MAGDA · 2019. 3. 18. · Title: MAGDA_ Created Date: 2/28/2019 12:06:02 PM
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The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius
© 2016 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Off ice of Education
This work is licensed under the Creative Commons Attribution CC BY 4.0
MODULE 8
Connecting Algebra & Geometry
SECONDARY
MATH ONE
An Integrated Approach
SECONDARY MATH 1 // MODULE 8
CONNECTING ALGEBRA & GEOMETRY
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
MODULE 8 - TABLE OF CONTENTS
CONNECTING ALGEBRA AND GEOMETRY
8.1 Go the Distance – A Develop Understanding Task
Using coordinates to find distances and determine the perimeter of geometric shapes (G.GPE.7)
READY, SET, GO Homework: Connecting Algebra and Geometry 8.1
8.2 Slippery Slopes – A Solidify Understanding Task
Proving slope criteria for parallel and perpendicular lines (G.GPE.5)
READY, SET, GO Homework: Connecting Algebra and Geometry 8.2
8.3 Prove It! – A Practice Understanding Task
Using coordinates to algebraically prove geometric theorems (G.GPE.4)
READY, SET, GO Homework: Connecting Algebra and Geometry 8.3
8.4 Training Day – A Solidify Understanding Task
Writing the equation f(t) = m(t) + k by comparing parallel lines and finding k (F.BF.3, F.BF.1, F.IF.9)
READY, SET, GO Homework: Connecting Algebra and Geometry 8.4
8.5 Training Day Part II – A Practice Understanding Task
Determining the transformation from one function to another (F.BF.3, F.BF.1, F.IF.9)
READY, SET, GO Homework: Connecting Algebra and Geometry 8.5
8.6 Shifting Functions – A Practice Understanding Task
Translating linear and exponential functions using multiple representations (F.BF.3, F.BF.1, F.IF.9) READY, SET, GO Homework: Connecting Algebra and Geometry 8.6
SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.1
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8.1 Go the Distance
A Develop Understanding Task
TheperformancesofthePodunkHighSchooldrillteamareverypopularduringhalf-timeattheschool’sfootballandbasketballgames.WhenthePodunkHighSchooldrillteamchoreographsthedancemovesthattheywilldoonthefootballfield,theylayouttheirpositionsonagridliketheonebelow:
Inoneoftheirdances,theyplantomakepatternsholdinglong,wideribbonsthatwillspanfromonedancerinthemiddletosixotherdancers.Onthegrid,theirpatternlookslikethis:
Thequestionthedancershaveishowlongtomaketheribbons.Gabriela(G)isstandinginthecenterandsomedancersthinkthattheribbonfromGabriela(G)toCourtney(C)willbeshorterthantheonefromGabriela(G)toBrittney(B).
1. Howlongdoeseachribbonneedtobe?
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CONNECTING ALGEBRA & GEOMETRY – 8.1
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2. Explainhowyoufoundthelengthofeachribbon.
Whentheyhavefinishedwiththeribbonsinthisposition,theyareconsideringusingthemtoformanewpatternlikethis:
3. WilltheribbonstheyusedinthepreviouspatternbelongenoughtogobetweenBritney(B)andCourtney(C)inthenewpattern?Explainyouranswer.
Gabrielanoticesthatthecalculationssheismakingforthelengthoftheribbonsremindsherofmathclass.Shesaystothegroup,“Hey,Iwonderifthereisaprocessthatwecoulduselikewhatwehavebeendoingtofindthedistancebetweenanytwopointsonthegrid.”Shedecidestothinkaboutitlikethis:
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CONNECTING ALGEBRA & GEOMETRY – 8.1
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(x2,y2)
(x1,y1)
B
A
“I’mgoingtostartwithtwopointsanddrawthelinebetweenthemthatrepresentsthedistancethatI’mlookingfor.Sincethesetwopointscouldbeanywhere,InamedthemA(x1,y1)andB(x2,y2).Hmmmmm....whenIfiguredthelengthoftheribbons,whatdidIdonext?”
4. Thinkbackontheprocessyouusedtofindthelengthoftheribbonandwritedownyourstepshere,intermsof(x1,y1)and(x2,y2).
5. Usetheprocessyoucameupwithin#4tofindthedistancebetweentwopointslocatedfarenoughawayfromeachotherthatusingyourformulafrom#4ismoreefficientthangraphingandcounting.Forexamplefindthedistancebetween(-11,25)and(23,-16)
6. Useyourprocesstofindtheperimeterofthehexagonpatternshownin#3.
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8.1
READY Topic:FindingthedistancebetweentwopointsUsethenumberlinetofindthedistancebetweenthegivenpoints.(ThenotationABmeansthedistancebetweenthepointsAandB.)
