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mathalivey
VISUALMATHEMATICSCOURSEIIIBLACKLINEMASTERS
Thispacketcontainsonecopyofeach
displaymasterandstudentactivitypage.
MathAlive!
VisualMathematics,CourseIII
byLindaCooperForemanandAlbertB.BennettJr.
BlacklineMasters
Copyright?1998TheMathLearningCenter,POBox12929,Salem,Oregon97309.Tel.503370-8130.Allrightsreserved.
ProducedfordigitaldistributionNovember2016.
TheMathLearningCentergrantspermissiontoclassroomteacherstoreproduceblacklinemasters,includingthoseinthisdocument,inappropriatequantitiesfortheirclassroomuse.
Thisprojectwassupported,inpart,bytheNationalScienceFoundationGrantESI-9452851.OpinionsexpressedarethoseoftheauthorsandnotnecessarilythoseoftheFoundation.
PreparedforpublicationonMacintoshDesktopPublishingsystem.PrintedintheUnitedStatesofAmerica.
DIGITAL2016
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BlacklineMasters
LESSON1ConnectorMasterA
ConnectorMasterBConnectorMasterCConnectorMasterDFocusMasterA
FocusStudentActivity1.1
FocusStudentActivity1.2
Follow-upStudentActivity1.3
LESSON2FocusMasterA
FocusMasterBFocusMasterCFocusMasterDFocusMasterEFocusMasterFFocusMasterGFocusMasterHFocusMasterIFocusMasterJ
FocusStudentActivity2.1
FocusStudentActivity2.2
FocusStudentActivity2.3
Follow-upStudentActivity2.4
LESSON3ConnectorMasterA
ConnectorMasterBFocusMasterA
FocusMasterBFocusMasterCFocusMasterD
FocusStudentActivity3.1
FocusStudentActivity3.2
FocusStudentActivity3.3
FocusStudentActivity3.4
Follow-upStudentActivity3.5
LESSON4ConnectorStudentActivity4.1
FocusMasterAFocusMasterBFocusMasterC
FocusStudentActivity4.2(optional)Follow-upStudentActivity4.3
LESSON5ConnectorMasterA(optional)
ConnectorMasterBConnectorMasterCConnectorMasterDFocusMasterA
FocusMasterBFocusMasterCFocusMasterD
FocusStudentActivity5.1
Follow-upStudentActivity5.2
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BlacklineMasters(continued)
LESSON6ConnectorMasterA
FocusMasterAFocusMasterBFocusMasterC
FocusStudentActivity6.1
FocusStudentActivity6.2
FocusStudentActivity6.3
FocusStudentActivity6.4
FocusStudentActivity6.5
Follow-upStudentActivity6.6
LESSON7ConnectorMasterA
FocusMasterAFocusMasterBFocusMasterCFocusMasterDFocusMasterE
FocusStudentActivity7.1
Follow-upStudentActivity7.2
LESSON8ConnectorMasterA
ConnectorMasterBConnectorMasterCConnectorMasterD
ConnectorStudentActivity8.1FocusMasterA
FocusMasterBFocusMasterCFocusMasterD
FocusStudentActivity8.2
Follow-upStudentActivity8.3
LESSON9ConnectorMasterA
ConnectorMasterBConnectorMasterC
ConnectorStudentActivity9.1FocusMasterA
FocusMasterBFocusMasterCFocusMasterDFocusMasterE
FocusStudentActivity9.2
FocusStudentActivity9.3
Follow-upStudentActivity9.4
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viii/MathAlive!VisualMathematics,CourseIII
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LESSON
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ConnectorMasterAFocusMasterA
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FocusMasterBFocusMasterCFocusMasterDFocusMasterE
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LESSON
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FocusMasterAFocusMasterBFocusMasterCFocusMasterDFocusMasterEFocusMasterFFocusMasterG
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LESSON
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ConnectorStudentActivity12.1
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FocusMasterAFocusMasterBFocusMasterCFocusMasterDFocusMasterE
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MathAlive!VisualMathematics,CourseIII/ix
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BlacklineMasters(continued)
LESSON
14
FocusMasterAFocusMasterBFocusMasterCFocusMasterD
FocusStudentActivity14.1
FocusStudentActivity14.2
Follow-upStudentActivity14.3
LESSON
15
ConnectorStudentActivity15.1
FocusMasterAFocusMasterBFocusMasterCFocusMasterD
FocusStudentActivity15.2
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LESSON
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ConnectorMasterAConnectorMasterBConnectorMasterC
ConnectorStudentActivity16.1FocusStudentActivity16.2
FocusStudentActivity16.3
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Follow-upStudentActivity16.5
LESSON
17
ConnectorMasterAConnectorMasterB
ConnectorStudentActivity17.1
FocusMasterAFocusMasterBFocusMasterC
FocusStudentActivity17.2
FocusStudentActivity17.3
Follow-upStudentActivity17.4
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x/MathAlive!VisualMathematics,CourseIII
ExploringSymmetryLesson1
ConnectorMasterA
ROTATIONS/TURNS
a)Completethisprocedure:
?Positionyournotecardsothatitfitsinitsframewithnogapsoroverlaps.
