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“Transformations” High School Geometry

“Transformations” High School Geometry. By C. Rose & T. Fegan. Links. Teacher Page. Student Page. Benchmarks Concept Map Key Questions Scaffold Questions Ties to Core Curriculum Misconceptions. Key Concepts Real World Context Activities & Assessment Materials & Resources

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“Transformations” High School Geometry

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  1. “Transformations”High SchoolGeometry By C. Rose & T. Fegan

  2. Links Teacher Page Student Page

  3. Benchmarks Concept Map Key Questions Scaffold Questions Ties to Core Curriculum Misconceptions Key Concepts Real World Context Activities & Assessment Materials & Resources Bibliography Acknowledgments Teacher Page Student Page

  4. Student Page • Interactive Activities • Classroom Activities • Video Clips • Materials, Information, & Resources • Assessment • Glossary home

  5. 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

  6. Concept Map Teacher Page

  7. Key Questions • What is a transformation? • What is a pre-image? • What is an image? Teacher Page

  8. 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

  9. 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

  10. 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

  11. 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

  12. Real World Context • Sports: golf, table tennis, billiards, & chess • Nature: leaves, insects, gems, & snowflakes • Art: paintings, quilts, wall paper, & tiling Teacher Page

  13. 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

  14. Materials & Resources • Computers w/speakers & Internet connection • Pencil, paper, protractor, and coordinate grids Teacher Page

  15. 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/self-check_quizzes.html http://glencoe.mcgraw-hill.com/sites/0078738181/student_view0/chapter9/lesson2/self-check_quizzes.html http://glencoe.mcgraw-hill.com/sites/0078738181/student_view0/chapter9/lesson3/self-check_quizzes.html http://www.unitedstreaming.com/index.cfm http://www.freeaudioclips.com Teacher Page

  16. 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

  17. 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

  18. 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

  19. 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 AXYW. • The x-coordinates stay the same, but the y-coordinates are opposite. • That is, (a, b)  (a, -b). Activity #2

  20. 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

  21. 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 ABCD. Activity #3

  22. 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

  23. 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.DEF is the image DEF under a45° clockwise rotation about the origin. Student Page

  24. Video Clips • Reflection • Translation • Rotation Student Page

  25. Material, Information, & Resources • Computers w/speakers & Internet connection • Pencil, paper, protractor, and coordinate grids Student Page

  26. Assessment • Self-Quiz on Reflections • Self-Quiz on Translations • Self-Quiz on Rotations Student Page

  27. 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.

  28. 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

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