1.AE 2.CF 3.GB 4.CA 5.BF 6.EG
7.Describeawaytofindthedistancebetweentwopointsonanumberlinewithoutcountingthespaces.
8. a.FindAB. b.FindBC. c.FindAC.9.WhyisiteasiertofindthedistancebetweenpointAandpointBandpointBandpointCthanitistofindthedistancebetweenpointAandpointC?
10.ExplainhowtofindthedistancebetweenpointAandpointC.
–4 –2 2 40
A B C D E F G
READY, SET, GO! Name PeriodDate
CB
A
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8.1
SET Topic:Slopetrianglesandthedistanceformula
TriangleABCisaslopetriangleforthelinesegmentABwhereBCistheriseandACistherun.NoticethatthelengthofsegmentBChasacorrespondinglengthonthey-axisandthelengthofAChasacorrespondinglengthonthex-axis.Theslopeformulaiswrittenas! = !!!!!
!!!!!wheremistheslope.
11.a.Whatdoesthevalue !! − !! tellyou?b.Whatdoesthevalue !! − !! tellyou?InthepreviousunityoufoundthelengthofaslantedlinesegmentbydrawingtheslopetriangleandthenusingthePythagoreantheoremonthetwosidesofthetriangle.Inthisexercise,trytodevelopamoreefficientmethodofcalculatingthelengthofalinesegmentbyusingthemeaningof !! − !! and!! − !! combinedwiththePythagoreantheorem.12.FindAB. 13.FindAB.14.FindAB. 15.FindAB.
10
8
6
4
2
5
x2x1
y1
y2 B
A C
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8.1
GO Topic:RectangularcoordinatesUsethegiveninformationtofillinthemissingcoordinates.Thenfindthelengthoftheindicatedlinesegment.16.a)FindHB.b)FindBD.17.a)FindDBb)FindCF
E ( , -4)
K ( , )
C ( , )G ( , ) A (0 , 0)
B ( , 6)H ( , )
F (-10, ) D (10 , )
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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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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.
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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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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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CONNECTING ALGEBRA & GEOMETRY – 8.2
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Nowyou’regoingtotrytousethisexampletodevelopaproof,likeyoudidwiththeperpendicular
lines.Herearetwolinesthathavebeenconstructedtobeparallel.
16.Showhowyouknowthatthesetwoparallellineshavethesameslopeandexplainwhythis
provesthatallparallellineshavethesameslope.
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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
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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.!"__________!"
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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.! −!",!" , ! !",!"
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CONNECTING ALGEBRA & GEOMETRY – 8.3
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8.3 Prove It!
A Practice Understanding Task
Inthistaskyouneedtouseallthethingsyouknowaboutquadrilaterals,distance,andslopetoprovethattheshapesareparallelograms,rectangles,rhombi,orsquares.Besystematicandbesurethatyougivealltheevidencenecessarytoverifyyourclaim.
1.
a.IsABCDaparallelogram?Explainhowyouknow.
b.IsEFGHaparallelogram?Explainhowyouknow.
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2.
a.IsABCDarectangle?Explainhowyouknow.
b.IsEFGHarectangle?Explainhowyouknow.
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CONNECTING ALGEBRA & GEOMETRY – 8.3
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3.
a.IsABCDarhombus?Explainhowyouknow.
b.IsEFGHarhombus?Explainhowyouknow.
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CONNECTING ALGEBRA & GEOMETRY – 8.3
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4.
a.IsABCDasquare?Explainhowyouknow.
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8.3
READY Topic:Interpretingtablesofvalueasorderedpairs.
Findthevalueof! ! forthegivendomain.Write!and! ! asanorderedpair.1.! ! = 3! − 2 2.! ! = !! 3.! ! = 5!
! ! ! !, ! !
-2
-1
0
1
2
! ! ! !, ! !
-2
-1
0
1
2
! ! ! !, ! !
-2
-1
0
1
2
SET Topic:Identifyingspecificquadrilaterals
4.a)Isthefigureattherightarectangle?Justifyyouranswer.b)Isthefigureattherightarhombus?Justifyyouranswer.c)Isthefigureattherightasquare?Justifyyouranswer.
