Geometry Unit

8/4/2019 Geometry Unit

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TransformationsTransformations

High SchoolHigh School

GeometryGeometry

ByBy

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


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LinksLinks

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Teacher PageTeacher Page

BenchmarksBenchmarks

Concept MapConcept Map

Key QuestionsKey Questions

Scaffold QuestionsScaffold Questions

Ties to CoreTies to Core

CurriculumCurriculum

MisconceptionsMisconceptions

Key ConceptsKey Concepts

Real World ContextReal World Context

Activities &Activities &

AssessmentAssessmentMaterials &Materials &ResourcesResources

BibliographyBibliography

AcknowledgmentsAcknowledgments

StudentPage


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Student PageStudent Page

Interactive ActivitiesInteractive Activities

Classroom ActivitiesClassroom Activities

Video ClipsVideo Clips

Materials, Information, & ResourcesMaterials, Information, & Resources

AssessmentAssessment

GlossaryGlossary

home


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BenchmarksBenchmarks

G3.1G3.1 DistanceDistance--preserving Transformations: Isometriespreserving Transformations: Isometries

G3.1.1 Define reflection, rotation, translation, & glide reflectionG3.1.1 Define reflection, rotation, translation, & glide reflection

and find the image of a figure under a given isometry.and find the image of a figure under a given isometry.

G3.1.2 Given two figures that are images of each other underG3.1.2 Given two figures that are images of each other 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 twoG3.1.3 Find the image of a figure under the composition of twoor more isometries & determine whether the resulting figure isor more isometries & determine whether the resulting figure is

a reflection, rotation, translation, or glide reflection image of thea reflection, rotation, translation, or glide reflection image of 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 preWhat is a pre--image?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 typesWhat 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 aHow is a glide reflection different than areflection?reflection?

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Ties to Core CurriculumTies to Core Curriculum

A.2.2.2 Apply given transformations to basic functionsA.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 haveL.1.2.3 Use vectors to represent quantities that havemagnitude of a vector numerically, and calculate the summagnitude of a vector numerically, and calculate the sum

and difference of 2 vectors.and difference of 2 vectors.

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MisconceptionsMisconceptions

Misinterpretation of coordinates:Misinterpretation of coordinates:

Relating xRelating x--axis as horizontal & yaxis as horizontal & y--axis asaxis asverticalvertical

+ &+ & -- directions for x & y (up/down or left/right)directions for x & y (up/down or left/right)

Rules of isometric operators (+ &Rules of isometric operators (+ & -- values)values)

and (x, y) verses (y, x)and (x, y) verses (y, x)

The origin is always the center of rotation (notThe origin is always the center of rotation (not

true)true)

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Key ConceptsKey Concepts

Students will learn to transform images on a coordinate plane according toStudents will learn to transform images on a coordinate plane according tothe 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 glideStudents will learn to identify a reflection, rotation, translation, and glidereflection.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 andStudents will relate the corresponding points of two identical images andidentify the points using ordered pairs.identify the points using ordered pairs.

Students will transform images on the coordinate plane using multipleStudents will transform images on the coordinate plane using multipleisometries.isometries.

Students will recognize when a composition of isometries is equivalent to aStudents will recognize when a composition of isometries is equivalent to areflection, 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, & chessNature: 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 visitStudents will visit

several interactiveseveral interactive

websites for activitieswebsites for activities

& quizzes.& quizzes.Students can view aStudents can view a

video clip to learn morevideo clip to learn more

about reflections.about reflections.

Students will createStudents will createtransformations using penciltransformations 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.com

http://illuminations.nctm.org/LessonDetail.aspx?ID=L467

http://illuminations.nctm.org/LessonDetail.aspx?ID=L466

http://illuminations.nctm.org/LessonDetail.aspx?ID=L474http://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.htmlhttp://glencoe.mcgraw-hill.com/sites/0078738181/student_view0/chapter9/lesson3/self-

check_quizzes.html

http://www.unitedstreaming.com/index.cfm

http://www.freeaudioclips.com

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AcknowledgmentsAcknowledgments

Thanks to all of those that enabled us toThanks to all of those that enabled us totake this class.take this class.

These include:These include:Pinconning & StandishPinconning & Standish--Sterling School districts,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 JoeThanks also to our instructor JoeBruessow for helping us solve issues whileBruessow for helping us solve issues whilecreating this presentation.creating this presentation.

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Interactive ActivitiesInteractive Activities

Interactive Website for Rotating FiguresInteractive Website for Rotating Figures

Interactive Website Describing RotationsInteractive Website Describing Rotations

Interactive Website for Translating FiguresInteractive Website for Translating Figures

Interactive Website with Translating ActivitiesInteractive Website with Translating ActivitiesInteractive Symmetry GamesInteractive Symmetry Games

Interactive Rotating ActivitiesInteractive Rotating Activities (Click on Play Activity)(Click on Play Activity)

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Classroom Activity #1Classroom Activity #1

Reflection on a Coordinate PlaneReflection on a Coordinate Plane

QuadrilateralQuadrilateral AXYWAXYWhas verticeshas vertices

AA((--2, 1),2, 1), XX(1, 3),(1, 3), YY(2,(2, --1), and1), and WW((--1,1, --2).2).

GraphGraphAXYWAXYWand its image under reflection in theand its image under reflection in the

xx--axis.axis.

