SECONDARY MATH III // MODULE 5

MODELING WITH GEOMETRY – 5.1

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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

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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

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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

4

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.

5

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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