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GeoGebra in the Math Classroom

Module by: Kelly Huenerberg. E-mail the author

GeoGebra in the Math Classroom

Introduction

According to the software’s own description, GeoGebra is “a dynamic mathematics software for schools that joins geometry, algebra and calculus.” Its interactive coordinate plane allows the user to manipulate entries both algebraically and geometrically. Students can select a point, line, vector, segment, circle, polygon or other feature and directly place their selection on the plane. From there the student may manipulate key attributes of their selection, such as the location of the point or the radius of the circle. The user may also enter his or her desired selection algebraically with a command, allowing for more complex equations and advanced functions, such as finding the derivative and integral.

GeoGebra is designed to be an educational resource and is useful to teachers and students alike. It was started in 2001 by Marcus Hohenwarter as a master’s thesis project and continued as a PhD thesis and now is supported by the Florida Center for Research in Science, Technology, Engineering, and Mathematics. GeoGebra is completely free and can be downloaded off its website. It has won several international awards, such as the European and German educational software awards and is in use all over the world.

Using GeoGebra

GeoGebra is very user-friendly. The author has created a user manual which introduces new users to the basic functions of the software and pauses with “Practice Blocks” to ensure user familiarity with each command. There is also a GeoGebra Wiki that users may post materials and ideas. The GeoGebra User Forum is a place for users to ask questions and get technical help.

GeoGebra Website

User Manual

GeoGebra Wiki

User Forum

A basic GeoGebra screen might look like the following. Launching the program opens a window with a dynamic coordinate plane. At the bottom the user can input commands, or, alternatively, he or she can click on a particular type of object he or she would like to add onto the plane. The window at the left shows all the objects on the plane. Some are considered “free,” meaning that they do not depend on the size or location of another object. Dependent Objects have been defined by independent or other dependent objects. For example, my line d is defined to be a line passing through the free point A and the dependent point C2C2 size 12{C rSub { size 8{2} } } {}. C2C2 size 12{C rSub { size 8{2} } } {} is defined as the intersection of the circle c and the line b.

Students can manipulate the free objects and learn how the dependent objects will be affected. For example, in the following shot I have moved the center of circle c (which is point A) from (2,2) to (1,1). This changes the radius of circle c, the location of points C1 and C2, and the equation of line d. Only the other free objects – point B and line b – remain unchanged.

GeoGebra certainly offers much more advanced functions, as described in depth in the manual, but this will give you an initial grasp on how to use the program.

GeoGebra in the Classroom

The GeoGebra Wiki is an excellent resource. Instructors and math fanatics alike have uploaded interactive teaching tools available for public use. Some include worksheets to guide students through manipulations and some merely call for students to experiment. Many ask students to use the “slider” function, which allows for the values of specific variables to change at the user’s command. Some have students adjust particular free objects and observe the effect on the dependent objects. The wiki is incredibly organized and teachers can find a resource on almost any topic of their choice. Here is the table of contents, which demonstrates the broad range of resources available.

The following are links to a few of my favorite resources:

Geometry: Triangle Congruence – Angle-Side-Side

This link allows students to adjust an angle measure, corresponding side, and opposite side with the intent to build a closed triangle. The angle and two sides are duplicated. Students have opportunity to prove to themselves by adjusting the second angle measure that a congruent angle, side, and opposite side ensure triangle congruence.

Algebra: Domain and Range

This link provides several resources for students to learn about domain and range with different kinds of functions such as the absolute value function, exponential function, rational function, and radical function. The domain and range are highlighted in red on the plane. The applet allows students to adjust the parameters of the function and see the affect that changing those parameters takes on the domain and range. The link also provides the lesson plan that this particular educator used in implementing the resource.

PreCalculus: Trig Functions

This resource allows students to experiment with the sine, cosine, and tangent functions. They can choose which and how many of these functions they wish to appear on the plane at any given time and learn how adjusting certain variables adjusts the frequency, amplitude, and y-intercept of the graph. No guided worksheet exists with this particular resource; the teacher would need to give students some guidelines.

Calculus: Upper & Lower Sum of a Function – Intro to Integrals

This particular resource, created by the GeoGebra creater himself, is an excellent introduction into the study of integrals. A function’s integral can be shown graphically as the area “under the curve.” This interactive tool shows two ways of approximating the area, how to increase/decrease the accuracy of those approximations, and sets the class up for the study of integrals.

Affordances/Constraints

The Good

At the high school level, especially once students have passed geometry, the use of physical manipulatives in the classroom is limited because the material is too advanced and dense to be modeled by tangible objects. GeoGebra functions as a virtual manipulative and still allows students to “play with” functions easily.

In comparison to a graphing calculator, GeoGebra is more user-friendly. Students do not need to switch between screens and re-enter equations simply to see the affect that changing a coefficient from 1 to -1 will have on a graph.

Students can manipulate variables easily by simply dragging “free” objects around the plane or by using sliders.

Students can create work for the teacher to view later.

GeoGebra provides good opportunity for students to work in pairs and talk through the material together. “Why does adjusting h have that affect on the graph?” “How would I increase the frequency of this sine function?” Students have good opportunity to learn from one another.

Students can personalize their own creations with color, line thickness, line style, shading, window size, and other features.

The Not-So-Good

Several of the applets I tried to load took far too long or would tell me an error had occurred. (Granted, my computer is very old so the problems may be more on my end.) Teachers must be prepared if the technology doesn’t come through.

It would be hard for a student coming in with no previous programming experience to enter algebraic commands in the input box. While the basic commands are not difficult to learn, students might feel intimidated or lost.

Students need to have some kind of parameters to work in or guidance in what to do. Telling them to “play with the variables” and “see what happens” is too vague for many students. It would help to have an interactive worksheet that asks more specific questions.

Tips

• Model GeoGebra in front of the class so they are familiar with the program before giving them their own computers.
• Be specific in your instructions, but allow for experimentation. Students who are less familiar with computers will need more guidance, but more competent ones will want to explore.
• Have students write down their observations so they have a physical record to refer to later.
• Print out student creations – many will consider their work a form of art.
• Think about the possibility of working in partners to help those less confident with technology and to encourage students to learn by talking through what they observe.

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