ConTEnTs Shape Makers l v

ConTEnTs

DoWnLoADInG sKETCHPAD DoCuMEnTs iii

PHILosoPHy AnD InsTRuCTIonAL ovERvIEW

BAsIC TEnETs oF THE InsTRuCTIonAL APPRoACH 3Mathematics as Reasoning 3Geometry as the Study of Structure 3Learning Mathematics in a Culture of Inquiry and Sense Making 3

Learning 3Teaching 3Establishing a Culture of Inquiry 4Responsibilities in a Culture of Inquiry 4

Understanding Students’ Geometric Thinking 5The Van Hiele Levels 5At What Levels Are Middle-Grade Students Functioning? 7

THE SHAPE MAKERS CoMPuTER MICRoWoRLD 8Dynamic Mental Models 8The Shape Makers Computer Microworld 9The Shape Makers CD 10Accessing Shape Maker Sketches 10

InsTRuCTIonAL ovERvIEW 12Goals 12Classroom Setup 12Computer Management 12The Importance of Student Predictions 13Cultivating Conjectures 13Justifying Ideas 14Assessment 14

oRGAnIZATIon oF THE BooK 15General Organization 15Glossary of Geometric Concepts 15Overview of Explorations 15

QuADRILATERAL EXPLoRATIons

QuADRILATERAL EXPLoRATIon 1 Shape Maker Pictures 19Session 1 20

Explaining the Operation of the Shape Makers 20Activity:MakeYourOwnPicture 21

Sessions 2–4 23Activity:CanYouMakeIt? 23

TeachingNote:ShapesVersusShapeMakers 25TeachingNote:EarlyReasoningAboutShapeMakers 26

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TeachingNote:NurturingProgressbyEncouragingReflection 27MathematicalNote:ICan’tDoIt,orIsItImpossible? 29

QuADRILATERAL EXPLoRATIon 2 Shape Maker Games 30Session 1 31Activity:ShapeMakerChallengeGame 31

Session 2 34Activity:IdentifytheHidingShapeMakers1 34

TeachingNote:PerturbationandProgress 35TeachingNote:EstablishingClassroomConsensusonLanguage 36TeachingNote:ControlPoints,Shapes,andProperties 37

QuADRILATERAL EXPLoRATIon 3 Predict and Check 39Sessions 1–3 40

Introducing Shape Maker Measurements 40Activity:PredictandCheck 41Activity:IdentifytheHidingShapeMakers2 42

MathematicalNote:MeasuringSidesandAngles 43MathematicalNote:TheMeaningofDegree 45TeachingNote:TheRectangleMakerCan’tMakeaSlant 45TeachingNote:CultivatingClarity 49

QuADRILATERAL EXPLoRATIon 4 Shape Maker Properties 51Session 1 52

Introducing Measured Shape Makers to Students 52Sessions 2–4 56Activity:WhichShapeMakersCanYouUse? 56Activity:ConjecturesandQueries 57

Session 5 58Activity:HowAreTheytheSame? 58

Sessions 6 and 7 59Activity:ShapeMakerRiddles 59

Session 8 60Activity:AnglesinIntersectingandParallelLines 60

TeachingNote:WhatDoesParallel Mean? 62TeachingNote:MirrorImages 65MathematicalNote:Symmetry 67MathematicalNote:PrecisionofMeasurements 69TeachingNote:AssessingStudents’IdeasAbouttheStudent

Sheet “How Are They the Same?” 71

QuADRILATERAL EXPLoRATIon 5 Shape Maker Mysteries 73Sessions 1–3 74Activity:TheMysteryofPolygonFlats 74

Sessions 4 and 5 76Activity:MakeYourOwnMysteryofPolygonFlats 76

TeachingNote:RightAnswers,IncompleteReasons 78

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QuADRILATERAL EXPLoRATIon 6 Relationships Between Shape Makers 80Sessions 1 and 2 81Activity:RelatingShapeMakers 81

Sessions 3 and 4 82Activity:MathematicalDebates 82

Sessions 5 and 6 83Activity:MakingaQuadrilateralClassificationChart 83Activity:QuadrilateralsAssessment 85

TeachingNote:ShapeMakersasRepresentationsofClasses of Shapes 85TeachingNote:SomeExampleDebates 86MathematicalNote:TheMathematician’sView 91

TRIAnGLE EXPLoRATIons

TRIAnGLE EXPLoRATIon 1 Making Pictures 95Sessions 1 and 2 96Activity:CanYouMakethePicture? 96

TRIAnGLE EXPLoRATIon 2 Measured Triangle Makers 98Sessions 1 and 2 99Activity:InvestigatingMeasuredTriangleMakers 99

Session 3 100Activity:TheSumoftheAnglesinaTriangle 100

Session 4 102Activity:HowAretheTrianglestheSame? 102

TeachingNote:InvestigatingMeasuredTriangleMakers 103TeachingNote:DiscussingMeasuredTriangleMakers 106

TRIAnGLE EXPLoRATIon 3 Other Kinds of Triangles 111Session 1 112Activity:OtherKindsofTriangles 112

TeachingNote:ItTakesMoreThanaPrettySmiletoBe(an) Acute Triangle 113

TRIAnGLE EXPLoRATIon 4 Making Shapes from Triangle Pairs 115Session 1 116

Introduction to Making Shapes from Triangle Pairs 116Activity:MakingShapesfromTrianglePairs 120

Sessions 2–4 121Activity:MakingSpecialQuadrilateralsfromTrianglePairs 121

Session 5 123Activity:WhatKindofTriangle?WhatKindof Quadrilateral?(Assessment) 123

TeachingNote:MakingaRectanglefromTwoRightTriangles 125TeachingNote:ProvingIt’saSquare;ProvingIt’saParallelogram 128

TRIAnGLE EXPLoRATIon 5 Tessellations 132Sessions 1 and 2 133Activity:TriangleTessellations 133

TeachingNote:DescribingOurTessellations 136

EXTEnsIons 141Polygon Flats with Verbal Descriptions 141Mystery Shape Makers Again 141Minimal Definitions for Shapes 141ExploringDiagonalsofQuadrilaterals 142Exploring Similarity 143InscribingQuadrilateralsandTrianglesinCircles 143Students Construct Their Own Shape Makers 145

GEoMETRIC GLossARy 147

sTuDEnT sHEETs 151

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