Overlapping circles and area of sectors

Today was a bit rough. With this being the last full week of school, it was showing in student effort. I started by asking students to brainstorm ideas of how they could find the area of overlap between two circles. I was being much more active in cycling through the class in an attempt to keep them focused and on task. This seemed to help, but the brainstorming was not producing much.

First, several groups came up with the idea of creating an equation. Good thought, but what would you include in your equation and what would the equation attempt to model. Second, students would suggest different calculations. Okay, what do the calculations you suggest actually represent in the problem situation.

It is discouraging to see that a large portion of the class continue to just slap two numbers together in the hope it may lead to something without thinking about what their equation or calculation actually represents. Several groups looked at triangles they could form. I like the idea. Invariably, the triangles they created were not actually helpful in modeling the situation.

I used the triangle idea to identify the sectors that overlapped. I wanted to focus on finding the area of sectors and this seemed like a good way to acknowledge their ideas while putting the discussion on a better course.

The sector piece moved along slowly. First, students couldn't remember how to calculate the area of a circle. I reminded them that we had dealt with circle area before and that they should know this. Next came the issue of central angles and the corresponding arc measurement. Again, students claimed they had never seen this.

I briefly touched on their intuition as to how central angles corresponded to arc measurement. I also tried to address the percentage of the circle covered by the sector. Eventually, we were able to do things, with some help, like what is the area of a sector if the circle radius is 3 units and the central angle formed by the sector is 55^o. We could also look at what the area of the sector is when the circle radius is 3 units and the arc measurement of the sector is 120^o.

I left as homework the problem of reversing the process. Given a sector has an area of 2π and the circle has a radius of 3 units, what is the central angle formed by the sector?

We'll practice this a bit more and move into forming triangles to subtract off the non-overlapping piece of the sector. This should bring us back into using some trigonometry.