Lesson Study Lesson Proposal – Decoding De Moivre
“DecodingDeMoivre”
YearGroup:TransitionYearHigherLevel
Topic:ComplexNumbers–DeMoivre’sTheorem
LessonStudyandResearchGroup:
DavidCrowdleandPaudgeBrennan,teachersinLoretoSS,Wexford,developedthislessonproposal.
Paudgetaughtthelessonon10thFebruary2017with17studentsfromhisTransitionYearclass.
Briefdescriptionofthelesson:
Theresearchlessonisthesecondstructuredproblemsolvingclassesleadingthestudentstoappreciateand
developanunderstandingofpolarmultiplicationandraisinganumbertoapower(DeMoivre’sTheorem)
intermsofthemoduliandargumentsofthenumbers.Studentswillengagewithaproblemworkingout
the value of a given complex number raised to the power of 5. Together they will present approaches
leadingtoanunderstandingofdeMoivre’sTheoremandtheneedforit.
1
Lesson Study Lesson Proposal – Decoding De Moivre
AimsoftheLesson:
Shorttermaim:Studentswill:
• appreciatethattherearedifferentcorrectapproachestoexpandingacomplexnumber,
• seetheadvantagestousingthe“Modulus-argument”formofacomplexnumberwhenitcomesto
multiplying,
• developaconceptualunderstandtheprincipleofdeMoivre’stheorem“toexpandanumberraised
toapoweryouraisethemodulustothatpowerandmultiplytheargumentbythepower.”1
Longtermaims:Wewouldlikeourstudents:
• togainconfidenceindealingwithabstractconcepts,
• tobecomeindependentlearnersthroughstructuredproblemsolving,
• toconnectandreviewtheconceptsthatwehavestudiedalready,
• todeveloptheirliteracyandnumeracyskillsthroughdiscussingideas.1
• todevelopkeyskillssuchas“ManagingInformation&Thinking:Thinkingcreativelyandcritically”,2
LearningOutcomes:
Asaresultofstudyingthistopicstudentswillbeableto:
• Understandindifferentwaysthemeaningofmultiplicationofwholenumbersandusethistomake
senseofcomplexnumbermultiplicationandexpansion.
• Understandandbeabletoexpressacomplexnumbersbothinrectangularformandintermsofits
modulusandargument.
• RecogniseanumberonanArganddiagramintermsofitsmodulusandargument
• Developtheinsightthatwhennumbersaremultipliedtheirmoduliaremultipliedandtheir
argumentsareaddedtogether.
• Beabletomultiplynumbersgiventheirargumentsandmoduli.
• Usethistodiscoverthatwhenanumberisraisedtoapoweritsmodulusisraisedtothatpower
anditsargumentismultipliedbythatpowerandexperiencethatthisapproachismuchmore
efficientthatusingrepeatedmultiplicationinrectangularform.
BackgroundandRationale
ThistopicwaschosenbecausestudentsfinddealingwiththepolarformofcomplexnumbersandDe
Moivre’sTheoremachallengingpartoftheHigherLevelCourse.Itishopedthatthisapproachwillhelp
studentsappreciatetheefficiencyofusingthepolarformformultiplicationandapplyingpowersto
numbers.
1
Department of Education & Skills (2011). Literacy and Numeracy for Learning and Life: the National Strategy to
Improve Literacy and Numeracy among Children and Young People 2011-2020.
2
NCCA (2015). Junior Cycle Key Skills: Managing Information and Thinking,
http://www.juniorcycle.ie/NCCA_JuniorCycle/media/NCCA/Documents/Key/Managing-information-andthinking_April-2015.docx, Accessed 12 February 2017
2
Lesson Study Lesson Proposal – Decoding De Moivre
Research
•
•
•
•
•
•
•
NCCA(2014).LeavingCertificateMathematicsSyllabus–Foundation,OrdinaryandHigherLevelfor
Examinationfrom2015
(ProjectMathsDevelopmentTeam(2015).TeacherHandbook,SeniorCycleOrdinaryLevel.
http://www.projectmaths.ie/documents/handbooks/LCOLHandbook2015.pdf
(ProjectMathsDevelopmentTeam(2015).TeacherHandbook,SeniorCycleHigherLevel.
http://www.projectmaths.ie/documents/handbooks/LCHLHandbook2015.pdf
ToshiakiraFUJII(2013).TheCriticalRoleofTaskDesigninLessonStudy.ICMIStudy22:TaskDesign
inMathematicsEducation
MathsDevelopmentTeam,ReflectionsonPracticefromMathsCounts2016:
http://www.projectmaths.ie/for-teachers/conferences/maths-counts-2016/
DepartmentofEducation&Skills(2011).LiteracyandNumeracyforLearningandLife:theNational
StrategytoImproveLiteracyandNumeracyamongChildrenandYoungPeople2011-2020
NCCA,JuniorCycleKeySkills:ManagingInformationandThinking,
http://www.juniorcycle.ie/NCCA_JuniorCycle/media/NCCA/Documents/Key/Managinginformation-and-thinking_April-2015.docx.
