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1. New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org

2. 8th Grade Math 2D Geometry: Transformations 2013-06-25 www.njctl.org

3. Setting the PowerPoint View • Use Normal View for the Interactive Elements • To use the interactive elements in this presentation, do not select the Slide Show view. Instead, select Normal view and follow these steps to set the view as large as possible: • On the View menu, select Normal. • Close the Slides tab on the left. • In the upper right corner next to the Help button, click the ^ to minimize the ribbon at the top of the screen.  • On the View menu, confirm that Ruler is deselected. • On the View tab, click Fit to Window. • Use Slide Show View to Administer Assessment Items • To administer the numbered assessment items in this presentation, use the Slide Show view. (See Slide 21 for an example.)

4. Table of Contents Click on a topic to go to that section • Transformations • Translations • Rotations • Reflections • Dilations • Symmetry • Congruence & Similarity • Special Pairs of Angles Common Core Standards: 8.G.1, 8.G.2, 8.G.3, 8.G.4, 8.G.5

5. Transformations Return to Table of Contents

6. Any time you move, shrink, or enlarge a figure you make a transformation. If the figure you are moving (pre-image) is labeled with letters A, B, and C, you can label the points on the transformed image (image) with the same letters and the prime sign. B B' A' A pre-image image C C'

8. There are four types of transformations in this unit: • Translations • Rotations • Reflections • Dilations • The first three transformations preserve the size and shape of the figure. • In other words: • If your pre-image is a trapezoid, your image is a congruent trapezoid. • If your pre-image is an angle, your image is an angle with the same measure. • If your pre-image contains parallel lines, your image contains parallel lines.

9. Translations Return to Table of Contents

10. A translation is a slide that moves a figure to a different position (left, right, up or down) without changing its size or shape and without flipping or turning it. You can use a slide arrow to show the direction and distance of the movement.

11. This shows a translation of pre-image ABC to image A'B'C'. Each point in the pre-image was moved right 7 and up 4.

12. Click for web page

13. To complete a translation, move each point of the pre-image and label the new point. Example: Move the figure left 2 units and up 5 units. What are the coordinates of the pre-image and image? A' B' C' D' A B C D Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent.

14. Translate pre-image ABC 2 left and 6 down. What are the coordinates of the image and pre-image? A C B Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent.

15. Translate pre-image ABCD 4 right and 1 down. What are the coordinates of the image and pre-image? A B D C Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent.

16. Translate pre-image ABCD 5 left and 3 up. What is the rule and what are the new coordinates of the image. A B C D Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent.

17. A rule can be written to describe translations on the coordinate plane. Look at the following rules and coordinates to see if you can find a pattern. 2 Left and 5 Up A (3,-1) A' (1,4) B (8,-1) B' (6,4) C (7,-3) C' (5,2) D (2, -4) D' (0,1) 2 Left and 6 Down A (-2,7) A' (-4,1) B (-3,1) B' (-5,-5) C (-6,3) C' (-8,-3) 5 Left and 3 Up A (3,2) A' (-2,5) B (7,1) B' (2,4) C (4,0) C' (-1,3) D (2,-2) D' (-3,1) 4 Right and 1 Down A (-5,4) A' (-1,3) B (-1,2) B' (3,1) C (-4,-2) C' (0,-3) D (-6, 1) D' (-2,0)

18. Translating left/right changes the x-coordinate. Translating up/down changes the y-coordinate. 2 Left and 5 Up A (3,-1) A' (1,4) B (8,-1) B' (6,4) C (7,-3) C' (5,2) D (2, -4) D' (0,1) 2 Left and 6 Down A (-2,7) A' (-4,1) B (-3,1) B' (-5,-5) C (-6,3) C' (-8,-3) 5 Left and 3 Up A (3,2) A' (-2,5) B (7,1) B' (2,4) C (4,0) C' (-1,3) D (2,-2) D' (-3,1) 4 Right and 1 Down A (-5,4) A' (-1,3) B (-1,2) B' (3,1) C (-4,-2) C' (0,-3) D (-6, 1) D' (-2,0)

19. Translating left/right changes the x-coordinate. • Left subtracts from the x-coordinate • Right adds to the x-coordinate • Translating up/down changes the y-coordinate. • Down subtracts from the y-coordinate • Up adds to the y-coordinate

20. A rule can be written to describe translations on the coordinate plane. click to reveal 2 units Left … x-coordinate - 2 y-coordinate stays rule = (x - 2, y) 5 units Right & 3 units Down… x-coordinate + 5 y-coordinate - 3 rule = (x + 5, y - 3) click to reveal

21. Write a rule for each translation. 2 Left and 5 Up A (3,-1) A' (1,4) B (8,-1) B' (6,4) C (7,-3) C' (5,2) D (2, -4) D' (0,1) 2 Left and 6 Down A (-2,7) A' (-4,1) B (-3,1) B' (-5,-5) C (-6,3) C' (-8,-3) click to reveal (x, y) (x-2, y-6) (x, y) (x-2, y+5) click to reveal 5 Left and 3 Up A (3,2) A' (-2,5) B (7,1) B' (2,4) C (4,0) C' (-1,3) D (2,-2) D' (-3,1) 4 Right and 1 Down A (-5,4) A' (-1,3) B (-1,2) B' (3,1) C (-4,-2) C' (0,-3) D (-6, 1) D' (-2,0) click to reveal (x, y) (x-5, y+3) (x, y) (x+4, y-1) click to reveal

