GeoGebra
Visualize Algebra & Geometry
Concepts with the Greatest of Ease
A tool to help meet the
California Mathematics Standards
presentation by Bill Lombard
some graphics from http://en.wikipedia.org and /http://commons.wikimedia.org
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
“Building mathematical skill on a foundation of understanding”
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GeoGebra
Overview
 Introduction
to GeoGebra
 User community
 Using GeoGebra in the classroom
 Hands-on Activities
 GeoGebra’s Web Export
 Open Source & GeoGebra Support
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Why use Technology?
“Technology is essential in teaching and
learning mathematics;
it influences the mathematics that is
taught and enhances students’ learning.”
(NCTM Principles and Standards, 2000)
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Technology or No Technology?
“What we have learned from all the media
comparison research is that it’s
not the medium, but rather the instructional
methods that cause learning.
When the instructional methods remain
essentially the same, so does the learning, no
matter how the instruction is delivered.”
(Clark & Mayer, e-Learning and the
Science of Instruction, 2002)
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
What is GeoGebra?




bill.lombard@gmail.com
http://mrlsmath.com
Dynamic Mathematics
Software
For Learning and Teaching
in Schools
Geometry, Algebra and
Calculus
Freely available from
www.geogebra.org
www.tttpress.com
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GeoGebra
GeoGebra = Geometry + Algebra
Algebra Window
Geometry & Graphics Window
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
GeoGebra is Innovative
Combination of elements of
 dynamic
geometry software (Sketchpad, Cabri)
 computer algebra systems (Derive, Maple)
High technical portability
 runs
on Windows, Linux,
Solaris, MacOS X
 dynamic worksheets (html)
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
2001-2006: Development



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Created by Markus Hohenwarter
Mar 2002: Master’s thesis in Computer Science
& Mathematics Education
Nov 2002: European Academic Software Award
Mai 2005: User Forum & GeoGebraWiki
Feb 2006: PhD project in Mathematics
Education funded by Austrian Academy of
Sciences
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
World Wide User Community
40 languages
 190 countries
 400,000 visitors
per month

bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Visitors on www.geogebra.org
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
GeoGebra as a Pedagogical Tool
 Presentation
Tool
dynamic visualizations
 Modeling
Tool
dynamic constructions
 Authoring
Tool
interactive online material
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Educational Principles
 Student Activity
 Modeling
 Multiple
Representations
 Experimental and
Guided Discovery Learning
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Examples of Using GeoGebra

Visualizing
students can “see” abstract concepts

Representations
students can make connections

Experiments
students can discover mathematics
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Creating Teaching Materials

Static materials
 Static
pictures
 Construction protocol


Allows you to reconstruct steps of a
construction
Dynamic materials
 Construction
files (ggb)
 Dynamic worksheets (html)
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Ways to use GeoGebra
 Tool
for students
 Tool
for teachers
for doing constructions from scratch
Visualisation
dynamic demonstrations
Creating
Teaching Materials
that foster student activity
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
What does GeoGebra offer?
 points,
vectors, segments, polygons, lines, all
conic sections and functions in x
 dynamic
constructions (mouse)
 coordinates,
equations, vectors, numbers and
commands (keyboard)
 easy-to-use
interface
 multilingual
menus, commands, help
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Using Dynamic Worksheets

Student opens html file with a web browser

Requirements on student‘s computer:
 Web
browser
 Java
1.4.2 or later
e.g. Internet Explorer, Firefox, Safari
free download from www.java.com
 Note:
bill.lombard@gmail.com
GeoGebra is NOT needed to use worksheet
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Combined Representations
There is no true understanding in
mathematics for students who do not
incorporate into their cognitive
architecture the various registers of
semiotic representations used to do
mathematics.
(Raymond Duval 1999)
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Age 6-10: Adding Integers
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Age 11-14: Multiplying Fractions
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Age 15-18: Geometry & Algebra
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Age 17 up: Functions & Calculus
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Visualizing Parameters
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Construction Protocol -
a built-in way to replay constructions
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
25
GeoGebra
GeoGebra as a Pedagogical Tool
 Using
for Geometry
Midpoints
Angle
bisectors
Perpendicular bisectors
Transformations
Conjecture / proof
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Geometry Examples
Mr. L demos using GeoGebra
at the workshop
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
GeoGebra as a Pedagogical Tool
 Using
for Algebra 1
Graphing
lines & parabolas
Modeling with parameters
Solutions of equations
Families of functions
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
28
GeoGebra
Algebra 1 Examples
Mr. L demos using GeoGebra
at the workshop
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
GeoGebra as a Pedagogical Tool
 Using
for Algebra 2
Graphing
complex functions
Conic sections (non-functions)
Exponential curves and logarithms
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
30
GeoGebra
Algebra 2 Examples
Mr. L demos using GeoGebra
at the workshop
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
GeoGebra as a Pedagogical Tool
 Using
for Trig & Calculus
Graphing
complex functions
Derivatives (slope of functions)
Area under a curve (sum of
rectangles
Integrals
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
32
GeoGebra
Trig/Calculus Examples
Mr. L demos using GeoGebra
at the workshop
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
33
GeoGebra
GeoGebra’s Web Export

Every GeoGebra Construction can be
exported as a Web Page (html)
Dynamic Worksheet
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
GeoGebra is Open Source
Available
in schools and at home
 Online collaboration

 International
community
 User forum – users help users
 GeoGebraWiki – pool of shared materials

Expandable
by using other open source libraries
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
International
User Community
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
GeoGebra User Forum
www.geogebra.org/for
um
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
GeoGebraWiki – Free Materials
www.geogebra.org/
wiki
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Status Quo

What’s there?
 Free
software and material
 Extensive user community
 Immense expertise developed by enthusiasts

What’s missing?
 Not
enough training
 Not enough support for teachers new to
technology
 Limited research on the impact of GeoGebra
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
39
GeoGebra
Future Plans
 Make
it Easier, Easier, Easier!
 Spreadsheet Window
 Symbolic Algebra Window (CAS)
 3D extension
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
40
GeoGebra
Additional GeoGebra Information

Download these slides:
www.geogebra.org/workshop

GeoGebra Webpage: www.geogebra.org

Introduction to GeoGebra (Book)
 Print:
 Online:

www.lulu.com/content/2650846
www.geogebra.org/book/intro-en/
Contact: judith@geogebra.org
markus@geogebra.org
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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GeoGebra
Teaching:
The hardest job you’ll ever love
just got easier with GeoGebra! !
bill.lombard@gmail.com
http://mrlsmath.com
www.tttpress.com
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