SECONDARY MATHEMATICS III // MODULE 5

MODELING WITH GEOMETRY – 5.4

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5.4 You Nailed It!

A Practice Understanding Task

Tatianaishelpingherfatherpurchasesuppliesforadeckheis

buildingintheirbackyard.Basedonhermeasurementsfortheareaof

thedeck,shehasdeterminedthattheywillneedtopurchase24decking

planks.Theseplankswillbeattachedtotheframingjoistswith16dnails.(Tatianathinksitis

strangethatthesenailsarereferredtoas“16pennynails”andwonderswherethatwayofnaming

nailscomesfrom.Afterdoingsomeresearchshehasfoundthatinthelate1700sinEnglandthe

sizeofanailwasdesignatedbythepriceofpurchasingonehundrednailsofthatsize.Shedoubts

thatherdadwillbeabletobuyonehundred16dnailsfor16pennies.)

Nailsaresoldbythepoundatthelocalhardwarestore,soTatiananeedstofigureouthow

manypoundsof16dnailstotellherfathertobuy.Shehasgatheredthefollowinginformation.

• Thedeckrequires24deckingplanks

• Eachplankrequires9nailstoattachittotheframingjoists

• 16dnailsaremadeofsteelthathasadensityof4.67oz/in3

• Thereare16ouncesinapound

Tatianahasalsofoundthefollowingdrawingofacrosssectionofa16dnail.Sheknowsshecanuse

thisdrawingtohelpherfindthevolumeofthenail,treatingitasasolidofrevolution.(Note:The

scaleonthex-andy-axisisininches.)

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SECONDARY MATHEMATICS III // MODULE 5

MODELING WITH GEOMETRY – 5.4

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

1. Deviseaplanforfindingthevolumeofthenailbasedonthegivendrawing.Describeyourplaninwords,andthenshowthecomputationsthatsupportyourwork.

2. Deviseaplanforfindingthenumberofpoundsof16dnailsTatiana’sfathershouldbuy.Describeyourplaninwords,andthenshowthecomputationsthatsupportyourwork.

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SECONDARY MATHEMATICS III // MODULE 5

MODELING WITH GEOMETRY – 5.4

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5.4 You Nailed It! – Teacher Notes A Practice Understanding Task

Purpose:Thistaskprovidesanopportunityforstudentstosolidifytheirunderstandingofsolidsof

revolutioninaproblem-solving,modelingcontext.Studentswilldecomposeageometricsolidof

revolutionintofamiliarthree-dimensionalobjectswhosevolumescanbecalculated.Studentswill

alsoneedtodrawupontheirabilitytoreasonwithunitswhensolvingproblems.

CoreStandardsFocus:

G.MG.1Usegeometricshapes,theirmeasures,andtheirpropertiestodescribeobjects(e.g.,

modelingatreetrunkorahumantorsoasacylinder).��

�

G.MG.2Applyconceptsofdensitybasedonareaandvolumeinmodelingsituations(e.g.,persons

persquaremile,BTUspercubicfoot).��

�

G.MG.3Applygeometricmethodstosolvedesignproblems(e.g.,designinganobjectorstructureto

satisfyphysicalconstraintsorminimizecost;workingwithtypographicgridsystemsbasedon

ratios).��

RelatedStandards:N.Q.1.

StandardsforMathematicalPractice:

SMP2–Reasonabstractlyandquantitatively

SMP4–Modelwithmathematics

SECONDARY MATHEMATICS III // MODULE 5

MODELING WITH GEOMETRY – 5.4

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Vocabulary:Studentsneedtounderstandthatthequantitydensityisaderivedquantitythat

representstheweightperunitofvolumeofthenails,inthiscasemeasuredinouncespercubic

inch.

TheTeachingCycle:

Launch(WholeClass):

Discusstheproblem-solvingscenarioofthistask,andmakesurethatstudentsunderstandthe

questiontheyaretryingtoanswer.Thisisamulti-stepproblem,soemphasizetheneedwith

studentstodeviseaplanforcarryingouttheirwork.

Explore(SmallGroup):

Listenforhowstudentsplantoapproachtheworkofproblems1and2:

Forproblem1,studentswillneedtodecomposethenailintothreefamiliarshapes:thedisk(“flat”

cylinder)thatformsthetopofthenail,thecylinderthatformsthelengthofthenail,andthecone

thatformsthepointofthenail.Theywillneedtocalculatethevolumeofeachoftheseshapesby

usingthegridtogivethempertinentinformationabouttheradiusandheightofeachobject.

