CCM6+/7+ - Unit 13 - Page 1 UNIT 13. Transformations CCM6+/7+ Name: Math Teacher: Projected Test Date:

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1 CCM6+/7+ - Unit 13 - Page 1 UNIT 13 Transformations CCM6+/7+ Name: Math Teacher: Projected Test Date: Main Idea Pages Unit 9 Vocabulary 2 Translations 3 10 Rotations Reflections Transformations Notes & Rules 23 Dilations Composition of Transformations Unit 9 Study Guide Page 1

2 CCM6+/7+ - Unit 13 - Page 2 Common Core Math 7 Plus Unit 9 Vocabulary Definitions of Critical Vocabulary and Underlying Concepts coordinate plane the plane formed by two lines intersecting at their zero points-the horizontal line is the "x-axis" and the vertical line is the "y-axis transformation when a figure or point is changed in size and position on a coordinate plane rigid transformation when a figure changes position on the coordinate plane but maintains the same size and shape translation a transformation that moves points right-left or up-down or a combination of these (slide) reflection a transformation that flips a figure over a given line of reflection-each point moves an equal distance from the line of reflection but on the opposite side rotation a transformation where a given figure rotates around a given point dilation a transformation which enlarges or reduces a figure using a given scale factor scale factor the factor by which you multiply original numbers to increase or decrease size prime coordinates the coordinates that come from applying a scale factor of original points coordinates the pair of numbers used to describe points on a coordinate plane composition of A composition of two transformations is a transformation in which a transformation second transformation is performed on the image of a first transformation glide reflection A composition of a translation and a reflection in a line parallel to the direction of the translation Page 2

3 CCM6+/7+ - Unit 13 - Page 3 TRANSLATIONS When working with any TRANSFORMATIONS the original points create the PRE-IMAGE. You can name the points using letters. For example A(4, 5) tells you that point A is located at position 4, 5 on the graph. Once the point is moved to its new position it is called a prime point and named like this: A - read this as A prime the figure is now called the IMAGE TRANSLATIONS involve moves that are either right-left, up-down, or a combination of these Create a pre-image by graphing and labeling the following points: A(-3, 2), B(-3, 6), C(-7, 2) Now take each point and move it 8 units right and then label the new points as primes. You have modeled a TRANSLATION. Name the new prime coordinates below: A (, ), B (, ), C (, ) Did the shape or size of the figure change? Look at the new x numbers. What do you notice happened to the x part of each ordered pair? Why do you think it was the x affected and not the y? What type of move do you think would affect the y? Page 3

4 CCM6+/7+ - Unit 13 - Page 4 Look at the graphed image below. Write in the coordinates for the pre-image and the image. Make sure that you label the coordinates that you list. A B Pre-Image Coordinates Image Coordinates C D A B C D How did you determine which was the pre-image and which is the image? What happened to the y part of the coordinates? Why? If you are moving UP or DOWN the part of your ordered pair will change. If you go up you will the number of units to the original y. If you go down you will the number of units from the original y. If you are moving RIGHT or LEFT the part of your ordered pair will change. If you go right you will the number of units to the original x. If you go left you will the number of units to the original x. Page 4

5 CCM6+/7+ - Unit 13 - Page 5 Graph and label the following points and then translate 3 units left and 2 units up. Label and list your new prime points. M(5, 8) A(0, 6) P(-3, -2) M (, ) A (, ) P (, ) Now describe what happened to each part of the ordered pairs: (x, y ) Describe the translation that you see below. 2 1 Can you give the new prime points without creating the graphs for these two translations. Example 1: Translate 2 units left and 4 units down A(5, -2) (5, -2 ) A (, ) M(2, 6) (2, 6 ) M (, ) B(0, -3) (0,-3 ) B (, ) Example 2: Translate 5 units right and 3 units down C(3, 5) (3, -2 ) C (, ) W(-2,5) (-2, 6 ) W (, ) T(1, -7) (1,-3 ) T (, ) Page 5

6 CCM6+/7+ - Unit 13 - Page 6 Translation Practice Plot the coordinates given for each pre-image. Translate the figure as instructed. Draw the image on the coordinate plane. List the coordinates of the new image. Describe what happened to both the x and y 1. Slide the triangle 5 units left and 3 units up A (3, 7) A (, ) B (3, 2) B (, ) C (7, 2) C (, ) (x, y ) Page 6

