比值法与非常规解题(Ratiomethodandunconventionalsolution)

1、比值法与特别规解题（Ratiomethodandunconventionalsolution）CollectedbyoneselfMistakesareunavoidableForreferenceonlyIncaseoferrorPleasecorrectme!ThankyouRatiomethodandunconventionalsolutionGuangzhousevenDuHoushengClassroomteachingisthemainchannelofqualityeducation,andthekeytoqualityeducationistooptimizetheproces

2、sofclassroomteaching,guidestudentstoparticipateactivelyinthelearningprocess,learntolearnandbehappytolearnTrulybecomethesubjectoflearningInmathematicsclassroomteachingProblemsolvingteachingisthemostimportantandmostimportantcontentofteachingThinkingactivitiesarefullyreflectedintheprocessofsolvingprobl

3、emsAndfullydevelopedStudentsshouldbeinproblemsolvingteachingLearntothink,learntomakeWedontrequireeveryteacherandstudenttodoadvancedresearch,saysPolya,thegreatestmatheducationthinkerofthecenturyHoweverMathematicalproblems,unconventionalsolutions,andrealcreativeworkWhatweneedtomasterisnotknowledgegain

4、edbymemoryaloneItistherealknowledgethatisreadilyavailableinsolvinginterestingproblemsThatistosayInproblemsolvingteachingUnconventionalsolutionscanbeusedtostimulatecreativityImprovetheenthusiasmandinitiativeofstudentsInthesolutionteachingofplanegeometryinjuniorhighschoolRatiomethodCanachievesomegeome

5、tricproblemsofunconventionalsolutionEffectivelyimprovestudentsproblem-solvingabilityArightangledtrianglewith30degreesand45degreesanglesafterstudyingrighttriangleSolvesuchaproblem:RtDeltaABCAngleC=90degreesAngleB=30degreesIfBC=50SeekthelengthofthebevelABMoststudentswillsetAB=xAC=ThereforeListequation

6、s:XieAB=DonttrytoseethePythagoreantheoremThestudentthoughtforamomentIfABisset,XCOS30=thismethodthanbyusingthePythagoreantheoremofgoodBecausethereisnoneedtosolvetheequationoftwovariablesAskthestudentagain:canyouseeAB=atonce?NoBecausethenumberistoobigChangeasmallernumberSuchasBC=3IthinkIcanusemymental

7、arithmetictofigureoutAB=2AdlocumThePythagoreantheoremandtrigonometricfunctionsaretheconventionaltextbooksolutionrequirementsByusingtheratiorule,averysimpleandunconventionalmethodcanbeobtainedSoIcametofindthescalewiththestudentsItsreallyeasyFromsmalltolargeTheratioofthethreesidesis1:2Ifashortstraight

8、edgeisknown,aisusedYes,a:a:2aThereforeJustknowtheshortstraightedgeaTherestofthetwosidesshouldbeaand2A;knowthatthebevelisa?TheothertwosidesareandInordertomasterthismethodJusttenminutesofpractice:shortandlongMultipliedbymultiples(and2);knownlength;shortdurationDividedbymultiples(and2)SimilarproblemsWa

9、nttoseeitallatonceAseasyasbreathingSowecallthisexpirationComeassoonasyoucallOpenyourmouth(asbelow)OfcourseTheisoscelesrighttriangleisthesameMultipliedbyordividedbymultiplesSimpler!(picturebelow)Howusefulisthisexhalation?Letshaveatry:Example1.,asshowninFig.KnownDC=3SeekthelengthofeachlinesegmentSolut

10、ion:AX30degrees,45degreesB50,D,X,C,BC=XCalloutall!Example2.,asshowninFig.KnownBD=50FindthelengthoflineACSolution:letAC=xThenDC=xBC=xSo50+x=xX=50/(-1)=25(+1)ThisisobviouslybetterthanusingtrigonometricfunctionsAAx30degrees45degreesB,x,D,x,CExample3.,asshowninFig.KnownBC=3+FindthelengthoflineABandACSol

