N
ew Point
Two Points
Line Between
Move
Line
Perpendicular
Through Point
Circle w/ Center
Angle
At Point
Mirror Object
Insert Text
Pad
Move Drawing
GeoGebra
Transformation Activities
If needed:
• Go to www.geogebra.org • Click on Download
• Click on GeoGebra WebStart
• Click on the button for GeoGebra WebStart
GeoGebra WebStart
Click on the button below to start GeoGebra WebStart. This will download and install GeoGebra on your computer. You will either get a GeoGebra icon or a "geogebra.jnlp" file on your desktop. Double click this icon or file to start GeoGebra - this will also work when you are offline. Please see GeoGebra Help to find out how to use GeoGebra.
Now open up GeoGebra!
Activity A - Reflection
1. Select the Polygon tool.
• Use the mouse to place the three vertices of a triangle A, B, and C on the screen. (Note: You will need to click on Point A AGAIN after you place Point C on the screen in order to close the triangle.)
2. Notice that Points A, B, and C are all free objects. Thus, they can be moved around. • The area as well as side lengths a, b, and c are all dependent objects. These values will
change only when the free objects move. 3. Select the Move tool.
• Use the mouse to move Point A. Notice how the coordinate for A changes as well as the are and the length of sides b and c.
4. Still using the Move tool, drag the interior of the triangle to the upper portion of the screen. 5. Select the Segment Between Two Points tool.
• Place a horizontal segment DE across the middle of the screen. 6. Select the Mirror Object at Line tool.
• Click on the interior triangle and then click on segment DE. 7. Take a moment to look at the free objects and dependent objects. 8. Sketch a picture of what you see here:
Drag and Discover for Activity A - Reflection:
Discovery #1: Using the Move (arrow) tool, drag one vertex of the original triangle. • What do you notice?
Discovery #2: Drag the interior of the original triangle. • What do you notice?
Discovery #3: Drag the line segment DE (line segment of reflection). • What do you notice?
Discovery #4: Drag an endpoint that makes up the line segment of reflection. • What do you notice?
Moving on…the rest of Activity A
9. Select the Segment Between Two Points tool. • Click on vertex A and then vertex A’ • Click on B and then B’
• Click on C and then C’
10. Discovery #5. Select the Move tool. Drag and discover again. • What do you notice about the new line segments (f, e, and g)?
11. Select the Intersect Two Objects tool.
• Click on the intersection between the line of reflection and AA’ (called F) • Click on the intersection between the line of reflection and BB’ (called G)
12. Select the Angle tool. Click on G followed by F followed by A • What is the size of angle alpha?
13. Discovery #6. Select the Move tool. Drag and discover again. • What do you notice about angle alpha?
Check it Out! Reflection (You must be online to do this!)
o Go to Geogebra.org o Click on GeoGebraWiki o Click on English
o Click on Middle School Geometry o Click on Transformations
Activity B - Translation
14. Select the Delete Object tool.
15. Click on everything EXCEPT the original ABC triangle and the interior of ΔABC. 16. Select the Vector Between Two Points tool.
• Use the mouse to place Point D somewhere on the screen and then move the mouse to create a vector. Use the mouse to place Point E to “lock” the vector down.
17. Select the Translate Object by Vector tool.
• Now select the interior of triangle ABC followed by vector u. 18. Sketch a picture of what you see (include labels of the points) here:
19. Take a moment to look at the coordinates of Triangle ABC (Free Objects), the coordinates of the translated triangle (Dependent Objects), and components for vector u (Dependent Objects).
• Write down any relationships that you see.
Drag and Discover for Activity B - Translation:
Discovery #7: Using the Move (arrow) tool, drag one vertex of the original triangle. • What do you notice?
Discovery #8: Drag the interior of the original triangle. • What do you notice?
Discovery #9: Drag vector u (but not the endpoints) • What do you notice?
Discovery #10: Drag either endpoint of vector u • What do you notice?
20. Select the Segment Between Two Points tool.
• Click on vertex A and then vertex A′ (automatically labeled d) • Click on B and then B′ (automatically labeled e)
• Click on C and then C′ (automatically labeled f)
21. The next task is to compare the lengths as well as the slopes of the vector and new line segments d, e, and f. However, GeoGebra does not allow the user to measure the slope of a vector. Thus to make things easier, use the Segment Between Two Points and click on the two points of the vector. You are placing a segment over the vector (You should see a label of g.)
