6.2 “sine” language...secondary math iii // module 6 modeling periodic behavior – 6.2...

SECONDARY MATH III // MODULE 6

MODELING PERIODIC BEHAVIOR – 6.2

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

6.2 “Sine” Language

A Solidify Understanding Task

Intheprevioustask,GeorgeW.Ferris’DayOff,you

probablyfoundCarlos’heightatdifferentpositionsontheFerriswheelusingrighttriangles,as

illustratedinthefollowingdiagram.

Recallthefollowingfactsfromthe

previoustask:

• TheFerriswheelhasaradiusof25feet• ThecenteroftheFerriswheelis30feetabovetheground

Carloshasalsobeencarefullytimingthe

rotationofthewheelandhasobserved

thefollowingadditionalfact:

• TheFerriswheelmakesonecompleterevolutioncounterclockwiseevery20seconds

1. HowhighwillCarlosbe2secondsafterpassingpositionAonthediagram?

2. Calculatetheheightofariderateachofthefollowingtimest,wheretrepresentsthenumberofsecondssincetheriderpassedpositionAonthediagram.Keeptrackofanyregularitiesyounoticeinthewaysyoucalculatetheheight.Asyoucalculateeachheight,plotthepositiononthediagram.

CC

BY

Em

ma

Cra

ig

http

s://f

lic.k

r/p/

Kw

Raf

r

5




ElapsedtimesincepassingpositionA Calculations

Heightoftherider

1sec

2sec

2.5sec

3sec

6sec

8sec

14.5sec

18sec

23sec

28sec

36sec

37sec

40sec

3. Examineyourcalculationsforfindingtheheightoftheriderduringthefirst5secondsafterpassingpositionA(thefirstfewvaluesintheabovetable).Duringthistime,theangleofrotationoftheriderissomewherebetween0°and90°.Writeageneralformulaforfindingtheheightoftheriderduringthistimeinterval.

4. Howmightyoufindtheheightoftheriderinother“quadrants”oftheFerriswheel,whentheangleofrotationisgreaterthan90°?

6




6.2 “Sine” Language – Teacher Notes A Solidify Understanding Task

Purpose:Thepurposeofthistaskistoextendthestrategiesusedintheprevioustaskforfinding

theheightofarideronastationaryFerriswheel,tofindingtheheightofariderafteranintervalof

timehaselapsedsincetheriderpassedthepointfarthesttotherightofthewheel.Thismotionof

theridercanbemodeledbyanangleofrotationdrawnin“standard”position(i.e.,withtheinitial

raypointingtotherightandwithapositiveanglerepresentingcounterclockwiserotation).

Studentswillidentifythatthefunctionheight=30+25sin(18t)givestheheightofarideraftert

seconds,atleastfor0<t<5seconds—timeswhere18tgivesananglebetween0°and90°,and

thereforesin(18t)canbefoundusingrighttriangletrigonometry.Fort>5seconds,studentswill

needtoconsidertherelatedrighttrianglesineachquadrant,andmodifytheirformulasothatthe

definitionofsineasaratioofsidesofarighttriangleholds.Thisleadstoapiecewise-defined

functionfortheheightoftherider—adilemmathatwillberesolvedinfuturetaskswhenthe

definitionofsineisextended.

CoreStandardsFocus:

F.TF.5Choosetrigonometricfunctionstomodelperiodicphenomenawithspecifiedamplitude,

frequency,andmidline.�

RelatedStandards:G.SRT.8

StandardsforMathematicalPractice:

SMP4–Modelwithmathematics

SMP7–Lookforandmakeuseofstructure




Vocabulary:Studentswillgenerateanewquantity,angularspeed,tofindtheangleofrotationasa

functionofelapsedtime.TheangularspeedoftheFerriswheelisfoundbydividing360°bythe

timeittakestomakeonecompleterevolution.TheangularspeedfortheFerriswheelinthetaskis

18°persecond.

TheTeachingCycle:

Launch(WholeClass):

Beginbyreviewingtheworkfromthe

previoustaskbyaskingastudenttodescribe

howthetriangledrawnonthediagramofthe

Ferriswheel(seethefirstpageofthetask)

wasusedtodeterminetheheightofthe

associatedpointfromtheground.Then,point

outthenewinformation—thewheelmakes

onecompleterevolutioncounterclockwise

every20seconds.Withthisadditional

information,askstudentstocalculateCarlos’

height2secondsafterhepassespointA.Give

studentsafewminutestoworkonthis

problem.Helpstudentsrecognizethatan

importantrelatedfactisthatthewheel

rotates18°persecond.Ifstudentsarefindingitdifficulttonoticethis,ask,“Ifittakes20secondsto

makeonecompleterevolution,howmanysecondswouldittaketorotatetothispositionwherethe

spokeis36°fromthehorizontal?”Studentsmightsetupaproportion,!"$%&'("° = +$%&'(° ,ordivide360°

by20tofindtheangularspeedof18°/sec.Alternatively,youmightask,“Howmanysecondswould

ittakefortheridertomovefrompositionAtopositionB?”

Sincetheyhavealreadycalculatedthisheightintheprevioustask,studentscanrecordthisheight

directlyontheirchart(seequestion2).Tellstudentstowatchforheightstheyhavealready




calculatedorrelatedworktheymightuseastheycompletethetableinquestion2.Havestudents

labelthepointat36°ast=2seconds,h=44.7feet;thenhavethemaddandlabelthehighestpoint

ontheFerriswheeldiagramast=5,h=55feet.Tellstudentstheyshouldplotandlabelother

pointsontheFerriswheelastheyworkonquestion2.

