geogebra ipg pri 092010

GeoGebra Institute of Malaysia

Introduction

We, as mathematics educators, need to teach our students to think critically and to do mathematics as mathematicians do. We need to let them construct their mathematical knowledge themselves. Students need mathematical abilities to solve problem. These abilities include being able to detect patterns, make conjectures, make generalizations, perform inductive and deductive reasoning, and make analogies. These skills can be nurture through tasks that require them to solve problems, engage in exploratory investigation, experience classroom discourse and using the information and communication technology.The multi-platform, open-source dynamic mathematics software GeoGebra (Hohenwarter & Preiner 2007) is one of the mathematics software that can be used to encourage investigation, discovery and experimentation in classrooms and their visualization features can be effectively employed in teaching to generate conjectures (Lavicza 2006, Kreis, 2004). By utilizing Geogebra in our teaching, students learn through exploring, investigating and discovering. GeoGebra enhances students’ higher order thinking skills in mathematics. It is and effective tool for both teacher and students.

Geogebra combines geometry, algebra, calculus, and statistics into a single easy-to-use package for learning and teaching mathematics from elementary through university level. It is available free of charge on the internet and has been translated into 36 languages and is estimated that more than 100 000 educators use this software for teaching and learning in more than 192 countries (Hohenwarter, M., Hohenwarter, J., Kreis & Lavicza, 2008). The open-source nature of the software has important implications both for educators and students. Unlike with commercial products, students are certainly not constrained to use the software only in schools or universities allowed by site licenses, but they can download and install GeoGebra on their private computers.

About the Creator: Markus Hohenwarter

Markus Hohenwarter created GeoGebra in 2001/2 during his master's thesis project and later PhD thesis in Mathematics Education at the

University of Salzburg, Austria. From 2006-2008, his work was supported by an NSF Math & Science Partnership project of Florida

Atlantic University and Broward County Schools in Florida. At present, he is leading the GeoGebra project at the Florida Center for Research in Science, Technology, Engineering, and Mathematics Education (FCR-STEM) at Florida State University in Tallahassee, USA.

Table of Contents

1. Installation of GeoGebra.............................................................................................................4Introduction to GeoGebra – A Workshop for GeoGebra Usersby Dr. Daniel Leong, GeoGebra Institute of Malaysia, SEAMEO RECSAM on 25 September 20101


Activity 1: Installing GeoGebra.....................................................................................................................4Installation WITH Internet access..................................................................................................................4Installation WITHOUT Internet access..........................................................................................................4

2. GeoGebra’s User Interface.........................................................................................................4

3. Basic Use of Tools...................................................................................................................... 5

Activity 2: Familiarizing with some of the Menu/Tools................................................................................6Menu...............................................................................................................................................................6Construction Tools.........................................................................................................................................7

4. Drawing with GeoGebra..............................................................................................................8

Activity 3: Drawing Geometric Figures and Other Objects...........................................................................8Activity 4: What to practice............................................................................................................................9

5. Saving Files............................................................................................................................... 10

Activity 5: Saving GeoGebra files................................................................................................................10Activity 6: Opening GeoGebra Files............................................................................................................10

6. Drawings, Constructions, and Drag Test................................................................................10

Discussion.....................................................................................................................................................10

7. Rectangle Construction............................................................................................................11

Activity 7: Constructing a rectangle.............................................................................................................11

8. Navigation Bar and Construction Protocol.............................................................................12

Activity 8: Checking your construction........................................................................................................12

9. GeoGebra’s Properties Dialog.................................................................................................12

How to access the Properties Dialog............................................................................................................12Activity 9: What to practice..........................................................................................................................13

10. Square Construction.................................................................................................................13

Activity 10: Constructing squares................................................................................................................13

11. Create custom tool.................................................................................................................... 14

Activity 11: Creating a Square tool..............................................................................................................14

12. Measuring Length, Area and Angles.......................................................................................14

13. Equilateral Triangle Construction............................................................................................15

Activity 12: Constructing an Equilateral Triangle.......................................................................................15

14. Export of Pictures to the Clipboard.........................................................................................16

Activity 13: Exporting Pictures to the Clipboard.........................................................................................16

15. Inserting pictures into a text processing document..............................................................16

16. Inserting Pictures into the Graphics Window.........................................................................16

Activity 14: Drawing Tools for Symmetric Figures.....................................................................................16

17. Basic Algebraic Input, Commands, and Functions................................................................17

Introduction to GeoGebra – A Workshop for GeoGebra Usersby Dr. Daniel Leong, GeoGebra Institute of Malaysia, SEAMEO RECSAM on 25 September 20102


Activity 15: Tips and Tricks.........................................................................................................................18Activity 16: Visualizing Integer Addition on the Number Line...................................................................19

18. Static and Dynamic Text...........................................................................................................20

Activity 17: Inserting Static Text.................................................................................................................20Activity 18: Inserting Dynamic Text............................................................................................................20

19. Conditional Formatting.............................................................................................................21

Activity 19: Conditional Formatting – Inserting Checkboxes....................................................................21

20. Fraction with Circles................................................................................................................. 22

Activity 20: Constructing Circular Sectors with sliders...............................................................................22

