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Transformations ppt

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Neil MacIntosh
Neil MacIntosh

The document discusses different types of geometric transformations including translations, rotations, reflections, and dilations. It provides examples and rules for each type of transformation. Translations "slide" an object to a new position without changing its shape or size. Rotations turn a figure about a fixed point by a certain number of degrees. Reflections produce a mirror image across a line of reflection. Dilations or stretches change the size of an object but preserve its shape and orientation.

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Transformations ppt
To transform something is to change. The main transformations include: •Translation •Rotation •Reflection •Dilatation (Enlargement)
Renaming Transformations • The original shape is labelled with capital letters: • The transformed shape or “image” is labelled with the same letters with a “prime” symbol attached.
A translation "slides" an object a fixed distance in a given direction. The original object and its image have the: •Same shape •Same size •Same orientation •Same translation vector between them t(4, -3) t(x,y)  (x+4, y-3)
In this example, the slide" moves the figure 7 units to the left and 3 units down. Vector t (-7,-3) Mapping t t(x,y)  (x-7, y-3)
Let's examine some translations related to coordinate geometry. The example shows how each vertex moves the same distance in the same direction.
Translation Vector • What are the coordinates for: A, B, C? A’, B’,C’? Copy and fill out the chart. Original (x,y) Image (x,y) A A’ B B’ C C’
• What are the coordinates for • A, B, C, D? • A’, B’, C’, D’? • Book 1, p. 97 • Q. 1a, 2, a,e,i Original (x,y) Image (x,y) A A’ B B’ C C’ D D’
Rotations and Art The concept of rotations can be seen in wallpaper designs, fabrics, and art work.
ROTATION A rotation is a transformation that turns a figure about a fixed point. It is described in terms of: •The centre of rotation, •The direction of rotation: cw (-) or ccw (+) •The degrees of rotation An original shape and its rotation image have the: •Same size •Same shape, •Same orientation
Rotation Rules
Direction This rotation is 90° clockwise about the point B. Book 1, p. 97, Q. 1b, 2b,c,h Clockwise Counterclockwise A B B’ C C’ A’
REFLECTION A reflection is described by the line of reflection. An object and its reflection have the: •Same shape •Same size •Opposite orientation
The line (where a mirror may be placed) is called the line of reflection. The distance from a point to the line of reflection is the same as the distance from the point's image to the line of reflection. A reflection can be thought of as a "flipping" of an object over the line of reflection. If you folded the two shapes together line of reflection the two shapes would overlap exactly!
What happens to points in a Reflection? • Name the points of the original triangle. • Name the points of the reflected triangle. • What is the line of reflection? • How did the points change from the original to the reflection?
Reflection Rules Page, 97, Q. 1c, 2d, f, g
DILATATION A dilatation is a transformation that produces an image that has the •Same shape •Same orientation •Different size. •A centre of Dilatation •A scale factor A dilatation used to create an image larger than the original is called an enlargement. A dilatation used to create an image smaller than the original is called a reduction.
Dilatation
Dilations always involve a change in size. N.B. EVERY (x,y) coordinate of the original triangle has been multiplied by the scale factor (x2).
Dilatation Rules • Given an enlargement with a scale factor S, and the center of dilatation at the origin, • (x,y)  (Sx, Sy) • E.g. Centre at (0,0), Scale Factor 2 • (x,y)  (2x, 2y) • (3,4)  (2x3, 2x4) = (6,8) • Page 109, Q. 1a, c, f
Stretch Rules • A stretch is a dilatation in which only one of the coordinates is affected. • E.g. the x value is stretched by a factor of 2 • (x,y)  (2x, y) • (5,8)  (10, 8) • E.g. the y value is compressed by a factor of 4 • (x,y)  (x, y/4) • (5,8)  (5, 2) Page 111, Q. 5 a,b,c
Stretch
REVIEW: Answer each question……………………….. Does this picture show a translation, rotation, dilation, or reflection? How do you know? Rotation
Does this picture show a translation, rotation, dilation, or reflection? How do you know? Dilation
Does this picture show a translation, rotation, dilation, or reflection? How do you know? (Line) Reflection
Which of the following lettered figures are translations of the shape of the purple arrow? Name ALL that apply. Explain your thinking. Letters a, c, and e are translations of the purple arrow.
Has each picture been rotated in a clockwise or counter-clockwise direction? The birds were rotated clockwise and the fish counterclockwise.
Basically, a tessellation is a way to tile a floor (that goes on forever) with shapes so that there is no overlapping and no gaps.
Dutch graphic artist M. C. Escher (1898-1972) is known for his creative use of tessellations in his work. What transformations can you see in this picture? The birds and fish have been translated here.
What transformations can you see in this Escher print? Some birds have been translated and some have been rotated.
Can you name examples in real life of each transformation? •Translation •Rotation •Reflection •Dilation
Check out these sites: http://www.farraguttn.com/fhs/math/nctm/index.htm http://www.mathsnet.net/transformations/index.html http://www.mcescher.com/

