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. I generally like to show at least one more angle that requires reflecting the triangle, like the 135-degree angle. I like doing the 60-degree angle because students don’t always notice that they can turn their triangle that way. Next, I would Think Aloud through an angle that doesn’t have 30 degrees as the reference angle. I can see that’s the other side of my triangle which is ½ so y is ½." So the x value is square root of 3 over 2! Now what about the y? That is how high up we have gone. 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 how far over from the origin have I gone? How long is this width? (Show with your fingers what length you’re talking about. Well, I know that x represents the horizontal distance from the origin. Now I need to fill in the ordered pair, so I need to find the x and y. I can see that my 30-60-90 triangle will fit in here if I put the 30-degree angle at the origin. If possible, project your handout and triangle to show how you are maneuvering the triangle while you explain your thinking. You’ll want to model how this works for at least two of the angles, if not three. 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. Students would need 4 different colors to do this. Color coding the sides is really helpful for seeing the patterns. 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. ( Use this unit circle and set of triangles). ![]() 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. 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. 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. It’s important to know what prerequisite knowledge students need for this lesson. Spotlight Lesson: Algebra 2 Lesson 9.6: The Unit Circle 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. ![]() 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. So how are we going to make this happen? Special right triangles. Students will be able to reason their way through identifying all the angles and coordinates BEFORE they ever see a completed unit circle. You can teach the unit circle so that students don’t have to memorize AND they develop a conceptual understanding of the unit circle. The good news is you don’t have to choose between understanding and accuracy. But what is the point of memorizing a bunch of fill-in-the-blank answers if there is no understanding of what they mean? I will admit, showing students these tricks does help them to memorize a unit circle. There are videos with millions of views, teaching students how to blindly complete a unit circle. Don’t believe me? Search Unit Circle Tricks on YouTube. I don’t think there are any topics in high school math with more tricks and gimmicks than completing a blank unit circle.
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