Getting A “Feel” for Trig Functions

My good friend Steve Kelly always talks about getting creative ideas in the shower and to be honest I’ve always been jealous. It’s not frequent that I get a good idea in the shower. However, that changed a couple weeks ago. (Having been entrenched in learning about creativity for the last several months I realized that maybe the problem was I was trying to find ideas in the shower as opposed to letting my subconscious do the work. So, I stopped doing that, not only during showers but also workouts, driving, etc. and the ideas have come more frequently.) Prior to the idea I had spent a lot of time thinking about how to help students understand the connection between trig ratios on the unit circle and the graphs of trig functions on the Cartesian plane. The idea I had was to somehow make a massive unit circle and Cartesian plane and show students the connections in a life size context. Below is the outline of the activity and below that, my reflections.

The basic layout can be seen in the images below. Basically I used note cards for tick marks with some of them marked with angle measures and some not (the only labeled cards on the unit circle were “0” and “pi”, all tick marks were labeled on the Cartesian plane). I then wrote five different trigonometric expressions (like sin (pi/3)) on several different note cards. They could work in pairs and the first round was practice. They could ask me questions, use notes, etc. In fact, I let anybody that wanted to hangout in the hallway during the practice rounds ask questions as well. This was a tremendous learning experience as one student was required to find the correct location on the unit circle (probably the easier part) and the other student had to find the correct location on the Cartesian plane. For some reason the conversations seemed more authentic then when they occur over a book assignment or even a whiteboard. For instance, students were really trying to understand the reasoning behind where each partner would have to stand. This may have been driven by the fact that there was going to be an assessment portion, where all they were allowed to use was a whiteboard, their brains, and each other to solve five on their own. Even with the motivation from an assessment, it’s difficult for me to relay how much was learned from these conversations. Something about the context of working with a life size unit circle and Cartesian plane shifted the engagement level significantly.

I also wanted to reflect on the assessment portion. As you can read in the activity write up, students were tested over this activity (here is the rubric I used during the assessment). As mentioned above they could not bring notes and I could not help them during this phase. I noticed a few things, first of which was that students made a major effort to memorize the special right triangles (45-45-90 and 30-60-90). They also finally dug into understanding radians deeper (for instance, where would 5pi/6 be located and what is the reference angle at the at location). They had to rely on each other to confirm their reasoning, not me. Students become accustomed to being able to get constant affirmation or redirection from their teacher so this was uncomfortable for many of them. But they had rich discussions around the mathematics as they tried to convince each other of their reasoning! It was one of my favorite parts of the whole unit.

This was a solid activity because I think it effectively integrated embodied cognition (which I talk more about in this post). This is the idea that we can gain a deeper understanding of a concept if we can physically interact with it. Also, it gets students up and moving around, as well as makes them see mathematics in a different context. I will definitely be doing this again next year and also looking for other places to integrate embodied cognition. As usual, please hit me up with your feedback and ideas!

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