This year I’ve been doing a lot of work with my geometry kids to get them to build up a deeply conceptual understanding of trigonometry. Right now we’re still in the part of the unit where the the terms sine/cosine/tangent haven’t been introduced, and kids are building up their understanding by thinking of ratios in specific triangles. But soon we are going to introduce the terms, and I’m afraid they are going to go to their calculator and use it blindly, and forget precisely what sine, cosine, and tangent really mean.
For my kids, at this level, I want each term to be a ratio generates a class of similar triangles — which all look the same, but have different sizes. And I want kids to conjure that up, when they think of . But I fear that 0.6428 will stop losing meaning as a ratio of sides… that 0.6428 won’t mean anything geometric or visual to them. Why? Because the words “sine” “cosine” and “tangent” start acting as masks, and kids start thinking procedurally when using them in geometry.
So here’s the setup for what we’re going to do.
Kids are going to be placed in pairs. They are going to be given the following scorecard:
They will also be given the following sheet, with a clever title (the Platonic part refers to something we’ve talked about before… don’t worry ’bout it) (.docx form). This sheet has a bunch of right triangles, with 10, 20, 30, … , 80 degree angles.
Then with their first partner, on the front board, I project:
The kids will have 3 minutes to discuss how they’re going to figure out which two triangles/angles best “fit” these trig equations. (I’m hoping they are going to say, eventually, something like “well the hypotenuse should be about twice the length of the opposite leg, so that looks a lot like triangle C in our placemat” for the first equation.)
They write down their answers. If they finish early, I have additional review questions from the beginning of the year that will be worth some number of points — to work on individually.
When time is up, they move to a new chair (in a particular way) so that everyone has a new partner. I throw some other equations up. And have them discuss and respond. Then they move again, and have new equations up.
I’ve scaffolded the equations I’m putting up in a particular way — so I’m hoping they lead to some good discussions. And I’m hoping as soon as a few people catch onto the whole “let’s compare side lengths” approach, the switching will allow for more discussion — so soon everyone will have caught on.
At the end of the game, we’ll have some discussion, and through those discussions we’ll reveal the answers. And of course, the student with the most correct answers will win some sort of fabulous prize.
The questions I’m going to ask are here:
The discussion questions are here:
I’m super excited to try this out on my kids next week sometime.