3
Lesson Study Lesson Proposal – Decoding De Moivre
AbouttheUnitandtheLesson
AccordingtothesyllabuswithregardtoComplexNumbersstudentsarerequiredto:
ORDINARYLEVEL
• Investigatetheoperationsofaddition,multiplication,subtractionanddivisionwithcomplexnumbersC
inrectangularforma+ib.
• IllustratecomplexnumbersonanArganddiagram.
• InterpretthemodulusasdistancefromtheoriginonanArganddiagramandcalculatethecomplex
conjugate.
HIGHERLEVEL
• Geometricallyconstructroot2androot3.
• Calculateconjugatesofsumsandproductsofcomplexnumbers.
• UsetheConjugateRootTheorem.
• Workwithcomplexnumbersinrectangularandpolarformtosolvequadraticandotherequations.
• UseDeMoivre’sTheorem.
• ProveDeMoivre’sTheorembyinductionfornanelementofN.
• Useapplicationssuchasnthrootsofunity,nanelementofN,andidentitiessuchasCos3θ=4Cos3θ
–3Cosθ
WiththisTransitionYeargrouponlytheOrdinaryLevelmaterialandaninformalapproachtothepolar
formofacomplexnumberandrelatedmultiplicationanddeMoivre’sTheoremwillbecovered.
FlowoftheUnit:
No.
Lesson
Time
1
IntroductiontoComplexNumbersandtheArganddiagram
40min.
2,3
Additionandsubtractionofcomplexnumbers
40min.
4,5
ComplexMultiplication
40min.
6
Modulusofacomplexnumber
40min.
7,8
ComplexDivision
40min.
9
Expressingnumbersintermsofargumentandmodulus.Consider
integermultiplicationintermsof“scalingandrotation.”
40min.
10
Multiplication(Structuredproblemsolvinglesson):considerscalingin
termsofthemoduliandtherotationintermsofthearguments.
(AppendixII)
40min.
11
ReserchLesson:“DecodingDeMoivre”
40min.
4
Lesson Study Lesson Proposal – Decoding De Moivre
FlowoftheLesson
1.Introduction(5minutes)
PriorKnowledge:
Studentsarequestionedontheirunderstandingofprevious
classesconcerning:
a) Arganddiagram
b) Rectangularformofacomplexnumber
c) Describingtheargumentandmodulusofacomplex
number
d) Indices
e) Multiplicationbyexpansion
f) Multiplicationintermsofargumentsandmoduli
2.PosingtheTask(2minutes)
Thetaskispresentedontheboard.Theyareinstructedtofind
thevalueofz5in“asmanywaysasyoucan”in“15minutes”.
Studentsarequestionedontheirunderstandingofthetask.
Eachstudentisgiven3copiesofthehandoutpagewiththe
questionandtheyareaskedto“explaintheirthinking”
(AppendixI).
3.Individualproblemsolving(15minutes)
AnticipatedStudentResponse1
Itisexpectedthatmoststudentswillrecognisethatthe
rectangularformofz=√3+iandthentheymighttryto
evaluate(√3+i)5byexpandingitinatraditionalway–
repeatedmultiplication.
AnticipatedStudentResponse2
Itishopedthatstudentsmighttrytodescribethemodulusand
argumentofz.|z|=2andArg(z)=30o
Theymightthendescribetheexpansionofz5intermsof
repeatedmultiplication:
|z5|=(2)(2)(2)(2)(2)=32
Arg(z5)=30o+30o+30o+30o+30o=150o
5
Lesson Study Lesson Proposal – Decoding De Moivre
AnticipatedStudentResponse3
Studentsmightbuildon“AnticipatedResponse2”toexpress
theexpansionofz5inamoreefficientway:
|z5|=(2)5=32
Arg(z5)=5(30o)
4PresentingStudentResponsesandCeardaíocht
Thestudentswillpresenttheirsolutionsandexplaintheir
thinkingattheboard.Anystudentsmaybeaskedtore-explain
whathasbeenpresented.Misconceptionswillbeaddressed.
Thesolutionswillbepresentedintheinorderofthe
AnticipatedResponses(outlinedabove).
StudentswillseeonanArganddiagramhowtheRectangular
formandtheArgument/Modulussolutionsfortheexpansion
describethesamenumber.
