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“Transformations” High School Geometry By C. Rose & T. Fegan Links Teacher Page Student Page Teacher Page Benchmarks Concept Map Key Questions Scaffold Questions Ties to Core Curriculum Misconceptions Key Concepts Real World Context Activities & Assessment Materials & Resources Bibliography Acknowledgments Student Page Student Page Interactive Activities Classroom Activities Video Clips Materials, Information, & Resources Assessment Glossary home Benchmarks G3.1 Distance-preserving Transformations: Isometries G3.1.1 Define reflection, rotation, translation, & glide reflection and find the image of a figure under a given isometry. G3.1.2 Given two figures that are images of each other under an isometry, find the isometry & describe it completely. G3.1.3 Find the image of a figure under the composition of two or more isometries & determine whether the resulting figure is a reflection, rotation, translation, or glide reflection image of the original figure. Teacher Page Concept Map Teacher Page Key Questions What is a transformation? What is a pre-image? What is an image? Teacher Page Scaffold Questions What are reflections, translations, and rotations? What is isometry? What are the characteristics of the various types of isometric drawings on a coordinate grid? What is the center and angle of rotation? How is a glide reflection different than a reflection? Teacher Page Ties to Core Curriculum A.2.2.2 Apply given transformations to basic functions and represent symbolically. Ties to Industrial Arts through Building Trades and Art. L.1.2.3 Use vectors to represent quantities that have magnitude of a vector numerically, and calculate the sum and difference of 2 vectors. Teacher Page Misconceptions Misinterpretation of coordinates: Relating x-axis as horizontal & y-axis as vertical + & - directions for x & y (up/down or left/right) Rules of isometric operators (+ & - values) and (x, y) verses (y, x) The origin is always the center of rotation (not true) Teacher Page Key Concepts Students will learn to transform images on a coordinate plane according to the given isometry. Students will learn the characteristics of a reflection, rotation, translation, and glide reflections. Students will learn the definition of isometry. Students will learn to identify a reflection, rotation, translation, and glide reflection. Students will identify a given isometry from 2 images. Students will describe a given isometry using correct rotation. Students will relate the corresponding points of two identical images and identify the points using ordered pairs. Students will transform images on the coordinate plane using multiple isometries. Students will recognize when a composition of isometries is equivalent to a reflection, rotation, translation, or glide reflection. Teacher Page Real World Context Sports: golf, table tennis, billiards, & chess Nature: leaves, insects, gems, & snowflakes Art: paintings, quilts, wall paper, & tiling Teacher Page Activities & Assessment Students will visit several interactive websites for activities & quizzes. Students can view a video clip to learn more about reflections. Students will create transformations using pencil and coordinate grids. Teacher Page Materials & Resources Computers w/speakers & Internet connection Pencil, paper, protractor, and coordinate grids Teacher Page Bibliography http://www.michigan.gov/documents/Geometry_167749_7.pdf http://www.glencoe.com http://illuminations.nctm.org/LessonDetail.aspx?ID=L467 http://illuminations.nctm.org/LessonDetail.aspx?ID=L466 http://illuminations.nctm.org/LessonDetail.aspx?ID=L474 http://nlvm.usu.edu/en/nav/frames_asid_302_g_4_t_3.html?open=activities http://www.haelmedia.com/OnlineActivities_txh/mc_txh4_001.html http://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/transformationshrev4.shtml http://glencoe.mcgraw-hill.com/sites/0078738181/student_view0/chapter9/lesson1/selfcheck_quizzes.html http://glencoe.mcgraw-hill.com/sites/0078738181/student_view0/chapter9/lesson2/selfcheck_quizzes.html http://glencoe.mcgraw-hill.com/sites/0078738181/student_view0/chapter9/lesson3/selfcheck_quizzes.html http://www.unitedstreaming.com/index.cfm http://www.freeaudioclips.com Teacher Page Acknowledgments Thanks to all of those that enabled us to take this class. These include: Pinconning & Standish-Sterling School districts, SVSU Regional Mathematics & Science Center, Michigan Dept. of Ed. Thanks also to our instructor Joe Bruessow for helping us solve issues while creating this presentation. Teacher Page Interactive Activities Interactive Website for Rotating Figures Interactive Website Describing Rotations Interactive Website for Translating Figures Interactive Website with Translating Activities Interactive Symmetry Games Interactive Rotating Activities (Click on Play Activity) Student Page Classroom Activity #1 “Reflection on a Coordinate Plane” Quadrilateral AXYW has vertices A(-2, 1), X(1, 3), Y(2, -1), and W(-1, -2). Graph AXYW and its image under reflection in the x-axis. Compare the coordinates of each vertex with the coordinates of its image. Activity 1 Answer Activity #1 – Answer Use the vertical grid lines to find a corresponding point for each vertex so that the x-axis is equidistant from each vertex and its image. A(-2, 1) A(-2, -1) X(1, 3) X(1, -3) Y(2, -1) Y(2, 1)W(-1, -2) W(-1, 2) Plot the reflected vertices and connect to form the image AXYW. The x-coordinates stay the same, but the y-coordinates are opposite. That is, (a, b) (a, -b). Activity #2 Classroom Activity #2 “Translations in the Coordinate Plane” Quadrilateral ABCD has vertices A(1, 1), B(2, 3), C(5, 4), and D(6, 2). Graph ABCD and its image for the translation (x, y) (x - 2, y - 6). Activity 2 Answer Activity 2 – Answer This translation moved every point of the preimage 2 units left and 6 units down. A(1, 1) A(1 - 2, 1 - 6) or A(-1, -5) B(2, 3) B(2 - 2, 3 - 6) or B(0, -3) C(5, 4) C(5 - 2, 4 - 6) or C(3, -2) D(6, 2) D(6 - 2, 2 - 6) or D(4, -4) Plot the translated vertices and connect to form quadrilateral ABCD. Activity #3 Classroom Activity #3 “Rotation on the Coordinate Plane” Triangle DEF has vertices D(2, 2,), E(5, 3), and F(7, 1). Draw the image of DEF under a rotation of 45˚ clockwise about the origin. Activity 3 Answer Activity #3 - Answer First graph DEF. Draw a segment from the origin O, to point D. Use a protractor to measure a 45° angle clockwise Use a compass to copy onto . Name the segment . Repeat with points E and F. DEF is the image DEF under a 45° clockwise rotation about the origin. Student Page Video Clips Reflection Translation Rotation Student Page Material, Information, & Resources Computers w/speakers & Internet connection Pencil, paper, protractor, and coordinate grids Student Page Assessment Self-Quiz on Reflections Self-Quiz on Translations Self-Quiz on Rotations Student Page Glossary Transformation – In a plane, a mapping for which each point has exactly one image point and each image point has exactly one preimage point. Reflection - A transformation representing a flip of a figure over a point, line, or plane. Rotation - A transformation that turns every point of a preimage through a specified angle and direction about a fixed point, called the center of rotation. Translation – A transformation that moves all points of a figure the same distance in the same direction. Isometry – A mapping for which the original figure and its image are congruent Glossary Cont. Glossary Continued Angle of Rotation – The angle through which a preimage is rotated to form the image. Center of Rotation – A fixed point around which shapes move in circular motion to a new position. Line of Reflection – a line through a figure that separates the figure into two mirror images Line of Symmetry – A line that can be drawn through a plane figure so that the figure on one side is the reflection image of the figure on the opposite side. Point of Symmetry – A common point of reflection for all points of a figure. Rotational Symmetry – If a figure can be rotated less that 360o about a point so that the image and the preimage are indistinguishable, the figure has rotated symmetry. Student Page