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TransformationsHigh SchoolGeometry

• By
• C. Rose T. Fegan

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Links
Teacher Page
Student Page
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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
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Student Page

• Interactive Activities
• Classroom Activities
• Video Clips
• Materials, Information, Resources
• Assessment
• Glossary

home
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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
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Concept Map
Teacher Page
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Key Questions

• What is a transformation?
• What is a pre-image?
• What is an image?

Teacher Page
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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
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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
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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
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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
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Real World Context

• Sports golf, table tennis, billiards, chess
• Nature leaves, insects, gems, snowflakes
• Art paintings, quilts, wall paper, tiling

Teacher Page
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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
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Materials Resources

• Computers w/speakers
• Internet connection
• Pencil, paper, protractor,
• and coordinate grids

Teacher Page
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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?openactivities
• http//www.haelmedia.com/OnlineActivities_txh/mc_t
  xh4_001.html
• http//www.bbc.co.uk/schools/gcsebitesize/maths/sh
  apeih/transformationshrev4.shtml
• http//glencoe.mcgraw-hill.com/sites/0078738181/st
  udent_view0/chapter9/lesson1/self-check_quizzes.ht
  ml
• http//glencoe.mcgraw-hill.com/sites/0078738181/st
  udent_view0/chapter9/lesson2/self-check_quizzes.ht
  ml
• http//glencoe.mcgraw-hill.com/sites/0078738181/st
  udent_view0/chapter9/lesson3/self-check_quizzes.ht
  ml
• http//www.unitedstreaming.com/index.cfm
• http//www.freeaudioclips.com

Teacher Page
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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
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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
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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
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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
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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
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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
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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
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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 a 45 clockwise rotation about the
  origin.

Student Page
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Video Clips

• Reflection
• Translation
• Rotation

Student Page
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Material, Information, Resources

• Computers w/speakers
• Internet connection
• Pencil, paper, protractor,
• and coordinate grids

Student Page
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Assessment

• Self-Quiz on Reflections
• Self-Quiz on Translations
• Self-Quiz on Rotations

Student Page
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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.
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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