The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius
© 2018 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Office of Education
This work is licensed under the Creative Commons Attribution CC BY 4.0
MODULE 5
Rational Functions & Expressions
ALGEBRA II
An Integrated Approach
Standard Teacher Notes
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
MODULE 5 - TABLE OF CONTENTS
RATIONAL EXPRESSIONS AND FUNCTIONS
5.1 Winner, Winner – A Develop Understanding Task
Introducing rational functions and asymptotic behavior (F.IF.7d A.CED.2, F.IF.5)
Ready, Set, Go Homework: Rational Functions 5.1
5.2 Shift and Stretch – A Solidify Understanding Task
Applying transformations to the graph of !(#) = &' . (F.BF.3, F.IF.7d, A.CED.2)
Ready, Set, Go Homework: Rational Functions 5.2
5.3 Rational Thinking – A Solidify Understanding Task
Discovering the relationship between the degree of the numerator and denominator and the
horizontal asymptotes. (F.IF.7d A.CED.2, F.IF.5)
Ready, Set, Go Homework: Rational Functions 5.3
5.4 Are You Rational? – A Solidify Understanding Task
Reducing rational functions and identifying improper rational functions and writing them in
an equivalent form. (A.APR.6, A.APR.7, A.SSE.3)
Ready, Set, Go Homework: Rational Functions 5.4
5.5 Just Act Rational – A Solidify Understanding Task
Adding, subtracting, multiplying, and dividing rational expressions. (A.APR.7, A.SSE.3)
Ready, Set, Go Homework: Rational Functions 5.5
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
5.6 Sign on the Dotted Line – A Practice Understanding Task
Developing a strategy for determining the behavior near the asymptotes and graphing rational
functions. (F.IF.4, F.IF.7d)
Ready, Set, Go Homework: Rational Functions 5.6
5.7 We All Scream – A Practice Understanding Task
Modeling with rational functions, and solving equations that contain rational expressions.
(A.REI.A.2, A.SSE.3)
Ready, Set, Go Homework: Rational Functions 5.7
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.1
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5.1 Winner, Winner
A Develop Understanding Task
Oneofthemostinterestingfunctionsinmathematicsis
!(#) = &'becauseitbringsupsomemathematicalmind
benders.Inthistask,wewillusestorycontextand
representationsliketablesandgraphstounderstandthis
importantfunction.
Let’sbeingbythinkingabouttheinterval[1,∞).
1.Imaginethatyouwonthelotteryandweregivenonebigpotofmoney.Ofcourse,youwould
wanttosharethemoneywithfriendsandfamily.Ifyousplitthemoneyevenlybetweenyourself
andonefriend,whatwouldbeeachperson’sshareoftheprizemoney?
2.Ifthreepeoplesharedtheprizemoney,whatwouldbeeachperson’sshare?
3.Modelthesituationwithatable,equation,andgraph.
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RATIONAL EXPRESSIONS & FUNCTIONS -5.1
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4.Justincaseyoudidn’tthinkaboutthereallybignumbersinyourmodel,howmuchofthepot
wouldeachpersongetif1000peoplegetashare?If100,000peoplegetashare?If100,000,000
peoplegetashare?
5.Usemathematicalnotationtodescribethebehaviorofthisfunctionas# → ∞.
Next,let’slookattheinterval(0,1]andconsideranewwaytothinkaboutsplittingtheprize
money.
6.Imaginethatyouwanteachperson’ssharetobe1 2. oftheprize.Howmanypeoplecouldsharetheprize?
7.Ifyouwanteachperson’ssharetobe1 3. oftheprize,howmanypeoplecouldsharetheprize?
8.Modelthissituationwithatable,graphandequation.
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RATIONAL EXPRESSIONS & FUNCTIONS -5.1
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9.Whatdoyounoticewhenyoucomparethetwomodelsthatyouhavewritten?
Now,let’sputitalltogethertographtheentirefunction,!(#) = &'.
10.Createatablefor!(#) = &'thatincludesnegativeinputvalues.
11.Howdothevaluesof!(#)intheintervalfrom(−∞, 0)comparetothevaluesof!(#)from(0,∞)?Usethiscomparisontopredictthegraphof!(#) = &
'.
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RATIONAL EXPRESSIONS & FUNCTIONS -5.1
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12.Graph!(#) = &'.
13.Describethefeaturesof!(#) = &',includingdomain,range,intervalsofincreaseordecrease,#-
and=-intercepts,endbehavior,andanymaximum(s)orminimum(s).
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RATIONAL EXPRESSIONS & FUNCTIONS -5.1
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5.1 Winner, Winner – Teacher Notes A Develop Understanding Task Purpose:Thepurposeofthistaskistointroducestudentstorationalfunctionsandtoexplorethe
behaviorofthefunction,!(#) = &',thatproducestheverticalandhorizontalasymptotes.Students
willuseastorycontexttothinknumericallyaboutthevaluesof!(#) = &'inquadrantIandextend
thatthinkingtographthefunctionovertheentiredomain.Studentswillconsiderallofthefeatures
ofthefunctionincluding,domain,range,intervalsofincreaseanddecrease,andintercepts.
CoreStandardsFocus:
F.IF.7dGraphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandin
simplecasesandusingtechnologyformorecomplicatedcases.*
d.Graphrationalfunctions,identifyingzeroswhensuitablefactorizationsareavailable,and
showingendbehavior.
A.CED.2(Createequationsthatdescribenumbersorrelationships)Createequationsintwoor
morevariablestorepresentrelationshipsbetweenquantities;graphequationsoncoordinateaxes
withlabelsandscales.
F.IF.5Relatethedomainofafunctiontoitsgraphand,whereapplicable,tothequantitative
relationshipitdescribes.Forexample,ifthefunctionh(n)givesthenumberofpersonhoursittakesto
assemblenenginesinafactory,thenthepositiveintegerswouldbeanappropriatedomainforthe
function.*
Related:F.IF.6,A.CED.3
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.1
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StandardsforMathematicalPractice:
SMP2–Reasonabstractlyandquantitatively
Vocabulary:Horizontalasymptote
TheTeachingCycle:
Launch(WholeGroup):
Beginbytellingstudentsthattoday’sworkwillfocusonthefunction!(#) = &'.Toactivatetheir
backgroundalittle,askstudentswhattheynoticeaboutthefunction,justbylookingatthe
equation.Theymayrecognizethingslike:
• Thefunctionisundefinedat0.
• Thefunctionvaluesaregoingtobealotoffractions.
• Thefunctionisthereciprocalof= = #.Atthispoint,thereisnoneedtopursueanyoftheirobservations,justkeeptheminmindduring
theclassdiscussiontoseehowtheycomeintoplayasstudentsworkwiththefunction.
Explainthecontextofthetask,thattheyhavewonthelotteryandhaveabigprizetoshare.The
amountofmoneyisnotimportant,theyshouldthinkabouttheprizeasonebig“potofmoney”and
howtheycandivideitupevenly.Askstudentsaboutquestion1.Besurethatstudentsunderstand
thatthepotwouldbesharedequallybetweentwopeoplesothateachpersongetshalfoftheprize.
Askstudentstodoquestion2anddiscussitbrieflybeforeaskingthemtoworkontherestofthe
task.
