After a hiatus: circular functions

It has been a busy time in gealgerobophysiculus land. By land, I of course mean school, and by busy, I mean what results when you have multiple exciting projects going on, school functions to organize, and the normal operations of a classroom to sort through and organize.

I haven’t taught the unit circle in three years. Before that, I took the approach of throwing a definition of the radian up on the board and discussing it as this strange thing that mathematicians decided would be a good idea. When I learned this in high school, we did some cool activities involving string and wrapping functions. At that time, it wasn’t clear to me how the string wrapping around a circular object really related to measuring an angle around it. I was always relating the idea of the radian angle back to degrees, because the angle part never made sense.

After some thinking and coding, I put together an activity that I thought would make this concept more concrete for the students in my tenth grade class. You can check it out at

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It starts with the premise of moving around a circle at distances of integer multiples of the radius. Looking at your own work doesn’t really establish how this relates to measuring angles at all. When you look at what happens when many people do the same thing to differently sized circles, the result makes clear that this could be a fairly natural way to measure out angles:
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I didn’t have the networked part of this applet working when I did this with students, so I collected screenshots of students and their different circles together. I asked students what they expected would be different about the locations of these six points for circles of different sizes, and there was pretty solid agreement that they would be in roughly the same point around the circle, but this was still too abstract to establish the idea that these points measure out angles. The students weren’t too surprised by the result, either, but I think the activity in this form still left me as the teacher to connect the dots.

I wish I had an extra day to configure the final screen of this activity. I wouldn’t have had to work so hard.

The rest of the unit walked the line from this concrete idea of moving around the circle up the ladder of abstraction to what we ask students to typically do with these functions. We went from identifying points around the circle for a given angle measured in radians, to using our knowledge of 30-60-90 triangles to find the coordinates of some of these points, to formal definitions of sine, cosine, and tangent functions using these points. Every time I could, I related this idea back to the first activity of moving around the circle, but by the time we got to graphing these functions, I think I was demanding a high level of abstraction without also demanding the deliberate practice necessary to connect the angles and coordinates to each other. Students struggled to evaluate the trigonometric functions at different angles not because they couldn’t piece it together with time, but because they always felt compelled to go all the way back to the circle. I suppose it’s the trigonometric equivalent of going back to counting on your fingers.

I also was a bit disappointed to see that only a third of the class answered this question correctly on the unit exam:

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To those that recognized the similarity to our opening activity, it was quite easy. The bulk did not see it this way though.

At this stage, however, I’m not too concerned. Many students admitted immediately after the exam that they did not practice the unit circle as much as they should have. They reported that they understood much of the unit up to the graphing part, where I think I pushed them a bit more quickly to piece together the graphs than would have been ideal for them to get an intuitive sense for them. I’m confident that a second and more rigorous look at these functions next year in IB year one will help solidify some of these concepts for them.

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