21. Visualizing Multiplication of Natural Numbers.......................................................................24

Activity 21: Constructing a Model for Visualizing Multiplication..............................................................24

22. Reconstruct ggb files...............................................................................................................26

Activity 22: Reconstructing “Thumb-up.ggb”.............................................................................................26

23. Creating Dynamic Worksheets................................................................................................26

Activity 23: Exporting a Dynamic Worksheet.............................................................................................27Activity 24: Enhancing Dynamic Worksheets.............................................................................................27

24. Providing Dynamic Worksheets to Students..........................................................................28

25. GeoGebra’s Spreadsheet View................................................................................................28

Spreadsheet Cells Input................................................................................................................................28Activity 25: Recording to Spreadsheet Feature............................................................................................29Activity 26: Relative Copy and Linear Equations........................................................................................30Activity 27: Creating a histogram................................................................................................................31

26. Useful Web Links...................................................................................................................... 32

GeoGebra Homepage...................................................................................................................................32GeoGebraWiki..............................................................................................................................................32GeoGebra User Forum..................................................................................................................................32GeoGebra Documentation............................................................................................................................32GeoGebra Institute of Malaysia....................................................................................................................32



1. Installation of GeoGebra

Activity 1: Installing GeoGebra

Preparations Create a new folder called “GeoGebra_your name” on your desktop. Hint: During the session, save all files into this folder so they are easy to find later on.

Installation WITH Internet accessInstall GeoGebra WebStart • Open your Internet browser and go to http://www.geogebra.org/cms/en/download• Click on the button called GeoGebra WebStart. Note: The software is automatically installed on your computer. You only need to confirm all messages that might appear with OK or YES. Hint: Using GeoGebra WebStart has several advantages for you provided that you have an Internet connection available for the initial installation: You don’t need to have special user permissions in order to use GeoGebra WebStart,

which is especially useful for computer labs and laptop computers in schools. Once GeoGebra WebStart was installed you can use the software off-line as well. Provided you have Internet connection after the initial installation, GeoGebra WebStart

frequently checks for available updates and installs them automatically. Thus, you are always working with the newest version of GeoGebra.

Installation WITHOUT Internet access • Your workshop presenter will provide the GeoGebra installer files via USB drives or CDs. • Copy the installer file from the storage device into the created “GeoGebra_your name” folder on your computer. Hint: Make sure you have the correct version for your operating system Examples: MS Windows: GeoGebra_3_0_0_0.exe

MacOS: GeoGebra_3_0_0_0.zip • Double-click the GeoGebra installer file and follow the instructions of the installer assistant.

2. GeoGebra’s User Interface The GeoGebra’s basic interface is divided into four sections: a graphics view, an algebra view, an input bar and a spreadsheet view. On the one hand you can operate the provided geometry tools with the mouse in order to create geometric constructions on the drawing pad of the graphics window. On the other hand, you can directly enter algebraic input, commands, and functions into the input field by using the keyboard or data into the spreadsheet view. While the graphical representation of all objects is displayed in the graphics window, their algebraic numeric representation is shown in the algebra window.

The user interface of GeoGebra is flexible and can be adapted to the needs of your students. If you want to use GeoGebra in the elementary or early middle school, you might want to hide the algebra window, input field, worksheet view and coordinate axes and just work with the drawing pad and geometry tools. Later on, you might want to introduce the coordinate system using a grid to facilitate working with integer coordinates. Introduction to GeoGebra – A Workshop for GeoGebra Usersby Dr. Daniel Leong, GeoGebra Institute of Malaysia, SEAMEO RECSAM on 25 September 20104


3. Basic Use of Tools • Activate a tool by clicking on the button showing the corresponding icon. • Open a toolbox by clicking on the lower part of a button and select another tool from this toolbox. Hint: You don’t have to open the toolbox every time you want to select a tool. If the icon of the desired tool is already shown on the button it can be activated directly. Hint: Toolboxes contain similar tools or tools that generate the same type of new object. • Check the toolbar help in order to find out which tool is currently activated and how to operate it.


ALGEBRA VIEW: Show and edit all the created objects and functions. Just double click on equation to edit it.

GRAPHIC VIEW: Show and construct objects and the graphs of the function.

INPUT BAR: Create new objects, equations and functions

SPREADSHEET VIEW


Activity 2: Familiarizing with some of the Menu/Tools

Explore some of the tools on your own.

Menu



Construction Tools



4. Drawing with GeoGebra

Activity 3: Drawing Geometric Figures and Other Objects

• Hide the algebra and spreadsheet window, and coordinate axes (View menu). • Show the coordinate grid (View menu).

Use the mouse and the following selection of tools in order to draw figures on the drawing pad (e.g. square, rectangle, house, tree, human figure…).



Introduction of new toolsNew pointHint: Click on the drawing pad or an already existing object to create a new point.MoveHint: Drag a free object with the mouse.Line through two pointsHint: Click on the drawing pad twice or on two already existing points.Segment between two pointsHint: Click on the drawing pad twice or on two already existing points.Delete objectHint: Click on an object to delete it.Undo / RedoHint: Undo / redo a construction step by step.Move drawing padHint: Click and drag the drawing pad to change the visible part.Zoom in / Zoom outHint: Click on the drawing pad to zoom in / out.

