An Activity for Pi Day

Pi Day (3/14) is fast approaching. Here’s a simple Geogebra activity you can do with your math students to help them get a sense of what pi is.

In Geogebra, construct a segment, then construct the midpoint. Construct a circle using the midpoint as center, and an endpoint of the segment as a point on the circle. In the Input pane, enter Circumference(<your circle>). When you divide the circumference measurement by the diameter length, you get pi! Geogebra is dynamic, so you can change the size of the circle, but the ratio will remain constant at pi.

Here’s an interactive version for you to play with at Geogebra’s site.

Here’s a screenshot:

All of my students have TI-84 graphing calculators, so I am going to have them use their Geogebra constructions to generate 10-12 different diameters and circumferences. Then we will enter the diameters in L1, the circumferences in L2, and plot the data. We will calculate the regression line, the slope of which should approximate pi.

Happy Pi Day!

Update: This went really well in my geometry class. Here’s the data we collected –

Here’s the regression line equation (check out the value of the slope):

And here is the plot of the data and the regression line:

 

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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

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:

Lazenby: Sierpinski Triangle

Trabue: Johnson’s Circle Theorem

Horner: Circle Inscribed in a Triangle

McCarthy: Tessellation

Nutter: Tesseract

Bennett: The Golden Ratio

Compton: Sierpinski Triangle

Cobb: Fibonacci Numbers

Webber: Borromean Rings

Boyd: Co-Eutrignon Theorem

Farrington: Pythagorean Theorem

Burkart: Exterior Angles of a Polygon

Hulme: Parallel Planes

Lund: Congruent Angles Formed by Parallel Lines and a Transversal

Jenkins: Properties of Parallel Lines and Transversals

Harrison: Exterior Angles in a Polygon

Triplett: Midpoint of Hypotenuse Properties

Johnson: Angle Bisector Theorem

Howard: The Four-Color Theorem

Draughn: Circle Inscribed in a Triangle

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:

Now 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:

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:

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).

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:

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.

Using your three sets of x and y-values, you can set up a system of three equations in the form of . 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:

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!