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PATTERNS OF ROTATIONS AND REFLECTIONS OF A TRIANGLE PRIOR KNOWLEDGE: Understanding the meaning of a reflection and a rotation Understanding the meaning of angular measure PURPOSE Build on the knowledge of reflections and rotations to construct results of sequences of simple transformations. Recognize and discover patterns and structure Recognize when commutative and associative properties are valid. INTRODUCTION Mathematics, natural science, art, music, and other subjects often focus on patterns. We will be investigating patterns that result when transformations are combined. FORMATIVE ASSESSMENTS After teams of 3 students have worked on #2 for about 10 minutes, ask individuals to indicate labels on vertices for transformations A, D, F. After teams of 3 students have worked on #3b for about 10 minutes, ask individuals to explain their reasoning for the composite transformation. SUMMATIVE ASSESSMENT: Students should have available their definitions of the six transformations. Ask individuals to 1. Indicate the result of the reference triangle being subjected to transformation C. 2. Indicate the result of the reference triangle being subjected to transformation F. 3. Explain why D followed by E is the same as E followed by D. 4. Is A followed by D the same as D followed by A? ANSWERS 1. Vertices A and Z are interchanged. 2. The result is that all three vertices retain their locations. 3. Both result in a net rotation by 360 degrees: 4. NO EXTENSIONS: Students could investigate a square which has four reflections (relative to vertical, horizontal and two diagonal lines) and four rotations (90o, 180o, 270o, 360o). The table is now 8x8. the number of transformations of a regular tetrahedron; there are 24 corresponding to the 24 rearrangements of the four symbols A,B,C,D representing the vertices. For example with D "at the top", there are 3 rotations and 3 reflections. Note that the threedimensional manipulative cannot easily represent reflections since one cannot move out of the three dimensions: implementation of reflections in two dimensions is accomplished by "flipping" into the third dimension. PATTERNS OF ROTATIONS AND REFLECTIONS OF A TRIANGLE This activity will focus on transformations -- reflections and rotations. Each part will begin with an equilateral triangle whose original position is indicated here: 1. Cut out the large triangle at the end of the worksheet. Label the vertices X,Y,Z on the reverse side, being careful that the label for each vertex matches the label on the front side. You may wish to color one or both sides in order to make it easier to keep track of which side is up. 2. We will investigate six transformations of triangle XYZ, each of which will leave the triangle looking the same as ABC but with vertices possibly moved. a. TRANSFORMATION "A": Indicate the LABELS for the vertices on the triangle at the right after a reflection around the vertical line through Z. ANSWER: Interchange X & Y b. TRANSFORMATION "B". Indicate the LABELS for the vertices on the triangle at the right after a reflection around the line through X and the midpoint of Y and Z. ANSWER: Interchange Y & Z c. TRANSFORMATION "C". Indicate the LABELS for the vertices on the triangle at the right after a reflection around the line through Y and the midpoint of X and Z. ANSWER: Interchange X & Z d. TRANSFORMATION "D". Indicate the LABELS for the vertices on the triangle at the right after a 120o counter-clockwise rotation of the triangle around its center. ANSWER: (X,Y,Z) become (Z,X,Y) e. TRANSFORMATION "E". Indicate the LABELS for the vertices on the triangle at the right after a 240o counter-clockwise rotation of the triangle around its center. ANSWER: (X,Y,Z) become (Y,Z,X) f. TRANSFORMATION "F". Indicate the LABELS for the vertices on the triangle at the right after a 360o counter-clockwise rotation of the triangle around its center. ANSWER: The vertices remain in their original positions. 3. Now we investigate what happens if one implement two transformations in succession, beginning with the original position. a. Show that the result of performing transformation A, then transformation D, has Z at the bottom left, Y at the bottom right, and X at the top. b. Indicate which of the six transformations from #2 is equivalent to the successive transformations in 3a. ANSWER: Transformation C. c. In your group complete the table below. Each square in the table should indicate the result of two successive transformations, corresponding to the row (first transformation) and column (second transformation) labels. If there are three members in your group, one should complete rows 1-4, another should complete rows 2-5, and the third rows 3-6. Column heading is the second of the transformations A B C D E F A F E D C B A B D F E A C B C E D F B A C D B C A E F D E C A B F D E F A B C D E F d. Discuss the patterns that are evident in the table and prepare to participate in a whole class discussion of the results After the group is confident regarding the accuracy of the results in the table, a color code can be chosen to represent the table. POSSIBLE ANSWERS Each row and each column contains all 6 possible transformations. The upper left and lower right 3x3 blocks contain only D,E,F. The upper right and lower left 3x3 blocks contain only A,B,C. Two successive reflections or two consecutive rotations are equivalent to a rotation. Successive transformations containing one each of a rotation and a reflection are equivalent to a reflection. e. Suppose your first transformation is "B" and after the second transformation the vertices are rotated 120o from their original positions. What second transformation was used? ANSWER: A f. IF TIME. Let the notation A▫B mean "perform transformation B, then transformation A". The first transformation to be performed is listed last in order to conform to notation that is used in high school. Represent each of the following in terms of a single symbol (A-F): (i) (A▫B)▫D and A▫(B▫D) ANSWER: Both are (ii) (B▫C)▫E and B▫(C▫E) ANSWER: Both are (iii) (A▫B)▫C and A▫(B▫C) ANSWER: Both are (iv) Is the operation ▫ commutative? ANSWER: In general NO. What do the results of these parts suggest? ANSWER: The operation ▫ is associative but not necessarily commutative