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GRADE5SUPPLEMENT
SetA11Number&Operations:Multiplying&DividingDecimals
Includes
★Activity1:MultiplyingbyPowersofTenA11.1
★Activity2:DividingbyPowersofTenA11.7
★Activity3:UsingDecimalstoCalculateSalePricesA11.15
★Activity4:MultiplyingDecimalsA11.21
★Activity5:BuildingaDeck,UsingPartialProducts&ArraysforDecimalMultiplicationA11.29
★Activity6:MultiplyingDecimals,More/LessA11.35
★Activity7:DividingDecimalswithMoney&MenusA11.45
★Activity8:UsingModels&StrategiestoDividewithDecimalsA11.53
★IndependentWorksheet1:ThinkingaboutTenths,Hundredths&ThousandthsA11.59
★IndependentWorksheet2:VeryLarge&VerySmallNumbersinContextA11.61
★IndependentWorksheet3:Multiplying&DividingbyPowersofTenA11.63
★IndependentWorksheet4:UsingLandmarkFractions&PercentstoMultiplybyDecimalsA11.65
★IndependentWorksheet5:MultiplyingTwoDecimalNumbersA11.67
★IndependentWorksheet6:Comparing&MultiplyingFractions&DecimalsA11.69
★IndependentWorksheet7:OlympicSwimmersA11.71
★IndependentWorksheet8:OlympicTrackStarA11.73
Skills&Concepts
★roundnumberstothenearest0.1,0.01,and0.001
★multiplyanddividebypowersof10,including0.01,0.1,1,10,100,and1,000
★multiplywholenumbersanddecimalnumbersbydecimalnumbers★applyfraction,decimal,andpercentequivalenciestosolveproblems
★describetheeffectofplacevaluewhenmultiplyingwholenumbersanddecimals★multiplydecimalusingconcretemodels,drawingandstrategies
★estimatesolutionstoarithmeticproblemsinordertoassessreasonablenessofresults
★dividedecimalsusingconcretemodelsordrawingsandstrategiesbasedonplacevalue,propertiesofoperations.
★relatethestrategytoawrittenmethodandexplainthereasoningused.
CJalem,
TheMATHLEARNINGCENTER
egOn
palliShed
P201503
BridgesinMathematicsGrade5Supplement
SetA11Number&Operations:Multiplying&DividingDecimals
TheMathLearningCenter,POBox12929,Salem,Oregon97309.Tel.1800575–8130.
©2013byTheMathLearningCenterAllrightsreserved.
PreparedforpublicationonMacintoshDesktopPublishingsystem.PrintedintheUnitedStatesofAmerica.
P201503
TheMathLearningCentergrantspermissiontoclassroomteacherstoreproduceblacklinemastersinappropriatequantitiesfortheirclassroomuse.
BridgesinMathematicsisastandards-basedK–5curriculumthatprovidesauniqueblendofconceptdevelopmentandskillspracticeinthecontextofproblemsolving.Itincorpo-ratestheNumberCorner,acollectionofdailyskill-buildingactivitiesforstudents.
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Ourmissionistoinspireandenableindividualstodiscoveranddeveloptheirmathematicalconfidenceandability.Weofferinnovativeandstandards-basedprofessionaldevelopment,curriculum,materials,andresourcestosupportlearningandteaching.Tofindoutmore,
visitusat.
SetA11Number&Operations:Multiplying&DividingDecimals
SetA11★Activity1
ACTIVITY
MultiplyingbyPowersofTen
Overview
Studentscompleteastringofcalculationswithfractionsanddecimalsandthendiscusstherelationshipsamongthosecalculationstobuildgreatercomputationalfluencyandastrongernumbersensewithdecimals.Thentheyexplorewhathappens,andwhy,whentheymultiplybypowersof10(0.01,0.1,1,10,etc.).