Explore(IndividualandthenSmallGroups):
Asstudentsareworking,listenforstudentsthatarediscussingideasaboutthebehaviorofthe
functionas# → ∞.Theyarelikelytoargueaboutwhetherornotthevalueofthefunctioniseverzero,sincepracticallyspeaking,theshareoftheprizewouldbecomesosmallthateachpersonis
gettingnearlynothing.Studentswhohaveweakunderstandingoffractionsmaynotimmediately
understandthatasthedenominatorofthefractionincreases,thevalueofthefractiondecreases.
Thestorycontextcanbeusedtosupporttheminthinkingaboutthisidea.
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.1
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Theremaybesomediscussionofwhetherornottomodelthissituationasadiscretefunctioninthe
interval[1,∞)becausethenumberofpeoplecan’tbeafraction.Sincethisisnotthefocusofthetask,youmaywishtoaskstudentstomakeacontinuousmodel,withtheunderstandingthatany
answerusingthemodelwillrequireinterpretationtobesurethatitisreasonableinthecontext.
Asstudentsmovetoconsidertheinterval(0,1),theymayneedmoresupportforthinkingabout
dividingbyafraction.Thestorycontextisprovidedforthispurposetohelpthemtoseeasituation
wheredividingbyafractionmakessense.Thisunderstandingisimportantforstudentstoconsider
thebehaviorofthefunctionneartheverticalasymptote.Studentsshouldalsonoticethattheyare
workingwiththe#and=variablesswitchedfromtheprevioussection,andyet,thefunctionisstill!(#) = &
'.
Discuss(WholeGroup):
Beginthediscussionwithquestion#3.Askastudenttopresenthis/hertableandtodescribehow
thetablemodelsthecontext.Then,asktheclasshowthetableisconnectedtotheequation,
!(#) = &'.Askasecondstudenttopresentthegraphofthefunctionintheinterval[1,∞).Focuson
whythefunctionapproacheszeroforlargevaluesof#andintroducetheterm“horizontalasymptote”.Becarefulnottodescribeahorizontalasymptoteasalinethatthefunction
approachesbutnevercrosses.Thereareseveralfunctionsthatoccurlaterinthemodulewherethe
graphactuallycrossesthehorizontalasymptote.Ahorizontalasymptotedescribestheend
behaviorofafunction,eitheras# → ∞or# → −∞.
Turnthediscussiontoquestion8.Again,askapreviously-selectedstudenttoshareatableandto
connectthevaluesinthetabletothestorycontext.Asktheclasswhathappenstothenumberof
peoplethattheprizecanbesharedwithiftheportionoftheprizethateachpersongetsis
increasinglysmaller.Helpstudentstounderstandhowthisbehaviorcreatesaverticalasymptote
at# = 0.Shareagraphofthefunctionintheinterval(0,1).Askstudentsabouttherelationshipbetweenthetwodifferenttablesthattheycreatedinquestions3and8.Theyshouldnoticethatthe
variableshavebeenswitched,andthus,thefunctionsareinverses.Askhowthiscouldbethecaseif
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.1
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theequationis!(#) = &'inbothsituations.Emphasizethatthetwodifferentwaysofthinkingabout
thestorycontextarepresentedheresothattheycanseethesimilaritiesintheverticaland
horizontalasymptotes.
Askastudenttopresenthis/herworkonquestion10andthenaskanotherstudenttopresenta
graphoftheentirefunction.AsktheclasswhattherelationshipisbetweenthegraphinQuadrantI
andthegraphinQuadrantIII.Theyshouldrecognizethatitisarotationof180degreesaroundthe
origin,so!(#) = &'isanoddfunction.Finally,discussthefeaturesof!(#) =
&'.Besurethat
studentscanusepropernotationtodescribethedomainandrange,aswellasrecognizingthatthe
functionisdecreasingthroughtheentiredomain.
AlignedReady,Set,Go:RationalExpressionsandFunctions5.1
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS – 5.1
5.1
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READY Topic:Recallingtransformationsonquadraticfunctions
Describethetransformationofeachfunction.Thenwritetheequationinvertexform.
1.Description:Equation:
2.Description:Equation:
3.Description:Equation:
4.Description:Equation:
5.Description:Equation:
6.Description:Equation:
READY, SET, GO! Name PeriodDate
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ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS – 5.1
5.1
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SET Topic:Exploringarationalfunction
ChileiscelebratingherQuinceañera.HannahknowstheperfectgifttobuyChile,butitcosts$360.Hannahcan’taffordtopayforthisonherownsothinksaboutaskingsomefriendstojoininandsharethecost.
7.Howmuchwouldeachpersonspendifthereweretwopeopledividingthecostofthegift?Howmuchwouldeachpersonspendiftherewerefivepeopledividingthecost?Tenpeople? Onehundred?
8.Thefunctionthatmodelsthissituationis!(#) = &'() .Definethemeaningofthenumeratorand
thedenominatorwithinthecontextofthestory.
9.Createatableandagraphtoshowhowtheamounteachpersonwouldcontributetothegiftwouldchange,dependingonthenumberofpeoplecontributing.
10.Hannahcreatedafundraisingsiteontheinternet.Within5days,enoughpeoplehadregisteredsothateachfriend,includingHannah,onlyneededtodonate$0.50.
a.Howmanypeoplehadregisteredin5days?
b.Bythedayoftheevent,enoughpeoplehadregisteredthateachfriend,includingHannah,onlydonated10¢.Howmanyfriendshadregistered?
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RATIONAL EXPRESSIONS & FUNCTIONS – 5.1
5.1
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GO Topic:Reviewingthehorizontalasymptoteinanexponentialfunction
Allexponentialfunctionshaveahorizontalasymptote.Allofthegraphsbelowshowexponentialfunctions.
Matchthefunctionrulewiththecorrectgraph.Thenwritetheequationofthehorizontalasymptote.
11.!(#) = 2) Equationofhorizontalasymptote:
12.+(#) = 2) − 3Equationofhorizontal
asymptote:
13.ℎ(#) = 2)/&Equationofhorizontalasymptote:
14.0(#) = −(2)) − 3Equationofhorizontalasymptote:
15.1(#) = 2(/)) + 3Equationofhorizontal
asymptote:
16.3(#) = −2(/))Equationofhorizontalasymptote:
a.
b. c.
d.
e. f.
17.Use!(#) = 45()/6) + 7toexplainwhichvaluesaffectthepositionofthehorizontalasymptoteinanexponentialfunction.Beprecise.
18.Whydoesanexponentialfunctionhaveahorizontalasymptote?
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ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.2
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5.2 Shift and Stretch
A Solidify Understanding Task
In4.1Winner,Winneryouwereintroducedtothefunction! = #
$.Beforeexploringthefamilyofrelatedfunctions,let’sclarifysomeofthefeaturesof! = #
$thatcanhelpwithgraphing.
Here’sagraphof! = #$.
1.Usethegraphtoidentifyeachofthefollowing:HorizontalAsymptote:_______________________VerticalAsymptote:__________________________AnchorPoints:(1,______)and(-1,______)&#' , ______+and&−
#' , ______+
(2,______)and(-2,______)
Nowyou’rereadytousethisinformationtofigureouthowthegraphof! = #$canbetransformed.
Asyouanswerthequestionsthatfollow,lookforpatternsthatyoucangeneralizetodescribethe
transformationsof! = #$.
