5.4 you nailed it! - utah education network · 2018-08-02 · secondary mathematics iii // module 5...
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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
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
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
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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
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
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SECONDARY MATH III // MODULE 5
MODELING WITH GEOMETRY - 5.4
5.4
Needhelp?Visitwww.rsgsupport.org
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
Needhelp?Visitwww.rsgsupport.org
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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.
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SECONDARY MATH III // MODULE 5
MODELING WITH GEOMETRY - 5.4
5.4
Needhelp?Visitwww.rsgsupport.org
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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) → _____
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SECONDARY MATH III // MODULE 5
MODELING WITH GEOMETRY - 5.4
5.4
Needhelp?Visitwww.rsgsupport.org
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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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