GO
Topic:Calculatingperimetersofgeometricshapes
READY, SET, GO! Name PeriodDate
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8.3
Findtheperimeterofeachfigurebelow.Roundanswerstothenearesthundredth.5.
6.
7.
8.
9.
10.
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CONNECTING ALGEBRA & GEOMETRY CONNECTING ALGEBRA & GEOMETRY
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8.4 Training Day A Develop Understanding
Task
FernandoandMariaharetrainingforsixweekstoruninamarathon.Totrain,theyrunlapsaroundthetrackatEastlandHighSchool.Sincetheirschedulesdonotallowthemtoruntogetherduringtheweek,theyeachkeeparecordofthetotalnumberoflapstheyrunthroughouttheweekandthenalwaystraintogetheronSaturdaymorning.ThefollowingarerepresentationsofhoweachpersonkepttrackofthetotalnumberoflapsthattheyranthroughouttheweekplusthenumberoflapstheyranonSaturday.
Fernando’sdata:
Time(inminutesonSaturday)
0 10 20 30 40 50
Distance(inlaps)
60 66 72 78 84 90
Mariah’sdata:
Time(inminutesonSaturday)
cc b
y ht
tps:
//flic
.kr/
p/qq
dfiN
/
Distance(inlaps)
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1. Whatobservationscanbemadeaboutthesimilaritiesanddifferencesbetweenthetwo
trainers?
2. Writetheequation,m(t),thatmodelsMariah’sdistance.
3. FernandoandMariahbothsaidtheyranthesamerateduringtheweekwhentheyweretrainingseparately.ExplaininwordshowFernando’sequationissimilartoMariah’s.Usethesentenceframe:Therateofbothrunnersisthesamethroughouttheweek,however,
Fernando______________________________________________________.
4. Inmathematics,sometimesonefunctioncanbeusedtobuildanother.WriteMariah’sequation,m(t),bystartingwithFernando’sequation,f(t).
f(t)=
5. Usethemathematicalrepresentationsgiveninthistask(tableandgraph)tomodeltheequationyouwrotefornumber4.Writeinwordshowyouwouldexplainthisnewfunctiontoyourclass.
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8.4
READY Topic:Verticaltransformationsongraphs
1.Usethegraphbelowtodrawanewgraphthatistranslatedup3units.
2.Usethegraphbelowtodrawanewgraphthatistranslateddown1unit.
3.Usethegraphbelowtodrawanewgraphthatistranslateddown4units.
4.Usethegraphbelowtodrawanewgraphthatistranslateddown3units.
READY, SET, GO! Name PeriodDate
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8.4
SET Topic:Graphingtransformationsandwritingtheequationofthenewgraph
Youhavebeengiventheequationsof! ! andthetransformation! ! = ! ! + !.Graphboth! ! and! ! .Thenwritethelinearequationfor! ! inthespaceprovided.
5.! ! = 2! − 4; ! ! = ! ! + 3 6.! ! = 0.5!;! ! = ! ! − 3
!(!) = ____________________________
!(!) = ____________________________
Basedonthegivengraph,writetheequationof! ! intheformof! ! = ! ! + !.Thensimplifytheequationof! ! intoslope-interceptform.Theequationsof! ! isgiven.7.!( ! ) = ¼ ! – 3 8.! ! = −2! + 5
a.! ! =__________________________________Translationformb.! ! =__________________________________Slope-interceptform
a.! ! =__________________________________Translationformb.! ! =__________________________________Slope-interceptform
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8.4
GO
Topic:Convertingunitsandmakingdecisionsbasedondata9.FernandoandMariaharetrainingforahalfmarathon.Thechartbelowdescribestheirworkoutsfortheweekjustbeforethehalfmarathon.Ahalfmarathonisequalto13.1miles.Iffourlapsmakeuponemile,doyouthinkMariahandFernandoarepreparedfortheevent?
Describehowyouthinkeachpersonwillperformintherace.Includewhoyouthinkwillfinishfirstandpredictwhatyouthinkeachperson’sfinishtimewillbe.Usethedatatoinformyourconclusionsandtojustifyyouranswers.
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CONNECTING ALGEBRA & GEOMETRY—8.5
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8.5 Training Day Part II A Solidify Understanding Task
FernandoandMariahcontinuedtraininginpreparationforthehalfmarathon.Forthe
remainingweeksoftraining,theyeachseparatelykepttrackofthedistancetheyranduring
theweek.SincetheyrantogetheratthesamerateonSaturdays,theytookturnskeepingtrack
ofthedistancetheyranandthetimeittook.Sotheywouldbothkeeptrackoftheirown
information,theotherpersonwouldusethedatatodeterminetheirowntotaldistanceforthe
week.