Skills&Concepts
★multiplybypowersof10,including0.01,0.1,1,10,100,and1,000
★describetheeffectofplacevaluewhenmultiplyingwholenumbersanddecimalsby0.01,0.1,1,10,100,and1,000
★applyfractionanddecimalequivalenciestosolveproblems
You’llneed
★PatternsinMultiplyingbyPowersofTen(pagesA11.4
andA11.5,run1copyfordisplay,plusaclassset)
★MultiplyingbyPowersofTenPractice(pageA11.6,run
1copyfordisplay,plusaclassset)
★basetenpiecesforeachpairofstudents,plusasetfordisplay
★GreatWallofBaseTensavedfromUnitSix
AdvancePreparationTrytofindsomecopiesof
BridgesStudentBookpages160and161,Fraction&Deci-malEquivalents,whichstudentscompletedinUnitSix,
Session10.YoumightalsofillinDisplayMaster6.10,Frac-tion&DecimalEquivalencies,whichyouusedinSession12.Bothoftheseresourcesmayjogstudents’memoryofthefractionequivalentsofcommondecimalsinsteps1and3below.
InstructionsforMultiplyingbyPowersofTen
1.Explaintostudentsthatthey’regoingtobemultiplyingdecimalnumbersinthenextfewdaysandthatthey’llbeginwithpowersof10,like0.1,10,and100.Writethefollowingproblemsoneatatime
wherestudentscanseethem(answersincludedinparenthesesforyourreference).Askstudentsto
workinpairsforaminuteortwotosolveoneproblematatime,andthenhavestudentssharetheiran-swersandstrategiesasawholegroup.
•1⁄2×10(5)
•0.5×10(5)
•1⁄4×10(2.5)
•0.25×10(2.5)
•0.75×10(7.5)
Whentheyhavesolvedallfiveproblems,askstudentstodiscusstherelationshipstheynoticedamongtheproblems.Studentsarelikelytonoteequivalenciesbetween1⁄2and0.5,andbetween1⁄4and0.25.Theymayalsohavenoticedthattheycouldhalvehalfof10tofindone-fourthof10,andthatthree-
fourths(0.75)isthreetimesone-fourth.Theymightalsonoticethatwhenmultiplyingadecimalnum-berby10,youmovethedecimalpointoneplacetotheright(e.g.,0.25×10=2.5).
©TheMathLearningCenterBridgesinMathematicsGrade5Supplement•A11.1
SetA11Number&Operations:Multiplying&DividingDecimalsActvity1MultiplyingbyPowersofTen(cont.)
Describingtherelationshipsamongtheproblemsshouldhelpstudentsbegintodevelopefficientstrate-giesforcomputingwithdecimalnumbers.Studentswillsolvesimilarsetsofproblemsatthebeginningofeachactivityinthisset.
2.PlacePatternsinMultiplyingbyPowersofTenondisplayandgiveeachstudentacopy.Reviewthesheetwiththeclass.Discussthesampleequationsineachtableandhavestudentsconnecttheelementsofeachequationtotheproblemsituation.Alsobesurestudentsrememberhowtowriteeachdecimal
(0.01and0.1)asafraction.InvitethemtorefertoBridgesStudentBookpages160and161,FractionandDecimalEquivalents,orafilledincopyofDisplayMaster6.10,FractionandDecimalEquivalencies,ifyouwereabletoretrievetheseresourcesfromUnitSix.
SetA11Number&Operations:Mutpying&DivdingDecmasBlacklneRun1copyfordspay,plusaclassset
NAMEDATEPatternsinMultiplyingbyPowersofTen,page1of2
1aThepostofficesellsone-centstamps.Filloutthetablebelowtoshowhowmuchitwouldcosttobuydifferentquantitiesofone-centstamps.
NumberofStamps
DecimalEquation
FractionEquation
TotalCost
1stamp
1×0.01=0.01
1×1/100=1/100
$0.01
2stamps
2×0.01=0.02
2×2/100=2/100
$0.02
10stamps
20stamps
45stamps
321stamps
404stamps
bWhatdoyounoticeaboutmultiplyingby0.01?
3.Givestudentstimetocompletethesheetinpairs.Thenreconvenetheclassasawholegroupand
openthediscussionbyaskingwhattheynoticedaboutmultiplyingby0.01,0.1,and10.Discusseach
multiplieroneatatime,andencouragestudentstoexplainwhythepatternstheysee(e.g.,“Whenyoumultiplyby0.01,thedecimalpointmovestwoplacestotheleft”)makesense.Encouragestudentstore-fertotheGreatWallofBaseTenandtousethebasetenpiecestoexplainthepatternstheysee.Remem-berthatwhenmodelingdecimals,thematrepresents1,thestrip0.10,andtheunit0.01.
