1. GeoGebra Dynamic Mathematics for Schools Auston B. Cron
PSJA North High School
South Texas College
CAMT 2009

2. What is GeoGebra?
Dynamic Mathematics software built by and for Mathematics instructors and students
Open Source, meaning
It‘s free!
It‘s supported by a community of like-minded people
Short History of GeoGebra
GeoGebra was created by Markus Hohenwarter in 2001/2002 as part of his master's thesis in mathematics education and computer science at the University of Salzburg in Austria. Supported by a DOC scholarship from the Austrian Academy of Sciences he was able to continue the development of the software as part of his PhD project in mathematics education. During that time, GeoGebra won several international awards, including the European and German educational software awards, and was translated by math instructors and teachers all over the world to more than 25 languages.
Since 2006 GeoGebra is supported by the Austrian Ministry of Education to maintain the free availability of the software for mathematics education at schools and universities. In July 2006, GeoGebra found its way to the US, where its development continues at Florida Atlantic University in the NSF project Standard Mapped Graduate Education and Mentoring.

3. Why GeoGebra? For more on this go to:
In the “Principles & Standards for School Mathematics” published by the NCTM, one of the principles states that:
“Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning.”
“Students can learn more mathematics more deeply with the appropriate and responsible use of technology.”
For more on this go to:
GeoGebra was created by Markus Hohenwarter in 2001as part of his master's thesis in mathematics education and computer science at the University of Salzburg in Austria. Supported by a DOC scholarship from the Austrian Academy of Sciences he was able to continue the development of the software as part of his PhD project in mathematics education. During that time, GeoGebra won several international awards, including the European and German educational software awards, and was translated by math instructors and teachers all over the world to more than 25 languages.
Since 2006 GeoGebra is supported by the Austrian Ministry of Education to maintain the free availability of the software for mathematics education at schools and universities. In July 2006, GeoGebra found its way to the US, where its development continues at Florida Atlantic University in the NSF project Standard Mapped Graduate Education and Mentoring.

4. GeoGebra is Innovative
Combination of elements of
dynamic geometry software (Sketchpad, Cabri)
computer algebra systems (Derive, Maple)
High technical portability
GeoGebra is fully platform
independent (Windows,
MacOS, Solaris, Linux)
dynamic worksheets (html)
Geogebra is free and runs on all computers

5. GeoGebra
Past & Present

6. Computer Algebra Systems
Mathematica
Derive
There are many types of Computer Algebra Systems the are being used by many. This are just of few that you may know.
Scientific Notebook

7. Dynamic Geometry Software
There are many dynamic geometry software programs including Geometer’s SketchPad.

8. Graphing Calculators TI-92: DGS and CAS HP 49g TI-84
Graphing calculators have progressed from initial tools like the TI-92;
to the HP-49g and TI-84;
Into the new TI-Nspire family.
HP 49g
TI-84
TI-Nspire/CAS System

9. GeoGebra has received many awards.

10. GeoGebra as a Pedagogical Tool
Drawing Tool
Geometric drawings for handouts & activities
Graphs for worksheets, quizzes & tests
In-class Presentation Tool
Dynamically show relationships
Visualization of abstract concepts
Authoring Tool (Web Export)
Interactive virtual manipulatives
Exploratory mathemathics
Geogebra meets a wide variety of educational needs
Aiding development of materials
In-class use by the instructor to demonstrate concepts and relationships
Serve as an interactive component of assignments

11. World Wide User Community
43 languages
190 countries
400,000 visitors per month

12. Visitors on www.geogebra.org
As of June 13, 2009.
User Forum: Total posts 26994
| Total topics 5796 | Total members 3155

13. GeoGebra Home Page www.geogebra.org/
The Geogebra home is your gateway to:
Installation
Examples
Forum: Ask questions, get answers, fast!
Much, much more…

14. Start up with GeoGebra Webstart
Click Help to get access to the GeoGebra
Home page – connect to all things GeoGebra
User Forum - ask questions, get answers!
Wiki – free pool of teaching materials
Before doing anything else, let’s see where the help is!

15. GeoGebra User Forum www.geogebra.org/forum
The support for Geogebra is second to none.
Ask questions, get answers, fast!

16. GeoGebraWiki – Free Materials
A great many geogebra materials are available on the Wiki.
Available in many languages.
You will need to explore (after you are settled).