?Markapointanywhereonyourcardwithadot,andlabelthispointP.
?PlaceapencilpointonyourpointPandholdthepencilfirmlyinaverticalpositionatP.
?RotatethecardaboutPuntilthecardfitsbackintoitsframewithnogapsoroverlaps.
b)HowmanydifferentrotationsofthecardaboutyourpointParepossiblesothatthecardfitsbackinits
framewithnogapsoroverlaps?Assumethatrotationsaredifferentiftheyresultindifferentplacementsof
thecardinitsframe.
c)Ifonlya360。(or0。)rotationaboutyourpointP
bringsthecardbackintoitsframe,findanotherposi-tionforPonthecardsothatmorethanonedifferentrotationaboutthispointispossible.Whatarethemea-suresoftherotationsandhowdidyoudetermine
them?
?1998,TheMathLearningCenterBlacklineMasters,MA!CourseIII
Lesson1ExploringSymmetry
ConnectorMasterB
REFLECTIONS/FLIPS
Figure1belowshowstheframeforarectangularcardwithalineldrawnacrosstheframe.InFigure2,the
cardhasbeenplacedintheframe.Figure3showstheresultofreflecting,orflipping,thecardoverlinel.No-ticethatafterthereflectionoverlinel,thecarddoesnotfitbackinitsframe.
l
Figure1
l
A
D
B
C
Figure2
l
Figure3
Determineallthedifferentpossibleplacementsoflinelsothatwhenyouflipyourcardonceoverl,thecardfitsbackinitsframewithnogapsoroverlaps.
HINT:Asaguideforflippingthecardaboutaline,youcouldtapeapencilorcoffeestirrertothecardalongthepathoflinel,as
shownbelow.Thenkeepthepencilorcoffeestirreralignedwithlinelasyouflipthecard.
..
l
A
D
C
B
BlacklineMasters,MA!CourseIII?1998,TheMathLearningCenter
ExploringSymmetryLesson1
ConnectorMasterC
a)Discussyourgroup’sideasandquestionsaboutthemeaningsofthefollowingterms.Talkaboutways
thesetermsrelatetoanonsquarerectanglesuchasyournotecard.Recordimportantideasandquestionstosharewiththeclass.
i)reflectionalsymmetry
ii)axisofreflection(alsocalledlineofreflection)
iii)rotationalsymmetry
iv)centerofrotation
v)frametestforsymmetry
b)Ifashapeissymmetrical,itsorderofsymmetryisthenumberofdifferentpositionsfortheshapeinitsframe,wheredifferentmeansthesidesoftheshapeandthesidesoftheframematchindistinctlydifferentways.Developaconvincingargumentthatyourrect-angularnotecardhassymmetryoforderfour.
?1998,TheMathLearningCenterBlacklineMasters,MA!CourseIII
ConnectorMasterD
a)
c)
b)
d)
f)
e)
g)
i)
h)
j)
?1998,TheMathLearningCenterBlacklineMasters,MA!CourseIII
FocusMasterA
OurGoalsasMathematicians
Weareacommunityofmathematicians
workingtogethertodevelopour:
a)visualthinking,
b)conceptunderstanding,
c)reasoningandproblemsolving,
d)abilitytoinventproceduresandmakegeneralizations,
e)mathematicalcommunication,
f)opennesstonewideasandvariedapproaches,
g)self-esteemandself-confidence,
h)joyinlearninganddoingmathematics.
?1998,TheMathLearningCenterBlacklineMasters,MA!CourseIII
FocusStudentActivity1.1
NAMEDATE
1Foreachshapebelow,determinementallyhowmanywaysonesquareofthegridcanbeaddedtotheshapetomakeitsymmetrical.Assumenogapsoroverlapsandthatsquaresmeetedge-to-edge.
A
B
C
D
凸
F
2Foreachshapebelow,determinementallyhowmanywaysonetriangleofthegridcanbeaddedtotheshapetomakeitsymmetri-cal.Assumenogapsoroverlapsandthattrianglesmeetedge-to-
edge.
/A/
/B
C
D
F\
(Continuedonback.)
