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 Kristin Ambrose. Her topic, from Precalculus: finding the foci of an ellipse.

How can you as a teacher create an activity or project that involves your topic?

In order to help my students visualize what foci are and the role they play in ellipses, I could do an activity that involves my students constructing ellipses given the foci. This can be done with two thumbtacks, string (tied into a loop), paper, and a pencil. What you do is place a piece of paper on top of a cork board, then stick two thumbtacks into the board and put a loop of string around them. Then take a pencil and pull the string tight, so that it makes a triangle. Then draw an ellipse by moving the pencil around the two thumbtacks, while keeping the string pulled tight to make a triangle shape. The picture below depicts how the activity should work.

I would give my students time to change the distance between the thumbtacks and create other ellipses, so that they could see how the distance between the two thumbtacks affects the shape of the ellipse. In keeping with the style of ‘discovery’ based learning, only after my students had created a few different ellipses would I explain that the thumbtacks are actually the ‘foci’ of the ellipse. I think this activity would help my students have a better visual of what foci actually are and how they affect the shape of ellipses. It would also help my students to understand why the sum of the distance between each foci and any point on the ellipse is always constant. I believe this would be a good segue into discussing how to find the foci of an ellipse.

How can this topic be used in your students’ future courses in mathematics and science?

Ellipses tend to come up in topics like Physics and Astronomy. Specifically in Astronomy, ellipses become important when learning about orbits. An orbiting satellite follows an elliptical shape around an object called the primary. The primary simply means the body being orbited and is typically located at one of the two foci of an ellipse. A good website that describes this phenomenon is http://www.braeunig.us/space/orbmech.htm. This website explains different types of orbits and how they relate to different conic sections, ellipses being one of them. In our solar system, the Earth orbits the sun, with the sun lying at one of the foci on the ellipse. In elliptical orbits, the center of mass is located at a focus of the ellipse, but since the sun contains most of the mass in our solar system, the center of mass is located almost at the sun; therefore the planets orbit the sun. Below is an illustration of this concept.

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

A really neat tool to use in mathematics is a computer application called GeoGebra. It is free to download and useful for a lot of mathematical topics. For the topic of foci and ellipses, I would have students create an ellipse in GeoGebra using the ellipse tool. Once the ellipse is created, students can grab the foci points and pull them around to change the shape of the ellipse. Students can also grab the point ‘C’ and move it around to change the shape of the ellipse. The nice thing about GeoGebra is that not only does it show the shapes and points on the graph it also states the coordinates of the points in the ‘Algebra’ section. As students are exploring the different ways they can change the shape of the ellipse, they can also see how the coordinates change. On my GeoGebra ellipse, I also added a point ‘D’ which is the center of the ellipse. I created this point by typing D = (A+B)/2 in the ‘input’ section. Once ‘D’ is created, as students move the foci around, the location of ‘D’ will change as well, so students can see how the center of the ellipse and the location of the foci are interconnected. I think this tool would be a great way to get started on the topic of how to find foci, and it helps students to visualize how the shape of the ellipse, the foci, and the center of the ellipse are all interconnected. Below are some pictures of different ellipses I created in GeoGebra.