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RATIONAL EXPRESSIONS & FUNCTIONS -5.2
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Ineachofthefollowingproblems,youaregiveneitheragraphoradescriptionofafunctionthatis
atransformationof! = #$.Useyouramazingmathskillstofindanequationforeach.
2.
Equation:
3.
Equation:
4.Thefunctionhasaverticalasymptoteat
- = −3andahorizontalasymptoteat! = 0.Itcontainsthepoints(-2,1)and(-4,-1).They-interceptis(0, #1).
Equation:
5.
Equation:
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ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.2
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6.Thefunctionhasaverticalasymptoteat
- = 0andahorizontalasymptoteat! = 0.Itcontainsthepoints(1,2),(-1,-2),(2,1),(-2,1),
31 26 , 48and3−1 26 ,−48.
Equation:
7.
Equation:
8.
Equation:
9.Thefunctionhasaverticalasymptoteat
- = −6andahorizontalasymptoteat! = −3.Itcrossesthex-axisat−6 #1.Itcontainsthepoints(-5,-4)and(-7,-2).
Equation:
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ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.2
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10.Matcheachequationtothephrasethatdescribesthetransformationfrom! = #$.
! = #$=>_________
A)Reflectionoverthe--axis.
! = ? + #$_________
B)Verticalshiftof?,makingthehorizontalasymptote! = ?.
! = >$_________
C)Horizontalshiftleft?,makingtheverticalasymptote- = −?.
! = A#$ _________
D)Verticalstretchbyafactorof?
! = #$A>_________
E)Horizontalshiftright?,makingtheverticalasymptote- = ?.
11.Grapheachofthefollowingequationswithoutusingtechnology.
! = −2 + 3- − 4
! = 1 − 1- + 3
12.Describethefeaturesofthefunction:
! = B + ?- − ℎ
VerticalAsymptote: HorizontalAsymptote:
VerticalStretchFactor: AnchorPoints:
Domain: Range:
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ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.2
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5.2 Shift and Stretch – Teacher Notes A Solidify Understanding Task
Purpose:
Thepurposeofthistaskistoextendstudents’understandingofrationalfunctionsbyapplying
transformationstothegraphofD(-) = #$.Thetaskhelpsstudentstofocusontheverticaland
horizontalasymptotesandhowtheycanbeusedtoeasilygraphsimplerationalfunctions.This
taskpreparesstudentsformorecomplicatedrationalfunctionsinupcomingtasks.
CoreStandardsFocus:
F.IF.7dGraphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandin
simplecasesandusingtechnologyformorecomplicatedcases.
d.Graphrationalfunctions,identifyingzeroswhensuitablefactorizationsareavailable,and
showingendbehavior.
F.BF.B.3Identifytheeffectonthegraphofreplacingf(x)byf(x)+k,kf(x),f(kx),andf(x+k)for
specificvaluesofk(bothpositiveandnegative);findthevalueofkgiventhegraphs.Experiment
withcasesandillustrateanexplanationoftheeffectsonthegraphusingtechnology.Include
recognizingevenandoddfunctionsfromtheirgraphsandalgebraicexpressionsforthem.
A.CED.2(Createequationsthatdescribenumbersorrelationships)Createequationsintwoor
morevariablestorepresentrelationshipsbetweenquantities;graphequationsoncoordinateaxes
withlabelsandscales.
StandardsforMathematicalPractice:
SMP5–Useappropriatetoolsstrategically
SMP8–Lookforandexpressregularityinrepeatedreasoning
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.2
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TheTeachingCycle:
Launch(WholeGroup):
Startthetaskbyremindingstudentsoftheirpreviousworkin4.1Winner,Winner.Discussquestion
#1together.Clarifyforstudentsthattheterm“anchorpoints”inthiscontextsimplymeansafew
usefulpointsforsketchingthegraph.Inthiscase,theanchorpointshelptogettheshapeofthe
curveright.Tellstudentsthatintheupcomingproblems,theywillbegivenagraphoradescription
ofafunctionandtheywillusetheirknowledgeofD(-) = #$towriteanequationforthefunction.
Oncetheyhavewrittentheequation,theyshouldusegraphingtechnologytochecktheirworkand
makeadjustmentstotheirequations,ifnecessary.Askstudentstopaycarefulattentiontohow
changesintheequationcorrespondtochangesinthegraphsothattheywillbereadytogeneralize
bytheendofthetask.
Explore(Individual,followedbysmallgroup):
Monitorstudentsastheyworktobesurethattheyareidentifyingthechangesinthegraphsand
matchingthemtoanequation.Themostdifficultproblemforstudentsisusuallythehorizontal
shiftbecausestudentsdon’tseethattheargumentofthefunctionisinthedenominator.Support
studentsinthinkingaboutinputsandoutputs,specificallythatwheneveranumberisaddedtothe
-,itisaffectingtheinputofthefunction.Whenthechangeoccursafterthefunctionoperationhasbeenapplied,inthiscase,takingthereciprocaloftheinput,thenthechangeaffectstheoutputof
thefunction.Changesthataffecttheoutputs,ory-values,areverticalshiftsorstretches.Changes
totheinputsarehorizontalshiftsandstretches.Inthistask,horizontalstretchesarenot
considered.Theseideasarenotnewtostudents,butthistypeoffunctioncanpressstudents
becausetheformseemsdifferent.
Discuss:
Ifstudentshavestruggledwithapplyingthetransformationsduringtheexplorationperiod,begin
thediscussionwithproblem#2andhavestudentspresenttheirworkoneachoftheproblems,2-8.
Ifmostoftheclasshasbeenabletoseethetransformationsduringtheexplorationperiod,then
beginwithquestion#5andthendiscussquestion#8.Ineithercase,besuretoasktheclasswhy
thechangesinthegraphresultfromthechangesintheequation.
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.2
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Afterdiscussingtheexamples,helpstudentstogeneralizebyworkingproblem#10andbeingsure
thateachstudentknowstheverticalandhorizontalshifts,alongwiththeverticalstretches.Then,
askapreviouslyselectedstudenttodemonstratehowhe/shegraphedeitheroftheproblemson
#11.
AlignedReady,Set,Go:RationalExpressionsandFunctions5.2
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS – 5.2
5.2
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READY Topic:Connectingthezeroesofapolynomialwiththedomainofarationalfunction
Findthezeroesofeachpolynomial.
1.!(#) = (& + ()(& − *)(& − +) 2.!(&) = (*& − ,)(-& − .)(& − /)
3.!(#) = (0& + 1)2&* − 03 4.!(&) = &* + */
Findthedomainofeachoftherationalfunctions.
5.4(&) = .(&5()(&6*)(&6+)
6.4(&) = .(*&6,)(-&6.)(&6/)
7.4(&) = .(0&51)(&*60)
8.4(&) = .&*5*/
SET Topic:Practicingtransformationsonrationalfunctions
Identifytheverticalasymptote,horizontalasymptote,domain,andrangeofeachfunction.Thensketchthegraphonthegridsprovided.(Gridsonnextpage.)9.8(9) = :
;
V.A.H.A.
Domain:Range:
10.8(9) = <;+ 2
V.A.H.A.
Domain:Range:
READY, SET, GO! Name PeriodDate
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RATIONAL EXPRESSIONS & FUNCTIONS – 5.2
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11.8(9) = − >
;6<
V.A.H.A.
Domain:Range:
12.8(9) = ?
(;5>)− 4
V.A.H.A.
