“decoding de moivre” - pdst post primary maths team€¦ · lesson study lesson proposal...

Lesson Study Lesson Proposal – Decoding De Moivre

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“DecodingDeMoivre”

YearGroup:TransitionYearHigherLevelTopic:ComplexNumbers–DeMoivre’sTheorem

LessonStudyandResearchGroup:

DavidCrowdleandPaudgeBrennan,teachersinLoretoSS,Wexford,developedthislessonproposal.Paudgetaughtthelessonon10thFebruary2017with17studentsfromhisTransitionYearclass.

Briefdescriptionofthelesson:

Theresearchlessonisthesecondstructuredproblemsolvingclassesleadingthestudentstoappreciateanddevelopanunderstandingofpolarmultiplicationandraisinganumbertoapower(DeMoivre’sTheorem)intermsof themoduliandargumentsof thenumbers.Studentswillengagewithaproblemworkingoutthe value of a given complex number raised to the power of 5. Together they will present approachesleadingtoanunderstandingofdeMoivre’sTheoremandtheneedforit.


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AimsoftheLesson:

Shorttermaim:Studentswill:

• appreciatethattherearedifferentcorrectapproachestoexpandingacomplexnumber,• seetheadvantagestousingthe“Modulus-argument”formofacomplexnumberwhenitcomesto

multiplying,• developaconceptualunderstandtheprincipleofdeMoivre’stheorem“toexpandanumberraised

toapoweryouraisethemodulustothatpowerandmultiplytheargumentbythepower.”1Longtermaims:Wewouldlikeourstudents:

• togainconfidenceindealingwithabstractconcepts,• tobecomeindependentlearnersthroughstructuredproblemsolving,• toconnectandreviewtheconceptsthatwehavestudiedalready,• todeveloptheirliteracyandnumeracyskillsthroughdiscussingideas.1• todevelopkeyskillssuchas“ManagingInformation&Thinking:Thinkingcreativelyandcritically”,2

LearningOutcomes:

Asaresultofstudyingthistopicstudentswillbeableto:

• Understandindifferentwaysthemeaningofmultiplicationofwholenumbersandusethistomakesenseofcomplexnumbermultiplicationandexpansion.

• Understandandbeabletoexpressacomplexnumbersbothinrectangularformandintermsofitsmodulusandargument.

• RecogniseanumberonanArganddiagramintermsofitsmodulusandargument• Developtheinsightthatwhennumbersaremultipliedtheirmoduliaremultipliedandtheir

argumentsareaddedtogether.• Beabletomultiplynumbersgiventheirargumentsandmoduli.• Usethistodiscoverthatwhenanumberisraisedtoapoweritsmodulusisraisedtothatpower

anditsargumentismultipliedbythatpowerandexperiencethatthisapproachismuchmoreefficientthatusingrepeatedmultiplicationinrectangularform.

BackgroundandRationale

ThistopicwaschosenbecausestudentsfinddealingwiththepolarformofcomplexnumbersandDeMoivre’sTheoremachallengingpartoftheHigherLevelCourse.Itishopedthatthisapproachwillhelpstudentsappreciatetheefficiencyofusingthepolarformformultiplicationandapplyingpowerstonumbers.

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-and-thinking_April-2015.docx, Accessed 12 February 2017


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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/Managing-information-and-thinking_April-2015.docx.


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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 WiththisTransitionYeargrouponlytheOrdinaryLevelmaterialandaninformalapproachtothepolarformofacomplexnumberandrelatedmultiplicationanddeMoivre’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.Considerintegermultiplicationintermsof“scalingandrotation.”

40min.

10 Multiplication(Structuredproblemsolvinglesson):considerscalingintermsofthemoduliandtherotationintermsofthearguments.(AppendixII)

40min.

11 ReserchLesson:“DecodingDeMoivre” 40min.


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FlowoftheLesson

1.Introduction(5minutes)

PriorKnowledge:Studentsarequestionedontheirunderstandingofpreviousclassesconcerning:

a) Arganddiagramb) Rectangularformofacomplexnumberc) Describingtheargumentandmodulusofacomplex

numberd) Indicese) Multiplicationbyexpansionf) Multiplicationintermsofargumentsandmoduli

2.PosingtheTask(2minutes) Thetaskispresentedontheboard.Theyareinstructedtofindthevalueofz5in“asmanywaysasyoucan”in“15minutes”.Studentsarequestionedontheirunderstandingofthetask.Eachstudentisgiven3copiesofthehandoutpagewiththequestionandtheyareaskedto“explaintheirthinking”(AppendixI).

3.Individualproblemsolving(15minutes)

AnticipatedStudentResponse1Itisexpectedthatmoststudentswillrecognisethattherectangularformofz=√3+iandthentheymighttrytoevaluate(√3+i)5byexpandingitinatraditionalway–repeatedmultiplication.

