5.1 any way you slice it - utah education network · 2018-08-02 · secondary math iii // module 5...
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SECONDARY MATH III // MODULE 5
MODELING WITH GEOMETRY – 5.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
5.1 Any Way You Slice It
A Develop Understanding Task
StudentsinMrs.Denton’sclassweregivencubesmadeof
clayandaskedtosliceoffacornerofthecubewithapieceof
dentalfloss.
Jumalslicedhiscubethisway.
Jabarislicedhiscubelikethis.
1. Whichstudent,JumalorJabari,interpretedMrs.Denton’sinstructionscorrectly?Whydoyousayso?
Whendescribingthree-dimensionalobjectssuchascubes,prismsorpyramidsweuse
preciselanguagesuchasvertex,edgeorfacetorefertothepartsoftheobjectinordertoavoidthe
confusionthatwordslike“corner”or“side”mightcreate.
Acrosssectionisthefaceformedwhenathree-dimensionalobjectisslicedbyaplane.It
canalsobethoughtofastheintersectionofaplaneandasolid.
2. DrawanddescribethecrosssectionformedwhenJumalslicedhiscube.
3. DrawanddescribethecrosssectionformedwhenJabarislicedhiscube.
4. Drawsomeotherpossiblecross-sectionsthatcanbeformedwhenacubeisslicedbyaplane.
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SECONDARY MATH III // MODULE 5
MODELING WITH GEOMETRY – 5.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
5. Whattypeofquadrilateralisformedbytheintersectionoftheplanethatpassesthroughdiagonallyoppositeedgesofacube?
Explainhowyouknowwhatquadrilateralisformedbythiscrosssection.
Crosssectionscanbevisualizedindifferentways.OnewayistodowhatJumalandJabari
did—cutaclaymodelofthesolidwithapieceofdentalfloss.Anotherwayistopartiallyfillaclear
glassorplasticmodelofthethree-dimensionalobjectwithcoloredwaterandtiltitinvariousways
toseewhatshapesthesurfaceofthewatercanassume.
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SECONDARY MATH III // MODULE 5
MODELING WITH GEOMETRY – 5.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Experimentwithvariouswaysofexaminingthecrosssectionsofdifferentthree-
dimensionalshapes.
6. Partiallyfillacylindricaljarwithcoloredwater,andtiltitinvariousways.Drawthecrosssectionsformedbythesurfaceofthewaterinthejar.
7. Trytoimagineacubicaljarpartiallyfilledwithcoloredwater,andtiltedinvariousways.Whichofthefollowingcrosssectionscanbeformedbythesurfaceofthewater?Whichareimpossible?
• asquare
• arhombus
• arectangle
• aparallelogram
• atrapezoid
• atriangle
• apentagon
• ahexagon
• anoctagon
• acircle
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SECONDARY MATH III // MODULE 5
MODELING WITH GEOMETRY – 5.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
5.1 Any Way You Slice It – Teacher Notes A Develop Understanding Task
Purpose:Thepurposeofthistaskistosurfaceavarietyofstrategiesforvisualizingtwo-
dimensionalcrosssectionsofthree-dimensionalobjects,andtoidentifyand/ordrawsuchcross
sections.Studentsencountercrosssectionswhentheyslicealoafofbread,apieceofcake,ora
hard-boiledegg,orwhentheytiltaglassofwaterindifferentwaysandexaminethesurfaceofthe
water.Thistaskaimstoformalizetheseobservationsbydefiningacrosssectionastheintersection
ofaplaneandathree-dimensionalobject.
CoreStandardsFocus:
G.GMD.4Identifytheshapesoftwo-dimensionalcross-sectionsofthree-dimensionalobjects,and
identifythree-dimensionalobjectsgeneratedbyrotationsoftwo-dimensionalobjects.
StandardsforMathematicalPractice:
SMP7–Lookforandmakeuseofstructure
Vocabulary:Studentswillneedtounderstandthatacrosssectionistheshapeofthesurface
formedwhenageometricsolidisslicedbyaplane.
TheTeachingCycle:
Launch(WholeClass):
Givestudentsafewminutestorespondtoquestions1-4individually,andthendiscussthemasa
class.Studentsshouldnotethat“corner”isanambiguousterm,sinceitcanrefertothevertexpoint
wheretheedgesofthecubemeet,ortothethreedimensionalregionwheretwofacesofthecube
meet,suchaswhenwesay,“Gostandinthecorneroftheroom.”Encouragestudentstousemore
preciselanguageastheyworkthroughthis,andsubsequenttasks.
Question4shouldhighlightthestrategyofdrawinginthe“edges”onthefacesofthecubewhere
theplaneintersectsthefaces,suchasinthefollowingdiagrams.
SECONDARY MATH III // MODULE 5
MODELING WITH GEOMETRY – 5.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Followingthisintroductorydiscussion,setstudentstoworkontheremainderofthetask.Setup
somestationsintheclassroomwherestudentscanaccessthematerialsneededtoworkon
questions6and7.
