Geogebra

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

GeoGebra and Spreadsheets 24

Representations of functions… f(x)=x² 25

Spreadsheet calculationCreate a GeoGebra sheet…•… visualizing the costs for mobile phone usage witha spreadsheet and functions.•… visualizing the volume of a cube as a function ofits side length.•… visualizing a functional dependence of yourchoice! 26

The Rabbit Problem!You have fence material with the length of 18 m, and youwant to build a rectangular enclosure for your rabbit. Howdo you have to choose the side length of the rectangle thatthe area for your rabbit is maximal?1.Use the spreadsheet calculation program to try outsome different solutions.2.Solve the problem formally on a sheet of paper!3.Create a dynamic sheet in GeoGebra showing the graphand the enclosure in order to visualize the solution.Be creative! 27

How to get Geogebra 28

Java http://www.java.com/en/download 29

How to get Geogebra http://www.geogebra.org 30

How to get Geogebra http://www.geogebra.org 31

Assessment Task• Write a detailed draft for a teaching unit of a math lesson with GeoGebra (ca. 10 pages)• Parts: – Target of the lesson (mathematical content or skills to learn) – Context (class, students, necessary equipment, …) – Why and how Geogebra is used (detailed rationale!) – Description of the phases of the lesson (including lesson plan) – Screenshots of the geogebra sheets• Add the GeoGebra files 32

Geogebra

by Christian Spannagel on Oct 11, 2012

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