GeoGebra

Thank you again for attending the GeoGebra presentation at the 2009 ORMATYC conference!

It is my hope that you saw enough to get a vision of how GeoGebra could serve you in your classrooms. I have concern that when you sit down with it for the first time it may well seem much less user-friendly than it first appeared.

I was asked how long it took me do put together one of the demos (was it the Riemann sums one?) and I quipped about 5 minutes, but someone else in the room spoke of the significant amount of time he sometimes spends building a construction in GeoGebra.

It certainly takes me quite a while also, when I am building a construction that needs something I have not done before. Browsing the help file and experimenting, and of course asking questions on the not done before. Browsing the help file and experimenting, and of course asking questions on the GeoGebra forum takes time.

I CAN assure you, however, that putting together a construction that involves only skills that I have mastered goes rather quickly (well, except that I can never leave it alone; there is usually some newly envisioned feature that suggests itself that sends me back down the rabbit hole).

…but! Try this: type “f(x)=x^2” into the input line, then type “UpperSum[f,0,2,8]” and you will see the Riemann sum example with 8 rectangles! Your assignment is to add a slider to allow you to change the number of rectangles.

Finally, I wish to reaffirm my desire to hear from you – I truly would enjoy interacting with you on your GeoGebra projects!

Tom Gettys email: gettyst@lanecc.edu

GeoGebra

If you have a computer with you:

� Install GeoGebra right now!� Go to www.geogebra.org� Click WebStart� Click WebStart

third item in the left menu� Click button WebStart

GeoGebra is installed automatically

GeoGebraFree Dynamic GeoGebraFree Dynamic Mathematics Software

Tom GettysLane Community College

ORMATYC 2009

GeoGebra

What is GeoGebra?

� Dynamic Mathematicssoftware built by and for Mathematics instructors

� Open Source, meaning�It‘s free!�It‘s supported by a

community of like-minded people

GeoGebra

Why GeoGebra?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.”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:http://standards.nctm.org/document/chapter2/index.htmhttp://standards.nctm.org/document/chapter2/techn.htm

GeoGebra

A short history of GeoGebra� GeoGebra was created in 2001 by Markus Hohenwarter as part of his

master's thesis in mathematics education and computer science at the University of Salzburg in Austria.

� Supported by a scholarship from the Austrian Academy of Sciences he was able to continue its development 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 including the European and German educational software awards, and was translated by math instructors and teachers all over the world to more than 40 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 Standards Mapped Graduate Education and Mentoring.

For more on this go to:

http://www.maa.org/joma/Volume7/Hohenwarter/index.html

GeoGebra

GeoGebra

World Wide User Community

� 40 languages� 190 countries� 400,000 visitors

per month

GeoGebra

� Combination of elements of�dynamic geometry software (Sketchpad, Cabri)�computer algebra systems (Derive, Maple)

GeoGebra is Innovative

� High technical portability�GeoGebra is fully platform

independent (Windows, MacOS, Solaris, Linux)

�dynamic worksheets (html)

GeoGebra

GeoGebra as a Pedagogical Tool� Drawing Tool

� Geometric drawings for handouts & activities� Algebraic graphs for worksheets, quizzes & tests

� In-class Presentation Tool� Dynamically show relationships � Visualization of abstract concepts

� Spontaneous demonstrations� Prebuilt demonstrations

� Authoring Tool (Web Export)� Interactive virtual manipulatives� Exploratory mathematics

GeoGebra

Geometric Drawings for your documents

GeoGebra

Algebraic graphs for your documents

GeoGebra

Examples of Using GeoGebra

� Visualizingstudents can “see” abstract concepts

� Representationsstudents can make connections

� Experimentsstudents can discover mathematics

GeoGebra

Visualizing Adding Integers

GeoGebra

Visualizing Multiplying Fractions

GeoGebra

Visualizing Parameters

GeoGebra

Visualizing Parameters

GeoGebra

Visualizing Movement

GeoGebra

Visualizing Slope Functions

GeoGebra

Examples of Using GeoGebra

� Visualizingstudents can “see” abstract concepts

� Representationsstudents can make connections

� Experimentsstudents can discover mathematics

GeoGebra

Representations: Sketch & Graph

GeoGebra

Unexpected Discoveries

GeoGebra

Examples of Using GeoGebra

� Visualizingstudents can “see” abstract concepts

� Representationsstudents can make connections

� Experimentsstudents can discover mathematics

GeoGebra

Experiments: What happens when ... ?

GeoGebra

Experiments: Connections

GeoGebra

Experiments: Approximation

GeoGebra

Experiments: Statistics

GeoGebra

Start up GeoGebra� Click HelpHelp to get access to the GeoGebra

�Home page – connect to all things GeoGebra

�User Forum - ask questions, get answers!

�Wiki – free pool of teaching materials�Wiki – free pool of teaching materials

GeoGebra

GeoGebra User Forum

www.geogebra.org/forum

GeoGebra

GeoGebraWiki – Free Materials

www.geogebra.org/wiki

GeoGebra

GeoGebra = Geometry + Algebra

Geometry Window

Algebra Window

GeoGebra

Getting to know the user interface

Help for theselected tool

The Move toolthe most important tool on the menu!

Tool bar“Toolboxes” group

Similar tools

tool on the menu!

Input field Input field assists

GeoGebra

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 point

� Create a second point to complete the action

� Observe that the points and line also appear in the algebra window!

