Training Material_Introduction to GeoGebra

7/25/2019 Training Material_Introduction to GeoGebra

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Training Material: Introduction to GeoGebra 1/60

Training material

Introduction to GeoGebra

Content:A. IntroductionB. First steps with GeoGebraC. Introduction drillD. ExportE. Represent and drag objects F.

Algebra window

G. Some proposed exercisesH. Functions II. Functions IIJ. How to insert the random function in a GeoGebra appletK. References

A. Introduct ion

The goal of this document is to facilitate the teaching staff the suitable knowledge

which will enable them to handle and/or modify the proposed exercises and to adapt them totheir didactic necessities.In general, an exercise or activity aimed at to the pupils of a specific level is taken as

the starting point to know some of the possibilities that the GeoGebra software offers.Among other facilities, it allows to generate htmlfiles from GeoGebra documents.

This free software application, General GNU Public License, has been developed byMarkus Hohenwarter, from the University of Salzburg (Austria). It combines the benefits ofthe dynamic geometry software with those of the algebraic calculation systems. TheGeoGebra objects are dynamically considered under these two aspects -graphicalrepresentation and analytical definition.

The last stable version -GeoGebra 3.RC1 (Release Candidate- can be downloadedfrom the Website of this software:

http://www.geogebra.organd, whenever possible, the WebStar versions in the already mentioned website (WebStar)can be used to ensure the usage of the last stable release.

It should be clarified that we are not dealing with a course on GeoGebra, so we willnot exhaust the possibilities of this software. We will try to focus on those aspects that allowus to modify and/or to generate exercises similar to the ones displayed.

Users will have at their disposal a Helpfile (in pdfand html formats), after installingthe program,. Besides, a Quick Reference Guide and Help can be downloaded from theWebsite (Help). In this very section there are other three GeoGebra documents:

GeoGebra Applet parameterGeoGebra Applets and JavaScript

GeoGebra XML file format

that can be very useful for all those interested in knowing the features of the GeoGebra filesand their possible ways to use. We will dedicate some chapters to explain how this
http://www.geogebra.org/http://www.geogebra.org/cms/index.php?option=com_content&task=blogcategory&id=70&Itemid=57http://www.geogebra.org/cms/index.php?option=com_content&task=blogcategory&id=75&Itemid=61http://www.geogebra.org/source/program/applet/geogebra_applet_param.htmlhttp://www.geogebra.org/source/program/applet/geogebra_applet_javascript.htmlhttp://www.geogebra.org/source/program/applet/geogebra_applet_javascript.htmlhttp://www.geogebra.org/source/program/applet/geogebra_applet_javascript.htmlhttp://www.geogebra.org/source/program/applet/geogebra_applet_javascript.htmlhttp://www.geogebra.org/source/program/applet/geogebra_applet_param.htmlhttp://www.geogebra.org/cms/index.php?option=com_content&task=blogcategory&id=75&Itemid=61http://www.geogebra.org/cms/index.php?option=com_content&task=blogcategory&id=70&Itemid=57http://www.geogebra.org/


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application works in a general way and also to the generation of interactive exercises in html

format.We suggest create 3 folders: activities, solutionsand workand unzipping:

all the files of the file activities.zipin the folder activities.

all the files of the file solutions.zip in the folder solutions.

The folder workwill contains the proposed activities.

A1. Icons used

We will use the following icons:

Makes reference to an existing external link which requires an Internetconnection

Opensa file in the folder activities

Saves(or Opens) a file in the folder work

One file in the folder solutions

A2. Conf igurations and programs

To enhance GeoGebrawork the virtual Java machine is required. If you dont haveany installed you can download it here:

Sun Microsystems. Virtual Java machine corresponding to Java Runtime EnvironmentVersion 6.0 Update 5.

The eventual modification of the html files generated by GeoGebra can be made byusing any web page editor. For these modifications we will use Microsoft FrontPage 2003though any page editor could do. We will assume that the user knows how to use thisprogram, at least, at a basic level (open, save, edit).

B. First steps with GeoGebra

B1. Document window

To start the GeoGebra application double click on the corresponding icon in the desk

Or follow the path: StartAll ProgramsGeogebraGeogebra.

http://java.com/en/download/windows_manual.jsphttp://java.com/en/download/windows_manual.jsp


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Figure 1 Start GeoGebra

You can do it either ways to get to the application window where the basic elementshave been remarked: Toolbar, Input field,Algebra windowand Drawing pad.

Figure 2 The basic elements of the GeoGebra application

To Hide/show the axes, select the commandAxesfrom the Viewmenu.By selecting one of the modes in the tool bar you can do constructions on the drawing

pad. Coordinates or equations of the objects appear in the algebra window.In the input field you can write coordinates, equations, commands and functions that

are displayed at the drawing pad after pressing the Enterkey (or equivalent).



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B2. Representation of points

Click on New Pointbutton. Place the cursor on the drawing pad and left click. PointAis

displayed as represented here:

Figure 3 Representation of points

The last selected mode remains active until another one is selected, that is to say, ifwe scroll the mouse and left click again a new point will be created. The two arrows on thetop right area (tool bar) allow us to undo or redo the last actions.

B3. Change name

GeoGebraassigns names to the objects alphabetically. In the case of the points: A,B, C

To change the nameof an object:



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Click on Movebutton.

Place the cursor on the object and right click.In the context menu select

Rename

In the Rename window, type the new name

of the object and press the key.

In this case we have changed the pointAtoAndrs.

B4. Modify properties

As it can be seen in the previous picture, after right clicking on an object, the context

menu with several commands is displayed. The last one, Properties allows access toall the properties of an object.

Figure 4 Modify properties



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Lets change the colour and the size of that point.

To change the colour, click on the tab and choose a new one by clicking onthe suitable cell. To finish, click on the button .

Figure 5 Change the colour

To change the size of the point , click on the tab and move the slider until themeasure you want. To finish, click on the button .

Figure 6 Change the size of the point

After the described modifications, the appearance of the point renamed asAndrsispresented this way:



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Figure 7 The appearance of the point Andrs

C. Introduction drill

The image shows a height (in meters) - weight (in kilograms) graph corresponding toa group of 5 friends as it can be seen in the records made due to a check-up (notice the lackof reference to the interruption of the axes).

Look at the chart from the Figure 7 and answer the following questions:- How tall is Alberto?- Which is Mnicas weight?

- Who is the shortest person? And the tallest?- Who is the most obese person? And the thinnest?- How tall is David and which is his weight?- What happens if we join the points?- Do you find the axis scale appropriate?



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Figure 8 Exercise

C1. Basic commands

Our goal is to get a representation similar to the previous one. We have alreadyexplained how to start GeoGebraand how to change a points colour, size and position.

Lets see now how to change the background colour, the axis colour and the gridcolour, insert a text and modify its properties, and how to hide the algebra window so that theentire window will be occupied by the drawing pad. Besides, we will describe the way toexport the constructions to htmlformat.

For that, we will modify a GeoGebra file, which we will save later.

C2. Open a file

Toopen a filewe can:

select the command Openfrom the Filemenu, in case the application is alreadystarted, or,

double click on the name of the file.

In either ways, we try to opentheactiv01.ggbfile.



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Figure 9 Open a file

Figure 10 A GeoGebra file

If the grid is not visualized, select the command Gridfrom the Viewmenu.



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C3. Hide elements

Lets hide the elements that, in this moment, are not going to be used (algebrawindow and edit line or input field).

To close the algebra window: Select the commandAlgebra window from the Viewmenu.

To hide the input line select the command Input fieldfrom the Viewmenu.The ticks shown in the image, in these items, must disappear.

Figure 11 The View menu

C4. Background colour

To change the background colour:select Drawing Padfrom the Optionsmenuor, place the cursor on any free area (from the drawing pad), right click and choose

Properties.



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aa)) bb))

Figure 12 a) Options menu and b) context menu

In the Drawing Padwindow, click on the Background colour button and proceed as inthe case of the points (First steps).

Figure 13 The Drawing Pad window


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Figure 14 Change the background colour to green

C5. Axis colour

To change the Axis colour:select Drawing Padfrom the Optionsmenuor, place the cursor on any free area (from the drawing pad), right click and choose

Properties.

a


a)) bb))

Figure 15 a) Options menu and b) context menu

In the Drawing Pad window, click on the Colour button and proceed as in theprevious case.


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Figure 16 The Drawing Pad window. The Axes tab

Figure 17 Change the axis colour to green.

C6. Grid colour

To change the grid colour proceed as in the previous cases until visualising theDrawing Padwindow.

Click on the tab, and then on the Colour button and proceed as in any of the

previous cases.Change the grey colour to dark blue.



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Figure 18 The Drawing Pad window. The Grid tab

C7. Save a file

To save a file:

Select the command Savefrom the Filemenu to save it with the same name,

or, Select the command Save as from the Filemenu, to save it with a differentname.

Save the file with the modifications made up to that moment with the name

activ01_01.ggb.



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Figure 19 Save a file

Change the name, colour and size of the points so that they are similar to the figure in

the introduction drill (to copy the shape of a point you can use the command Copy

visual style last button in the tool bar) and save the new file with the previous name(activ01_1.ggb).

C8. Texts

To insert a text you have to use the Insert texttool (penultimate button in the tool bar).

Once selected, place the cursor on the drawing pad and left click.The Text window appears in the one we have to type the corresponding text and click

on the button.

Figure 20 The text window



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To complete the introduction drill we still need the texts 1.80 in the X axis and 80 in

the Y axis. We type the texts (place the cursor, nearly at the same height that the alreadyexisting ones) and then, we modify its properties so that the style will be similar to that of theoriginal figure.

We can use the Copy visual styletool, above mentioned, to copy some of the featuresin the existing texts. Once this operation is made, we place the cursor on the text, right click

and in the context menu (Text T13) we choose Properties.

Figure 21 The Properties window

In the Properties window, we select tab and we deactivate the Absolute

position on screen cell.In the Starting point field we write the coordinates of the text, (6.8, -0.5) and click on

the button.

We repeat this operation with the Y axis. The coordinates in this case will be (-0.5, 7).

Save the new file with the name activ01_2.ggb.

D. Export

GeoGebraallows to export the constructions (the drawing pad or just a part of it) asgraphics formats (png, eps, svg, emf), copy them in the Windows clipboard and turn theminto an interactive html page.

Open the last saved activ01_2.ggbfile in the previous session,

or

the solution fileactiv01_2Sol.ggb.



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It must look like the following figure:

Figure 22 The activ01_2Sol.ggb file

The export menu is activated by the command Exportfrom the Filemenu.

Figure 23 Export a file in GeoGebra

By selecting the Drawing Pad as Picture (png, eps) option, the drawing pad iscopied in a chosen file.

The Export: Drawing Pad window allows to choose the format from the output file(png, eps, svg, emf), the scale and the resolution (only in the case of png format).



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Figure 24 - Export: Drawing Pad window

By clicking on the button we get to the Save window, where we canchoose the name and placement for the output file.

Figure 25 The Save window



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The Dynamic Worksheet as Webpage (html) option allows to generate an html

file which contains the construction and, at the same time, the movement of the free objects.It is possible to activate the application through a button and to allow the execution of the fullprogram by double clicking on it.

In the window Export: Dynamic Worksheet (html)we can write the title, author anddate on the corresponding fields which will appear as head and footnote of the html file. It isalso possible to include a text above (Text above the construction) and after (Text after theconstruction) in the application window and to include html code in them.

a)



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b)

Figure 26 The Export: Dynamic Worksheet (html) window. a) Advanced tab, b) General tab

The radio button Button to open application window with construction allows to hidethe visualization of the construction until this button is pressed.

The tab allows to choose among the program options available for theuser.

If the Double click opens application window cell is activated, the double click on theconstruction in the Web page generated allows to open GeoGebrawith the same featuresthat the Windows application.

In any case, several files are generated. If the name of the file is activ01_03.html, thefiles activ01_03.ggb, geogebra.jar, geogebra_cas.jar, geogebra_export.jar, geogebra_gui.jar,geogebra_properties.jar. are kept in the same folder.

Next, two images are presented as a result of the exportation of each of the options

found in the tab:



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Figure 27 Exportation result

Figure 28 Exportation result



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E. Represent and drag objects

E1. Representation of the points

1. In the following figure you can see the representation of the point P (3, 2).Remember that the first number, the abscise or x-coordinate, is represented on the horizontalaxis (abscises); to the right of its origin if it is positive and to the left if negative. However, theordinate or y-coordinate, the second number, is represented on the vertical axis (ordinates);above its origin if positive and below if negative.

Figure 29 Example

If you drag the point P with the mouse you can see how the coordinates change. Youcan check that if the point is in the first or fourth quadrants, the x-coordinate (abscise) ispositive, but if it is found in the third or in the second, the x-coordinate is negative. As regards

the y-coordinate (ordinate), it is positive for the quadrants first and second and negativewhen the points are in the third or fourth quadrants.

2. The following figure has 4 points: two in the first quadrant, B and D, and two in thesecond, A and C. The thing is to drag the points in such a way that there is one per quadrantregarding the attached table:

Point Quadrant

A 1

B

2



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C 3

D

4

Figure 30 Example

Complete now the following table:

Point Signofx Signofy

A

B

C

D

E2. Dragging objects, segments, texts

Lets see how to make drills as the ones described in the previous paragraphs. Wealready know how to represent points and to change their names. Lets see now how to limittheir movement. We will explain later the way to define segments and to add texts withproperties dealing with the existing objects.



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E3. Restraint the dragging to the grid

We open the activ02.ggb file, which already has the labelled axis and it is placedin the centre of the window. The thing is to incorporate a point P, which can be dragged withthe mouse and whose coordinates are to be shown together with the name, to present thecorresponding segments to the abscise and ordinate values (see figure in Representation ofthe points, drill 1). Besides, we want this dragging limited to the points of the quadrant.

Lets remember the way to define a point: Choose New Point, drag the cursor to theestimate position (3, 2) and left click. Change the name, size and colour until it looks like theoriginal figure in this activity (see First steps).

To limit the dragging of the points to the grid we have to select the command Pointcapturing from the Options menu and choose on. If we drag now the point P well see that

when it is close to any of the points on the grid it is attracted by it. If we want it to jump fromone point on the grid to another without allowing any mid-positions, we have to select on(Grid).

Figure 31 Selecting On the grid

We can deal now with the distance between the units of the X and Y axis togetherwith the ones defined for the grid. The definition of these distances is made in the Propertieswindow in the Drawing Pad (see First steps).

The distance between the X and Y axis is set by default at 2 units and the distancebetween the points on the grid at 1 for both axis. If we want the points on the grid to becorrespondent with 0.5 unit intervals, we will have to write that value in the unit field in the

tab.Lets see these considerations with two figures. The one on the left shows the window

without the grid, whereas the one on the right with the 0.5 distance grid. If we have the onoption (Grid) selected, the point P will change its coordinates in 0.5 unit intervals in both axis.



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Figure 32 Example

E4. Show the coordinates of one point

If in the Properties window of the point P, in the field Show label (in the tab),we select Name & Value, the P label adopts the shape we can see next in the image.

Figure 33 Show the coordinate of one point

Another possibility is to hide the label (deactivate the Show label cell) and insert a textin the position of the point. That way we can get texts different in colour and printing to those

in the point.



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We define a text (see First steps) writing P + P. What it is under quotation appears

as it is; GeoGebra substitutes the P that follows + for the value of the point (in this case itscoordinates). Then, in the Properties window click on T1 (in the list of objects), tab, and

select bold ( ).

Figure 34 Define a text


In the tab choose a green shade and, finally, in the tab deactivatethe Absolute position on screen cell.



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Figure 36 The Properties window. The Position tab

E5. Coordinates of a point

In the last figure it can be seen, in the Starting point field, a chunk of the originalexpression:

(x(P)) + 0.25, y(P) + 0.25)x(P) is the value of the x-coordinate of the P point and y(P) corresponds to the value

of the y-coordinate. These are dynamic values, that is to say, the values change regardingthe position of the P point.

Save the file as activ02_01.ggb..

E6. Segments

The only thing left is to lay out the segments corresponding to the coordinates of the Ppoint and put the appropriate labels.

We choose the Segment between two points tool (third button to the left in the tool bar).We draw the segment between P and the X axis, corresponding to the ordinate. In order toget this segment to always show the value of the ordinate, we have to modify its properties

(right click on the segment and choose ) and assign the point (x(P), 0) as itsend. Next, we can delete the point A drawn by the program to lay the segment out.



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And the same way to draw the segment corresponding to the y-cooordinate. In thiscase the ends will be P and (0, y(P)).

In a similar way to the insertion of texts in the case of the point P, we can insert thelabels of the segments (to show xP you have to write x_P).

E7. Text with parameters

With the cursor surrounding the segment a, we choose the Insert text tool

(penultimate button in the tool bar) and we writey_P = + y(P)

Remember that what it is under quotation will appear in the same way (but for thesubscript) and the expression under after + is replaced by the value of the object y(P), that isthe y-coordinate of the point.


And the same holds good for labelling the segment corresponding the x-coordinate.



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Remember that in order to drag the text with the point you have to deactivate the

Absolute position on screen cell and write in the Starting point field the correspondingcoordinates (that must make reference, in any way, to the coordinates of the point).

Save the file as activ02_02.ggb.

Use the command Export (see Export) with the option Dynamic Worksheet asWebpage (html) and save the generated file as activ02_03.html. Open it with your browserand check how it works.

Reproduce drill 2 at the beginning of this section. Save the file as activ02_04.ggb.

Use the command Export (see Export) with the option Dynamic Worksheet asWebpage (html) and save the generated file as activ02_05.html. Open it with your browserand check how it works.

F. Algebra window

In the previous exercises we disregarded the algebra window. Lets see now theinformation and the edition possibilities this window provides.

To open the Algebra window, click on the command named the same in the Viewmenu.

The attached image corresponds to the exercise described in Represent and dragobjects.

As you can see, theres a free object, the point P, and two others dependent, thesegments aand b, built to represent the x-coordinate and the y-coordinate of the point P.

Figure 39 Example

The same considerations already mentioned dealing with the context menu in theDrawing Padhold good for the algebra window. For instance, if we place the cursor on the



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point P, in the Algebra window, and right click on it, we reach the context menu

corresponding that point.

Figure 40 - The context menu of the P point

Any change made on any of the two windows immediately occurs in the other. So, forinstance, the dragging of the point P to a new position causes the corresponding change in

the algebra window regarding the definition of P and the measure of the segments. Andreciprocally, all changes made in the algebra window are translated into changes in theobjects represented in the drawing pad.

Figure 41 Example



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F1. Construction Protocol

By selecting the command Construction Protocol from the View menu, theConstruction Protocol window is displayed where all the existing objects appear.

In the View menu of this window we can choose the fields listed there.

Figure 42 The Construction Protocol

Figure 43 The Construction Protocol

And with the arrows in the lower part of the window, we can regenerate the building

from the very beginning, step by step.The command Export as Webpage (html) in the File menu in this window allows us toexport this protocol and decide whether to include the drawing of the construction or not.

This command is the same we had previously seen, command Export in the Filemenu.



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Figure 44 The Export as Webpage (html) window

The next figure shows how this export browser looks like.

Figure 45 - The export browser



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F2. Export

Before, see Export, we mentioned that the use of this command to obtain a Web pagegenerates several files. For instance, if the html file is named test.html, this one is alsocreated unless the test.ggb file already exists.

Well, to modify the test.html file regarding the geometric construction, you only haveto modify and save with the same name and in the same folder the test.ggb file. You can getthis file from the browser if the option Double click opens application window is selected bydefault; otherwise you can open it from GeoGebra provided it is accessible in your computer.

Another possibility is to re-export the construction again, once all the proper changeshave been made. But if we have made unrelated modifications with the construction in thehtml file (add and/or modify texts and/or graphics), those will be lost with the new export.

Due to this considerations, when we have to open an html file that contains a

GeoGebra construction, and we want to save it with a different name, we will have to followthe next steps:

1. Open the html file (lets suppose that it is named test.html).2. Open the GeoGebra construction by double clicking on the corresponding window.3. Export the file with a different name (for instance, testNuevo.html).4. Make the proper changes in the GeoGebra file and:

Save it as testNuevo.ggb in the same folder that testNuevo.html,or,

Export the new construction with the same name testNuevo.html. In this case, the filetestNuevo.ggb is automatically created.

G. Some proposed exercises

Next, some exercises which can be useful for working with the students, simply asthey are, or after the modifications suggested. You must bear in mind that the usual way ofworking with didactic applications consists in re-using the existing material made by oneselfor other partners.

Open the activ03.html file. You can get the GeoGebra file by double clicking on theDrawing Pad.

Export the construction to the activ03_1.html file. (Remember to activate the cellDouble click opens the application window in the Drawing Pad, so that GeoGebracan beopened from your browser). Close GeoGebra and open now the exported file. You canaccess the application by double clicking on the window.



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Figure 46 Exercise

The coordinates of the points only accept integer numbers. Remove the restrictionswhich make the coordinates to take only integer numbers (use intervals of 0.25 units).

Save the modified file with the same name activ03_1.ggb.Check the changes made by opening again the activ03_1.html file.

Key: The command Point capturing in the Options menu is placed in on (Grid). To get

displacements of 0.25 units in both axes, activate the Distance cell in the tab in theDrawing Pad and place the 0.25 value in both axes. The activ03_1Sol.html file contains the

solution.

Modify the activ03_1.html file to get an exercise like the one in the figure. The points A,B, C and D have two coordinates in the interval [-5, 5] and they can only accept integernumbers.



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Figure 47 Exercise

Save the modified file with the name activ03_2.html.

Key: Undo the changes made in the previous exercise by placing again the distancein the grid in 1. To limit the movement of the points to the interval [-5, 5] you only have toadjust the size of the GeoGebra window to these dimensions and check that he commandPoint capturing in the Options menu is placed in on (Grid). The activ03_2Sol.html filecontains the solution.

The goal is to design an exercise like the one in the figure with the following features:The coordinates of the point P can only take integer numbers in the interval [-5, 5].



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Figure 48 Exercise

Note: To draw the circumference, activate the input line (command Input field in the Viewmenu) and write the circumference equation in the way x^2+y^2=x(P)^2+y(P)^2, where x(P)and y(P) are the coordinates of the point P. Another possibility is to draw the centrecircumference the origin of the coordinates through P:

Circle with centre through point

Export the file with the name activ03_3.html.

Key: To limit the movement of the point P to the interval [-5, 5] it is made like in theprevious exercise. The activ03_3Sol.html file contains a solution.

Obtain an exercise like the one in the figure. The points A and C move on the x-axis,(to place these points on the x-axis, open the algebra window and move the pointer aroundthe x-axis until the Line xAxis) label appears, while B and D can only be moved on the y-axis.The label this time must indicate Line yAxis). The 4 points have their movement restrained tothe interval [-5, 5] and the coordinates vary every 0.1 units.



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Figure 49 Exercise

We can use the activ03_2.html (or activ03_2Sol.html in folder solutions) file as basis. In

order to limit the movement of the points to the axes, the tab in the Propertieswindow must show, in the Definition field Point[xAxis] for the points A and C; and Point[yAxis]in the case of B and D.

Save the modified file with the name activ03_4.html.

Key: To limit the movement of the points to the interval [-5, 5], make it like in theprevious exercises. The activ03_4Sol.html file contains a solution.



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H. Functions I

H1. Functions. Tables

In these activities tables are related to associated graphics and formulae. They areaimed to students in the first years of secondary school, so we will just see degree onepolynomial functions.

H2. Graphics and tables

Lets see how to represent functions and to modify the relation and aspect of the axesand the grid. We will also explain how to use the sliders to drag objects.Graphic-Table (1). The goal is to create a GeoGebra document which fits the problem

shown in the following figure:

Figure 50 Exercise

We start by defining the function y = 2 x using the input field. For that we only have towrite f(x)=2 * x (or simply f(x) = 2x) and click on enter:

GeoGebra automatically represents the function with the assigned name and its

definition appears in the algebra window, within the Free objectscategory.



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Figure 51 Exercise

In the Properties window of the function f, tab, we have increased the Line

thickness to improve visualization, and in the tab we have chosen red.



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In the Drawing Pad window, tab, subtab, we have activated the

Numbers cell and inserted 3 in the Distance field. In the subtab, the same operationshave been made but for inserting 4 in the Distance field instead. In both cases the texts forthe axes have been placed in the Label field.


Finally, in the tab, we have activated the Distance cell with the values x = 1, y =1.



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Figure 54 Set de distance

We add a slider so that the position of the point P is still to be determined by the valueof this parameter.

To define a Slider, place the cursor on the drawing pad and click on the correspondingbutton (button 6 in the tool bar, option 6). We change the name to photocopies (you can alsouse the command Rename for this purpose). From the Propertieswindow of the definedslider we modify some of its features by selecting the corresponding tab.

Figure 55 Slider features

Intervalmin: minimum value of the associated number.max: maximum value.Increment: number to be added to the present value of the slider when it moves.

Sliderfixed: it avoids the control movement through the window.



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vertical/horizontal: control orientation.

Width: control width.


The properties related to colour and style are modified by activating the correspondingtabs in the same way we have already explained for other objects.

In order to move the point P according to the value of the slider and on the line, wedefine it in the input line:

P=(photocopies,f(photocopies))

That is to say, the x-coordinate is provided by the value placed in the slider and the y-coordinate is calculated by the defined function.

A P point can also be defined in any position, and change its definition later.If instead of dragging the point with the slider we want to move it freely on the line,

you just have to choose the tool New Point and move the cursor around the graphic until thelabel Function f appears. If we check the definition now we will see P = Point [f].Thus, thepoint is linked to the line and it is not possible to drag it outside it.

To limit the definition command of the function, like in the one described, we have touse the command Function(from the input line)

Function[ f, a, b]

where a and b are the limits of the definition interval of the function. Next, we are going tohide the original function f ( ).

GeoGebra defines a new function g equal to f in the definition interval. The propertiesof this new function can be modified as usual.

To finish, we can see that in the GeoGebra file the x-coordinate and the y-cooordinatevalues on the corresponding axes appear in red. To achieve this effect you just have todefine two pointsA(x(P), 0) y B(0, f(x(P))and change their properties (colour and style).

The labels attached to these two points show the abscise and ordinate values of Pand they are created with the command Insert text defining its position according to thecoordinates of the point P.

Points Textofthelabel Positionofthelabel



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A(x(P), 0) +x(P) (x(P) - 0.25, -1)

B(0, f(x(P)) +y(P)

(-1, y(P) - 0.25)

Key: The activ04_1.html file contains the previous exercise completely made (theassociated GeoGebra file is named, as usual, activ04_1.ggb).

Graphic-Table (2). Check the GeoGebra file in the previous activity (or the fileactiv04_1Sol.ggb in the folder solutions) and modify it so that it fits the situation in thefollowing figure.

Figure 57 Exercise

Export the modified file with the name activ04_2.html.

Key: The activ04_2Sol.html file contains the key (the associated Geogebra filed isnamed activ04_2Sol.ggb).Graphic-Table (3)

Open the activ04_3.html file and check the associated GeoGebra file.Our aim is to illustrate with this example the way to define a point which is moving on

a line when the ordinate varies. In this case, we have used a non-visible slider (values arenot shown, the increments or decreases are made by dragging the point) which we havenamed PriceWithTaxes.

The activity prioritizes the ordinates, calculating the corresponding abscises, insteadof proceeding as usual. In order not to repeat the calculations we have defined the function

f(x) = x / 1.16. The point P will be determined by (f(x), x), being x the associated number tothe slider, that is PriceWithTaxes.



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Graphic-Table (4)

Modifies the GeoGebra file making that every increment of the slider corresponds to a5 unit movement in the abscise axis (see figure). Change the name of the slider forPriceWithoutTaxes and add the value of the slider in the label.

Remember that the function to be used in this case is the one called g (g(x) = 1.16 x).

Save the modified file with a different name and export it with the name activ04_4.html.

Figure 58 Exercise

Key : The activ04_4Sol.html file contains a key. The associated Geogebra file is namedactiv04_4Sol.ggb.

H3. Tables and graphics

Open the activ04_5.html file. The thing is to identify the function corresponding to thestatement.



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Figure 59 Exercise

The way to limit the command of the functions, the representation of the points on theaxes and the insertion of texts have already been dealt, so we will not add anything else.

Among the possible options for the axes scales, we have decided to fix a 1 to 2.54relation (see fields xAxis: yAxis= in the following image), coincident with the function to berepresented -we will explain how to make similar representations keeping the 1:1 relation inanother exercise. Besides, the numbers in the axes have been removed (although they couldhave been kept instead of adding the texts).

As regards the grid, the following values have been established for the field Distancex: 2.5, y: 7.62, which seem to be the most appropriate for the problem to represent.



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We have defined the functions f, g, h (and the corresponding ones which limit thecommand f1, g1, h1, the ones shown in the graphic)

f(x) = 2.54 x, g(x) = f(x) + f(14), h(x) = f(x) f(14)

As you can see in the algebra window, these new functions belong to the Dependent

objects category since another object is involved in its definition.Points on the axes

X_1 = (14,0), X_2 = (20,0) X_5=(35,0)

Y_1 = (0, f(x(X_1))), , Y_4 = (0, f(x(X_4)))

Remember that X_1 turns into X1in the GeoGebra window. On the other hand, x(X_1)represents the abscise of the point X1, so f(x(X_1)) will be the y-coordinate in the function f.

For the texts, proceed as in the previous exercises: once the Insert text tool isselected, place the cursor around the point and write the text. For instance, for the label35.56 in the yAxis we should type +y(Y_1), while for the label 14 in the xAxis, we shouldtype +x(X_1). Both the defined points and the labels have turned into fixed objects to avoid

movement.We must remark that the functionality of the exercise is kept without the need of

adding the points and texts already mentioned. We could activate the field Numbers in theDrawing Padwindow and use the appearing values.



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Figure 61 Exercise

H4. Tables and identification of graphics

In this exercise a slider is used to show a set of graphics so that the student will beable to identify the one corresponding to a given situation.

Open the activ04_6.html file. The slider allows to draw 4 lines, the ones corresponding

to the family f(x) = 4 - sx, where s can take the values 1, 2, 3, 4.

Figure 62 Exercise

For this activity an a slider has been defined (hidden label, green colour, style 5, min:1, max: 4, Increment: 1) and a function f(x) = 4 ax. The labels of the axes have beendefined by means of the Insert text command fitting the values to the conditions of theproblem.

Design an activity to identify the graphic of the family y = 2 + bx, where b takes valuesamong -2 and 2 (increment 0.5). Use a slider to start changing the graphic.



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Save the file naming it as activ04_7.ggb. Use the command export to generate an htmlfile named as activ04_7.html.

Key : It is enough to modify the definitions of the slider and the one of the function toadapt them to the new situation. The activ04_7Sol.html file contains a key (the associatedGeoGebra file is named activ04_7Sol.ggb).

H5. Some exercises about lines

The files activ04_8.html, activ04_9.html and activ04_A.html set out exercises wheretables, formulae, graphics and verbal description of functions are used. They can be easily

solved with what has been previously stated.The activ04_8.html file presents a modification of the function to be represented, very

usual when using the blackboard or the notebook to represent the functions. The function tobe represented is f(x) = 0.6 x, but the one that is really represented is f(x) = 1.2 x. Thissituation holds good as an example to insist on the graphical representation of functionswhen the divisions of the axes are not adapted to the visualization necessities.

In general, computer applications need adjustments so that the representation fits thedata visualized. If we had used the original function it would have been impossible to matchthe used divisions in the axes and the graphic of the function.

The following image shows the point P under the restriction of being dragged on theline.

f(x) = 0.6 x,

The segments corresponding to the coordinates have been drawn to ease thevisualization of the difference among the reality of the function and what we haverepresented.

As expected, GeoGebra represents the point P(3,1.8) corresponding to f(x) = 0.6 x.

Figure 63 Point P under restriction

http://../html/Actividad08_2.htmlhttp://../html/Actividad08_3.htmlhttp://../html/Actividad08_3.htmlhttp://../html/Actividad08_2.html


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Figure 64 - Exercise

But what we really want to represent is the point P(3,3.6) belonging to the graphicalfunction

g(x) = 1.2 x

You can always define the original function and use it for the function which fits the

measures we want to represent on the axes. In the example,

g(x) = 2 * f(x)

I. Functions II

I1. Polynomial functions and their graphics

In these activities we will analyze the polynomial functions and the shapes of theirgraphics. The GeoGebra functionalities to carry out these exercises have already beenexplained. Remember that the usual way to introduce the functions is by means of the editline or input field.

Open the activ05_1.html file. The thing is to compare the graphics of the functions

belonging to the type f(x) = xn, depending on whether nis evenor odd.

Compare the graphics of the functions y = x2, y = x4and y = x6with the graphics ofy = x 3and y = x 5and write the similarities and/or differences between the ones in thefirst group and the ones in the second.

Note: The slider nallows you to visualize the graphics of the functions above.



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unity Collaborating Space for Science Education


As it has been shown in the graphics above,

if n is any positive integer, describe as detailedas possible the aspect of the graphic of thefunctions of the type

f(x) = xn,according to whether nis evenor odd.

Which of the following graphics correspond to the functions of the type f(x) = xnwith neven? And with nodd?

We have defined an n slider (for integer values between 2 and 6) and the functionf(x) =xnis represented with the corresponding text.



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Openthe activ05_2.htmlfile. The thing is to draw conclusions on the influence of the

sign kin the graphics of the function belonging to the type f(x) = (x k)2.

Describe in detail the characteristics of the graphics of the functions y = (x - k)2, being kanynonzero real number.

Note: The slider kallows you to visualize the graphics of the functions above for the valuesof kamong -4and 4.

Describe in detail the influence which the signk has in the graphic of the functions f(x) = (x - k)2Identify which of the following graphics correspond to the functions of the type f(x) = (x - k)2withkpositiveand with k negative.

We have defined a kslider (for integer values among -4and 4), the function f(x) = (x-k)2and the corresponding text.

To avoid the visualization of f(x) = (x-0)2 we have included two texts. The first,adequate for all values of k but for 0 - "f(x) = " + f and the second which can only be

visualized when k = 0- f(x) = x^2. To get this effect you just have to impose the conditionsshown in the tab in the Properties window of the corresponding object.

Figure 65 The Advanced tab in the Properties window



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sign kin the graphics of the functions belonging to the type f(x) = kx2.

Describe in detail the characteristics of the graphics of the functions y = kx2, beingkany nonzero real number.

Note: The slider kallows you to visualize the graphics of the functions above forthe values of kbetween -3and 3.

Describe in detail the influence which the signkhas in the graphic of the functions f(x)= kx 2.

Identify which of the following graphics correspond to the functions of the type f(x) =kx 2with kpositiveand with k negative.

We have defined a kslider (for integer values among -3and 3 with0.5 increments),the function f(x) = kx2and the corresponding text. In this case the exception is applicableboth to the text and the functionf.



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sign kin the graphics of the functions belonging to the type f(x) = x2+ k.We have defined a kslider (for values betweens -4and 4), the function f(x) = x2+ k

and the corresponding text with similar restrictions to the previous exercise (see image).

Describe in detail the characteristics of the graphics of the functions y = x2+ k , being kany nonzero real number.

Note: The slider k allows you to visualize the graphics of the functions above for thevalues of kbetween -4and 4.

Describe in detail the influence which the signkhas in the graphic of the functions f(x) = x2+kIdentify which of the following graphics correspond to the functions of the type f(x) = x2+ kwith kpositiveand with k negative.



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Openthe activ05_5.htmlfile. The thing is, according to the previous exercises, to draw

conclusions on the influence of the signs a, band c in the graphics of the functions of thetype f(x) = a (x b)2+ cWe have defined two sliders bandc(for values between -4and 4), the function

f(x) = a (x b)2+ c

and the corresponding texts similar restrictions to the previous exercises.

Describe in detail the characteristics which the graphics of the functionsy = a(x - b)2+ c,

should have being a, b and c real numbers witha nonzero.Note: The sliders band callows you to visualize the graphics of the functions above for

the values band cbetween -4and 4.

Describe in detail the influence the parameters a, b and c have on the graphic of the

functions f(x) = a(x - b)2

+ creferring to the sign.Identify which of the following graphics corresponds to the function f(x) = (x -2)2 + 2 andwhich corresponds to f(x) = -(x +2)2- 2.



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We have defined a number nusing the GeoGebra function randomwhich generates

a random number between 0and 1: n = random()The coefficient acan only take the values 1 and -1. To define it we have used the

auxiliary number npreviously generated jointly with the conditional:

If [n < 0.5, 1, -1]

The syntax of the conditional is the following: If [condition, action_1, action_2],where action_1is performed when conditionis trueand action_2when it is false.

The symbol reset -mark the cell Show icon to reset construction, in the tab in

the Export: Dynamic Worksheet (html) window- generates again the number nand, thus, thevalue of a.


A different way to make this exercise: Define band cas random integers with valuesbetween -4and 4and write their values. To change the values of the coefficients we mustuse then the symbol reset.

The definitions of band c, would be:b = ceil (8 random() - 4)c = ceil (8 random() - 4)

The same procedure could have been used in the previous exercises.

Figure 67 Exercise



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J. How to insert the random function in a GeoGebra applet

Although you can use the randomfunction, GeoGebra turns the objects in which thisfunction is involved into dependent objects so that it is not possible for them to be draggedwith the mouse.

As an alternative, lets see how to insert random objects by modifying the code in anhtml page containing a GeoGebraapplet.

We will start by using the activ03.htmlas basis.

You can use any text editor, but we will use FrontPage 2003. In the Filemenu, select

the commandOpenor just click on the Openicon.The Open File window is displayed, search the folder which contains the file and

select it.

Figure 68 The Open window

Click on the Codetab at the bottom of the window and we will see how the web pagecode is.



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Figure 69 The web page code

The part of the code corresponding the GeoGebraappletis:

Figure 70 The GeoGebra applet

This method consists on 3 steps:Step 1:Name the appletto be easily found.

Lets place the cursor of the mouse between the words appletand code. Then write:

name= applet_ticmates

as it can be seen in the following figure.



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Figure 71 Name the applet

Step 2:Just under the applet code we will insert the JavaScriptcode which will enable us tocalculate a integer number by random and pass it to the applet to place the points A, B, Cand Dregarding the random coordinates, generated every time you load or refresh the webpage.

Copy the following code:

//function to obtain an integer among -6 and 6

function entero_azar(){

var el_entero = Math.floor(Math.random()*(13)-6);

return el_entero;

}

//Function to assign ramdom values to points A, B, C & D

//it will be called from the "body" tag after loading the page

function coordenadas_puntos_azar(){

var applet=document.ticmates;

applet.evalCommand("A = (" + entero_azar() + ", " + entero_azar() + ")");

applet.evalCommand ("B = (" + -entero_azar() + ", " + entero_azar() + ")");

applet.evalCommand ("C = (" + -entero_azar() + ", " + -entero_azar() + ")");

applet.evalCommand ("D = (" + entero_azar() + ", " + -entero_azar() + ")");

}

And we paste it under the applet code as shown in the following figure:



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Figure 72 Insert the JavaScript code

Step 3: In order that the web page can execute the function coordenadas_puntos_azar() from the added code, we must modify the content of the tag body

Write (in between the quotation marks) onload="coordenadas_puntos_azar()" as

shown in the following figure:

Once these three steps have been accomplished we can save the file,

activ03Random.html.

Key: The activ03RandomSol.html file contains a key.

To check the result we will open the folder with the browser to ensure that the points areplaced by random every time we open or refresh the web page. In the way we have definedthe function coordenadas_puntos_azar, the coordinates of the points A, B, C and D takevalues in the intervals that are shown in the following table:

A B C D

x [1, 5] [-1, -5] [-1, -5] [1, 5]

y [1, 5] [1, 5] [-1, -5] [-1, -5]

You can check some more examples about the use of JavaScript in this url:http: //www.geogebra.org/source/program/applet/geogebra_applet_javascript.html

http://www.geogebra.org/source/program/applet/geogebra_applet_javascript.htmlhttp://www.geogebra.org/source/program/applet/geogebra_applet_javascript.html


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K. References

[1] The Geogebra web site: http://www.geogebra.org/[2] Hohenwarter Markus, Hohenwarter Judith, Introduction to Geogebra, second

edition, Unites States, 2008;[3] Geogebra,http://notropis.net/geogebra/
http://www.geogebra.org/http://notropis.net/geogebra/http://notropis.net/geogebra/http://notropis.net/geogebra/http://www.geogebra.org/

Documents

Training Material_Introduction to GeoGebra