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.

Transformations of Functions

Well, it’s time to start cranking up the online lectures for my Honors Precalculus classes, and the first one of the year is all about transforming graphs of functions. That means shifting left, right, up, and down, reflecting on x and y-axes, and scale changes. I hope you enjoy my explanation of this exciting topic!

Vectors and Tessellations

This quarter’s project for my geometry students was to create a tessellation using vectors and Geometer’s Sketchpad. It’s a lot of fun, and they made some beautiful patterns. I have lots of M. C. Escher prints in my classroom, so when they finally get a chance to see the geometry behind his designs, they really enjoy it.

Here’s an example:

You can see all of my students’ tessellations here.