Engaging students: Parabolas

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 Claire McMahon. Her topic, from Precalculus: parabolas.

green line

The parabola took a long time to get to us and took a few thinkers to really get the idea down.  This website that I found really nailed the dates and also simplified the rational that led up to the parabola as we know it today.  The history of the parabola is as follows:

The parabola was explored by Menaechmus (380 BC to 320 BC), who was a pupil of Plato and Eudoxus. He was trying to duplicate the cube by finding the side of the cube that has an area double the cube. Instead, Menaechmus solved it by finding the intersection of the two parabolas x2=y and y2=2x. Euclid (325 BC to 265 BC) wrote about the parabola. Apollonius (262 BC to 190 BC) named the parabola. Pappus (290 to 350) considered the focus and directrix of the parabola. Pascal (1623 to 1662) considered the parabola as a projection of a circle. Galileo (1564 to 1642) showed that projectiles falling under uniform gravity follow parabolic paths. Gregory (1638 to 1675) and Newton (1643 to 1727) considered the properties of a parabola.

This really got me to thinking what it really took to figure out the derivation of the formula and even for the graph of the parabola.  I find it interesting that the idea had to travel through seven genius minds to come to all of the properties that the parabola holds to this day.

green line

This same website led me to another use of the parabola, other than to describe a projectile’s path.  The use of suspension bridges relies heavily on a parabolic model.  Other parabolic models would include the satellite dishes and even all types of lights.  Have you ever thought that every single place that light bulb reflects is a reflection off a point from the focus to the parabola to create your beam of light!!  Pretty cool!!  So you might ask why do I need to know anything about parabolas?  There is your answer; it’s used in everyday life.  Here are a couple of examples from the website that I found interesting:

One of the “real world” applications of parabolas involves the concept of a 3D parabolic reflector in which a parabola is revolved about its axis (the line segment joining the vertex and focus). The shape of car headlights, mirrors in reflecting telescopes, and television and radio antennae (such as the one below) all utilize this property.

Antenna of a Radio Telescope

All incoming rays parallel to the axis of the parabola are reflected through the focus.

Flashlights & Headlights

In terms of a car headlight, this property is used to reflect the light rays emanating from the focus of the parabola (where the actual light bulb is located) in parallel rays.

Here are the specs on the suspension bridge:

Hold up a chain by both ends and you’ll get a curve. What kind of curve is it? You might say it is a parabola – Galileo Galili believed it was a parabola. Yet, Galileo was wrong!!!! That curve is NOT a parabola. It is a catenary.It makes sense that you would think that the curved chain is a parabola. Both the catenary and the parabola have similar properties. Both curves have a single low point. They both have a vertical line of symmetry, they at least appear to be continuous and differentiable throughout, and the slope is steeper as we move away from the low point, but it never becomes vertical.So, how is the curve of the cable in a suspension bridge a parabola? When the structure is being built and the main cables are attached to the towers, the curve is a catenary. But when the cables are attached to the deck with hangers, it is no longer a catenary. The curve of the cables become the curve of a parabola. Unlike the catenary, which is curving under its own weight, the parabola is curving not just under its own weight, but also curving from holding up the weight of the deck. The cable of a suspension bridge is under tension from holding up the bridge.Therefore, the cables of a suspension bridge is a parabola, because the weight of the deck is equally distributed on the curve.

I never really knew that there was a difference between the two and now I know that there are certain properties that made it down through the ages that hold true today.  This was a very enlightening subject matter.

Website used: http://www.carondelet.pvt.k12.ca.us/Family/Math/03210/page2.htm

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