7 CCM6+/7+ - Unit 13 - Page 7 2. Translate the parallelogram 10 units left and 2 units down. H (3, -2) H (, ) I (1, -5) I (, ) J (6, -5) J (, ) (x, y ) K (8, -2) K (, ) Page 7

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11 CCM6+/7+ - Unit 13 - Page 11 ROTATIONS You will be exploring a TRANSFORMATION called a ROTATION. A ROTATION is a movement of a figure that involves rotating in 90 degree increments around the origin. The new prime points will be in the quadrant that is the given number of degrees clockwise or counterclockwise from the original figure. The following activities will help you discover what happens when a point, line, or figure is rotated a given number of degrees. In a rotation, the original shape does not change in size or shape but does move to a new position on the coordinate plane. You will need to remember the names of the quadrants: II I III IV Page 11

12 CCM6+/7+ - Unit 13 - Page 12 EXAMPLE #1: STEP 1: The following is a 90 0 clockwise rotation: STEP 2: List the pre-image points and the image points below. B A(, ) A (, ) A C B(, ) B (, ) C(, ) C (, ) A B In what quadrant is the Pre-Image? ; Image? Compare each set of PRE-IMAGE coordinates with the IMAGE coordinates. What is the relationship between the pre-image coordinates and the image coordinates? EXAMPLE #2: STEP 1: The following is a 90 0 clockwise rotation: C STEP 2: List the pre-image points and the image points below. A A(, ) B(, ) A (, ) B (, ) B C C C(, ) C (, ) In what quadrant is the Pre-Image? ; Image? A B Compare each set of PRE-IMAGE coordinates with the IMAGE coordinates. What is the relationship between the pre-image coordinates and the image coordinates? Using the two examples above, describe what happens to the coordinates in a 90 0 clockwise rotation? Using the rule you have discovered, find the prime coordinates for a line with pre-image points at (2, -6) (, ) (7, -1) (, ) Page 12

13 CCM6+/7+ - Unit 13 - Page 13 EXAMPLE #3: STEP 1: The following is a 90 0 counter-clockwise rotation: STEP 2: List the pre-image points and the image points below. C B A(, ) B(, ) A (, ) B (, ) B A A C C(, ) C (, ) In what quadrant is the Pre-Image? ; Image? Compare each set of PRE-IMAGE coordinates with the IMAGE coordinates. What is the relationship between the pre-image coordinates and the image coordinates? EXAMPLE #4: STEP 1: The following is a 90 0 counter-clockwise rotation: STEP 2: List the pre-image points and the image points below. A(, ) B(, ) C(, ) A (, ) B (, ) C (, ) C C B In what quadrant is the Pre-Image? ; Image? A B A Compare each set of PRE-IMAGE coordinates with the IMAGE coordinates. What is the relationship between the pre-image coordinates and the image coordinates? Using the two examples above, describe what happens to the coordinates in a 90 0 counter-clockwise rotation? Using the rule you have discovered, find the prime coordinates for a line with pre-image points at (2, -6) (, ) (7, -1) (, ) Page 13

14 CCM6+/7+ - Unit 13 - Page 14 EXAMPLE #5: STEP 1: The following is a rotation: STEP 2: List the pre-image points and the image points below. B A(, ) A (, ) A C B(, ) C(, ) B (, ) C (, ) C A In what quadrant is the Pre-Image? ; Image? B Compare each set of PRE-IMAGE coordinates with the IMAGE coordinates. What is the relationship between the pre-image coordinates and the image coordinates? EXAMPLE #6: STEP 1: The following is a clockwise rotation: STEP 2: List the pre-image points and the image points below. B A A(, ) B(, ) A (, ) B (, ) C C C(, ) C (, ) In what quadrant is the Pre-Image? ; Image? A B Compare each set of PRE-IMAGE coordinates with the IMAGE coordinates. What is the relationship between the pre-image coordinates and the image coordinates? Using the two examples above, describe what happens to the coordinates in a rotation? Using the rule you have discovered, find the prime coordinates for a line with pre-image points at (2, -6) (, ) (7, -1) (, ) Page 14

