Begin with GeoGebra 1
Core Elements
Table of Contents
Getting Started
Installing/enabling Geogebra
Installation WITH Internet access
Installation WITHOUT Internet access
GeoGebra version 4
GeoGebra version 5
Learning Phases
Introduction of GeoGebra
User Interface/Main Window of GeoGebra
Basic Use of GeoGebra Toolbar
Drawing without Mathematics
Construction Protocol
CheckBox to Show/Hide Objects
Numeric Foundations
Creating dynamic worksheets, mathlets
Using GeoGebra Animation
Geometry buttons/tools: characteristics and concepts
Basic geometric constructions, connection between geometry and algebra
Linear functions, polynomials of 1st degree
Quadric functions, polynomials of 2nd degree
Spreadsheet view - statistics
Using a powerful markup language
Famous patterns and problems: Sierpinski triangle, Fibonacci series, normal distribution and others.
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Getting started
Create a new folder called GeoGebra_Intro (or similar) on your desktop or as a folder in your filestructure. It is a good strategy to save all files in a separate folder so they are easy to find later on.
Installation/enabling GeoGebra
Different ways are available to install and start GeoGebra. Go to the page http://www.geogebra.org/cms/en/download to find those ways.
The current version (October 2011) of GeoGebra is GeoGebra 4. The latest release of GeoGebra 4 is still a Beta version. The release notes are found on the link http://www.geogebra.org/en/wiki/index.php/Release_Notes_GeoGebra_4.0
Installation WITH Internet access
There are a couple of webbased versions available: WebStart, AppleStart and GeoGebraPrim.
GeoGebra WebStart
Open the scrollist in the upper right corner of the download webpage and select the preferred tool/installation language.
Click on the button called WebStart. In this case the Java Network Launching Protocol (JNLP)or Java Web Start functionality is used. Java Web Start provides a platform-independent, secure, and robust deployment technology. It enables developers to deploy full-featured applications to end-users by making the applications available on a standard Web server.
The software is automatically installed on your computer. You only need to confirm all messages that might appear with OK or YES.
Using GeoGebra WebStart has several advantages for you provided that you have an Internet connection available for the initial installation:
You don’t have to deal with different files because GeoGebra is installed automatically on your computer.
You don’t need to have special user permissions in order to use GeoGebra WebStart, which is especially useful for computer labs and laptop computers in schools.
Once GeoGebra WebStart was installed you can use the software off-line as well.
Provided you have Internet connection after the initial installation, GeoGebra WebStart frequently checks for available updates and installs them automatically. Thus, you are always working with the newest version of GeoGebra.
When the installation is ready you have got the following short-cut icon on the desktop.
GeoGebra AppletStart
Open the scrollist in the upper right corner of the download webpage and select the preferred tool/installation language.
Click on the button called Applet Start. GeoGebra is opened and run as an ordinarie Java Applet.
GeoGebra GeoGebraPrim
Open the scrollist in the upper right corner of the download webpage and select the preferred tool/installation language.
Click on the button called GeoGebraPrim. A corresponding .jnlp-file is then available. This is a “stripped” version of GeoGebra and the restrictions can be found on the link http://www.geogebra.org/en/wiki/index.php/Release_Notes_GeoGebra_4.0
Installation WITHOUT Internet access: Offline Installers
You need to have the installation media or installer file “GeoGebra.exe” (Windows platform), “One click installers” or “GeoGebra.zip” (Linux platform).
Open the link offline installer and the address http://www.geogebra.org/cms/en/installers will open.
Copy the installer file for your preferred platform from the storage device into a created folder (with a suitable name) on your computer. Current version for Windows platform: GeoGebra-Windows-Installer-4-X-X-X.exe.
Double-click the GeoGebra installer file and follow the instructions of the installer wizard.
When the installation is ready you have got the following short-cut icon on the desktop.
GeoGebra version 5
Find the latest notes about the currently developed GeoGebra 5.0 beta version ( June 2012) in the document
http://wiki.geogebra.org/en/Release_Notes_GeoGebra_5.0
You can run the GeoGebra 5.0 beta version directly here:
http://www.geogebra.org/webstart/5.0/ge ... jogl1.jnlp
If you have trouble with that, try this one which uses JOGL2
http://www.geogebra.org/webstart/5.0/ge ... jogl2.jnlp
Java OpenGL (JOGL) is a wrapper library that allows OpenGL to be used in the Java programming language.
