Begin with GeoGebra 1Core Elements

Table of Contents

Getting StartedInstalling/enabling GeogebraInstallation WITH Internet accessInstallation WITHOUT Internet accessGeoGebra version 4GeoGebra version 5Learning PhasesIntroduction of GeoGebraUser Interface/Main Window of GeoGebraBasic Use of GeoGebra ToolbarDrawing without Mathematics

Construction ProtocolCheckBox to Show/Hide Objects

Numeric FoundationsCreating dynamic worksheets, mathletsUsing GeoGebra AnimationGeometry buttons/tools: characteristics and conceptsBasic geometric constructions, connection between geometry and algebraLinear functions, polynomials of 1st degreeQuadric functions, polynomials of 2nd degreeSpreadsheet view - statisticsUsing a powerful markup languageFamous patterns and problems: Sierpinski triangle, Fibonacci series, normal distribution and others.

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Getting started

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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 GeoGebraDifferent 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 accessThere are a couple of webbased versions available: WebStart, AppleStart and GeoGebraPrim.

GeoGebra WebStartOpen 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 AppletStartOpen 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

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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 InstallersYou 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 5Find the latest notes about the currently developed GeoGebra 5.0 beta version ( June 2012) in the documenthttp://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 JOGL2http://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 PhasesAn 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 GeoGebraGeoGebra 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.

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User Interface/Main Window of GeoGebraStart 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 5.0This 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 ToolbarActivate 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

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Tool Menu – Menu tabs for navigation for GeoGebra Toolbar – Specific for GeoGebra Views

Algebra View

Geometry View. Drawing Pad

Input Field

Input Options

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”.

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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

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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 ObjectsA 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.

ExerciseConstruct a stick man.

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ExerciseMake a pentagram. Start with the tool Regular Polygon (Pentagon). Connect all the corners on the pentagon.Hide the pentagon object.

Congruent constructionsTwo 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 lineA 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.

TranslationThe 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.

RotationIn 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.

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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.

Parallel and perpendicularIn the Euclidean geometrical theory (Euclidean geometry, see chapter “Geometry buttons/tools: characteristics and concepts” later on), there is only a small collection of self-evidently true axioms and derive, in a logically sound manner, the consequences of these, known as theorems. In GeoGebra we have a collection of buttons/tools which correspond to theorems among those “Parallel Line” and “Perpendicular Line”.

Create three optional points with the tool “Points”.Choose the tool “Move” . Move the three points!Select the tool “Parallel Line” . Click on the point C and then on the blue line; a black line appears. Select the “Move” tool again and move the three points. Describe in detail how the two lines are related.Click on “Reset Construction” in the upper right corner.

Select the tool “Perpendicular Line” . Click on the point C and then on the blue line; a black line appears. Select the “Move” tool again and move the three points. Describe in detail how the two lines are related.Click on “Reset Construction” in the upper right corner.Select the tool “Perpendicular Line” . Click on the point B and then on the blue line; a black line appears. Move the points!

Using the Input FieldGeoGebra offers algebraic input and commands in addition to the geometry tools. Every tool has a matching command and therefore, could be applied without even using the mouse. GeoGebra offers more commands than geometry tools. Therefore, not every command has a corresponding geometry tool!Check out the list of commands next to the input field (in the lower right corner of the main window) and look for commands whose corresponding tools were already introduced so far.

Use the Input Field and construct the chair again with commands.Input: A = (0, 0). The point A is created.Input: (0, 2). The point B is created. If a specific name is not given the objects are not named in alphabetical order. Input: C = (2, 2)Input: D = (0,-2)Input: E = (2, -2)Input: Segment[A,B]Input: Segment[B,C]Input: Segment[A,D]Input: Segment[B,E]

The same “chair” as before is constructed. The commands are instantly visualized in the geometry view.

Properties of objectsChange properties of objects in order to improve the construction’s appearance (e.g. colors, line thickness, auxiliary objects dashed,…). Right click on the chair objects (points, segments) and select the Objects Properties alternative. A properties window is opened.

Draw text – Tool Slider -> ABC – Insert Text

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Enter the desired text into the appearing window.

Numeric FoundationsMathematics is mainly about digits and numbers and their connections, patterns and changes. So let us return to this main track, the numbers.

Visualizing Integer Addition with the Number LineDouble-click on any of the GeoGebra WebStart icon, GeoGebra Installer icon or links to GeGebra4/GeoGebra5. The GeoGebra tool opens the standard window.

