Teaching Math Connections Between a Circle and an Ellipse in a Conics Unit

Whether teaching conics in Algebra I or Algebra II, teachers have at their disposal a process for comparing circles with ellipses. Realistically, a circle is a type of ellipse in which the two foci have merged together into one point. Therefore, if a student becomes comfortable working with a circle equation, it is a fairly simple conceptual leap to have them understand ellipses. This article explores the circle and ellipse equations, and presents a teaching strategy for increasing student depth of knowledge about these conics.

Circles
First of all, the equation of a circle at center, (h,k), is (x-h)^2 + (y-k)^2 = r^2. Given a center and a radius value, students typically can substitute correctly to obtain the equation of a specific circle. They then can graph the circle by plotting the center point, and counting by the radius in the four major directions of up, right, down, and left. Finally, students can draw rounded lines connecting these four points to obtain their circle.

Pitfalls to watch out for with circle concepts
Typically where students go wrong with setting up circle equations is they incorrectly subtract a center's point if one or both of the point's coordinates are negative. Reminding students that subtraction means to "add the opposite" often helps rectify this problem.

General ellipse concept
If possible, show a video from Discovery Learning or another video website to illustrate the major conics concept of an ellipse. As a hollow double napped cone is intersected with a laser in various directions, conic sections are formed. When the laser is perpendicular to the vertical axis of the cones, a circle is formed. However, when the laser is moved slightly from perpendicular, an ellipse occurs. It also helps to remind students that ellipses are important when studying astronomy as many planets and space bodies follow elliptical orbits.

Details of ellipses
The specific equation of an ellipse centered at the origin (easiest to teach students) is (x^2 / a^2) + (y^2 / b^2) = 1 or (x^2 / b^2) + (y^2 / a^2) = 1. 'a' and '-a' are the endpoints of the major axis. 'b' and '-b' are the endpoints of the minor axis. The foci, denoted by 'c', are found by the positive and negative values of the square root of a^2 minus b^2. Foci are placed on the major axis. Yes, this sounds detailed and difficult. These formulas typically seem complicated to students as well.

Making the connection
To better understand these conic sections, begin by teaching students to graph the circle or ellipse from an equation. Show students how to square root the ellipse's denominators to find the vertices. The x-intercept is underneath the x^2, and the y-intercept is underneath the y^2. Similarly, teach students to square root the number at the right of a circle equation to find the radius. Thus, they can see how a square root is involved with calculations of both types of conics.

Another way to connect the mathematics between the circle and ellipse is to write a circle's general equation. Divide all terms by the number on the right of the equal sign to make the equation's right side equal to one. This visually shows students that the denominators of a^2 and b^2 of an ellipse would be the same in a circle, hence becoming a radius.

To finalize the conceptual understanding of a circle and an ellipse, have students work backwards. Have them generalize an equation from a graph. A great assignment includes students accurately drawing five circles, five vertical ellipses, and five horizontal ellipses. Then students trade papers and find the equations of their partners' conics. They can trade back and the original conic designer can grade the equations for accuracy. This process helps foster a great dialogue between students about conics and how to accurately work with them.

Overall, conics do not need to be a difficult concept to teach. Also, students tend to enjoy drawing quadratic relations that are difficult for calculators to draw. Sketching conics such as the circle and ellipse encourages students to actively participate in their graphing, as it is easier than using their calculator.

Have you ever learned about ellipses and circles? Have you used a similar technique for teaching the relation between a circle and ellipse? Please share your thoughts below.