1. TRACE'ONE'SECTOR'from'each'of'these'circles'and'then'cut'it'out.''' 2. Label'the'central'angle'(90,'60,'or'30 )'to'distinguish'the'sectors.
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- Κρέων Κωνσταντόπουλος
- 5 χρόνια πριν
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1 Module3Lesson5Activity Name: Date: Hour: DEVELOPINGAREAFORMULAS:AREAOFACIRCLE Aninformalmethodcanbeusedtodeterminetheareaofacircle.Todothiswearegoingtouseadissection technique. 1. TRACEONESECTORfromeachofthesecirclesandthencutitout. 2. Labelthecentralangle(90,60,or30 )todistinguishthesectors. Circle#1(4Sectorsof90 each) Circle#2(6Sectorsof60 each) Circle#3(12Sectorsof30 ) Module3Lesson5Page28
2 1)CIRCLE#1 Continuetotracetheothertwosectorsinthesamepatternaswasbegunhere. Whatshapedoesthisfigureseemtoresemble? 2)CIRCLE#2 Continuetotracetheotherfoursectorsinthesamepatternaswasbegunhere. Whatshapedoesthisfigureseemtoresemble? Howdoesthisfiguredifferfromtheonecreatedfromcircle#1above? Whatistherelationshipoftheareabetweenthefigurein#1andthisfigurein#2? Whatvaluewouldapproximatethe base ofthisfigure? Whatvaluewouldapproximatethe height ofthisfigure? Module3Lesson5Page29
3 3)CIRCLE#3 Continuetotracetheothertensectorsinthesamepatternaswasbegunhere. Whatshapedoesthisfigureseemtoresemble? Howdoesthisfiguredifferfromtheonescreatedfromcircle#1above? Whatistherelationshipoftheareabetweenthesethreefigures? Whatvaluewouldapproximatethe base ofthisfigure? Whatvaluewouldapproximatethe height ofthisfigure? Findaformulatoapproximatetheareaofthisfigureusingtheapproximatebaseandheightyouhavefound above. Iftheangleofthesectorwas1,insteadof30,howwouldthisaffecttheshapeofthefigure? 4)InformalLimitArgument Ifwesplitthecircleintoinfinitelymanypiecesandalternatedthemlikewe vedoneabove,whatshapewould italmostexactlyresemble? Howdoesthisactivityhelpusunderstandthattheareaofacircleisπr 2? Thisiscalledan.Whenthecircleissplitintomore andmorepieces,itwilleventuallyresembleaparallelogramwhenwealternateandputthepiecestogether.if thenumberofsides,n,getsclosertoinfinity,theshapewillbecloserandclosertoaparallelogram.thus,we canusethemeasurementsoftheparallelogramwe vecreatedtofindtheareaofthecircle. Module3Lesson5Page30
4 Module3Lesson5InYClassNotes Name: Date: Hour: 1.Recallourpreviousactivitywherewediscoveredtheformulafortheareaofacircle.Describehowwe useaninformallimitargumenttoexplainwheretheformulafortheareaofacirclecomesfrom.drawa diagramtohelpyourexplanation. 2.Determinethemissingmeasureofthecircle.Rememberthatwefoundtheareaofacircletobe =. (E)meansleaveanexactanswer. a)r=3cm A= (E) b)d=12cm A= (E) c)a= " cm2 r= d)a=16cm 2 d= 3.Determinetheareaofthecircle. a) b) c) 4"cm 16#cm 32#cm Area= (E) Area= (E) Area= (E) Module3Lesson5Page31
5 4.Determinetheareaofthecirclesector. a) b) c) 30 4%cm 270 3&cm 83 8$cm Area= (E) Area= (E) Area= (2dec.) 5.Determinetheareaofthefollowingfigures.(Linesthatappeartobeperpendicularareperpendicularand linesthatappeartobeparallelare.)(e)meanstoleaveanexactanswer. a) b) Area= (E) Area= (E) c) Area= (E) Module3Lesson5Page32
6 6.DeterminetheareaoftheSHADEDregion. (Linesthatappeartobeperpendicularareperpendicular.Linesthatappeartobeparallelareparallel.) a)asquarewithasideof6cm.theinscribedcircle alsohasadiameterof6cm. b)arectanglewithdimensionsof6cmand10cm. Areaofshadedregion= (E) Areaofshadedregion= (E) c)twoconcentriccircleswithradiiof2cmand3cm. Areaofshadedregion= (E) Module3Lesson5Page33