Prepare a horizontal number line.Right click on the Graphics View/Drawing Pad and select the Graphics View properties. Hide the yAxis (uncheck the “Show yAxis” checkbox) and give the xAxis the range from min = -10 to max = 10.

Use sliders to show and modify a variable and a variable value.Activate/click on the Slider tool in the toolbar and locate and click the mouse cursor on the geometry view/drawing pad.

Right click on the slider icon and open the Object Properties. Set the slider range with the Min and Max values, e.g. the default values -5 and 5 and Increment 1 (to get integer values). As all objects the slider variable name is given the small letter a in alphabetic order. Activate/click the Move tool in the toolbar

and then use the mouse to change the value for the slider/variable with the knob/pin on the slider. Click on the slider with the mouse cursor and drag the slider to an optional location on the drawing pad.

Create another slider/variable b with the same value range as the slider/variable a. The algebra view is open so the sliders/variables a and b with the current values are presented as Free Objects.

Create the point Origo for the 0 on the number line as a reference for the start of the number system. Use the command

Origo = (0, 0)

in the Input field or activate and drag the tool to the value 0 on the number line.

Visualize the variable a with the startpoint A and the endpoint B with an arrow (vector) with the name aVector along the number line. Put those objects one unit above the numberline, y-coordinate = 1, to get a better visualibility. Write the following commands in the Input field:

A = (0,1)

B = A + (a,0)

aVector = Vector[A,B] or use the tool “Vector between two Points”, subtool to “Line between to Points”

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The purpose with the arrow /vector is to get a visualization of the signed number a so the name aVector can be hidden. Activate the Move tool and right click on the aVector object and uncheck “Show Label”.

Use the mouse to change the value for the slider/variable a with the knob/pin on the slider and observe that the value is visualized with the arrow/vector length. Don’t forget to activate the “Move” tool.

Visualize the variable b with the startpoint C and the endpoint B with an arrow (vector) with the name bVector along the number line. Put those objects one more unit above, y-coordinate = 2, to keep the visualibility. Write the following commands in the Input field:

C = B + (0,1)

D = C + (b,0)

bVector = Vector[C,D] or use the tool “Vector between two Points”, a subtool to “Line between two Points”

The purpose with the arrow /vector is to get a visualization of the signed number b so the name bVector can be hidden. Activate the Move tool and right click on the bVector object and uncheck “Show Label”.

Use the mouse to change the value for the slider/variable b with the knob/pin on the slider and observe that the value is visualized with the arrow/vector length.

Visualize the sum a + b. The slider/variable a and b can now both be changed with the mouse to visualize the sum of a + b. The x coordinate for the point D is the sum of a and b, sum = a + b. Project the x coordinate of D, Sum, and a projection line of D on the number line. The name Sum must have a capital letter S because it is the name of a point in GeoGebra. Use the commands

Sum = (x(D),0)

Segment[D, Sum] or use the tool” Segment between two Points”, a subtool to “Line between two Points”

Uncheck the label for the segment (default) name c. There is no need to visualize the name c for the moment.

To get a continuation of the visualized addition, make a segment line also between B anc C

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Segment[B, C]

Uncheck the label for the segment (default) name d. There is no need to visualize the name d for the moment.

The get an even more visualization make the segment lines between B and C and D and Sum. Activate the “Move” tool and wright click on the segment objects B and C and D and Sum. Open “Object Properties”, select the “Style” tab and choose a “dashline style” in the scrollist.

Insert the algebraic expression of the addition

Use the subtool “Insert Text” to the tool “Slider”

Activate the “Insert Text” tool and click on the drawing pad. A separate Edit window is opened in which (already defined) objects can be choosed with the Object button. Click the Object button and select the variable a from the scrollist. A dynamic textfield for a is shown in the Edit window and the value is shown in the Preview window. Press OK button. The dynamic value of the variable a is shown on the drawing pad. Activate the “Move” tool and drag the dynamic value of a to a suitable location on the drawing pad. Change the slider/variable a value to see that the dynamic text value is following.

Now we want the + operator in the a + b expression. This is a static text and this is made with a quote expression in GeoGebra (like a static String in Java).

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Use the till tool “Insert Text” again. Write the string “ + “ in the Edit Window. Press OK. Activate the “Move” tool and drag the static “value”, operator +, to a suitable location on the drawing pad.

Use the tool “Insert Text” again for the dynamic value of the object/variable b. Activate the “Move” tool and drag the dynamic value of b to a suitable location on the drawing pad. Change the slider/variable b value to see that the dynamic text value is following.

Use the tool “Insert Text” again for the static text for the assignement operator =.

