Last year my kids struggled with visualizing the placement of radian angles in standard position. They either had to convert (say 7 pi / 6) into degrees and then draw the angle accurately, or they just shook their heads at the crazy things their teacher asked them to do.
Over the summer my retired teacher friend mentioned that he used to teach the pi/6 and pi/3 parts by having them look at the clock and noticing that (say) from 3 to 9 (moving counter clockwise as a positive angle does) there are 30 minutes. Break that into 6 parts. They notice that's on "the hours" 2, 1, 12, 11, 10. Then the kids go back to their unit circle and place dots all around signifying the "hours" and those are the pi/6 separations. Then they go from there. Similarly, from 3 to 9 counterclockwise (a full pi), if we want to break it into 3 parts (pi / 3), that's every 10 minutes.
I had great success on Wednesday with them accurately placing the angles without converting. Woot. Woot. Hopefully, it'll stick in their heads better this year.
I also didn't do a great job last year with having them memorize thoroughly (and "get") sine and cosine of any multiple of 45 degrees or 30 / 60 degrees. They still had to work at it without immediately being able to answer. This year, I'm going to laminate page size unit circles. Then everyone gets laminated 30-60-90 and 45-45-90 triangles that have the side lengths and angles in degrees and radians on both sides and that fit nicely on the circle. Then we'll do an activity where they will place the reference triangles on the circle for various angles I give them. Hopefully, this will allow them to later on picture in their heads the height/placement of the reference triangle and quickly be able to recreate sin 135 or cos 5pi/6. We'll see.
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