?1998,TheMathLearningCenterBlacklineMasters,MA!CourseIII
Lesson1ExploringSymmetry
FocusStudentActivity(cont.)
3Createashapethatismadeofsquaresjoinededge-to-edge(nooverlaps)andhasexactly3waysofaddingoneadditionalsquaretomaketheshapesymmetrical.
4Createashapethatismadeoftrianglesjoinededge-to-edge(nooverlaps)andhasexactly4waysofaddingoneadditionaltriangletomaketheshapesymmetrical.
BlacklineMasters,MA!CourseIII?1998,TheMathLearningCenter
ExploringSymmetryLesson1
FocusStudentActivity1.2
NAMEDATE
Writeawell-organized,sequentialsummaryofyourinvestigationofoneofProblems1or2.Includethefollowinginyoursummary:
?astatementoftheproblemyouinvestigate
?thestepsofwhatyoudo,includinganyfalsestartsanddead-ends
?relationshipsyounotice(smalldetailsareimportant)
?questionsthatoccurtoyou
?placesyougetstuckandthingsyoudotogetunstuck
?yourAHA!sandimportantdiscoveries
?conjecturesthatyoumake—includewhatsparkedandwaysyoutestedeachconjecture
?evidencetosupportyourconclusions.
1Anonsquarerectangleandanonsquarerhombuseachhave2
reflectionalsymmetries.However,the2linesofsymmetryareof2differenttypes—thelinesofsymmetryofarectangleconnectthe
midpointsofoppositesidesandthelinesofsymmetryofarhombusconnectoppositevertices.Investigateotherpolygonswithexactly2linesofsymmetryofthese2types.Generalize,ifpossible.
2What,ifany,istheminimumnumberofsidesforapolygon
with3rotationalsymmetriesandnoreflectionalsymmetry?What,ifany,isthemaximumnumberofsides?What,ifany,isthemini-mumnumberofsidesforpolygonswith4rotationalsymmetries
andnoreflectionalsymmetries?5rotationalandnoreflectionalsymmetries?nrotationalandnoreflectionalsymmetries?Investi-gate.
?1998,TheMathLearningCenterBlacklineMasters,MA!CourseIII
ExploringSymmetryLesson1
Follow-upStudentActivity1.3
NAMEDATE
1Traceandcutoutacopyofeachoftheaboveregularpolygons.Usethecopiesandoriginalpolygons,butnomeasuringtools(norulers,protractors,etc.),tohelpyoucompletethefollowingchart:
No.ofdifferent
positionsinframe
No.ofreflectionalsymmetries
No.ofrotationalsymmetries
Measuresofallanglesofrotation
Measureofeachinteriorangle*
*Interioranglesaretheangles“inside”thepolygonandareformedbyintersectionsofthesidesofthepolygon.
Completethefollowingproblemsonseparatepaper.BesuretowriteaboutanyAHA!s,conjectures,orgeneralizationsthatyoumake.
2Explainthemethodsthatyouusedtodeterminetheanglesofrotationandtheinterioranglemeasuresforthechartabove.Re-member,noprotractors.
3LabelthelastcolumnofthechartinProblem1“Regularn-gon”andthencompletethatcolumn.Foreachexpressionthatyouwriteinthelastcolumn,drawadiagram(onaseparatesheet)toshow
“why”theexpressioniscorrect.
(Continuedonback.)
?1998,TheMathLearningCenterBlacklineMasters,MA!CourseIII
Lesson1ExploringSymmetry
Follow-upStudentActivity(cont.)
4Discussthesymmetriesofacircle.Explainyourreasoning.
5Locatearesourcethatshowsflagsofthecountriesoftheworld.Foreachofthefollowing,ifpossible,sketchandcoloracopyofadifferentflag(labeleachflagbyitscountry’sname)andciteyour
resource.
a)rotationalsymmetrybutnoreflectionalsymmetry,
b)reflectionalsymmetryacrossahorizontalaxisonly,
c)nosymmetry,
d)bothrotationalandreflectionalsymmetry,
e)180。rotationalsymmetry.
6Sortandclassifythecapitallettersofthealphabetaccordingtotheirtypesofsymmetry.
7Attachpicturesof2differentcompanylogosthathavedifferenttypesofsymmetry.Describethesymmetryofeachlogo.
8Createyourpersonallogosothatithassymmetry.Recordtheorderofsymmetryforyourlogo,showthelocationofitsline(s)of
symmetry,and/orrecordthemeasuresofitsrotationalsymmetries.
9Jamaalmadeconjecturesa)andb)below.Determinewhether
youthinkeachconjectureisalways/sometimes/nevertrue.Give
evidencetoshowhowyoudecidedandtoshowwhyyourconclu-sioniscorrect.Ifyouthinkaconjectureisnottrue,edititsothatitistrue.