Now we need the dynamic value for the a + b sum. We need to create a variable for that, e.g. sum. The value for sum is the x coordinate for D, see above. Be aware, the object Sum above (with a capital letter S) is a point! Now we need a variable sum (with a small letter s). Write the following command in the Input field:

sum = x(D)

Use the tool “Insert Text” again for the dynamic value of the object/variable sum. Activate the “Move” tool and drag the dynamic value of sum to a suitable location on the drawing pad. Change the slider/variable a and b value to see that the dynamic text values are following.

Decorate the algebraic expression and the geometric visualization.To even more increase the visualization decorate the a slider/variable and attached arrow/vector with a blue color, the b slider/variable and attached arrow/vector with a red color and the point Sum and the variable sum with a green color.Activate the tool Move and wright click on the specific objects. Open the Color tab and select the color from the color palett. Also for the dynamic values for a, b and sum open the Text tab for those Object Properties and select suitable font (e.g. Very Large, B, …). For the arrow/vector objects a higher Style/Line Thickness (e.g. 7) can be chosen.

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Visualizing Integer Subtraction with the Number Line The calculus rules for addition and subtraction are of course built in GeoGebra for the commands including the operator + and -. (Geogebra uses the development software Java and all its foundations.)Delete the object D from the Algebra View and write the subtraction command

D = C – (b,0) (The main change.)

Rewrite the commands

bVector = Vector(C, D) (Hide the bVector object label.)

Difference = (x(D), 0) (Point Difference.)

Segment(D, Difference) (Hide the Segment object label.)

difference = x(D) (Variable difference.)

Change the text object + to the static subtraction operation - and update the dynamic text object for the variable difference.

Decorate the objects with color and style as the for the visualize addition case.

Visualizing Integer Multiplication of Natural NumbersDouble-click on any of the GeoGebra WebStart icon, GeoGebra Installer icon or links to GeGebra4/GeoGebra5. The GeoGebra tool opens the standard window.

Use sliders to show and modify a variable and a variable value.Hide the Axes view. Activate/click on the Slider tool in the toolbar and locate and click the mouse cursor on the geometry view/drawing pad.

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Right click on the slider icon and open the Object Properties. Set the slider range with the Min and Max values, e.g. the default values 1 and 10 and Increment 1 (to get integer values). Set the slider width = 500. As all objects the slider variable name is given the small letter a in alphabetic order. Change the slider variable name to factor1. Click on the slider with the mouse cursor and drag the slider to an optional location at the bottom of the drawing pad.

Create a point A. Put in the left lower corner on the drawing pad.

Create a horizontal segment from A with the tool “Segment with a Given Length from Point”. The segment object is named a. Give the segment length the slider variable name factor1. The endpoint of the segment is automatically called B in alphabetic order. With the slider variable factor1 the segment a (segment AB) can now be given different length values from 1 to 10.

Draw vertical lines through A and B, perpendicular to segment a. Use the tool “Perpendicular Line”. Activate the segment object a and then the point object A and B with the mouse. The perpendicular line objects are named b and c.

Create another slider/variable factor2 with the same value range as the slider/variable factor1. Set this slider orientation to Vertical. Click on the slider with the mouse cursor and drag the slider to an optional location to the left side of the drawing pad.

Use the tool “Circle with Center and Radius” to connect the line b to the slider variable factor2. Activate the tool “Circle with Center and Radius” with the mouse and click the mouse on the point A. The circle object is named d. Give the circle the slider variable factor2.

Use the tool “Intersect Two Objects” to create a point C as the intersection between the circle object d and line object b. Activate the tool “Intersect Two Objects” with the mouse and click the mouse on this intersection. The intersection point will be called C automatically.

Use the tool “Parallel Line” to create a line through C parallel to segment a. Activate the tool “Parallel Line” with the mouse and click the mouse on the segment a and then on the point C. This line object is named e.

Use the tool “Intersect Two Objects” to create a point D as the intersection between the line object e and line object c. Activate the tool “Intersect Two Objects” with the mouse and click the mouse on this intersection. The intersection point will be called D automatically.

Use the tool “Polygon” to create the polygon/square ABDC.

Right click on the specific object, uncheck “Show Object” in order to hide all line objects b, c e, circle object d and segment object a. Uncheck “Show Label” in order to hide labels of the (polygon) segment objects.

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b (Visible in the drawing pad. The picture is cut down here.)

c (Visible in the drawing pad. The picture is cut down here.)

Change the sliders to factor1 = 10 and factor2 = 10.

Divide the polygon/square into 10x10 segments/parts using the slider variable factor1 and factor2.