Hint: Don’t forget to read the toolbar help if you don’t know how to use a tool

Activity 4: What to practice

• How to select an already existing object. Hint: When the pointer hovers above an object, it highlights and the pointer changes its shape from a cross to an arrow. Clicking selects the corresponding object. • How to create a point that lies on and object. Hint: The point is displayed in a light blue color. Always check if the point really lies on the object by dragging it with the mouse. • How to correct mistakes step-by-step using the Undo (Ctrl-Z) and Redo (Ctrl-Y)buttons. Tip: Several tools allow the creation of points “on the fly”. Therefore, no existing objects are required in order to use the tool. Example: The tool Segment between two points can be applied to two already existing points or to the empty drawing pad. By clicking on the drawing pad the corresponding points are created and a segment is drawn in between them.

5. Saving Files

Activity 5: Saving GeoGebra files

• Open the File menu and select Save. • Select the folder “GeoGebra_your name” in the appearing dialog window.



• Type in a name for your GeoGebra file. • Click Save in order to finish this process. Hint: A file with the extension ‘.ggb’ is created. This extension identifies GeoGebra files and indicates that they can only be opened with GeoGebra. Hint: Name your files properly: Avoid using spaces or special symbols in a file name since they can cause unnecessary problems when transferred to other computers. Instead you can use underscores or upper case letters within the file name (e.g. First_Drawing.ggb).

Activity 6: Opening GeoGebra Files • Open a new GeoGebra window (menu File – New window). • Open a blank GeoGebra interface within the same window (menu File – New) Hint: If you didn’t save the existing construction yet GeoGebra will ask you to do so before the blank screen is opened. • Open an already existing GeoGebra file (menu File – Open).• Navigate through the folder structure in the appearing window. • Select a GeoGebra file (extension ‘.ggb’) and Click Open.

6. Drawings, Constructions, and Drag Test Open the dynamic worksheet 01_Drawing_Construction_Squares.html The dynamic figure shows several squares constructed in different ways. • Examine the squares by dragging ALL their vertices with the mouse. • Find out which of the quadrilaterals are real squares and which ones just happen to look like squares. • Try to come up with a conjecture about how each square was created.

Q1. Write down your conjectures.

……………………………………………………………………………………………………………………………………..........

DiscussionQ2. What is the difference between a drawing and a construction?

………………………………………………………………………………………………………………………………………………Q3. What is the “drag test” and why is it important?

………………………………………………………………………………………………………………………………………………Q4. Why is it important to construct figures instead of just drawing them in interactive geometry software?

………………………………………………………………………………………………………………………………………………Q5. What do we have to know about the geometric figure before we are able to construct it using dynamic mathematics software?

………………………………………………………………………………………………………………………………………………



7. Rectangle Construction

Activity 7: Constructing a rectanglePreparations • Summarize the properties of a rectangle before you start the construction. • Open new GeoGebra file. • Hide algebra window, input field and coordinate axes (View menu). • Change the labeling setting to New points only (menu Options – Labeling).

Introduction of new tools • Perpendicular line and Parallel line tools Hint: Click on an already existing line and a point in order to create a perpendicular line / parallel line through this point. • Intersect two objects tool Hint: Click on the intersection point of two objects to get this one intersection point. Successively click on both objects to get all intersection points. • Polygon tool Hints: Click on the drawing pad or already existing points in order to create the vertices of a polygon. Connect the last and first vertex to close the polygon! Always connect vertices counterclockwise! Hint: Don’t forget to read the toolbar help if you don’t know how to use a tool. Hint: Try out all new tools before you start the construction.

Construction Steps:

8. Navigation Bar and Construction Protocol

Activity 8: Checking your construction Apply the drag test to check if the construction is

correct.



Show the Navigation bar (View menu) to review your construction step-by step using the buttons.

Show the construction protocol (View menu) and use it to review your rectangle construction step-by-step. Try to change the order of some construction steps by dragging a line with the mouse.

Q1. Why does this NOT always work?

……………………………………………………………………………………………………………………………………………… Group several constructions steps by setting breakpoints: Show the column Breakpoint (View menu of the construction protocol window) Group construction steps by checking the Breakpoint box of the last one of the group Change setting to Show only breakpoints (View menu of the construction protocol

window)

Q2. Use the navigation bar to review the construction step-by-step. Did you set the breakpoints correctly?

………………………………………………………………………………………………………………………………………………Q3 How do you intend to use this construction for teaching?

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9. GeoGebra’s Properties Dialog

How to access the Properties Dialog Right-click an object OR Select Properties… from the Edit menu

OR Double click an object in Move mode

and click on button “Properties…” in the appearing Redefine dialog window.

Activity 9: What to practice

Select different objects from the list on the left hand side and explore the available properties tabs for different types of objects

Select several objects in order to change a certain property for all of them at the same time

Hint: Hold the Ctrl-key (MacOS: Open Apple-click) pressed and select all desired objects. Select all objects of one type by clicking on the corresponding heading. Show the value of different objects and try out different label styles. Introduction to GeoGebra – A Workshop for GeoGebra Usersby Dr. Daniel Leong, GeoGebra Institute of Malaysia, SEAMEO RECSAM on 25 September 201012


Change the properties of certain objects (e.g. color, style,…)

10.Square Construction

Activity 10: Constructing squares Construct a square using the above approach. Think of another way to construct a square. Describe it.………………………………………………………………………………………………………………………………………………

………………………………………………………………………………………………………………………………………………

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……………………………………………………………………………………………………………………………………………… Construct the square using the approach you have just described.Q1. How can you use this square to guide your students to find the formula for the area of a square?

