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 Christine Gines. Her topic, from Precalculus: finding the equation of a parabola from the focus and directrix.

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

Beginning the class with a short clip involving a certain topic is a great way to start and engage a classroom for several reasons. First of all, videos can achieve things that a teacher can’t in a limited classroom. Also, videos save preparation time for the teacher and students just like watching videos in general! Youtube.com is, if not the, one of the largest video sharing websites. You can find videos on just about any topic and for this reason, I recommend it.

On youtube I found a great introductory video parabola involving the focus of a parabola. This video does a fantastic job of engaging viewers by demonstrating the effects of concentrated sunlight by melting metal, stone, and setting wood on fire. Not only does this video grasp students’ attention, but it also raises a sense of curiosity by not explaining what is happening. After watching, students will ideally be eager to find answers at which point the teacher could introduce the topic and let students explore their questions.

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

Deriving the equation of a parabola may seem like a procedural concept, but it doesn’t have to be. The following activity is an example of how you can let students explore this concept visually and kinesthetically.

The only materials you will need are wax paper and pencils for each student. The instructions are as follows:

- Draw a line about 2cm above the edge of the wax paper.
- Fix a point above the line
- Draw several point on the fixed line
- Fold each point on the line so that it touches the fixed point above the line

This is what the activity should look like:

This activity lets students explore the relationship between the directrix and the focus. A good follow-up to this activity is a peer-to-peer discussion of why a parabola was created. Ask them questions like, “Where is the vertex of this parabola in relation to the line and fixed point?” or “Can you find a relationship between this activity and the video that melted stone?” The activity benefits all types of learners and challenges students to find a deeper understanding, rather than simply following algebraic steps

How did people’s conception of this topic change over time?

The discovery of the conics section can be traced back to Ancient Greece, when Menaechmus (pupil of Eudoxus and tutor of Alexander the Great) was puzzled with mathematical problem of doubling a cube. While attempting to solve this problem, Menaechmus discovered the conics section. This happened around 360-350 B.C. He was also the first to demonstrate that parabolas can be obtained by cutting a cone in a plane that was not parallel to the base, like so:

Parabolas at this time were only a mathematical concept to be studied and not put to use in the real world. It wasn’t until Pappus came along and discovered the focus and directrix property, that parabolas were noticed for their practical use. This discovery led to many applications of parabolas. Just a few examples include telescopes, satellites, microphones, and even bridges.