GeoGebra Institute of Malaysia

Transcription

1 Introduction GeoGebra Institute of Malaysia 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-touse 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 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 , his work was supported by an NSF Math & Science Partnership project of Florida Atlantic University and Broward County Schools in Florida. A t p r e s e n t, h e i s a Professor for Mathematics Education, Johannes Kepler University Linz, Austria. Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 1

2 GeoGebra Institute of Malaysia Table of Contents Introduction Installation of GeoGebra... 4 Installation WITH Internet access... 4 Installation WITHOUT Internet access GeoGebra s User Interface Basic Use of Tools... 5 Activity 1: Familiarising with some of the Menu/Tools... 5 Construction Tools Drawing with GeoGebra... 7 Activity 2: Drawing Geometric Figures and Other Objects... 7 Activity 3: What to practice Saving and Opening File... 8 Activity 4: Saving GeoGebra files... 8 Activity 5: Opening GeoGebra Files Drawings, Constructions, and Drag Test... 9 Discussion Rectangle Construction... 9 Activity 6: Constructing a Rectangle Navigation Bar and Construction Protocol Activity 7: Checking your construction GeoGebra s Properties Dialog Activity 8: What to practice Measuring Length, Area and Angles Square Construction Activity 9: Constructing a square Discussion Create Custom Tool Activity 10: Creating a Custom Square Tool Equilateral Triangle Construction Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 2

3 Activity 11a: Constructing an Equilateral Triangle Activity 11b: Checking your construction Theorem of Thales Activity 12: Visualising the Theorem of Thales Export of Pictures to the Clipboard Activity 13: Exporting Pictures to the Clipboard Inserting Pictures into the Graphics View Activity 14: Drawing Tools for Symmetric Figures Resizing, Reflecting, Distorting and Translating a Picture Activity 15: Resizing and Reflecting a Picture Discussion Activity 16: Distorting a Picture Activity 17: Exploring the Properties of Reflection Discussion Activity 18: Translating a Picture Rotation Activity 19: Rotating Polygons Basic Algebraic Input and Commands Activity 20: Visualising Properties of a Linear Equation Activity 21: Exploring Quadratic Functions Activity 22: Exploring Trigonometric Functions Activity 23: The Slope Function Activity 24: Visualising Area of a Circle Activity 25: Recording to Spreadsheet Feature Creating Dynamic Worksheets Activity 26: Exporting a Dynamic Worksheet Useful Web Links Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 3

4 1. Installation of GeoGebra Create a new folder called GeoGebra_your name on your desktop. Hint: During the session, save all files to this folder so they are easy to find later on. Installation WITH Internet access Install GeoGebra WebStart Open your Internet browser and go to 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. 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-Windows-Installer exe MacOS: GeoGebra_GeoGebra-Windows-Installer 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. 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 View, Input Bar, Spreadsheet View, and coordinate axes and just work with the Graphic View and geometry tools. You can customise your preference using the Perspectives Menu. 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. SPREADSHEET VIEW INPUT BAR: Create and to redefine mathematical objects Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 4

5 3. Basic Use of Tools Activate a tool by clicking on the button showing the corresponding icon. Open a toolbox by clicking on the small triangle and select the you want to use from this toolbox. Check the toolbar help in order to find out which tool is currently activated and how to operate it. Activity 1: Familiarising with some of the Menu/Tools Explore some of the tools on your own. Menu Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 5

6 Construction Tools GeoGebra Institute of Malaysia Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 6

7 10 GeoGebra Institute of Malaysia Drawing with GeoGebra Activity 2: Drawing Geometric Figures and Other Objects Hide the Algebra and Spreadsheet View, 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 Graphic View (e.g. square, rectangle, house, tree, animal, human figure ). Introduction of new tools New point Hint: Click on the Graphic View or an already existing object to create a new point. Move Hint: Drag a free object with the mouse. Line through two points Hint: Click on the Graphic View twice or on two already existing points. Segment between two points Hint: Click on the Graphic View twice or on two already existing points. Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 7

8 GeoGebra Institute of Malaysia Delete object Hint: Click on an object to delete it. Undo / Redo Hint: Undo / redo a construction step by step. Circle with Centre through Point Semicircle through Two Points Semicircular Arc through Three Points Circular Sector with Centre between Two points Move Graphic View Hint: Click and drag the Graphic View to change the visible part. Zoom in / Zoom out Hint: Click on the Graphic View to zoom in / out. Hint: Don t forget to read the toolbar help if you don t know how to use a tool Activity 3: 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. 5. Saving and Opening File Activity 4: 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 (for example use Act.2 ). 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. Activity 5: 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. Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 8

