“Decoding De Moivre” - PDST Post Primary Maths Team€¦ · Lesson Study Lesson Proposal...
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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