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17 CCM6+/7+ - Unit 13 - Page 17 Translations and Rotations Practice Translations: 1. Translate ΔQRS (x+4, y-2). List the new coordinates below. Q 2. Translate ΔQRS (x-3, y+1). List the new coordinates below. S R 3. ΔABC has vertices at A (-1, -1), B (4,-1), C(1, 3). Where are the new vertices located after a move 3 units up and 5 units left? 4. Create your own translation. List the preimage coordinates, the translation movements, and the image coordinates. Rotations: 1. Translate ΔQRS 180 degrees. List the new coordinates below. 2. Translate ΔQRS 90 degrees counterclockwise. List the new coordinates below. Q 3. ΔABC has vertices at A (-1, -1), B (-4,-1), C(-2, -3). Where are the new vertices located after a 90 degree clockwise rotation? S R 4. Create your own rotation. List the preimage coordinates, the direction and degrees of rotation, and the image coordinates. Page 17

18 CCM6+/7+ - Unit 13 - Page 18 REFLECTIONS You will need: a straight edge, pencil, and several pieces of patty paper You will be exploring a TRANSFORMATION called a REFLECTION. When working with any TRANSFORMATIONS the original points create the PRE-IMAGE. You can name the points using letters. For example A(4, 5) tells you that point A is located at position 4, 5 on the graph. Once the point is moved to its new position it is called a prime point and named like this: A - read this as A prime the figure is now called the IMAGE REFLECTIONS involve moves that flip over a given line. The following activities will help you discover what happens when a point, line, or figure is reflected over a given line of reflection. This can also be called the LINE OF SYMMETRY. In a reflection, the original shape does not change in size or shape but does move to a new position on the coordinate plane. Page 18

19 CCM6+/7+ - Unit 13 - Page 19 EXPLORE: STEP 1: Create a pre-image by graphing, labeling, and connecting the following points: A(-3, 2), B(-3, 6), C(-7, 2) STEP 2: Using a piece of patty paper, and a straight edge, trace the original figure. STEP 3: FLIP the piece of patty paper over, lining up AB an equal distance from the y-axis but on the opposite side of the y-axis. STEP 4: Record the new coordinates below after the flip and then add this figure to the graph. A(-3, 2) B(-3, 6) C(-7, 2) A (, ) B (, ) C (, ) STEP 1: Create a pre-image by graphing, labeling, and connecting the following points: R(3, 2), T(3, 6), N(8, 1), Q(8, 8) STEP 2: Using a piece of patty paper, and a straight edge, trace the original figure. STEP 3: FLIP the piece of patty paper over, lining up RT an equal distance from the y-axis but on the opposite side of the y-axis. STEP 4: Record the new coordinates below after the flip and then add this figure to the graph. Compare each set of PRE-IMAGE coordinates with the IMAGE coordinates after the move. What do you notice? R(3, 2) R (, ) T(3, 6) N(8, 1) T (, ) N (, ) Compare each set of PRE-IMAGE coordinates with the IMAGE coordinates after the move. What do you notice? Q(8, 8) Q (, ) DISCOVERY: When a figure is reflected over the y-axis, the y part of each coordinate and the x part of each coordinate. Page 19

20 CCM6+/7+ - Unit 13 - Page 20 STEP 1: Create a pre-image by graphing, labeling, and connecting the following points: A(-3, 2), B(-3, 6), C(-7, 2) STEP 2: Using a piece of patty paper, and a straight edge, trace the original figure. STEP 3: FLIP the piece of patty paper over, lining up AB an equal distance from the x-axis but on the opposite side of the x-axis. STEP 4: Record the new coordinates below after the flip and then add this figure to the graph. A(-3, 2) B(-3, 6) C(-7, 2) A (, ) B (, ) C (, ) STEP 1: Create a pre-image by graphing, labeling, and connecting the following points: R(3, 2), T(3, 6), N(8, 1), Q(8, 8) STEP 2: Using a piece of patty paper, and a straight edge, trace the original figure. STEP 3: FLIP the piece of patty paper over, lining up RT an equal distance from the x-axis but on the opposite side of the x-axis. STEP 4: Record the new coordinates below after the flip and then add this figure to the graph. Compare each set of PRE-IMAGE coordinates with the IMAGE coordinates after the move. What do you notice? R(3, 2) R (, ) T(3, 6) N(8, 1) T (, ) N (, ) Compare each set of PRE-IMAGE coordinates with the IMAGE coordinates after the move. What do you notice? Q(8, 8) Q (, ) DISCOVERY: When a figure is reflected over the x-axis, the x part of each coordinate and the y part of each coordinate. Page 20