Learning Phases
An important idea in the material “Begin with GeoGebra” is built on three learning phases:
- collaboration phase with construction protocol and jointly adapted worksheets and work-outs with step-by-step guidance in discussion with teacher/instructor
- elaboration phase with discovery/self-study/self-reviewed worksheets and work-outs, typically performed as investigation of additional concepts, parallel concepts or attack concept/problem from another angle
- exploration phase/”e-learning” supported by interactively modifyable worksheets/work-outs to foster experimental as well as discovery learning to strengthen and confirm the understanding and use of concepts, patterns and models. Every GeoGebra construction can be exported as a Web Page (html), known as a Dynamic Worksheet. Computer on local base or access to the internet is all that is needed to interact with it!
Those three phases will be practiced through the “Begin with GeoGebra” material.
Introduction of GeoGebra
GeoGebra is a user-friendly and interactive software for mathematics learning that dynamically combines geometry, algebra, and calculus and also CAS (Computer Algebra System) in the latest versions GeoGebra 4 and 5.
On the one hand, GeoGebra is an interactive geometry system, the geometry view. You can do constructions with points, vectors, segments, lines, and conic sections as well as functions while changing them dynamically afterwards.
On the other hand, commands, equations and coordinates can be entered directly, the algebra view. Thus, GeoGebra has the ability to deal with variables for numbers, vectors, and points. It finds derivatives and integrals of functions and offers commands like Root or Vertex. The algebra view is connected to an Input field to make the direct textual input.
These two views are characteristic of GeoGebra: an expression in the algebra view corresponds to an object in the geometry view and vice versa and the views are toggled in real time.
The third view is the calculus spreadsheet and its functionality. This will be handled later on.
And there is even a fourth view: CAS – Computer Algebra System for symbol handling introduced in GeoGebra 4 and 5. This will also be handled in a separate chapter later on.
User Interface/Main Window of GeoGebra
Start GeoGebra with a double-click on the GeoGebra WebStart icon, GeoGebra Installer icon, link to Applet Start or link to GeoGebra4/GeoGebra5. The GeoGebra tool opens the following standard/main window with a common type of layout for user interface and main page window. GeoGebra’s user interface/standard main page consists of a graphics window and an algebra view opened for usage. The calculus view and the CAS view are hidden when the GeoGebra interface/main page is opened. Those views are open from the toolbox, examined later on.
The User Interface/Main Window for GeoGebra 5.0
This interface/main windows is started with a Perspective Menu Window from which you can easily switch between different views, without selecting each individually.
You can choose between 5 different standard perspectives:
Algebra & Graphics: The Algebra View and the Graphics View with axes are shown.
Basic Geometry: Only the Graphics View without axes or grid is displayed.
Geometry: Only theGraphics View with grid is shown.
Spreadsheet & Graphics: The Spreadsheet View and theGraphics View are displayed.
CAS & Graphics: The CAS View and the Graphics View are displayed.
Basic Use of GeoGebra Toolbar
Activate a tool by clicking on the button showing the corresponding icon.
Open a toolbox by clicking on the lower part of a button and select another tool from this toolbox.
You don’t have to open the toolbox every time you want to select a tool. If the icon of the desired tool is already shown on the button it can be activated directly.
Toolboxes contain similar tools or tools that generate the same type of new object.
Check the toolbar help in order to find out which tool is currently activated and how to operate it.
Drawing without Mathematics
Double-click on any of the GeoGebra WebStart icon, GeoGebra Installer icon or links to GeGebra4/GeoGebra5. The GeoGebra tool opens the following standard window.
If you don’t have Swedish as the tool language in the GeoGebra window (the right picuture above), click on the toolbar tab Option and activate Languge -> R-Z -> Swedish in the opened scrollist. Then the tool language is changed to Swedish as in the left picture above.