Use the Sequence command. Open the built-in Command list with the tab to the right of the Input field. Select “All Commands” and “Sequence”. Open the “Show Online help”.

Select “Segment” and open the “Show Online help”.

Make a sequence list of segments between A and C (vertical segments) and then between A and B (horizontals segments). Write the following commands in the Input field.

Sequence[Segment[A+i*(1,0), C+i*(1,0)], i, 1, factor1]

Sequence[Segment[A+i*(0,1), B+i*(0,1)], i, 1, factor2]

The following segment grid is created. Change the slide variables factor1 and factor2 to check that the number of segments in the grid is changed.

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Insert the algebraic expression of the additionUse the subtool “Insert Text” to the tool “Slider” Activate the “Insert Text” tool and click on the drawing pad. A separate Edit window is opened in which (already defined) objects can be choosed with the Object button. Click the Object button and select the variable factor1 from the scrollist.A dynamic textfield for a is shown in the Edit window and the value is shown in the Preview window. Press OK button. The dynamic value of the variable a is shown on the drawing pad. Activate the “Move” tool and drag the dynamic value of factor1 to a suitable location on the drawing pad. Change the slider/variable a value to see that the dynamic text value is following.

Now we want the * operator in the factor1 * factor2 expression. This is a static text and this is made with a quote expression in GeoGebra (like a static String in Java).

Use the till tool “Insert Text” again. Write the string “ * “ in the Edit Window. Press OK. Activate the “Move” tool and drag the static “value”, operator *, to a suitable location on the drawing pad.

Use the tool “Insert Text” again for the dynamic value of the object/variable factor2. Activate the “Move” tool and drag the dynamic value of factor2 to a suitable location on the drawing pad. Change the slider/variable factor2 value to see that the dynamic text value is following.

Use the tool “Insert Text” again for the static text for the assignement operator =.

Now we need the dynamic value for the factor1 * factor2 product. We need to create a variable for that, e.g. product. Write the following command in the Input field:

product = factor1 * factor2

The following picture visualize the multiplication 7 * 8 = 56

Decorate the algebraic expression and the geometric visualization.To even more increase the visualization decorate the factor1 slider/variable with a blue color, the factor2 slider/variable with a red color and the variable product with a green color.Activate the tool Move and wright click on the specific objects. Open the Color tab and select the color from the color palett. Also for the dynamic values for factor1, factor2 and product open the Text tab for those Object Properties and select suitable font (e.g. Very Large, B, …).

The points A, B, C and D can optional be hidden.

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FractionsA fraction is a number that describes part of a whole number. Because fractions are numbers just like 7, 2 or 99, they can live on a number line.A fraction is made up of a numerator and a denominator:numeratordenominatorThe numerator tells how many of those parts you have.The denominator tells how many parts each unit interval has been cut into.

GeoGebraCreate sliders for the nominator n and denominator d.Open the tool “Insert Text” and use the Latex Formula , see chapter “Using a powerful markup language” to visualize the fraction nd

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Visualize the denominator d with a blue arrow (Vector object) and the fraction nd with w red arrow (Vector object).

If the nominator > denominator we have an improper fraction. This fraction can be changed into a mixed number with a whole part and a part fraction with a nominator < denominator.A fraction can be "reduced", like 21 and 7 in the picture above and have at least one common factor (other than 1). This GCD, Greatest Common Divisor, can be calculated in GeoGebra with the function GCD(), g = GCD(n,d). Reduce the nominator n and denominator d with n1 = n/g and d1 = d/gIn GeoGebra the whole part can be calculated with the built-in function floor(), whole = floor(n1/d1), and the remaining nominator part = n1 – whole * d1.When those expressions are written in the Input field, use the “Keep Input”, Alt+Enter to get this feature. The value of g will then not effect the value of n and calculate a new value for g.See the “GeoGebra Documentation”:“Enter: evaluates the current row depending on the selected tool in the toolbar: =, numeric, keep input. Ctrl+Enter switches between Numeric and Evaluate. Alt + Enter switches between Keep Input and Evaluate.”Use the Latex Formula to visualize the origin fraction, the reduced fraction and the mixed fraction.

Adding fractions1. Find common denominator for the fractions2. Rename one or both fractions with the common denominator

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3. Add nominators.4. Reduce and get mixed form.

Subtracting fractionsTo subtract fractions similar steps as for addition fractions are required:1. Find common denominator for the fractions2. Rename one or both fractions with the common denominator3. Subtract nominators.4. Reduce and get mixed form.

