Additional practice with circle equations

Today started with me passing back the self assessment problems I had collected last class. I discussed the two major issues that I saw while going through the work: switching signs for the circle's center and not adding additional value when completing the square.

With that, I had students work on the last two problems on the self-assessment sheet, since no one had got that far. The first problem dealt with showing why two circles with given equations were externally tangent. The second dealt with the idea of picking a random point inside a circle and determining how large a radius it could have while still being completely enclosed inside the circle.

Students had questions about the wording of the first problem. Specifically, they were having trouble understanding what the problem meant by externally tangent. A drawing helped clarify this point. A second issue arose from how to determine that the circles were externally tangent. It took a while for many students to realize they needed to compare the distance between the two centers to the sum of the radii of the two circles. Some students still did not understand this point.

I wanted students to consider what they would need to use or to do to help determine the size of an internal circle centered at a random point A. Many students recognized that the radius for a circle centered at A would have to become smaller as A moved toward any point lying on the circle. They also recognized that the radius would become larger as point A moved closer to the center of the original circle.

At this point, I wanted students to practice more with circle equation problems. I gave students the seven Circle Challenge problems from the MVP curriculum. Students worked on these problems in class. I left the completion of the problems as homework, which I'll collect at the start of next class.

We'll start looking at the area of overlapping circles next class. This will enable students to build on the ideas of externally tangent circles and circles lying within circles. We'll also be able to look at solving quadratic equations, use some trigonometry, central angles, area of sectors, and review area of triangles.