geogebra dynamic mathematics for schools auston b. cron psja north high school south texas college...
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GeoGebraDynamic Mathematics for Schools
Auston B. CronPSJA North High School
South Texas CollegeCAMT 2009
GeoGebra
What is GeoGebra?
Dynamic Mathematicssoftware built by and for Mathematics instructors and students
Open Source, meaningIt‘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.”
“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.htmhttp://www.maa.org/joma/Volume7/Hohenwarter/index.html
GeoGebra
Combination of elements ofdynamic geometry software (Sketchpad, Cabri)computer algebra systems (Derive, Maple)
High technical portabilityGeoGebra is fully platform
independent (Windows,
MacOS, Solaris, Linux)dynamic worksheets (html)
GeoGebra is Innovative
GeoGebra
GeoGebraPast & Present
GeoGebra
Computer Algebra Systems
Derive
Mathematica
Scientific Notebook
GeoGebra
Dynamic Geometry Software
GeoGebra
TI-92: DGS and CAS
Graphing Calculators
HP 49g
TI-84
TI-Nspire/CAS System
GeoGebra
GeoGebra
GeoGebra as a Pedagogical Tool Drawing Tool
Geometric drawings for handouts & activitiesGraphs for worksheets, quizzes & tests
In-class Presentation ToolDynamically show relationshipsVisualization of abstract concepts
Authoring Tool (Web Export) Interactive virtual manipulativesExploratory mathemathics
GeoGebra
World Wide User Community
43 languages 190 countries 400,000 visitors
per month
GeoGebra
Visitors on www.geogebra.org
User Forum: Total posts 26994 | Total topics 5796 | Total members 3155
GeoGebra
Start up with GeoGebra Webstart Click Help to get access to the GeoGebra
Home page – connect to all things GeoGebraUser Forum - ask questions, get answers!Wiki – free pool of teaching materials
GeoGebra
GeoGebra = Geometry + Algebra
Geometry Window
Algebra Window
GeoGebra
Getting to know the user interface
Tool bar“Toolboxes” group
Similar tools
Help for theselected tool
The Move toolthe most important tool on the menu!
Input field Input field assists
GeoGebra
Spreadsheet View has being added
Algebra View
Toolbar
Graphics View
Input Bar
Spreadsheet View
GeoGebra
Spreadsheet in GeoGebra Dynamic Table of Values
Regression Lines & PolynomialsBinomial Distribution
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
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
GeoGebra
Context Menu
Context menu
Right-click any object in either window Notice the Properties… item at the bottom
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 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 – Constructing a Parabola(Part 1: create basic interactive graph of a parabola)
Type the following into the Input fielda=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:
GeoGebra
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:
GeoGebra
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:
GeoGebra
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:
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
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 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]” 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”
Right click on a, then Show objectThe 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]
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 valuesAccess the ‘Slider’ properties for nSet min, max and increment appropriately
GeoGebra
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
GeoGebra
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
GeoGebra
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!
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]
GeoGebra
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
GeoGebra
GeoGebraFuture
GeoGebra
Future Plans
Make it Easier, Easier, Easier! Symbolic Algebra Window (CAS) 3D Extensions Split views to allow rearranging
Independent sizingPossible 4 corner model
GeoGebra
Next year: CAS, 3D
GeoGebra
Resources Sites
GeoGebra Homewww.geogebra.com
PSJA GeoGebra Pagewww.psja.k12.tx.us/~abcronGeoGebra Institute of Ohiohttp://giohio.pbworks.com
E-Mail [email protected]