Combining Constructions and Radians

We ran out of time in our last topic of circles to go over circle constructions and radians. I felt it was best use of time to give a test on a day I had to be out, so I cut those two topics from the test. I have some more freedom with our next unit, so I had put together student notes for each of those cut concepts, treating them as separate ideas with radians coming first and then constructions. A couple of days before the lesson, I was struck with an idea. Part of the constructions that students were doing was to find the center of a circle. Then for two of the constructions they also had to work with the radius. Why not combine practice finding the center of a circle and the radius with explaining the concept of radians?

I ripped apart their notes and just gave them the construction instructions first. Several students asked why there were staple holes in the one sheet of paper I gave them. I explained that I had changed my mind about how we were learning our last two topics. Once we had gone over how to find the center of a circle, and constructing inscribed squares, equilateral triangles, and regular hexagons, I gave them a cardboard circle from the top of a pie tin. I asked them to use constructions to find the center of this circle. Then I explained that a radian is another way to measure an angle and that one radian is the size a central angle has to open to intercept an arc the length of the radius.

I had them use a measuring tape to measure the length of the radius and then curve the measuring tape around the outside of their circle, repeatedly marking the length of the radius around circle until they ran out of room. I asked each group how many radii they could fit around the circle. Fortunately, they all said 6 with a little bit of room left over.

We talked about the circumference of a unit circle and that it would be 2𝜋. Then we estimated what 2𝜋 was--about 6.28.  Light bulbs started for some, that the little bit of left over on their circle represented the .28 of 6.28.  Then I handed out their radian notes and we talked about how this idea related to the formula for arc length with radians.

I threw a lot at them today, but I am hoping that it was the start of a solid understanding of radians as they prepare to go into Algebra 2 next year.

Here are the notes that I gave them to go along with this lesson.  I gave them the constructions page first though, and then the radians and arc length pages.  The questions included are from old Geometry Regents exams.