9 6. Drawings, Constructions, and Drag Test Open the dynamic worksheet 01_Drawing_Construction_Squares.html. You can find this file in the folder named 01_Drawing_Construction_Squares. The dynamic figure shows several squares constructed in different ways. Examine the squares by dragging ALL their vertices with the mouse. Discussion Q1. Find out which of the quadrilaterals are real squares and which ones just happen to look like squares.... Q2. What do you think is the difference between a drawing and a construction? Q3. What is the drag test and why is it important?.. Q4. 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 6: Constructing a Rectangle Summarize the properties of a rectangle before you start the construction. Open new GeoGebra file. Hide Algebra View, Input Bar and coordinate axes (View menu). Change the labelling setting to New points only (menu Options Labelling). Introduction of new tools Perpendicular Line and Parallel Line Tools Intersect Two Objects Tool Polygon Tool : 1 Segment AB 2 Perpendicular line to segment AB through point B 3 New point C on perpendicular line 4 Parallel line to segment AB through point C Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 9

10 5 Perpendicular line to segment AB through point A 6 Intersection point D 7 Polygon ABCD Hint: To close the polygon click on the first vertex again 8 Save the construction 8. Navigation Bar and Construction Protocol Activity 7: 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. 9. GeoGebra s Properties Dialog How to access the Properties Dialog Right-click an object OR Select Object Properties from the Edit menu OR Double click an object in Move mode and click on button Object Properties in the appearing Redefine dialog window. Activity 8: 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 Select all objects of one type by clicking on the corresponding heading. Show the value of different objects and try out different label styles. Change the properties of certain objects (e.g. color, style, ) and hide all unnecessary lines. 10. Measuring Length, Area and Angles Introduction of New Tools Distance or Length Tool Area Tool Angle Tool 1 Select the Distance or Length Tool Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 10

11 2 Click one side of the rectangle to measure its length 3 Using the same tool, click the other three sides to measure the length of each side 4 Click on the rectangle to measure the perimeter. 5 Select the Area Tool 6 Click on the rectangle to measure the area 7 Select the Angle Tool and measure all the angles 8 Save your file 11. Square Construction Activity 9: Constructing a square Discussion Q1. Think of a way to construct a square. Describe it. Q2. Construct the square using the approach you have just described. What are the preparations needed? A sample of 1 Create segment AB 2 Construct a perpendicular line to AB through point B 3 Construct a circle with center B through point A 4 Intersect the circle and the perpendicular line (created in step 2) to get an intersection point C 5 Construct parallel line to BC and passes through point A Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 11

12 6 Construct parallel line to AB and passes through point C 7 Intersect both lines created in steps 5 & 6 to intersection point D 8 Construct polygon ABCD 9 Construct segments for all sides of the square except side AB 10 Save the file 12. Create Custom Tool Activity 10: Creating a Custom Square Tool Continue from the previous activity. Introduction of New Tools Create New Tool Hide / Show Object Tool Hide / Show Label Tool 1 Hide auxiliary objects (lines and circle) 2 Hide labels of all objects 3 4 Create your Square Tool (menu Tools - Create New Tool ). Output objects: square, and all segments Input objects: point A and B Name: Square Toolbar help: Click on two points. Click Finish. Save your Square Tool as file Square_Tool.ggt Hint: Menu Tools - Manage tools - Save as 13. Equilateral Triangle Construction Activity 11a: Constructing an Equilateral Triangle Summarize the properties of an equilateral triangle before you start the construction. Open new GeoGebra file. Hide Algebra View, Input Bar and coordinate axes (View menu). Change the labelling setting to New points only (menu Options Labelling). Discussion Q1. Think of a way to construct an equilateral triangle. Describe it. Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 12

13 Q2. Construct the Equilateral triangle using the approach you have just described. A sample of 1 Segment AB 2 3 Circle with center A through B Hint: Drag points A and B to check if circle is connected to them Circle with center B through A Hint: Drag points to check if circle is connected to them 4 Intersect both circles to get point C 5 Polygon ABC 6 Hide circles 7 Measure interior angles of triangle 8 Save the construction Activity 11b: Checking 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. 14. Theorem of Thales Activity 12: Visualising the Theorem of Thales Open a new file. Hide Algebra View, Input Bar and the coordinate axes Change the labelling setting to New points only (menu Options Labelling). Discussion Q1. Find out who discovered the Theorem of Thales and what is it about? Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 13