Multiplying fractionsIt is a three-step process to multiply mixed numbers:1. Convert mixed numbers into fractions2. Multiply across3. Simplify: reduce and rename

Dividing fractionsIt is a four-step process to divide mixed numbers:1. Convert mixed numbers to impropers2. Flip second fraction and change divison to multiplication3. Multiply across4. Simplify: reduce and rename

Geometry buttons/tools: characteristics and conceptsNavigate to this link and read an overview http://en.wikipedia.org/wiki/Euclid about Euclid of Alexandria, 300 BC, a great mathematician whose life we know very little about but whose work has give us a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.He constructed a geometrical theory using a minimum of assumptions. That is to say he wanted to assume only a small collection of self-evidently true axioms and derive, in a logically sound manner, the consequences of these, known as theorems. His assumptions were that it is possible to1. draw a straight line from any point to any point,2. extend a finite straight line indefinitely in a straight line,3. draw a circle with any center and any diameter.

In Geogebra the first of these assumption is implemented by the button , “Segment between two points”, and

the first two combine to provide the button , “Line through two points”. We shall mostly consider indefinite

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straight lines. The third is the button , “Circle with center through point”. When Euclid talks of "any diameter" he means any previously constructed length. That is, he may use two existing points to open his compasses against. He does not mean any diameter we can imagine or perhaps define algebraically.Furthermore, he assumed that4. all right angles are equal to one another, and5. that if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which the angles are less than the two right angles.This last assumption is sometimes known as the parallel postulate.We note in passing that one strength of GeoGebra is the conjunction of algebraic and geometric views of mathematics. For example, one can type in the equation of a parabola as y = x2, then place a point on the curve and drag it around. The tangent line can be illustrated using the button “Tangents”, and the equation of this line recovered from the algebra window.

GeoGebra provides many buttons/tools which have nothing to do with geometry. For example buttons about measurement (“Distance”, “Angle”, “Polygon (area of) , “Relation between two objects”. Those concepts are not considered within Euclidean Constructions.

Then we try to identify a set of axiomatic buttons/tools from which all the others can be constructed. If the functionality of another button/tool can be constructed by axioms or previously constructed buttons then it can be called a theorem button. Consider the following collection of buttons/tools: “Line through two points”, “Circle with center and through point”, “Conic through five points”. These three provide one way to create all three of the classic identifiable objects in plane geometry, the line, circle and a conic section. Notice that the first two are special cases of the last.

In addition we may place a "New point" either unconstrained in space or to be constrained on one of these

objects. We may also find the intersection of two objects and this button places new point(s) there. This latter operation applies to any two of line, circle and conic and it returns between zero and two new points, which are automatically assigned names. The intersection button is needed so that points can be created which are related to two existing objects. Without this, or something similar, there is no way to establish relationships between objects. Notice that implicit in identifying a point of intersection is an assumption of continuity.

Then we have a collection of buttons/tools which correspond to theorems. Those are: “Perpendicular Bisector”, “Perpendicular Line”, “Parallel Line”, “Midpoint or Centre”, “Angular Bisector”, “Circle with Centre and Radius”, “Semicircle through Two Points”, “Circle through Three Points”, “Mirror Object in Point”, “Mirror Object at Line”,

“Tangents”, “Polar or Diameter Line”. In the case of the Circle with center and radius, , GeoGebra expects the user to type in an algebraic distance. Hence the status of this button as a purely geometric theorem is questionable. In order to include this button we would need to include the "Distance tool", or use GeoGebra's "Segment between two points" which returns the length of the segment.

In addition it might be advantageous to build a basic arithmetic system by identifying the length of a line segment with a number, starting with an arbitrary agreed unit. Then, in a systematic way, to construct the geometric counterparts of the arithmetic operations such as addition, multiplication, and so on. It is not at all clear which operations and hence numbers are constructible in this way. This combination of algebra and geometry is a classical topic and the basic geometric constructions for addition, multiplication.

In GeoGebra a facility for user-defined “buttons/tools" is provided. This allows a construction to be encapsulated as a new button, providing the opportunity for the above, or other, constructions to be implemented. Furthermore, an interface is provided in which existing buttons can be "switched off" or the order rearranged. This allows an application in GeoGebra to be configured with a web page containing only a small number of buttons, and from

these the task to demonstrate a particular construction. For example, given only , and ,

show how we can make the button equivalent to . While this removes the complaint "we already have a button for this" it also removes the freedom to make choices of their own about the any mutual dependencies.

Basic geometric constructs, connection between geometry and algebra

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Bisect an AngleThis is a most common geometric construction. Use GeoGebra to do it.Double-click on any of the GeoGebra WebStart icon, GeoGebra Installer icon or links to GeGebra4/GeoGebra5. The GeoGebra tool opens the standard window.Hide the Axes view.

