I’ve never been wild about how Riemann sums go down in my classroom.
Part of the problem is that I don’t remember doing a whole lot with Riemann sums – if anything! – when I was in AP Calculus, so I can’t pull on what my teacher did. I looked through my notes and found nothing, so I’m not entirely sure what level is appropriate. (Hey, calc teachers – input on this appreciated!)
The bigger problem is that I feel rushed. An hour just hasn’t been enough time, but it doesn’t feel right to break it up over two days. Not sure why.
This year, though, I get a two hour block with Calculus once a week – today. I made sure we did Riemann sums on this day so that we could really slow down and sink our teeth into it. It made such a difference!
Last time, we did left and right endpoint rectangles, and we talked about how more rectangles gives a better approximation. We played around with mathworld.wolfram.com/RiemannSums to see that more clearly.
We spend some time developing the Riemann sum and all the notation. This is the heavy lifting, in a lot of ways: you’re taking a very simple idea (add up the areas!) and burying it in a lot of notation. I remind everyone over and over again what each expression in the sum represents. Color-coding things helps, too.
Then I asked, “Ok, if more rectangles is better, how many is best?”
“Infinitely many!” they cried.
“But we can’t have infinitely many. Infinity isn’t a number; it’s a philosophical concept. How can we get close to that?”
Eventually, someone suggests a limit, and we take the limit of our Riemann sum.
I set up a Riemann sum, then have them set up a Riemann sum. I evaluated my Riemann sum (which scared them), then had them evaluate their Riemann sum (which scared them more). This is where having a long period was key, because they had a lot of time to delve into it and no one felt rushed. Not everyone finished, but I gave them some time at the end to go back and think about it. It was fun to watch the ones that got it – they were so proud of themselves. (I was, too! I didn’t give them an easy problem!)
We then stepped back and talked about the definite integral, and how the limit of Greek letters is Roman letters. We used some geometry to actually evaluate our definite integrals, which was easier.
A new twist this year is that I gave them some Riemann sums and asked them to write them as definite integrals. I liked this problem. Thinking about things backwards is good, and helped them clarify a few things. I was actually surprised with how easily they figured things out. The bounds, in particular, were very easy for them to see, which was a pleasant surprise!
So: working out the Riemann sum was new (no time in the past), and the backwards problem was new. The main difference, though, was the two hour block. The extra time went a long way.
All in all, it was a really satisfying class. Things clicked, no one panicked, and we did some hard mathematical work today.