Domain:Range:
13.Writeafunctionoftheform8(9) = A
;6B+ Cwithaverticalasymptoteat9 = −15anda
horizontalasymptoteatF = −6.
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RATIONAL EXPRESSIONS & FUNCTIONS – 5.2
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GO Topic:Findingtherootsandfactorsofapolynomial
Usethegivenroottofindtheremainingroots.Thenwritethefunctioninfactoredform.
14.8(9) = 9< − 9H − 179 − 15
9 = −1
15.8(9) = 9< − 39H − 619 + 63
9 = 1
16.8(9) = 69< − 189H − 609
9 = 0
17.8(9) = 9< − 149H + 579 − 72
9 = 8
18.Arelationshipexistsbetweentherootsofafunctionandtheconstanttermofthefunction.
Lookbackattherootsandtheconstanttermineachproblem.Makeastatementaboutanythingyounotice.
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ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.3
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5.3 Rational Thinking A Solidify Understanding Task
Thebroadcategoryoffunctionsthatcontainsthefunction!(#) = &'is
calledrationalfunctions.Arationalnumberisaratioofintegers.A
rationalfunctionisaratioofpolynomials.Sincepolynomialscomein
manyforms,constant,linear,quadratic,cubic,etc.,wecanexpect
rationalfunctionstocomeinmanyformstoo.Someexamplesare:
!(#) = 1# !(#) = # + 1
# − 3 !(#) = # − 4#- + 2#/ − 4# + 1 !(#) = #/ − 4
(# − 3)(# + 1)
Degreeofthenumerator=0Degreeofthedenominator=1
Degreeofthenumerator=1Degreeofthedenominator=1
Degreeofthenumerator=1Degreeofthedenominator=3
Degreeofthenumerator=2Degreeofthedenominator=2
Intoday’stask,youaregoingtolookforpatternsintheformssothatyoucancompletethefollowingchart: Howtofindthe
verticalasymptote:Howtofindthehorizontalasymptote:
Howtofindtheintercepts:
Degreeofthenumerator<Degreeofthedenominator
Degreeofthenumerator=Degreeofthedenominator
Youaregivenseveraldifferentrationalfunctions.Startbyidentifyingthedegreeofthenumeratoranddenominatorandusingtechnologytographthefunction.Asyouareworking,lookforpatternsthatwillhelpyoucompletethetable.Youneedtofindaquickwaytoidentifythehorizontalandverticalasymptoteswhenyouseetheequationofarationalfunction,aswellasnoticingotherpatternsthatwillhelpyouanalyzeandgraphthefunctionquickly.Thelasttwographsaretheresoyoucanexperimentwithyourownrationalfunctionsandtestyourtheories.
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ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.3
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1.0 = '1-'2/
2.0 = '('1/)('2-)
Deg.ofNum._____Deg.ofDenom._____HorizontalAsymptote___________VerticalAsymptote___________Intercepts___________________________________
Deg.ofNum._____Deg.ofDenom._____HorizontalAsymptote___________VerticalAsymptote___________Intercepts___________________________________
3.0 = &('13)4
4.0 = -'2&'13
Deg.ofNum._____Deg.ofDenom._____HorizontalAsymptote___________VerticalAsymptote___________Intercepts___________________________________
Deg.ofNum._____Deg.ofDenom._____HorizontalAsymptote___________VerticalAsymptote___________Intercepts___________________________________
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RATIONAL EXPRESSIONS & FUNCTIONS -5.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
5.0 = '42-'('2/)('1&)('1-)
6.0 = /'41-(/'15)(/'25)
Deg.ofNum._____Deg.ofDenom._____HorizontalAsymptote___________VerticalAsymptote___________Intercepts___________________________________
Deg.ofNum._____Deg.ofDenom._____HorizontalAsymptote___________VerticalAsymptote___________Intercepts___________________________________
7.YourOwnRationalFunction:
8.YourOwnRationalFunction:
Deg.ofNum._____Deg.ofDenom._____HorizontalAsymptote___________VerticalAsymptote___________Intercepts___________________________________
Deg.ofNum._____Deg.ofDenom._____HorizontalAsymptote___________VerticalAsymptote___________Intercepts___________________________________
9.Nowthatyouhavetriedsomeexamples,it’stimetodrawsomeconclusionsandcompletethetworowsofthetableonthefirstpage.Bepreparedtodiscussyourconclusionsandwhyyouthinkthattheyarecorrect.
17
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RATIONAL EXPRESSIONS & FUNCTIONS -5.3
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5.3 Rational Thinking – Teacher Notes A Solidify Understanding Task Purpose:Thepurposeofthistaskistointroducetheterm“rationalfunction”andtouseseveral
examplesofrationalfunctionstodiscovertherelationshipbetweenthedegreeofthenumerator
anddenominatorandthehorizontalasymptotes.Studentswillalsofindverticalasymptotesand
interceptsforrationalfunctionsandusethisinformationtocompleteagraphicorganizer.Graphing
technologyisusedasanimportanttoolforexploringrationalfunctionsinthistask.
CoreStandardsFocus:
F.IF.7dGraphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandin
simplecasesandusingtechnologyformorecomplicatedcases.
d.Graphrationalfunctions,identifyingzeroswhensuitablefactorizationsareavailable,and
showingendbehavior.
A.CED.2(Createequationsthatdescribenumbersorrelationships)Createequationsintwoor
morevariablestorepresentrelationshipsbetweenquantities;graphequationsoncoordinateaxes
withlabelsandscales.
F.IF.5Relatethedomainofafunctiontoitsgraphand,whereapplicable,tothequantitative
relationshipitdescribes.Forexample,ifthefunctionh(n)givesthenumberofpersonhoursittakesto
assemblenenginesinafactory,thenthepositiveintegerswouldbeanappropriatedomainforthe
function.
Related:F.IF.6,A.CED.3
StandardsforMathematicalPractice:
SMP3–Constructviableargumentsandcritiquethereasoningofothers
SMP7–Lookforandmakeuseofstructure
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Vocabulary:Rationalfunction
TheTeachingCycle:
Launch(WholeGroup):
Launchthetaskbydefiningtheterm,“rationalfunction”anddiscussingtheexamplesgivenatthe
beginningofthetask.Makesurethatstudentsareabletoidentifythedegreeofthepolynomialsin
thenumeratoranddenominator.Showstudentsthechartthattheywillbecompletingwhenthey
getthroughthetask.Tellthemthattheywillusegraphingtechnologytofigureouthowtofindthe
horizontalandverticalasymptotesandtheinterceptsforrationalfunctions.Inthistasktheywill
workwithrationalfunctionswherethedegreeofthenumeratorislessthanorequaltothe
denominator.Remindthemthattheyalreadyknowalotaboutonerationalfunction:!(#) = &'.
Explore(Individual,FollowedbySmallGroup):
Supportstudentsastheyworkinproperlyidentifyingthedegreeofthenumeratorand
denominatorsothatitiseasierforthemtodrawconclusions.Astheyaregraphingtherational
functions,youmaywishtoencouragethemtographtheasymptotesalongwiththefunction,sothat
itiseasiertoseethattheyarecorrect.Manystudentswillusethetechnologytofindtheintercepts,
soyoumayneedtopromptthemtothinkabouthowtheycouldfindtheinterceptswithout
technology.Itwillhelpifstudentsarerequiredtowritetheinterceptsaspoints,sothattheyare
remindedthata0-interceptoccurswhen# = 0andan#-interceptoccurswhen0 = 0.