Explore(SmallGroup):

Initially,studentsmayneedtothinkaboutthenumberofdegreesofrotationassociatedwitheach

time.Theymayreasonthatsincetheangularspeedis18°persecond,Carloswillrotate9°inahalf-

secondor54°in3seconds.Listenforstudentswhorecognizethattheycanusetheexpression18t

insidethesinefunction.Also,watchforstudentswhomakeuseofrelatedtrianglestoreducethe

numberofcalculationstheyneedtocomplete.Forexample,theworkusedtocalculatetheheightat

t=2secondscanbeusedtocalculatetheheightatt=8secondsandt=18secondsduetothe

symmetryofthecircle.

Asstudentsmoveintoother“quadrants”,suchaswhent=6secondsandtheangleofrotationis

108°,theymaycalculatesin(108°)ontheircalculatorwithoutrecognizingthata108°angledoesn’t

makesenseasanangleinrighttriangletrigonometry.Acknowledgethatthecalculatorcando

somethingwedonotyetunderstand,andthereforewearenotgoingtousethese“mysterious,

obtuse”values.Instead,askthemtodowhattheydidintheprevioustask:drawarelatedright

triangle—inthiscaseatrianglewithanacuteanglemeasuring72°—andusethattriangleto

calculatetheheightatt=6secondsoranangleofrotationof108°.Thisisimportantworkfor

developingunderstandingoftrigonometricrelationships,sodon’tskipoveritbyallowingstudents

tousethecalculatormindlessly.Thisdilemmawillberesolvedinfuturetaskswhenthedefinition

ofsineisextended.

Discuss(WholeClass):

Beginthediscussionwithquestion3,wheretheangleofrotationisbetween0°and90°andt,the

elapsedtime,isbetween0and5seconds.Selectastudentwhocanpresenthowwemight

generalizethecomputationalworkinthistimeintervalusingtheformulah(t)=30+25sin(18t).




Makesureallstudentsunderstandthedilemmaofusingthisformulafort>5sincewehavenot

establishedmeaningforthesineofananglegreaterthan(orevenequalto)90°.Ifthereare

studentswhohaverespondedtoquestion4,havethempresenttheirformulasforquadrantsII,and

haveotherstudentspresenttheirworkforpositionsinquadrantsIIIandIV.Eveniftheirformulas

areinitiallyinaccurate,theworkofresolvingwhattodoineachquadrantisimportantworkto

discuss.Thisdiscussionshouldleadtothefollowingpiecewise-definedfunctionsforonerevolution

ofthewheel.

AlignedReady,Set,Go:ModelingPeriodicBehavior6.2

!!

h(t)=

30+25sin(18t), 0< t <530+25sin(180−18t), 5< t <1030−25sin(18t −180), 10< t <1530−25sin(360−18t), 15< t <2055, t =530, t =0or t =10or t =205, t =15

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪




6.2

Needhelp?Visitwww.rsgsupport.org

READY Topic:Describingintervalsfromgraphs

Foreachgraph,writetheinterval(s)where!(#)ispositiveandtheinterval(s)whereitisnegative.

1.

Positive_____________________________________________

Negative____________________________________________

2.

Positive_____________________________________________

Negative____________________________________________

3. (Thescaleonthex-axisisinincrementsof45°.)

Positive_____________________________________________

Negative____________________________________________

4. (Thescaleonthex-axisisinincrementsof45°.)

Positive_____________________________________________

Negative____________________________________________

READY, SET, GO! Name PeriodDate

7




6.2


4

2

–2

–4

–6

5

Writethepiece-wiseequationsforthegivengraphs.

5.

6.

Equation:

Equation:

SET

Topic:Calculatingsineasafunctionoftime

Recallthefollowingfactsfromtheclassroom

task:

• TheFerriswheelhasaradiusof25

feet

• ThecenteroftheFerriswheelis30

feetabovetheground

Duetoasafetyconcern,themanagementof

theamusementparkdecidestoslowthe

rotationoftheFerriswheelfrom20seconds

forafullrotationto30secondsforafullrotation.

8




6.2


7. Calculatehowhighariderwillnowbe2secondsafterpassingpositionAonthediagram.

8. Calculatetheheightofariderateachofthefollowingtimest,wheretrepresentsthenumberofsecondssincetheriderpassedpositionAonthediagram.Asyoucalculateeachheight,plotthepositiononthediagram.Connectthecenterofthecircletothepointyouplotted.ThendrawaverticallinefromtheplottedpointontheFerriswheeltothelinesegmentAFinthediagram.Eachtimeyoushouldgetarighttrianglesimilartotheoneinthefigure.

ElapsedtimesincepassingpositionA

CalculationsHeightoftherider

(infeet)

1sec

3sec

5sec

7sec

8sec

11sec

14sec

15sec

16sec

20sec

22sec

23sec

25sec

27sec

30sec

9




6.2


9. Howdidthepositionofthetrianglesyoudrewchangebetween7secondsand8seconds?

10. Howdidthetrianglesyoudrewchangebetween14,15,and16seconds?

11. Howdidthetrianglesyoudrewchangebetween22secondsand23seconds?

12. Describearelationshipbetweentheorientationoftherighttrianglesaroundthecircleandtheangle

ofrotation.Usethediagramtohelpyouthink

aboutthequestion.(Thedottedarcshowsthe

angleofrotation.)

GO Topic:Findingmissinganglesintriangles

FindthemeasureofeachacuteangleofrighttriangleABCwithM∠O = QR°.

Roundyouranswerstothenearestdegree.

13.T = 3UVW = 5UV 14.T = 5XYW = 10XY

15.T = 9.1W[W = 12.3W[ 16.T = 14.1W[W = 18. W[

17.T = 9.7UV\ = 12.7UV 18.T = 14.6XYW = 20.3XY

10

6.2 “sine” language...secondary math iii // module 6 modeling periodic behavior – 6.2...

Documents