I don’t think there are any topics in high school math with more tricks and gimmicks than completing a blank unit circle. Don’t believe me? Search Unit Circle Tricks on YouTube. There are videos with millions of views, teaching students how to blindly complete a unit circle. I will admit, showing students these tricks does help them to memorize a unit circle. But what is the point of memorizing a bunch of fill-in-the-blank answers if there is no understanding of what they mean?
The good news is you don’t have to choose between understanding and accuracy. You can teach the unit circle so that students don’t have to memorize AND they develop a conceptual understanding of the unit circle. Students will be able to reason their way through identifying all the angles and coordinates BEFORE they ever see a completed unit circle.
So how are we going to make this happen? Special right triangles.
Turns out the Unit Circle is just a whole bunch of special right triangles! Did you know that? I didn’t, and it truly blew my mind when I realized it. But why does that matter? Well, with a couple of carefully chosen special triangles, you can find all angles and coordinates of the unit circle. Let’s look at how.
Spotlight Lesson:
It’s important to know what prerequisite knowledge students need for this lesson. In our Algebra 2 Trigonometry unit, students have just gone over special right triangles and angles in standard position on the coordinate plane in the previous two lessons. They have not learned about radians yet so we only fill in the unit circle with degrees today and will come back to fill in the radians later.
Students will start out the activity by finding sides lengths for a 30-60-90 triangle and 45-45-90 triangle that both have a hypotenuse of 1. (Use this unit circle and set of triangles). Once they’ve done that, they should cut out the set of triangles that you give them and label all sides and all angles on both sides of the paper. Color coding the sides is really helpful for seeing the patterns. Students would need 4 different colors to do this.
Students will use these triangles to fill in the angles and coordinates for all of the points marked on the unit circle by fitting the angles of the triangle into the reference angle of the circle and then using the side lengths of the triangle to determine the x and y coordinates. You’ll want to model how this works for at least two of the angles, if not three. If possible, project your handout and triangle to show how you are maneuvering the triangle while you explain your thinking.
"Let’s look at this first angle. I can see that my 30-60-90 triangle will fit in here if I put the 30-degree angle at the origin. So this angle must be 30 degrees. Now I need to fill in the ordered pair, so I need to find the x and y. Well, I know that x represents the horizontal distance from the origin. So how far over from the origin have I gone? How long is this width? (Show with your fingers what length you’re talking about. Try to get students to notice that we already have this width.) Oh, well the distance over is the same as this leg of my triangle which I know is square root of 3 over 2. So the x value is square root of 3 over 2! Now what about the y? That is how high up we have gone. I can see that’s the other side of my triangle which is ½ so y is ½."
Next, I would Think Aloud through an angle that doesn’t have 30 degrees as the reference angle. I like doing the 60-degree angle because students don’t always notice that they can turn their triangle that way.
I generally like to show at least one more angle that requires reflecting the triangle, like the 135-degree angle. During your modeling, you’ll want to talk through how you found the angle by being 45 degrees short of 180 so the angle must be 135. You’ll also should mention that we are now to the left of the origin so the x value is negative.
After modeling the thinking for these examples, give students time to manipulate their triangles and find all of the missing sides and angles. By the end of the activity, they’ll not only have a beautiful unit circle, but they’ll also be able to reason their way to finding any of the angles and coordinates. No memorizing required!
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