Teaching Circles

Inspired by this post, I wanted to share a little bit about how I teach circles. When I pose the questions that follow, I usually use PollEverywhere to survey the entire class at the same time. I find this encourages table discussion. Keep in mind, when I show the students these equations, students HAVE seen the Pythagorean theorem, but other than that, this is the first time they have seen anything resembling the equation for a circle. The answer is not at all immediately clear and many students take a "guess and check" approach to finding the solution

Question 1

Which of the following points satisfies the above equation?

Once I have collected responses from students (the majority do get this question correct) I ask how they came up with it and I also ask if there is a "quick way" of determining the answer, without any calculation or multiple choice. Then I give this question. Notice how the multiple choice contains only those which are horizontally or vertically positioned from the center.

Question 2

Which of the following points satisfies the above equation?

A. I only

B. II only

C. III only

D. II and IV only

E. I, II, and IV only

Again, we discuss how students came up with their response and then I ask,

Does this equation resemble anything you have seen before?

Eventually, we are able to see it is something like the Pythagorean theorem. Soon, we talk through how the equation for a circle is like the equation for all points that would be the endpoint of the hypotenuse of a triangle whose base is parallel to the x-axis. But I think that discussion is covered well by carmelschettino