A Small Change: Unit Circle & Trigonometric Functions

I wrote nearly a year ago about my adjustment to what I had done previously to develop the topic. The idea was based on what my own pre-Calculus teacher did in high school, a series of activities related to a ‘wrapping function’ moving around the unit circle. This lesson is for a group of Algebra 2 level students that will likely move into the IB program for next year. Mastery of trigonometric functions isn’t necessary, but I do want students to feel comfortable converting between radians and degrees, locating angles on the unit circle, and evaluating trigonometric functions.

In the last class, we talked about 30-60-90 and 45-45-90 triangles and the fact that we can evaluate trigonometric functions exactly using our knowledge of ratios and the Pythagorean theorem. We also did a series of exercises having students locate angles on the unit circle during the last class.

Today’s warm-up was a continuation of these ideas through these sets of questions:

Normally at this stage, I show a development using similar triangles of finding what these coordinates are. Though I bring up this goal in a number of different ways, whether students are doing this at their seats, or I’m doing it for them, I can never the sense of understanding that I want. This development is also not what I want them to do when they are evaluating trigonometric functions either – I want them to figure out where they are on the unit circle, and then evaluate based on the x and y-coordinates of the point.

Today I made a subtle change to my sequence. I directly told students that the coordinates of these points were some combination of a set of five lengths. Two of these lengths we found in a previous lesson, but I never made a connection to it here. I asked them to put the numbers in order from least to greatest

Then I asked them to complete the coordinates in this blank unit circle. Here’s a student’s work, corrected by a classmate when it was shared:

All of the conversations about sign and value that I had to force previously happened naturally this time. The handout was folded so that as students finished, I could then nudge them into the next step of finding angles that match to particular coordinates, an exercise on the other side.

For most of the students, this wasn’t a problem. Some even looked like they were enjoying it.

It was only in the last few minutes of the class that I introduced the sine, cosine, and tangent as a shorthand way of asking the question of finding the x-coordinate, y-coordinate, or the ratio of the two. My students are pretty trusting, but they have also become used to asking why [Statement A] is true once they have the basic idea of what [Statement A] means. This lesson was just a continuation of this process. Almost every student was able to evaluate a cosine function of a different angle during the exit activity.

I felt a little bad about giving the coordinates and putting off the understanding to later. This short bit of mathematical fact, however, was followed immediately by a task that required them to reason about what they mean. It builds the need to show why those coordinates are what they are, and this process of looking at 45-45-90 and 30-60-90 triangles on the unit circle will make much more sense in the context of the student experiences here.

One student summed up my motivation for doing this beautifully as she was packing up – I love that I’m not making this quote up:

It’s good that you don’t have to memorize it because you can just see the picture in your head and know what the answer is.

Jackpot.