Introducing basic trigonometric ratios

I take no credit for the idea, but I am absolutely sharing the results. Thanks to Mr. Shah, a rock-star teacher blogger, and his Virtual Filing Cabinet, I had a wealth of ideas at my disposal when planning a trigonometry unit for the second time this year.

Last year, I started with special right triangles. We then for some reason detoured into arc/sector measurements before returning to me telling them the patterns that these “special triangles” could make in the unit circle. YAWN!  I asked my students to memorize the unit circle, which I am now not convinced that it is that important anyway (Although Ms. Gruen’s blog gave me a great memory tool to help students remember unit circle values). Then we used a great activity from TI Education to build the sine and cosine waves. I stand behind this one, although I already know this year it will make SO MUCH MORE SENSE to them. 

The reason I am optimistic about the TI activity is now all rooted in the new approach I took this year. Courtesy of Mr. Lark, my students were able to find their own patterns and shortcuts. Mr. Lark’s activity takes a lot of the heady, abstract nature of trig and returns to a bunch of simple calculations. Students measure angles, draw radii, and write the x- and y-coordinates of the points at which those radii end. By the time they get really tired of doing this over and over again, they can start to see patterns that they can latch on to. I introduced my version of the activity (see below) with a very explicit exhortation to use the patterns when you find them. After a couple days, we discussed and students suggested patterns that could be later tied into:

  • reference angles
  • why sin, cos, and tan are positive and negative in different quadrants
  • symmetry between the various quadrants

I’m drooling at this point. My students are thinking like mathematicians and all while working on an activity that really only required knowledge of how to use a protractor and how to read the coordinates of a lot of points. I then make the big reveal: “I know you’ve heard how miserable trigonometry is, but YOU’VE BEEN DOING IT ALL ALONG!” The stares I get back say, “Who is this crazy person?” but then I explain how the x-coordinate they were finding over and over is cosine, and the y-coordinate is sine, and the ratio of y/x is tangent (which I somehow didn’t realize until this year is the slope of the radii in the unit circle formed by that angle!).

We moved forward and derived the SOH CAH TOA general trig ratios by showing how any right triangle was proportional to a unit circle triangle. I am really impressed with how well my students have latched onto the sin, cos, and tan since then.

It’s also pretty cool when you hear “I didn’t know SOH CAH TOA actually meant anything! I thought it was just something people said…”

Now that I am getting to introducing the sine and cosine waves this week, I’m looking forward to lots of A-ha! moments from students remembering that cosine is the x-coordinate and sine is the y-coordinate on the unit circle.

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