““Transformations”Transformations”High SchoolHigh SchoolGeometryGeometry

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C. Rose & T. FeganC. Rose & T. Fegan

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Teacher PageTeacher Page

Benchmarks

Concept Map

Key Questions

Scaffold Questions

Ties to Core CurriculumMisconceptions

Key ConceptsReal World ContextActivities & AssessmentMaterials & ResourcesBibliographyAcknowledgments

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Student PageStudent Page

Interactive Activities

Classroom Activities

Video Clips

Materials, Information, & Resources

Assessment

Glossary

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BenchmarksBenchmarks

G3.1G3.1 Distance-preserving Transformations: IsometriesDistance-preserving Transformations: Isometries

G3.1.1 Define reflection, rotation, translation, & glide G3.1.1 Define reflection, rotation, translation, & glide reflection reflection and find the image of a figure under a given and find the image of a figure under a given isometry.isometry.

G3.1.2 Given two figures that are images of each other G3.1.2 Given two figures that are images of each other under under an isometry, find the isometry & describe it completely.an isometry, find the isometry & describe it completely.

G3.1.3 Find the image of a figure under the composition of G3.1.3 Find the image of a figure under the composition of two two or more isometries & determine whether the resulting figure or more isometries & determine whether the resulting figure is is a reflection, rotation, translation, or glide reflection image of a reflection, rotation, translation, or glide reflection image of the the original figure.original figure.

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Concept MapConcept Map

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Key QuestionsKey Questions

What is a transformation?What is a transformation?

What is a pre-image?What is a pre-image?

What is an image?What is an image?

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Scaffold QuestionsScaffold Questions

What are reflections, translations, and rotations?What are reflections, translations, and rotations?

What is isometry?What is isometry?

What are the characteristics of the various types What are the characteristics of the various types of isometric drawings on a coordinate grid?of isometric drawings on a coordinate grid?

What is the center and angle of rotation?What is the center and angle of rotation?

How is a glide reflection different than a How is a glide reflection different than a reflection?reflection?

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Ties to Core CurriculumTies to Core Curriculum

A.2.2.2 Apply given transformations to basic functions A.2.2.2 Apply given transformations to basic functions and represent symbolically.and represent symbolically.

Ties to Industrial Arts through Building Trades and Art.Ties to Industrial Arts through Building Trades and Art.

L.1.2.3 Use vectors to represent quantities that have L.1.2.3 Use vectors to represent quantities that have magnitude of a vector numerically, and calculate the sum magnitude of a vector numerically, and calculate the sum and difference of 2 vectors.and difference of 2 vectors.

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Misconceptions Misconceptions

Misinterpretation of coordinates:Misinterpretation of coordinates: Relating x-axis as horizontal & y-axis as Relating x-axis as horizontal & y-axis as

vertical vertical + & - directions for x & y (up/down or left/right)+ & - directions for x & y (up/down or left/right) Rules of isometric operators (+ & - values) Rules of isometric operators (+ & - values)

and (x, y) verses (y, x)and (x, y) verses (y, x) The origin is always the center of rotation (not The origin is always the center of rotation (not

true)true)Teacher Page

Key Concepts Key Concepts

Students will learn to transform images on a coordinate plane according to Students will learn to transform images on a coordinate plane according to the given isometry.the given isometry.Students will learn the characteristics of a reflection, rotation, translation, Students will learn the characteristics of a reflection, rotation, translation, and glide reflections.and glide reflections.Students will learn the definition of isometry.Students will learn the definition of isometry.Students will learn to identify a reflection, rotation, translation, and glide Students will learn to identify a reflection, rotation, translation, and glide reflection.reflection.Students will identify a given isometry from 2 images.Students will identify a given isometry from 2 images.Students will describe a given isometry using correct rotation.Students will describe a given isometry using correct rotation.Students will relate the corresponding points of two identical images and Students will relate the corresponding points of two identical images and identify the points using ordered pairs.identify the points using ordered pairs.Students will transform images on the coordinate plane using multiple Students will transform images on the coordinate plane using multiple isometries.isometries.Students will recognize when a composition of isometries is equivalent to a Students will recognize when a composition of isometries is equivalent to a reflection, rotation, translation, or glide reflection.reflection, rotation, translation, or glide reflection.

