One of the VCE Maths Methods topics I don’t think I’ve covered particularly well in the past has been the estimation of definite integrals using rectangles. By the time I get up to it, we’ve usually had so many interuptions during the first half of the year (this year is no exception) that I’m trying to catch up and often rush through the material with some short board notes and a few textbook questions.
This year I made a conscious decision to improve the way I introduced it. As much as sketched diagrams help a little, I think they still leave the concepts pretty abstract. I wanted to make my examples more concrete. I wanted to set the stage for explaining the definite integral as the limit of a sum. Hopefully this will also later help draw attention to the significance of the Fundamental Theorem of Calculus.
My solution (which I’m starting to realise is the solution to pretty much everything!) was cutting and pasting. Initially my students weren’t terribly keen on it, and thought it was a waste of time when they could just draw it, but I think they came round to the idea – kind of. One of them asked an off-topic question, which they started by saying, “Since we’re not really doing maths today…”. That’s OK, I’ve still got a semester to convince them that cutting and pasting absolutely is maths. (Actually, the question was a really interesting one, but I’ll wait until another post to write about it.)
I gave the students a sheet with the same graph of y = -x2 + 4 repeated four times, on grids where 1 cm = 0.5 units. I also gave them coloured paper to cut into strips 2 cm and 1 cm, to use as the rectangles. The result is below:
With more time, I would have liked to have also done left and right endpoint rectangles, as well as midpoint rectangles and trapezia (I stuck to the boring smartboard for these). Personally I prefer upper and lower rectangles because you can talk about how they, besides being an estimation, form upper and lower bounds on the actual value, and you can demonstrate that the bounds get closer as the number of rectangles increases. But that could just be my pure maths background – I’m sure I’d prefer a faster method for calculating if I had an applied background.
Thinking ahead to next year, this could be followed up with some sort of technology to quickly demonstrate with even more rectangles – maybe using Excel or Geogebra, or maybe even the CAS has some way of doing that. I’ll need to investigate those options. That way we could experimentally find the limit, and hopefully comfirm it matches the definite integral we find using calculus.
EDIT: I meant to include a link to the worksheets here, I’ll add them once I’m near a computer.
EDIT (23-6-2015): A year later (to the day, apparently), I finally got around to adding those worksheets!