I started Trig in my precalculus classes last week and decided to do things very differently than I have before, seeing it hasn’t gone especially well ever before.
I must admit to largely stealing what the CME Algebra 2 and Precalculus books do. The Algebra 2 book does a nice introduction to using the unit circle, and the Precalculus book does a great job explaining how to use radians. Since my students haven’t seen anything about trig in a few years, and what they did see was very basic triangle stuff, the Precalculus material on its own was too advanced for them.
As my ‘velcro,’ as one of my grad school teachers referred to the idea of introducing new material as an extension of something that students already know, I asked kids to use the Pythagorean theorem to find the missing sides of some triangles.
Here was the really crucial question of the do now:
Crucial, I say, because it makes question #4 on the classwork handout much easier.
Speaking of, here’s the handout:
What I think went well is that it doesn’t even mention degrees until the second page. Rather than having my students constantly converting back to degrees, I’d like them to think fluently in radians. Of course conversion is an important ability.
The idea of a person walking around a circle seemed to work okay. Sometimes they had some goofy interpretations of what that could mean — though not in a ‘how cute are these teens and their creativity’ sort of way. More in a what-the-heck-are-you-thinking way. I can’t even remember what they were — everybody figured out the correct way of thinking about it by the time they handed in their work, so I don’t have any copies left.
Somehow it only occurs to me now that a good way of dealing with this would have been to have my students actually walk around a circle on the ground. Suddenly this seems very obvious. I was trying to think up something that involved wrapping hot pink rope around a paper plate. Silly.
I was pretty impressed with how quickly students did pick up conversions from radians to degrees. I didn’t give any more help than sometimes suggesting:
“So if pi is 180, what would that make pi/6?”
I expecting that converting back would be more difficult, but it seemed to go pretty well.
The next thing that I did was reintroduce SOHCAHTOA. Then we used this worksheet:
I did stick to multiples of pi/6, pi/4, and pi/6. I want them to get used to these angles. We’ll talk about multiples of pi/2 next.
Rather than just announce that the (x, y) value where an angle hits the unit circle is (cos(z), sin(z)), the hope is that they simply notice it. I haven’t had many kids finish this up, so I can’t be sure yet. One girl does keep thinking she’s making mistakes because the same numbers keep showing up. Little does she know, that’s exactly how I knew she was doing it correctly.
One thing that I have not done yet is make sure that students are using precise values, including square root notation. So far they’ve been giving approximate decimal answers. It’s important that they use the square root ones so that all sorts of things make sense in the future, but for now I like the decimals because I think they’re more meaningful to my students. I don’t think that answers like “square root 2 over 2” are very satisfying ending points for them. So I have to figure out how to introduce those.
Next up for the dear kids: filling in the whole unit circle.
Then after that: tangent. You may notice that I have completely neglected it. That’s on purpose.