JosieIsawwhenyoumultiplyanumberby0.01,likeinthefirstproblem,youcanjustmovethedecimalpointtwoplacestotheleftlikethis.Itworkseverytime.
45×0.01=0.45
45.0becomes0.45
TeacherWhydoesitwork?CanyouusetheGreatWallofBaseTenorthesebasetenpiecestoexplain?
JosieWell,45timesone-hundredthis45hundredths.40hundredthsisthesameasfour-tenths.
That’sthe.4partoftheanswer.And5hundredthsisjust5hundredths.Soit’slikeeachpartofthefirstnumbergetsahundredtimessmaller:40becomesfour-tenthsand5becomesfive-hundredths.Oryoucouldjustthink45hundredths,really.That’sahundredtimessmallerthan45.
A11.2•BridgesinMathematicsGrade5Supplement©TheMathLearningCenter
SetA11Number&Operations:Multiplying&DividingDecimalsActvity1MultiplyingbyPowersofTen(cont.)
40hundredthsis4tenths
5hundredthsisjust5hundredths
4.Afterstudentshavediscussedthepatternsthatemergedwhenmultiplyingby0.01,0.1,and10,give
eachstudentacopyofMultiplyingbyPowersofTenPractice.Explainthatthey’llcompleteitindepen-dently,andthenselectacoupleofproblemsfromthesheettodotogetherbeforeaskingstudentstoworkontheirown.
Extensions
•Ifstudentsfinishearly,askthemtoturntheirpapersoverandwriteproblemsforeachotherinthisform:
45×=0.04545×=4,50045×=4.5
Thentheycantradepapersandfillinthemissingpowersof10ineachequation.
•Clarifytheterm“poweroften”usingtheGreatWallofBaseTen,andintroduceexponentnotation.Apoweroftenisanumberresultingfrommultiplying10byitselfanynumberoftimes.Weuseexpo-nentstoshowhowmanytimesanumber,inthiscase10,ismultipliedbyitself.Anegativeexponentindicatesanumberlessthan1(afractionoradecimal).
1000=103100=10210=1011=1000.1=10–10.01=10–2
©TheMathLearningCenterBridgesinMathematicsGrade5Supplement•A11.3
SetA11Number&Operations:Multiplying&DividingDecimalsBlacklineRun1copyfordisplay,plusaclassset.
NAMEDATE
PatternsinMultiplyingbyPowersofTen,page1of2
1aThepostofficesellsone-centstamps.Filloutthetablebelowtoshowhowmuchitwouldcosttobuydifferentquantitiesofone-centstamps.
NumberofStamps
DecimalEquation
FractionEquation
TotalCost
1stamp
1×0.01=0.01
1×
$0.01
2stamps
2×0.01=0.02
2×
$0.02
10stamps
20stamps
45stamps
321stamps
404stamps
bWhatdoyounoticeaboutmultiplyingby0.01?
2aAmeliafeedsherpetlizardcrickets.Thepetstoresellscricketsfortencentseach.Filloutthetablebelowtoshowhowmuchitwouldcosttobuydifferent
quantitiesofcrickets.
NumberofCrickets
DecimalEquation
FractionEquation
TotalCost
1cricket
1×0.10=0.10
1×=
$0.10
2crickets
2×0.10=0.20
2×=
$0.20
10crickets
20crickets
(Continuedonnextpage.)
A11.4•BridgesinMathematicsGrade5Supplement©TheMathLearningCenter
SetA11Number&Operations:Multiplying&DividingDecimalsBlacklineRun1copyfordisplay,plusaclassset.
NAMEDATE
PatternsinMultiplyingbyPowersofTen,page2of2(cont.)
2a(cont.)
NumberofCrickets
DecimalEquation
FractionEquation
TotalCost
45crickets
321crickets
404crickets
bWhatdoyounoticeaboutmultiplyingby0.10?
3aAlfonso’scompanysellsT-shirtstosoccerteams.EachT-shirtcoststendol-lars.Filloutthetablebelowtoshowhowmuchitwouldcosttobuydifferent
quantitiesofT-shirts.