22. 1 What rule describes the translation shown? E' (x,y) (x - 4, y - 6) A D' F' (x,y) (x - 6, y - 4) B E D (x,y) (x + 6, y + 4) C F G' D (x,y) (x + 4, y + 6) G

23. 2 What rule describes the translation shown? (x,y) (x, y - 9) A E (x,y) (x, y - 3) B D (x,y) (x - 9, y) C F D (x,y) (x - 3, y) G E' D' F' G'

24. 3 What rule describes the translation shown? (x,y) (x + 8, y - 5) A E' (x,y) (x - 5, y - 1) B D' (x,y) (x + 5, y - 8) C F' D (x,y) (x - 8, y + 5) E D G' F G

25. 4 What rule describes the translation shown? (x,y) (x - 3, y + 2) A E (x,y) (x + 3, y - 2) B D F (x,y) (x + 2, y - 3) C E' D' D (x,y) (x - 2, y + 3) F' G G'

26. 5 What rule describes the translation shown? (x,y) (x - 3, y + 2) A E' (x,y) (x + 3, y - 2) B D' F' E (x,y) (x + 2, y - 3) C F D D (x,y) (x - 2, y + 3) G' G

27. Rotations Return to Table of Contents

28. A rotation (turn) moves a figure around a point. This point can be on the figure or it can be some other point. This point is called the point of rotation. P

29. Rotation The person's finger is the point of rotation for each figure.

30. When you rotate a figure, you can describe the rotation by giving the direction (clockwise or counterclockwise) and the angle that the figure is rotated around the point of rotation. Rotations are counterclockwise unless you are told otherwise. A B Click for answer Click for answer This figure is rotated 90 degrees counterclockwise about point A. This figure is rotated 180 degrees clockwise about point B.

31. How is this figure rotated about the origin? In a coordinate plane, each quadrant represents 90. B A D C B' C' A' D' Click to Reveal This figure is rotated 270 degrees clockwise about the origin or 90 degrees counterclockwise about the origin. Check to see if the pre-image and image are congruent.

32. The following descriptions describe the same rotation. What do you notice? Can you give your own example?

33. The sum of the two rotations (clockwise and counterclockwise) is 360 degrees. If you have one rotation, you can calculate the other by subtracting from 360.

34. 6 How is this figure rotated about point A? (Choose more than one answer.) C D B clockwise A B' counterclockwise A, A' B E C' 90 degrees C E' D 180 degrees D' E 270 degrees Check to see if the pre-image and image are congruent.

35. 7 How is this figure rotated about point the origin? (Choose more than one answer.) A B D C clockwise A D' C' counterclockwise B 90 degrees C A' B' D 180 degrees E 270 degrees Check to see if the pre-image and image are congruent.

36. 8 What is the turn that will turn this hexagon onto itself? A B C D

37. A B Now let's look at the same figure and see what happens to the coordinates when we rotate a figure. Write the coordinates for the pre-image and image. What do you notice? D C B' C' A' D' When rotated 90 counter-clockwise, the x-coordinate is the opposite of the pre-image y-coordinate and the y-coordinate is the same as the pre-image of the x-coordinate. In other words: (x, y) (-y, x) Click to Reveal

38. What happens to the coordinates in a half-turn? Write the coordinates for the pre-image and image. What do you notice? A B D C C' D' A' B' When rotated a half-turn, the x-coordinate is the opposite of the pre-image x-coordinate and the y-coordinate is the opposite of the pre-image of the y-coordinate. In other words: (x, y) (-x, -y) Click to Reveal

39. Can you summarize what happens to the coordinates during a rotation? 90 Counterclockwise: Half-turn: 90 Clockwise: (x, y) (-y, x) Click to Reveal (x, y) (-x, -y) Click to Reveal (x, y) (y, -x) Click to Reveal

40. 9 What are the new coordinates of a point A (5, -6) after a 90 rotation clockwise? (-6, -5) A (-6, 5) B (-5, 6) C D (5, -6)

41. 10 What are the new coordinates of a point S (-8, -1) after a 90 rotation counterclockwise? (-1, -8) A (1, -8) B (-1, 8) C D (8, 1)

42. What are the new coordinates of a point H (-5, 4) after a 180 rotation counterclockwise? 11 (-5, -4) A (5, -4) B (4, -5) C D (-4, 5)

43. What are the new coordinates of a point R (-4, -2) after a 270 rotation clockwise? 12 (4, -2) A (-2, 4) B (2, 4) C D (-4, 2)

44. What are the new coordinates of a point Y (9, -12) after a Half-turn? 13 (-12, 9) A (-9,12) B (-12, -9) C D (9,12)

45. Reflections Return to Table of Contents

46. Example

47. A reflection (flip) creates a mirror image of a figure.

48. A reflection is a flip because the figure is flipped over a line. Each point in the image is the same distance from the line as the original point. t A and A' are both 6 units from line t. B and B' are both 6 units from line t. C and C' are both 3 units from line t. Each vertex in ABC is the same distance from line t as the vertices in A'B'C'. A' A C B B' C' Check to see if the pre-image and image are congruent.

49. Reflect the figure across the y-axis. Check to see if the pre-image and image are congruent.

50. What do you notice about the coordinates when you reflect across the y-axis? Tap box for coordinates y A (-6, 5) A' (6, 5) B (-4, 5) B' (4, 5) C (-4, 1) C' (4, 1) D (-6, 3) D' (6, 3) When you reflect across the y-axis, the x-coordinate becomes the opposite. So (x, y) (-x, y) when you reflect across the y-axis. A B A' B' D' D C' C x Check to see if the pre-image and image are congruent.