1. Unit 1: Transformations, Congruence, and Similarity

2. Basic Types of Transformations:
Translations
Reflections
Rotations

3. Quadrant II
Quadrant I
(x,y)
x-axis
Quadrant III
Quadrant IV
y-axis

4. Object to Image (Before) (After)
Before Transformation:
After Transformation:
(‘ = PRIME) A A’ C’ B C B’

5. Translations…

6. A translation "slides" an object a fixed distance in a given direction
A translation "slides" an object a fixed distance in a given direction.  The original object and its translation have the same shape and size, and they face in the same direction. Objects that are translated are congruent. *The word "translate" in Latin means "carried across".

7. Example 1:
Translate the object down 2 and right 3 units.

8. Example 1 Solution:
Translate the object down 2 and right 3 units.

9. Example 2:
Translate the object (-3, 4)

10. Example 2 Solution:
Translate the object (-3, 4)

11. Translations are SLIDES!!!
Remember: Translations are SLIDING on a graph!!! The shape doesn’t change at all.
Translations are SLIDES!!!
                                                                                                             

12. Reflections…

13. A reflection “flips” an object and can be seen in water, in a mirror, in glass, or in a shiny surface.  An object and its reflection have the same shape and size, but the figures face in opposite directions.  In a mirror, for example, right and left are switched.

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.
The object ABCD is being reflected over the x-axis.

15. Example 3:
Reflect the object over the y-axis.

16. Example 3 Solution:
Reflect the object over the y-axis.

17. Example 4:
Reflect the object over x = 2.

18. Example 4 Solution:
Reflect the object over x = 2.

19. Rotations…

21. A rotation is a transformation that turns a figure about a fixed point called the center of rotation.  An object and its rotation are the same shape and size, but the figures may be turned in different directions.

22. Rotations Graphically…
Physically rotate the graph paper and use the original points – just graphed in the different quadrants.

23. Rules:
When students complete rotating the original figure clockwise 90˚, 180˚, and 270˚, you can have them come up with the rules on their own.
If you have time, you can have them use the same figure and rotate counter clockwise and come up with the rules of those too.

24. Rotation Rules: Clockwise Counter Clockwise 90˚ (y, -x) (-y, x) 180˚
270˚

25. Dilations

26. What is a Dilation? Dilated PowerPoint Slide
Dilation is a transformation that produces a figure similar to the original by proportionally shrinking or stretching the figure.
You are probably familiar with the term ‘dilate’ in reference to the eye doctor.
When doctors dilate your eyes, they are making the pupils larger for a better view of the back if the eye.
Dilated PowerPoint Slide

27. Proportionally
Let’s take a look…
And, of course, increasing the circle increases the diameter.
When a figure is dilated, it must be proportionally larger or smaller than the original.
So, we always have a circle with a certain diameter. We are just changing the size or scale.
Decreasing the size of the circle decreases the diameter.
We have a circle with a certain diameter.
Same shape, Different scale.

28. Which of these are dilations??
HINT: SAME SHAPE, DIFFERENT SIZE
Which of these are dilations?? C A B D
Discuss slides before clicking for the answer.
A is not a dilation because the milk containers are not the same shape.
B is a dilation because it is the same image, same size just a different scale or size.
C is a dilation because it is the same image, same size just a different scale or size.
D is not a dilation because the baby duck is not just a smaller replication as the mother duck since it is not the same size.

29. Scale Factor and Center of Dilation
When we describe dilations we use the terms scale factor and center of dilation.
Scale factor
Center of Dilation
Here we have Igor. He is 3 feet tall and the greatest width across his body is 2 feet.
He wishes he were 6 feet tall with a width of 4 feet.
Scale Factor - change in size of the object.
Center of Dilation – the very central point of the object.
His center of dilation would be where the length and greatest width of his body intersect.
He wishes he were larger by a scale factor of 2.

30. The Object and the Image
The original figure is called the object and the new figure is called the image.
The object is labeled with letters.
The image may be labeled with the same letters followed by the prime symbol.
Image
Object

31. Determining Scale Factor:

32. Scale factor > 1 0 < Scale Factor < 1 Scale Factor
If the scale factor is larger than 1, the figure is enlarged.
If the scale factor is between 1 and 0, the figure is reduced in size.
Scale factor > 1
The length of each side of the new image is equivalent to the length of the corresponding side of the original figure multiplied by the scale factor.
0 < Scale Factor < 1

33. Reduction or Enlargement
Ratio
Fraction
Decimal
Percentage
Reduction or Enlargement
 1:2
1/2
 .5
50%
Reduction
3/4
0.9
400%
2:5
1/8

34. Are the following enlarged or reduced??
Scale factor of 1.5
Discuss slides before clicking for the answer.
A is enlarged.
B is reduced.
C is enlarged.
D is reduced. D
Scale factor of 3 B
Scale factor of 0.75
Scale factor of 1/5

35. Dilations Used Everyday
World’s Largest Arcade Game – enlarged in size for fun.
Toy cars - reduced in size for play.
Large Chair – enlarged to capture attention as a furniture store sign.
Maps – reduced in sized for practicality or use.
Models – reduced in size for the sight of the overview of the building for modeling.
Large Burger – enlarged in size for fun.

36. Practice Dilation Quiz
Remember
Dilations are enlargements or reductions.
What are some things that you would not
mind dilating to make larger or smaller?
Practice Dilation Quiz
Answer: Larger: maybe a house, one’s favorite food, a bed, a television.
Smaller: maybe favorite animal like an elephant, a car to fit into pocket, items needed to be put in storage.

37. Dilation
A transformation that changes the size of an object, but not the shape.
A Dilation will be a similar figure, but not a congruent figure.
Example:

38. Dilate the object by a scale factor of ½
(2,2)
(-2,2)
(2,-2)
(-2,-2)

39. (-6,6) (6,6) (2,2) (-2,2) (2,-2) (-2,-2) (6,-6) (-6,-6)
Dilate the object by a scale factor of 3
(6,6)
(2,2)
(-2,2)
(2,-2)
(-2,-2)
(6,-6)
(-6,-6)

40. A spider has taken up residence in a small cardboard box which measures 2 inches by 4 inches by 4 inches. What is the length, in inches, of a straight spider web that will carry the spider from the lower right front corner of the box to the upper left back corner of the box?
A in.
B in.
C. 5 in.
D. 6 in.