Unit 2 - Transformations
moving the old unit 12 up in the batting order (baseball has taken over my department).
Table of Contents
- 12.1 Reflections
- Copy of Reflections
- Flip Flop Reflection
- Reflections and Translations
- Reflections Introduction
- Reflections in the Coordinate Plane
- Test yourself: Reflection
- Manipulating a Reflected Image
- 12.2 Translations
- Translations and Rotations
- Translating Triangle by a Vector
- Parallel Translation
- 12.3 Rotations
- Rotations: Introduction
- Exploring Rotations in the Coordinate Plane
- 12.4 Compositions of Transformations
- Tool for compositions of rigid transformations and dilations
- Applet for Multiple Transformations
- Messing With Lisa
- Copy of Compositions of Transformations
- Glide Reflection
- Composition of Transformations 3
- 12.5 Symmetry
- Point Symmetry: Another Perspective
- Rotational Symmetry
- Lines of Symmetry - Rectangle
- Folding Test: investigating lines of reflection symmetry
- assignments/tasks
- Transformations: Exercise 1
- Transformations: Exercise 2
- Transformations: Exercise 3
Copy of Reflections
Opposite Isometries
Any isometry that reverses the orientation of a triangle is called an [i]opposite isometry[/i]. When you read off the vertices of triangle ABC in cyclic order, they are read in a counterclockwise direction. When you read off the vertices of triangle A'B'C' in cyclic order, they are read in a clockwise direction.
[math]\sigma_m[/math] is the notation we will use for a reflection in line [i]m[/i]. [br]In the sketch below, we would write [math]\sigma_{DE}\left(A\right)=A'[/math].
Reflection
Translations and Rotations
Direct Isometries
Translations and rotations are [i]direct isometries[/i]. If you read the names of the vertices in cyclic order (A-B-C and A'-B'-C'), both would be read in the counterclockwise (or clockwise) direction.
Translation
Rotation
Rotations: Introduction
[color=#000000]The applet below was designed to help you better understand what it means to rotate a point about another point. [br][br]In the applet below, feel free to change the locations of point [i]A[/i] and[/color] [color=#1e84cc]point [i]B[/i][/color]. [br][color=#000000]Interact with this applet for a few minutes, then answer the questions that follow.[/color]
[color=#000000][b]Questions: [/b][br][br]1) Regardless of the [/color][color=#1e84cc][b]amount of rotation[/b][/color][color=#000000], how does the distance [i]AC [/i]compare to the distance [i]AB[/i]? [br][br]2) Notice how, in the applet above, the [/color][color=#1e84cc][b]angle of rotation[/b][/color][color=#000000] could be [/color][color=#1e84cc][b]positive[/b][/color][color=#000000] or [/color][color=#1e84cc][b]negative[/b][/color][color=#000000].[br] From what you've observed, what does it mean for a [/color][color=#1e84cc][b]rotation angle[/b][/color][color=#000000] to have positive orientation? [br] What does it mean for an [/color][color=#1e84cc][b]angle of rotation[/b][/color][color=#000000] to have negative orientation? [br] Explain. [/color]
Tool for compositions of rigid transformations and dilations
Tool for compositions of rigid transformations and dilations. Click the challenge when you are ready.
Tool for compositions of rigid transformations and dilations.
Point Symmetry: Another Perspective
Reflect points [i]B[/i], [i]C[/i], and [i]D[/i] about [color=#1e84cc][b]point [i]A[/i][/b][/color]. Then, slide the slider to see another alternative way to describe what it means to reflect any object (point, polygon, whatever) [color=#1e84cc][b]about a point[/b][/color]. (Feel free to move points [i][color=#1e84cc][b]A[/b][/color][/i], [i]B[/i], [i]C[/i], and/or [i]D[/i] anywhere you'd like at any time.)
Transformations: Exercise 1
[color=#000000]In the applet below, a [/color][b][color=#ff00ff]pink rectangle[/color][/b][color=#000000] and an unfilled rectangle are shown. [br][br][/color][color=#000000]Your job is to use the transformational tools of GeoGebra to superimpose (map) the [/color][color=#ff00ff][b]pink rectangle[/b][/color][color=#000000] perfectly onto the blank rectangle. [br][br][/color][color=#000000]To see your work presented in a different context (problem) at any time, select the [/color][b]New Context[/b][color=#000000] button. [br][/color][color=#000000]To remove the [/color][b][color=#ff00ff]pink shading[/color][/b][color=#000000] from the [/color][color=#ff00ff][b]original rectangle[/b][/color][color=#000000], slide the given slider to the left. [br][br][/color][color=#000000]Feel free to move any of the white points around (at any time) to change the size of the original rectangle. [br][br]To create a new exercise, select the [b]refresh[/b] icon in the upper right hand corner. [/color]
[b][color=#000000]Questions:[/color][/b][br][br][color=#000000]1) What transformation(s) did you use in your mapping? [br]2) What is common about all these transformations you've listed? [/color]