Practicing with equations of circles

My intent for the next couple of classes is to allow students to get comfortable working with equations of circles. To this end, I had students work on problems 1-4 and 9-14 of the practice sheet (pages 6 and 7 from Michelle Bousquet's Equation of a Circle lesson plan).

Before working through these problems, I displayed the answers to the four examples we worked through last class. The main question focused on example one; how do you find the equation of a circle when given just the end points of a diameter. One student presented their approach, which was to graph the points out and work from there. We revisited how to find mid-points and worked through the example again.

There were a few common issues that arose on the assigned problems; this was not unexpected. First, when given a value for the square of the radius, many students initially treated this as the radius. I had students reference the general equation of a circle that they had recorded yesterday. For the few that were still unclear, I reminded them that the equation says that we have the square of the radius. This seemed to be enough of a nudge to get them going.

Some students wanted to pull the center coordinates directly from the equation, so (x - 2)² + (y + 3)² would have a center of (-2, 3). I worked with these students on relating how to translate the new center back to (0, 0). So, if the center were at (-2, 3), would subtracting 2 from -2 and adding 3 to 3 move the center back to (0, 0). Most students seemed to understand this idea, but a few continued to struggle through a couple more problems.

Problem 11 presented the same situation as example 1 from yesterday. I would point out these were the same and that we had already discussed how to work through the problem.

I had to explain what tangent meant in the sense of problems 12 and 13. Students were trying to make sense of the opposite side/ adjacent side definition that was used in right triangle trig. After explaining the idea of a line tangent to a circle, I would tell them that the line was acting, in essence, as a boundary for the circle. Students seemed comfortable with this and appeared to be able to proceed ahead successfully.

The final question came up on problem 4 where the square of the radius is 14. Students didn't feel comfortable with the radius being √14, but understood that this was what the radius must be.

Questions 5-8 require students to complete the square, which we have not covered yet. We'll get there, but I still want the class to become even more comfortable with what they are doing with the equations of circles. It will be easier for them to reverse the process if they know what form they are trying to achieve.