Themostchallengingrelationshipforstudentstodiscoveristhatwhenthedegreeofthenumerator
isthesameasthedegreeofthedenominator,thehorizontalasymptoteistheratioofthelead
terms.Basedontheexamples,studentsmaybegintothinkthatthehorizontalasymptoteinthis
caseisalways0 = 1.Youmayneedtochallengetheirthinkingbypressingthemtolookcloselyatproblem#6,wheretheasymptoteis0 = 1/2.Ifstudentsarenotcomingupwithaconjectureforthehorizontalasymptoteinthecasewherethedegreeofthenumeratoristhesameasthedegree
ofthedenominator,encouragethemtochoosefunctionsofthistypeforthelasttwoproblems.
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.3
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Discuss(WholeGroup):
Beginthediscussionwiththegraphicorganizertable,askingstudentswheretheyfoundthevertical
asymptotes.Askstudentstomakeargumentsaboutwhytheverticalasymptotesoccurwherethe
denominatoriszero.Besuretomakethepointthatthefunctionisundefinedatthevertical
asymptotesbecausedivisionbyzeroisundefined.
Next,asktheclasstodescribehowtofindthehorizontalasymptotewhenthedegreeofthe
numeratorislessthanthedegreeofthedenominator.Askpreviously-selectedstudentstoexplain
howtheydrewthisconclusionbasedonproblems#2,#3,and#5.Inaddition,remindstudentsof
theirexperiencewith0 = 1/#confirmstheconclusionthatthehorizontalasymptoteswillbeat0 = 0.Askstudentswhythisoccurs.Presstheclasstothinkabouttheirpreviousworkwiththeendbehaviorofpolynomials.Theylearnedthatasvaluesof#becomeverylarge,thehigherdegreepolynomialsgrowmuchfasterthanalowerdegreepolynomial.Ifthehigherdegreepolynomialis
inthedenominatorofafraction,thenthatfractionwilleventuallyapproachzero.
Next,asktheclasstoshareideasforhowtofindthehorizontalasymptotewhenthedegreeofthe
numeratoranddenominatorarethesame.Useproblems#1,#4,and#6toconfirmordenythe
conjecturesthattheclasshasmade.Studentsmayneedhelpinrecognizinganddescribingtheidea
thatthehorizontalasymptotewillbetheratiooftheleadterms.Again,askstudentswhythismay
occurandreferbacktotheirworkwiththeendbehaviorofpolynomials.Pressfortheideathatfor
largevaluesof#thedifferenceintherateofchangebetweenthetwopolynomialsiscausedbytheleadcoefficients.Usetechnologytoshowanumericexampleofhowthisoccurs,usingproblem#4
asanexample.
Finallydiscusshowtogettheintercepts.Moststudentsshouldbeabletosaythatthe#-interceptsarefoundbysubstituting0 = 0intotheequation.Selectastudenttosharethathasextendedthatideaandnoticedthatthisturnsouttobethesameaslookingfortherootsofthenumerator.
Findingthey-interceptisnodifferentthananyotherfunction,butquickmethodsfordoingshould
behighlighted.Closethelessonbyaskingstudentstobesurethattheirgraphicorganizerisnow
completelycorrectandgivingtimetomakemodifications,ifneeded.
AlignedReady,Set,Go:RationalExpressionsandFunctions5.3
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS – 5.3
5.3
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READY Topic:Doingarithmeticwithrationalnumbers
Performtheindicatedoperation.Bethoughtfulabouteachstepyouperformintheprocedure.Showyourwork.
1.!"# +
%"#
2.&"' +
"()! 3.
*! +
#""
4.+)&, ∙ +*!, 5.+)&, ∙ +
."(, 6.+"'&&, ∙ +
"""!,
7.Explaintheprocedureforaddingtwofractions. a.Whenthedenominatorsarethesame: b.Whenthedenominatorsaredifferent:8.Explaintheprocedureformultiplyingtwofractions.9.Whenmultiplyingtwofractions,isitbettertoreducebeforeyoumultiplyorafteryoumultiply?Explainyourreasoning.
SET Topic:Identifyingkeyfeaturesofarationalfunction
READY, SET, GO! Name PeriodDate
18
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RATIONAL EXPRESSIONS & FUNCTIONS – 5.3
5.3
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Fillinthespecifiedfeaturesofeachrationalfunction.Sketchtheasymptotesonthegraphandmarkthelocationoftheintercepts.10./ = 12
(14()(16*)
Degreeofnum.__________Degreeofdenom.__________
Equationofhorizontalasymptote:
Equationofverticalasymptote(s):
y-intercept:(writeasapoint)
x-intercept(s):(writeaspoints)
11./ = (16()14&
Degreeofnum.__________Degreeofdenom.__________
Equationofhorizontalasymptote:
Equationofverticalasymptote(s):
y-intercept:(writeasapoint)
x-intercept(s):(writeaspoints)
12./ = "'1(14&)2
Degreeofnum.__________Degreeofdenom.__________
Equationofhorizontalasymptote:
Equationofverticalasymptote(s):
y-intercept:(writeasapoint)
x-intercept(s):(writeaspoints)
13./ = (14")(14))(16!)
Degreeofnum.__________Degreeofdenom.__________
Equationofhorizontalasymptote:
Equationofverticalasymptote(s):
y-intercept:(writeasapoint)
x-intercept(s):(writeaspoints)
19
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RATIONAL EXPRESSIONS & FUNCTIONS – 5.3
5.3
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GO Topic:Reducingfractions
Reducethefollowingfractionstolowestform.Thenexplainthemathematicsthatmakesitpossibletorewritethefractioninitsnewform.(Improperfractionsshouldnotbewrittenasmixednumbers.)Ifafractioncan’tbereduced,explainwhy.14.
")"!Explanation:
15.)(""Explanation:
17.!""#Explanation:
18.("&Explanation:
19.""*)# Explanation:
20.6"*,!)."*,!). Explanation:
20
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
5.4 Are You Rational?
A Solidify Understanding Task
BackinModule3whenwewereworkingwithpolynomials,
itwasusefultodrawconnectionsbetweenpolynomialsand
integers.Inthistask,wewilluseconnectionsbetween
rationalnumbersandrationalfunctionstohelpustothink
aboutoperationsonrationalfunctions.
1.Inyourownwords,definerationalnumber.
Circlethenumbersbelowthatarerationalandrefineyourdefinition,ifneeded.
3 − 5 <=<>
=142.7√52=3C=DEF<9
H>
2.Theformaldefinitionofarationalfunctionisasfollows:
AfunctionL(N)iscalledarationalfunctionifandonlyifitcanbewrittenintheformL(N) = Q(N)R(N)
whereQSTURarepolynomialsinNandRisnotthezeropolynomial.
Interpretthisdefinitioninyourownwordsandthenwritethreeexamplesofrationalfunctions.
3.Howarerationalnumbersandrationalfunctionssimilar?Different?
CC
BY
Tho
ught
Cat
alog
http
s://f
lic.k
r/p/
244g
Kgz
21
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.4
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Nowwearegoingtousewhatweknowaboutrationalnumberstoperformoperationsonrational
expressions.Thefirstthingweoftenneedtodoistosimplifyor“reduce”arationalnumberor
expression.Thenumbersandexpressionsarenotreallybeingreducedbecausethevalueisn’t
actuallychanging.Forinstance,2/4canbesimplifiedto½,butasthediagramshows,thesearejust
twodifferentwaysofexpressingthesameamount.
