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GeoGebraCombining Dynamic Geometry,

Computer Algebra and Spreadsheet Calculation

Christian SpannagelUniversity of EducationHeidelberghttp://cspannagel.wordpress.com

Twitter: @dunkelmunkel

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Dynamic Geometry Systems

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Accurate Constructions

Construct accurately:1.a regular triangle2.a square3.a rectangle4.a regular pentagon5.a regular hexagon6.A regular octagon

Find at least three ways to construct a perpedicular bisector.

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Draw a nice…

… ethnic pattern, Mandala, picture….… which is completely resizable!

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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 the angle at the circumference.4.… that a triangle’s perpendicular bisectors always intersect in one single point, the circumcenter. Do this also for its median 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!

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Loci of points

sine, cosine, and tangent1. 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.

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Loci of Points

Construct cycloids andepicycloids!

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Loci of Points

Construct a pantograph!

C. Scheiner, Book Pantographice seu ars delineandi, 1631

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Loci of Points

Construct a pantograph!

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Loci of Points

Given a point P and a line l. Construct the the loci of all points which have the same distance from P and l. What is the result?

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Loci of Points

Construct the garderner‘s ellipse!

Les Dioptriques de Descartes, 1636

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Defining macros

Define a macro for…

1.a regular triangle2.a square3.a regular hexagon4.your nice pattern/mandala/…5.… whatever you may need in future!

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Patterns and Tesselations

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Dynamic Geometry Systems

Characteristics 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!

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DGS in schools: Brainstorming!

Would you use DGS in schools? Why? Why not?

How can DGS be used in schools? Any ideas?

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GeoGebra and Functions

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Invent a Bathtube Story!

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Draw the Filling Graphs!

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Exploring functions

Show the effects of functional parameters on graphs of different functions. Create dynamic worksheets using sliders!1.… for different forms of quadratic functions

1. f(x)=ax²+bx+c2. 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…

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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

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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

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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?

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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 whenyou change the borders.

3. … where you can see the tangent at a given point on the curve of a given function.

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GeoGebra and Spreadsheets

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Representations of functions…

f(x)=x²

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Spreadsheet calculation

Create a GeoGebra sheet…•… visualizing the costs for mobile phone usage with a spreadsheet and functions.•… visualizing the volume of a cube as a function of its side length.•… visualizing a functional dependence of your choice!

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The Rabbit Problem!

You have fence material with the length of 18 m, and you want to build a rectangular enclosure for your rabbit. How do you have to choose the side length of the rectangle that the area for your rabbit is maximal?

1.Use the spreadsheet calculation program to try out some different solutions.2.Solve the problem formally on a sheet of paper!3.Create a dynamic sheet in GeoGebra showing the graph and the enclosure in order to visualize the solution. Be creative!

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How to get Geogebra

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Java

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

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How to get Geogebra

http://www.geogebra.org

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How to get Geogebra

http://www.geogebra.org

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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