Presentation on theme: "GeoGebra Dynamic Mathematics for Schools Auston B. Cron"— Presentation transcript:
1 GeoGebra Dynamic Mathematics for Schools Auston B. Cron PSJA North High SchoolSouth Texas CollegeCAMT 2009
2 What is GeoGebra?Dynamic Mathematics software built by and for Mathematics instructors and studentsOpen Source, meaningIt‘s free!It‘s supported by a community of like-minded peopleShort History of GeoGebraGeoGebra was created by Markus Hohenwarter in 2001/2002 as part of his master's thesis in mathematics education and computer science at the University of Salzburg in Austria. Supported by a DOC scholarship from the Austrian Academy of Sciences he was able to continue the development of the software as part of his PhD project in mathematics education. During that time, GeoGebra won several international awards, including the European and German educational software awards, and was translated by math instructors and teachers all over the world to more than 25 languages.Since 2006 GeoGebra is supported by the Austrian Ministry of Education to maintain the free availability of the software for mathematics education at schools and universities. In July 2006, GeoGebra found its way to the US, where its development continues at Florida Atlantic University in the NSF project Standard Mapped Graduate Education and Mentoring.
3 Why GeoGebra? For more on this go to: In the “Principles & Standards for School Mathematics” published by the NCTM, one of the principles states that:“Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning.”“Students can learn more mathematics more deeply with the appropriate and responsible use of technology.”For more on this go to:GeoGebra was created by Markus Hohenwarter in 2001as part of his master's thesis in mathematics education and computer science at the University of Salzburg in Austria. Supported by a DOC scholarship from the Austrian Academy of Sciences he was able to continue the development of the software as part of his PhD project in mathematics education. During that time, GeoGebra won several international awards, including the European and German educational software awards, and was translated by math instructors and teachers all over the world to more than 25 languages.Since 2006 GeoGebra is supported by the Austrian Ministry of Education to maintain the free availability of the software for mathematics education at schools and universities. In July 2006, GeoGebra found its way to the US, where its development continues at Florida Atlantic University in the NSF project Standard Mapped Graduate Education and Mentoring.
4 GeoGebra is Innovative Combination of elements ofdynamic geometry software (Sketchpad, Cabri)computer algebra systems (Derive, Maple)High technical portabilityGeoGebra is fully platformindependent (Windows,MacOS, Solaris, Linux)dynamic worksheets (html)Geogebra is free and runs on all computers
6 Computer Algebra Systems MathematicaDeriveThere are many types of Computer Algebra Systems the are being used by many. This are just of few that you may know.Scientific Notebook
7 Dynamic Geometry Software There are many dynamic geometry software programs including Geometer’s SketchPad.
8 Graphing Calculators TI-92: DGS and CAS HP 49g TI-84 Graphing calculators have progressed from initial tools like the TI-92;to the HP-49g and TI-84;Into the new TI-Nspire family.HP 49gTI-84TI-Nspire/CAS System
10 GeoGebra as a Pedagogical Tool Drawing ToolGeometric drawings for handouts & activitiesGraphs for worksheets, quizzes & testsIn-class Presentation ToolDynamically show relationshipsVisualization of abstract conceptsAuthoring Tool (Web Export)Interactive virtual manipulativesExploratory mathemathicsGeogebra meets a wide variety of educational needsAiding development of materialsIn-class use by the instructor to demonstrate concepts and relationshipsServe as an interactive component of assignments
11 World Wide User Community 43 languages190 countries400,000 visitors per month
12 Visitors on www.geogebra.org As of June 13, 2009.User Forum: Total posts 26994| Total topics 5796 | Total members 3155
13 GeoGebra Home Page www.geogebra.org/ The Geogebra home is your gateway to:InstallationExamplesForum: Ask questions, get answers, fast!Much, much more…
14 Start up with GeoGebra Webstart Click Help to get access to the GeoGebraHome page – connect to all things GeoGebraUser Forum - ask questions, get answers!Wiki – free pool of teaching materialsBefore doing anything else, let’s see where the help is!
15 GeoGebra User Forum www.geogebra.org/forum The support for Geogebra is second to none.Ask questions, get answers, fast!