1. Week2:Mariahhadcompleted15morelapsthanFernandobeforetheytrainedonSaturday.
a. CompletethetableforMariah.
Time(inminutes
onSaturday)
0 10 20 30 40 50 60
Fernando:Distance
(inlaps)
50 56 62 68 74 80 86
Mariah:
Distance(inlaps)
b. WritetheequationforMariahasatransformationofFernando.EquationforMariah:m(t)=f(t)_________
2. Week3:OnSaturdaymorningbeforetheystartedrunning,FernandosawMariah’stable
andstated,“Myequationthisweekwillbef(t)=m(t)+30.”a. WhatdoesFernando’sstatementmean?
b. BasedonFernando’stranslatedfunction,completethetable.
Time(inminutes
onSaturday)
0 20 40 60 70
Fernando:Distance
(inlaps)
Mariah:
Distance(inlaps)
45 57 69 81 87
©w
ww
.flic
kr.c
om/p
hoto
s/ab
lem
an/
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CONNECTING ALGEBRA & GEOMETRY—8.5
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c. Writetheequationforbothrunnersinslope-interceptform:
d. WritetheequationforMariah,transformedfromFernando.
e. Whatrelationshipdoyounoticebetweenyouranswerstopartscandd?
3. Week4:Themarathonisonlyacoupleofweeksaway!
a. UseMariah’sgraphtosketchf(t).f(t)=m(t)–10
Time(inminutesonSaturday)
b. Writetheequationsforbothrunnersinslope-interceptform.
c. Whatdoyounoticeaboutthetwographs?Wouldthisalwaysbetrueifonepersonran
“k”lapsmoreorlesseachweek?
Distance
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4. Week5:Thisisthelastweekoftrainingtogether.NextSaturdayisthebigday.Whentheyarrivedtotrain,theynoticedtheyhadbothrun60lapsduringtheweek.
a. WritetheequationforMariahonSaturdaygiventhattheyrunatthesamerateastheweekbefore.
b. WriteFernando’sequationasatransformationofMariah’sequation.
5. Whatconjecturescanyoumakeaboutthegeneralstatement:“g(x)=f(x)+k”whenitcomestolinearfunctions?
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8.5
READY Topic:Describingspread.
1.Describethespreadinthehistogrambelow.
2.Describethespreadinthelineplotbelow.3.Describethespreadintheboxandwhiskerplot.
READY, SET, GO! Name PeriodDate
https://commons.wikimedia.org/wiki/File:Black_cherry_tree
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8.5
SET Topic:Writingfunctionsintranslationform.
Youaregiveninformationabout! ! and! ! .Rewrite! ! intranslationform:! ! = ! ! + !
4.! ! = 7! + 13! ! = 7! − 5
!(!) = ____________________
Translationform
5.! ! = 22! − 12! ! = 22! + 213
!(!) = ____________________ Translationform
6.! ! = −15! + 305! ! = −15! − 11
!(!) = ____________________ Translationform
7.!(!) = ____________________ Translationform
x f(x) g(x)3 11 26
10 46 61
25 121 136
40 196 211
8.!(!) = ____________________ Translationform
x f(x) g(x)-4 5 -42
-1 -1 -48
5 -13 -60
20 -43 -90
9.!(!) = ____________________ Translationform
x f(x) g(x)-10 4 -15.5
-3 7.5 -12
22 20 0.5
41 29.5 10
GO
Topic:Verticalandhorizontaltranslations.10.Usethegraphof! ! = !"todothefollowing:a.Sketchthegraphof! ! = !" − !onthesamegrid.b.Sketchthegraphof! ! = ! ! − ! c.Describehow! ! ,! ! ,!"# ! ! aredifferentandhowtheyarethesame.d.Explaininwhatwaytheparenthesesaffectthegraph.Whydoyouthinkthisisso?
f (x)
30
SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
8.6 Shifting Functions
A Practice Understanding Task
PartI:Transformationofanexponentialfunction.ThetablebelowrepresentsthepropertyvalueofRebekah’shouseoveraperiodoffouryears.Rebekah’sHome
Rebekahsaysthefunction! ! = 150,000 1.06 !representsthevalueofherhome.