Use the tool “New Point” and create a point AUse the tool “Segment between Two Points” and create segments to a point B and a point C. In order to hide the segments name (a and b), uncheck “Show Labels”. Those segments are the angles legs.Use the tool “Angle” and draw the angle BAC by clicking on the points B, A and C. Right click on the angle object, open “Objects Properties”, “Basic” and select “Show Label”: Name

Use the circle tool “Compas”. Activate the tool, click on point A and point B and click on point A to fix the the center of the circle to point A. The circle with the radius = length of segment AB (optional made shorter than segment AC) is intersecting the segment AC.Use the tool “Intersect Two Object” to create/fix the point D.

Create a segment object between C and D and decorate the segment as a dashed line. Create a point E on the segment CD with an optional location but nearer C than D.

Use the circle tool “Compass”. Activate the tool and click with the mouse on point D, point E and the again on point D to fix the center of the circle to point D.Use the circle tool “Compass” again. Activate the tool and click with the mouse on point D, point E and the again on point C to fix the center of the circle to point C.Use the tool “Intersect Two Object” to create/fix the intersection point F between the circles.Use the tool “Ray through Two Points” and create a line between A and F.

Hide all the circles objects, segment CD and point E with a right click with the mouse on the objects and uncheck “Show Object”.

Use the tool “Angle” to create the angels DAF and FAC. Activate the “Angle” tool and click with the mouse on point D, A and F. Activate the “Angle” tool and click with the mouse on point F, A and C.Right click on the name (α and β) and value for the angels, open “Object Properties” and make some decoration for the angels.

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The values of the angles α and β are the same.Use the tool “Move” and move the point B and C. As an example you can get a right angel divided into two angles with the value 45º.

Eqvilateral TrianglesAnother classic geometry construction is eqvilateral triangels. Here is an abstract from http://en.wikipedia.org/wiki/Euclid%27s_Elements that gives an overview of Euclid's Elements a collection of 13 books written by the Greek mathematician Euclid in Alexandria about 300 BC.

A proof from Euclid's Elements that, given a line segment, an equilateral triangle exists that includes the segment as one of its sides. The proof is by construction: an equilateral triangle ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle.

Again GeoGebra is perfect for this construction.Double-click on any of the GeoGebra WebStart icon, GeoGebra Installer icon or links to GeGebra4/GeoGebra5. The GeoGebra tool opens the standard window.Hide the Axes view.

Use the “New Point” tool to create a point A and point B.

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Use the Circle tool “Compass” to draw a circle with the radius = distance between A and B. Activate the tool “Compass” and click with the mounse on point A and B and then on A again to make A as the circle center.Repeat the same scenario for a similar circle but center on the point B.Create/fix the upper intersection point between the circles with the tool “Intersect Two Objects”.

Create segment AB, AC and BC with the tool “Segment between Two Points”. Hide the Circle Object.Use the tool “Angle” to control the eqvilateral triangle angels. Activate the toll “Angle” and click on the point B, A and C to get the angle BAC.

Theorem of PythagorasThis is a classic connection between a geometry construction and algebraic expression.

Make a right-angled triangle using these tools: “Line through Two Points”, “Perpendicular Line”, “New Point” and “Polygon”. Use the tool “Angle” to show the right angle. Move the points! The triangle should remain right-angled. The points of the polygon should be placed in a counterclockwise order.

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In order to demonstrate Pythagoras' theorem we must show the square of the hypotenuse and the sum of the squares of the shorter sides. We hence introduce two variables to store these values.There is a standard way of writing subscripts and superscripts in GeoGebra; this way of writing is used in a number of mathematics programs.

Superscripts are written using ^     x^2 is shown like this x2 or use the symbol table opened by the α icon in the right end of the Input field

Subscripts are written using _     c_1 is shown like this c1

You write a ^ by pressing ShiftAlt^, the character ^ may not show up until you press the next character or spacebar.

Use the Input field at the bottom of the window to store the square of the hypotenuse in the variable hypKvad1=b2 and the sum of the squares of the shorter sides in hypKvad2=a2+c2. Observe the values of hypKvad1 and hypKvad2 I the algebra view as you move the points of the triangle.

Area of a triangleAnother fundamental connection between geometry and algebra is calculation of the area of a triangle. Open GeoGebra in standard view with algebra view, input field and coordinate axes (View menu).