Let’stryusingwhatweknowaboutsimplifyingrationalnumberstosimplifyrationalexpressions.
Fillinanymissingpartsinthefractionsbelow.
Given: 24
30 4.
j< − j − 6j< − 4
5.j< + 8j + 15j< + 9j + 18
Lookforcommonfactors:
2 ∙ 2 ∙ 2 ∙ 32 ∙ 3 ∙ 5
(j + 2)(j − 2)
Dividenumeratoranddenominatorbythesamefactor(s):
o ∙ 2 ∙ 2 ∙ po ∙ p ∙ 5
Writethesimplifiedform:
45
j − 3j − 2
j + 5j + 6
6.Whydoesdividingthenumeratoranddenominatorbythesamefactorkeepthevalueoftheexpressionthesame?
7.Ifyouweregiventheexpressionr
rsCt,woulditbeacceptabletoreduceitlikethis:
NNo − 1
=1
j − 1
Explainyouranswer.
22
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.4
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In4.3RationalThinking,welearnedtopredictverticalandhorizontalasymptotes,andtofindinterceptsforgraphingrationalfunctions.
8.Givenv(j) = rsCrCwrsCx
,predicttheverticalandhorizontalasymptotesandfindtheintercepts.9.Usetechnologytoviewthegraph.Wereyourpredictionscorrect?Whatoccursonthegraphatj = −2?Rationalnumberscanbewrittenaseitherproperfractionsorimproperfractions.10.Describethedifferencebetweenproperfractionsandimproperfractionsandwritetwoexamplesofeach.Arationalexpressionissimilar,exceptthatinsteadofcomparingthenumericvalueofthenumeratoranddenominator,thecomparisonisbasedonthedegreeofeachpolynomial.Therefore,arationalexpressionisproperifthedegreeofthenumeratorislessthanthedegreeofthedenominator,andimproperotherwise.Inotherwords,improperrationalexpressionscanbewrittenas{(r)
|(r),where}(j)}~�Ä(j)arepolynomialsandthedegreeof}(j)isgreaterthanorequaltothe
degreeofÄ(j).11.Labeleachrationalexpressionasproperorimproper.
(rÅt)
(rC<)(rÅ<) rÇC=rsÅÉrCt
rsCxrÅx (rÅ=)(rÅ<)
rÑCx rÅ=
rÅÉr
ÇCÉrÅ<rCt>
Aswemayremember,improperfractionscanberewritteninanequivalentformwecallamixed
number.Ifthenumeratorisgreaterthanthedenominatorthenwedividethenumeratorbythe
denominatorandwritetheremainderasaproperfraction.Inmathtermswewouldsay:
23
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RATIONAL EXPRESSIONS & FUNCTIONS -5.4
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If} > Ä, thenthefraction {|canberewritenas {
|= Ü + á
|,whereqrepresentsthequotientandr
representstheremainder.
12.Rewriteeachimproperfractionasanequivalentmixednumber.
a)=HÉ= b)tÉ>
t<=
Rationalexpressionsworktheverysameway.Iftheexpressionisimproper,thenumeratorcanbe
dividedbythedenominatorandtheremainderiswrittenasafraction.Inmathematicalterms,we
wouldsay:{(r)|(r)
= Ü(j) + á(r)|(r)
whereÜ(j)representsthequotientandâ(j)representstheremainder.
Tryityourself!Labeleachrationalexpressionasproperorimproper.Ifitisimproper,thendivide
thenumeratorbythedenominatorandwriteitinanequivalentform.
13.rsÅÉrÅHrÅ<
14. CÉrÅt>rÇÅwrsÅ=rCt
15.rsÅ<rÅÉrÅ=
16.=rÅãrCt
24
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.4
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In4.3RationalThinking,whenwelookedatthegraphsofrationalfunctions,wedidnotconsider
thecasewhenthenumeratorofthefractionisgreaterthanthedenominator.So,let’stakeacloser
lookattherationalfunctionfrom#13.
16.Letv(j) = j2+5j+7j+2 .Wheredoyouexpecttheverticalasymptoteandtheinterceptsto
be?
17.Usetechnologytographthefunction.Relatethegraphofthefunctiontotheequivalent
expressionthatyouwrote.Whatdoyounotice?
18.Let’strythesamethingwith#15.Letv(j) = j2+2j+5j+3 .Findtheverticalasymptote,the
intercepts,andthenrelatethegraphtotheequivalentexpressionforv(j).
19.Usingthetwoexamplesabove,writeaprocessforpredictingthegraphsofrationalfunctions
whenthedegreeofthenumeratorisgreaterthanthedegreeofthedenominator.
25
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RATIONAL EXPRESSIONS & FUNCTIONS -5.4
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5.4 Are You Rational? – Teacher Notes A Solidify Understanding Task Purpose:
Inthistask,rationalfunctionsareformallydefinedandconnectedtorationalnumbers.Students
willusetheseconnectionstoreducerationalfunctionsandtoidentifyrationalfunctionsthatare
improperandwritetheminanequivalentform.Studentswillgraphrationalfunctionsthatcanbe
reduced,seeingthatthegraphisequivalenttothereducedfunctionexceptthatthereisaholein
thegraph,producedbythefactorthatisreduced.Theywillalsographimproperrationalfunctions
andidentifytheslantasymptote.
CoreStandardsFocus:
F.IF.7dGraphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandin
simplecasesandusingtechnologyformorecomplicatedcases.*
d.Graphrationalfunctions,identifyingzeroswhensuitablefactorizationsareavailable,and
showingendbehavior.
A.APR.6Rewritesimplerationalexpressionsindifferentforms;write{(r)|(r)
intheformsuchthat
{(r)|(r)
= Ü(j) + á(r)|(r)
where}(j), Ä(j), Ü(j)}~�â(j)arepolynomialswithdegreeofâ(j)lessthan
thedegreeofÄ(j),usinginspection,longdivision,orforthemorecomplicatedexamples,a
computeralgebrasystem.
A.APR.7Understandthatrationalexpressionsformasystemanalogoustotherationalnumbers,
closedunderaddition,subtraction,multiplicationanddivisionbyanonzerorationalexpression;
add,subtract,multiplyanddividerationalexpressions.
A.SSE.3Chooseandproduceanequivalentformofanexpressiontorevealandexplainproperties
ofthequantityrepresentedbytheexpression.
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.4
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StandardsforMathematicalPractice:
SMP2–Reasonabstractlyandquantitatively
SMP8–Lookforandexpressregularityinrepeatedreasoning
Vocabulary:Slantasymptote
TheTeachingCycle:
Launch--Part1(WholeGroup):
Beginthetaskbyaskingstudentsquestion#1,“Whatisarationalnumber?”Askthemtolookatthe
numbersgiveninquestion#1andtodecidewhicharerational.Discussresponsesanddefine
rationalnumbersas“anynumberthatcanbeexpressedasthequotientorratio,p/qoftwo
integers,anumeratorpandanon-zerodenominatorq.”Then,discusstheformaldefinitionof
rationalfunctionsthatisgiveninquestion#2andconnectittotheinformationdefinitiongivenin
task4.3,thatrationalfunctionsarearatioofpolynomials.Shareafewoftheexamplesthat
studentshavewrittenandtellstudentsthatrationalfunctionsbehavemuchlikerationalnumbers.