Open the View tab and uncheck the Axis, check the Grid alternatives. Close the Algebra View. Then you get a Geomety View with a Drawing Pad
Select the Geometry tool “New Point”
to create a Point A
Select the Geometry tool “Line through two points” and click the red triangle in the low right corner of the tool icon. Select the “Segment between two points”.
Use the mouse cursor (cross mark) to draw a “chair”.
Activate the points of the chair with a right-click and select “Show Label”. The chair is labeled like this
Open the Algebra View. The coordinates for the five points on the chair are presented as Free Objects.
Activate the seat on the chair with the mouse cursor. This is named as the segment a or Segment [A,B]. In the same way the back, left and right leg are presented as Segment name and value as Dependent Objects in the Algebra View.
GeoGebra distinguishes between free and dependant objects. While free objects can be directly modified either using the mouse or the keyboard, dependant objects adapt to changes of their parent objects. Thereby, it is irrelevant in which way (mouse or keyboard) an object was initially created!
Construction Protocol
Select Construction Protocol under the View tool.
In the construction protocol you can see in what order the objects have been constructed. At the bottom of the construction protocol table there are a set of navigation button that can be used to display the construction sequence of the objects. The buttons are easily recognized from a common recorder.
CheckBox to Show/Hide Objects
A common use of the CheckBox tool in GeoGebra is to allow objects to hidden or revealed. We connect a checkbox “Show chair” to the chair.
Select the tool “CheckBox to Show/Hide Object”. Click on the Grahics View on a optional position. A checkbox dialog window is opened.
Write “Show chair” in the Caption textfield and select all objects for the chair in the “Select objects in construction or choose from list” scrollist. Press “Apply” button.
The checkbox “Show chair” is checked. When you uncheck the checkbox with the mouse, the chair is hidden!
In the algebra view there is a Boolean variable created with a variable name in alphabetic order and a value true. When the checkbox “Show chair” is unchecked, the Boolean variable gets the value false.
Exercise
Construct a stick man.
Exercise
Make a pentagram. Start with the tool Regular Polygon (Pentagon). Connect all the corners on the pentagon.
Hide the pentagon object.
Congruent constructions
Two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. An isometry of the plane is a linear transformation which preserves length.
The Euclidean geometry (Euclidean geometry, see chapter “Geometry buttons/tools: characteristics and concepts”
later on) include five types of isometrics: translation, rotation, reflection, glide reflection, identity. Reflection or mirror isometrics can be combined to produce any isometrics.
Mirroring in a line
A point and its mirror point have the same perpendicular distance to the line.
Open the View tab and uncheck the tool Axes or right-click anywhere in the drawing pad and uncheck Axes.
Enter a line between the points A and B.
Enter a free point C.
Use the tool “Reflect Object in Lin”. Click on the point C and then on the line. The mirror point C' is created.
In order to distinguish between the free point C (the point you can drag) and the dependent point C', you can change the look of the points. Right-click on C and choose Object Properties. Change colour under the tab Colour. Change the size and the appearance under the tab Style.
Put a trace on both points by right-clicking on them and checking Trace On. Draw a picture by dragging the point C.
You can erase the picture drawn by zooming in or out, use the mouse wheel or the tools in the tool bar.
An image has a rotational symmetry if you can rotate the image around some point and get the same image. An image has a reflection symmetry if you can reflect the image in some line and get the same image.
Translation
The red arrow is called a vector. A vector has a direction and a length. If you check the check box you can see that all the vertices of the polygon are translated along the same vector. The gray arrows are all parallel.
You make a vector in GeoGebra by using the tool Vector between Two Points .
You make a translation by using the tool Translate Object by Vector . Click on the object you want to translate and then on the vector. The object itself is not translated but a translated copy of the object is created.
Rotation
In order to rotate an object you need an angle.
Make two segments with one common endpoint A. Use the tool Segment between Two points .
Use the tool Angle . Click on one of the segments, then on the other segment. An angle called α appears (α is the first letter in the Greek alphabet).
Create a geometrical object, a circle or a polygon.
Use the tool Rotate Object around Point by Angle . Click on the geometrical object; then on the point A; then on the angle α. You can click either in the drawing pad or in the algebra view.