibrary%2FCalculus_Made_Easy_Thompson.pdf

  6. I also have been toying showing them what a change in coordinates *does* with u-substitution. If we have \int 2x(x^2+1)^3 dx (evaluated from 0 to 4), and \int u^3 du (1 to 17), I wonder about making a worksheet that shows what f(x)=2x(x^2+1)^3 looks like (and shaded in), and what g(u)=u^3 looks like (and shaded in), and then literally show how one rectangle in the riemann sum for f(x) morphs to something else in g(u). Sorry if this doesn’t make sense…

    I only toyed with doing something like that this year when I did u-substitution, but decided against it and only taught is as procedure (and not conceptually). But who knows – maybe I’ll do something like this next year…

    • That sounds pretty neat. It may or may not be helpful, but there’s no reason not to go through it as a worksheet thing at least once right? This is something I did in calculus 3 with a couple of my kids this past fall when we were doing substitutions in multiple integrals (same thing as u-sub in a single though!) and we did all these transformations. I actually had them graph the u and v transformation functions and sketch the areas for the “transformed” domain. We were taking weirdly shaped domain regions in x and y and transforming them into nice rectangular domain regions in u and v. We talked about how the axes are an arbitrary construct (abstraction!) and we could orient it however we want to fit the region/shape we want to work with, etc. It was pretty cool, I don’t think any of them remember it right now though…

  7. whoops, those limits of integration are wrong, but you get the point…

    • yeah that makes total sense. i have the same reservation though that you seem to have (perhaps it’s just too much? and would it really accomplish more?). cool idea, let me know if you pull something together for that next year. also, i can’t figure out what’s wrong with those limits of integration??

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