So last year I began to experiment with sectors to build up arguments that arc-length and radius determine the angle. I saw some success, but I had left it too unstructured for the students to reach a meaningful conclusion. When I tried to make a jump to radians I got a lots of looks that said, "I got what we just did, but what does THIS have to do with THAT?"
At this point I asked them to create an equation relating arc-length, radius and the angle. A few arguments broke out (Awesome!), a few questions were asked and virtually all of the students wrote down something akin to:
From this I asked the question "if you know the arc-length and the radius can you tell me the angle?" I had the students draw a sector whose arc-length and radius ratio was 2 and then measure the angle... Each student had a different sized sector but all had the same angle. This lead to a conversation about ratios where a student pointed out that:
With a little math teacher magic, and no mention of radians, by the end of the 65-minute period I had most of the students agreeing that:
The next class we talked about the "arc-length" of a full circle. We worked our way around the unit circle talking about half and quarter circles before jumping into smaller angles and coming up with radian equivalents for the typical 30-45-60 angles. Only at the end did I finally mention the name "radian."
I've followed this up with a few "Pop Quizzes" that were not graded, but just a bit of practice estimating and sketching angles in radians. Quiz 1 and Quiz 2.
The result is the least confused class I've ever had. Dealing with radians still takes some effort, but they seem to get it.
My handout with the sectors can be found here via Google Docs. The rest of my trig bits (a work in progress - isn't everything?) are in a Google Doc/Drive collection.