Itishopethatstudentsmightarticulate:“Averyefficientway
ofraisinganumbertoapoweristoraisethemodulusbythe
powerandmultiplytheargumentbythepower.”
5.Summingup
Itwillbeexplainedthattheideasdiscussedintodaysclass
relatetodeMoivre’sTheorem
Studentswillbeaskedtowritedownwhettheylearnedin
today’sclass.
6
Lesson Study Lesson Proposal – Decoding De Moivre
PlanforObservingtheStudents
Theteacherandtwoobserverswillbepresentfortheresearchlesson.Aseatingchartwillbeusedto
identifystudentsandtheirresponses.Theteacherwilluseaclip-boardandrecordsheetthatidentifies
whichstudentsattemptedandcompletedeachofthethreeanticipatedresponses.Acamerawillbeused
torecordtheboardwork.Duringtheresearchlessontheteachersarepreparedtoconsiderthefollowing
Post-LessonDiscussionquestions:Whathappenedthatyouexpected?Whathappenedthatyoudidn’t
expect?Towhatdegreewerethegoalsofthelessonachieved?
Introduction,posingthetask
Canstudentsrecallpriorknowledgerelationtocomplexnumbers?
Wasthetaskclear?
Questionsaskedbystudents
Individualwork
Canstudentscorrectlydescribezinrectangularform?
Canstudentscorrectlyexpandzinrectangularform?
5
Canstudentscorrectlyexpandz intermsofitsargumentandmodulus?
5
Canstudentcorrectlyexpandz intermsofitsargumentandmodulus?
Whatproblemsdothestudentsencounter?Arepromptsrequired?
Whatstrategiesdotheyemploywhendrawingcongruenttriangles?
Whatkindofquestionsdostudentsask?
Dotheypersistwiththetask?
PresentationandDiscussion
Arestudentsattentivetowhatishappeningontheboard?
Areclarificationsneededtopresenters’boardwork?
Didthediscussionandboardworkpromotestudentlearning?
DuringeachpartoftheclassevidencewillberecordedonpaperandusingtheLessonNoteApp.
Theanswersheetsusedbythestudentwillbecollectedalongwiththestudents’reflectiontheirown
learningattheendofthelesson.
7
Lesson Study Lesson Proposal – Decoding De Moivre
TheBoardPlan
PlannedBoardwork
ActualBoardwork
8
Lesson Study Lesson Proposal – Decoding De Moivre
PostLessonReflection
•
•
•
•
•
15studentscorrectlyinterpretedandwrotedowntherectangularformofz.
14studentstriedtoexpandz5usingrepeatedmultiplication,butnostudentcompletedthis
correctly.Discussionaroundproblemsrelatingtothisexpansionsupportedthelearninginthe
class.
Anumberofstudentsintheireffortstocompletethismethodcorrectlydidnothavetimeto
considerotherapproaches.
2studentshadthemisconceptionthat(√3+i)5=((√3)5+(i)5
Anumberofstudentsspenttimeworkingoutthemodulusandtheargumenteventhoughthis
informationcouldbegleanedfromthediagram.Whenastudentpointedthisout,therewasa
collective“Oooh!”fromtheclass.
•
•
4studentsusedthemodusandtheargumenttocorrectlyexpandz5.
Thestudentswereabletoderiveandexplainallthreeanticipatedresponses.Although
questioningwasusedtoleadstudentstowritethat30o+30o+30o+30o+30ocouldmore
efficientlybedescribedas5(30o).
•
DuringCeardaíochtastudentasked:“Whatifthepowerisn’t5?”Otherstudentswereableto
generalisehowtoapplytheprincipleofdeMoivre’sTheorem.
Thelearninggoalsoftheclassseemedtobeachieved.10studentreflectionresponsesstatedin
one-wayoranotherthat:“tomultiplycomplexnumbersyoumultiplythemoduliandaddthe
arguments”orto“raiseacomplexnumberbyapoweryouraisethemodulusbythepowerand
multiplytheargumentbythepower”.
Manyexpressedanappreciationforthismethod:“Ilearnedthatapageofmathscouldjustas
easilybesolvedinntwolines-|z5|=(2)5=32andArg(z5)=5x30o”orthat“itiseasierto
multiplycomplexnumbersnow".
•
•
9
Lesson Study Lesson Proposal – Decoding De Moivre
AppendixI–StudentHandoutwiththeProblem
Student:
This argand diagram shows the complex number z.
Im (i)
2
z
1
-2
30o
-1
1
√3 2
Re
-1
-2
5
Work out the value of z .
Explain your thinking...
10
Lesson Study Lesson Proposal – Decoding De Moivre
AppendixII–StudentTasksPriortothisLesson
11