I was really happy with my lesson plan today, but it was only something of a success in one of my three classes. Reasons why are more focused on student behavior which is not something I wanted to document here, so I'll focus on the ideas.
In my experience teaching geometry, students have no real understanding for what an angle is. Asked to solve for an angle of a right triangle and they'll provide the length of a side (and vice versa). Asked to provide units for an angle and they'll say "feet." When I created this new curriculum over the summer, one of my primary goals was the importance of angles as measures of rotation.
So I started class with a demo involving 8 students in a circle around me. I used meter sticks to divide up the circle and talked about fractions. From there I expand the idea of fractions to describe how far I would turn to face certain people. And finally I had the circle enlarged to emphasize the idea that rotational measures are independent of distance measures.
Then we discussed distance measures and their units, since most students are familiar with those ideas. We then went back to rotational measures and probed the idea of units. Some students suggested percents (which was awesome), but it took a LOT of probing to get the students to say "degrees" (exactly my point). They've all heard of angles & degrees before, but had seemingly never connected those terms to the idea of rotation.
We then worked through some examples of setting up proportions using first fractional turns, then degrees in a circle, and finally converting to percents.
I had a nice side story prepared about why there are 360 degrees in a circle, but we were unable to get to it in two of the classes today. When simply asked about it, most students said "because 1/4 turn is a right angle which is 90 deg, so 4 * 90 = 360." Ok, so where did 90 deg come from? At that point, they were happy to reason in circles without seeing the missing explanation of where 360 even comes from. Oh well.