1. GeoGebra Dynamic Geometry, Algebra and Calculus Markus Hohenwarter, 2005 www.geogebra.at www.geogebra.at

2. GeoGebra What is GeoGebra?  dynamic mathematics software  for students and teachers  dynamic geometry, algebra and calculus  open source (free of charge)

3. GeoGebra GeoGebra =GeometryAlgebra+

4. GeoGebra DGS and CAS Dynamic Geometry Software  drag mode (dynamic)  graphical representation Cabri, Cinderella, Euklid Dynageo, Geometer‘s Sketchpad, Geonext, Ruler and Compass, … Computer Algebra Systems  algebraic and numerical computations  symbolic representation Derive, Mathematica, Maple, MuPad, MathCAD, …

5. GeoGebra Combined Representations „ There is no true understanding in mathematics for students who do not incorporate into their cognitive architecture the various registers of semiotic representations used to do mathematics.“ (Duval 1999)

6. GeoGebra Bidirectional Combination Bidirectional change of representation symbolic ⇔ iconic (visual) Examples  coordinates ⇔ point in coordinate system  function‘s equation ⇔ graph of function  circle‘s equation ⇔ circle drawn in coordinate system

7. GeoGebra New Possibilities 3: GeoGebra equation of the circle ⇔ dynamic plot of the circle (drag the circle with the mouse) 1: DGS Construct a circle ⇒ equation is shown 2: CAS Enter the equation of a circle ⇒ circle is plotted

8. GeoGebra Didactic Principles using Wittmann (1981) Student activity orientation principle Interaction of representations principle (iconic and symbolic representations) Genetic principle (experimental learning of mathemtics)

9. GeoGebra Activity based learning Iconic and Symbolic representations (Bruner)

10. GeoGebra What does GeoGebra offer?  points, vectors, segments, polygons, lines, all conic sections and functions in x  dynamic constructions (mouse)  coordinates, equations, vectors, numbers and commands (keyboard)  intuitive notation c: (x - 3)² + (y + 2)² = 25  easy-to-use interface  multilingual menus, commands, help

11. GeoGebra GeoGebra is Innovative Bidirectional combination of elements of  dynamic geometry software  computer algebra High technical portability  runs on Windows, Linux, Solaris, MacOS X  dynamic worksheets (html)

12. GeoGebra Dynamic Geometry and Algebra ruler and compass geometry analytic geometry vectors

13. GeoGebra

17. Dynamic Calculus functions in x derivatives integrals

18. GeoGebra

24. Dynamic Mathematics for School GeoGebra – the dynamic mathematics software www.geogebra.at www.geogebra.at GeoGebraWiki – the interactive pool of materials www.geogebra.at/wiki www.geogebra.at/wiki GeoGebra User Forum www.geogebra.at/forum www.geogebra.at/forum

25. GeoGebra GeoGebra Markus Hohenwarter © 2001-2005