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