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Real World ContextReal World Context

Sports: golf, table tennis, billiards, & chessSports: golf, table tennis, billiards, & chess

Nature: leaves, insects, gems, & Nature: leaves, insects, gems, & snowflakessnowflakes

Art: paintings, quilts, wall paper, & tilingArt: paintings, quilts, wall paper, & tiling

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Activities & AssessmentActivities & Assessment

Students will visit Students will visit several interactive several interactive websites for activities websites for activities & quizzes.& quizzes.

Students can view a Students can view a video clip to learn more video clip to learn more about reflections.about reflections.

Students will create Students will create transformations using pencil transformations using pencil and coordinate grids.and coordinate grids.

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Materials & ResourcesMaterials & Resources

Computers w/speakers & Computers w/speakers &

Internet connectionInternet connection

Pencil, paper, protractor, Pencil, paper, protractor,

and coordinate gridsand coordinate grids

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BibliographyBibliography

http://www.michigan.gov/documents/Geometry_167749_7.pdf http://www.glencoe.comhttp://illuminations.nctm.org/LessonDetail.aspx?ID=L467http://illuminations.nctm.org/LessonDetail.aspx?ID=L466http://illuminations.nctm.org/LessonDetail.aspx?ID=L474http://nlvm.usu.edu/en/nav/frames_asid_302_g_4_t_3.html?open=activitieshttp://www.haelmedia.com/OnlineActivities_txh/mc_txh4_001.htmlhttp://www.bbc.co.uk/schools/gcsebitesize/maths/shapeih/transformationshrev4.shtmlhttp://glencoe.mcgraw-hill.com/sites/0078738181/student_view0/chapter9/lesson1/self-

check_quizzes.htmlhttp://glencoe.mcgraw-hill.com/sites/0078738181/student_view0/chapter9/lesson2/self-

check_quizzes.htmlhttp://glencoe.mcgraw-hill.com/sites/0078738181/student_view0/chapter9/lesson3/self-

check_quizzes.htmlhttp://www.unitedstreaming.com/index.cfmhttp://www.freeaudioclips.com

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AcknowledgmentsAcknowledgments

Thanks to all of those that enabled us to Thanks to all of those that enabled us to take this class.take this class.These include:These include:

Pinconning & Standish-Sterling School districts, Pinconning & Standish-Sterling School districts, SVSU Regional Mathematics & Science Center, SVSU Regional Mathematics & Science Center, Michigan Dept. of Ed.Michigan Dept. of Ed.

Thanks also to our instructor Joe Thanks also to our instructor Joe Bruessow for helping us solve issues while Bruessow for helping us solve issues while creating this presentation.creating this presentation.

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Interactive ActivitiesInteractive 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) (Click on Play Activity)

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Classroom Activity #1Classroom Activity #1

““Reflection on a Coordinate Plane”Reflection on a Coordinate Plane”

Quadrilateral Quadrilateral AXYWAXYW has vertices has verticesAA(-2, 1), (-2, 1), XX(1, 3), (1, 3), YY(2, -1), and (2, -1), and WW(-1, -2). (-1, -2).

Graph Graph AXYWAXYW and its image under reflection in the and its image under reflection in the xx-axis.-axis.

Compare the coordinates of each vertex with the Compare the coordinates of each vertex with the coordinates of its image.coordinates of its image.

Activity 1 Answer

Activity #1 – AnswerActivity #1 – Answer

Use the vertical grid lines to find a corresponding point Use the vertical grid lines to find a corresponding point for each vertex so that the for each vertex so that the xx-axis is equidistant from -axis is equidistant from each vertex and its image.each vertex and its image.

AA(-2, 1) (-2, 1) AA(-2, -1) (-2, -1) XX(1, 3) (1, 3) XX(1, -3) (1, -3) YY(2, -1) (2, -1) YY(2, (2, 1)1)WW(-1, -2) (-1, -2) WW(-1, 2)(-1, 2)

Plot the reflected vertices and connect to form the image Plot the reflected vertices and connect to form the image AAXXYYWW..The The xx-coordinates stay the same, but the -coordinates stay the same, but the yy-coordinates -coordinates are opposite. are opposite.

That is, (That is, (aa, , bb) ) ( (aa, -, -bb).).

Activity #2

Classroom Activity #2Classroom Activity #2

““Translations in the Coordinate Plane”Translations in the Coordinate Plane”

Quadrilateral Quadrilateral ABCDABCD has vertices has vertices

AA(1, 1), (1, 1), BB(2, 3), (2, 3), CC(5, 4), and (5, 4), and DD(6, 2). (6, 2).

Graph Graph ABCDABCD and its image for the translation and its image for the translation

((xx, , yy) () (xx - 2, - 2, yy - 6). - 6).

