I'm teaching four classes this semester, which is a lot for me. That's embarrassing to admit - I know most math bloggers are high school teachers, and teach way more hours a week than I do, with more responsibilities for their students. But for me it's a heavy load. So I'm not prepping as much as usual. I've taught calc and pre-calc dozens of times, so I can usually get by with winging it. And, once in a while, I'm able to conduct a better class by improvising than I ever could have with a tight plan.

That's what happened yesterday in pre-calc. The day before that I had worked hard to get their tests graded, so in the morning I printed out the new unit sheet, and walked into class not particularly sure how I wanted to get us started. I had grabbed a problem from my computer, and asked them to start thinking about it while I handed back tests.

The problem:

Consider three circles, all tangent (externally). Their radii are 4 in, 5in, and 6in. What is the area between them?

I had asked the students to draw a picture. After they had had plenty of time, I drew my picture on the board. Then I asked them how we might start thinking about the problem. A student suggested finding the area of the triangle formed by connecting the centers. I asked if that triangle's sides actually went through the points of tangency. No one answered. Unlike in a math circle, I rescued them be showing a picture of one circle with a tangent line, and reminding them that they likely proved in geometry that the tangent is perpendicular to the radius (the one that ends at the point of tangency). I don't know what that proof would look like. To me, it seems obvious because of the symmetry. (In the afternoon class, they didn't think it needed proving. It already looked necessary to them.)

To find the area of the triangle, one student suggested drawing in the height. We drew it in, but couldn't yet see how to find its length. One of the students suggested that we could find the measures of the angles. They first suggested using law of sines. That didn't work, so we used law of cosines. Sine of that angle gave the height over a triangle side, so we got the height, which gives us area of the triangle. Then we got the other angles and found the sector areas. The afternoon class did it without the height, so they got to use law of sines.

It was a lot of steps for them, but it was a great review of the triangle trig we'd done earlier in the semester. And maybe they got a small taste of what problem-solving looks like.

When we were done, I had just enough time to explain radians to the morning class. The afternoon class had more time, so we worked out the new circle-based definitions of the trig functions.