Engaging students: Graphing an ellipse

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 Donna House. Her topic, from Precalculus: graphing an ellipse.

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

A great hands-on activity for learning about an ellipse is created with some cardboard, a string, some tape, a yarn needle (or something to make a hole in the cardboard), and a marker.

To create the “ellipse boards,” take a piece of cardboard about one foot by one foot. You do not have to use a square piece and it can be larger or smaller. Just make certain the cardboard is large enough for the graph to be clearly seen. (I prefer white cardboard because it is easier to see the marks, but regular cardboard will also work if you use dark markers.)

Next, using the marker, make two marks for the foci. Thread the string (or yarn) through the yarn needle and poke a hole through one of the foci, pulling the string to the back side of the cardboard. Tie a knot in the string and tape it to the back of the board.

Now, thread the other end of the string through the yarn needle and poke a hole through the other focus. Decide how long the string needs to be to create a nice ellipse. (Remember the string must be 2a long – whatever length that is. Unless you really want the ellipse to be a certain size, the length of the string can vary. The farther apart the foci are, the more elongated the ellipse will be. This can also lead to a discussion about what happens to the shape of the ellipse as the foci get very close to each other!) Make certain the drawing will not fall off the edge of the board. Then tie a knot in this end of the string and tape it down. Each ellipse board will have a different sized ellipse unless you VERY carefully measure the foci and the string. I think having different sizes is better (and much easier to do) and shows the students that the formula for an ellipse works. Now the boards are ready for the students! (The students can put together their ellipse boards in class or you can have them pre-made to save time.)

The fun part is the actual drawing of the ellipse. This, however, is not as easy as it looks! To draw the ellipse, use the marker to stretch the string taut and let the string guide your drawing. Be sure to draw one before class so you will be able to give the students suggestions as they draw their own ellipses.

On their boards, the students can find the center, draw the major and minor axes, can find the vertices, and can easily see that the foci are on the major axis. Using the string, you can prove that the sum of the distances from any point on the ellipse to each of the foci is always 2a, and, using the Pythagorean Theorem, the students can see how to find the foci.

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

            Since an ellipse is created when a cylinder is cut at an angle, ellipses are commonly encountered in construction. An example is creating a right angle while joining two pipes to build the corner of a fence. One joining method is to cut each pipe at a 45° angle then weld them together. Students could be asked to determine the length of the major and minor axes of the resulting ellipse when a 2” diameter pipe is cut at a 45° angle.

This same idea is used to make holes in walls or tile for some light fixtures, plumbing fixtures (like shower heads), vent pipes, etc.

I also found the following class project. This could be done in small groups by giving each group the main problem and letting them brainstorm to come up with the solution. I think this would be wonderful to stimulate creativity in the classroom.

http://www.pleacher.com/mp/mlessons/calculus/appellip.html

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

To engage the students, begin by showing the first 3 1/2 minutes or so of this video from YouTube:

https://www.youtube.com/watch?v=Yl8Qy79aLk0

Note that the doctor actually touches the peppermint while the sound waves are on!

But what does this have to do with an ellipse?

A unique characteristic of the ellipse is that shockwaves emitted from one focus will
reflect off the ellipse and go through the other focus. Using this characteristic, medical engineers have created a device called a lithotripter (as shown in the video) which can break up kidney and gall stones with minimal damage to the surrounding tissue. This eliminates the need for traditional surgery. Mathematics continues to make life easier!

As illustrated in the diagram above, when an energy ray reflects off a surface, the angle of incidence is equal to the angle of reflection.

Here is a short article explaining how the medical device works. (The above illustration comes from this article.) Using the computer, project the article onto the screen to show to the class.

http://mathcentral.uregina.ca/beyond/articles/Lithotripsy/lithotripsy1.html

This not only shows how technology can be used to engage students, it also shows how this topic is used in technology!