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# Geogebra

## by Christian Spannagel, Professor at Pädagogische Hochschule Heidelberg on Oct 11, 2012

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Combining Dynamic Geometry, Computer Algebra and Spreadsheet Calculation

Combining Dynamic Geometry, Computer Algebra and Spreadsheet Calculation
Course held by Christian Spannagel in Beira, Mozambique, 2-11 October 2012

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## GeogebraPresentation Transcript

• GeoGebra Combining Dynamic Geometry,Computer Algebra and Spreadsheet Calculation Christian Spannagel University of EducationHeidelberg http://cspannagel.wordpress.com Twitter: @dunkelmunkel 1
• Dynamic Geometry Systems 2
• Accurate ConstructionsFind at least three waysto construct aperpedicular bisector.Construct accurately:1.a regular triangle2.a square3.a rectangle4.a regular pentagon5.a regular hexagon6.A regular octagon 3
• Draw a nice…… ethnic pattern, Mandala, picture….… which is completely resizable! 4
• Exploring Geometry„Proof“…1.… Thales’ Theorem2.… that the sum of a triangle’s angles is always 180°.3.… that in a circle the angle at the center is double of theangle at the circumference.4.… that a triangle’s perpendicular bisectors always intersectin one single point, the circumcenter. Do this also for itsmedian lines (centroid) and for the heights of its sides(orthocenter). How are these three points related?5.… Pythagoras’ Theorem„Proof“ some other theorems you know! 5
• Loci of pointssine, cosine, and tangent 1. Create an interactive GeoGebra sheet where the connection between the unit circle and sine is shown. 2. Do the same for cosine. 3. Do the same for tangent. 6
• Loci of Points Construct cycloids and epicycloids! 7
• Loci of PointsConstruct a pantograph! C. Scheiner, Book Pantographice seu ars delineandi, 1631 8
• Loci of PointsConstruct a pantograph! 9
• Loci of PointsGiven a point P and a line l. Construct the the lociof all points which have the same distance from Pand l. What is the result? 10
• Loci of PointsConstruct the garderner‘s ellipse! Les Dioptriques de Descartes, 1636 11
• Defining macrosDefine a macro for…1.a regular triangle2.a square3.a regular hexagon4.your nice pattern/mandala/…5.… whatever you may need in future! 12
• Patterns and Tesselations 13
• Dynamic Geometry SystemsCharacteristics of a DGS:•accurate drawings•dragging • exploring the dynamic behavior of a construction•loci (traces of objects)•extracting construction texts•defining macros Computers are great in making things dynamic! 14
• DGS in schools: Brainstorming!Would you use DGS in schools?Why? Why not?How can DGS be used in schools?Any ideas? 15
• GeoGebra and Functions 16
• Invent a Bathtube Story! 17
• Draw the Filling Graphs! 18
• Exploring functionsShow the effects of functional parameters on graphsof different functions. Create dynamic worksheetsusing sliders!1.… for different forms of quadratic functions 1. f(x)=ax²+bx+c 2. f(x)=a(x-xs)²+ys 3. f(x)=a(x-x1)(x-x2)2.…(co)sine, tangent, …3.… exponential functions…4.… “crazy” functions, whatever… 19
• Derivates and Integrals• Try to create derivatives• Again, change the parameters a,b,c, … of a function and see… – …how the derivative changes – …how integrals change 20
• Be dynamic!• static aspects of functions – f(x)=y• dynamic aspects of functions – What happens to y if I make x larger/smaller…?• Thinking results from acting – questioning „what happens if…“• Creating hypotheses – Computer as a cognitive tool / thinking tool – „outsourcing“ of „stupid“ calculations 21
• What happens to…• … the height of an isoceles triangle with an constant area of 10 cm² when I change the length of the base?• … the height of a triangle with an constant circumference of 30 cm when I change the length of the base? 22
• Build a dynamic GeoGebra sheet…1. … where you can find the intersections of the functions f(x)=ax³-bx and g(x)=(-a)x³+(b+2)x+c. (use sliders for a, b, and c). Explore!2. … where you can see how the integral of a function between two borders changes when you change the borders.3. … where you can see the tangent at a given point on the curve of a given function. 23