GeoGebra

1. GeoGebra Dynamic Mathematics for Schools Auston B. Cron PSJA North High School South Texas College RGVCTM2009

2. What is GeoGebra? Dynamic Mathematicssoftware built by and for Mathematics instructors and students Open Source, meaning It‘s free! It‘s supported by a community of like-minded people

3. 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.” “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/techn.htm http://www.maa.org/joma/Volume7/Hohenwarter/index.html

4. Combination of elements of dynamic geometry software (Sketchpad, Cabri) computer algebra systems (Derive, Maple) High technical portability GeoGebra is fully platform independent (Windows, MacOS, Solaris, Linux) dynamic worksheets (html) GeoGebra is Innovative

5. GeoGebra Past & Present

6. Computer Algebra Systems Mathematica Derive Scientific Notebook

7. Dynamic Geometry Software

8. Graphing Calculators TI-92: DGS and CAS HP 49g TI-84 TI-Nspire/CAS System

9. GeoGebra as a Pedagogical Tool • Drawing Tool • Geometric drawings for handouts & activities • Graphs for worksheets, quizzes & tests • In-class Presentation Tool • Dynamically show relationships • Visualization of abstract concepts • Authoring Tool (Web Export) • Interactive virtual manipulatives • Exploratory mathemathics

10. World Wide User Community 43 languages 190 countries 400,000 visitors per month

11. Visitors on www.geogebra.org User Forum: Total posts 26994 | Total topics 5796 | Total members 3155

12. GeoGebra Home Page www.geogebra.org/

13. Start up with GeoGebra Webstart • Click Help 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

14. GeoGebra User Forum www.geogebra.org/forum

15. GeoGebraWiki – Free Materials www.geogebra.org/wiki

16. GeoGebra = Geometry + Algebra Algebra Window Geometry Window

17. The Move tool the most important tool on the menu! Input field Getting to know the user interface Help for the selected tool Tool bar “Toolboxes” group Similar tools Input field assists

18. Spreadsheet View has being added Toolbar Spreadsheet View Algebra View Graphics View Input Bar

19. Spreadsheet in GeoGebra Dynamic Table of Values Regression Lines & Polynomials Binomial Distribution

20. 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

21. Add 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

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

23. Properties Dialog –object customization • The Propertiesdialog is accessed via the context menu (or the Edit menu) • Notice that this is a tabbed dialog; each tab 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

24. Student Constructions on Web On the bottom of the GeoGebra selection page, you will find a link to these Student Constructions. More constructions of this type will be available by summer.

25. Activity – Constructing a Parabola(Part 1: create basic interactive graph of a parabola) • Type the following into the Input field • a=1 • f(x) = a x^2 • You now see the parabola f(x) = x2 • Right click on a=1 in Algebra window and select “Show Object” • Use the Move Tool to change ‘a’ Tools Used:

26. Activity – Constructing a Parabola(Part 2: create a family of curves) • Right click on a=1 in Algebra window and select “Properties” • Click on the “Slider” tab and change the increment to 1 • Click on the Function f, select “Show Trace” and close • Move the slider left and right. Tools Used:

27. Activity – construct a rhombus(a rhombus is a quadrilateral with all sides equal in length) METHOD 1 (File | New “Don’t Save” 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 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:

28. Activity – construct a rhombus(a rhombus is a quadrilateral with all sides equal in length) METHOD 2 (File | New “Don’t Save” 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 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:

29. 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:

30. 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 • 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:

31. 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 • 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?

32. 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]” • 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]”

33. Parameters & Animation A 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” • 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, thensweep the slider from -5 to 5; a family of curves!

34. 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] • 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

35. 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

36. GeoGebra’s Web Export Every GeoGebra Construction can be exported as a Web Page (html), known as a Dynamic Worksheet Try 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 Go to:http://www.psja.k12.tx.us/~abcron Click on GeoGebra Tutorials

37. Dynamic Worksheet Example:Graphing Linear Equations • GeoGebra was used to create this page • Access to the internet is all that is needed to interact with it! • User is limited to just the actions you design in

38. Dynamic Worksheet Example:Basic Construction Kit • A limited subset of the construction tools are available • The algebra and input windows are turned off • The menu bar is on so students can save their work!

39. 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]

40. New Features to Mention(Version 3.2) • Record Spreadsheet • Best Fit Line • Compass • Easy Quadratic Functions • Ellipse (Two foci and a point on it) • Hyperbola (Two foci and a point on it) • Parabola (A focus and directrix) • Mirror point at circle

41. GeoGebraFuture

42. Future Plans Make it Easier, Easier, Easier! Symbolic Algebra Window (CAS) 3D Extensions Split views to allow rearranging Independent sizing Possible 4 corner model

43. Next year: CAS, 3D

44. Resources Sites GeoGebra Home www.geogebra.com PSJA GeoGebra Page www.psja.k12.tx.us/~abcron GeoGebra Institute of Ohio http://giohio.pbworks.com E-Mail me abcron@psja.k12.tx.us