# GeoGebra

I use GeoGebra a lot in my classroom, a free mathematics visualization software that has a nice blend of Geometric and Algebraic capabilities. I use it mostly for modeling purposes, with its powerful abilities to fit a variety of functions to data, but occasionally I use it to do math labs to explore various mathematical relationships. GeoGebra calls this idea a “dynamic worksheet” but I think it can be a whole lot more that that.

My post on getting started with GeoGebra.

Guillermo Bautista’s healthy collection of GeoGebra resources, including over 50 awesome tutorials.

My integration drawing project with GeoGebra.

My Related Rates unit, heavily involved with GeoGebra. Kate Nowak’s take on the same ideas.

Dave Pugh’s GeoGebra resources, including instructions on how to combine a Google Form and a GeoGebra applet.

GeoGebra Tube, a place to get tons and tons of great applets made by talented educators.

Also, I wanted to share a few GeoGebra applets that I constructed myself not because I think that these are absolutely awesome, but I wanted to show you how easy it is to do very powerful things very quickly with GeoGebra.

[Click on the image to open up the GeoGebra applet on a new page.]

 POINTS OF CONCURRENCY IN A TRIANGLE This GeoGebra applet has a map of our school’s campus. The students can place the three points of a triangle at three points on campus and then see the points of concurrency (centroid, incenter etc) and see where they are. The goal is to try to figure out where the optimal meeting point is (however the student defines it). You could also have them construct the points instead of having them constructed for you.Another version of this problem is if you have three cities, where do you build an airport in between themif you want [each city to be the same distance from each one, the sum of the distance to be a minimum etc]. DERIVATIVE TRACER Plug in a function to f(x) and then drag a slider to trace out its slope at each point. I used this before I officially defined the derivative and had students determine some simple differentiation rules on their own. MIRROR IMAGES This applet shows the images created by a concave mirror of an object placed at various places in front of it. The applet connects the physical situation to the mirror equation and gives various attributes of the image. Students could use this in conjunction with a physical setup or with calculations. ACCUMULATION FUNCTION Many of my students had trouble imaging that if one bank’s rate of earnings were modeled by and anothers was , the latter would earn more money over the first month/year/unit of time. This applet shows the money accumulating over time to visualize why one bank earns more money than the other. MODELING A BALL THROWN IN THE AIR This applet shows a ball in vertical free fall, and charts its height over time. All parameters can be modified for a particular situation, the equation of the height vs. time can be shown, and the whole thing can be animated. INVERSE TRACER Plug in a function for f(x) and the applet traces out its inverse function. This could be used as a lab for students to determine what it means to be an inverse function and how to construct one. A great challenge would be to have the students animate the applet themselves.

# Winplot

GeoGebra can’t do everything. Most notably, there aren’t really any 3D capabilities, and it can’t handle slope fields and differential equations very easily. For those, I use a program called Winplot (also free, and actually tiny). I know this isn’t revolutionary, many educators use this, but it took me a year to stumble on this program, so I thought I would throw it out there.

A project some of my students did modeling 3D objects in Winplot.

Here’s what Winplot can do that GeoGebra cannot. Note that all these things took less than a minute to construct, and note that the 3D images are dynamic (you can zoom and rotate).

 SOLIDS OF KNOWN CROSS SECTIONS This is the solid whose base is the area bounded between $y=x^2$ and $y=\sqrt{x}$ and whose cross sections are squares (viewed from the bottom). VOLUMES OF REVOLUTION This is the region bounded between $y=x^2$ and $y=\sqrt{x}$ rotated around the y-axis. SLOPE FIELDS AND SOLUTIONS TO DIFFERENTIAL EQUATIONS EQUATIONS DEFINED IMPLICITLY GeoGebra can handle implicitly defined functions as long as they are conics, but Winplot can do anything.