11、ution:letAD=DC=xThenBD=xGotx+x=3+x=1AB=2x=2=AC=xExample4.,asshowninFig.RtDeltaABCAngleC=90degreesAngleB=15degreesRequest:tg15degreesvalueSolution:asADTheangleADC=30degreesLetAC=1ThenAD=BD=2DC=BC=2+tg15=2-*Cantheybeextended?OtherspecialrightangledtrianglesseemtobeokForexample,therearemorethanPythagor

12、eannumbertrianglevalue3:4:5;5:12:13;7:24:25WaitAnyrightangledtriangle?No,istherearatio?Giveitatry:Setbevelto1Yes,thebeveledgeisaYes：Setthestraightedgeto1Yes,thestraightedgeisaYes：Example5.takeadvantageofthegraphaboveBythePythagoreantheoremProvablesin2A+cos2A=1(squarerelation)DefinedbytangentsProvabl

13、etgA=(ratiorelation)ByA+B=90degreesGetsinA=cosBTgA=ctgB(mutualredundancy)Example6.,asshowninFig.RtDeltaABCAngleD=90degreesAlpha/B=Beta/ACD=BC=mForAD(thisisthepromotionofexample6)Solution:letAD=xCD=xctgbetaBD=xctgalpham+xctg=xctgalphabetax=InthemiddleschoolmathematicsteachingmaterialTherearesomanysub

14、jectswith30degreesand45anglesBreatheoutmakesusfast,accurateandrelaxedTwo.Theunconventionalsolutionofarclengthandsectorialarea1.Theteacheraskedthestudent,length?Yes.whatistheformulaforarcHowdidyougetit?Becauseofthecenterofthecircleat360degrees,thearcisthecircumferenceofthecircleTherefore,thecenteroft

15、hearcat1degreesisthelengthofthearcThus,thecenterofthearcatndegreesisthelengthofthearcTheteacheraskedthestudent,whatstheformulaforsectorialarea?Therearetwo.Sfan=andSfan=LRThelatterformulaisusedwhenthearclengthandradiusareknownWiththesetwosetsofformulasStudentsknowthecenterofthecircleandtheradiusThere

16、isnodifficultyinfindingthelengthofthearcandtheareaofthefanBut.VeryboringItsnotfunnyatallThestudentsaidAndImafraiditllsoonbeforgottenTheteacherthenasked,canyouexportthisresultinanyotherway?Thestudentsaid,howcouldthatbe?CurvilinearcalculationInadditiontothecircleperimeterandthecircleareaformulaWedonth

17、aveanyotherwayYoucanexportformulaslikethisIthinkitsverycleverTheteachersaid,ifyouknow,thecircumferenceofacircleis16PI.Thearcofthefanpairis2piCanyoutellthecenterofthefan,theangle,thedegree,andthefan-shapedareaatonce?Canyoubreatheout?Itsabitdifficult.ThereforeTheteacherandthestudentgobacktothedefiniti

18、onAllrightThedefinitionisactuallytousetheratiotoobtaintheformulaItstheratioagain!ThismeansInasectorThecenteranglenispartPerigonisallispartofthearclength;Thecircumferenceisall;thefanareaispartTheroundareaisthewholeIfyouset=kthatInthen,L,sthesepartsTherespartof=kallThiskiseasytobreatheoutAndthiswholei

19、salsoveryfamiliarThereforeThetitleabovecanbeexhaled:thearclengthis2piThecircumferenceofthecircleis16piSok=?Thecenterangleis=45(degree)FanareashouldbefirstroundareaTheperimeteris16piRadiusis8Sotheareaofthecircleis64piThefanareais*64PI=8piJustalittlemoreskilledItseasiertobreatheoutTheteacherspent10mor

20、eminuteswithhisstudents:AnglenRatioKN:360RadiusrArclengthLK.2RpiFanareasK.R2pidegreesRdegreesRdegreesRdegreesRdegreesRdegreesRdegreesRdegreesRdegreesRdegreesRItsmoreinterestingnowAndWedonthavetorelyonformulasJustrememberthepartandthewholerelationshipWhenwillitbecalculated?ThestudentsaidTheteachersai

21、d,actually.TheformulaisalsoamongtheseratiosItseasytosee,thestudentsaid:Thatis,s=lrActualcalculationtimeUsingn=k*360L=k*2piRS=k*PIR2ObviouslyfastandaccurateTheteachersaid,canyouapplythisresulttootherproblems?Thestudentlookedatthefanagain.theresachordhere,ABItshouldbeapartofaregularpolygoninscribedina