Create 2 horizontal lines: Write e.g. y= -1 (line name a) and y = 4 (line name b) in the Input field.Create two points (tool “New Point”) on line y = -1 (point name A and B) and one point on the line y = 4 (point name C). Connect the points with the tool “Segment between Two Points 2” (sepment name c, d and e). Draw a perpendicular line between the point on line y = 4 (point C) and the line y = -1. Use the tool “Intersect Two Objects” for the intersection point between the perpendicular line and line y = -1 (point D).Use the tool “Segment between Two Points 2” between point C and D. Open the “Objects Properties” for the segment CD and make the style as “dashed line”.Hide the lines y = -1 (line a), y = 4 (line b) and the perpendicular line. Right click the objects and uncheck the “Show Object”.Rename the segment between A and B to the name b or base and the segment between C and D to the name h or height. Check the “Object Properties”/“Show label”/Name & Value” for the segment b and h. Write the triangle area formula in the Input field: Area = base*height/2.

Use the “Insert Text” tool to create a text “Area =” and the Area objects value as a Latex Formula. Move the point C along the (now hidden) line b (y = 4) and the points A and B along the (now hidden) line a (y = -1). Observ how the “height” is following and can be both “inner” and “outer” of the triangle.

Area of a parallelogram

Area of rectangular prisma

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Area of a cylinder

Area of a cone

Area of a pyramide

Area of a sphere

Making a demonstration

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Linear functions, polynomial of 1st degreeThe term linear function is sometimes used to mean a first-degree polynomial function of one variable. These functions are known as "linear" because they are precisely the functions whose graph in the Cartesian coordinate plae is a straight line.Such a function can be written asy = kx + l(y – y1) = k(x − x1)ax + by + c = 0The form y = kx + l is called slope-intercept form, where k and l are real constants and x is a real variable. The constant k is often called the slope or gradient, while l is the y-interept, which gives the point of intersection between the graph of the function and the y-axis. Changing k makes the line steeper or shallower, while changing l moves the line up or down.The form y – y1 = k(x – x1) is called point-slope form where k is the is the slope and (x1,y1) is a given point on the line.The form ax + by + c = 0 is called the standard form where a, b and c are real values, coefficients.

Slope-intercept formOpen GeoGebra in standard view with algebra view, input field and coordinate axes (View menu). Create sliders for k and l.Write y = k*x + l in the input field.Exercise the line with different values for k and l on the sliders.

Point-slope formOpen GeoGebra in standard view with algebra view, input field and coordinate axes (View menu). Create a point A = (x1, y1)Write x1 = x(A) in Input fieldWrite y1 = y(A) in Input fieldCreate a slider for k.Write y – y1 = k*(x – x1) in the Input field.

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Standard formOpen GeoGebra in standard view with algebra view, input field and coordinate axes (View menu). Create 3 sliders a, b and c.Write a*x + b*y + c = 0 in the Input field.Make a checkbox for the line ax + by + c = 0

Write the “slope-intercept form” translation y = -ab

x - ca

of the standard form ax + by + c = 0 in the Input field.

Make a checkbox for the line y = -ab

x - ca

.

Change the values of a, b and c and toggle between the two checkboxes.

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Check the algebraic expression for y = -ab

x - ca

for the value b = 0!

Quadratic functions, polynomial of 2nd degreeA quadratic equation is an equation of a polynomial of degree two. When graphed, a quadratic equation makes a parabola with a vertical “symmetric axis” or “mirror line”.

A quadratic function can be expressed in three formats:

f(x) = ax2 + bx + c is called the general form,

f(x) = a(x – x1)(x – x2) is called the factored form, where x1 and x2 are the roots of the quadratic equation,

f(x) = a(x – x0)2 + y0 is called the vertex form (or standard form), where x0 and y0 are the x and y coordinates of the vertex, respectively.

To convert the general form to factored form, one needs only the quadratic formula to determine the two roots x1 and x2. To convert the general form to standard form, one needs a process called completing the square. To convert the factored form (or standard form) to general form, one needs to multiply, expand and/or distribute the factors.

General formThe general form of a quadratic equation is f(x) = ax2 + bx + c or y = ax2 + bx + c where a, b and c are constant coefficients and a≠0.

Open GeoGebra in standard view with algebra view, input field and coordinate axes (View menu). Create 3 sliders a, b and c.Write f(x) = a*x^2 + b*x + c or y = a*x^2 + b*x + c in the Input field.Exercise the line with different values for a, b and c l on the sliders.In this example a = 1, b = -2 and c = -3.

