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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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GeoGebra What is GeoGebra? Dynamic Mathematics software built by and for Mathematics instructors and students Open Source, meaning Its free! Its supported by a community of like- minded people

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GeoGebra Why GeoGebra? 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:

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GeoGebra 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 Innovative

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GeoGebra Past & Present

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GeoGebra Computer Algebra Systems Derive Mathematica Scientific Notebook

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GeoGebra Dynamic Geometry Software

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GeoGebra TI-92: DGS and CAS Graphing Calculators HP 49g TI-84 TI-Nspire/CAS System

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GeoGebra

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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

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

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GeoGebra Visitors on User Forum: Total posts | Total topics 5796 | Total members 3155

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GeoGebra GeoGebra Home Page

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GeoGebra Start up with GeoGebra Webstart Help 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

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GeoGebra GeoGebra User Forum

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GeoGebra GeoGebraWiki – Free Materials

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GeoGebra GeoGebra = Geometry + Algebra Geometry Window Algebra Window

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GeoGebra Getting to know the user interface Tool bar Toolboxes group Similar tools Help for the selected tool The Move tool the most important tool on the menu! Input field Input field assists

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GeoGebra Spreadsheet View has being added Algebra View Toolbar Graphics View Input Bar Spreadsheet View

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GeoGebra Spreadsheet in GeoGebra Dynamic Table of Values Regression Lines & Polynomials Binomial Distribution

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GeoGebra Lets 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! SymbolicGeometric Extra credit: On the menu bar Select View, then Grid

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GeoGebra Add a Drawing to a Document Click on File, Export, Drawing Pad to Clipboard Start Word, then Edit, Paste – Youre done! Resize and position as desired

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GeoGebra Context Menu Context menu Right-click any object in either window Notice the Properties… item at the bottom

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GeoGebra 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) Extra credit: Modify the labeling of point A to Value

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GeoGebra ActivityActivity – 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) = x 2 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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GeoGebra 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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GeoGebra ActivityActivity – construct a rhombus (a rhombus is a quadrilateral with all sides equal in length) METHOD 1 (File | New Dont 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 2 nd circle centered at B out to point A Draw a 3 rd 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 Tools Used:

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GeoGebra Activity – construct a rhombus (a rhombus is a quadrilateral with all sides equal in length) METHOD 2 (File | New Dont 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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GeoGebra ActivityActivity – 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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GeoGebra 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 Tools Used:

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GeoGebra 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 Now the fun part! Move point C and what (doesnt) happen?

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GeoGebra 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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GeoGebra 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 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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GeoGebra RiemannRiemann Approximations - part 1 Type the following into the Input field: f(x)=4x-x^2 a=0 b=4 Area=Integral[f,a,b] Type the following into the Input field: n=10 UpperSum=UpperSum[f,a,b,n] LowerSum=LowerSum[f,a,b,n] Adjust the colors of Upper & LowerSum

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GeoGebra 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 GeoGebras 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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GeoGebra 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

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GeoGebra 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!

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GeoGebra 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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GeoGebra 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 GeoGebra Future

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GeoGebra 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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GeoGebra Next year: CAS, 3D

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

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