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23 CCM6+/7+ - Unit 13 - Page 23 Transformations Notes & Rules Translations Reflection Rotations If sliding down: Subtract from y (y decreases in value) If sliding up: Add to y (y increases in value) If sliding left: Subtract from x (x decreases in value) If sliding right: Add to x (x increases in value) Slide Flip Turn Over x axis: Change all y coordinates to their opposites ( x, oy ) Over y axis: Change all x coordinates to their opposites ( ox, y ) 90 degrees clockwise: (270 degrees counterclockwise) Flip x and y coordinates Change x coordinate to its opposite ( y, ox ) 180 degrees clockwise: (180 degrees counterclockwise): Change x and y coordinate to their opposites ( ox, oy ) 270 degrees clockwise: (90 degrees counterclockwise): Flip x and y coordinates Change y coordinates to its opposite ( oy, x ) 360 degrees/0 degrees clockwise: (360 degrees/0 degrees counterclockwise) Original position ( x, y ) Page 23

24 CCM6+/7+ - Unit 13 - Page 24 DILATIONS Dilate: Scale Factor A scale factor greater than one means A scale factor less than one means A dilated image will always be to its original image. TRY THIS: Looking at the coordinates below, can you identify what scale factor was used? EXAMPLE 1: EXAMPLE 2: A (5, 3), B(2, 5) X(-2, 4), Y(2, 8) A (10, 6), B (4, 10) X (-1, 2), Y (1, 4) SCALE FACTOR: SCALE FACTOR: Now use the attached graph page to graph the lines formed by both the original points and the prime points in examples 1 and 2 to see what happens. Can you guess which would be larger? Which will be smaller? Now use a straight edge to create a line that goes through A and A and another line that goes through B and B in your graph for example 1. Extend your line through the whole graph shown. Do the lines go through the origin? Repeat the process for example 2. Page 24

25 CCM6+/7+ - Unit 13 - Page 25 You have just dilated your first figures in a coordinate plane. Now try the ones listed below and watch what happens. EXAMPLE 3: Plot the following points for a triangle and then draw the triangle. Next apply a scale factor of 3. Write the new prime points below and then plot them on the provided graph. R(1, 1) S(1, 4) T(4, 1) EXAMPLE 4: R (, ) S (, ) T (, ) What happens if you go to the graphed figures and draw lines extended from each given point and its prime? How could you tell that the graphed triangle would get larger or smaller? Plot the following points and draw the figure. Next apply a scale factor of 1/3. Write the new prime points below and then plot and draw them on the provided graph. M(-3, -3) N(0, -9) Q(-9, -3) D(-12, -9) M (, ) N (, ) Q (, ) D (, ) What happens if you go to the graphed figures and draw lines extended from each given point and its prime? How could you tell that the graphed triangle would get larger or smaller? EXAMPLE 5: What do you think may happen if the scale factor is negative? Try it with this figure using a scale factor of -2: D(-2, 1) A(-5, 1) B(-2, 6) D (, ) A (, ) B (, ) What happens with this figure if you go to the graphed figures and draw lines extended from each given point and its prime? Does it still go through the origin? Describe what happened to the figure: EXAMPLE 6: You create your own figure, apply a scale factor of your choice, and draw the prime figure. You will trade with another student to see if they can determine your scale factor. Mark blanks on your sheet for them to fill out points, prime points, and scale factor. Page 25

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27 CCM6+/7+ - Unit 13 - Page 27 Dilations Practice 1. Given the following points for an image, list the prime points after a dilation using a scale factor of 3: W(1, 1), S(2, 1), T(1, 2) G(2, 2) W (, ), S (, ), T (, ), G (, ) 2. For a given dilation, the point (5, 0) has a prime point of (35, 0). What are the coordinates of the prime point using the same dilation for (10, 2)? (, ) 3. What are the coordinates for the prime point given an original point of (4, 6) after a scale factor of -6 is applied? (, ) 4. For the following, identify the scale factor that was used. D(7, -3) D (14, -6) E(-2, -5) E (-4, -10) SCALE FACTOR: M(8, 2) M (16, 4) 5. Graph the following image then apply a scale factor of ¼. Draw the new image on the same graph and list the prime points. E(4, -2) X(4, -8) A(12, -4) L(12, -8) E (, ) X (, ) A (, ) L (, ) 6. How do you know if an image will get larger or smaller based on the scale factor? Page 27