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Transformations ppt

  • 2. To transform something is to change. The main transformations include: •Translation •Rotation •Reflection •Dilatation (Enlargement)
  • 3. Renaming Transformations • The original shape is labelled with capital letters: • The transformed shape or “image” is labelled with the same letters with a “prime” symbol attached.
  • 4. A translation "slides" an object a fixed distance in a given direction. The original object and its image have the: •Same shape •Same size •Same orientation •Same translation vector between them t(4, -3) t(x,y)  (x+4, y-3)
  • 5. In this example, the slide" moves the figure 7 units to the left and 3 units down. Vector t (-7,-3) Mapping t t(x,y)  (x-7, y-3)
  • 6. Let's examine some translations related to coordinate geometry. The example shows how each vertex moves the same distance in the same direction.
  • 7. Translation Vector • What are the coordinates for: A, B, C? A’, B’,C’? Copy and fill out the chart. Original (x,y) Image (x,y) A A’ B B’ C C’
  • 8. • What are the coordinates for • A, B, C, D? • A’, B’, C’, D’? • Book 1, p. 97 • Q. 1a, 2, a,e,i Original (x,y) Image (x,y) A A’ B B’ C C’ D D’
  • 9. Rotations and Art The concept of rotations can be seen in wallpaper designs, fabrics, and art work.
  • 10. ROTATION A rotation is a transformation that turns a figure about a fixed point. It is described in terms of: •The centre of rotation, •The direction of rotation: cw (-) or ccw (+) •The degrees of rotation An original shape and its rotation image have the: •Same size •Same shape, •Same orientation
  • 12. Direction This rotation is 90° clockwise about the point B. Book 1, p. 97, Q. 1b, 2b,c,h Clockwise Counterclockwise A B B’ C C’ A’
  • 13. REFLECTION A reflection is described by the line of reflection. An object and its reflection have the: •Same shape •Same size •Opposite orientation
  • 14. The line (where a mirror may be placed) is called the line of reflection. The distance from a point to the line of reflection is the same as the distance from the point's image to the line of reflection. A reflection can be thought of as a "flipping" of an object over the line of reflection. If you folded the two shapes together line of reflection the two shapes would overlap exactly!
  • 15. What happens to points in a Reflection? • Name the points of the original triangle. • Name the points of the reflected triangle. • What is the line of reflection? • How did the points change from the original to the reflection?
  • 16. Reflection Rules Page, 97, Q. 1c, 2d, f, g
  • 17. DILATATION A dilatation is a transformation that produces an image that has the •Same shape •Same orientation •Different size. •A centre of Dilatation •A scale factor A dilatation used to create an image larger than the original is called an enlargement. A dilatation used to create an image smaller than the original is called a reduction.
  • 19. Dilations always involve a change in size. N.B. EVERY (x,y) coordinate of the original triangle has been multiplied by the scale factor (x2).
  • 20. Dilatation Rules • Given an enlargement with a scale factor S, and the center of dilatation at the origin, • (x,y)  (Sx, Sy) • E.g. Centre at (0,0), Scale Factor 2 • (x,y)  (2x, 2y) • (3,4)  (2x3, 2x4) = (6,8) • Page 109, Q. 1a, c, f
  • 21. Stretch Rules • A stretch is a dilatation in which only one of the coordinates is affected. • E.g. the x value is stretched by a factor of 2 • (x,y)  (2x, y) • (5,8)  (10, 8) • E.g. the y value is compressed by a factor of 4 • (x,y)  (x, y/4) • (5,8)  (5, 2) Page 111, Q. 5 a,b,c
  • 23. REVIEW: Answer each question……………………….. Does this picture show a translation, rotation, dilation, or reflection? How do you know? Rotation
  • 24. Does this picture show a translation, rotation, dilation, or reflection? How do you know? Dilation
  • 25. Does this picture show a translation, rotation, dilation, or reflection? How do you know? (Line) Reflection
  • 26. Which of the following lettered figures are translations of the shape of the purple arrow? Name ALL that apply. Explain your thinking. Letters a, c, and e are translations of the purple arrow.
  • 27. Has each picture been rotated in a clockwise or counter-clockwise direction? The birds were rotated clockwise and the fish counterclockwise.
  • 28. Basically, a tessellation is a way to tile a floor (that goes on forever) with shapes so that there is no overlapping and no gaps.
  • 29. Dutch graphic artist M. C. Escher (1898-1972) is known for his creative use of tessellations in his work. What transformations can you see in this picture? The birds and fish have been translated here.
  • 30. What transformations can you see in this Escher print? Some birds have been translated and some have been rotated.
  • 31. Can you name examples in real life of each transformation? •Translation •Rotation •Reflection •Dilation
  • 32. Check out these sites: http://www.farraguttn.com/fhs/math/nctm/index.htm http://www.mathsnet.net/transformations/index.html http://www.mcescher.com/