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

Activity 1Answer


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Activity #1Activity #1 AnswerAnswer

Use the vertical grid lines to find a corresponding pointUse the vertical grid lines to find a corresponding point

for each vertex so that thefor each vertex so that the xx--axis is equidistant fromaxis is equidistant from

each vertex and its image.each vertex and its image.

AA((--2, 1)2, 1)pp

AAdd

((--2,2, --1)1)XX

(1, 3)(1, 3)pp XXdd

(1,(1, --3)3)YY

(2,(2, --1)1)pp

YYdd(2, 1)(2, 1)WW((--1,1, --2)2)ppWWdd((--1, 2)1, 2)

Plot the reflected vertices and connect to form the imagePlot the reflected vertices and connect to form the image

AAddXXddYYddWWdd..

TheThe xx--coordinates stay the same, but thecoordinates stay the same, but the yy--coordinatescoordinates

are opposite.are opposite.

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

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Translations in the Coordinate PlaneTranslations in the Coordinate Plane

QuadrilateralQuadrilateral ABCDABCD has verticeshas vertices

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

GraphGraphABCDABCD and its image for the translationand its image for the translation

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

Activity 2Answer


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Activity 2Activity 2 AnswerAnswer

This translation moved every point of the preimage 2This 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) pp AAdd(1(1 -- 2, 12, 1 -- 6) or6) orAAdd((--1,1, --5)5)

BB(2, 3)(2, 3) pp BBdd(2(2 -- 2, 32, 3 -- 6) or6) orBBdd(0,(0, --3)3)CC(5, 4)(5, 4) pp CCdd(5(5 -- 2, 42, 4 -- 6) or6) orCCdd(3,(3, --2)2)

DD(6, 2)(6, 2) pp DDdd(6(6 -- 2, 22, 2 -- 6) or6) orDDdd(4,(4, --4)4)

Plot the translated vertices and connect to formPlot the translated vertices and connect to formquadrilateralquadrilateral AAddBBddCCddDDdd..

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Rotation on the Coordinate PlaneRotation on the Coordinate Plane

TriangleTriangle DEFDEFhas verticeshas vertices DD(2, 2,),(2, 2,), EE(5, 3), and(5, 3), and FF(7, 1).(7, 1).

Draw the image ofDraw the image ofUUDEFDEFunder a rotation of 45under a rotation of 45clockwise about the origin.clockwise about the origin.

Activity 3Answer


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Activity #3Activity #3 -- AnswerAnswer

First graphFirst graphUUDEFDEF..

Draw a segment from the originDraw a segment from the originOO

, to point, to pointDD

..Use a protractor to measure a 45Use a protractor to measure a 45 angle clockwiseangle clockwise

Use a compass to copy onto .Use a compass to copy onto .Name the segment .Name the segment .

Repeat with pointsRepeat with points EEandand FF..UUDDddEEddFFdd is the imageis the imageUUDEFDEFunder aunder a4545 clockwise rotation about the origin.clockwise rotation about the origin.

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Video ClipsVideo Clips

ReflectionReflection

TranslationTranslation

RotationRotation

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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

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AssessmentAssessment

SelfSelf--Quiz on ReflectionsQuiz on Reflections

SelfSelf--Quiz on TranslationsQuiz on Translations

SelfSelf--Quiz on RotationsQuiz on Rotations

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GlossaryGlossary

TransformationTransformation In a plane, a mapping for which each point hasIn a plane, a mapping for which each point has

exactly one image point and each image point has exactly oneexactly one image point and each image point has exactly one

preimage point.preimage point.

ReflectionReflection -- A transformation representing a flip of a figure over aA transformation representing a flip of a figure over a

point, line, or plane.point, line, or plane.

RotationRotation -- A transformation that turns every point of a preimageA transformation that turns every point of a preimage

through a specified angle and direction about a fixed point, calledthrough a specified angle and direction about a fixed point, called

the center of rotation.the center of rotation.

TranslationTranslation A transformation that moves all points of a figure theA transformation that moves all points of a figure the

same distance in the same direction.same distance in the same direction.

IsometryIsometry A mapping for which the original figure and its imageA mapping for which the original figure and its imageare congruentare congruent

GlossaryCont.


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Glossary ContinuedGlossary Continued

Angle of RotationAngle of Rotation The angle through which a preimage is rotatedThe angle through which a preimage is rotatedto form the image.to form the image.

Center of RotationCenter of Rotation A fixed point around which shapes move inA fixed point around which shapes move incircular motion to a new position.circular motion to a new position.

Line of ReflectionLine of Reflection a line through a figure that separates the figurea line through a figure that separates the figure

into two mirror imagesinto two mirror imagesLine of SymmetryLine of Symmetry A line that can be drawn through a plane figureA line that can be drawn through a plane figureso that the figure on one side is the reflection image of the figure onso that the figure on one side is the reflection image of the figure onthe opposite side.the opposite side.

Point of SymmetryPoint of Symmetry A common point of reflection for all points of aA common point of reflection for all points of afigure.figure.

Rotational SymmetryRotational Symmetry If a figure can be rotated less that 360If a figure can be rotated less that 360oo

aboutabouta point so that the image and the preimage are indistinguishable, thea point so that the image and the preimage are indistinguishable, thefigure has rotated symmetry.figure has rotated symmetry.

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Geometry Unit

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