Foundations of Math II Unit 2: Transformations in the Coordinate Plane

Transcription

1 Foundations of Math II Unit 2: Transformations in the Coordinate Plane Academics High School Mathematics

2 2.1 Warm Up 1. Draw the image of stick-man m when translated using arrow p. What motion will take stick-man m back onto man m? 2. Which of the figures shown is the image of figure a, if figure a were rotated using center X? Explain why or why not for each figure. Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 1

3 Transformations: Examples and Counterexamples The images are indicated using gray. 1. Reflection Examples Counterexamples 2. Rotation Examples Counterexamples before after 3. Translation Examples Counterexamples 2

4 2.1 Show What You Know! For each set of points, apply the rule and graph the pre-image and the image. Describe what transformation occurred. 1. A(2, -2), B(4, 5), C(-5, 1) 3. B(-3, 2), R(0, 4), A(3, 0), D(1, -3) (x, y) (x + 2, y 4) (x, y) (x, -y) 2. M(-6, -2), A(4, 6), T(-4, 4) (x, y) 4. H(4, 1), O(-1, 2), T(-3, -3) (x, y) (y, x) 3

5 2.2 Warm Up 1. Carrie is watching a display of the solar system. She notices that the figures rotate around center T as shown by the given arrow r. Draw the following using a single drawing: a. Draw the image of figure v when rotated around center T using arrow r. Label your image. b. Draw the image of figure s when rotated around center T using arrow r. Label your image. c. Draw the image of figure m when rotated around center T using arrow r. Label your image. d. Which figure rotated most? Explain your answer. 2. Quadrilateral M A R T is the image of quadrilateral MART when rotated using center A and arrow n. Draw quadrilateral MART. Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 4

6 2.2 Worksheet 1 5

7 2.2 Worksheet 2 6

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12 2.3 Worksheet 1 11

13 2.3 Worksheet 2 12

14 2.3 Worksheet 3 13

15 2.3 Worksheet 4 Graph the points to form a figure. Reflect each figure over the x-axis. Draw the image in a different color. Then write the coordinates of the image points. What pattern do you notice? x y x y x y x y x y x y Write the algebraic rule for a reflection over the x-axis: 14

16 Graph the points to form a figure. Reflect each figure over the y-axis. Draw the image in a different color. Then write the coordinates of the image points. What pattern do you notice? x y x y x y x y x y x y Write the algebraic rule for a reflection over the y-axis: 15

17 2.3 Show What You Know! Graph the image of the figure using the indicated reflection. 1. across the x-axis 3. across the y-axis A A B C M T H 2. across the y-axis 4. across the x-axis C B O A R T A Use the algebraic rule to find the coordinates of the vertices of the image of each figure after the given reflection. 5. D(2, -3), E(1, 4), N(9, -12); over the y-axis 6. T(-4, 19), H(3, 13), A(5, -6), W(-6, -8); over the x-axis 16

18 2.4 Warm Up 17

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22 2.4 Worksheet 1 Graph the points to form a figure. Reflect each figure over the line y = x (drawn for you on the first graph). Draw the image in a different color. Then write the coordinates of the image points. What pattern do you notice? x y x y x y x y x y x y Write the algebraic rule for a reflection over the line y = x: 21

23 Graph the points to form a figure. Reflect each figure over the line y = -x (drawn for you on the first graph). Draw the image in a different color. Then write the coordinates of the image points. What pattern do you notice? x y x y x y x y x y x y Write the algebraic rule for a reflection over the line y = -x: 22

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25 Draw the line of reflection. Describe the reflection (for example, reflect across the line y = 3 ). Draw the line of reflection. Describe the reflection (for example, reflect across the line y = 3 ). Challenge: Write the algebraic rule that describes each reflection. 24

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27 Draw the line of reflection. Describe the reflection (for example, reflect across the line y = 3 ). Challenge: Write the algebraic rule that describes each reflection. 26

28 2.5 Warm Up 1. a. Accurately draw a rotation arrow so that figure d is the image of figure d. b. Accurately draw a different rotation arrow so that figure d is the image of figure d. c. Compare your two arrows. What do you notice? 2. Complete figure P Q R S T, the image of PQRST using the given motion. Describe the motion used. C T P Q P S R Q R Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 27

29 2.5 Worksheet 1 28

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31 Describe each transformation verbally (e.g. rotate 90 CCW). 30

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33 Use the algebraic rule to find the coordinates of the vertices of the image for each rotation. 32

34 2.6 Warm Up 1. Draw the image of using translation arrow b shown. Describe the relationship between and. b A G 2. Answer each of the following on the same drawing. a. Draw x, the image of figure m using rotation arrow a and center R. b. Draw y, the image of figure m using rotation arrow b and center R. c. Draw z, the image of figure m using rotation arrow c and center R. d. What do you notice? Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 33