16 GeoGebraWiki – Free Materials A great many geogebra materials are available on the Wiki.Available in many languages.You will need to explore (after you are settled).
17 GeoGebra = Geometry + Algebra Algebra WindowThe algebra and geometry windows are tightly coupled,Pretty much 1-1!Geometry Window
18 Getting to know the user interface The Move toolthe most importanttool on the menu!Input fieldHelp for theselected toolTool bar“Toolboxes” groupSimilar toolsThe 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 smallTriangle turns red, indicating that there is a drop-down list of tools.Input field assists
19 Spreadsheet View has being added ToolbarSpreadsheet ViewAlgebra ViewGraphics ViewThe 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 HohenwarterPretty much 1-1Input Bar
20 Spreadsheet in GeoGebra Dynamic Table of ValuesRegression Lines & PolynomialsBinomial DistributionWith the release of GGB 3.2, these feature have been added.
21 Extra credit: On the menu bar Select View, then Grid Let’s get busy!Select the Line through two points tool (notice the tool tips!)Move the cursor into the geometry window (AKA the drawing pad)Hold the mouse button down and notice that coordinates are displayed; release to set pointCreate a second point to complete the actionObserve that the points and line also appear in the algebra window!Symbolic↔GeometricNotice the menu bar; View and Options allow customization of most aspects of the user interface.This duality of representations is typical in geogebra – every object has an algebraic and a geometric representation.Extra credit: On the menu bar Select View, then Grid
22 Add a Drawing to a Document Click on File, Export, Drawing Pad to ClipboardStart Word, then Edit, Paste – You’re done!Resize and position as desiredAlso use the View menu to enable the grid.Many aspects of the user-interface are selectable in the View and Options menus.
23 Context Menu Right-click any object in either window Notice the Properties… item at the bottomNotice that when an object is selected in the geometry window it is also highlighted in the algebra window.Notice the Properties item at the bottom of the context menu.Context menu
24 Properties Dialog – object customization The Properties dialog is accessed via the context menu (or the Edit menu)Notice that this is a tabbed dialog; each tab contains a set of related propertiesTry this:Change the color of your line (Color tab)Change the style of your line (Style tab)The properties dialog is where you can change various properties of the object.Extra credit: Modify the labeling of point A to Value
25 Activity – Constructing a Parabola (Part 1: create basic interactive graph of a parabola) Type the following into the Input fielda=1f(x) = a x^2You now see the parabola f(x) = x2Right click on a=1 in Algebra window and select “Show Object”Use the Move Tool to change ‘a’Tools Used:
26 Activity – Constructing a Parabola (Part 2: create a family of curves) Right click on a=1 in Algebra window and select “Properties”Click on the “Slider” tab and change the increment to 1Click on the Function f, select “Show Trace” and closeMove the slider left and right.Tools Used:
27 Activity – construct a rhombus (a rhombus is a quadrilateral with all sides equal in length) METHOD 1 (File | New “Don’t Save” and turn off the grid & axes)Begin by drawing a circle (center A and point B)Create a point C on the circleDraw a 2nd circle centered at B out to point ADraw a 3rd circle centered at C out to point ACreate a point D at the intersection of these two circlesABCD are the vertices of the rhombusFinish by hiding the circles and drawing the line segmentsManipulate your rhombus and observe the behaviorWe need to erase the drawing pad – Select All using control-A, then delete.Notice that this is the same key sequence used in Word & many any other programs!Note the difference between a “drawing” and a “construction”.Tools Used:
28 Activity – construct a rhombus (a rhombus is a quadrilateral with all sides equal in length) METHOD 2 (File | New “Don’t Save” and turn off the grid & axes)Begin by drawing a circle (center A and point B)Create a point C on the circleDraw line segments AB and ACConstruct a line through B parallel the ACConstruct a line through C parallel the ABCreate a point D at the intersection of these two linesABCD are the vertices of the rhombusFinish by hiding the lines and drawing the line segmentsManipulate your rhombus and observe the behaviorTools Used:
29 Activity – construct a rectangle (a rectangle is a quadrilateral with all interior angles 90°) Begin by drawing a line segment ABDraw a line through A and perpendicular to ABDraw a line through B and perpendicular to ABCreate a point C on the line through ADraw a line through C perpendicular to ACCreate a point D at the intersection of these two linesABCD are the vertices of the rectangleFinish by hiding the lines and drawing the line segmentsManipulate your rhombus and observe the behaviorTools Used:
30 Activity - The Pythagorean Theorem Draw a right triangleDraw a line segment ABDraw perpendicular to AB at APoint C on perpendicularDraw line segment CBHide perpendicular lineConstruct squaresUse Regular polygon tool(Notice the polygon areas are shown in the algebra window)Label each square with its areaProperties - Show label - ValueThis drawing is pretty familiar to most teachers of mathematics, and so the dynamic aspect can be appreciated without getting bogged down in a discussion of the “meaning” of the drawing.Tools Used:
31 Area of a Triangle Create 2 horizontal lines Input: y=0 and y=6 Create a triangle ABCMake 3 points on these linesConnect using line segmentsCreate the altitudeLine thru C and perpendicularMake intersection point DHide line; make line segmentCreate textLabel & Value of height & baseText showing computed areaHere we use the Input window for the first time.Some people are not aware of the implication of the formula for the area of a triangle(all triangles with the same base and height must have the same area).Now the fun part! Move point C and what (doesn’t) happen?
32 Tools & Commands – Two ways of doing things Input y=x^2 (parabola c is created)Create point A on cUse the New Point tool or Input “Point[c]”Create the line tangent to c at AUse the Tangents tool or Input “Tangent[A,c]”Create a Slope objectUse the Slope tool or Input “Slope[a]”
33 Parameters & Animation A Slider is a graphical representation of a numberInput “a=2”Input “y=a*x^2”Note in the algebra window that a is “off”Right click on a, then Show objectThe variable a has become a graphical object!Manipulate the slider and observe its effectThere is also a tool in the Slider toolbox for creating checkboxes.Extra credit: set a to -5 and then right-click on the parabola &turn trace on, then sweep the slider from -5 to 5; a family of curves!
34 Riemann Approximations - part 1 Type the following into the Input field:f(x)=4x-x^2a=0b=4Area=Integral[f,a,b]n=10UpperSum=UpperSum[f,a,b,n]LowerSum=LowerSum[f,a,b,n]Adjust the colors of Upper & LowerSumMore use of the Input window, and the first use of commands.
35 Riemann Approximations - part 2 Right click on ‘a’ and select “Show object”The variable has become a graphical object!Manipulate the slider and observe its effectRepeat for the variables ‘b’ and ‘n’‘n’ should take only positive integer valuesAccess the ‘Slider’ properties for nSet min, max and increment appropriately
36 GeoGebra’s Web ExportEvery GeoGebra Construction can be exported as a Web Page (html), known as a Dynamic WorksheetTry it with your current drawingFile, Export, Dynamic Worksheet as WebpageExamplesGo toClick on GeoGebra DemonstrationsExample 1: Graphing Linear EquationsExample 2: Basic Construction KitGo to:Click on GeoGebra Tutorials
37 Dynamic Worksheet Example: Graphing Linear Equations GeoGebra was used to create this pageAccess to the internet is all that is needed to interact with it!User is limited to just the actions you design inThe focus is limited to the components you create.The user is not distracted by extraneous features.
38 Dynamic Worksheet Example: Basic Construction Kit A limited subset of the construction tools are availableThe algebra and input windows are turned offThe menu bar is on so students can save their work!Notice that there are no “tool boxes”, every tool that is needed is visible.
39 Other Features to Mention BooleansCheck boxes, used to reveal/hide partsVectorsu=(1,2); v=(2,1); w=u+vPolar Coordinatesa=0; r=1; P=(r;a)Parametric Equationsc=Curve[2cos(3t), sin(4t), t, 0, 2 pi]
40 New Features to Mention (Version 3.2) Record SpreadsheetBest Fit LineCompassEasy Quadratic FunctionsEllipse (Two foci and a point on it)Hyperbola (Two foci and a point on it)Parabola (A focus and directrix)Mirror point at circle