SECONDARY MATH I // MODULE 8

CONNECTING ALGEBRA & GEOMETRY – 8.2

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8.2 Slippery Slopes

A Solidify Understanding Task

Whileworkingon“IsItRight?”inthepreviousmoduleyoulookedatseveralexamplesthatleadto

theconclusionthattheslopesofperpendicularlinesarenegativereciprocals.Yourworkhereisto

formalizethisworkintoaproof.Let’sstartbythinkingabouttwoperpendicularlinesthatintersect

attheorigin,likethese:

1. Startbydrawingarighttrianglewiththesegment!" asthehypotenuse.Theseareoftencalledslopetriangles.Basedontheslopetrianglethatyouhavedrawn,whatistheslopeof

!"?

2. Now,rotatetheslopetriangle90°abouttheorigin.Whatarethecoordinatesoftheimage

ofpointA?

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https://flic.kr/p/kFus4X

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SECONDARY MATH I // MODULE 8

CONNECTING ALGEBRA & GEOMETRY – 8.2

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3. Usingthisnewpoint,A’,drawaslopetrianglewithhypotenuse!"′ .Basedontheslopetriangle,whatistheslopeoftheline!"′?

4. Whatistherelationshipbetweenthesetwoslopes?Howdoyouknow?

5. Istherelationshipchangedifthetwolinesaretranslatedsothattheintersectionisat

(-5,7)?

Howdoyouknow?

Toproveatheorem,weneedtodemonstratethatthepropertyholdsforanypairofperpendicular

lines,notjustafewspecificexamples.Itisoftendonebydrawingaverysimilarpicturetothe

exampleswehavetried,butusingvariablesinsteadofnumbers.Usingvariablesrepresentsthe

ideathatitdoesn’tmatterwhichnumbersweuse,therelationshipstaysthesame.Let’strythat

strategywiththetheoremaboutperpendicularlineshavingslopesthatarenegativerecipricals.

8

SECONDARY MATH I // MODULE 8

CONNECTING ALGEBRA & GEOMETRY – 8.2

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• Lineslandmareconstructedtobeperpendicular.

• StartbylabelingapointPonthelinel.

• LabelthecoordinatesofP.

• DrawtheslopetrianglefrompointP.

• Labelthelengthsofthesidesoftheslopetriangleusingvariableslikeaandbforthe

runandtherise.

6. Whatistheslopeoflinel?

RotatepointP90°abouttheorigin,labelitP’andmarkitonlinem.Whatarethe

coordinatesofP’?

7. DrawtheslopetrianglefrompointP’.Whatarethelengthsofthesidesoftheslope

triangle?Howdoyouknow?

8. Whatistheslopeoflinem?

9. Whatistherelationshipbetweentheslopesoflinelandlinem?Howdoyouknow?

10. Istherelationshipbetweentheslopeschangediftheintersectionbetweenlinelandlinem

istranslatedtoanotherlocation?Howdoyouknow?

11. Istherelationshipbetweentheslopeschangediflineslandmarerotated?

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SECONDARY MATH I // MODULE 8

CONNECTING ALGEBRA & GEOMETRY – 8.2

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12. Howdothesestepsdemonstratethattheslopesofperpendicularlinesarenegative

reciprocalsforanypairofperpendicularlines?

Thinknowaboutparallellinesliketheonesbelow.

13.DrawtheslopetrianglefrompointAtotheorigin.Whatistheslopeof!"?

14.Whattransformation(s)mapstheslopetrianglewithhypotenuse!"ontotheotherlinem?

15.Whatmustbetrueabouttheslopeoflinel?Why?

m

l

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SECONDARY MATH I // MODULE 8

CONNECTING ALGEBRA & GEOMETRY – 8.2

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

Nowyou’regoingtotrytousethisexampletodevelopaproof,likeyoudidwiththeperpendicular

lines.Herearetwolinesthathavebeenconstructedtobeparallel.

16.Showhowyouknowthatthesetwoparallellineshavethesameslopeandexplainwhythis

provesthatallparallellineshavethesameslope.

11

SECONDARY MATH I // MODULE 8

CONNECTING ALGEBRA & GEOMETRY – 8.2

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8.2 Slippery Slopes – Teacher Notes

A Solidify Understanding Task

Purpose:Thepurposeofthistaskistoprovethatparallellineshaveequalslopesandthatthe

slopesofperpendicularlinesarenegativereciprocals.Studentshaveusedthesetheorems

previously.Theproofsusetheideasofslopetriangles,rotations,andtranslations.Bothproofsare

precededbyaspecificcasethatdemonstratestheideabeforestudentsareaskedtofollowthelogic

usingvariablesandthinkingmoregenerally.

CoreStandardsFocus:

G.GPEUsecoordinatestoprovesimplegeometrictheoremsalgebraically.

G.GPE.5Provetheslopecriteriaforparallelandperpendicularlinesandusethemtosolve

geometricproblems(e.g.,findtheequationofalineparallelorperpendiculartoagivenlinethat

passesthroughagivenpoint).

RelatedStandards:G.CO.4,G.CO.5

StandardsforMathematicalPracticeofFocusintheTask:

SMP3–Constructviableargumentsandcritiquethereasoningofothers.

SMP6-Attendtoprecision.