Discovery #11: Select the Move tool. Drag and discover again. Pay close attention to the lengths of the new segment s (d, e, f, and g) in the dependent object list.
• What do you notice about the lengths?
22. In the Input: bar at the bottom of the screen type in slope[d], slope [e], slope[f], and slope[g].
Discovery #12. Select the Move tool. Drag and discover again. • What do you notice about the slopes?
23. Confirm the results for Discovery #11 & #12 algebraically?
Check it Out! Translation (You must be online to do this!)
o Go back to the GeoGebraWiki
o Middle School Geometry / Transformations o Click on Translated Point Practice
Activity C - Rotation
24. Select the Move Drawing Pad tool and change to Delete Object.
25. Click on everything EXCEPT the original ABC triangle and the interior of ΔABC. 26. Select the New Point tool.
• Use the mouse to click a new Point D.
27. Select the Move (arrow) tool and then double click on Point D. You should see:
28. Rename Point D to Point Center by clicking here: (Don’t forget to click on Apply) 29. Select the Slider tool.
• Click on the lower left hand of the screen • Choose Angle
• From 0 to 360
• With 5 degree increments • Click on Apply
30. Select the Rotate Object Around Point By Angle tool.
• Now select the interior of triangle ABC followed by Point Center and then enter alpha (from the drop down menu) for the angle.
• Click on Apply.
31. Select the Move (arrow) tool and move the slider point to 45 degrees. (WAIT to Drag and Discover!)
32. Sketch a picture of what you see (include labels of the points) here:
Drag and Discover for Activity C - Rotation:
Discovery #13: Using the Move (arrow) tool, drag one vertex of the original triangle. • What do you notice?
Discovery #14: Drag the interior of the original triangle. • What do you notice?
Discovery #15: Drag the Center point. • What do you notice?
You have to click on this menu and then click on alpha again
Discovery #16: Drag the point on the Slider bar. • What do you notice?
33. Using the Segment Between Two Points tool,
• Click on vertex A and then Center point (automatically labeled d) • Click on vertex A’ and then Center point (automatically labeled e) • Note: you could do this for the other 4 vertices as well!
34. Select the Angle tool.
• Click on A then Center then A’
• Use the Move (arrow) tool to double click on the green Beta angle • Change the drop down menu “Name” to “Value” and then click Apply. Discovery #17. Using the Move (arrow) tool, drag and discover again.
• What do you notice about the angle AS WELL AS the lengths of segments d & e? 35. Using the Move tool, double click on any vertex.
• The Properties window should pop up
• Click on A’ and then check the box for Show Trace • Click on Apply
Discovery #18. Drag the point on the slider bar only…and discover again. • What do you notice?
Check it Out! Rotation (You must be online to do this!)
o Go back to the GeoGebraWiki
o Middle School Geometry / Transformations o Click on Rotated Point Practice
Activity D - Dilation
36. Select the Eraser (Delete Object) tool.
37. Click on everything EXCEPT the original ABC triangle and it’s interior. 38. Select the Slider tool.
• Click on the lower left hand of the screen • Choose Number
• Interval from 0 to 5 • With 0.25 increments • Click on Apply •
39. Select the Dilate Object From Point By Factor tool.
• Now select the interior of triangle ABC followed by Point A (you will have to specifically choose Point A) and then type in d for the slider bar value
• Click on Apply.
40. Select the Move (arrow) tool and move the slider point to d=2. (WAIT to Drag and Discover!)
BY THE WAY…YOU CAN ALSO DOUBLE CLICK ON THE d= VALUE IN THE FREE OBJECTS MENU AND PHYSICALLY TYPE IN A 2. OR YOU CAN CLICK ON THE d= VALUE AND USE THE ARROW KEYS TO CHANGE THE VALUE. YOU CAN ADJUST COORDINATES AS WELL AS MANY FREE OBJECTS THIS WAY…VERY POWERFUL!
44. Sketch a picture of what you see (include labels of the points) here:
Drag and Discover for Activity D - Dilation:
Discovery #19: Using the Move (arrow) tool, drag vertex A of the original triangle. • What do you notice?
Discovery #20: Using the Move (arrow) tool, drag vertex B or C of the original triangle. • What do you notice?
Discovery #21: Drag the point on the Slider bar. • What do you notice?