NumberofShirts
Equation
TotalCost
1shirt
1×10=10
$10
2shirts
2×10=20
$20
10shirts
20shirts
45shirts
321shirts
404shirts
bWhatdoyounoticeaboutmultiplyingby10?
©TheMathLearningCenterBridgesinMathematicsGrade5Supplement•A11.5
SetA11Number&Operations:Multiplying&DividingDecimalsBlacklineRun1copyfordisplay,plusaclassset.
NAMEDATE
MultiplyingbyPowersofTenPractice
Completethefollowingequations.
106×0.01=
47×0.01=
3×0.01=
0.6×0.01=10×0.01=
0.32×0.01=
0.1×0.01=
452×0.1=
302×0.1=
64×0.1=
0.9×0.1=
0.57×0.1=
0.04×0.1=
0.1×0.1=
360×10=
23×10=
4×10=
0.7×10=
0.54×10=
0.01×10=
0.32×100=
4.3×100=
4×100=
45×100=
309×100=
0.1×100=
0.17×1,000=603×1,000=
0.34×1,000=0.01×1,000=
9.6×1,000=
A11.6•BridgesinMathematicsGrade5Supplement©TheMathLearningCenter
SetA11Number&Operations:Multiplying&DividingDecimals
SetA11★Activity2
ACTIVITY
DividingbyPowersofTen
Overview
Studentscompleteastringofcalculationswithfractionsanddecimalsandthendiscusstherelationshipsamongthosecalculationstobuildgreatercomputationalfluencyandastrongernumbersensewithdecimals.Thentheyexplorewhathappens,andwhy,whentheydivideby
powersof10(0.01,0.1,1,10,etc.).
Skills&Concepts
★dividebypowersof10,including0.01,0.1,1,10,100,and1,000
★describetheeffectofplacevaluewhendividing
wholenumbersanddecimalsby0.01,0.1,1,10,100,and1,000
★applyfractionanddecimalequivalenciestosolveproblems
You’llneed
★PatternsinDividingbyPowersofTen(pagesA11.10–A11.12,run1copyfordisplay,plusaclassset)
★DividingbyPowersofTenPractice(pageA11.13,run1copyfordisplay,plusaclassset)
★basetenpiecesforeachpairofstudents,plusasetfordisplay
★GreatWallofBaseTensavedfromUnitSix
InstructionsforDividingbyPowersofTen
1.Writethefollowingproblemsoneatatimewherestudentscanseethem(answersincludedinparen-thesesforyourreference).Askstudentstoworkinpairsforaminuteortwotosolveoneproblematatime,andthenhavestudentssharetheiranswersandstrategiesasawholegroup.
•10×0.1(1)
•10×0.6(6)
•600×0.01(6)
•600×0.04(24)
•40×0.8(32)
2.Whentheyhavesolvedallfiveproblems,askstudentstodiscusstherelationshipstheynoticed
amongtheproblems.Studentsarelikelytonotethatmultiplyingby0.1islikedividingby10,justas
multiplyingby0.01islikedividingby100.Withthisinmind,theycansolve600×0.04,forexample,inthefollowingway:600÷100=6and6×4=24.
3.Nowexplaintostudentsthattodaythey’regoingtobedividingbypowersof10,like0.1,10,and100.
PlacePatternsinDividingbyPowersofTenondisplayandgiveeachstudentacopy.Reviewthesheetwiththeclass.Discussthesampleequationsineachtableandhavestudentsconnecttheelementsofeachequa-tiontotheproblemsituation.Alsobesurestudentsrememberhowtowriteeachdecimalasafraction.
©TheMathLearningCenterBridgesinMathematicsGrade5Supplement•A11.7
SetA11Number&Operations:Multplying&DvidingDecimalsRun1copyfordisplay,plusaclasssetPatternsinDividingbyPowersofTen,page2of3
2bFilloutthetablebelowtoshowhowmuchitwouldcosttobuydifferentquantitiesofcrickets.
TotalCost
DecimalEquation
FractionEquation
NumberofCrickets
$0.10
0.10÷0.10=1
1⁄10÷1⁄10=1
1cricket
$0.20
0.20÷0.10=2
2⁄10÷1⁄10=2
2crickets
$1.00
$2.00
$3.30
$5.20
cWhatdoyounoticeaboutdividingby0.10?
