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**GeoGebra Dynamic Mathematics for Schools Auston B. Cron**

PSJA North High School South Texas College CAMT 2009

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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.

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**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.

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**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

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GeoGebra Past & Present

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**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

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**Dynamic Geometry Software**

There are many dynamic geometry software programs including Geometer’s SketchPad.

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**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

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**GeoGebra has received many awards.**

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**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

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**World Wide User Community**

43 languages 190 countries 400,000 visitors per month

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**Visitors on www.geogebra.org**

As of June 13, 2009. User Forum: Total posts 26994 | Total topics 5796 | Total members 3155

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**GeoGebra Home Page www.geogebra.org/**

The Geogebra home is your gateway to: Installation Examples Forum: Ask questions, get answers, fast! Much, much more…

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**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!

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**GeoGebra User Forum www.geogebra.org/forum**

The support for Geogebra is second to none. Ask questions, get answers, fast!

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**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).

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**GeoGebra = Geometry + Algebra**

Algebra Window The algebra and geometry windows are tightly coupled, Pretty much 1-1! Geometry Window

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**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

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**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

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**Spreadsheet in GeoGebra**

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

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**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

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**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.

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**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

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**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

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**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:

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**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:

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**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:

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**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:

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**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:

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**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:

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**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?

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**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]”

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**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!

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**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.

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**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

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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

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**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.

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**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.

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**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]

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**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

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GeoGebra Future

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**Future Plans Make it Easier, Easier, Easier!**

Symbolic Algebra Window (CAS) 3D Extensions Split views to allow rearranging Independent sizing Possible 4 corner model

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Next year: CAS, 3D

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**GeoGebra Institute of Ohio**

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

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