In the past I've introduced radians by having students measure angles of a circle graph with these weird protractors and compare that to arc length and degree measure until we finally tease out what these things are. It's never smooth and it doesn't result in any major insights. The minor insight comes from this lovely geogebra applet and the understanding comes from continued practice using radians. So I reorganized this year and I'm much happier with how things went. All the notes I took during this lesson are at the bottom. I'm too lazy to give them to you page by page so they go with the correct paragraph, I'm confident in your ability to match things up.
We'd just finished As the World Turns which uses both linear and angular speed. So we compare linear and angular distance for some turns a figure skater makes. They want to know the radius, I tell them to just use r. They find a nice pattern - add one revolution and you add 360 degrees or 2πr. This pattern is not nearly so convoluted. We realize this makes sense.
I tell a nice little story about the history of 360 and how it's nice but arbitrary and the radian measure is not! Then we watch the lovely geogebra applet. It's still not obvious to them that it will take ~6.28 radii to get around but there are some aha moments when someone shares that idea. Someone asks what if the circle was a different size, I love these children! I change the size of the circle and it looks the same, a few more aha moments. We add this to our notes.
Great, now we have a thing that doesn't seem totally crazy, we really just need to practice with it. But before we jump into graphing in standard position, why not solve a little puzzle? I start out facing East (note - that's the positive x-axis aka standard position) and I make a quarter turn to the left. If you start out facing East, what other moves could you make to end up in the same position? Hm, now I'm wondering if it would have been better to just tell them I end up facing North, then ask, how did that happen? Only benefit of my initial scenario is it sets counter-clockwise as the original direction. Either way, they come up with -3/4 turn pretty easily. We express both as radians. Then I ask for more, we eventually get around to making Ms. Cardone dizzy and see that lovely 2π pattern again. A student describes the pattern beautifully and I make a big deal of it and write it nice and big on its own slide. I kept using the word co-terminal enough times and it ended up in the nice generalization.
Now we all practice drawing a few angles in standard position - start to the East/positive x-axis and turn in the direction that the quadrants are numbered. This is where I find out who was fully engaged in that class discussion we concluded just moments ago and who was nodding and smiling without taking any notes (mental or on paper). Once I'm sure that everyone can draw an angle and recognizes a fraction of π as a fraction of a half turn, then we're ready for our pi(e) eating contest!
A while back I saw this post (possibly linked by someone who was doing this activity with radians?). Having them start in standard position and then add the angles together meant they were labeling angles around the unit circle without even knowing it! I gave them the first quadrant angles with some extra π/6's so no one won too fast. Having them make the angles with the radian protractors means they're building their own wedges a la Shireen and Meg, which we'll be able to cut out and use next class. They needed the practice adding fractions so most groups were still on their first round when the bell rang (five minutes early! Possibly because we were having an assembly during advisory block or maybe just because the bells were messed up on our first advisory day). That means this whole thing took about 75 minutes (I was half way through passing out quizzes when the bell went off unexpectedly so we didn't get to the final two slides below).
Next class we'll build the first quadrant of the unit circle together (similar to second photo here) and then they'll work in pairs to extend it.
My pi(e) eating contest boards and radians measures.