3aThepostofficesellsone-centstamps.Ifyouspent$2.08,howmanyone-centstampscouldyoubuy?
bFilloutthetablebelowtoshowhowmanystampsyoucouldbuywithdiffer-entamountsofmoney.
TotalCost
DecimalEquation
FractionEquation
NumberofStamps
$0.01
0.01÷0.01=1
1⁄100÷1⁄100=1
1stamp
$0.02
0.02÷0.01=2
2⁄100÷1⁄100=2
2stamps
$0.10
$0.40
(Continuedonnextpage.)
SetA11Number&Operations:Multiplying&DividingDecimalsActivity2DividingbyPowersofTen(cont.)
SetA11Number&Operatons:Multiplyng&DividingDecimasBlacklineRun1copyfordisplay,plusaclassset
NAMEDATE
PatternsinDividingbyPowersofTen,page1of3
1aAlfonso’scompanysellsT-shirtstosoccerteams.EachT-shirtcoststendol-lars.Ifyouspent$1030,howmanyshirtscouldyoubuy?
bFilloutthetablebelowtoshowhowmanyT-shirtsyoucouldbuywithdiffer-entamountsofmoney.
TotalCost
Equation
NumberofShirts
$10
10÷10=1
1
$20
20÷10=2
2
$100
$200
$450
$3210
$1020
cWhatdoyounoticeaboutdividingby10?
2aAmeliafeedsherpetlizardcrickets.Thepetstoresellscricketsfortencentseach.IfAmeliaspent$1.30oncricketslastweek,howmanycricketsdidshebuy?
(Continuedonnextpage.)
Asyoureviewthesheet,discusshowtowritethenumbersthataregreaterthan1asafraction.Inthiscase,studentswillprobablyfinditmostusefultowritethemasimproperfractions.Forexample,theywould
write2.47as247⁄100inthefirsttable.Thiswillprobablymakedividingby1⁄100moresensibletothem.
4.Nowaskstudentstocompletethesheetinpairs.Encouragethemtousethebasetenpiecestothinkabouttheproblemsifthathelps.Thenreconvenetheclassasawholegroupandopenthediscussionbyaskingwhattheynoticedaboutdividingby0.01,0.1,and10.Discusseachdivisoroneatatime,anden-couragestudentstoexplainwhythepatternstheyseemakesense.(e.g.,“Whenyoudivideby0.01,thedecimalpointmovestwoplacestotheright.That’swhathappenswhenyoumultiplyby100too!”)In-
vitestudentstorefertotheGreatWallofBaseTenandtousethebasetenpiecestoexplainthepatternstheysee.Rememberthatwhenmodelingdecimals,thematrepresents1,thestrip0.10,andtheunit0.01.
SydneyWhenyoudividebyadecimalnumber,it’slikemultiplyingbythereversewholenumber,soyoumovethedecimalpointthatmanyplacestotheright.
TeacherPleaseusethebasetenpiecestoshowuswhatyoumeanandwhythisistrue.
SydneyWell,thinkaboutthesestrips.Theyshow40.Soifyoudivideby0.1,it’slikeasking,howmanytenthsin40?Thereare10tenthsineachlittleunitand40unitsaltogether,soyougo10×40=400.So40÷0.1=400.400islike40withthedecimaloneplacetotheright.
A11.8•BridgesinMathematicsGrade5Supplement©TheMathLearningCenter
SetA11Number&Operations:Multiplying&DividingDecimalsActivity2DividingbyPowersofTen(cont.)
Thereare10tenthsineachlittleunit,and40
unitsaltogether.10×40=400.
Students’verbalexplanationswillvaryconsiderablyintheirclarity,soencouragethemtoshowtheirthinkingwithbasetenpiecesandequations.Thiswillallowyoutogetaclearersenseofwhattheyun-derstandandwillmaketheirexplanationsmorecomprehensibletootherstudents.
5.Afterstudentshavediscussedthepatternsthatemergedwhendividingby0.01,0.1,and10,giveeachstu-dentacopyofDividingbyPowersofTenPractice.Explainthatthey’llcompleteitindependently,andthenselectacoupleofproblemsfromthesheettodotogetherbeforeaskingstudentstoworkontheirown.
