Writing equations of circles can be taught many different ways in high school Geometry classrooms. However, in my opinion, the best way to teach the standard equation of a circle is through student’s discovering it for themselves. The Bill and Melinda Gates foundation has poured a countless amount of money into different ways to educate America’s future and one lesson that was funded from this money involved writing equations of circles.
The lesson begins with a circle with a center at the origin of a graph (0,0) and it has a radius of 5. It clearly intersects the circle at the points (5,0), (-5,0), (0,5), and (0,-5). The students are asked to come up with points on the circle other than these obvious points. It should be noted that coming into this lesson, students should have a firm understanding of the distance formula and the Pythagorean Theorem. Most students will not have a problem identify the point (3,4) on the circle using the common 3,4,5 Pythagorean Triple. If students have trouble seeing this have a student come to the front of the classroom and demonstrate their work to their classmates. After seeing an example have students come up with several other points on the circle. It should be plain for them to see after finding enough points that the equation of the circle is x^2 + y^2 = 25.
At this point, the circle’s center should be shifted to another whole number point on the grid although the size of the circle remains unchanged. Either the teacher or the students need to identify that the equation x^2 + y^2 = 1 no longer works for all of the points on the circle. It may take a little bit of time, but students will eventually begin to see the how the new equation will be formed. If students do not see it on their own, then it will be a good idea to finally show them the standard equation of a circle is indeed: (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the coordinates of the center of the circle. This is a perfect lesson that any teacher can make student centered, rather than dictating to them what is so.