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GeoGebra Workshop

(Short Version)

Contents Introduction ..................................................................................................................................... 2

What is GeoGebra? ..................................................................................................................... 2

Get GeoGebra ............................................................................................................................. 2

Workshop Format ....................................................................................................................... 2

GeoGebra Layout ........................................................................................................................ 3

Examples ......................................................................................................................................... 5

The Incenter and Incircle ............................................................................................................ 5

The Sine Function Family ........................................................................................................... 7

Graphing the Derivative Using the Slope: Example 1 ................................................................ 9

Net Area Exploration ................................................................................................................ 11

Generalized Parabolas ............................................................................................................... 13

User Defined Tools and Saving Sketches as Applets ................................................................... 15

User Defined Tools ................................................................................................................... 15

Making Applets from your Sketches ........................................................................................ 18

Don Spickler Department of Mathematics and Computer Science

Salisbury University

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Introduction

What is GeoGebra?

GeoGebra is dynamic mathematics software that joins geometry, algebra and calculus. It is developed for learning and teaching mathematics in schools by Markus Hohenwarter and an international team of programmers. As one reviewer put it, "Think of it as a free Geometer's Sketchpad, just twin-turbocharged and on steroids..."

GeoGebra is much like Geometer's Sketchpad (GSP) but as with any two software packages that do similar things there are going to be some things that one does better than the other and vice-versa. In my opinion geometry and geometric measurements is much easier in GSP but if you want to move to other classes, like Calculus, GeoGebra is far better.

One of the biggest differences between GeoGebra and GSP is the base philosophy of their use. GSP was designed to emulate straightedge and compass constructions/ Now GSP does not restrict itself to purely straightedge and compass constructions but it is very close. This is the primary reason that GSP has so few tools in comparison to GeoGebra. GeoGebra does not restrict itself to straightedge and compass constructions in the least. In fact, GeoGebra has many features that are not constructions. The base philosophy with GeoGebra is to present the user with a large set of visualization features.

Get GeoGebra

GeoGebra is a free multiplatform Java application; moreover, there are several ways that it can be installed. You can install it like any other program, run it through either Java WebStart or as an applet or in a portable format that can be run directly off of a network drive or thumb drive. The software can be downloaded from the GeoGebra web site (www.geogebra.org). The GeoGebra web site also contains documentation, workshop materials, and user forum. http://www.geogebra.org/cms/

Workshop Format

This document was constructed from the online GeoGebra workshop that is on my web site. Instead of going through all of the features of the GeoGebra program I decided to make this workshop example oriented. The online version has applets for all example constructions so that you can change the sketch dynamically to see how it works, with this printed version all we can do is include graphical images of the applets. The online version also has a resources page of links and documentation on the GeoGebra tools and menu options as well as GeoGebra files for each of the constructions in the workshop.

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GeoGebra Layout

When you start up GeoGebra you should see something like the image below.

There are four main sections to the GeoGebra window.

1. The Menu and Toolbar - These are at the top, as usual. The toolbar is customizable by selecting Tools > Customize Toolbar... from the main menu. This allows you to restrict the student to use only the tools you want them to. For example, you could remove all tools except for the straightedge and compass tools if you are beginning a study of Euclidean Geometry.

2. The Algebra View - This window on the left. The Algebra View is a tree list of all objects in the current construction. You can alter the properties of any object in the construction using this list.

3. The Graphics View - This is the main window on the right. This where the construction can be viewed and manipulated.

4. The Input Bar - The input bar is at the bottom of the GeoGebra window. The Input Bar is where the user can add objects and calculations that are beyond the capabilities of the tools.

There is also a spreadsheet view in the GeoGebra program. We will look at this later in the workshop. When viewed it appears to the right of the Graphics View.

All views, except for the graphics view can be turned off or on using the View option in the main menu. You will also note that the tools in the toolbar have small triangles pointing down in the lower right of the icon. If you click on this triangle a drop-down menu will appear with other tools that are related to the one you clicked. The toolbar also contain two icons to the right for undo and redo. Furthermore, there are tool hints between the tools and the undo/redo icons. These hints give you quick instructions on how to use the tool that is currently selected.

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The View option in the main menu also allows you turn the axes and grid off and on. GeoGebra has pop-up menus for all of its objects. If you want to change any of the properties of an object simple right-click on that object either in the Algebra View or the Geometry View and a pop-up menu will appear for that object.

If you are familiar with GSP then the use of the GeoGebra program should be fairly natural to you. The big difference is that the general process is reversed. In GSP you usually select one or more objects and then select what you want to do with them, in GeoGebra this is reversed and you select what you want to do and then select the objects. For example, if you want to create a line that goes through a point and perpendicular to a given line in GSP you would select the point and line first and then tell the program to construct a perpendicular. In GeoGebra you would select the perpendicular tool and then select the line and point.

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Examples

The Incenter and Incircle

Angle Bisector Concurrence Theorem: If ABC is any triangle, the three bisectors of the interior angles of ABC are concurrent. The point of concurrency is equidistant from the sides of the triangle.

Definition: The point of concurrency of the bisectors is called the incenter of the triangle. The distance from the incenter to sides of the triangle is the inradius. The circle that has its center at the incenter and is tangent to each of the sides of the triangle is called the inscribed circle, or simply the incircle of the triangle.

The construction of the incenter and incircle is fairly simple.

1. Construct a triangle. 2. Construct the angle bisectors of each of the interior angles. These bisectors will intersect

in a single point, the incenter. 3. Construct the intersection point of these lines. 4. Drop a perpendicular from the incenter to any side. This gives a radius. 5. Construct the point of intersection between the side and perpendicular. 6. Construct a circle that has center of the incenter and the intersection point on the circle

itself. This is the incircle.

The GeoGebra construction follows the same pattern.

1. Start a new sketch and hide the axes. 2. Select the line segment tool and create a triangle. 3. Do a drag-test. 4. Select the Angle Bisector tool. This is one of the options under the Perpendicular Line

tool. One big difference between GSP and GeoGebra is that when you construct something in GSP you select the objects first and then go to the menu to do the construction but in GeoGebra you select the type of construction first and then select the objects.

5. Click on all three vertices of the triangle, one at a time. You will see that an angle bisector is created at the vertex of the second point clicked.

6. Repeat for the other two angles. 7. Select the Perpendicular Line tool and construct a perpendicular to one side passing

through the incenter. 8. Construct the point of intersection of the perpendicular and the side.

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9. Select the circle tool and construct a circle that has center of the incenter and the intersection point on the circle itself.

10. Do a drag-test. 11. Clean up the sketch.

Another neat feature of GeoGebra is the construction step interface. There are two ways to view how the constructions were done. First you can select View > Construction Protocol... from the main menu. When you do you will get the following dialog box displaying each step in the construction. If you click on the controls at the bottom the graphics view will display the construction from the first step to the position of the highlighted line.

The second way is to select View > Navigation Bar for Construction Steps from the main menu. This will place the step selector at the bottom of the graphics view window. Again clicking on the controls moves you through the construction steps. GeoGebra also has an automatic construction animation feature.

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The Sine Function Family

This construction is geared to the exploration of the family of curves f(x) = a sin(bx+c)+d. We will implement the constants a, b, c and d as sliders.

1. Start a new sketch. 2. Select the slider tool in the right side of the toolbar. 3. Click on the graphics view. When you do the slider dialog will appear. This dialog allows

you to set the minimum and maximum values for the slider as well as the increment for the slider.

If you click on the Slider tab you will see a few other options. This tab allows you to select the orientation of the slider as well as the length of the slider in pixels. The fixed option will lock its position in the graphics view, if this is not selected the Move tool can move it freely around the window.

4. Select Apply and you should have a slider on the screen with the label a. 5. Do this three more times to get sliders b, c, and d. 6. Move the sliders to a convenient position on the screen. 7. To input a function in GeoGebra we use the Input Bar at the bottom. The input bar, in

general, is used for inputting commands that are more difficult than simple constructions. Although this requires the user to learn some syntax the functionality trade-off is worth it. Type the following into the input box and hit enter.

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f(x) = a sin(b x + c) + d

Note that there must be a space between the a and sin as well as the b and x. At this point you should see the graph of the sine function and when you move the sliders you should see the graph change dynamically.

8. We will add one more thing to our sketch, a function label. This will require the Insert Text tool which is under the slider tool. Select this tool and click on the graphics window. This will bring up the text tool dialog.

Input

"f(x) = " + f

and select OK. The text in "" will be displayed verbatim, the + is a string concatenation and the last f will be substituted with the current value of the function, it will be dependent on a, b, c, and d. If you move the sliders at this point not only will the graph change but so will the displayed function label.

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Graphing the Derivative Using the Slope: Example 1

This construction is a simple setup for your students to see the graphical relationship between a function and its derivative.

1. Start a new sketch. 2. In the Input Bar at the bottom type in

f(x) = x^3 - 5 x^2

at this point you should see the graph of the cubic.

3. If you look at the applets below you will see that the scale on the x and y axes are different. The default for GeoGebra is to keep them the same since this is what you want when doing geometry. There are several ways to change the axes. One way is to right-click on the graphics view and select the x-axis : y-axis option from the pop-up menu and then select the ratio you want. Another way is to select the Move tool, which allows you grab the entire image and move it statically, put your cursor over one of the axis markings and then click and drag. Use either method to bring both relative extrema into view.

4. Select the point tool and put a point on the function. This should be A but if it is not on your sketch go into the properties and change it to A. Do a drag-test to make sure the point is attached to the function.

5. Now we will create the tangent line to the cubic that passes through the point you just put on the line. Select the Tangents tool, this is under the perpendicular tool icon. Click on the point and then click on the cubic, or vice-versa. At this point you should see the tangent line to the cubic passing through the point.

6. Now we will get the slope of the line. Select the Slope tool, it is under the measurements icon that probably has the picture of an angle in it. Now just click on the tangent line. This will produce the standard rise over run triangle and a slope measurement assigned to the variable m.

7. At this point we want a point that has the same x value as A and has a y value that is equal to m. To do this simply type the following into the Input Bar and hit enter.

(x(A), m)

The x(A) extracts the x value of A and of course m is m. This should produce the point B on the sketch that is on the graph of the derivative.

8. At this point we could simply use the locus or trace on B and get the graph of the derivative but we will add in a little more to help the user see how B was constructed. Construct a perpendicular to the x-axis through B.

9. Plot the point of intersection of the perpendicular and the x-axis. 10. Hide the perpendicular and construct a line segment from B to this point of intersection.

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11. Since the length of this line is m we would like it to be labeled as m. Select the properties for the segment and in the drop-down box for the label select caption. In the caption box above this type in m. Finally, close the properties box.

12. The last step is to do the locus. Select the locus tool, click on A and then click ob B. At this point you should see the derivative function. Do a drag-test.

Notes:

1. GeoGebra has a built-in computer algebra system so it is capable of calculating a symbolic derivative. So we could have simply typed

f'(x)

into the Input Bar to get the graph of the derivative instead of using the locus.

2. In GeoGebra you can move functions around with the mouse. So if you click and drag the cubic function it will translate, the nifty part is that so will the rest of the sketch. If you want to lock the function, right-click on the function, select the properties option and then under the Basic tab click the Fix Object check box.

If you would rather use point tracing in place of the locus just make the following changes,

1. Right click on the locus and select the delete option. 2. Right click on the tracing point and select Trace On. 3. Do a drag-test.

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Net Area Exploration

This exploration is the counterpart to the ones from the differential section which used the slope of the tangent lines to graph the derivative. This one uses net area to graph the integral of the function. Another new thing in this sketch is to put LaTeX code with dynamic values into a text box.

1. Start a new sketch. 2. Plot the function, f(x) = x^2 - 3 x 3. Plot points A and B on the x-axis. 4. Construct perpendicular lines to the x-axis through A and B. 5. Construct the points of intersection of these perpendiculars to the function. 6. Hide the perpendiculars, construct line segments in their place and then hide the

intersection points. 7. Input the following into the Input Bar,

NetArea = Integral[f(x), x(A), x(B)]

This is obviously the definite integral of f(x) from A to B. Just as a side note GeoGebra can take the indefinite integral of a function as well, the syntax here is simply Integral[f(x)].

8. Now we will plot a point on the graph of the integral. Input the following into the Input Bar,

(x(B), NetArea)

This will produce a new point on the sketch, we will call it D.

9. Locus time, create a locus with B and D in that order. 10. Just for pedagogical reasons put in a segment from B to D, hide the label of D, change the

label for the integral to Name and Value, make the caption for the segment BD say Net Area, and set the label of the function to Name and Value as well.

11. Last thing to do is put that nifty label on the point D. Create a text box and put the following in for the text.

"Net \; Area \; = \int_A^B f(x) \; dx \; = \; " + NetArea

The stuff inside the "" is LaTeX (the \; just puts in a little more space) the + is string concatenation and NetArea is the value to print out. Do a drag-test. Now select the text box properties, the Position tab and select the starting point as point D.

12. Do some color changes and font size changes.

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Generalized Parabolas

I got this from a talk by Greg Hartman and Dan Joseph of VMI at one of our MAA section meetings. The idea was simple, as we all know the parabola is defined as the locus of points that are equal distant from a fixed straight line and fixed point. What happens if you let the line be a curve instead of a straight line? This series of three applets will examine this question.

First we will construct the parabola so that we have a method for the more general case.

1. Start a new sketch. 2. Plot an infinite line. 3. Plot a point not on the line (C) and plot a point on the line (D). 4. Plot a perpendicular to the line through D. 5. Plot the segment CD. 6. Plot the perpendicular bisector to CD. 7. Plot the point of intersection of the two perpendiculars (E). 8. Create a locus using D as the driver and E as the point on the locus. 9. Do a drag-test and clean things up a bit. This should give you a sketch like the one below.

The authors then did something neat, they replaced the straight line with a circle. They wanted to be able to increase the radius of the circle indefinitely, thus producing a straight line when the circle's radius was infinite. To do this we will use a circle through three points.

1. Start a new sketch. 2. Plot three points. 3. Select the circle through three points tool and then click on the three points you just

created. 4. Plot a point not on the circle (D) and plot a point on the circle (E). 5. Plot a tangent line to the circle through E and plot a perpendicular to that line through E. 6. Plot the segment DE. 7. Plot the perpendicular bisector to DE. 8. Plot the point of intersection of the two perpendiculars (G). 9. Create a locus using E as the driver and G as the point on the locus. 10. Do a drag-test and clean things up a bit. This should give you a sketch like the one below.

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What was so nifty about this was that when D was inside the circle the locus was an ellipse, when the D was outside the circle the locus was a hyperbola and when the radius of the circle was infinite the locus was a parabola. I authors proved this to be the case and showed that the point D was always one of the foci. So the parabola construction gives all three conic sections when you use a circle.

The authors then went on to investigate other curves. We will simply use a parabola here.

1. Start a new sketch. 2. Plot f(x) = 1/10 x^2 3. Plot a point not on the parabola (A) and plot a point on the parabola (B). 4. Plot a tangent line to the parabola through A and plot a perpendicular to that line through

A. 5. Plot the segment AB. 6. Plot the perpendicular bisector to AB. 7. Plot the point of intersection of the two perpendiculars (C). 8. Create a locus using B as the driver and C as the point on the locus. 9. Do a drag-test and clean things up a bit. This should give you a sketch like the one below.

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User Defined Tools and Saving Sketches as Applets

User Defined Tools

In this example we will take a construction, make two user defined tools and save them to a tools file. Download the file CircumcenterTools.ggb to your computer and open it up in GeoGebra. This is simply a construction of the circumcenter and circumcircle of a triangle. The sketch should look like the figure below.

We will create two tools, the first will be the circumcenter. Select Tools > Create New Tool. At this point the following dialog box will appear.

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Using the drop-down box select the object to be produced, in this case the circumcenter point. The point will be listed in the box below when you select it. Note that although we are selecting only one object we can select as many as we would like. Simply select a second, third, ... from the drop-down box. If you select one you do not want you can remove it by highlighting it and clicking the X button.

Click the Next button. At this point you will see the objects that your final object depends on. In this case, it is the three points A, B and C. Click the Next button.

Finally, give the tool a name, command and tool tip. You can also load in an icon for it if you wish. We will call the tool Circumcenter, the command will be the same and the tool tip will be to select three points. Click Finish and you are done.

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Now you have a circumcenter tool that when selected you can click on three points and the circumcenter for that triangle will be produced. Furthermore, you have also created a new command. If you create three new points D, E and F you could create the circumcenter of that triangle using the command Circumcenter[D,E,F] in the input bar.

Now do the same process to create a tool for the Circumcircle. Notice that once you do the user defined tools will have a small triangle in the lower right, indicating that there are several tools in this menu.

Now if we save this file we will also save the new tools we created, but what if we want to use these tools in a new sketch? Well all we need to do is save the tools as a GeoGebra Tools file (*.ggt). To do this select Tools > Manage Tools... from the main menu. When you do, the following dialog box will appear.

To save the tools first select all of the tools in the list, GeoGebra will only save the ones that are highlighted. Then click the Save as... button, give the file a name and click OK. At this point your tools are saved. To load them into another sketch all you need to do is select File > Open... from the main menu and select the file you just saved.

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Making Applets from your Sketches

Making applets from your sketches is easy with GeoGebra. After you create your sketch set up the views the same way you want the applet to appear to the user. That is, if you do not want the applet to display the algebra view, hide the algebra view. Then select File > Export > Dynamic Worksheet as Webpage (html) from the main menu. When you do the following dialog box will appear.

Under the General tab you can set the title, author, date, the text to be placed above the applet and the text to be placed below the applet. Furthermore, you can set the style of how the applet starts. The options here are to embed the applet in the page or to display a button that when pressed opens up an applet window.

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Under the Advanced tab are options for what you want the user to see, the interface functionality and the way that the files will be saved. These options are up to you but for most applications you will want to select the ggb file and jar files at the bottom of the dialog box. What this will do is save the current sketch as a standard GeoGebra file, create an HTML file that loads the applet and copies the files

geogebra.jar geogebra_cas.jar geogebra_export.jar geogebra_gui.jar geogebra_main.jar geogebra_properties.jar to the same folder. Now if you copy all of these files to the same folder on a web server you are done. The html file with the applet is on your web site.

Note that if you are placing several applet pages on your site you only need to copy the jar files one time. Just make sure that all of your html files, GeoGebra files and these jar files are in the same folder. Also note that if you simply export the files without filling out the options you can extract the applet tag and place it into another web page. Just remember to move the GeoGebra file to the same folder as the jar and HTML files.