35 2.6 Activity 1 1. On a clean sheet of graph paper, each group member should draw a triangle. 2. Record the coordinates of the vertices of the triangle in the table below. 3. Trace the figure with patty paper and rotate 90 counterclockwise. 4. Record the coordinates of the vertices of the image in the table. 5. Look at the coordinates of corresponding vertices. What patterns do you notice? 6. Discuss your findings with your group members. Check to see if the same pattern works for each group member s figure. 7. Write an algebraic rule for a 90 counter-clockwise rotation. 8. Now rotate the figure 90 clockwise. 9. Record the coordinates of the vertices of the image in the table. 10. Look at the coordinates of corresponding vertices. What patterns do you notice? 11. Discuss your findings with your group members. Check to see if the same pattern works for each group member s figure. 12. Write an algebraic rule for a 90 clockwise rotation. Preimage 90 counterclockwise rotation 90 clockwise rotation x y x y x y Algebraic Rules: 90 counter-clockwise rotation (x, y) 90 clockwise rotation (x, y) 34

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37 The Basics of Transformations All rectangles in the grid below are congruent. Follow the instructions and then write the number of the rectangle that matches the figure you moved ) Reflect Rectangle 1 over the y-axis. Then translate down three units and rotate 90 counterclockwise around the point (3, 1). (Hint: redraw the axes so that the origin corresponds to (3, 1).) Which rectangle is the final image? 2) Translate Rectangle 2 down one unit and reflect over the x-axis. Then reflect over the line x = 4. Which rectangle is the final image? 36

38 3) Reflect Rectangle 3 over the y-axis and then rotate 90 clockwise around the point (-2, 0). Finally, glide five units to the right. Which rectangle is the final image? 4) Rotate Rectangle 4 90 clockwise around the point (-3, 0). Reflect over the line y =2 and then translate one unit left. Which rectangle is the final image? 5) Translate Rectangle 5 left five units. Rotate 90 clockwise around the point (-2, 2) and glide up two spaces. Which rectangle is the final image? 6) Rotate Rectangle 6 90 clockwise around the point (4, 4) and translate down three units. Which rectangle is the final image? 7) Rotate Rectangle 7 90 clockwise around (-4, 4) and reflect over the line x = -4. Which rectangle is the final image? 8) Reflect Rectangle 8 over the x-axis. Translate four units left and reflect over the line y = 1.5. Which rectangle is the final image? Adapted from Geometry Teacher s Activities Kit by Muschla & Muschla 37

39 2.6 Show What You Know! Perform the given rotation for each figure. 1. rotate 90 clockwise 3. rotate 90 counterclockwise 2. rotate 180 clockwise 4. rotate 90 clockwise 38

40 2.7 Warm Up 1. Lori wants to cover a portion of her backyard patio with L-shaped bricks like the one below. a. Sketch her patio floor. b. Sketch another possibility for her patio floor. 2. Which of the figures shown are images of figure a using a rotation? Find the center and arrow of rotation for each rotation image. Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 39

41 Writing Directions for Transformations Write a set of directions to reflect, rotate, and translate each figure below. Make an answer key on graph paper showing where the image of the transformed figure will be. When you are done, exchange your sets of directions with those of another student. Using graph paper, follow each other s directions for transforming the figures. Rewrite any directions that are unclear or confusing Adapted from Geometry Teacher s Activities Kit by Muschla & Muschla 40

42 TransmoGrapher Exploration Questions 1. Pick a partner. Go to the following link: Each of you should perform 2 transformations (translation, reflection, or rotation) on a triangle. Make sure you keep track of the transformations you use. Also, be sure to click on Show Original Polygon and unclick Show Pre-image. Now switch computers and see if you can get your partner s triangle back to its home position (also keeping track of the transformations you make). Now compare the transformations you made to move the triangle with your partner's moves to get it back to its original position. Are they the same? Explain. 3. Now each of you should perform 3 transformations (translation, reflection, or rotation) on a square. Make sure you keep track of the transformations you use. Now switch computers and see if you can get your partner s square back in to its home position (also keeping track of the transformations you use). Now compare the moves you did to move the square with your partner's moves to get it back. Are they the same? Explain. 4. Now you and your partner should perform 2 transformations (translation, reflection, or rotation) on a parallelogram. Make sure you keep track of the transformations you use. Now switch computers and see if you can get your partners parallelogram back in to its home position also keeping track of the transformations you use). Now compare the moves you did to move the parallelogram with your partner's moves to get it back. Are they the same? Explain. 41