Activity 2 Answer

Activity 2 – AnswerActivity 2 – Answer

This translation moved every point of the preimage 2 This translation moved every point of the preimage 2 units left and 6 units down.units left and 6 units down.

AA(1, 1)(1, 1) AA(1 - 2, 1 - 6) or (1 - 2, 1 - 6) or AA(-1, -5)(-1, -5)

BB(2, 3)(2, 3) BB(2 - 2, 3 - 6) or (2 - 2, 3 - 6) or BB(0, -3)(0, -3)

CC(5, 4)(5, 4) CC(5 - 2, 4 - 6) or (5 - 2, 4 - 6) or CC(3, -2)(3, -2)

DD(6, 2)(6, 2) DD(6 - 2, 2 - 6) or (6 - 2, 2 - 6) or DD(4, -4)(4, -4)

Plot the translated vertices and connect to form Plot the translated vertices and connect to form quadrilateral quadrilateral AABBCCDD..

Activity #3

Classroom Activity #3Classroom Activity #3

““Rotation on the Coordinate Plane”Rotation on the Coordinate Plane”

Triangle Triangle DEFDEF has vertices has vertices DD(2, 2,), (2, 2,), EE(5, 3), and (5, 3), and FF(7, 1).(7, 1).

Draw the image of Draw the image of DEFDEF under a rotation of 45˚ under a rotation of 45˚ clockwise about the origin.clockwise about the origin.

Activity 3 Answer

Activity #3 - AnswerActivity #3 - Answer

First graph First graph DEFDEF..

Draw a segment from the origin Draw a segment from the origin OO, to point , to point DD..Use a protractor to measure a 45° angle clockwiseUse a protractor to measure a 45° angle clockwise

Use a compass to copy onto .Use a compass to copy onto .Name the segment .Name the segment .Repeat with points Repeat with points EE and and FF..DDEEFF is the image is the image DEFDEF under a under a45° clockwise rotation about the origin.45° clockwise rotation about the origin.

Student Page

Video ClipsVideo Clips

Reflection

Translation

Rotation

Student Page

Material, Information, & ResourcesMaterial, Information, & Resources

Computers w/speakers & Computers w/speakers &

Internet connectionInternet connection

Pencil, paper, protractor,Pencil, paper, protractor,

and coordinate gridsand coordinate grids

Student Page

AssessmentAssessment

Self-Quiz on ReflectionsSelf-Quiz on Reflections

Self-Quiz on TranslationsSelf-Quiz on Translations

Self-Quiz on RotationsSelf-Quiz on Rotations

Student Page

GlossaryGlossary

Transformation – In a plane, a mapping for which each point has Transformation – In a plane, a mapping for which each point has exactly one image point and each image point has exactly one exactly one image point and each image point has exactly one preimage point.preimage point.

Reflection - A transformation representing a flip of a figure over a Reflection - A transformation representing a flip of a figure over a point, line, or plane.point, line, or plane.

Rotation - A transformation that turns every point of a preimage Rotation - A transformation that turns every point of a preimage through a specified angle and direction about a fixed point, called through a specified angle and direction about a fixed point, called the center of rotation.the center of rotation.

Translation – A transformation that moves all points of a figure the Translation – A transformation that moves all points of a figure the same distance in the same direction.same distance in the same direction.

Isometry – A mapping for which the original figure and its image Isometry – A mapping for which the original figure and its image are congruentare congruent

Glossary Cont.

Glossary ContinuedGlossary Continued

Angle of Rotation – The angle through which a preimage is rotated Angle of Rotation – The angle through which a preimage is rotated to form the image.to form the image.Center of Rotation – A fixed point around which shapes move in Center of Rotation – A fixed point around which shapes move in circular motion to a new position.circular motion to a new position.Line of Reflection – a line through a figure that separates the figure Line of Reflection – a line through a figure that separates the figure into two mirror imagesinto two mirror imagesLine of Symmetry – A line that can be drawn through a plane figure 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 so that the figure on one side is the reflection image of the figure on the opposite side.the opposite side.Point of Symmetry – A common point of reflection for all points of a Point of Symmetry – A common point of reflection for all points of a figure.figure.Rotational Symmetry – If a figure can be rotated less that 360Rotational Symmetry – If a figure can be rotated less that 360oo about a point so that the image and the preimage are about a point so that the image and the preimage are indistinguishable, the figure has rotated symmetry.indistinguishable, the figure has rotated symmetry.

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