Monday, February 24, 2014
Here Is Some Great Slow Motion Of A Cat Falling
We're starting a unit on average/instantaneous rates of change in Precalculus. Roughly, my idea is to give kids lots of scenarios where they can find speeds at various intervals and moments and use that to give them time to develop ways to represent rates of change. Once they have a firm grasp on using secant and tangent lines to represent rates of change we'll start measuring the slopes of tangent lines along curves, and then we'll notice some cool things.
A sticking point is how to explain those cool things. For instance, we'll notice that the slopes of tangent lines along y = x^2 vary as y = 2x. How do we make sense of that observation? Some curricula explain this using polynomial division, but my kids don't have that skill and I don't want to give it to them. I could do it like Newton and use infinitesimals -- so that we're looking at (x+o)^2 - x^2 over o, where o is ridiculously small -- but I'm a little woozy about just being like "here's some magic children!" (Though, I guess it's sort of OK since that magic turns out to be Calculus.)
I've got graphingstories.com and the Shell Centre's Functions and Graphs book. Anyone have advice or resources as I embark on this unit?
Thanks in advance!