Working with sectors

Today was focused on having students become more comfortable working with sectors. As usual, this took a lot longer than I had hoped. The good news was that most of the class seemed to be getting more comfortable working with sectors.

We started by looking at the homework problem: if a sector has an area of 2π and the circle has a radius of 3, what is the arc measurement of the circle? The issue students faced was how to find the portion of the circle covered by the sector and how to convert this portion to the corresponding degrees.

Fundamentally, students were not recognizing that the problem 9π x ___ = 2π, was the same as solving for x when you have the equation ax = b. Students knew they should divide to find x = a/b. For some reason, having values with π in the expression baffled them.

Once they realized these were the same problem, they found the sector covered 2/9 of the circle. The next struggle was to translate the 2/9 of the circle into an equivalent number of degrees. The first suggestion was to divide 360^o by 2/9. I asked students to do this and to see if this made sense. Students quickly saw that the result didn't make sense. When students multiplied 360^o by 2/9 they got an answer of 80^o, which did make sense.

I then gave the class two problems to work on, one in which the radius and arc measurement were given and they needed to find the area of the sector, and one in which the radius and the area of the sector were given and they needed to find the arc measurement.

These problems went slowly with different struggles occurring for different students. Simplifying expressions such as 5π / 25π to 5 / 25π were not uncommon. There is also a tendency to take two given values and either multiply or divide the values. When asked what the calculation represents, the typical answer is, "I don't know." I had to continue to reinforce the idea that a calculation should represent the physical reality.

After working through these two problems, we had some time left. I wanted students to get more comfortable with the work they were doing, so I provided two additional problems, changing the radii, area sizes, and arc measurement values. I provided easily divisible values, for example a sector with area 3π and radius 6 or a circle with radius 2 and a sector with arc measurement of 45^o.

Many students finished one or both of the questions in the last few minutes of class. We'll go through the answers and then focus on the overlap of the two sectors and dissecting those into component parts.

We're supposed to have an assessment on the circle unit tomorrow. I think I am going to make this a take-home assessment in order to allow time to work on overlapping.

There are only two more classes before the final exam. I won't get through transitioning from circles to functional transformations. I'll outline that once I finish my work with circles.