7
SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.2
Whileworkingon“IsItRight?”inthepreviousmoduleyoulookedatseveralexamplesthatleadto
theconclusionthattheslopesofperpendicularlinesarenegativereciprocals.Yourworkhereisto
formalizethisworkintoaproof.Let’sstartbythinkingabouttwoperpendicularlinesthatintersect
attheorigin,likethese:
1. Startbydrawingarighttrianglewiththesegment!" asthehypotenuse.Theseareoften
calledslopetriangles.Basedontheslopetrianglethatyouhavedrawn,whatistheslopeof
!"?
2. Now,rotatetheslopetriangle90°abouttheorigin.Whatarethecoordinatesoftheimage
ofpointA?
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A Solidify Understanding Task
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8.2 Slippery Slopes
8
SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.2
3. Usingthisnewpoint,A’,drawaslopetrianglewithhypotenuse!"′ .Basedontheslope
triangle,whatistheslopeoftheline!"′?
4. Whatistherelationshipbetweenthesetwoslopes?Howdoyouknow?
5. Istherelationshipchangedifthetwolinesaretranslatedsothattheintersectionisat
(-5,7)?
Howdoyouknow?
Toproveatheorem,weneedtodemonstratethatthepropertyholdsforanypairofperpendicular
lines,notjustafewspecificexamples.Itisoftendonebydrawingaverysimilarpicturetothe
exampleswehavetried,butusingvariablesinsteadofnumbers.Usingvariablesrepresentsthe
ideathatitdoesn’tmatterwhichnumbersweuse,therelationshipstaysthesame.Let’strythat
strategywiththetheoremaboutperpendicularlineshavingslopesthatarenegativerecipricals.
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.2
•
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
12. Howdothesestepsdemonstratethattheslopesofperpendicularlinesarenegative
reciprocalsforanypairofperpendicularlines?
Thinknowaboutparallellinesliketheonesbelow.
m
l
13.DrawtheslopetrianglefrompointAtotheorigin.Whatistheslopeof!"?
14.Whattransformation(s)mapstheslopetrianglewithhypotenuse!"ontotheotherlinem?
15.Whatmustbetrueabouttheslopeoflinel?Why?
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.2
Nowyou’regoingtotrytousethisexampletodevelopaproof,likeyoudidwiththeperpendicular
lines.Herearetwolinesthathavebeenconstructedtobeparallel.
16.Showhowyouknowthatthesetwoparallellineshavethesameslopeandexplainwhythis
provesthatallparallellineshavethesameslope.
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.2
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,
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CONNECTING ALGEBRA & GEOMETRY – 8.2
emphasizethatthereasonthatweknowthatthelinesareperpendicularisthattheywere
constructedbasedupona90degreerotation.
Explore(SmallGroup):
Theproofthattheslopesofperpendicularlinesarenegativereciprocalsfollowsthesamepattern
astheexamplegiveninthepreviousproblem.Monitorstudentsastheywork,allowingthemto
selectapoint,labelthecoordinatesandthenthesidesoftheslopetriangles.Referstudentsbackto
thepreviousproblem,askingthemtogeneralizethestepssymbolicallyiftheyarestuck.When
studentsarefinishedwithquestions6-12,discusstheproofasawholegroupandthenhave
studentscompletethetask.
Discuss(WholeClass):
Thesetupfortheproofisbelow:
y
(a, b)
l
m
(-b,a)
b
a
a
-b
!
!
!
!!
Theslopeoflinelis andtheslopeoflinemis
!
or- .Theproductofthetwoslopesis-1,
!
thereforetheyarenegativereciprocals.Ifthelinesaretranslatedsothattheintersectionisnotat
theorigin,theslopetriangleswillremainthesame.Discusswiththeclasshowquestions6-12help
ustoconsiderallthepossiblecases,whichisnecessaryinaproof.Afterstudentshavefinishedthe
task,gothroughthebriefproofthattheslopesofparallellinesareequal.
AlignedReady,Set,Go:ConnectingAlgebraandGeometry8.2
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SECONDARY MATH I // MODULE 8
8.2
CONNECTING ALGEBRA & GEOMETRY – 8.2
READY, SET, GO!
Name
PeriodDate
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.
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.2
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.
Completethemathematicalstatementsaboutthekiteusing
thegivensymbols.Proveeachstatementalgebraically.
(Asymbolmaybeusedmorethanonce.)
≅ ⊥ ∥ < > =
4.!"__________!"
5.!"__________!"
6.!"__________!"
Proof
______________________________________________________________________________
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.2
7.∆!"#______ ∆!"#
8.!!__________!"
9.!"__________!"
10.!"__________!"
8.2
GO
Topic:Writingequationsoflines
Usethegiveninformationtowritetheequationofthelineinstandardform. !" + !" = ! !
12.! !!, −! , ! !, ! 11.!"#$%: −
!"#$% !", ! !
13.! − !"#$%&$'#: − !; ! − !"#$%&$'#: − !
14.!"" ! !"#$%& !"# −! . ! !" !"# !"#$%&.
!
16.! −!", !" , ! !", !" 15.!"#$%: ; ! − !"#$%&$'#: !
!
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