About a week ago my older son asked me a really fun question about parabolas -> in the Cartesian plane, is there more area inside a parabola or outside of the parabola? His hypothesis was that since both of the areas were infinite, the two areas would actually be the same.
As excited as I was by the question, talking through the answer isn’t as simple as it seems. First, playing around with infinity produces all sorts of unusual results. See, for example, Bertrand’s Paradox (which I accidentally called “Russell’s paradox” in the 5th video):
Second, there are all sorts of little details you can either skip over or dive way, way, way into along the way, so finding the right set of ideas to bring together to talk about this problem was proving to be hard. Luckily there are folks on Twitter who can think about teaching math way better than me, and this time Earl Samuelson (@earlsamuelson) provided exactly the idea I was looking for :
So, with that idea in place, we began talking. The first talk was just recapping the problem itself and talking through a couple of non-intuitive things about infinity. I use the example of comparing the number of positive integers to the number of positive even integers to show that some aspects of infinity are a little strange:
In the next talk I brought up Earl Samuelson’s idea and looked at the area inside of y = | x | since that area is a little bit easier to get your arms around. I always tell my kids that a really important step in problem solving is trying to start with a problem that is similar to, but a little bit easier than the problem you are trying to solve. Thanks again to Earl for reminding me of my own lesson!
Next we start down the path of finding the area inside of a parabola. We begin by looking at the various pieces of the area we need to add up to figure out the area outside of the parabola (and I sort of botch the geometry at the beginning, but that was fortunately a minor point). That leads us pretty naturally to the problem of finding a way to calculate the area under the parabola. That, of course, leads to the basic idea of a Riemann sum.
So, having touched on the concept of a Riemann sum in the last talk, I actually really struggled to think about what to do next. The problem with really rolling up your sleeves and digging into Riemann sums is that you are heading down a black hole – so much material and no easy place to stop. I’m not trying to really cover calculus anyway, so I just decided to show the geometry and save the details for a few more years! Outside of the video, though, we did play around on a Riemann sum app we found online just to see how the process works.
As an aside, in thinking about how to talk through Riemann sums, I read through the treatment that Spivak gives in his Calculus text. As with just about everything of his that I’ve read, I was really impressed with his presentation. Particularly the end of the section where he says simply:
“The moral of this tale is that anything which looks like a good approximation to an integral really is . . . .”
Anyway, on to the next talk (and thankfully I realized I’d gotten the geometry wrong in the 3rd video before we made this one):
Now with the Riemann sum talk out of the way, all we had to do was add up the areas. So, off we went:
And that’s the story. As I mentioned in the beginning of this post, playing around with infinity can produce odd results, and looking at finite areas that aren’t squares might lead to a different result (take rectangles centered at (0,0) that are 2x wide and 2x^2 tall, for example) but I didn’t really want to let those sorts of complexities get in the way of the discussion here.
All in all talking through the answer to my son’s question was really fun. Thanks again to Earl Samuelson for helping me see how to put these talks together.
As always, I’m happy that I have the time and flexibility to have talks like these with my kids.