What I've learnt by teaching - Mathematical Association€¦ · What I have Learnt by Teaching Jim...
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What I have Learnt by Teaching Jim Simons
Transcript of What I've learnt by teaching - Mathematical Association€¦ · What I have Learnt by Teaching Jim...
WhatIhaveLearntbyTeaching
JimSimons
AnALevelMechanicsProblem
Ignoringhealthandsafetyadvice,Georgeleansoutfromtheedgeofacliffofheighth,andthrowsastoneupwardsatspeedu.Aimanstandsonthebeachdirectlybelow,andatthesameinstantfiresoffafireworkrocketwhichacceleratesstraightupwardswithacceleraFona.Atwhatspeeddothestoneandtherocketcollide,assumingtheydo?
suvatKnowsnoPhysics
• ThesuvatequaFonshavenothingtodowithphysics
• TheyarejustthesoluFontothedifferenFalequaFon
• Withvasashortandfords/dt,andiniFalcondiFonss=0,andv=uatt=0
suvatKnowsnoPhysics
• Sotdoesn’thavetobeFme.sdoesn’thavetobedistance.Themeasurementdoesn’thavetobeinaninerFalframeofreference.
• Considerthestoneintherestframeoftherocket,orleth–sbethedistancebetweenrocketandstone:
!!!!!! = ! + !
TheLanguageofMathemaFcs
• …isEnglishwithgreaterprecision• 37=x• Talktome!Don’tjustmanipulatesymbols.• …butitisnotjustEnglish• DangerofliteraltranslaFon
TrigonometricRaFos
ScalarsandVectors
ScalarSize
VectorSize&Direc/on
Distance DisplacementSpeed Velocity
ScalarsandVectors
ScalarSize
VectorSize&Direc/on
Distance DisplacementSpeed VelocityAcceleraFon
ScalarsandVectors
ScalarSize
VectorSize&Direc/on
Distance DisplacementSpeed Velocity? AcceleraFon
ScalarsandVectors
ScalarSize
VectorSize&Direc/on
Distance DisplacementSpeed Velocity? AcceleraFonForce
ScalarsandVectors
ScalarSize
VectorSize&Direc/on
Distance DisplacementSpeed Velocity? AcceleraFon
Force
ForceisnotaVector
ScalarSize
VectorSize&Direc/on
Size,Direc/on&Posi/on
Distance DisplacementSpeed Velocity? AcceleraFon
Force
ForceisnotaVector
ScalarSize
VectorSize&Direc/on
?Size,Direc/on&Posi/on
Distance Displacement ?Speed Velocity ?? AcceleraFon ?? ? Force
ForceisnotaVector
ScalarSize
VectorSize&Direc/on
RootedVector?Size,Direc/on&Posi/on
Distance Displacement Journey?Speed Velocity ?? AcceleraFon ?? ? Force
• SowhatdoesF=mamean?• Fisthevectoroftheforce,iejustitsmagnitudeanddirecFon,notthewholeforce.
• IftwodifferentforceshavethesamevectorthentheyproducethesameacceleraFon…
• ...buttheymayhavedifferenteffectsapartfromacceleraFon.
ForceisnotJustaVector
• ForceswithequalvectorsbutdifferentlinesofacFonproducedifferentangularacceleraFons.
• ForceswithequalvectorsonthesamelineofacFoncanproducedifferentdeformaFons.
• ForceislessabstractthatspeedoracceleraFonetc
ForceisnotJustaVector
StaFsFcsCounts
• Mystudentsareverypoliteabouttheirteachers• …exceptwhentheyareteachingstaFsFcs.• SomemathsteachersfindstaFsFcsboring,anditshows.
• Butformanystudents,itthemostpracFallyusefulpartsoftheAlevelsyllabus.
• AndusingitisreallyinteresFng.• SogoandlistentoSidneyTyrrell,andgetinspired!
VanAubel’sTheorem
TheRemarkableWineRackProperty
3≤w≤4
TheRemarkableWineRackProperty
AWidespreadMeme
• TheCosinerulesaysa2=b2+c2–2bccosA• Mostproofsstart,ineffect:Eitherallthreeanglesareacute,orAisobtuse.
Kronecker’sQuotaFon
• Godmadetheintegers,allelseistheworkofman
• Vertebratadiscoveredtheintegers,allelseistheworkofhomo
ExponenFaFon
• WeinventthenotaFonanfornaposiFveintegerasashorthandforrepeatedmulFplicaFon
• Wediscoversomerules,egan+m=an×am• Wethendiscoverthatifwedefineaxfornon-integralxappropriately,theseruleswillsFllwork
• Sowedodefineitthatway• Anewconceptemerges,exponenFaFon,whichisdifferentfromrepeatedmulFplicaFon
• Andithastobeunderstoodinadifferentway
MulFplicaFon
• WeinventthenotaFona×bwhereaandbareposiFveintegersasashorthandforrepeatedaddiFon
• Wediscoversomerules,ega×(b+c)=a×b+a×c• Wethenrealisethatifwedefinea×bforraFonalaandbappropriately,theseruleswillsFllwork
• Sowedodefineitthatway• Anewconceptemerges,mulFplicaFon,whichisdifferentfromrepeatedaddiFon
• Andithastobeunderstoodinadifferentway
SineandCosineforAngles>90°
RememberingFormulae
• “Examsshouldn’tbememorytests”• “Youdon’tneedtorememberfactsbecauseyoucanalwayslookthemup”
• ThisisonlytrueifyouwantonefactataFme,andknowthekeytolookitupby
RememberingFormulae
• Forproblemsolving,youneedtoaccessallthefactsinparallel,andwithoutthekey–thesubconsciousmindisaparallelprocessor
• Soyouneedtorememberenoughabouttheoremstobeabletosolveproblems,andyouneedtounderstandthem
• Tomakeaseriousbreakthrough,youhavetofillyourbrainwithrelevantfacts,andthinkaboutthemforages
ReconstrucFngtheCosineRule
• Itmustinvolve4quanFFes,socosα,a,b,c• Whenα=π/2,cosα=0,soitmustreducetoarelaFonshipbetweena,bandc,whichmustbea2=b2+c2,andinparFcularitisquadraFc
• Whenα=0,a=b–cora=c–b,soitmustreducetoa2=b2+c2–2bc
• Whenα=π,a=b+c,soitmustreducetoa2=b2+c2+2bc
• Soitmustbea2=b2+c2–2bccosα
FracFonFricFon12+14= 2
9x = 4
9
x = 49÷29
=49×92
=3618
112=
MisappliedAlgorithms
1+ /2x/2
s/in (π /2)s/in(π /4)
cos2 θ + 45°( ) = cos2(θ )+ cos2(45°)
TheRemarkableWineRackProperty
TheRemarkableWineRackProperty
• If3≤w≤2+√3(≈3.73),thetheoremholdswhereverthemiddlebooleofthebooomrowisplaced.
• If2+√3<w≤1+2√3(≈4.46),thetheoremholdswheneverthemiddlebooleofthebooomrowissufficientlyclosetothemiddle.
• Sothereisnothingspecialaboutw=4