43 2.7 Practice Draw and label the image after each transformation. Write a rule for each transformation. 1. Translate 5 units left. 5. Rotate 90 counter-clockwise. 2. Reflect over the x-axis. 6. Translate 2 units up. 3. Rotate 180 clockwise around the origin. 7. Rotate 90 clockwise. 4. Reflect over the y-axis. 8. Translate 4 units right and 7 units down. 42

44 2.7 Show What You Know! Practice Quiz Part 1: Vocabulary Fill in the blank with the appropriate term. 1. A(n) is a change in position, orientation, or size of a figure. 2. A(n) is a transformation in which all points of a figure move the same distance in the same direction. 3. A(n) is a transformation in which the preimage and the image are congruent. 4. A(n) is a transformation in which a figure and its image have opposite orientations. 5. A(n) is a transformation in which a figure is turned around a fixed point. Part 2: Graphing Transformations on the Coordinate Plane Graph each figure. Then find the image after the given transformation. 1. HIJ with vertices H(-2, 1), I(2, 3), and J(0, 0) translated right two and up three. 3. with endpoints J(-3, -2) and (2, 4) rotated 90 clockwise. 2. Quadrilateral QRST with vertices Q(1, 0), R(2, -3), S(0, -3), and T(-3, -1) reflected over the y-axis. 4. ABC with vertices A(-4, -2), B(-1, -4), C(2, -2) reflected over the x-axis. 43

45 5. STR with vertices S(-2, 0), T(0, -1), and R(-3, -3) rotated 180 clockwise. 7. with endpoints P(4, 2) and Q (-1, 5) reflected across the line y = x. 6. EFG with vertices E(2, 0), F(-1, -1), and G(1, 3) translated left three and up one. 8. TAM with vertices T(0, 5), A(4, 1) and M(3, 6) rotated 90 counter-clockwise. Part 3: Writing a Rule 1. Write a rule to describe each translation. a. D O G b. V W W O D V G 44

46 2.8 Warm Up 1. Sammy cuts out S-shaped pieces of paper to cover his locker, as shown below. a. Describe his pattern. b. Help Sammy finish covering his locker by adding several more S s to his pattern. c. One of Sammy s friends decides to cover her locker with her initial as well. Show what she may have done by creating a pattern using another initial. 2. Complete the image of quadrilateral ABCD below. Explain your method. X A C A B C D a) Describe the motion as precisely as you can. b) Make at least four observations about the image. Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 45

47 2.9 Warm Up 1. Marilyn s mother asks her to tile the bathroom floor. She begins laying the tiles as shown below. When her mother walks into the bathroom, she says, That just won t do, Marilyn! Explain why Marilyn s mother is dissatisfied with her work. 2. The distance between Gardenville and Mt. Airy is 300 miles; it is 200 miles between Mt. Airy and Portertown. Find the air distance from Gardenville to Portertown. Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 46

48 2.10 Warm Up 1. Draw a tessellation using this tile. Describe the motions you used to make your tessellation. 2. Draw an image of MEL using a dilation with center R. Explain your method. What scale factor did you use? M R L E a) Draw a different image of MEL using a dilation with center R. What scale factor did you use? b) Draw a very different image of MEL using a dilation with center R. What scale factor did you use? Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 47

49 2.10 CCW about the origin. reflected across the x-axis. 48

50 2.11Warm Up 1. Draw a dilation of quadrilateral CHRS using X as the center of dilation and the following scale factors: a. 50%; label the image LOVY. b. 200%; label the image FIJU. c. 100%; label the image DEGA. C X S H R 2. Sean connects a rubber band ( ) between two pegs on the hands of a clock as shown, where BE = 4 units and EN = 3 units. a) What happens to the length of his rubber band ( ) as the hands move through a full hour? b) What lengths are possible for BN? N E B Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 49

51 Vocabulary Word Definition Characteristics Picture and/or Symbol Real Life Examples congruence motion image preimage reflection rotation 50

52 Vocabulary Word Definition Characteristics Picture and/or Symbol Real Life Examples tessellation transformation translation 51