1. Explainhowthisfunctioniscorrectbyusingthetabletoshowtheinitialvalueandthecommonratiobetweenterms.
JeremylivesclosetoRebekahandsaysthathishouseisalwaysworth$20,000morethanRebekah’shouse.Jeremycreatedthefollowingtableofvaluestorepresentthepropertyvalueofhishome.Jeremy’sHome
WhenRebekahandJeremytriedtowriteanexponentialfunctiontorepresentJeremy’spropertyvalue,theydiscoveredtherewasnotacommonratiobetweenalloftheterms.
2. UseyourknowledgeoftransformationstowritethefunctionthatcouldbeusedtodeterminethepropertyvalueofJeremy’shouse.
Time(years)
PropertyValue
CommonRatio
0 150,000 1 159,0002 168,5403 178,6524 189,372
Time(years)
PropertyValue
RelationshiptoRebekah’stable
0 170,000 1 179,0002 188,5403 198,6524 209,372
CCBY
https://flic.kr/p/bFcAW
q
31
SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Part2:Shiftyfunctions.Giventhefunctiong(x)andinformationaboutf(x),
• writethefunctionforf(x),• graphbothfunctionsonthesetofaxes,and• showatableofvaluesthatcomparesf(x)andg(x).
3. !" ! ! = 3 2 ! !"# ! ! = ! ! − 5, !ℎ!" ! ! = ________________________________
x f(x) g(x)
4. !" ! ! = 4 . 5 ! !"# ! ! = ! ! + 3, !ℎ!" ! ! = ________________________________
x f(x) g(x)
5. !" ! ! = 4! + 3 !"# ! ! = ! ! + 7, !ℎ!" ! ! = ________________________________
x f(x) g(x)
32
SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
6. !" ! ! = 2! + 1 !"# ! ! = ! ! − 4, !ℎ!" ! ! = ________________________________
x f(x) g(x)
7. !" ! ! = −! !"# ! ! = ! ! + 3, !ℎ!" ! ! = ________________________________
x f(x) g(x)
PartIII:Communicateyourunderstanding.8. Iff(x)=g(x)+k,describetherelationshipbetweenf(x)andg(x).Supportyouranswers
withtablesandgraphs.
33
SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.6
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
8.6
READY Topic:Findingpercentages
Mrs.Gonzaleznoticedthathernewchorusclasshadalotmoregirlsthanboysinit.Therewere32girlsand17boys.(Roundanswerstothenearest%.)1.Whatpercentoftheclassaregirls?2.Whatpercentareboys?3.68%ofthegirlsweresopranos.a.Howmanygirlssangsoprano?b.Whatpercentoftheentirechorussangsoprano?4.Only30%oftheboyscouldsingbass.a.Howmanyboyswereinthebasssection?b.Whatpercentoftheentirechorussangbass?5.Comparethenumberofgirlswhosangaltotothenumberofboyswhosangtenor.Whichmusicalsectionislarger? Justifyyouranswer.
SET Topic:Graphingexponentialequations.
6.Thinkaboutthegraphsof! = 2! and! = 2! − 4.a.Predictwhatyouthinkisthesameandwhatisdifferent.b.Useyourcalculatortographbothequationsonthesamegrid.Explainwhatstayedthesameandwhatchangedwhenyousubtracted4.Identifyinwhatwayitchanged.(Ifyoudon’thaveagraphingcalculator,thiscaneasilybedonebyhand.)
READY, SET, GO! Name PeriodDate
-10
-10
10
10
34
SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.6
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
8.6
7.Thinkaboutthegraphsof! = 2! and! = 2 !!! .a.Predictwhatyouthinkisthesameandwhatisdifferent.b.Useyourcalculatortographbothequationsonthe
samegrid.Explainwhatstayedthesameandwhatchanged.Identifyinwhatwayitchanged.
GO
Topic:VerticaltranslationsoflinearequationsThegraphof! ! andthetranslationformequationof! ! aregiven.Graph! ! onthesamegridas! ! andwritetheslope-interceptequationof! ! and! ! .8.! ! = ! ! − 5a.
9.! ! = ! ! + 4a.
10.! ! = ! ! − 6a.
b.! ! =_______________________
b.! ! =_______________________
b.! ! =_______________________
c.! ! =_______________________Slope-interceptform
c.! ! =_______________________Slope-interceptform
c.! ! =_______________________Slope-interceptform
f (x)
f (x)
f (x)
-10
-10
10
10
35