Ifashapehasexactly2axesofreflection,then
a)thoseaxesmustbeatrightanglestoeachother.
b)theshapealsomusthave2rotationalsymmetries.
BlacklineMasters,MA!CourseIII?1998,TheMathLearningCenter
IntroductiontoIsometriesLesson2
FocusMasterA
Investigatewaystouseslides,flips,and/orturnstomoveSquareFexactlyontoSquareD.Usewordsand/ormarkdiagramstoexplainthemovements
F
thatyouuse.
D
?1998,TheMathLearningCenterBlacklineMasters,MA!CourseIII
Lesson2IntroductiontoIsometries
FocusMasterB
F
D
BlacklineMasters,MA!CourseIII?1998,TheMathLearningCenter
IntroductiontoIsometriesLesson2
FocusMasterC
2
A
1
3
5
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7
?1998,TheMathLearningCenterBlacklineMasters,MA!CourseIII
Lesson2IntroductiontoIsometries
FocusMasterD
PartI
ItispossibletomoveShapeAdirectlytoseveralofthenumberedpositionsusingexactlyoneoftheseisometriesonlyonce:translation,reflection,orrota-tion.Findeachpositionforwhichthisispossible,
andtellthesinglemotionthatmovesShapeAtothatposition.
PartII
DescribewaystomoveShapeAfromitsstarting
positiontoeachnumberedpositionusingacombi-nationofexactlytworeflections,rotations,and/ortranslations.Note:combinationsofmorethanonetypeofmotionareallowedaslongasnomorethantwomotionsareused.
BlacklineMasters,MA!CourseIII?1998,TheMathLearningCenter
IntroductiontoIsometriesLesson2
FocusMasterE
FriezeA
FriezeB
?1998,TheMathLearningCenterBlacklineMasters,MA!CourseIII
Lesson2IntroductiontoIsometries
FocusMasterF
FriezeA
FriezeB
BlacklineMasters,MA!CourseIII?1998,TheMathLearningCenter
IntroductiontoIsometriesLesson2
FocusMasterG
FriezeA
FriezeB
FriezeC
?1998,TheMathLearningCenterBlacklineMasters,MA!CourseIII
Lesson2IntroductiontoIsometries
FocusMasterH
FriezeA
FriezeB
FriezeC
BlacklineMasters,MA!CourseIII?1998,TheMathLearningCenter
IntroductiontoIsometriesLesson2
FocusMasterI
?1998,TheMathLearningCenterBlacklineMasters,MA!CourseIII
Lesson2IntroductiontoIsometries
FocusMasterJ
BlacklineMasters,MA!CourseIII?1998,TheMathLearningCenter
IntroductiontoIsometriesLesson2
FocusStudentActivity2.1
NAMEDATE
1Shownbelowareseveralpairsofcongruentshapes.Investigate
waystouseoneormoretranslations,reflections,rotations,orcom-binationsofthem,tomoveeachfirstshapeexactlyontothesecond.Foreachpairofshapes,writeanexplanationinwordsonlyofyour“favorite”motionorcombinationofmotions;explaininenough
detailthatareaderwouldbeabletoduplicateyourmotionswithoutadditionalinformation.
a)
b)
c)
d)
e)
2Challenge.Eachmotionorcombinationofmotionsthatyou
determinedforProblem1producesamappingofthefirstshape(thepre-image)exactlyontothesecond(theimage).Howmanydifferentmappingsarethereforeachofa)-e),ifdifferentmeansthesidesofthepre-imageandthesidesoftheimagematchindistinctlydiffer-entways.
3Recordyour“Iwonder…”statements,conjectures,orconclu-sions.
?1998,TheMathLearningCenterBlacklineMasters,MA!CourseIII
Lesson2IntroductiontoIsometries
FocusStudentActivity2.2
NAMEDATE
1Shownattherightare2congruentsquares.Determinewaystouseexactlyoneisometry(translation,reflection,rotation,orglidereflection)tomoveSquareFexactlyontoSquareD.
2RepeatProblem1forthe2equilateraltrianglesshownhere:
3SketchthereflectedimageofShapeAacrosslinem.Nexttoyoursketchwriteseveralmathematicalobservationsabout
relationshipsyounotice.ThenexplainhowyouverifiedthattheimageisareflectionofShapeAacrosslinem.
4Challenge.DevelopamethodofaccuratelyreflectingShapeBacrosslinen.Showanddescribeyourmethodoflocatingthe
reflectedimageofShapeBandtellhowyouverifiedthatyourmethodwascorrect.Canyougeneralize?
F
D
\m
A
\n
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