53 TESSELLATION PROJECT DIRECTIONS You are to create your own tessellation masterpiece. Your tessellation will be created based on specific criteria. You MUST follow the guidelines given in order to receive full credit. The tessellation will be constructed on white legal size graph paper (I will supply). Your grade will be based on the following. Tessellation WebQuest (20) Escher Essay (1-2 page paper and reference page) (20) Tessellation Creation Pattern Tessellates the plane (10) Template with modifications turned in (5) Appearance and Neatness (10) Creativity/Originality/Difficulty (10) Total Points (75) Part 1 Tessellation WebQuest (SAS Curriculum Pathways 736) The tessellation WebQuest is an introduction to the world of tessellations. You will define a tessellation, identify the properties of tessellations, and take a look at the history of tessellations. In this quest you will also have to find specific examples of tessellations in nature as well as man-made tessellations. Part 2 M.C. Escher Essay You are to write a 1-2 page essay on the Mathematical art of M.C Escher. Your essay should include His background. Who is M.C. Escher? Where he was born? What was his education? Etc. Escher s contributions to art and mathematics. How does he integrate Mathematics with art? Also give specific examples of his work. What is his nickname? and any additional interesting facts about him. The paper should be typed, 12 point font, Times New Roman, double spaced, and 1 inch margins. In addition to the 1-2 pages you are to have a reference page of the websites, or books you used to write your essay. Spelling and grammar count! Part 3 Create your own Tessellation The appearance of your tessellation should be neat! Your tessellation (pattern) should cover the ENTIRE page (no gaps or unintentional white spaces). You are to turn in the template figure you used to create your tessellation. Coloring should be in between the lines, and NO wrinkled or torn projects! Your project will also be based on Creativity and Difficulty. The more difficult and complex the tessellation is, then the higher the grade. You may not simply take a polygon and slide, rotate, or reflect 52

54 it over and over again to create your tessellation. You must use a figure created using one of the nibbling methods. What to do 1. Begin by creating a template using at least 1 of the nibbling methods. Be sure to label all the pieces. 2. Once the template has been created, place your template at any of the four corners of your graph paper and trace your first figure. (the squares on the graph paper should align with the original square you started with, and therefore your template may hang off the edge of the paper. 3. Once your figure has been traced slide it or reflect it and trace your next figure (whether you reflect or slide it depends on the nibbling technique you used) 4. Continue step 3 until the entire page is covered. All your figures on the paper should fit together so that it looks like a puzzle. Figures at the edge will only be partial images 5. Once you have the page covered in your tessellation, begin to add color in the figure. The figure can be animated as well (see attached examples for animations). 6. Be creative and have fun with the project! **Remember the more creative your tessellation is the more points! WHAT TO TURN IN Cover page (with Title, Name, Date, Period) Completed WebQuest Escher Essay (1-2 pages typed) Tessellation Creation with template This project is DUE 53

55 CREATING YOUR TEMPLATE METHOD 1: The nibbling technique for geometric transformations: 1. Begin with a square and design from one corner of the square to an adjacent corner. (Do not draw diagonally). Do not stop halfway across! 2. Cut on the design line, being sure to have 2 pieces when done -the nibble and the rest of the sheet. There should be no other pieces laying around. This is very important! No trimming allowed. 3. Slide the nibble across the sheet to the opposite side and tape the straight edges together. (Do not attach it to an adjacent edge. Do not flip the nibble around. Do not overlap the edges when taping.) The corners of the piece and the nibble should match perfectly. 4. Repeat the procedure for the other sides 54

56 METHOD 2: The Rotating and nibbling technique for geometric transformations: 1. Begin with a rectangle and design from one corner of the rectangle to an adjacent corner. (Do not draw diagonally). Do not stop halfway across! 2. Cut on the design line, being sure to have 2 pieces when done -the nibble and the rest of the sheet. There should be no other pieces laying around. This is very important! No trimming allowed. 3. Instead of sliding the nibble, rotate the nibble at its end point to an adjacent side of their square (not an opposite side). Mark your point of rotation and tape the straight edges together. (Do not overlap the edges when taping.) The corners of the piece and the nibble should match perfectly. 4. Repeat the procedure for the other sides 55

57 Creating a Tessellation Using Reflections: 1. Select any figure you want to use as your pattern (can use a figure from the nibbling Methods or a combination of both. 2. Then reflect or flip the figure repeatedly vertically (over the y-axis). 3. Once you have a row of flips, take that row and reflect it horizontally (over the x-axis) 4. Continue Steps 2 and 3 until your paper is covered. Example 56

58 EXAMPLES OF TEMPLTAES AND THEIR TESSELLATIONS Two specific silhouettes or illustrations of animate figures are shown below. They are examples of alterations of opposites sides of a regular grid by translations Tessellating shapes can be formed by applying the nibbling techniques to both pairs of parallel sides. An example is shown below. 57

59 Creating Animate Figures with Tessellations There are two approaches to altering tessellating polygons into animate figures 1. Have a specific object in mind and to alter the original polygons (rectangle, square, or parallelogram) sides to make the shape look like the object (nibbling techniques). This approach may require a bit of trial and error. 2. Create a new shape using the nibbling techniques and then use you imagination to see that you think it look like. Below are two examples Does the new shape look like anything special to you? Does it remind you of anything? There are many things it could be. It could be the head of a person wearing a feather in his cap Here is a different altered square. What does it look like to you? Maybe a flying owl? 58