I created some calculus Geogebra applet thingies last summer that I wanted to use last year. Alas, time ran short and we never got to use them. However since I’m no longer teaching calculus (at least not next year), I figured I’d throw them up in case anyone else out there finds them useful.
They deal with u-substitution. I’ve always had a problem with teaching it. Here’s how it goes… You have some integral in terms of . You convert all the s and s into s and s. And viola! It works out. It’s very powerful. And it’s procedural. And kids have throughout the years learned this “substitution”-y thing works . So kids tend to like it.
But here’s the thing. For my kids, it’s just a random method to evaluate an integral. They don’t conceptually understand what is going on… what this changing of variables is doing.
When I thought deeply about this, I realized what truly is happening is that we are transforming space… From the plane to a much convoluted plane. But it is through our particular choice of that makes the change in space beautiful, because it turns something that looks particularly nasty and converts it into something that looks rather nice. Ish.
Here is a screenshot from one of my geogebra applets illustrating this (you can click on the screenshot to be taken to the applet):
We start with a pretty ugly function that we’re integrating. But by using this substitution to morph space, we end up with a much nicer function. I mean, throw both of these up and ask your kids — which one of these would they rather find the integral of. They’ll say the one on the right! The u-substitution one. Although not perfect , it’s pretty kewl.
The applets are here:
And the applets are dynamic! You can change the lower and upper bounds on the graphs and the lower and upper bounds automatically change on the graph! But because math is awesome, the areas are preserved!
Some things I maybe would have done with the applets in my class:
- Let kids play with the applets and get familiar with them.
- For the first applet (starting simple), have kids count the boxes and estimate the area on one graph, and then do it on the other (careful though! the gridlines are different on the two graphs!). Whoa, they are always the same!
- For the first applet (again, starting simple), ask them to drag the upper limit to the left of the lower limit. Explain what happens and why.
- The second applet is my favorite! Put the lower limit at x=0. Drag the upper limit to the right. Explain what is happening graphically — and that tie that graphically understanding to the particular u-substitution chosen.
- In the second applet, can students find three different sets of bounds which give a signed area of 0?
- In the fourth applet, have students put the lower and upper bounds on x=6 and x=7. Have them calculate the average height of that function in that interval (the area is given!). Do they have visual confirmation of this average height for this interval?Now Looking at the u-graph, the bounds are now u=8 and u=10. Have them estimate the average height of that function in that interval (again, the area is given)! (The average height “halves” in order to compensate for the wider interval. It has to since the areas must be the same) Have students do this again for any lower and upper bounds for this graph. It will always work!
- In the fifth applet, have students put the lower bound at x=0, and have them drag the upper bound to the right. What can they conclude about the areas of each of the pink regions on the graph? (Alternatively, you can ask: you can see from the graph that the signed area on the original graph will never get bigger than 1, no matter what bounds you choose. Try it! It is impossible! Armed with that information, can you conclude about the pink regions in original graph?)
I’m confident I had more ideas about how to use these when I made them . But it was over a year ago and I haven’t really thought of them since. But anyway, I hope they are of some use to you. Even if you just show them to your kids cursorily to illustrate what graphically is going on when you are doing u-substitution.
 Though I bet if you asked a class why they can use “substitution” when solving a system of equations, what the reasoning is behind this method, they might draw a bit of a blank… But that’s neither here nor there…
 What would actually be perfect would be a copy of individual Riemann Sum rectangles from the graph “leaving” the first graph, then in front of the viewer stretching/shrinking their height and width for the appropriate graph, and then floating over to the graph and placing itself at the appropriate place on the axis. And then a second rectangle does that. And a third. And a fourth. You get the picture. But even though the height and width morph, the area of the original rectangle and the area of the new rectangle will be the same (or to be technical, very very close to the same, since we’re just doing approximations). In this sort of applet, you’d see the actual morphing. That’s what is hidden in my applets above. But that’s actually where the magic happens!
 I recall now I was going to make kids do some stuff by hand. For example: before they use the applets, kids would be given lower and upper x-bounds, and asked to calculate lower and upper u bounds. And then use the applets to confirm. Similarly, given lower and upper u-bounds, calculate lower and upper x-bounds. Use the applets to confirm.