Symbolic ↔Geometric

Extra credit: On the menu bar Select View , then Grid

GeoGebra

Adding a Drawing to a Document� Click on File , Export , Drawing Pad to Clipboard� Start Word, then Edit , Paste – You’re done!� Resize and position as desired

GeoGebra

Context Menu� Right-click any object in either window� Notice the Properties… item at the bottom

Context menu

GeoGebra

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 contains a set of related properties

� Try this:� Change the color of

your line (Color tab)� Change the style of

your line (Style tab)

Extra credit: Modify the labeling of point A to Value

GeoGebra

Activity – construct a rhombus(a rhombus is a quadrilateral with all sides equal in length)

METHOD 1 (delete all objects and turn off the grid & axes)

� Begin by drawing a circle (center A and point B)� Create a point C on the circle� Draw a 2nd circle centered at B out to point A� Draw a 2nd circle centered at B out to point A� Draw a 3rd circle centered at C out to point A� Create a point D at the intersection of these two circles� ABCD are the vertices of the rhombus� Finish by hiding the circles and drawing the line segments� Manipulate your rhombus and observe the behavior

Tools Used:

GeoGebra

Activity – construct a rhombus(a rhombus is a quadrilateral with all sides equal in length)

METHOD 2 (delete all objects and turn off the grid & axes)

� Begin by drawing a circle (center A and point B)� Create a point C on the circle� Draw line segments AB and AC� Construct a line through B parallel the AC� Construct a line through B parallel the AC� Construct a line through C parallel the AB� Create a point D at the intersection of these two lines� ABCD are the vertices of the rhombus� Finish by hiding the lines and drawing the line segments� Manipulate your rhombus and observe the behavior

Tools Used:

GeoGebra

Activity – construct a rectangle(a rectangle is a quadrilateral with all interior angles 90°)

� Begin by drawing a line segment AB� Draw a line through A and perpendicular to AB � Draw a line through B and perpendicular to AB� Create a point C on the line through A� Draw a line through C perpendicular to AC� Create a point D at the intersection of these two lines� ABCD are the vertices of the rectangle� Finish by hiding the lines and drawing the line segments� Manipulate your rhombus and observe the behavior

Tools Used:

GeoGebra

Activity - The Pythagorean Theorem

� Draw a right triangle� Draw a line segment AB� Draw perpendicular to AB at A� Point C on perpendicular� Draw line segment CB� Hide perpendicular line� Hide perpendicular line

� Construct squares� Use Regular polygon tool

(Notice the polygon areas are shown in the algebra window)

� Label each square with its area� Properties - Show label - Value

Tools Used:

GeoGebra

Area of a Triangle� Create 2 horizontal lines

� Input: y=0 and y=6� Create a triangle ABC

� Make 3 points on these lines� Connect using line segments

Create the altitude� Create the altitude� Line thru C and perpendicular� Make intersection point D� Hide line; make line segment

� Create text � Label & Value of height & base� Text showing computed area

Now the fun part! Move point C and what (doesn’t) happen?

GeoGebra

Tools & Commands –Two ways of doing things� Input y=x^2 (parabola c is created)� Create point A on c

�Use the New Point tool or Input “Point[c]”�Use the New Point tool or Input “Point[c]”� Create the line tangent to c at A

�Use the Tangents tool or Input “Tangent[A,c]”� Create a Slope object

�Use the Slope tool or Input “Slope[a]”

GeoGebra

Parameters & AnimationA Slider is a graphical representation of a number

� Input “a=2”� Input “y=a*x^2”

Note in the algebra window that a is “off”Note in the algebra window that a is “off”

� Right click on a, then Show object�The variable a has become a graphical object!�Manipulate the slider and observe its effect

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!

GeoGebra

Riemann Approximations - part 1� Type the following into the Input field:

� f(x)=4x-x^2� a=0� b=4� Area=Integral[f,a,b]� Area=Integral[f,a,b]

� Type the following into the Input field:� n=10� UpperSum=UpperSum[f,a,b,n]� LowerSum=LowerSum[f,a,b,n]� Adjust the colors of Upper & LowerSum

GeoGebra

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 effect

� Repeat for the variables ‘b’ and ‘n’

� ‘n’ should take only positive integer values�Access the ‘Slider’ properties for n�Set min, max and increment appropriately

GeoGebra

GeoGebra’s Web Export� Every GeoGebra Construction can be

exported as a Web Page (html),also known as a Dynamic Worksheet

� Try it with your current drawingTry it with your current drawing� File, Export, Dynamic Worksheet as Webpage

� Examples�Go to http://teach.lanecc.edu/gettyst

� Click on GeoGebra Demonstrations� Example 1: Graphing Linear Equations� Example 2: Basic Construction Kit

GeoGebra

Dynamic Worksheet Example:Graphing Linear Equations

� GeoGebra was used to createthis page

� Access to the � Access to the internet is all that is needed to interact with it!

� User is limited to just the actions you design in

GeoGebra

Dynamic Worksheet Example:Basic Construction Kit

� A limited subset of the construction tools are availabletools are available

� The algebra and input windows are turned off

� The menu bar is on so students can save their work!

GeoGebra

Other Features to Mention� Booleans

�Check boxes, used to reveal/hide parts

� Vectors� u=(1,2); v=(2,1); w=u+v

� Polar Coordinates� a=0; r=1; P=(r;a)

� Parametric Equations� c=Curve[2cos(3t), sin(4t), t, 0, 2 pi]