TheTeachingCycle:

Launch(WholeClass):

Ifstudentshaven’tbeenusingtheterm“slopetriangle”,startthediscussionwithabrief

demonstrationofslopetrianglesandhowtheyshowtheslopeoftheline.Studentsshouldbe

familiarwithperforminga90degreerotationfromthepreviousmodule,sobeginthetaskby

havingstudentsworkindividuallyonquestions1,2,3,and4.Whenmoststudentshavedrawna

conclusionfor#4,haveadiscussionofhowtheyknowthetwolinesareperpendicular.Sincethe

purposeistodemonstratethatperpendicularlineshaveslopesthatarenegativereciprocals,

SECONDARY MATH I // MODULE 8

CONNECTING ALGEBRA & GEOMETRY – 8.2

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emphasizethatthereasonthatweknowthatthelinesareperpendicularisthattheywere

constructedbasedupona90degreerotation.

Explore(SmallGroup):

Theproofthattheslopesofperpendicularlinesarenegativereciprocalsfollowsthesamepattern

astheexamplegiveninthepreviousproblem.Monitorstudentsastheywork,allowingthemto

selectapoint,labelthecoordinatesandthenthesidesoftheslopetriangles.Referstudentsbackto

thepreviousproblem,askingthemtogeneralizethestepssymbolicallyiftheyarestuck.When

studentsarefinishedwithquestions6-12,discusstheproofasawholegroupandthenhave

studentscompletethetask.

Discuss(WholeClass):

Thesetupfortheproofisbelow:

Theslopeoflinelis!! andtheslopeoflinemis !!!or-!!.Theproductofthetwoslopesis-1,

thereforetheyarenegativereciprocals.Ifthelinesaretranslatedsothattheintersectionisnotat

theorigin,theslopetriangleswillremainthesame.Discusswiththeclasshowquestions6-12help

ustoconsiderallthepossiblecases,whichisnecessaryinaproof.Afterstudentshavefinishedthe

task,gothroughthebriefproofthattheslopesofparallellinesareequal.

AlignedReady,Set,Go:ConnectingAlgebraandGeometry8.2

-b

y

m

l

b

a

a

(a, b)

(-b,a)

SECONDARY MATH I // MODULE 8

CONNECTING ALGEBRA & GEOMETRY – 8.2

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8.2

READY Topic:Usingtranslationstographlines

Theequationofthelineinthegraphis! = !.1.a)Onthesamegridgraphaparallellinethatis3unitsaboveit.

b)Writetheequationforthenewlineinslope-interceptform.

c)Writethey-interceptofthenewlineasanorderedpair.

d)Writethex-interceptofthenewlineasanorderedpair.

e)Writetheequationofthenewlineinpoint-slopeformusingthey-intercept.

f)Writetheequationofthenewlineinpoint-slopeformusingthex-intercept.g)Explaininwhatwaytheequationsarethesameandinwhatwaytheyaredifferent.

Thegraphattherightshowstheline! = −!".2.a)Onthesamegrid,graphaparallellinethatis4unitsbelowit.

b)Writetheequationofthenewlineinslope-interceptform.

c)Writethey-interceptofthenewlineasanorderedpair.

d)Writethex-interceptofthenewlineasanorderedpair.

e)Writetheequationofthenewlineinpoint-slopeformusing

they-intercept.

f)Writetheequationofthenewlineinpoint-slopeformusingthex-intercept.g)Explaininwhatwaytheequationsarethesameandinwhatwaytheyaredifferent.

READY, SET, GO! Name PeriodDate

12

SECONDARY MATH I // MODULE 8

CONNECTING ALGEBRA & GEOMETRY – 8.2

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8.2

Thegraphattherightshowstheline! = !! !.

3.a)Onthesamegrid,graphaparallellinethatis2unitsbelowit.

b)Writetheequationofthenewlineinslope-interceptform.

c)Writethey-interceptofthenewlineasanorderedpair.

d)Writethex-interceptofthenewlineasanorderedpair.

e)Writetheequationofthenewlineinpoint-slopeformusingthey-intercept.

f)Writetheequationofthenewlineinpoint-slopeformusingthex-intercept.g)Explaininwhatwaytheequationsarethesameandinwhatwaytheyaredifferent.

SET Topic:Verifyingandprovinggeometricrelationships

Thequadrilateralattherightiscalledakite.Completethemathematicalstatementsaboutthekiteusingthegivensymbols.Proveeachstatementalgebraically.(Asymbolmaybeusedmorethanonce.)

≅ ⊥ ∥ < > =

Proof

4.!"__________!" ______________________________________________________________________________

5.!"__________!"

6.!"__________!"

13

SECONDARY MATH I // MODULE 8

CONNECTING ALGEBRA & GEOMETRY – 8.2

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8.2

7.∆!"#______ ∆!"#

8.!!__________!"

9.!"__________!"

10.!"__________!"

GO

Topic:Writingequationsoflines

Usethegiveninformationtowritetheequationofthelineinstandardform. !" + !" = ! 11.!"#$%: − !

! !"#$% !",!

12.! !!,−! , ! !,!

13.! − !"#$%&$'#: − !; ! − !"#$%&$'#: − !

14.!"" ! !"#$%& !"# −! . ! !" !"# !"#$%&.

15.!"#$%: !! ; ! − !"#$%&$'#:! 16.! −!",!" , ! !",!"

14