22、circleAndbowItisacircleminusthepartoftheremainingareaoftheregularpolygonThereareaABCareaItispartoftheareaofaregularpolygoninscribedinacircleTheirratiosarealsoequaltoKIfconcentriccirclesareusedGetaringThentheratiooftheareaoftheshadowtotheareaoftheringisalsoKThereisaprobleminthebook(geometrythird,187p

23、ages,11questions)WetrytocomputewiththisresultcasesofknowntwoconcentriccircleswitharadiusoftwowastruncatedbytheAB(-)=10PIcmCD(-)=6PIcmAC=12cmagainTheareaoftheshadedpartSolution:settwocircleradiiofRandR,respectivelydoes:Xier=18R=30dreamsk=*(CM)AtthistimeOnestudentsaid,thefigureisabitlikeatrapezoid.Kno

24、wthebottom,bottom,andheightIfusedTheresultisexactlythesame!ThestudentsareveryhappyHowsimpleisthat?!Isitacoincidence?TheteacherdoesntbelieveitThestudentsaid,wecanprobablyproveit.Thestudentsproofisasfollows:Proof:setupWellthen:ThereisS=(R-r)indeedItsliketrapezoidalareaformula!Thearclengthandfanareacan

25、beexhaledSo,cantheareaofthebowbeexhaled?Theareaoftheisoscelesrighttriangleis=a2Theareaofanequilateraltriangle=Thearcuateareaatleast60degrees,90degrees,and120degreesarcscanalsobeexhaledAslongasthefanareaandtheareaofthetrianglearerespectivelyexhaled(inabowof120degrees)TheAOBareaisreplacedbytheequilate

26、ralDeltaOBCarea)Inthecalculationofthearchareaandtheareaofthecompositefigure(shadowpart)TheincidenceofthesearchareasHighTheconnotationofratioisbroadandprofoundTheratiomethodisconciseandharmoniousUtilizationratioTheresultsobtainedareoftenthemostintuitiveandpracticalMiddleschoolmathematicsteachingmater

27、ialMethodsandmaterialsthatenableustobecreativeItisfarmorethantheratiomethodasyou?MaterialcanbefoundalmosteverywhereProduceinspirationthattriggersourcreativityTheunconventionalsolutiontomathematicalproblemsThree.Understandingofunconventionalsolutionswhataretheunconventionalsolutionstomathematicalprob

28、lems?Itseemsthattheteachersfeelthatitisself-evidentButIcantsayitsdefinitionWhatdidnotreadthecompletediscussionofthisproblem(includingteachingsyllabus,teachingreference,journalpapers,relevantteachingworks)MypersonalunderstandingThisisbecauseunconventionalsolutionshaverelativityTherearethreesourcesofg

29、eneralproblem-solvingmethods:oneisthemethodthatmustbemasteredbyteachingmaterialsAsthePythagoreantheorem,quadraticequationwithoneunknowndistributionmethod,crossmultiplication;twoisthroughthemethodofsolvingaclassofproblemsSuchasformulaetc.Thethreeislogic,strictnessandprecisionMiddleschoolalgebraandgeo

30、metrytextbooksAlmosteveryunitoflearningprocessThemethodofsolvingthisunitisgivenbytheteachingmaterialAndaskstudentstomasteritTheseproblem-solvingmethodsItformstheso-calledregularproblem-solvingmethodsProblemsolvingmethodsthatvaryfromtextbookrequirementsItcanalsobecalledunconventionalproblem-solvingme

31、thodShouldseeConventionalandunconventionalcanbetransformedintoeachotherUnconventionalactuallystemsfromtheconventionalInthelegendofGauss,forexampleFrom1to100andGaussusedthemethodofarithmeticprogressionsummationThisisunconventionalforschoolchildrenAndinthechapterofaseriesofstudiesItsstandardpracticeAn

