One Good Thing: Triangles

We’re in the middle of reviewing for the AP exam – AB Calc. We talked about area and volume and how to find the volume of shapes with cross-sections. We’ve worked a lot of problems like this before, so the kids know what to expect.

I throw up an equilateral triangle problem. They groan. “Ugh…triangles!”

“Is there, like, a formula that works for all equilateral triangles to find the area?” a student asks.

“Well, there’s one half base times height…” I reply, not sure what he’s looking for.

“Yeah, but specifically for equilateral triangles!”

“Well, sure,” I answer. I draw a triangle on the board. “What do you think we should do?”

They start discussing. “Well, we could draw in the height and make a right triangle…and use the Pythagorean Theorem…” They eventually get to this: for equilateral triangles. $A = \frac{\sqrt{3}}{4} b^2$.

They’re over the moon. “I’m so excited about this!” “Why has no one ever taught us this?!” “This is so cool!”

I’m confused. Very confused. How is this new? We’ve done lots of triangle problems, and it’s the exact same process every time. “Does this surprise you?” I ask.

One student has been watching all this, as confused as I am. He shakes his head. “J, why does this not surprise you?” I ask.

“Because it’s a 30-60-90 triangle. That’s the ratio of the sides,” he answers.

“Or we could do sin(60), also. Which I did teach you, last year in Precal,” I reply.

The rest of the class doesn’t care. “Yeah, but we’ve never done this before! This is so cool! I’m so excited about this!”

There are ten minutes left in class. I decide that, clearly, we aren’t very interested in reviewing area and volume right now, so I just keep going with area of shapes. I remember Math With Bad Drawing’s Recent post on rectangles and area (see here) and go with it.

“You know you can find the area of anything with a rectangle, right?”

They’re hooked. We talk about how to turn triangles into rectangles, how to turn parallelograms into rectangles. They ask about circles, breathless. I ask them what they think. They come up with a couple of good ideas, one of which is the main idea behind polar area, a BC topic. (This is AB Calc.)

So, they learned about area. And better still, they were excited about it. I’ll take that!