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

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Light, Math, & Color: 2014 Edition

For several years now I’ve taught a three-week course on making artglass windows. After learning the basic technique, I ask each student to research a math-related topic and illustrate it with an artglass window. This year’s group did exceptionally good work! The projects ranged from perennial favorites like the Pythagorean Theorem and tessellations to some new topics –  tesseracts, Borromean Rings, and Johnson’s Circle Theorem.

Pictures of their projects are below:

Water, Water, Everywhere, and Parabolas to Boot

I’ve done this activity before, but never with GeoGebra, the dynamic geometry and function graphing program. So for those of you who would like to try it with that open-source (meaning, it’s completely free!) program, here are the steps I’ve found to work best.

First, you need a picture of water that is following a parabolic path. Fortunately, that happens at most drinking fountains:

Digital StillCameraNow we need to paste this into GeoGebra. You can drag it directly into the graphics pane of GeoGebra, or go to Edit/Insert Image From…File:

GeoGebra1

Now you can resize and drag the picture around any way you like. (Helpful tip: It really simplifies things if you make the water stream pass through the origin.) Hit the little Graphics toggle at the top left of the Graphics pane, and click on the grid button:

GeoGebra2

You’ll notice that you can’t see the axes or the grid behind the picture, so click on the picture to select it, then right-click on it to choose “Object Properties”. Under the Color tab, you can then change the opacity of the picture. Something around 50% usually works fine. (If you don’t see an Opacity slider, click on the pic1 label in the left pane of the Preferences window).

GeoGebra3

You can also change the color of the grid by going to Options/Advanced, select the “Preferences – Graphics” button (it has a little circle and triangle) and clicking on the Grid tab:

GeoGebra4

Now you’re ready to actually model the water’s path! Place the picture so that it passes as close as possible to a couple of grid intersections. Put three points on the water’s path using the point tool. In the Algebra pane on the left, you’ll notice that the points’ coordinates are automatically calculated.

Graph1

Using your three sets of x and y-values, you can set up a system of three equations in the form of y=a{{x}^{2}}+bx+c. Students can use any method they wish to solve that system. Mine were comfortable enough with matrices and graphing calculators to use them to solve the system:

Calc

Enter the quadratic function with the calculated a, b, and c values into the Input bar at the bottom of GeoGebra. Then, see your function model the water’s path perfectly!

Graph2

2013 Winterim: Light, Math & Color

I taught another three-week stained-glass mini-course this year. After my students learn the basic technique of copper-foil stained glass windows, they research a math topic, write a paper on it, and illustrate it with a window of their own design. Topics this year included systems of inequalities, the Fibonacci Sequence, corresponding angles formed by two lines and a transversal, the Four-Color Theorem, and the Pythagorean Theorem among others.

Here’s a gallery of their finished windows: