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1 Quickstart for Desktop Version What is GeoGebra? Dynamic Mathematics Software in one easy-to-use package For learning and teaching at all levels of education Joins interactive 2D and 3D geometry, algebra, tables, graphing, calculus, and statistics Open source software, freely available from Quick Facts GeoGebra facilitates the creation of mathematical constructions and models by students. It allows interactive explorations by dragging objects and changing parameters. GeoGebra is also an authoring tool for teachers to create interactive worksheets. Find free classroom materials and share your own on Getting Started Toolbar Algebra View Graphics View Undo / Redo Perspectives sidebar Input Bar GeoGebra Quickstart, Version

3 Preparations Open a new GeoGebra window. Click on the arrow on the right side of the Graphics View in order to open the Perspectives sidebar and select the Geometry Perspective. Familiarize yourself with the tools provided in the Graphics View Toolbar. Hint: You can open a toolbox by clicking on the small arrow in the lower right corner of the default tools. Construction Steps Hints Choose the tool Polygon from the Toolbar. Now counterclockwisely click three times on the Graphics View to create the vertices A, B, and C. Close the triangle by clicking on point A again. Select the tool Perpendicular Bisector and construct two line bisectors by consecutively clicking on two sides of the triangle. Hint: If a tool is not displayed in the Toolbar by default, you may want to look for it in the toolboxes. Using the tool Intersect you can successively click on the line bisectors to create the center of the triangle s circumcircle. Select the tool Circle with Center through Point. Click first on the center point, then on any vertex of the triangle to create the circumcircle. Using the Move tool you can now use your mouse to drag the triangle vertices - your construction will change dynamically while you move them. Hint: In order to move the whole construction within the Graphics View, select the tool Move Graphics View and simply drag it around. Try the Undo / Redo buttons on the right side of the Toolbar. Use the Show / Hide Object tool from the Graphics View Toolbar in order to hide an object. Using this tool, select all objects you would like to hide. Then, switch to another tool in order to apply your changes. You can change the appearance of objects (e.g. change color or line style, show / hide labels) easily using the Style Bar. Select the object you want to enhance and click on the little arrow in the upper left corner of the Graphics View in order to open the Style Bar. GeoGebra Quickstart, Version

4 You can also show / hide axes and a grid using the Style Bar. Click on the white background of the Graphics View to make sure that no object is selected. Then, click on the little arrow in the upper left corner of the Graphics View in order to open the Style Bar. The Construction Protocol (see View menu) provides a table listing all the construction steps. Using your arrow keys you can go through the construction steps again. Furthermore, you can change the order of construction steps by dragging rows up or down. Example 2: Parameters of a Linear Equation Task: Use sliders to dynamically change the parameters of a linear equation. Preparations Open a new GeoGebra window using the File menu. Click on the arrow on the right side of the Graphics View in order to open the Perspectives sidebar and select the Algebra Perspective. GeoGebra Quickstart, Version

5 Construction Steps 1 Enter y = 0.8 * x into the Input Bar and hit the Enter key. Exploration: Which parameters can be changed when you modify 2 the equation of the line in the Algebra View (use the arrow keys) or the line in the Graphics View (use the mouse)? Hint: Make sure the equation/line is selected before you try to modify it. In the Graphics View, use the Delete tool in order to delete the line. 3 Hint: You need to open a toolbox in order to select this tool. Use the Slider tool in order to create two sliders a and b using the 4 default settings of sliders. Hint: After clicking on the Graphics View, a window appears allowing you to specify the parameters of your slider. Click Apply in order to close the window and create a slider. 5 Enter y = a * x + b in the Input Bar and hit the Enter key. Select tool Intersect from the Graphics View Toolbar in order to 6 create the intersection point A between the line and the y-axis. Use the Point on Object tool in order to create a point B at the 7 origin. Use the Segment tool in order to create a segment between points 8 A and B. 9 Use the Slope tool in order to create a slope (triangle) of the line. Hint You might want to enhance the appearance of your construction using the Style Bar (e.g. increase the line thickness of the segment to make it visible on top of the y-axis, show the slope's value instead of its name). Select the object you want to enhance and then click on the little arrow in the upper left corner of the Graphics View in order to open the Style Bar. Example 3: Intersection of Polynomial Functions Task: Explore how a parabola can be intersected with a linear function by determining the roots of their difference function. Preparations Open a new GeoGebra window using the File menu. Click on the arrow on the right side of the Graphics View in order to open the Perspectives sidebar and select the CAS Perspective. GeoGebra Quickstart, Version

6 Construction Steps In the CAS View, create a quadratic polynomial by entering 1 f(x):= x^2 3/2 * x + 2 into the first row and hitting the Enter key. Select the Slider tool from the Graphics View Toolbar and create 2 two sliders a and b using the default settings of sliders. Hint: After clicking on the Graphics View, a window appears allowing you to specify the parameters of your slider. Click Apply in order to close the window and create a slider. In the CAS View, create a linear function by entering 3 g(x):= a * x + b into the next row and hitting the Enter key. In the Graphics View, use the Move tool in order to change the 4 value of slider a to 0.5 and the value of slider b to 2. In the CAS View, enter h(x):= f(x) g(x) in order to determine 5 the difference between those functions. Enter h(x), then select the tool Factor from the CAS View Toolbar in 6 order to factorize the polynomial. Hint: You can now use these factors to determine the roots of h(x). 7 Enter Solve[h(x)] in order to confirm the roots. Hint: GeoGebra creates a list of points which can be displayed in the Graphics View by clicking on the icon Show Object below the corresponding row number of the CAS View. Enter Intersect[f(x), g(x)] in order to create the intersection 8 points of functions f(x) and g(x). Hint: You can display the intersection points in the Graphics View by clicking on the Show / Hide icon below the corresponding row number in the CAS View. Exploration: Try to find out what the intersection points of f(x) and g(x) have in common with the roots of the difference function h(x). 9 Change the parameters of the linear function to find out for which values of a and b there are two, one, or no intersection point(s). GeoGebra Quickstart, Version

7 Hint: Use the Move tool in order to change the values of the sliders and create new functions to explore. Hints The CAS View allows you to work with fractions, equations and formulas (that may include variables that have not already been defined). Input and automatic completion = Enter y = 2 x + 1 in order to define an equation. := () [] Enter f(x):= 2x + 1 in order to assign the name f to the function. If you enter opening parentheses or brackets, GeoGebra will automatically insert the corresponding closing counterpart. While entering a command in the CAS View, GeoGebra tries to automatically complete your command in order to help you with the syntax. Hint: Select the command you want from the appearing list and hit Enter in order to accept the suggestion. The CAS Toolbar offers three different tools to evaluate your input with: 'Evaluate' calculates and simplifies the input in a symbolic way. 'Numeric' calculates the input numerically and yields the result in decimal notation. 'Keep Input' keeps and checks the input. This is very useful if you don t want your input to be simplified automatically, for example when introducing the manipulation of expressions. You can also manipulate only part of an expression by selecting it with the mouse and then clicking on a tool like Factor. Example 4: Scatter Plot and Best Fit Line Task: Create a Scatter Plot and conduct a Two Variable Regression Analysis for a given data set. Explore how outliers affect the best fit line. GeoGebra Quickstart, Version

9 Hints You may change the style (e.g. color, thickness) of the line and points. Select the object you want to enhance and then click on the little arrow in the upper left corner of the Graphics View in order to open the Style Bar. GeoGebra allows you to import data from other spreadsheet software into the Spreadsheet View: Select and copy the data you want to import. Hint: For example, use the keyboard shortcut Ctrl - C (Mac OS: Ctrl - C) in order to copy the data to your computer's clipboard. Open a GeoGebra window and show the Spreadsheet View. Click on the spreadsheet cell that should contain the first data value (e.g. cell A1). Paste the data from your computer s clipboard into GeoGebra's Spreadsheet View. Hint: For example, right - click (Mac OS: Ctrl - click) on the highlighted cell and select Paste. Example 5: Intersection of a Plane and a Line Task: Find out how you can intersect a plane and a line in the 3D Graphics View of GeoGebra. GeoGebra Quickstart, Version

10 Preparations Open a new GeoGebra window using the File menu. Click on the arrow on the right side of the Graphics View in order to open the Perspectives sidebar and select the 3D Graphics Perspective. Construction Steps Hints Enter p: x + y = z into the Input Bar and hit the Enter key in order to define a plane p. Create two points A and B by entering A = (3, 4, 3) and B = (-4, -2, -1) in the Input Bar, hitting the Enter key after each input. Select the Line tool from the 3D Graphics Toolbar and click on the points A and B respectively to create the line a through both points. Select the Intersect tool. Then, click on the plane and the line in order to create the intersection point C. Exploration: Use the Move tool in order to change the position of the points A and B. What happens to the intersection point if the line is parallel to the plane, or if both points lie within the plane? Hint: By clicking on a point you may switch between moving the point parallel to the x-y-plane or parallel to the z-axis. You might want to use the tool views on your construction. Rotate 3D Graphics View to get different You might want to enhance the appearance of your construction by using the Style Bar. Select the object you want to enhance and then click on the little arrow in the upper left corner of the 3D Graphics View in order to open the Style Bar. You can also hide the x-y-plane or the coordinate axes using the Style Bar. Click on the white background of the Graphics View to make sure that no object is selected before you open the Style Bar. Further Information You can find further information, materials, and help on our web pages at GeoGebra Quickstart, Version

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