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Mastering Geometry Transformations: Isometric Drawings & Real-world Applications

This comprehensive book explores concepts of reflection, rotation, translation, and glide reflection in geometry. Students learn to transform images on a coordinate plane, relate corresponding points, and identify isometries. Real-world applications in sports, nature, and art provide context. The book includes interactive activities, videos, quizzes, and teacher resources. Misconceptions and key concepts are addressed, improving understanding and application of transformational geometry.

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Mastering Geometry Transformations: Isometric Drawings & Real-world Applications

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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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