# Drawing Tangent Lines

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.

## 5 thoughts on “Drawing Tangent Lines ”

1. I love your explanation. I too remembering having trouble with getting some kids to “see” tangent lines. I have done that zooming in thing on Geogebra. And after it looks super straight, I ask kids if it’s “really” straight. They say no. Then I put a tangent line at a point on the graph (geogebra has the capability to draw a tangent line). And you can barely tell the difference between the tangent line and the function. Then I zoom out with BOTH things graphed – the function and the tangent line. The tangent line is a great approximation/model for the function at the point. But it becomes a pretty terrible approximation/model for the function far from that point!

Okay, calculus teacher geek mode off!

• Oh love!!!! I sort of go the other way around. I start with y=x^2 graphed on the screen, but zoomed WAY in on the origin so that everyone thinks I’ve actually graphed y=0. Then as I start to zoom out the kids decide that may the graph is actually not y=0. We do the same with a few other points to highlight local linearity. I love it so much.

Thanks for geeking out with me. ❤

2. Mr. Chase

I love the “zooming in” explanation for drawing tangent lines (cup your hands around the point of tangency and stick your eye right in there…does the line look like the curve?). It also emphasizes the important idea that differentiability implies local linearity.

Here are two more somewhat fun-loving methods that don’t have nearly as much mathematical depth:

(1) Imagine the curve is the top view of a curvy section of road and your car is at the point of tangency. Along what line do your headlights point? Taillights? Boom baby, draw your tangent line. More morbidly, you could ask, “in what direction would your car fly off the road?”

(2) Imagine the curve is the side view of a rollercoaster and you’re seated at the point of tangency–it’s only natural to accompany this explanation with cute pictures of a rollercoaster with you in it, hands raised and all. In which direction are you looking? That is, what’s your line of sight? Draw it!

So really the local linearity idea is the best, but you might enjoy these two “cute” ideas I’ve used over the years.

Thanks for sharing so consistently on this blog. You’re a constant encouragement!

• Ah! These are both such good analogies! Definitely stealing…especially the roller coaster one. 🙂

Thank you for your kind words–YOU are the encourager. ❤

3. suevanhattum

I’ve been using Sam’s packet this past week too! (college course)