………………………………………………………………………………………………………………………………………………

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An alternative construction process

11. Mencipta Custom Tool

Aktiviti 11: Mencipta Segi Empat

Proses Mencipta

1. Cipta Segi Empat ( menu Tools-create new tool..)

Objek Output : segi empat, sisi segi empat, titik C dan titik D Introduction to GeoGebra – A Workshop for GeoGebra Usersby Dr. Daniel Leong, GeoGebra Institute of Malaysia, SEAMEO RECSAM on 25 September 201013


Objek Input : Titik A dan B

Nama : Segi Empat

Toolbar Help : Klik pada 2 titik

2. Simpan segi empat sebagai fail Square_Tool.ggt

Petunjuk : Menu Tools-Manage tools…-Save as

Mengukur Panjang, Luas dan Sudut Memperkenalkan Peralatan Baru Jarak atau Panjang Petunjuk : Klik pada dua titil, garis, polygon, atau bulatan untuk mengukur panjang. Luas dan Petunjuk : Klik pada polygon, bulatan atau konik untuk mengukur luas. Sudut Petunjuk : Klik pada tiga titik atau dua garis untuk menentukan sudut.

Proses mencipta Bina segi empat.

1 Pilih Distance or Length tool.

2 Klik satu sisi pada segi empat untuk mengukur panjang sisi.3 Menggunakan kaedah yang sama, klik tiga sisi yang lain untuk mengukur

panjang setiap sisi. Klik pada segi empat untuk mengukur ukur lilit.

4 Plih Area Tool

5 Klik pada segi empat untuk mengukur luas segi empat.6 Pilih Angle Tool

7 Ukur semua sudut 8 Simpan file.

12.Equilateral Triangle Construction

Activity 12: Constructing an Equilateral Triangle

Preparations



• Summarize the properties of an equilateral triangle before you start the construction. • Use the buttons of the navigation bar in order to replay the construction process. • Open new GeoGebra file. • Hide algebra window, input field and coordinate axes (View menu). • Change the labeling setting to New points only (menu Options – Labeling).

Introduction of new tools • Circle with center through point tool Hint: First click creates center, second click determines radius of the circle. • Show / hide object tool Hints: Highlight all objects that should be hidden, then switch to another tool in order to apply the visibility changes! Hint: Don’t forget to read the toolbar help if you don’t know how to use a tool. Hint: Try out all new tools before you start the construction.

Construction Steps:

Check your construction

Apply the drag test to check if the construction is



correct. Show the navigation bar or use the construction protocol to review your construction step-

by-step.

13.Export of Pictures to the Clipboard GeoGebra’s drawing pad can be exported as a picture to your computer’s clipboard. Thus, they can be easily inserted into text processing or presentation documents allowing you to create appealing sketches for tests, quizzes, notes, or mathematical games.

Activity 13: Exporting Pictures to the Clipboard

Create your sketch: Open a saved GeoGebra document. GeoGebra will export the whole graphics window into the clipboard. Thus, you need to make the GeoGebra window smaller in order to reduce unnecessary space on the drawing pad: • Move your figure (or the relevant section) to the upper left corner of the drawing pad

using the Move drawing pad tool. You might want to use tools Zoom in and Zoom out

or use the mouse’s wheel in order to prepare your figure for the export process. • Reduce the size of the GeoGebra window by dragging its lower right corner with the mouse. The pointer will change its shape when hovering above an edges or corner of the GeoGebra window. Use the File menu to export the drawing pad to the clipboard:

Export – Drawing Pad to Clipboard. You could also use the key combination Ctrl – Shift – C.

Your figure is now stored in your computer’s clipboard and can be inserted into any word processing or presentation document.

14. Inserting pictures into a text processing documentAfter exporting a figure from GeoGebra into your computer’s clipboard you can now paste it into a word processing document (e.g. MS Word) Open a new text processing document From the Edit menu select Paste. The picture is inserted at the position of the cursorHint: You can use the key combination Ctrl-V instead.

15. Inserting Pictures into the Graphics Window

Activity 14: Drawing Tools for Symmetric Figures

Introduction of new tools TraceHint: Select the point to be traced and right-click. Select from the menu Trace on Reflect object in lineHint: Select object to reflect then line of reflection Insert ImageHint: Click on the drawing pad or on point to specify the location



Construction process

Note: The Trace feature has the following characteristics: A temporary phenomenon. Whenever the graphics are

refreshed, the trace disappears. The trace can’t be saved and is not shown in the Algebra

View. To delete the trace you need to refresh the views: Select

menu View - Refresh views (Ctrl-F).

16.Basic Algebraic Input, Commands, and Functions Using the Input bar you can directly enter algebraic expressions in GeoGebra. After hitting the Enter-key your algebraic input appears in the Algebra view while its graphical representation is automatically displayed in the Graphics view. For example, the input f(x) = x^2 gives you the function f in the Algebra view and its function graph in the Graphics view.In the Algebra view, mathematical objects are organized as free or dependent objects. If you create a new object without using any other existing objects, it is classified as a free object. If your newly created object was created by using other existing objects, it is classified as a dependent object. Hint: If you want to hide the algebraic representation of an object in the Algebra view, you can specify the object as an auxiliary object: Right click (MacOS: Ctrl-click) on the corresponding object in Introduction to GeoGebra – A Workshop for GeoGebra Usersby Dr. Daniel Leong, GeoGebra Institute of Malaysia, SEAMEO RECSAM on 25 September 201017


the Algebra view and select ‘Auxiliary Object’ from the appearing context menu. By default, auxiliary objects are not shown in the Algebra view, but you can change this setting by selecting the item ‘Auxiliary Objects’ from the View menu.

GeoGebra also offers a wide range of commands that can be entered into the Input bar. You can open the list of commands in the right corner of the Input bar by clicking on the button Command. After selecting a command from this list (or typing its name directly into the Input bar) you can press the F1-key to get information about the syntax and arguments required to apply the corresponding command.

Activity 15: Tips and Tricks Name a new object by typing in name = in front of its algebraic representation. Example : P = (3, 2) creates point P. Multiplication needs to be entered using an asterisk or space between the factors.Example : a*x or a x • GeoGebra is case sensitive! Thus, upper and lower case letters must not be mixed up.

Note : Points are always named with upper case letters Example : A = (1, 2)

Segments, lines, circles, functions… are always named with lower case letters. Example : f(x) = 3*x + 2

If you want to use an object within an algebraic expression or command you need to create the object prior to using its name in the input field.

The variable x within a function and the variables x and y in the equation always need to be lower case.

Examples : y = m x + b creates a line whose parameters are already existing values m and b (e.g. numbers / sliders).

Line[A, B] creates a line through existing points A and B. • Confirm an expression you entered into the input field by pressing the Enter key. • Open the help window for using the input field and commands by clicking the question mark to the left of the input field. • Error messages: Always read the messages – they could possibly help to fix the problem! • Commands can be typed in or selected from the list next to the input field. Hint : If you don’t know which parameters are required within the brackets of a certain command, type in the full command name and press key F1. A pop-up window appears explaining the syntax and necessary parameters of the command. • Automatic completion of commands: After typing in the first two letters of a command into the input field, GeoGebra tries to complete the command. If GeoGebra suggests the desired command, hit the Enter key in order to place the cursor

within the brackets. If the suggested command is not the one you wanted to enter, just keep typing until the

suggestion matches.

Activity 16: Visualizing Integer Addition on the Number Line

Preparations Open a new GeoGebra file. Hide the Algebra View (View menu).


b=2


In the Options menu set the Labeling to All new objects.

Introduction of new tools • Slider - Hint: Click on drawing pad to specify position. • Vector between two points - Hints: Select starting point, then end point.Construction Process


Drawing Pad.

Drawing Pad dialog for the Graphic View


17.Static and Dynamic Text

Introduction of new toolsInsert textHint: Click the drawing pad to specify the location of your text. Enter the desired text into the appearing window and click Apply.

Hints: Don’t forget to read the toolbar help if you don’t know how to use a tool. Try out new tools before you start the construction.

Activity 17: Inserting Static TextInsert a heading/title into the Graphic View. Activate the Text Tool and click on the upper part of the Graphic View. Type the following text into the appearing window:

Visualizing Integer Addition on the Number Line You can change the properties of the text in the Properties dialog Click Apply Adjust the position of the text using the Move tool. Fix the position of the text so it cannot be moved accidentally. (Properties dialog – tab

Basic – Fix object).

Activity 18: Inserting Dynamic Text

Enhance your interactive figure by inserting dynamic text that displays the corresponding addition problem. Dynamic text refers to existing objects and adapts automatically to modifications. In order to display the parts of the addition problem in different colors you need to insert the dynamic text step by step.

Construction Process



18.Conditional Formatting

Activity 19: Conditional Formatting – Inserting Checkboxes

Preparations Customizing your toolbar.

Insert checkboxIntroduction of new tools



Check Box to Show/Hide objectsHint: Click on the drawing pad to specify position.

Insert a checkbox into the Graphics View that allows you to show or hide the result of the addition problem.


19.Fraction with Circles

Activity 20: Constructing Circular Sectors with slidersPreparations Open a new file Show the Input Bar

Pengenalan kepada pekakas baru (new tools)



Putarkan Objek disekeliling Titik ( point ) melalui Sudut ( angle )Petunjuk: Pilih objek untuk diputarkan, kemudian pilih pusat bulatan (centre point) dan masukkan nilai sudut ( angle)Sektor bulatan (circular Sector) dengan pusat bulatan diantara dua titik. Petunjuk: Pilih pusat bulatan dan dua titik.

Proses Pembinaan (construction process)1 Bina gelongsor mendatar (horizontal slider) dilabel ‘den’ untuk menunjukkan

nombor dengan skala 4 ke 8, dengan peningkatan (increment) 12 Sudut zz=3600/den3 Gelongsor a bagi nombor dengan skala dari -5 ke 70, dengan peningkatan 0.1 4 Hasilkan satu titik A (katakan A=(-20, 23))5 Bina satu bulatan dengan berpusatkan A dan jejari (radius) a6 Bina satu garisan melalui A yang selari dengan paksi x7 Titik B dan C yang mana bersilang di titik c, b8 Bina gelongsor yang dilabel num untuk nombor dengan skala dari 1 hingga 8,

dengan peningkatan 1 9 Titik C’ terhasil daripada putaran C dengan sudut num zz pada A

(kamu boleh masukkan C’=Rotate[C,zz,A])10 Titik D terhasil daripada putaran C dengan sudut zz at A 11 Titik E terhasil daripada putaran C dengan sudut 2zz at A12 Titik F terhasil daripada putaran C dengan sudut 3zz at A13 Titik G terhasil daripada putaran C dengan sudut 4zz at A14 Titik H terhasil daripada putaran C dengan sudut 5zz at A15 Titik I terhasil daripada putaran C dengan sudut 6zz at A16 Titik J terhasil daripada putaran C dengan sudut 7zz at A17 Titik K terhasil daripada putaran C dengan sudut 8zz at A18 Bina tembereng (segment) C’A

(kamu boleh masukkan Segment[C’A])19 Bina tembereng DA20 Bina tembereng EA21 Bina tembereng FA22 Bina tembereng GA23 Bina tembereng HA24 Bina tembereng IA25 Bina tembereng JA26 Bina tembereng KA27 Bina satu bulatan dengan berpusatkan A dan jejari a28 Bina sektor bulatan ACC’29 Bulatan berpusatkan A dan jejari a (Syarat: tunjuk bulatan ini sekiranya tour>=1)30 Masukkan tour=num/den31 Perbaiki hasil kamu dan sertakan dynamic, static text, dan Checkbox jika perlu.



20.Menggambarkan Pendaraban Nombor Natural (Visualizing Multiplication of Natural Numbers).

Activiti 21: Bina Satu Model untuk Penggambaran Pendaraban

PersediaanBuka satu file GeoGebra yang baru. Sembunyikan paparan algebra ( Algebra View) dan paksi koordinat (coordinate axes)

[View menu]. Tunjukkan Input Bar [View menu]. dalam set menu pilihan (Options menu) pilih label

untuk semua objek baru ( Labeling to All New Objects).


1 Horizontal slider Columns for number with interval from 1 to 18, increment 1, and width 300

2 New point A

3 Segment a with given length Columns from point A

4 Move slider Columns to check the segment with given length.

5 Perpendicular line b to segment a through point A

6 Perpendicular line c to segment a through point B

7 Vertical slider Rows for number with interval from 1 to 15, increment 1, and width 300

8 Circle d with center A and given radius Rows

9 Move slider Rows to check the circle with given radius

10 Intersect circle d with line b to get intersection point C

11 Parallel line e to segment a through intersection point C

12 Intersect lines c and e to get intersection point D

13 Create polygon ABDC

14 Hide all lines, circle d, and segment a

15 Hide labels of segments

16 Set both sliders Columns and Rows to their maximum value



17 Create a list of vertical segmentsSequence[Segment[A+i*(1, 0), C+i*(1, 0)], i, 1, Columns]Note:A + i(1, 0) specifies a series of points starting at point A with distance 1 from each otherC + i(1, 0) specifies a series of points starting at point C with distance 1 from each otherSegment[A + i(1, 0), C + i(1, 0)] creates a list of segments between pairs of these points. Note, that the endpoints of the segments are not shown in the graphics window.Slider Column determines the number of segments created.

18 Create a list of horizontal segmentsSequence[Segment[A+i*(0, 1), B+i*(0, 1)], i, 1, Rows]

19 Move sliders Columns and Rows to check the construction.

20

Insert static and dynamic text that state the multiplication problem using the values of sliders Columns and Rows as the factors:text1: Columnstext2: *text3: Rowstext4: =

21 Calculate the result of the multiplication: Inputresult = Columns * Rows

22 Insert dynamic text5: result

23 Hide points A, B, C, and D

24 Enhance your construction using the Properties dialog.

Q1. Describe how you would use this file to enable your students learn multiplication.

………………………………………………………………………………………………………………………………………………

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Q2. What other topics that you can enable the students to learn using this file?

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21.Reconstruct ggb files

Activity 22: Reconstructing “Thumb-up.ggb”Preparations Open the ggb file named “Thumb-up”. Arrange the pictures in the correct drawing sequence by dragging the blue dot to the

correct box in the table. When it is arranged correctly, the “thumb-up” picture will appear. Click on the View menu and select Construction Protocol. On the Construction Protocol window click View. Make sure the following are selected: Name, Definition, Command, and Algebra. Try to reconstruct the “Thumb-up” GeoGebra file by following the construction

processes/steps displayed.

22.Creating Dynamic Worksheets

Dynamic Worksheets GeoGebra allows you to create your own interactive instructional materials, so called dynamic worksheets, by exporting dynamic figures into web pages. Usually, a dynamic worksheet consists of a heading, short explanation, interactive applet, as well as tasks and directions for your students. In order to work with dynamic worksheets your students don’t need to know how to operate GeoGebra at all. The interactive web pages are independent of the software and can be provided either online or on a local storage device.

Reducing the size of the GeoGebra window GeoGebra will export the algebra and graphics window into the dynamic figure of the worksheet. In order to save space for explanations and tasks on the dynamic worksheet you need to make the GeoGebra window smaller prior to the export. o If you don’t want to include the algebra window you need to hide it prior to the

export. o Move your figure (or the relevant section) to the upper left corner of the drawing pad

using the Move drawing pad tool (see left figure below). Hint : You might want to use tools Zoom in and Zoom out in order to prepare your figure for the export process.o Reduce the size of the GeoGebra window by dragging its lower right corner with the

mouse (see right figure below). Hint : The pointer will change its shape when hovering above an edge or corner of the GeoGebra window. Note: Although the interactive applet should fit on one screen and even leave some space for text on the worksheet you need to make sure that it is big enough to allow students manipulations and experiments.



Activity 23: Exporting a Dynamic Worksheet

Preparations Open the Thumbup.gbb file that you have created. Reduce the size of the GeoGebra

window as mentioned above. After adjusting the size of the GeoGebra window, you are now ready to export t h e f i g u r e a s a d y n a m i c w o r k s h e e t u s i n g t h e F i l e m e n u .

E x p o r t – D y n a m i c W o r k s h e e t a s W e b p a g e Hint : You could also use the key combination Ctrl – Shift – W. Fill in the text fields in the appearing window (title of the worksheet, name of the author,

and date). Type a short explanation of the dynamic figure into the text field Text above the

construction. Enter tasks and directions for students into the text field Text after the construction. Click Export and save your dynamic worksheet. Hint : GeoGebra will create several files which always need to stay together in order to maintain the functionality of the dynamic worksheet. We recommend creation of a new folder (e.g. Dynamic_Worksheets) within the GeoGebra_Introduction folder prior to saving your dynamic worksheet.

Activity 24: Enhancing Dynamic Worksheets

The export dialog window for dynamic worksheets consists of two tabs: General and Advanced. In the last activity you used tab General in order to add explanations, tasks, and directions to the dynamic figure prior to the export. You will now learn how to enhance your dynamic worksheet by including different features in the interactive figure using the tab Advanced.

Functionality Right click enabled: Your students will be able to right click objects or the drawing pad in

order to access the features of the context menu (e.g. show / hide object or label, trace on / off, Properties dialog).

Show icon to reset construction: A reset icon is displayed in the upper right corner of the interactive applet allowing your students to reset the interactive figure to its initial state.

Double click opens application window: Your students will be able to open a full GeoGebra window by double clicking the interactive applet.

User interface Show menu bar: The menu bar is

displayed within the interactive applet. Show toolbar: The toolbar is displayed Introduction to GeoGebra – A Workshop for GeoGebra Usersby Dr. Daniel Leong, GeoGebra Institute of Malaysia, SEAMEO RECSAM on 25 September 201027


within the interactive applet allowing your students to use the geometry tools. Show toolbar help: In combination with the toolbar, you can also display the toolbar help

within the interactive applet. If you want your students to use geometry tools they can use the toolbar help in order to find out how to operate the different tools on their own.

Show input field: The input field is displayed at the bottom of the interactive applet allowing your students to use algebraic input and commands for their explorations.

• Width and height of the interactive applet : You can modify the width and height of the interactive applet. Note : If you reduce the size of the applet important parts of the dynamic worksheets might be invisible for your students. Hint : If you include the menubar, toolbar, or input field you might want to adjust the height of the interactive applet.

23.Providing Dynamic Worksheets to Students You can provide your dynamic worksheets in several ways to your students. However, in all cases it is very important to keep all the files together which were created during the export process. Note: The files created have different file name extensions (.ggb, .html, .jar). If one of these files is missing your dynamic worksheet won’t function any more.

Task Use the dynamic figure you have created and export it as an enhanced dynamic worksheet. Use the Advanced tab to try out different options and check how the applet of your dynamic worksheet is changed accordingly.

24.GeoGebra’s Spreadsheet View

The dynamic mathematics software GeoGebra provides three different views of mathematical objects: a Graphics view, a, numeric Algebra view and a Spreadsheet view. They allow you to display mathematical objects in three different representations: graphically (e.g., points, function graphs), algebraically (e.g., coordinates of points, equations), and in spreadsheet cells. Thereby, all representations of the same object are linked dynamically and adapt automatically to changes made to any of the representations, no matter how they were initially created.

Spreadsheet Cells InputIn GeoGebra’s Spreadsheet view every cell has a specific name that allows you to directly address each cell. For example, the cell in column A and row 1 is named A1. Note: These cell names can be used in expressions and commands in order to address the content of the corresponding cell.

Into the spreadsheet cells you can enter not only numbers, but all types of mathematical objects that are supported by GeoGebra (e.g., coordinates of points, functions, commands). If possible, GeoGebra immediately displays the graphical representation of the object you enter into a Introduction to GeoGebra – A Workshop for GeoGebra Usersby Dr. Daniel Leong, GeoGebra Institute of Malaysia, SEAMEO RECSAM on 25 September 201028


spreadsheet cell in the Graphics view as well. Thereby, the name of the object matches the name of the spreadsheet cell used to initially create it (e.g., A5, C1).

Note: By default, spreadsheet objects are classified as auxiliary objects in the Algebra view. You can show or hide these auxiliary objects by selecting ‘Auxiliary Objects’ from the View menu.

Activity 25: Recording to Spreadsheet FeaturePreparations

Open a new GeoGebra file. Show the Spreadsheet view (View menu). Hide the Algebra view (View menu).


1

Create a slider with default interval and increment 1.Hint: Select tool Slider and click in the Graphics view to set the position for the slider. In the appearing dialog window change the increment to 1 and click the Apply button.

2 A = (a, 2a) Create point A by entering A = (a, 2a) into the Input bar.

Hint: The value of slider a determines the x-coordinate of point A while the y-coordinate is a multiple of this value.

3 Show the label of point A in the Graphics view.

4 Change the value of slider a to examine different positions of point A.

5Use tools Move Graphics View, as well as Zoom In and Zoom Out to adjust the visible part of the Graphics view and make point A visible in all positions.



6 Turn on the trace of point A.

Hint: Right click (MacOS: Ctrl-click) on point A and select ‘Trace On’ from the appearing context menu.

7 Change the value of slider a to examine the trace point A leaves for every slider position.

8 Set the value of slider a to -5.

9 Record the coordinates for different positions of point A to the spreadsheet:

(1) Select tool Record to Spreadsheet. Then, click on point A in order to highlight it. Note: The coordinates for the actual position of point A are immediately entered into cells A1 (x-coordinate) and B1 (y-coordinate) of the spreadsheet.(2) Now, change the value of slider a in order to record the coordinates of all other possible positions of point A to the spreadsheet as well.

Note: Do not switch to another tool before moving the slider.

Activity 26: Relative Copy and Linear EquationsPreparations

Open a new GeoGebra file. Show the Spreadsheet view (View menu). Hide the Algebra view (View menu).

Construction ProcessIntroduction to GeoGebra – A Workshop for GeoGebra Usersby Dr. Daniel Leong, GeoGebra Institute of Malaysia, SEAMEO RECSAM on 25 September 201030


1 Activate tool Move Graphics View and drag the origin of the coordinate system close to the lower left corner of the Graphics view.

2 (0, 0) In the Spreadsheet view, click on cell A1 enter the point coordinates (0, 0).

3 (1, 1) In the Spreadsheet view, click on cell A2 enter the point coordinates (1, 1).

4 Show the labels of both points in the Graphics view.

5 Relative copy the inserted point coordinates to other cells in column A:

(1) Highlight both cells A1 and A2 by using the mouse. (2) Click on the little square at the lower right corner of the highlighted cell range. (3) Hold the mouse button down and drag the pointer down to cell A11.

6 Use tools Move Graphics View, as well as Zoom In and Zoom Out to adjust the visible part of the Graphics view and make all points visible.

Activity 27: Creating a histogramYesterday, you gave a mathematics quiz to the 25 students of your 1st period math class. After the quiz, you asked your students to rate the difficulty of the quiz on a scale from 1 (‘very easy’) to 5 (‘very difficult’).

4 of your students rated the quiz ‘very easy’ (1)

6 students rated the quiz ‘easy’ (2) 6 other students rated the quiz ‘difficult’ (4) 1 student rated the quiz ‘very difficult’ (5) The rest of the students thought the difficulty

of the quiz was ‘ok’ (3).Enter the data into GeoGebra’s spreadsheet and create a histogram that visualizes this data. Hints:

If you don’t know how to use command Histogram, enter the command into the Input bar and press the F1 key.

Note : ‘Class boundaries’ determine the position and width of the bars of the histogram. The absolute number of students that rated the difficulty of the quiz for each item determines the height of the histogram bars.

Choose the class boundaries so that the actual rating score is displayed in the middle of each histogram bar.

You need to create a list of the data in each column before you can use the Histogram commandNote: Highlight all numbers in one column and right click (MacOS: Ctrl-click) on one of the highlighted cells. Select Create List from the appearing context menu.



25.Useful Web Links

GeoGebra Homepage

http://www.geogebra.org/cms/ Get the software as well as information about the GeoGebra project

GeoGebraWiki

http://www.geogebra.org/en/wiki/index.php/Main_Page Pool of free instructional materials created with GeoGebra

GeoGebra User Forum

http://www.geogebra.org/forum/ Search for answers or post your own questions. Teachers from all over the world will try to help you.

GeoGebra Documentation

http://www.geogebra.org/cms/en/help GeoGebra Help Document - the Official Manual GeoGebra Book GeoGebra Quickstart GeoGebra Video Tutorials


http://www.recsam.edu.my/geogebra/ http://www.geogebra.org/en/wiki/index.php/GeoGebra_Institute_of_Malaysia


http://www.geogebra.org/en/wiki/index.php/GeoGebra_Institute_of_Malaysia

http://www.recsam.edu.my/geogebra/

http://www.geogebra.org/cms/en/help

http://www.geogebra.org/en/wiki/index.php/Main_Page

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