17. GeoGebra = Geometry + Algebra
Algebra Window
The algebra and geometry windows are tightly coupled,
Pretty much 1-1!
Geometry Window

18. Getting to know the user interface
The Move tool
the most important
tool on the menu!
Input field
Help for the
selected tool
Tool bar
“Toolboxes” group
Similar tools
The toolbar has tool tips and fly-over feedback.
In the image above, notice that the selected tool has a blue outline,
The tool that the mouse cursor is near glows yellow and the small
Triangle turns red, indicating that there is a drop-down list of tools.
Input field assists

19. Spreadsheet View has being added
Toolbar
Spreadsheet View
Algebra View
Graphics View
The dynamic mathematics software GeoGebra provides three different views of mathematical objects: a Graphics view, a, numeric Algebra view and a Spreadsheet view. They allow you to display mathematical objects in three different representations: graphically (e.g., points, function graphs), algebraically (e.g., coordinates of points, equations), and in spreadsheet cells. Thereby, all representations of the same object are linked dynamically and adapt automatically to changes made to any of the representations, no matter how they were initially created.
Judith and Marcus Hohenwarter
Pretty much 1-1
Input Bar

20. Spreadsheet in GeoGebra
Dynamic Table of Values
Regression Lines & Polynomials
Binomial Distribution
With the release of GGB 3.2, these feature have been added.

21. Extra credit: On the menu bar Select View, then Grid
Let’s get busy!
Select the Line through two points tool (notice the tool tips!)
Move the cursor into the geometry window (AKA the drawing pad)
Hold the mouse button down and notice that coordinates are displayed; release to set point
Create a second point to complete the action
Observe that the points and line also appear in the algebra window!
Symbolic↔Geometric
Notice the menu bar; View and Options allow customization of most aspects of the user interface.
This duality of representations is typical in geogebra – every object has an algebraic and a geometric representation.
Extra credit: On the menu bar Select View, then Grid

22. Add a Drawing to a Document
Click on File, Export, Drawing Pad to Clipboard
Start Word, then Edit, Paste – You’re done!
Resize and position as desired
Also use the View menu to enable the grid.
Many aspects of the user-interface are selectable in the View and Options menus.

23. Context Menu Right-click any object in either window
Notice the Properties… item at the bottom
Notice that when an object is selected in the geometry window it is also highlighted in the algebra window.
Notice the Properties item at the bottom of the context menu.
Context menu

24. Properties Dialog – object customization
The Properties dialog is accessed via the context menu (or the Edit menu)
Notice that this is a tabbed dialog; each tab contains a set of related properties
Try this:
Change the color of your line (Color tab)
Change the style of your line (Style tab)
The properties dialog is where you can change various properties of the object.
Extra credit: Modify the labeling of point A to Value

25. Activity – Constructing a Parabola (Part 1: create basic interactive graph of a parabola)
Type the following into the Input field
a=1
f(x) = a x^2
You now see the parabola f(x) = x2
Right click on a=1 in Algebra window and select “Show Object”
Use the Move Tool to change ‘a’
Tools Used:

26. Activity – Constructing a Parabola (Part 2: create a family of curves)
Right click on a=1 in Algebra window and select “Properties”
Click on the “Slider” tab and change the increment to 1
Click on the Function f, select “Show Trace” and close
Move the slider left and right.
Tools Used:

27. Activity – construct a rhombus (a rhombus is a quadrilateral with all sides equal in length)
METHOD 1 (File | New “Don’t Save” and turn off the grid & axes)
Begin by drawing a circle (center A and point B)
Create a point C on the circle
Draw a 2nd circle centered at B out to point A
Draw a 3rd circle centered at C out to point A
Create a point D at the intersection of these two circles
ABCD are the vertices of the rhombus
Finish by hiding the circles and drawing the line segments
Manipulate your rhombus and observe the behavior
We need to erase the drawing pad – Select All using control-A, then delete.
Notice that this is the same key sequence used in Word & many any other programs!
Note the difference between a “drawing” and a “construction”.
Tools Used:

28. Activity – construct a rhombus (a rhombus is a quadrilateral with all sides equal in length)
METHOD 2 (File | New “Don’t Save” and turn off the grid & axes)
Begin by drawing a circle (center A and point B)
Create a point C on the circle
Draw line segments AB and AC
Construct a line through B parallel the AC
Construct a line through C parallel the AB
Create a point D at the intersection of these two lines
ABCD are the vertices of the rhombus
Finish by hiding the lines and drawing the line segments
Manipulate your rhombus and observe the behavior
Tools Used:

29. Activity – construct a rectangle (a rectangle is a quadrilateral with all interior angles 90°)
Begin by drawing a line segment AB
Draw a line through A and perpendicular to AB
Draw a line through B and perpendicular to AB
Create a point C on the line through A
Draw a line through C perpendicular to AC
Create a point D at the intersection of these two lines
ABCD are the vertices of the rectangle
Finish by hiding the lines and drawing the line segments
Manipulate your rhombus and observe the behavior
Tools Used:

30. Activity - The Pythagorean Theorem
Draw a right triangle
Draw a line segment AB
Draw perpendicular to AB at A
Point C on perpendicular
Draw line segment CB
Hide perpendicular line
Construct squares
Use Regular polygon tool
(Notice the polygon areas are shown in the algebra window)
Label each square with its area
Properties - Show label - Value
This drawing is pretty familiar to most teachers of mathematics, and so the dynamic aspect can be appreciated without getting bogged down in a discussion of the “meaning” of the drawing.
Tools Used:

31. Area of a Triangle Create 2 horizontal lines Input: y=0 and y=6
Create a triangle ABC
Make 3 points on these lines
Connect using line segments
Create the altitude
Line thru C and perpendicular
Make intersection point D
Hide line; make line segment
Create text
Label & Value of height & base
Text showing computed area
Here we use the Input window for the first time.
Some people are not aware of the implication of the formula for the area of a triangle
(all triangles with the same base and height must have the same area).
Now the fun part! Move point C and what (doesn’t) happen?

32. Tools & Commands – Two ways of doing things
Input y=x^2 (parabola c is created)
Create point A on c
Use the New Point tool or Input “Point[c]”
Create the line tangent to c at A
Use the Tangents tool or Input “Tangent[A,c]”
Create a Slope object
Use the Slope tool or Input “Slope[a]”

33. Parameters & Animation
A Slider is a graphical representation of a number
Input “a=2”
Input “y=a*x^2”
Note in the algebra window that a is “off”
Right click on a, then Show object
The variable a has become a graphical object!
Manipulate the slider and observe its effect
There is also a tool in the Slider toolbox for creating checkboxes.
Extra credit: set a to -5 and then right-click on the parabola &
turn trace on, then sweep the slider from -5 to 5; a family of curves!

34. Riemann Approximations - part 1
Type the following into the Input field:
f(x)=4x-x^2
a=0
b=4
Area=Integral[f,a,b]
n=10
UpperSum=UpperSum[f,a,b,n]
LowerSum=LowerSum[f,a,b,n]
Adjust the colors of Upper & LowerSum
More use of the Input window, and the first use of commands.

35. Riemann Approximations - part 2
Right click on ‘a’ and select “Show object”
The variable has become a graphical object!
Manipulate the slider and observe its effect
Repeat for the variables ‘b’ and ‘n’
‘n’ should take only positive integer values
Access the ‘Slider’ properties for n
Set min, max and increment appropriately

36. GeoGebra’s Web Export
Every GeoGebra Construction can be exported as a Web Page (html), known as a Dynamic Worksheet
Try it with your current drawing
File, Export, Dynamic Worksheet as Webpage
Examples
Go to
Click on GeoGebra Demonstrations
Example 1: Graphing Linear Equations
Example 2: Basic Construction Kit
Go to:
Click on GeoGebra Tutorials

37. Dynamic Worksheet Example: Graphing Linear Equations
GeoGebra was used to create this page
Access to the internet is all that is needed to interact with it!
User is limited to just the actions you design in
The focus is limited to the components you create.
The user is not distracted by extraneous features.

38. Dynamic Worksheet Example: Basic Construction Kit
A limited subset of the construction tools are available
The algebra and input windows are turned off
The menu bar is on so students can save their work!
Notice that there are no “tool boxes”, every tool that is needed is visible.

39. Other Features to Mention
Booleans
Check boxes, used to reveal/hide parts
Vectors
u=(1,2); v=(2,1); w=u+v
Polar Coordinates
a=0; r=1; P=(r;a)
Parametric Equations
c=Curve[2cos(3t), sin(4t), t, 0, 2 pi]

40. New Features to Mention (Version 3.2)
Record Spreadsheet
Best Fit Line
Compass
Easy Quadratic Functions
Ellipse (Two foci and a point on it)
Hyperbola (Two foci and a point on it)
Parabola (A focus and directrix)
Mirror point at circle

41. GeoGebra Future

42. Future Plans Make it Easier, Easier, Easier!
Symbolic Algebra Window (CAS)
3D Extensions
Split views to allow rearranging
Independent sizing
Possible 4 corner model

43. Next year: CAS, 3D

44. GeoGebra Institute of Ohio
Resources Sites
GeoGebra Home
PSJA GeoGebra Page
GeoGebra Institute of Ohio me