Unit 5: Butterflies, Pinwheels, & Wallpaper
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1 Unit 5: Butterflies, Pinwheels, & Wallpaper Directions: Please complete the necessary problems to earn a maximum of 10 points according to the chart below. Show all of your work clearly and neatly for credit- which will be earned based on completion rather than correctness. I can complete transformations of shapes on a coordinate plane. Lesson Practice problems Options Maximum Points Lesson 1: Flipping on a 1, 2, 3, 8 3 Points Grid Lesson 2: Sliding on a 4, 5, 9 2 Points Grid Lesson 3: Spinning on a 6, 7, 10 2 Points Grid Lesson 5: Parallel Lines, 11, 12, 13, 14 3 Points Transversals, and Angle Sums / 10 Points
2 For Exercises 1 7, use the figure below and find the image of the figure after each transformation. 1. Complete the table showing the coordinates of points A E and their images after a reflection in the y axis. Point A B C D E Original Coordinates (-5, 1) (-2, 5) Coordinates After a y-axis Reflection Coordinates After a x-axis Reflection Coordinates After a x-axis Reflection Followed By a y-axis Reflection a. Draw the Image. b. Write a rule relating coordinates of key points and their images after a reflection in the y-axis: (x, y) (, ). 2. Add a row to your table for Exercise 1 to show the coordinates of points A E and their images after a reflection in the x- axis. b. Write a rule relating coordinates of key points and their images after a reflection in the x-axis: (x, y) (, ). 3. Add another row to your table from Exercise 1 to show the coordinates of points A E and their images after a reflection in the x-axis, followed by a reflection in the y-axis. b. Write a rule relating coordinates of key points and their images after bother reflections: (x, y) (, ). c. What single transformation in this Investigation has the same effect as the sequence of two line reflections?
3 4. Complete the table showing the coordinates of points A E and their images after a translation that moves point B to point (3, 4). Point A B C D E Original Coordinates (-5, 1) (-2, 5) Coordinates After Translating B to (3, 4) Coordinates After Translating B to (-1, 0) b. Write a rule relating coordinates of key points and their images after the translation: (x, y) (, ). 5. Add a row to your table from Exercise 4 to show the coordinates of points A E and their images after the first translation, followed by a translation that moves points B to (-1, 0). b. Write a rule relating coordinates of key points and their images after the second translation: (x, y) (, ). c. Write a rule relating coordinates of key points and their images after the sequence of the two translations: (x, y) (, ). d. What single information is equivalent to the sequence of the two translations? 6. Complete the table showing the coordinates of points A E and their images after a counterclockwise rotation of 90 about the origin.
4 Point A B C D E Original Coordinates (-5, 1) (-2, 5) Coordinates After a 90 Rotation Coordinates After a 180 Rotation b. Write a rule relating coordinates of key points and their images after a rotation of 90 : ( x, y) (, ). 7. Add a row to your table from Exercise 6 to show the coordinates of points A E and their images after two counterclockwise rotations of 90 about the origin. b. Write a rule relating coordinates of key points and their images after both rotations: (x, y) (, ). c. What single transformation is equivalent to the sequence of the two rotations? 8. Complete the table showing the coordinates of points A C and their images after a reflection in the line y = x. Point A B C Original Coordinates Coordinates After a Reflection in y = x Coordinates After a Reflection of Triangle A B C b. Draw the image and label the vertices A, B, and C.
5 c. Add a row to your table to show the coordinates of points A C and their images after a reflection of triangle A B C in the axis. d. Draw the image and label the vertices A, B, and C. e. Draw the image of triangle ABC after the same two reflection, but in the reverse order. That is, reflect triangle ABC in the x-axis and then reflect its image, triangle A B C in the line y = x. What does the result suggest about the commutativity of a sequence of line reflections? 9. i. Translate ABC according to the rule (x, y) (x + 2, y 3). Label its image A B C. ii. Translate ABC according to the rule (x, y) (x 4, y 6). Label its image A B C. b. Use the coordinates of the vertices of triangle ABC and its two images to compare the slopes of each pair of line segments.
6 c. What do your results from parts (a) and (b) say about the effect of translations on the slopes of lines? About the relationship between a line and its image under a translation? 10. a. Use triangle ABC from Exercise 8. Draw triangle ABC on a coordinate grid and its image after a 180 rotation about the origin. Label the Image A B C. b. Use the coordinates of the vertices of triangle ABC and its image to compare the slopes of each pair of line segments. c. What do your results from part (a) and (b) say about the effect of half-turns or 180 rotations on the slopes of line? About the relationship between a line and its image under a 180
7 11. In the diagram below, line L1 and L2 are parallel. What are the measures of angles a g? 12. What are the measures of angles a and b in the triangle? 13. What is the value of x in the diagram? 14. The diagram shows parallelogram ABCD with one diagonal DB. Assuming only that opposite sides of any parallelogram are parallel: a. Which angles are congruent? How do you know?
8 b. How can you be sure that triangle ABD is congruent to triangle ADB? What are the corresponding vertices, sides, and angles? c. How does the congruence of triangles ABD and ADB guarantee that, in a parallelogram, opposite sides are the same length? Exit Ticket Level of Understanding After finishing this investigation you should be comfortable doing the following: -using coordinate methods to study flips, turns, and slides. -understanding the properties of parallel lines. -understanding the properties of angles in triangles
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