14 Introduction of New Tools Semicircle through Two Points tool 1 Segment AB GeoGebra Institute of Malaysia 2 Semicircle through points A and B 3 New point C on the semicircle (make sure the point really lies on the arc) 4 Polygon / Triangle ABC 5 Measure all the interior angles of the triangle ABC 6 Save your file 15. Export of Pictures to the Clipboard GeoGebra s G r a p h i c V i e w 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 Hide Algebra View, Input Bar and coordinate axes (View menu). Introduction of New Tools Regular Polygon Tool Open new GeoGebra file. 1 Create a square 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 Graphic View: Move your figure (or the relevant section) to the upper left corner of the Graphic View u sing the Move Graphic View Tool. You might want t o use tools Zoom In and Zoom Out or use the mouse s wh eel in ord er to prepare your f igure for the exp ort 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 Graphic View to the clipboard: Export Graphic View 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 Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 14

15 word processing or presentation 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 cursor Hint: You can use the key combination Ctrl-V instead. 2 Repeat for other regular polygons such as pentagon, hexagon, octagon, etc. 16. Inserting Pictures into the Graphics View Activity 14: Drawing Tools for Symmetric Figures Introduction of new tools Trace Tool Reflect Object in Line Tool Insert Image Tool Construction process 1 New point A 2 Show the label of point A 3 Line of reflection through two points 4 Mirror point at line to get image A 5 Segment between point A and its image A 6 Turn the Trace on for points A and A Hint: Right Click the point and select Trace On from the menu. Whenever point A is moved it leaves a trace on the drawing pad. 7 Move point A to draw a dynamic figure Insert picture flower.jpg. Hint: Click on the lower left corner of the Graphic View to insert the picture at this position. Adjust the position and set the image as background image (Properties dialog, Tab Basic) Reduce the filling of the image (Properties dialog, Tab Style). Hint: After specifying the picture as a background image you need to open the Properties Dialog using the Edit menu. You can t select a background image in the Graphic View any more. 11 Save the construction Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 15

16 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). 17. Resizing, Reflecting, Distorting and Translating a Picture Activity 15: Resizing and Reflecting a Picture Open a new GeoGebra file. Close the Algebra View and hide the coordinate axes (View menu). 1 Insert picture twintower.jpg into the left part of the Graphic View 2 Point A at the lower left corner of the picture 3 Set point A as Corner 1 of your picture 4 Point B along the length of the side of the picture 5 Set point B as Corner 2 of the picture. Hint: You can now resize the image by moving point B. You just changed the width of the picture. 6 Line through two points in the middle of the drawing pad 7 Mirror the picture at the line 8 Save the construction Discussion Q1. Move point A with the mouse. How does this affect the picture? Q2. Move the picture with the mouse and observe how this affects its image.. Q3. Move the line of reflection by dragging the two points with the mouse. How does this affect the image?.. Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 16

17 Activity 16: Distorting a Picture GeoGebra Institute of Malaysia Hint: You might want to reduce the filling of the image in order to be 1 able to better distinguish it from the original (Properties dialog). 2 Create a new point D at the upper left corner of the original picture 3 Set point D as Corner 4 of your picture Discussion Q1. Move Point D. How does moving point D affect the picture and its image?.. Activity 17: Exploring the Properties of Reflection In this activity you will create a dynamic figure that allows your students to explore the properties of reflection. You will now modify the construction created in the previous activity. Continue from the file you have created in the previous activity which 1 contains the distorted picture of the Petronas Twin Towers and its reflection at a line 2 Segment between points A and B 3 Segment between points A and D 4 Parallel line to segment AB through point D 5 Parallel line to segment AD through point B 6 Intersect the two lines to get intersection point C 7 Hide auxiliary objects 8 Reflect all four corner points A, B, C, and D at the line to get their images A, B, C, and D 9 Connect corresponding points with segments (e.g. points A and A ) 10 Create angles between the line of reflection and the segments 11 Save the construction Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 17

18 Discussion Q1. Move the corner points A, B, C, and D of the original picture. Are you able to drag all these points with the mouse? If no, which one can t be dragged and why? Q2. Move the line of reflection. What do you notice about the angles between the segments connecting the corresponding corner points and the line of reflection?.. Activity 18: Translating a Picture Open a new GeoGebra file. Show the Algebra View, Input Bar, coordinate axes, and grid (View menu). In the Options menu set the point capturing to on (Grid). Introduction of new tools Vector between Two Points Tool Point Capturing Fixed to Grid Translate Object by Vector Tool 1 Insert picture upin_ipin.jpg into the first quadrant 2 Create Points A = (1, 1), B = (3, 1), and D = (1, 4) 3 Set point A as Corner 1, B as Corner 2, and D as Corner 4 of the picture. Hint: Properties dialog tab Position 4 Create Triangle ABD 5 Points O = (0, 0) and P = (3, -2) 6 Click at point O then point P (Vector OP) 7 Translate the picture by vector u. Hint: You might want to reduce the filling of the image 8 Translate the three corner points A, B, and D by vector u. 9 Triangle A B D Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 18