Tellstudentsthatinthistask,theywillbeworkingwithrationalfunctionsthatcanbereducedor
areimproper.Youmaywishtogoovertheprocessforreducingrationalnumbersbeforeasking
studentstostartworkingwithrationalfunctions.Then,tellstudentsthattheyshouldrelyontheir
experienceswithrationalnumberstohelpthemthinkaboutrationalfunctions.Askstudentsto
workproblemsupto#9beforehavingaclassdiscussionandre-launchingtheremainderofthe
task.
Explore(Individually,FollowedbySmallGroup):
Monitorstudentsastheyworkonquestions#4and#5toseethattheyaremakingsenseofthe
necessarystepsinworkingwithrationalexpressions.Thescaffoldingisgiveninthetask,but
studentswillneedtobeabletofactorthefunctionsandreducethem.Theanswersaregiveninthe
tasksothatstudentscanfocusontheprocessthatwillgetthemtheanswer.Makesurethatthey
areworkingontheprocessandcorrectingerrorsiftheirworkisnotleadingthemtotheright
answers.Helpstudentstofocusontheideathatreducingisjustfindinganequivalentformby
dividingthesamefactoroutofthenumeratoranddenominator,whichisjustdividingbyone.
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.4
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Selectstudentstosharethathavearticulatedthisideainsomeway,sothatotherstudentscanhear
severalversionsoftheidea.
Asstudentsprogress,supporttheminusingtechnologyforquestion#9toexploretheareaofthe
graphnear-2wherethereisahole.Asthecurveistracedonsomegraphingtechnology,like
Desmos,thevalueat-2isshowntobeundefined.Onothertechnology,likesomegraphing
calculators,thereisnovalueshownat-2.Eitherway,studentswillneedsupportforinterpreting
thegraphduringthediscussion.Listenfortheirideasthatwillbeusefultoshare.
Discuss—Part1(WholeGroup):
Beginthediscussionbyaskingastudentsharehis/herworkonquestion#4.Focusonhowthe
numeratorwasfactoredandthecommonfactorsarereduced.Similarly,askastudenttoshare
question#5.Askpreviously-selectedstudentstoshareideasforquestion#6.Emphasizethat
commonfactorsmustbedividedfromthenumeratoranddenominatorsothevalueofthefraction
isunchanged.Then,askstudentsaboutquestion#7.Sincethisproblemisbaseduponacommon
misconception,theremaybesomecontroversy.Letstudentsmakeargumentsaboutwhetheror
notitiscorrect.Afterhearingarguments,bringtheclasstoconsensusthatitisnotcorrectbecause
jisnotafactorinthedenominator,so“reducing”itwillnotresultinanequivalentexpression.
Discussquestions#8and#9.Askstudentsfortheirpredictionsandthenprojectagraphofthe
function.Askstudentswhythereisnotanasymptoteatj = −2.Helpthemtoseethattheoriginal
functionisnotdefinedat-2,sothereisnovaluethere,andalltheothervaluesarethesameasthe
functionthatremainsafterthefactor(j + 2)isreduced.
Launch--Part2(WholeGroup):
Re-launchthesecondpartofthetaskbydiscussingquestions#10and#11together.Discuss
question#12,emphasizingtheprocessforwritingamixednumeralfromanimproperfraction.Tell
studentsthatthisisthesameprocessforimproperrationalfunctionsandthenaskthemtoworkon
theremainingquestionsinthetask.
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.4
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Explore(Individually,FollowedbySmallGroup):
Studentsmayneedalittlenudgetodecidetouselongdivisionofpolynomials.Remindthemthat
fractionsarejustanotherwaytowritedivision,sothehintofwhattodoisrightintheexpression
thattheyaregiven.
Asstudentsareworkingonquestion17,watchforastudentthatnoticesthatthefunctionis
approachingthelineå = j + 3.Thismaytakealittlepromptingwithquestionslike,“Whatisthe
endbehaviorofthefunction?Whatlinedoesitappeartobeapproaching?Howisthatlinerelated
totheequivalentfunctionthatyoufound?”
Discuss—Part2(WholeGroup):
Beginthediscussionwithquestion#13.Askastudenttosharehis/herworkinwritingan
equivalentexpression.Then,projectagraphofthefunctionandaskthepreviouslyselected
studenttodescribetherelationshipbetweentheequivalentexpressionandthegraph,specifically
howtoidentifytheendbehaviorofthefunction.Tellstudentsthatthistypeofendbehavioris
calledaslant(oroblique)asymptote.Repeatthesameprocesswithquestion#15.Discuss
question#19,bringingtheclasstoconsensusonhowtographarationalfunctionwherethedegree
ofthenumeratorisgreaterthanthedegreeofthedenominatorthatincludes:
• Dividingthenumeratorbythedenominatortofindanequivalentexpression.• Findingtheverticalasymptotebyfindingtherootsofthedenominator.• Usingtheequivalentexpressiontoidentifytheslantasymptote.• Findtheinterceptsusingthesameprocessasotherrationalfunctions.
Iftimeallows,askastudenttosharehis/herworkinwritinganequivalentexpressionforquestion
#16.Thenasktheclasswhattheywouldexpectofthegraphoftheoriginalfunction.Theyshould
noticethatthedegreeofthenumeratoristhesameasthedegreeofthedenominator,sotheycan
predicttheverticalandhorizontalasymptotes.Connecttheirpredictedasymptotestothe
equivalentform.Theyshouldbeabletoseethatthisfunctionturnsouttobeatransformationof
å = 1/j.
AlignedReady,Set,Go:RationalExpressionsandFunctions5.4
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS – 5.4
5.4
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READY Topic:Connectingfeaturesofpolynomialsandrationalfunctions
Findtherootsanddomainforeachfunction.
1.!(#) = (# + 5)(# − 2)(# − 7)
2.+(#) = #, + 7# + 6
3..(#) =/
(012)(03,)(034)
4.ℎ(#) =/
(0614017)
5.Makeaconjecturethatcomparesthedomainofapolynomialwiththedomainofthereciprocalofthepolynomial.(Notethatthereciprocalofapolynomialisarationalfunction.)
6.Dotherootsofthepolynomialtellyouanythingaboutthegraphofthereciprocalofthepolynomial?Explain.
7.Findthey-interceptfor#1and#2.Whatisthey-interceptfor#3and#4?
SET Topic:Distinguishingbetweenproperandimproperrationalfunctions.
Determineifeachofthefollowingisaproperoranimproperrationalfunction.8. 9. 10.
!(#) =#> + 3#, + 7
7#, − 2# + 1
!(#) = #> − 5#, − 4!(#) =
3#, − 2# + 7
#2 − 5
READY, SET, GO! Name PeriodDate
26
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RATIONAL EXPRESSIONS & FUNCTIONS – 5.4
5.4
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11.!(#) =0A1B061,0
/C014 12.!(#) =
2063B01B
40D3,01>
13.Whichoftheabovefunctionshavethefollowingendbehavior?
EF# → ∞, !(#) → 0EJKEF# → −∞, !(#) → 014.Completethestatement:
ALLproperrationalfunctionshaveendbehaviorthat___________________________________________
Determineifeachrationalexpressionisproperorimproper.Ifimproper,uselongdivisiontorewritetherationalexpressionssuchthatL(M)
N(M)= O(M) +
P(M)
N(M)whereO(M)representsthe
quotientandP(M)representstheremainder.