AnticipatedStudentResponse2Itishopedthatstudentsmighttrytodescribethemodulusandargumentofz.|z|=2andArg(z)=30o

Theymightthendescribetheexpansionofz5intermsofrepeatedmultiplication:|z5|=(2)(2)(2)(2)(2)=32Arg(z5)=30o+30o+30o+30o+30o=150o


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AnticipatedStudentResponse3Studentsmightbuildon“AnticipatedResponse2”toexpresstheexpansionofz5inamoreefficientway:|z5|=(2)5=32Arg(z5)=5(30o)

4PresentingStudentResponsesandCeardaíochtThestudentswillpresenttheirsolutionsandexplaintheirthinkingattheboard.Anystudentsmaybeaskedtore-explainwhathasbeenpresented.Misconceptionswillbeaddressed.ThesolutionswillbepresentedintheinorderoftheAnticipatedResponses(outlinedabove).StudentswillseeonanArganddiagramhowtheRectangularformandtheArgument/Modulussolutionsfortheexpansiondescribethesamenumber.Itishopethatstudentsmightarticulate:“Averyefficientwayofraisinganumbertoapoweristoraisethemodulusbythepowerandmultiplytheargumentbythepower.”

5.SummingupItwillbeexplainedthattheideasdiscussedintodaysclassrelatetodeMoivre’sTheoremStudentswillbeaskedtowritedownwhettheylearnedintoday’sclass.


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PlanforObservingtheStudents

Theteacherandtwoobserverswillbepresentfortheresearchlesson.Aseatingchartwillbeusedtoidentifystudentsandtheirresponses.Theteacherwilluseaclip-boardandrecordsheetthatidentifieswhichstudentsattemptedandcompletedeachofthethreeanticipatedresponses.Acamerawillbeusedtorecordtheboardwork.DuringtheresearchlessontheteachersarepreparedtoconsiderthefollowingPost-LessonDiscussionquestions:Whathappenedthatyouexpected?Whathappenedthatyoudidn’texpect?Towhatdegreewerethegoalsofthelessonachieved?Introduction,posingthetask Canstudentsrecallpriorknowledgerelationtocomplexnumbers?

Wasthetaskclear?Questionsaskedbystudents

Individualwork Canstudentscorrectlydescribezinrectangularform?Canstudentscorrectlyexpandzinrectangularform?Canstudentscorrectlyexpandz5intermsofitsargumentandmodulus?Canstudentcorrectlyexpandz5intermsofitsargumentandmodulus?Whatproblemsdothestudentsencounter?Arepromptsrequired?Whatstrategiesdotheyemploywhendrawingcongruenttriangles?Whatkindofquestionsdostudentsask?Dotheypersistwiththetask?

PresentationandDiscussion Arestudentsattentivetowhatishappeningontheboard?Areclarificationsneededtopresenters’boardwork?Didthediscussionandboardworkpromotestudentlearning?

DuringeachpartoftheclassevidencewillberecordedonpaperandusingtheLessonNoteApp.

Theanswersheetsusedbythestudentwillbecollectedalongwiththestudents’reflectiontheirownlearningattheendofthelesson.


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TheBoardPlan

PlannedBoardwork

ActualBoardwork


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PostLessonReflection

• 15studentscorrectlyinterpretedandwrotedowntherectangularformofz.• 14studentstriedtoexpandz5usingrepeatedmultiplication,butnostudentcompletedthis

correctly.Discussionaroundproblemsrelatingtothisexpansionsupportedthelearningintheclass.

• Anumberofstudentsintheireffortstocompletethismethodcorrectlydidnothavetimetoconsiderotherapproaches.

• 2studentshadthemisconceptionthat(√3+i)5=((√3)5+(i)5• Anumberofstudentsspenttimeworkingoutthemodulusandtheargumenteventhoughthis

informationcouldbegleanedfromthediagram.Whenastudentpointedthisout,therewasacollective“Oooh!”fromtheclass.

• 4studentsusedthemodusandtheargumenttocorrectlyexpandz5.• Thestudentswereabletoderiveandexplainallthreeanticipatedresponses.Although

questioningwasusedtoleadstudentstowritethat30o+30o+30o+30o+30ocouldmoreefficientlybedescribedas5(30o).

• DuringCeardaíochtastudentasked:“Whatifthepowerisn’t5?”OtherstudentswereabletogeneralisehowtoapplytheprincipleofdeMoivre’sTheorem.

• Thelearninggoalsoftheclassseemedtobeachieved.10studentreflectionresponsesstatedinone-wayoranotherthat:“tomultiplycomplexnumbersyoumultiplythemoduliandaddthearguments”orto“raiseacomplexnumberbyapoweryouraisethemodulusbythepowerandmultiplytheargumentbythepower”.

• Manyexpressedanappreciationforthismethod:“Ilearnedthatapageofmathscouldjustaseasilybesolvedinntwolines-|z5|=(2)5=32andArg(z5)=5x30o”orthat“itiseasiertomultiplycomplexnumbersnow".


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AppendixI–StudentHandoutwiththeProblem

Student:

1

2

-1

-2

1 2-1-230o

Re

Im (i)

z

√3

Explain your thinking...

This argand diagram shows the complex number z.

Work out the value of z5.


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AppendixII–StudentTasksPriortothisLesson