Building a Stronger FoundaFon for Understanding in ... · 7/15/16 5 Given that cos 80 = 0.173648 . . . , explain how to ﬁnd cos 100, cos 260, cos 280, and sin 190 without using

7/15/16

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JasonSlowbeGreatOakHighSchool

SanDiego,CA

@[email protected]#NCTMinst

BuildingaStrongerFoundaFonforUnderstandinginAdvanced

Trigonometry

•  SlidesandhandoutwillbeonIns1tutewebsite:h3p://nctm.org/hs16

•  Allresourceswillbeonmyownwebsite:h3p://theslowbe.weebly.com

•  Pleasetweetthroughouttoday’ssession

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@theslowbe #NCTMinst

7/15/16

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Goalsforthissession

•  Re-introducingtrigfunc1ons•  Makingsenseofamplitude,phaseshiG•  Makingsenseofall6trigfunc1ons•  Mo1va1ngnon-righttriangletrig

4 @theslowbe #NCTMinst

AcomplicaFon


HenriPiccioMo

h3p://blog.mathedpage.org/@hpiccio3o


7/15/16

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Findperimeterandareainunitcircle

h3p://blog.mathedpage.org/@hpiccio3o7 @theslowbe #NCTMinst

PhillipsExeterAcademy

h3p://www.businessinsider.com/what-its-like-to-a3end-phillips-exeter-academy-2014-11?op=1



7/15/16

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Sampleproblems,ExeterAcademy

Atconstantspeed,awheelrotatesoncecounterclockwiseevery10seconds.Thecenterofthewheelis(0,0)anditsradiusis1foot.Apaintspotisini1allyat(1,0);whereisittsecondslater?


Quinnisrunningaroundthecirculartrackx2+y2=10000,whoseradiusis100meters,at4meterspersecond.Quinnstartsatthepoint(100,0)andrunsinthecounterclockwisedirec1on.AGer30minutesofrunning,whatareQuinn’scoordinates?



Represen1ngoneunitbyatleastfivesquaresonyourgraphpaper,drawtheunitcircle,whichiscenteredattheoriginandgoesthroughpointA=(1,0).Useaprotractortomarkthethird-quadrantpointPonthecircleforwhicharcAPhasangularsize215degrees.Es1matethecoordinatesofP,readingfromyourgraphpaper.No1cethatbotharenega1venumbers.Turnonyourcalculatorandaskforthecosineandsinevaluesofa215-degreeangle.Dofurtherexplora1on,thenexplainwhysineandcosineareknownascircularfunc/ons.



7/15/16

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Giventhatcos80=0.173648...,explainhowtofindcos100,cos260,cos280,andsin190withoutusingacalculator.



A15-degreecounterclockwiserota1oncenteredat(2,1)sends(4,6)toanotherpoint(x,y).Findxandy,correcttothreedecimalplaces.



DiscoveringTrigFuncFons

Hands-OnDiscoveryofTrigonometricFunc1ons(adaptedfromRondaDavis,AlbuquerquePublicSchools,andRonStewart,BaylorSchool)Whenyouentery-valuesinL3andy/r-valuesinL5,enterthemusingthefewestkeystrokespossible


7/15/16

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Angle x-coordinate

0 1

10 0.9

… …

80 0.2

90 0

100 -0.2

… …

170 -0.9

180 -1

190 -0.9

… …

260 -0.2

270 0

280 0.2

… …

350 0.9

360 1

y-coordinate

0

0.2

…

0.9

1

0.9

…

0.2

0

-0.2

…

-0.9

-1

-0.9

…

-0.2

0

QI

QII QII

QIII QIII

QIV QIV

QV

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Angle x-coordinate

0 1

10 0.9

… …

80 0.2

90 0

100 -0.2

… …

170 -0.9

180 -1

190 -0.9

… …

260 -0.2

270 0

280 0.2

… …

350 0.9

360 1

y-coordinate

0

0.2

…

0.9

1

0.9

…

0.2

0

-0.2

…

-0.9

-1

-0.9

…

-0.2

0

QI

QII QII

QIII QIII

QIV QIV

QV

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sin θ( ) = cos θ −90( )cos θ( ) = sin θ +90( )sin θ( ) = −cos θ +90( )

7/15/16

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DiscoveringTrigFuncFons

Predict,thenSca3erplotwithpartner:•  L1vL2•  L1vL3•  L1vL4•  L1vL5


Predict,thenSca3erPlotwithpartner:•  L2vL3•  L4vL5

FoundaFonforfuturereference

•  PhaseshiGàDele1ng,Inser1ngquadrants•  AmplitudeàRadius


FerrisWheeltask–MARS


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Debrief

•  Seesineandcosineascircularfunc1ons•  Contextformo1va1ngandunderstandingbothamplitudeandphaseshiG

•  Otherobserva1ons,take-aways?