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29 CCM6+/7+ - Unit 13 - Page 29 Determine if the following scale factor would create an enlargement or a reduction / / /8 Given the point and its image, determine the scale factor. 17. A(3,6) A (4.5, 9) 18. G (3,6) G(1.5,3) 19. B(2,5) B (1,2.5) Page 29

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35 CCM6+/7+ - Unit 13 - Page 35 Composition of Transformations Composition of Transformation: A composition of two transformations is a transformation in which a second transformation is performed on the image of a first transformation Glide reflection: a composition of a translation and a reflection in a line parallel to the direction of the translation Guided Examples: Given DEF with D (3, 1), E (-3, 2), and F (-2, -2). Find the image points after: a. A reflection over the x-axis, then a dilation of 3 1 Complete one transformation at time IN ORDER. b. A translation of (x, y) (x - 5, y + 2), then c. A reflection over the y-axis, then a a rotation of 90 counter clockwise translation of (x, y) (x + 1, y 4) d. Triangle DEF has vertices D (3, -4), E (2, -2), and F (0, 1). Find the coordinates after a glide reflection composed of the translation (x, y) (x, y - 2) and a reflection in the y-axis. Page 35

36 CCM6+/7+ - Unit 13 - Page 36 You Try Given DEF with D (-5, 7), E (-3, 2), and F (-4, 8). Find the image points after: e. A rotation of 180 counter clockwise, then a dilation of 2. f. A reflection over the x-axis, then a rotation of 270 counter clockwise g. Triangle ABC has vertices A (3, 2), B (-1, -3), and C (2, -1). Find the coordinates after a glide reflection of (x, y) (x + 3, y) and a reflection over the y-axis. Page 36

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39 TRANSFORMATIONS practice Show me that you can: Translate a figure. Reflect a figure. Rotate a figure. The original figure: A(3, -1) B(5, -1) C(4, -3) Translate it 3 left and 4 up. A (, ) B (, ) C (, ) Reflect it across the x-axis. A (, ) B (, ) C (, ) Reflect it across the y-axis. A (, ) B (, ) C (, ) CCM6+/7+ - Unit 13 - Page 39 Now, let s start with a new shape: D(1, 1), E(2, 2), F(3, 2), G(4, 1) Rotate it 90 clockwise: D (, ), E (, ), F (, ), G (, ) Rotate it 180 either direction: D (, ), E (, ), F (, ), G (, ) Rotate it 90 counterclockwise: D (, ), E (, ), F (, ), G (, ) Page 39

40 CCM6+/7+ - Unit 13 - Page 40 Unit 13 Study Guide I. Matching: Match the terms in the left column with the correct definitions or examples in the right column. 1. Reflection a. (x, y) 2. Translation b. where the x and y axes intersect (0,0) 3. Rotation c. a turn that moves 1 quadrant 4. X axis d. the same direction as a clock 5. Y axis e. moving a figure by flipping it in a coordinate grid 6. Origin f. the vertical axis (up and down) 7. Coordinate plane g. a numbered grid with x and y axes degree rotation h. moving a figure by sliding it in a coordinate grid 9. Clockwise i. the horizontal axis (across) 10. Ordered Pair j. moving a figure by turning it in a coordinate grid Page 40

41 CCM6+/7+ - Unit 13 - Page 41 II. Application: On the coordinate grids provided, transform the figures as directed. Use prime notation to label each point on the coordinate grid. Write the ordered pairs for the coordinates of the new image below for each problem. Plane 1 - Translate triangle ABC x-4, y+1. A B C Plane 2 - Reflect trapezoid DEFG over the x axis. D E F G Page 41

42 CCM6+/7+ - Unit 13 - Page 42 Plane 3 - Rotate parallelogram HIJK over the 180 degrees. H I J K Plane 4 - Dilate square LMNO by a scale factor of 2. L M N O Page 42

43 CCM6+/7+ - Unit 13 - Page 43 Plane 5 - Rotate rectangle PQRS 90 degrees clockwise about the origin. P Q R S Plane 6 - Dilate square TUVW 180 by a scale factor of ½. T U V W Page 43

44 CCM6+/7+ - Unit 13 - Page 44 Plane 7 - Plot triangle XYZ on the coordinate grid using the following coordinates: X (-4, 4) Y (-4, -2) Z (-1, -2) Reflect the figure over the y-axis, then translate x-2, y+1. Plane 8 The pre-image and image have been graphed. Explain the transformations that were applied to get to the image. Page 44