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ìRightTrianglesandTrigonometryConstruc.onoftheunitcircle

Introduction

ì  DavidBergerì  HSteacherinMenomonie,WIì  Finishingmy16thyearatMHS

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IceBreaker

ì  Introduceyourselftoyourtable.ì  Whereareyoufrom?ì  Wheredoyouwork?ì  Favoritemathjoke?

UnderstandingtheUnitCircleì  Originalpresenta.onwasNovember2015atNCTMRegionalin

MinneapoliswithJakeLeibold.

ì  Builtfrompreviouslyunderstoodconcepts�

ì  Allowsstudentstoexperiencetheunitcirclethroughconstruc.on–greatforvisualandkinesthe.clearners�

ì  Doesn’trelyonmemoriza.ontricks�

ì  DevelopedbyateamofteachersduringalessonstudyattheParkCityMathIns.tutein2014�

ì  Dependingonthelevelofyourstudents–couldtakebetween60and120minutestocomplete

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Today’sAgenda

1.  Pre-requisiteOverview�

2.  UnitCircleConstruc.on�

3.  Whatwelearned…

4.  RadianExplora.on�

5.  Whatwelearned…

6.  Reflec.ons,Logis.cs,StudentWorkSamples�

7.  Ques.onandAnswerwithExitTicket

StudentPrerequisiteKnowledge

ì  SpecialRightTriangleMeasurements�

ì  ScalingShapes(toscaledownspecialrighttriangles)�

ì  RightTriangleTrigonometry�

ì  Ra.onalizingDenominatorswithsinglesquareroots�

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Studentsshouldnotknow

ì  Conversionbetweenradiansanddegrees

ì  Referenceangles

ì  UnitCircle

Goal:Createtheunitcirclebyusinganunderstandingofspecialrighttriangles

CommonCoreStandards�

ì  F-TF.3(+)Usespecialtrianglestodeterminegeometricallythevaluesofsine,cosine,tangentforπ/3,π/4andπ/6,andusetheunitcircletoexpressthevaluesofsine,cosines,andtangentforx,π+x,and2π–xintermsoftheirvaluesforx,wherexisanyrealnumber.�

ì  F-TF.4(+)Usetheunitcircletoexplainsymmetry(oddandeven)andperiodicityoftrigonometricfunc.ons.

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UnitCircleConstructionMaterials

ì  PictureofGraph

UnitCircleConstructionMaterials

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UnitCircleConstruction�LabelingtheTriangles

UnitCircleConstruction–PlottingPoints

1.  Grabonetriangle�

2.  Placethelabeledangleattheoriginandthedarkenedbasealongthex-axis�

3.  Plotthepointatthevertexoftheotheracuteangle�

4.  Labelthecoordinatesofthepoint

5.  Flipthetrianglealongthey-andx-axistoplotmorepoints�

6.  Repeatthestepswiththeother3triangles

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UnitCircleConstruction–PlottingPoints

1.  Grabonetriangle�

2.  Placethelabeledangleattheoriginandthedarkenedbasealongthex-axis�

3.  Plotthepointatthevertexoftheotheracuteangle�

4.  Labelthecoordinatesofthepoint

5.  Flipthetrianglealongthey-andx-axistoplotmorepoints�

6.  Repeatthestepswiththeother3triangles

UnitCircleConstruction�PlottingPoints

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UnitCircleConstruction�PlottingPoints

ì  Onceyouhavefinishedplojngyourpoints,grab3moreof1colorandgluethemdowntheywayyouplokedthem�

ì  Glueyourotherthreetrianglesonthebokomofthesheet�

ì  Turnthesheetoverandworkontheques.ons

UnitCircleConstruction�Degrees

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UnitCircleConstruction�Connections

UnitCircleConstruction�Connections

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UnitCircleConstruction�Connections

UnitCircleConstruction�Connections

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UnitCircleConstruction�Degrees

1.  Whyisthecircleyoucreatedcalledtheunitcircle?�

2.  Wherecanyoulookontheunitcircletofindthevalueofcos(30°)orcos(60°)?Why?Bespecific�

3.  Whataboutsin(45°)orsin(30°)?Bespecific�

4.  Anotherques.onasksyoutofindthesin(135°).Wheredoyouthinkyouwouldlooktofindthat?Why?�

5.  Finally,youareaskedtofindcos(-60°).Wheredoyouthinkthiswouldbelocated?Why?

UnitCircleConstruction�Recap

1.  Specialrighttrianglesputupontheboard�

2.  Scalehypotenuseto1(independent,group,orasaclass)forbothtriangles�

3.  Labelgiventrianglescut-outswiththisinforma.on�

4.  Usetrianglestoplotpoints�

5.  Usetrianglestolabelcoordinatesatthepoints�

6.  Gluesetof4ofonecoloronthecoordinategrid.Glueother3onbokomofsheet�

7.  Answerques.onsontheback(independent,group,orasaclass)�

8.  Walkthroughdegreeextension/connec.on�

9.  Labeldegreesoncreatedunitcircle

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WhatWeLearned

ì  Whataresomethingsyouno.cedaboutthisac.vity?�

ì  Thinkaboutyourstudents–howwouldthisac.vitygoforthem?Whatmightbesomeoftheirchallengesormisconcep.ons?�

ì  Whatlingeringques.onsdoyouhave?

RadianExplorationIntroduction

ì  Whataresomecommonmisconcep.onsaboutRadians?

ì  WhatexactlyisaRadian?

ì  HowwereyoutaughtRadians?

ì  HowdoyouteachRadians?

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RadianExploration�Standards

ì  F-TF.1.Understandradianmeasureofanangleasthelengthofthearcontheunitcirclesubtendedbytheangle. �

ì  F-TF.2.Explainhowtheunitcircleinthecoordinateplaneenablestheextensionoftrigonometricfunc.onstoallrealnumbers,interpretedasradianmeasuresofanglestraversedcounterclockwisearoundtheunitcircle.�

ì  G.C.5Deriveusingsimilaritythefactthatthelengthofthearcinterceptedbyanangleispropor.onaltotheradius,anddefinetheradianmeasureoftheangleastheconstantofpropor.onality.

RadianExploration�CircleCircumference

ì  Calculatethecircumferenceoftheunitcircle�

ì  C=2πr,r=1->C=2π�

ì  ThereareWikkiS.cksatyourtable–wrapthemaroundthecirclehavingthetwotappedendslineup.HowlongistheWikkiS.ck?�

ì  LabeleachendofthepipeWikkiS.ckas0and2πwiththemarkeratyourtable�

ì  No.cetheothertapedintervals–labelthesewiththecorrectmeasurements

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RadianExploration�WikkiStickRulers

ì  ApertheWikkiS.x/pipecleanersarelabeled,wrapaWikkiS.x/pipecleaneraroundtheunitcircle–no.cehoweachintervalbasicallymatchesupwithoneofthepointsploked�

ì  Labelthesemeasurementswithabluepencil

RadianExploration�Radians

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RadianExploration�RadianConnection

ì  Throughthisac.vityresisttheurgetouse“radian”–use“arclength”instead�

ì  Neartheendoftheac.vity,usetheCommonCoreStandardstodefineradian:“thelengthofthearcontheunitcirclesubtendedbytheangle”�

ì  Studentsexperiencethephysicalnatureofradians,notjustanotherwayofmeasuringanangle

WhatWeLearned

ì  Whataresomethingsyouno.cedaboutthisac.vity?�

ì  Thinkaboutyourstudents–howwouldthisac.vitygoforthem?

ì  Whatmightbesomeoftheirchallengesormisconcep.ons?�

ì  Whatlingeringques.onsdoyouhave?

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Reflections

ì  Studentsthatstrugglewiththepre-requisiteknowledgewillhavedifficulty�

ì  Caneasilybecometeachercentered,notteacherfacilitated�

ì  Timeconsuming–hardtocompletequickly�

ì  Studentsthatmissadaywillhaveahard.mecatchingup�

ì  Supplyprepcantakeawhile

Logistics

ì  DigitalResources�ì  hkp://www.wismath.org/2017-WMC-Annual-

Conference-Speaker-Materialsì  Print-readyversionsoftoday’sclassworksheet,triangles,

exit.ckets,andaddi.onalques.oningideas(forhigherlevelstudents)�

ì  LessonPlans,ClassroomPowerPointslides�

ì  Supplies�ì  Mul.-ColorCardStockPaper�ì  19’’/20’’pipecleaners�orWikkiS.xì  Tape/glues.cks/dryerasemarkers

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QuestionsandAnswers

ExitTicket

ì  Didyoucomeupwithanyneatideasduringthesession?Ifso,Iwouldlovetohearfromyou!�

ì  Whatrevisionswouldyousuggesttoeitherpor.onofthelesson?

�

ì  DigitalCopiesofallresourcesfromtoday,aswellasdetailedlessonplans,classroomPowerPoints,print-readymaterials(includingtrianglesheets):�ì  hkp://www.wismath.org/2017-WMC-Annual-Conference-

Speaker-Materialsì  Contact:berger.dave.m@gmail.com