Discovery #22: The Dependent Object menu contains both the Area of the original polygon (ABC) labeled as P and the transformed polygon (A’B’C’) labeled as P’. Drag the point on the
slider bar only…and discover again.
• What do you notice about the Areas?
41. Using the Move (arrow) tool, double click on the slider bar. • Change the interval to be -3 to 5
• Click on Apply
Discovery #23: Drag the point on the slider bar only…and discover again. • What do you notice?
42. To put closure to this series of activities, go to GeoGebra.org / Middle School /
Transformations and find the final topic “Transformations Overview” Take a moment to look through this applet
Check it Out! (You must be online to do this!)
o Go back to the GeoGebraWiki
o Middle School Geometry / Transformations Click on Transformations Overview
Activity E – Transformations on the Sinusoid
43. Go to the View menu and turn on the Grid. 44. Select the Slider bar tool.
• Place FOUR sliders bars in the upper left hand portion of the screen. • Slider bar a: -5 to 5 w/ increments of 0.1
• Slider bar b: -5 to 5 w/ increments of 1 • Slider bar c: -5 to 5 w/ increments of 0.5 • Slider bar d: -5 to 5 w/ increments of 0.1 45. Set a=1 and b=1 using any option listed below.
• Option 1: Use the Move tool to move the points on the slider bar.
• Option 2: Double-click on “a=” and “b=” in the Free Objects area and change the values.
• Option 3: Single-Click on the “a=” in the Free Objects and use the arrow keys. • Option 4: Single-Click on the point of the slider bar and then use arrow keys. 46. In the Input: bar, enter: f(x)=a*sin(b*(x-c))+d
• Note: asin(b(x-c))+d will give you an error with respect to b asin(b*(x-c))+d will graph the arcsin
You CAN just use “spaces” between the a and sin and between the b and ( f(x)=a sin(b (x-c))+d
47. Sketch a picture of what you see (include the slider bars) here:
Drag and Discover for Activity E – Transformations on the Sinusoid:
Discovery #24: Using the Move (arrow) tool, drag one vertex of the original triangle. • What do you notice?
Discovery #25: Drag the interior of the original triangle. • What do you notice?
Discovery #26: Change the a value. • What do you notice?
Discovery #27: Change the b value. • What do you notice?
Discovery #28: Change the c value. • What do you notice?
Discovery #29: Change the d value. • What do you notice?
48. Optional. Use your Applet to solve the following two problems:
A signal buoy in the Chesapeake Bay bobs up and down with the height h of its transmitter (in feet) above sea level modeled by h = a sin bt + 5. During a small squall, its height varies from 1 ft to 9 ft and there are 3.5 seconds from one 9-ft height to the next. What are the values of the constants a and b?
a ≈ _____ b ≈ _____
Enter g(x) = 2cos(3x)+sin(3x) into the Input Bar. What values a, b, c, and d for the sinusoid a*sin(b*(x-c))+d will “fit” g(x)?
a ≈ _____ b ≈ _____ c ≈ _____
Activity F – Transformations on Bart Simpson
49. Click on “File” “New Window”
50. Go to the View menu and turn on the Grid.
51. Use the Point tool and place Points A, B, and C in the upper left hand corner. 52. Open a web browser and type in:
• http://www.nebrwesleyan.edu/people/cminer/
• Click on GeoGebra found near the bottom of the screen
• Save the picture of Bart Simpson to your desktop (Mac users click and hold down on the picture and choose Save to disk)
53. Change to the Insert Image tool.
54. Click on the grid and navigate to find the picture of Bart Simpson you saved to the computer. Open the file.
55. Click on the Move (arrow) tool and then do one of the following: a) PC users: Right-click on the picture or
b) Mac users: Hold down Apple key and then click on the picture or c) Double click on the picture or
d) Go to “Edit” “Properties” with the goal of bringing the Properties to the forefront 56. Change the Corner settings A, B, and C as shown in the picture:
57. After you click on Apply, make sure the Move tool is highlighted. Discovery #30: Drag the Points A, B, and C around.
• What do you notice?
58. Double click on each of the coordinates in the Free Objects menu. Change A to (1,-1). Change B to (7, -1). And change C to (1, 6). This can be helpful in placing the picture exactly where you want it!
59. Optional. Use the Segment tool to draw line segments AB and AC.
60. Practice using the transformational tools you learned about in Activities A-D. Try to create one or more of these on your own!
• Reflection • Rotation • Translation • Dilation