Extensions
•Ifstudentsfinishearly,askthemtoturntheirpapersoverandwriteproblemsforeachotherinthisform:
45÷=0.04545÷=45045÷=4.5
Thentheycantradepapersandfillinthemissingpowersof10ineachequation.
•Youmightalsoconsideraskingthemtowritetheirproblemsinthisform:
45÷10=45×45÷0.10=45×45÷0.01=45×
•Helpstudentsunderstandpowersof10inagraphicway.TheMolecularExpressionswebsite(see
URLbelow)featuresaphotographicdisplaycalledSecretWorlds:TheUniverseWithinthatillustratespowersof10startingwiththeMilkyWay,10millionlightyears(1020meters)fromEarth.AseriesofphotosmovecloserandclosertoEarth,decreasingindistancebyapowerof10eachtime,untilyoureachatalloaktreeseenatadistanceof1meter(100meter).Thephotosdon’tstopthere,however.Thepowersof10gonegativeastheseriesmovesinthemicroscopicworldofanoakleaf,andfinallyintoasubatomicuniverseofelectronsandprotons.
/primer/java/scienceopticsu/powersof10/
INDEPENDENTWORKSHEET
UseSetA11IndependentWorksheets1–3onpagesA11.59–A11.64toprovidestudentswithmorepracticemultiplyinganddividingbypowersoften,aswellasroundinganddoingcalculationswithdecimals.
©TheMathLearningCenterBridgesinMathematicsGrade5Supplement•A11.9
SetA11Number&Operations:Multiplying&DividingDecimalsBlacklineRun1copyfordisplay,plusaclassset.
NAMEDATE
PatternsinDividingbyPowersofTen,page1of3
1Alfonso’scompanysellsT-shirtstosoccerteams.EachT-shirtcoststendollars.aIfyouspent$1030,howmanyshirtscouldyoubuy?
bFilloutthetablebelowtoshowhowmanyT-shirtsyoucouldbuywithdiffer-entamountsofmoney.
TotalCost
Equation
NumberofShirts
$10
10÷10=1
1
$20
20÷10=2
2
$100
$200
$450
$3210
$1020
cWhatdoyounoticeaboutdividingby10?
2Ameliafeedsherpetlizardcrickets.Thepetstoresellscricketsfortencentseach.aIfAmeliaspent$1.30oncricketslastweek,howmanycricketsdidshebuy?
(Continuedonnextpage.)
A11.10•BridgesinMathematicsGrade5Supplement©TheMathLearningCenter
SetA11Number&Operations:Multiplying&DividingDecimalsBlacklineRun1copyfordisplay,plusaclassset.
NAMEDATE
PatternsinDividingbyPowersofTen,page2of3(cont.)
bFilloutthetablebelowtoshowhowmuchitwouldcosttobuydifferentquanti-tiesofcrickets.
TotalCost
DecimalEquation
FractionEquation
NumberofCrickets
$0.10
0.10÷0.10=1
÷=1
1cricket
$0.20
0.20÷0.10=2
÷=2
2crickets
$1.00
$2.00
$3.30
$5.20
cWhatdoyounoticeaboutdividingby0.10?
3Thepostofficesellsone-centstamps.
aIfyouspent$2.08,howmanyone-centstampscouldyoubuy?
bFilloutthetablebelowtoshowhowmanystampsyoucouldbuywithdifferentamountsofmoney.
TotalCost
DecimalEquation
FractionEquation
NumberofStamps
$0.01
0.01÷0.01=1
÷=1
1stamp
$0.02
0.02÷0.01=2
÷=2
2stamps
$0.10
$0.40
(Continuedonnextpage.)
©TheMathLearningCenterBridgesinMathematicsGrade5Supplement•A11.11
SetA11Number&Operations:Multiplying&DividingDecimalsBlacklineRun1copyfordisplay,plusaclassset.
NAMEDATE
PatternsinDividingbyPowersofTen,page3of3(cont.)
3bCont.
TotalCost
DecimalEquation
FractionEquation
NumberofStamps
$0.86
$2.47
$3.05
cWhatdoyounoticeaboutdividingby0.01?
A11.12•BridgesinMathematicsGrade5Supplement©TheMathLearningCenter
SetA11Number&Operations:Multiplying&DividingDecimalsBlacklineRun1copyfordisplay,plus
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