Explore(SmallGroup):
Studentscandiscussquestion5insmallgroupswhilewaitingfortheirturnstoaccessthematerials
forquestions6and7.Studentsmayinitiallythinkthattheshadedcrosssectioninquestion5isa
parallelogram(orarhombus),sinceitlookslikeoneinthistwo-dimensionalimage.Listenfor
students’justificationastowhattypeofquadrilateraltheyclaimittobe.Askhowtheymight
justifythatonesidelengthislongerthanorthesamelengthasanother.Howmighttheyreason
abouttheanglesinthequadrilateral?
Ifpossible,forquestion6provideavarietyofsealedcontainers,includingacylinder,eachpartially
filledwithcoloredwater.Itmightbesurprisingtostudentstofindthattheycancreaterectangular
crosssectionsinacylinder,oratriangularcrosssectioninacone.Asanalternativeapproachto
thisquestion,allowstudentstopartiallysubmergeobjectsinwaterandtracethe“edges”wherethe
objectintersectsthesurfaceofthewater.Regardlessofhowstudentscollectthedata,theyshould
sketchthevarioustypesofcrosssectionsthatcanbeformedbyintersectingaparticularobject
withaplane.
Inquestion7watchforstudentswhofinditdifficulttovisualizehowtodrawcrosssectionswithin
atwo-dimensionaldrawingofathree-dimensionalobject.Howdotheyattendtothevertices,
SECONDARY MATH III // MODULE 5
MODELING WITH GEOMETRY – 5.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
edgesandfacesthatwouldbeintersectedbyasingleplane?Watchforstudentswhocreate
“impossible”crosssectionsbyusingpointsonthesameedgeorfacethatcouldnotpossiblylieina
singleplane.
Discuss(WholeClass):
Thisisanopen-endedtaskthatisintendedtosurfacedifferentwaysofthinkingaboutcross
sectionswhenwecan’tactuallyexperimentwithanobjectdirectly.Discussthestrategiesthathave
emergedforstudentsandrelatethesebacktotheideasthatwasintroducedinquestion4:
specifically,toimaginetracingthe“edges”ofthefigureoutliningthesurfacewheretheplane
intersectstheobject.
Havestudentsdrawanddescribesomeofthecrosssectionstheynotedinvariousthree-
dimensionalshapesthatwereunexpectedorsurprisingtothem,suchastherectangularcross
sectionsinacylinderorthehexagoncrosssectioninacube.
AlignedReady,Set,Go:ModelingwithGeometry5.1
SECONDARY MATH III // MODULE 5
MODELING WITH GEOMETRY – 5.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Cubedrawingsforusewithquestion#7
SECONDARY MATH III // MODULE 5
MODELING WITH GEOMETRY - 5.1
5.1
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READY Topic:Comparingperimeter,areaandvolume
Solveeachofthefollowingproblems.Makecertainyoulabeltheunitsoneachofyouranswers. 1. Calculatetheperimeterofarectanglethatmeasures5cmby12cm.2. Calculatetheareaofthesamerectangle.3. Calculatethevolumeofarectangularboxthat
measures5cmby12cm.andis8cm.deep.4. Lookbackatproblems1–3.Explainhowtheunitschangeforeachanswer.5. Calculatethesurfaceareafortheboxinproblem3.AssumeitdoesNOThaveacoverontop.
Identifytheunitsforthesurfacearea.Howdoyouknowyourunitsarecorrect?6. Calculatethecircumferenceofacircleiftheradiusmeasures8inches.(Useπ=3.14)7. Calculatetheareaofthecircleinproblem6.
8. Calculatethevolumeofaballwithadiameterof16inches.!" = %& '(&)
9. Calculatethesurfaceareaoftheballinproblem8.(+, = 4'(.)10. Ifameasurementweregiven,couldyouknowifitrepresentedaperimeter,anarea,ora
volume? Explain.
11. Intheproblemsabove,whichtypeofmeasurementwouldbeconsidereda“linearmeasurement?”
READY, SET, GO! Name PeriodDate
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SECONDARY MATH III // MODULE 5
MODELING WITH GEOMETRY - 5.1
5.1
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SET Topic:ExaminingthecrosssectionsofaconeConsidertheintersectionofaplaneandacone.
12. Iftheplanewereparalleltothebaseofthecone,whatwouldbetheshapeofthecross-section?Canthinkof2possibilities?Explain.
13. Howwouldaplaneneedtointersecttheconesothatitwouldcreateaparabola?
14.Describehowtheplanewouldneedtointersecttheconeinordertogetacross-sectionthatisatriangle.Wouldthetrianglebescalene,isosceles,orequilateral?Explain.
15.Woulditbepossiblefortheintersectionofaplaneandaconetobealine?Explain.
GO Topic:Findingtheareaofatriangle
CalculatetheareaoftriangleEFGineachexercisebelow.
16.
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SECONDARY MATH III // MODULE 5
MODELING WITH GEOMETRY - 5.1
5.1
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17. 18.
19.Calculatetheareasof∆123, ∆153, 678∆193.Justifyyouranswers.
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