7 Module3Lesson5Homework Name: Date: Hour: AREAOFCIRCLES 1.Findthemissingpartsofthecircle.Rememberthatwefoundtheareaofacircletobe =. (E)meanstoleaveanexactanswer. a)r=4cm Area= (E) b)d=9cm Area= (E) c)area=9cm 2 r= d)area=2cm 2 r= (E) 2.Determinetheareaofthecirclesector &cm Area= (E) 3.Determinetheareaofthefollowingfigures. a) b) Area= (E) Area= (E) Module3Lesson5Page34
8 Module3Lesson6In.ClassNotes Name: Date: Hour: LAUNCH: WatchtheVolumeSongvideoandanswerthequestionsbelow. a)howmanydimensionsdoesvolumehave? b)ifoneedgeofacubeis3cm,whatisthevolumeofthecube? CalculatingtheVolumeofaPrism Thevolumeofaprismisfoundbymultiplyingtheareaofthebaseshapetimesthemeasureoftheheight. wherebistheareaofthebaseandhistheheightoftheprism. INTRODUCTIONTOVOLUME Solid AthreeDdimensionalclosedspatialfigure. Polyhedron Ageometricsolidwithpolygonsasfaces. FaceofaPolyhedron Oneofthepolygonsthatformthe polyhedron.sometimesthesegetcalledsidesbutthebetter termisface. LateralFaceofaPolyhedron Afaceinaprismorpyramid thatisnotabase. Edge Theintersectionoftwofacesofapolyhedron. Vertex Theintersectionoftwoormoreedges. Prisms A isapolyhedronwithlateralfacesthatareparallelograms,andendfaces(bases)thatare parallelandequalinshapeandsize.prismsarenamedbytheshapesoftheirbases.forexample,a rectangular*prismisaprismwithrectangularbases.atriangular*prismisaprismwithtriangularbases.ifthe baseandthelateraledgesarenotperpendicularthentheprismiscalledan PRISM. Module3Lesson6Page35
9 1.Matchthefollowingtermstothediagram. GiventherectangularprismwithfaceBCFEasoneofitsbases.Useeach valueonlyonce. 1.Edge 2.LateralFace 3.Base 4.Vertex 5.Height A.RectangleADHG B. HF C. AD D.PointB E.RectangleHDCF 2.Afterlookingattherectangularprismtotheright,astudentintheclassraises herhandandsays, CouldIuserectangleADCBasmybaseinsteadofrectangle BHGC? Howshouldtheteacherrespond? 3.Properlynamethefollowingprisms.HINT:Considerthebaseofeachprism. a) b) c) Name: Name: Name: e) f) g) Name: Name: Name: Module3Lesson6Page36
10 4.Mikedoesn tunderstandhowvolumeworksforaprismandhenryistryingtoexplainittohim. It s whatisinsidetheshape forexample,ifyoucalculatedtheareaofonepieceofpaperandthenstacked 100piecesofpaperontopofeachotheritwouldcreateaprismandthevolumewouldbetheareaofthe onepieceofpapermultipliedbytheheightofthestack. Mikeisstillconfused,canyougiveanother exampletoexplainthisconcept. Cross.Sections InordertohaveabetterunderstandingofthreeDdimensionalshapes,let stalkaboutcrossdsections.firstlet s distinguishbetweenasliceandacrossdsection. Slice Theshapemadewhenasolidiscutbyaplane. Cross.Section Theshapemadewhenasolidiscutbyaplanethatisparalleltothebase. CrossDsectionofasquarepyramid CrossDsectionofalog CrossDsectionofabaseball 5.Identifythecross.section: a)cone b)hexagonalpyramid CrossDsection: CrossDsection: c)obliquetriangularprism CrossDSection: 6.Sketchthefigurefromwhichthecross.sectionwastaken. a)triangle b)trapezoid Module3Lesson6Page37
11 RotationsaboutanAxis SolidscanalsobeformedfromrotatingatwoDdimensionalshapeaboutanaxis,asdemonstratedbelow. 7.Thesolidbelowwasformedbyrotatingatwo.dimensionalshapeabouttheverticalaxis. Sketchthetwo.dimensionalshapethatitcamefromontheaxisbelow. Cavalieri sprinciple BonaventuraFransescoCavalieri(1598D1647)wasadiscipleofGalileoandheinvestigatedthestacking principle.hecametodefinewhatisnowknownasthecavalieriprinciple: Iftheareasofthecrosssectionsoftwosolidsbyanyplaneparalleltoagiven planearealwaysequal,thenthetwosolidshavethesame. Inotherwords,iftwoprismshavethesameheightandthesamebasethenobliqueandrightprismswillhave thesamevolumes.thisisalittleliketheshearingtechniquebutinthethirddimension. Volumesareequal. Volumesareequal. Module3Lesson6Page38
