Getting back into recognizing circle equations

Today was a bit of re-orientation to recognizing circle equations. I wanted students to try to re-engage on their own as much as possible. With that in mind, I had students focus on two problems that were previously assigned as homework but with which they had little success.

The two equations were:

1. 9 = 2y - y² - 6x - x²
2. 16 + x² + y² - 8x - 6y = 0

For the first problem, students were stumped about how to get started. I know that this equation looks different because of the negative in front of the squared terms. As I walked around, I talked with students about trying to re-format the equation into something that they would feel more comfortable working with. While students liked the idea, they were still stumped. I told them that I am more used to working with equations where the squared terms were positive and I would use that as my starting point. This helped some students, but not all. For these students I had to talk about ways to manipulate the equation so that equality was maintained, either by adding and subtracting terms to both sides of the equation or by multiplying both sides of the equation by the same value, such as -1.

Students ended up with a variation of the equation -9 = -2y + y² + 6x + x², which can be written as -9 = y² -2y + x² + 6x. For some students, this was enough to get them going. For others, we re-visited the area model.

Using the x² + 6x expression, we set up the area model as follows:

x2

The two areas adjacent to the x² area need to evenly split the 6x term. Students saw this led to:

x2	3x
3x

Students saw that they needed to square the expression (x + 3). But by doing so, they had the following:

x2	3x
3x	9

(x + 3)² produced the expression x² + 6x but also added an extra 9. This got the students rolling and they were able to successfully complete the square for the y terms and find the additional value being added in the equation.

Students worked on the second equation and most were able to complete this problem successfully.

I left 3 similar problems, although somewhat easier, as homework so that they can build confidence and practice the process.

We'll hopefully complete working through problems in the next class or two, at which point we'll look at finding the area of overlap of intersecting circles. Time permitting, we'll then explore connections between circle equations and ellipses.