# Engaging students: Finding the equation of a circle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Lucy Grimmett. Her topic, from Precalculus: finding the equation of a circle.

How could you as a teacher create an activity or project that involved your topic?

I love doing activities to teach topics, or solidify students knowledge after direct teaches. The link below describes an activity/project a teacher did with her students. The students were asked to create an image using circles. They can use other images along with circles; however, circles are the main focus of the project. The picture had to include at least four circles. Once they had drawn their image containing the circles they were asked to find the equations of each of the circles in their picture. As an extra challenge students were asked to create a question to go with one of their circles that would aid another student in finding the equation.  This is where students have to truly put two-and-two together to create an in depth connection between the lesson, word problems, and furthermore the idea of the center of a circle and the length of the radius.

http://secondarymissrudolph.blogspot.com/2012/04/equations-of-circles-update.html

How does this topic extend what your students should have learned in previous courses?

In Algebra II students learn about writing equations of graphs and how each piece of the equations manipulates the graph. This is a skill that continues into Pre-Calculus. Whether students are graphing circles, exponential functions, or trigonometric equations there is always variables that can be manipulated that manipulate the graph. Further, students will be required to find equations of hyperbolas, ellipses, and parabolas. These equations go hand and hand with one another when students are using a focus and directrix. As you know, mathematics constantly builds on itself. With students previous knowledge of quadratic equations specifically, they see how in the equation $y = a(x-h)^2 + k$, $(h,k)$ is the “center” of the graph. The same goes for the equation of a circle. The point $(h,k)$ is the literal center of the circle in the formula $(x-h)^2+(y-k)^2=r^2$.  With their previous knowledge of h affecting the x component and k affecting the y, students are able to grasp the concept more quickly, and more efficiently.

How does this topic extend what your students should have learned in previous courses?

In Algebra II students learn about writing equations of graphs and how each piece of the equations manipulates the graph. This is a skill that continues into Pre-Calculus. Whether students are graphing circles, exponential functions, or trigonometric equations there is always variables that can be manipulated that manipulate the graph. Further, students will be required to find equations of hyperbolas, ellipses, and parabolas. These equations go hand and hand with one another when students are using a focus and directrix. As you know, mathematics constantly builds on itself. With students previous knowledge of quadratic equations specifically, they see how in the equation $y = a(x-h)^2 +k$, $(h,k)$  is the “center” of the graph. The same goes for the equation of a circle. The point $(h,k)$ is the literal center of the circle in the formula  $(x-h)^2+(y-k)^2 = r^2$.  With their previous knowledge of h affecting the x component and k affecting the y, students are able to grasp the concept more quickly, and more efficiently.

How can be used to effectively engage students with this topic?

There are so many online resources to use for mathematics teaching. Students have easy access to an online graphing calculator called Desmos. This allows students to play with different numbers in equations to see how they affect the graph. For examples, students can manipulate the radius and the center point. This will allow students to visually see how each variable contributes to the equation. Below are other links that are beneficial when teaching equations of circles. Khan Academy is a tool that is used by many educators, not only does Khan Academy include instructional videos, but it also has mini quizzes after that check for student knowledge. The second link below is an online resource which student can insert equations of circles and the program will give them the radius, center, a graph, and it will even give an explanation. There is a second mode on this resource which student put the radius and center and the site will return the equation, again, the website will give an explanation if needed. These resources are quick ways for student to see how the equation of a circle can change different pieces of circles.