The Wonderful World of Conics

My Algebra Two students are in the thick of learning about conics – parabolas, ellipses, circles, and hyperbolas. I’ve just finished recording a series of screencasts about them, and I am posting them below for your enjoyment.

Why are they called conics? Well, we can construct them using cones, as illustrated by this demo, courtesy of  Irina Boyadzhiev via GeogebraTube (I wish I could embed it, but WordPress won’t allow it):

https://ggbm.at/T8TV2JqG

Here are my screencasts:

Conic Basics

The Parabola

The Ellipse

The Hyperbola

Conics Hide and Seek

We just wrapped up our study of conic sections, which can be a pretty dry topic. So to liven things up, I had my precalculus students go on a scavenger hunt. These days, everyone has either a digital camera or phone with a camera, so everybody could participate. Here is the handout I gave them outlining the rules:

Conics Hide and Seek

Names____________________________________

Math is all around us, and in this activity, you are going to find some places where it is hiding. We have finished our study of conic sections (parabolas, circles, ellipses, and hyperbolas), so it’s time to have a scavenger hunt! Using your phone or digital camera, you and a partner will explore the campus and take a picture of at least one example of each type of conic. The team with the most points will get a prize. You are on your honor not to share your finds with other teams.

Here’s how many points each type of conic is worth:

Circle: 1 pt (maximum of 5 examples)

Ellipse: 2 pts

Parabola: 3 pts (remember, a parabola is not the same as a “U”!)

Hyperbola: 5 pts

You can submit your photos via email. Happy hunting!

They spent more than half of an 80-minute block combing the campus for examples of conics. Did this activity involve rigorous mathematics? No, but it was a lot of fun for the girls, and it opened their eyes to some of the ways math can describe the world around them. When we reviewed the teams’ submissions, there was a lot of discussion about whether certain shapes actually were parabolic, or ellipsoid, etc. All in all, a very useful activity.

Here is a sampler of the best submissions. The winning team took over 100 photos!

When Are We Ever Gonna Use This?

How many times have math teachers heard that question? I question the assumption underlying it – that math should only be learned if it has “real-life” application. I wonder if my colleagues who teach literature have to deal with that! Of course, math is worth studying in and of itself, just as poetry is.

That said, I do try to make connections between abstract mathematical concepts and things my students encounter in their lives. So, as my precalculus students wrapped up their investigations into parabolas, ellipses, and hyperbolas, we looked at some examples of how they occur in the real world. We took photos of the water coming out of a drinking fountain, the fireplace of our school’s library, and a flashlight’s beam when it is next to a wall. Then, we pasted the photos into Geometer’s Sketchpad, placed a grid over them, and came up with functions that model each conic section. They got very excited as they saw their function plots match the photos so closely. Hmm, maybe there is something to these crazy conics after all….

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From an Ellipse To a Hyperbola (or, There and Back Again)

As my Precalculus students begin to explore conic sections, a related activity that is a lot of fun is to use the concept of locus to generate them. A locus is the set of all points satisfying some condition. For example, the locus of all points equidistant from a fixed point would be a circle. The locus of all points equidistant from a line and a point not on that line would be a parabola (but you already knew that if you had seen this earlier post!).

I used to use waxpaper folding to create repeated perpendicular bisectors that eventually resulted in conic sections. It was very tedious, but now Geometer’s Sketchpad can do it in seconds, as well as animate the “creases”.

Here’s the basic setup, if you want to do it for yourself:

The blue line is the perpendicular bisector of segment DE. Point E moves around the circle, while point D moves bidirectionally along segment BC. I had Sketchpad trace the blue line (as well as change its color based upon the length of segment DE). You can watch the results in the video below. By the way, Screencast-O-Matic has a new feature allowing the screencaster to add sound directly from your computer. I was in an ’80s mood, so I set this video to Jan Hammer’s song, Evan, from the Miami Vice soundtrack. It creates an ominous sense of menace as the ellipses transform themselves into hyperbolas.