12 8.Cavalieri sprinciplesaysthatthesetwoprismshaveequal volume.explainwhythatistrue? 9.IftheVolumeofthecubeis(4)(4)(4)=64 cm 3,whatisthevolumeoftheobliqueprism ifithasbeentiltedat60? 10.JennysaysthatthetwoprismsDONOThavethesamevolumebecausethecrosssectionsarenotthe same.reneedisagrees;shesaysthatitisn ttheshapethathastobethesame,itisthearea.reneethinks theyhavethesamevolume.whoisrightandwhy? Cylinders Acylinderisaclosedsolidthathastwoparallelcircularbasesconnectedbyacurvedsurface. 11.Randysays Cylindervolumeiseasy itisdonethesamewayasaprismexceptitsbaseisacircle. WhatdoesRandymeanbythis? Module3Lesson6Page39
13 12.Determinethevolumeoftheprisms.(Linesthatappearperpendicularareperpendicular.) a) b) Volume= c)regularhexagonalprism Volume= Volume= 13.Findthevolumeofthefollowingcylinders. a) b) Volume= (E) Volume= (E) 14.Determinethevolumeofthefollowingcompositefigures.(Linesthatappeartobeperpendicularare perpendicularandlinesthatappeartobeparallelare.) a) b) Volume= Volume= (E) Module3Lesson6Page40
14 Module3Lesson6Page41 Module3Lesson6Homework Name: Date: Hour: 1.Findthevolumeofthefollowingshapesandwritethecross.sectionalshapewhennecessary. (Linesthatappearperpendicularareperpendicular.)(E)meanstoleaveanexactanswer. a) b) c)regularhexagonalprism Volume= ShapeofcrossDsection: Volume= (E) ShapeofcrossDsection: Volume= (E) ShapeofcrossDsection: d) e) f) Volume= (E) ShapeofcrossDsection: Volume= ShapeofcrossDsection: Volume= (E) e) Volume= (E)
15 Module3Lesson7In.ClassNotes Name: Date: Hour: Part1 VolumeofPyramids WhatisaPyramid? Apyramidisapolyhedronwithapolygonal andtriangularlateral,whichmeetata pointatthetopcalledthe. 1.Matchthefollowingtermstothediagram.Fortheonesyoudon tknow,thinkaboutthemeaningofthe termstofigurethemout. Giventhesquarepyramid. 1.SlantHeight 2.Apex 3.Height 4.LateralEdge 5.Face 6.Vertex 2.JeffmissedclassandDillonisexplainingthenotes. Theslantheightandtheheight ofthepyramidbasicallymeanthesamething. Isthissummaryofheightcorrect? Explain. NamingPyramids Pyramidsarenamedbytheirbases.Forinstance,apyramidwithasquarebaseiscalledasquarepyramid.If thebaseshapeisahexagon,thepyramidiscalledahexagonalpyramid.apyramidwithatriangularbasehasa specialname:tetrahedron.a pyramidisonewheretheapexliesdirectlyabovethecenterofthe base.an pyramidisonewheretheapexisnotaligneddirectlyabovethecenterofthebase. 3.Properlynamethepyramid. a) b) c) d) Name: Name: Name: Name: Module3Lesson7Page42
16 4.Twopyramidswiththesamebasearesidebyside.Oneisarightpyramidandtheotherisanoblique pyramid.iftheobliquepyramidhasbeentiltedtoanangleof80,whatisthevolumerelationship betweenthetwopyramids? CalculatingVolumeofPyramids FromourLaunchactivity,wediscoverthatthevolumeofapyramidis. 5.Determinethevolumeofthepyramid.(E)meanstoleaveanexactanswer. a)rectangularpyramid b)equilateraltriangularpyramid c)regularhexagonalpyramid V= Volume= (E) Volume= (E) Volume= (E) 6.Determinethevolumeofthecompositefigure. Volume= Module3Lesson7Page43
17 Module3Lesson7 Part1Application Name: Date: Hour: Therearemanypyramidsoutsideoftheclassroom.TwoexamplesaretheGreatPyramidofGizaandthe LouvrePyramidinParis.Choosearealworldpyramidandresearchormeasureitsdimensionstocalculatethe volume.youmustshowyourwork. Whichpyramiddidyouchoose? Length: Width: Height: Namethetypeofpyramiditis: Websiteortoolyouusedtogetthedimensionsofthepyramid: VolumeofPyramid: Module3Lesson7Page44