Finally,theywillneedtoaddthevolumeofthethreeshapestogethertogetthetotalvolumeofthe

nail.Forproblem2,studentswillneedtocalculatetheweightofa16dnailbymultiplyitsvolume

bythedensityofthesteelofwhichitiscomposed.Oncetheweightofonenailhasbeenfound,they

needtomultiplybythetotalnumberofnailsneededfortheproject.Finally,theyneedtoconvert

thistotalweightinouncestototalweightinpounds.

Discuss(WholeClass):

Havestudentssharetheirproblem-solvingplansandtheircomputationalworkforquestions1and

2,asneeded.Aninterestingquestiontoaskstudentsmightbe,“Howmuchofthetotalweightof

thenailsisduetotheseeminglyinsignificantweightofthepointed-tipofthenails?”

AlignedReady,Set,Go:ModelingwithGeometry5.4

SECONDARY MATH III // MODULE 5

MODELING WITH GEOMETRY - 5.4

5.4

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READY Topic:FindingthetrigonometricratiosinarighttriangleUsethegivenmeasuresonthetriangletowritetheindicatedtrigvalue.Writethemasafraction.Thenwritethemasadecimalroundedtothethousandthsplace.

1.sin$ =

cos $ =

tan$ =

2.sin+ =

cos + =

tan+ =

3.sin, =

cos , =

tan, =

4.sin - =

cos - =

tan - =

5.Whichtrigonometricratioisexact,thefractionorthedecimal? Explain.

6.Mycalculatortellsmethat√//= 0.7071067812.Isonevaluemoreaccuratethantheother?

Explain.

READY, SET, GO! Name PeriodDate

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SECONDARY MATH III // MODULE 5

MODELING WITH GEOMETRY - 5.4

5.4

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SET Topic:Exploringapplicationsofvolume,weightanddensityAnswerthefollowingquestionsaboutthegrainstoredinthestoragesilos.

7.Thefigureattherightisof2grain,storagesilos.Thediameterofeachmeasures24feetandtheheightofthecylindermeasures51feet.Theheightoftheconeaddsanadditional12feet.Findthetotalvolumeofonesilo.

8.Howmanybushelsofgrainwilleachsilobeabletostore,ifabushelis1.244cubicfeet?(Assumeitcanbefilledtothetop.)

9.Densityrelatestothedegreeofcompactnessofasubstance.Acubicinchofgoldweighsagreatdealmorethanacubicinchofwoodbecausegoldismoredensethanwood.Thedensityofgrainsalsovaries.Usetheinformationbelowtocalculatehowmanytonsofeachgraincanbestoredinonesilo.(1ton=2000lbs.)

1bushelofoatsweighs32pounds

1bushelofbarleyweighs48pounds

1bushelofwheatweighs60pounds

10.A¾-tonpickuphasthecapacitytohaulalittlemorethan1500lbs.Ifthehaulingbedofthepickupmeasures4ft.by6.5ft.by2ft.,cana¾-tonpickupsafelyhaulafull(level)loadofoats,barley,orwheat?Justifyyouranswerforeachtypeofgrain.

21

SECONDARY MATH III // MODULE 5

MODELING WITH GEOMETRY - 5.4

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GO Topic:FormsoflinearandquadraticfunctionsWritewhatyouknowaboutthefunction(includingend-behavior)andthengraphit.

11. Equation:7(9) = (9 − 2)(9 + 3) Graph:

WhatIknowaboutthisfunction:Endbehavior:>?9 → −∞, 7(9) → _____>?9 → ∞, 7(9) → _____

12. Equation:D(9) = 9/ + 69 + 9 Graph:

WhatIknowaboutthisfunction:Endbehavior:>?9 → −∞, 7(9) → _____>?9 → ∞, 7(9) → _____

22

SECONDARY MATH III // MODULE 5

MODELING WITH GEOMETRY - 5.4

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13. Equation:F = −9/ − 4 Graph:

WhatIknowaboutthisfunction:Endbehavior:>?9 → −∞, 7(9) → _____>?9 → ∞, 7(9) → _____

14. Equation:ℎ(9) = 2(9 − 5) + 3 Graph:

WhatIknowaboutthisfunction:Endbehavior:>?9 → −∞, 7(9) → _____>?9 → ∞, 7(9) → _____

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