19 10 Hide point O so it can t be moved accidentally. 11 Change the color and size of objects to enhance your construction. 12 Save the construction 18. Rotation Activity 19: Rotating Polygons Open a new GeoGebra file. Hide Algebra View and Input Bar (View menu). Show the coordinate axes and the grid (View menu). Open the Properties dialog for Graphic View by right clicking the Graphic View o On tab Axes xaxis change the Distance for x-axis to 1 o On tab Axes yaxis change the Distance for y-axis to 1 Label all New Points Only Introduction of new tool Slider Tool Rotate Object around Point by Angle Tool Rename Object Tool Create an arbitrary triangle ABC in the second quadrant placing the 1 vertices on grid points. 2 New point D at the origin of the coordinate system 3 Rename the new point to O 4 Slider for angle α In the slider dialog window check Angle and set the increment to 90. Make sure you don t delete the symbol. 5 Rotate triangle ABC around point O by angle α 6 Construct Segments AO and A O 7 Measure Angle AOA 8 Save the construction Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 19

20 19. Basic Algebraic Input and Commands GeoGebra Institute of Malaysia The Input Bar allows you to 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. P = (3, 2) and Q = (4, 8) creates point P and Q respectively. Circle c: (x 2)^2 + (y 1)^2 = 16 Line[P, Q] creates a line through existing points P and Q. 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 Input Help button. You can press the F1-key to get information about the syntax and arguments required to apply the corresponding 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. o If GeoGebra suggests the desired command, hit the Enter key in order to place the cursor within the brackets. o If the suggested command is not the one you wanted to enter, just keep typing until the suggestion matches. Activity 20: Visualising Properties of a Linear Equation Open new GeoGebra file. Show the Algebra View, Input Bar, and coordinate axes (View menu). Introduction of new tools Slope Tool Insert Text Tool: Dynamic and Static text Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 20

21 GeoGebra Institute of Malaysia 1 Create sliders m and c using the default settings of sliders 2 3 Enter the equation: y=m x+c Hint: Don t forget to use asterisk or space to indicate multiplication! Intersection point A between the line and the y-axis Hint: You may also use the command Intersect [line, yaxis] 4 Point B at the origin 5 Segment between two points A and B Hint: You may want to increase the line thickness in order to make the segment visible on top of the y-axis. 6 Determine Slope (triangle) of the line 7 Enhance the appearance of the file using the Properties Dialog 8 Insert Text. Type y = m x + c (for dynamic text use m and c as Object) 9 Save the construction Activity 21: Exploring Quadratic Functions Open a new GeoGebra file. Show Algebra View, Grid, and Coordinate Axes. Show the Input Bar. 1 Create sliders a, b, and c using the default settings of sliders 2 Enter the function : f(x)=a x^2 + b x + c 3 Display the equation of the function with mixed text 4 Save the construction Activity 22: Exploring Trigonometric Functions Open a new GeoGebra file. Show Algebra View, Grid, and Coordinate Axes. Show the Input Bar Introduction of new tools Check Box to Show / Hide Objects Tool 1 Create sliders a, b, c and d using the default settings of sliders Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 21

22 2 GeoGebra Institute of Malaysia Configure interval in the x-axis from 1 to π/2 by Right-Click blank space in Graphic View, select Graphics... Check the Distance check box in the x- Axis tab, then select π/2 from the Distance drop-down list box. Click Close button 3 Enter the function: f(x) = a sin(b x +c) + d 4 Display the equation of the function 5 6 Activate Check Box Tool to Show/Hide Objects. In the Check Box dialog box, type Show / Hide Sine Function in the Caption box, then choose Function f(x) = a sin (bx + c) + d. Click the Apply button Move the Check Box to the desired position. Select object properties and check Fix Check Box. 7 Enhance the appearance of the file using the Properties Dialog 8 Save the construction 9 Repeat all the steps above for the functions g(x)= a cos(b x+c) + d and h(x) = a tan(b x + c) + d Activity 23: The Slope Function Open a new GeoGebra file. Show Algebra View, Input Bar, and Coordinate Axes. Introduction of new tools Tangents Tool 1 Enter the polynomial f(x)=x^2/ Create new point A on the function f 3 Create tangent t to the function f through point A 4 Create points S: S= ( x(a), slope[t]) 5 Connects points A and S 6 Enhance the appearance using the Properties Dialog 7 Save the construction Discussion Q1. Turn on the trace of point S. Move point A. Guess the equation of the resulting slope function. Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 22

