My calc kids are working on Sam Shah’s wonderful “Implicit Differentiation, Visually” packet. I just love this packet. If you teach calc and don’t know about this packet, please go here now.
So Sam is great about making kids go back and forth between the derivative and the graphical representation of various relations. Each problem is very attainable, but the connections made can be very deep. Low floor, high ceiling–as they say.
Anyway, one part of one of the questions asks the kids to pick three points on a given graph and then draw the tangent lines there. As I was looking over some kids’ packets, I was horrified at their attempts at drawing tangent lines. They were about as wrong as you can get (aka most looked more like normal lines than anything else…)
I realized that we’ve found tangent lines algebraically (over and over and over again), but never really graphically. Yeah, we may graph our answers on a calculator, but it’s not the same.
I’ve never felt I’ve been able to explain how to draw a tangent line well. I just kind of see them and draw them and I think my kids just accept it (not proud of this, but that’s the truth).
And then today when I saw their awful lines I had an epiphany.
I explained it something like this:
“You know how when we zoom in enough on a tangent line, it starts to look exactly like the original graph? You can’t tell the two apart. That’s what you need to draw. If you get close to the point you choose, your line and the given graph should look the same.”
And then I took a straightedge (aka a piece of paper) to the whiteboard and showed them an example.
And that was that. All the tangent lines I checked after that were just lovely.
I know this is a lot of math jargon in this post, but my good thing is this: I got better at explaining something today. I was able to make a concept that my kids were clearly struggling with much more lucid for them. And I love that feeling. Evey teacher knows exactly what I’m talking about…it’s the best.