# 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 Billy Harrington. His topic, from Precalculus: finding the equation of a circle.

1)     Word problems that have to do with landscaping always seem to engage the students that I am teaching. I would ask a question about a garden, a fountain, or even a gazebo. There are a variety of questions you can ask them about these topics, such as finding the total land (total area) this certain object of scenery would occupy on this piece of land pre-determined. You can even ask for different attributes of the circle if you give say the radius, diameter, or circumference, then the students can find the rest of the characteristics of the circle. Place the objects on a Cartesian coordinate plane and tell the students to identify the characteristics of the circle, and identify the radius and points of the circle to identify and discover the radius of the circle.

2)     For a full activity, I would give students a cut of out regular geometric shapes that represent different characteristics of a landscaping project. Each geometric figure represents an object that is being considered for the final product. The problem is below.

Lord Quintanilla request from the local landscaping firm called “Class of 4050”, that he want a new circular house to retire in and spend the rest of his life in with his family. His lot size is rectangular (represents the Cartesian plane), however, he wants his house to be circular. Help Lord Quintanilla find the dimensions of his house by finding the equation of the circular house on a Cartesian coordinate plane. He wants his new house to have at least 2500 square feet. Help him find the radius, and best location.

1)     Students use area in their curriculum in geometry and any upper level math classes that deal with shapes. A big topic in calculus that deals with circles is related rates. Students must understand each and every formula that deals with a circle, and they must know how to alter and manipulate each formula to fit the related rates problem. Another big section that circles are used is conics. Students must find the equation of a circle. When students are given the area of a circle, they must find the characteristics of the circle and label where the center is and find out exactly what the radius equals.

2)     Students learn the actual equation of a circle in algebra 2, however, once students learn the equation of a circle, then they can re-visit the circle sections of geometry and apply the topics to find the equations of all the different circles. To alter and make the topic more difficult, change the radius length or even change where the center of the circle is. This will help elicit higher level thinking to help students determine the changes to the equation.

1)     Students can use technology by using either their calculators or even by using their computers to graph and calculate the different characteristics of the circle. A great website to show circle characteristics is http://www.geogebra.com . This website is a great geometry website that shows many of the characteristics of the shape. Using this website, it can show how different characteristics of the circle, such as the radius or circumference are changed, when you increase or decrease the diameter. This website is a great website to visually show how a circle is altered when you change one of the measurements of the circle.

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