18 Lesson7Part2 VolumeofCones LAUNCH Theconeandcylindertothelefthavethesamebaseandheight.Whatdo youthinkistherelationshipbetweenthevolumeoftheconeandthevolume ofthecylinder? Let sfindoutwatchthevideo,thenthinkaboutwhatthevolumeformulaforaconemightbe.writeithere: THECONE Theconeandthecylinderrelationshipfollowthesamepatternasthe and justdid. Youcanpourthesandorthewaterfrom theconetothecylinderandyouwillfind thattherelationshipisexactlyone_third thevolumeofthecylinder. = CONECALCULATION Determinethevolumeofthecone. a) b) c) Volume= (E) Volume= (E) Volume= (E) Module3Lesson7Page45
19 ApplyingVolumeFormulasforPyramidsandCones Name: Hour: 1.Findthevolumeofeachfigurebelow. a) b) c) Volume= (E) d) Volume= (E) e) Volume= (E) Volume= (E) Volume= (E) 2.Explainwhy = "workstosolveforthevolumeofapyramidorcone. Module3Lesson7Page46
20 Lesson7Part3 VolumeofSpheres LAUNCH Watchthevideoandfillintheblanksforwhythevolumeofasphereis =. Let sstartwithacylinderwitharadiusofrandaheightof. Weknowtheformulaforthevolumeofacylinderis = h.sowhenwepluginourheightof2r,our formulabecomes = 2 =. Ifwehaveaspherewiththesameradiusasthecylinder,howmanyhemispheresareneededtofillthe cylinder? Thusthevolumeofthehemisphereis thevolumeofthecylinder. Thismeansthatthevolumeofthehemisphereis = 2 =. Sincethehemisphereis½ofthesphere,weneedtomultiplythevolumeofthehemisphereby. Thismeansthatthevolumeofthesphereis = 2 =. VolumeofaSphere Thenicethingabouttheformulaforasphereisthatthereisonlyonevariable involved,theradius.theradiusrepresentsallthreedimensions. 3 VSPHERE = π r CalculatingtheVolumeofaSphere 1.Determinethevolumeofthespheresorhemispheresbelow. a) b) c)volumeofaquarium 4 3 Volume= (E) Volume= (E) Volume= (E) Module3Lesson7Page47
21 2.Determinethevolumeofthecompositeshapesbelow. a) Volume= (E) 3.Findtheradiusofthesphere. b)twotennisballsfitsexactlyinthe48cm tallcylindericalcan.whatisthevolumeof airinthecan? 48cm Volume= (E) a) = 36cm 3 b) = 972cm 3 = = Module3Lesson7Page48
22 Module3Lesson8-Density Name: Date: Hour: Launch Ashippingcontainerisintheshapeofarightrectangularprism.Ithasalengthof12feet,widthof6feet,and heightof7feet.thecontaineriscompletelyfilledwithcontentsthatweigh,onaverage,2.3poundspercubic foot.whatistheweight,inpounds,ofthecontentsinthecontainer? Inordertoanswerthequestion,let sbreakitdowninsteps. 1. a)whatarethedimensionsoftheshippingcontainer? Length Width Height b)howmanycubicfeetdoestheshippingcontainerhold?(inotherwords,whatisitsvolume?) 2. Onaverage,howmanypoundspercubicfootdothecontentsweigh? lbs 3. Whatistheweightofthecontentsinthecontainer? lbs Density Densityistheratioofsomethingperunitareaorvolume.Someexampleswouldbethenumberofcattleper squaremile,gramsoflemonademixperliterofwater,numberoftreespersquareacre,etc. PopulationDensityisatypeofdensitythatisameasurementofpeopleperunitvolumeorarea.Anexample wouldbethenumberofpeoplepersquaremileinacity. Module3Lesson8Page 49
23 PracticewithDensityCalculations 1.In2014,thepopulationdensityofFlagstaffwas1,028peoplepersquaremile.ThepopulationofFlagstaffin 2014was65,660.HowmanysquaremilesisFlagstaff? 2.Agallonofpaintwillcoverapproximately450squarefeet.Anartistwantstopaintalltheoutsidesurfaces ofacubemeasuring12feetoneachedge.whatistheleast&numberofgallonsofpainthemustbuytopaint thecube? 3.ThecapacityatPepsiAmphitheatreinFlagstaffis3000peopleforfestivalstyleseating.Theareaofthe seatingspaceisapproximately3021squaremeters.iftheamphitheatreisatcapacityforamusicfestival,how manysquaremeterswilleachpersonhave? m 2 perperson 4.GuatemalaandZambiahaveverysimilarpopulationsizes.However,theareaofZambiaisalmost7times thesizeoftheareaofguatemala.whichcountryhasthehigherpopulationdensity?circleyouranswer. Zambia Guatemala Module3Lesson8Page 50
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