Roots of the equation ax2 + bx + c = 0 – Graphic solution

Use the tool “Intersect Two Objects” to find the intersection A and B between the parabola y = a*x^2 + b*x + c for the current values a = 1, b = -2 and c = -3: y = x2 – 2x -3 and the x-axis: y = 0. The x-coordinates for those intersections A and B are the roots to the equation: x2 – 2x – 3 = 0. x = -1 and x = 3

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The discriminant of a quadratic equation is used to determine if a quadratic equation has real or complex roots. The expression for the discriminant is b2 – 4acIf the discriminant is positive, the quadratic equation has two real roots. If the discriminant is zero, the quadratic equation has one real root. If the discriminant is negative, the quadratic equation has two complex roots. In “Begin with GeoGebra 3”, chapter “GeoGebra CAS solving equations” there is a complete description of the quadratic formula and the use of GeoGebra to solve equations.

ExerciseGive the sliders new values for a, b and c, giving the discriminant value zero (one real “double” root) and negative value (no real roots = no intersection between the parabola and the x-axis)

Intercept formThe intercept form or factored form of a parabolic equation is y = a(x-x1)(x-x2) where x1 is one x-intercept of the quadratic equation, x2 is the other x-intercept, and a indicates how steep the sides of the quadratic equation are. If x1 = x2, the quadratic equation intercepts the x-axis only once. Not all quadratic equations can be described using the x-intercept form.Open GeoGebra in standard view with algebra view, input field and coordinate axes (View menu). Create 3 sliders a, x1 and x2.In the Imput field, write the expression y = a*(x – x_1)*(x – x_2)

Vertex formThe vertex form of a parabolic equation is y-y0 = a(x-x0)2. The vertex of the quadratic equation is at the point (x0,y0). a shows how steep the sides of the quadratic equation are. Click on the points on the sliders in manipulative 4 and drag them to change the figure.Open GeoGebra in standard view with algebra view, input field and coordinate axes (View menu). Create 3 sliders a, x0 and y0.

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Using a powerful markup languageLaTeX is a document markup language and document preparation system for the TeX typesetting program. The term LaTeX refers only to the language in which documents are written, not to the editor used to write those documents. In order to create a document in LaTeX, a .tex file must be created using some form of text editor. While most text editors can be used to create a LaTeX document, a number of editors have been created specifically for working with LaTeX.In Geogebra there is a LaTeX Formula editing scrollist built in in the tool “Insert Text”. If the LaTeX Formula is checked the edit window can be used as a LaTeX enabled editor.

This textelement in the preview window can then be configured with tool “Insert Text” Object Properties in GeoGebra.

The \ (back-slash character) is heavily used in the LaTeX script language. Some important LaTeX commands are explained in following table. Please have a look at any LaTeX documentation for further information. LaTeX input Resulta \cdot b a⋅b\frac{a}{b} ab\sqrt{x} x√\sqrt[n]{x} x√n\vec{v} v⃗\overline{AB} AB−−−x^{2} x2a_{1} a1

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\sin\alpha + \cos\beta sinα+cosβ\int_{a}^{b} x dx ∫baxdx\sum_{i=1}^{n} i^2 ∑ni=1i2

This link http://en.wikipedia.org/wiki/Help:Displaying_a_formula#Basics gives more information about LaTeX.

LaTeX FormulaText[Object]A special feature in LaTeX is the command FormulaText[Object]. This command gives a LaTeX formula for the Object. In GeoGebra the Object can be an expression invoked in the Input field. e.g. a quadratic function. In this example we have the quadratic function y = ax2 + bx + c. This object will get an ordinary GeoGebra name in alphabetic order, e.g. e. The name can be renamed as all GeoGebra objects, e.g. to the name polynomial.The three coefficients a, b and c are implemented as sliders. In the Edit window in the “Insert Text” tool, check LaTeX Formula. Between the start-end $ $ symbol for the LaTeX expression, open the Objects scrolllist at the bottom of the Edit window and select the object e. The text object e is then translated to LaTeX and presented in the Graphics view where the “Insert Tool” has been positioned.The text object for y = ax2 + bx + c with the current values for a, b and c is presented in LaTeX format in Preview window.

The text object e: y = 2x2 – x – 1 can be configured with the properties Text and Color under “Object Properties”. The coefficients a, b and c can be given new values by the sliders and the current text object for the polynomial is presented. In the Edit window in the tool “Insert text” you can also create a LaTeX static text object written between “ “ characters, e.e “Polynomial 2nd degree:”.

Another way to create the LaTeX formula for the polynomial expression e, is to write the command “” + Formula[e] or “” + FormulaText[polynomial] in the tool “Insert text” Edit window.

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