32、othermethodofsolvingproblems,suchascombinationofnumbersandshapesInsomefunctionalproblemsandinthecomplexnumberproblemItsunconventionalInanalyticgeometryIsthemosttypicalroutineAgain,asmentionedabove,arectangulartrianglewithaspecialangleiscalculatedbytheratiomethodOnceinteaching,studentsshouldbevalueda

33、ndmasteredIthasbecomearoutineproblemsolvingmethodEventhefirstroutineItmaybebecausethereisnostrictlinebetweenregularsolutionsandunconventionalsolutionsTherefore,unconventionalproblem-solvingisdifficulttodefineItmaybeunderstoodthatasimplemethodofsolvingproblemsthatisnotrequiredbytheteachingmaterialand

34、thattheteacherorthestudentusuallydoesnotexpectisunconventionalSimplicityisemphasizedhereWithoutsimplicityImafraiditshardforeveryonetoadmitthattheyareunconventionalTheessenceofunconventionalproblemsolvingistheoptimizationofproblemsolvingGausssaid,goforamostbeautifulandconciseproof.Itwasthemainmotivat

35、ionthatattractedmetostudyitSimplicityisanimportantsymbolofmathematicalbeautySimplyreferstothefactthatmanyfactsaresummedupinsimpleformulasEuclidsgeometryoriginallystartedwith36definitionsand19axioms467propositionsareobtainedHavecompletedtheestablishmentofasubjectExactly100yearsagoInnineteenthCentury,

36、Hilbert,thegreatestmathematician,publishedhisepoch-makingessay,mathematicsproblematthesecondinternationalmathematicianCongressinParisThemainpartofthethesisputsforward23mathematicalproblemsthatshouldbestudiedandbrokendowninanewcenturyLessthan500wordsAparagonofbrevityKleinsaid:newwaystosolveoldproblem

37、s.CanpromotethedevelopmentofMathematicsUnconventionalsolutionsareactuallynewwaysofsolvingoldproblemsteachingofunconventionalproblemsolvingTheteachermustfirstpassthroughhimselfTheinnovationandspiritofthemathematicsteacheristhemodelofthestudentsSometeachersplacetoomuchemphasisonthemasteryandmasteryofB

38、ookMethodsNotonlydidnotthinkofunconventionalsolutionsNotevenstudentscanuseitQuiteafewstudentstalkedaboutsomemajorexamsSomeofthesubjectsaredifficultAlthoughsomesolutionshavebeenthoughtoutEvengottherightsolutionButthemethodsusedaredifferentfromthoseinbooksSoIdarenotwriteitoutSomehavebeenwrittenoutButs

39、ometeachersdonotgivepointsmarkingInTeachingSometeachersgivetheirstudentsananalysisoftheproblemanduseunconventionalproblem-solvingmethodsStudentsoftenask,doyougivepointsinthistest?Becausethatsnotwhattheteachingmaterialdoes!SometeacherstalkabouttrigonometricfunctionsTeachstudentstostudythepropertiesof

40、trigonometricfunctionsbyunitcircles(includingtheevaluationoftrigonometricfunctions,solutionsoftrigonometricequations,symbolsoftrigonometricfunctions,periodicity,definitions,domains,ranges,etc.)ButitwascriticizedbyotherteachersThatinterfereswithnormalteachingItisunfavorableforstudentstomastertheknowl

41、edgeoftextbooksDonotconformtotherequirementsofteachingmaterials,etc.ItshardtoimagineAmanwhostickstohisownrulesdoesnotdaretoteachmoreAteacherwhodoesnotdiscussmultiplesolutionsandunconventionalsolutionsAbletoteachcreativestudentsStudentscreativitycomesfromteachersdemonstrationandencouragementThekeytot

42、hedevelopmentofunconventionalproblem-solvingabilityistheteachertheformationofstudentsunconventionalproblem-solvingabilityShouldseeTheroutineisthefoundationIsthepremiseUnconventionalistoimproveAbreakthrough;aconventionisgeneralUnconventionalisparticularityOnlyproficiencyinavarietyofconventionalproblem-solvingmethodsCanbesynthesized,compared,summarizedGetunconventionalproblem-solvingmethodsTheessenceoftheunconventionalsolutionistooptimizetheproblemTheconfirmationoftheoptimizationisestablishedincomparisonThereisonlyonesolutiontoaquestionItdoe