23 (Enter the function into Input Bar and move point A. If it is correct, the trace of point S will match the graph. Change the equation of the initial polynomial f to produce a new problem.) Activity 24: Visualising Area of a Circle Open a new GeoGebra file. Hide the Spreadsheet View, Algebra View, Coordinate Grid, and Coordinate Axes. Select Options Labelling No New Objects. Select Fixed to Grid the Option/Point Capturing menu Show the Input Bar. Point A and Point B where length of AB = 1 unit. For e.g. : A= (0, 0); 1 B=(1,0). 2 Circle through B with centre A. Circle [A, B]. 3 Slider n with increment of 1, min 3 and max Rotate [B, /n, A]. 5 Polygon [B, B, n]. 6 Drag slider n 7 Save the construction 22. GeoGebra s Spreadsheet View Activity 25: Recording to Spreadsheet Feature Open a new file Show the Spreadsheet View Show Axes and Input Bar Hide Algebra View Introduction of new tools Record to Spreadsheet tool 1 Create a slider a with default interval and increment 1 2 Create point A by entering A = (a, a^2) into the Input bar. 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 Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 23

24 5 GeoGebra Institute of Malaysia Use Move Graphics View Tool, 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 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 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 10 Save the construction Discussion Q1. Suggest a prediction about a function graph that runs through all different positions of point A. Enter the corresponding function into the Input bar to check if your prediction is correct (e.g., enter f(x) = 2x to create a line through all points). Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 24

25 Activity 26: Relative Copy and Linear Equations Preparations Open a new GeoGebra file. Show the Spreadsheet view (View menu). Hide the Algebra view (View menu). Activate Move Graphics View Tool and drag the origin of the coordinate 1 system close to the lower left corner of the Graphics View 2 3 In the Spreadsheet View, click on cell A1 enter the point coordinates (0, 0) 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: Hint: Highlight both cells A1 and A2 by using the mouse. Click on the little square at the bottom right corner of the highlighted cell range Hold the mouse button down and drag the pointer down to cell A11 6 Use Move Graphics View Tool, 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 7 Save the construction Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 25

26 Discussion Explore the pattern in column B, which is created by the y-coordinates of different positions of point A. Q1. Suggest an equation that would create a graph going through all points of this sequence. Write down the equation and Enter this equation into the Input bar in order to check your prediction. Activity 27: Creating a Scatter Plot and Determining the Best Fit Line Preparations Open a new GeoGebra file. Show the Spreadsheet view (View menu). Hide the Algebra view (View menu). In the Options menu set the Labeling to New Points Only. Enter the following numbers into the spreadsheet cells of column A: 1 A1: 1 A2: 5 A3: 2 A4: 8 A5: Enter the following numbers into the spreadsheet cells of column B: B1: -1 B2: 2 B3: 3 B4: 4 B5: 1 Create a Scatter Plot from this data by using the mouse to highlight all cells of columns A and B that contains numbers. Right-Click on one of the highlighted cells and select Create List of Points from the appearing context menu 4 Activate Best Fit Line Tool and select all data using a selection rectangle 5 Save the construction With this construction you can easily demonstrate how outliers impact the best fit line of a data set: Drag one of the points with the mouse and explore how this modification influences the best fit line. Note: You can easily change the initial data in the Spreadsheet View as well. Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 26

27 20. Creating Dynamic Worksheets GeoGebra Institute of Malaysia 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. 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 26: Exporting a Dynamic Worksheet Open any gbb file that you have created. Reduce the size of the GeoGebra window. After adjusting the size of the GeoGebra window, you are now ready to export the figure as a dynamic worksheet using the File Menu: Export - Dynamic Worksheet as W e b p a g e (html) 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. Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 27

28 GeoGebra Institute of Malaysia 21. Useful Web Links GeoGebra Homepage Get the software as well as information about the GeoGebra project GeoGebraWiki Pool of free instructional materials created with GeoGebra 4. GeoGebra User Forum GeoGebra Manual Tutorials Events Tips & Tricks Publications GeoGebraTube Various materials Featured materials Newest Materials Best Rated Materials Uploaded Materials files and dynamic worksheets GeoGebra Institute of Malaysia Effective Use of GeoGebra in the Classroom: A Workshop for GeoGebra Users by Dr. Daniel Leong - 22 & 23 October SEAMEO RECSAM 28