7/15/16

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“Concepts_______Procedures”

Fill-intheblank,thendiscussatyourtables:beforeaGerbalancedwithinsteadof


“ConceptsBeforeProcedures”

FromNCTM’sPrinciplestoAc/ons(p42):•  “theimportanceofanintegratedandbalanceddevelopmentofconceptsandproceduresinlearningmathema1cs”

•  “NCTMandCCSSMemphasizethatproceduralfluencyfollowsandbuildsonafounda1onofconceptualunderstanding,strategicreasoning,andproblemsolving”


“ConceptsBeforeProcedures”

FromNCTM’sPrinciplestoAc/ons(p42):•  “Studentsmustbeabletodomuchmorethancarryoutprocedures.Theymustknowwhichprocedureisappropriateandmostproduc1veinagivensitua1on,whataprocedureaccomplishes,andwhatkindofresultstoexpect.Mechanicalexecu1onofprocedureswithoutunderstandingtheirmathema1calbasisoGenleadstobizarreresults.”


7/15/16

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Goalsforthissession

•  Re-introducingtrigfunc1ons•  Makingsenseofamplitude,phaseshiG•  Makingsenseofall6trigfunc1ons•  Mo1va1ngnon-righttriangletrig


MakingSenseofAll6TrigFuncFons


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Discussatyourtables:

Whydoweteachreciprocaltrigfunc1ons?

Howdoyouteachreciprocaltrigfunc1ons?

What’sthepurpose?



•  Isanymoredifficulttosolvethan?

•  Evengraphingcalculatorsdonothavebu3onsforreciprocalfunc1ons

•  Mathema1calneeds,applica1onsinfuturecourses

•  Otherra1onales?


sin 28°( ) = 4x

csc 28°( ) = x4

x = 4csc 28°( ) = 4sin 28°( )


Also,discussatyourtableshowyouintroduce:•  Pythagoreaniden11es•  Cofunc1oniden11es


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hMp://www.mathsisfun.com/algebra/trig-interacFve-unit-circle.html


www.slideshare.net/sirgibey/math-9-similar-triangles-intro




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hMp://www.mathsisfun.com/algebra/trig-interacFve-unit-circle.html


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RepresenFngAll6TrigFuncFons

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RepresenFngAll6TrigFuncFons

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RepresentaFonsandConnecFons

•  Reciprocaltrigfunc1ons•  Cofunc1oniden11es•  Pythagoreaniden11es


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MoFvaFngnon-righttriangletrig

•  Discussatyourtables:Howdoyouintroducenon-righttriangletrig?

•  Bepreparedtosharesomeideaswithwholegroup…


LaserLevel


LaserLevel

45

a

c

b

a2 − c2

b2 − c2

h= a2 − c2 + b2 − c2


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LaserLevel

46

a

c b

a2 − c2

b2 − c2

h= a2 − c2 − b2 − c2

h

b


LaserLevel

47

a

c

b

a2 − c2

b2 − c2

b


SSAAmbiguousCase

•  GGBappletwww.geogebra.org/m/ndmobwP•  MyGeoGebrapage,manymoreapplets:www.geogebra.org/theslowbe


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Amazingvideo,app

h3p://www.graphingstories.com/Axhh3p://www.vernier.com/products/soGware/video-physics/

JasonSlowbeGreatOakHighSchool

SanDiego,CA

@[email protected]#NCTMinst

BuildingaStrongerFoundaFonforUnderstandinginAdvanced

Trigonometry

7/15/16

18

Disclaimer The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.

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Building a Stronger FoundaFon for Understanding in ... · 7/15/16 5 Given that cos 80 = 0.173648 . . . , explain how to ﬁnd cos 100, cos 260, cos 280, and sin 190 without using