15.,0A340617
03/ 16.
(01/)
(03,)(01,)
17.0A3>061203/
063B01B 18.
0A3201,
03/C
27
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS – 5.4
5.4
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GO Topic:Findingthedomainofrationalfunctionsthatcanbereduced
Statethedomainofthefollowingrationalfunctions.
19.Q =(03,)
(03,)(012) 20.Q =
(017)
(03B)(017) 21.Q =
(034)(01/C)
(01/C)(03>)(034)
a)Eachofthepreviousfunctionshasonlyoneverticalasymptote.Writetheequationoftheverticalasymptotefor#19,#20,and#21below.
19a)V.A. 20a)V.A. 21a)V.A.
b)Thegraphsof#19,#20,and#21arebelow.Foreachgraph,sketchintheverticalasymptote.Putanopencircleonthegraphanywhereitisundefined.
19b)
20b)
21b)
28
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
5.5 Just Act Rational
A Solidify Understanding Task
In5.4AreYouRational?,yousawhowconnectingrationalnumberscanhelpustothinkaboutrationalfunctions.Inthistask,we’llextendthatworktoconsideroperationsonrationalexpressions.Let’sbeginwithmultiplication.Ineachofthetablesbelowtherearemissingdescriptionsandmissingpartsofexpressions.Yourjobistofollowtheprocessformultiplyingrationalnumbersanduseittocompletethedescriptionsoftheprocessandtoworktheanalogousproblemswithrationalexpressions.1.DescriptionoftheProcedure:
ExampleUsingNumbers
RationalExpressionA
RationalExpressionB
Given: 34∙56 C(C − 3)
(C + 1)∙5CI
(C + 1)(C − 2)(C + 2)
∙(C + 5)
(C − 2)(C + 2)
3 ∙ 54 ∙ 6
K ∙ 52 ∙ 2 ∙ 2 ∙ K
Writethesimplifiedform:
58
5(C − 3)C(C + 1)
Or5C − 15C(C + 1)
(C + 1)(C + 5)(C + 2)I
OrCI + 6C + 5(C + 2)I
2.Inmultiplication,doesitmatterinwhichstepthesimplifyingisdone?Why?
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RATIONAL EXPRESSIONS & FUNCTIONS -5.5
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Nowlet’strythesameprocesswithdivision.3.Completethetablebelowbyfillinginthemissingdescriptionsorstepsfordividingtherationalexpressions.Descriptionofthe
Procedure:ExampleUsingNumbers
RationalExpressionA
RationalExpressionB
Given: 34÷56
(C − 2)(C + 2)
÷(C + 5)C(C + 2)
(C + 1)(C − 6)
(C + 2)÷
(C + 1)(C − 3)(C + 2)
34∙65
Multiplyandsimplify:
3 ∙ 64 ∙ 5
=3 ∙ \ ∙ 32 ∙ \ ∙ 5
910
C(C − 2)(C + 5)
OrCI − 2C(C + 5)
(C − 3)(C − 6)
4.Howcouldyoucheckyouranswerafterperforminganoperationonapairofrationalexpressions?
30
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RATIONAL EXPRESSIONS & FUNCTIONS -5.5
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Areyoureadyforaddition?Tryit!5.Completethetablebelowbyfillinginthemissingdescriptionsorstepsforaddingtherationalexpressions.DescriptionoftheProcedure:
ExampleUsingNumbers
RationalExpressionA
Given 23+17
3
(C + 7)+
4(C − 4)
Determinethefactorsneededforacommondenominator
23d77e +
17d33e
3
(C + 7)dC − 4C − 4
e +4
(C − 4)dC + 7C + 7
e
1421
+321
14 + 321
Simplify 1721
7C + 16
(C + 7)(C − 4)
6.Afterwritingbothtermswithacommondenominator,aretheyequivalenttotheoriginalterms?Explain.
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RATIONAL EXPRESSIONS & FUNCTIONS -5.5
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7.Completethetablebelowbyfillinginthemissingdescriptionsandstepsforaddingtherationalexpressions.DescriptionoftheProcedure: RationalExpressionBGiven
2C − 3(C + 3)
+C + 5(C − 2)
Determinethefactorsneededforacommondenominator
Simplify 3CI + C + 21(C − 2)(C + 3)
8.Isitpossibletogetananswertoanadditionproblemthatstillneedstobereduced?Ifso,howcanyoutellifyouranswerneedstobefurthersimplified?
32
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RATIONAL EXPRESSIONS & FUNCTIONS -5.5
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9.Atlonglast,wehavesubtraction.Completethetablebelowbyfillinginthemissingdescriptionsandstepsforaddingtherationalexpressions.DescriptionoftheProcedure:
ExampleUsingNumbers
RationalExpressionA
Given: 78−35
3C + 1(C + 5)
−C − 4(C − 2)
78d55e −
35d88e
3540
−2440
35 − 2440
Simplify 1140
2CI − 6C + 18(C + 5)(C − 2)
Or
2(CI − 3C + 9)(C + 5)(C − 2)
10.Whatstrategieswillyouusetobesurethatyoudon’tmakesignerrorswhensubtracting?
33
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RATIONAL EXPRESSIONS & FUNCTIONS -5.5
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5.5 Just Act Rational – Teacher Notes A Solidify Understanding Task
Purpose:Thepurposeofthistaskisforstudentstolearntoadd,subtract,multiplyanddivide
rationalexpressions.Thetaskisdesignedsothatstudents,again,connectoperationswithrational
numberstooperationswithrationalexpressions.
CoreStandardsFocus:
A.APR.7Understandthatrationalexpressionsformasystemanalogoustotherationalnumbers,
closedunderaddition,subtraction,multiplicationanddivisionbyanonzerorationalexpression;
add,subtract,multiplyanddividerationalexpressions.
A.SSE.3Chooseandproduceanequivalentformofanexpressiontorevealandexplainproperties
ofthequantityrepresentedbytheexpression.
StandardsforMathematicalPractice:
SMP1–Makesenseofproblemsandpersevereinsolvingthem
SMP8–Lookforexpressregularityinrepeatedreasoning
TheTeachingCycle:
Launch(WholeGroup):
Beginthetaskbyremindingstudentsthatrationalnumbersandrationalexpressionsareanalogous
andthattheyoperatethesameway.Tellstudentsthatintoday’stasktheyaregoingtousewhat
theyknowaboutrationalnumberstoperformoperationsonrationalexpressions.Theyshouldn’t
worryifthey’renottoogreatwithfractions.Thereisalotofsupportinthetasktohelpthem.
Beginbydemonstratinghowtousethesupportgiven.Think-aloudforstudentsabouthowtoget
fromthefirststeptothesecondinthemultiplicationproblem.Itcouldsoundsomethinglikethis:
“IseethatI’mgivenhi∙ jk.Inthenextstep,Ihaveh∙j
i∙k,soIseethatthenumeratorshavebeen
multipliedandthedenominatorshavebeenmultipliedtogetasinglefraction.Iwillwrite thatintheboxprovided.Now,Icandothesamethingwiththerationalexpression,
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.5
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l(lmh)(lno)
∙ jlp.Iwillmultiplythenumeratorstogetherandthedenominatortogetherandmake
asinglefraction:jl(lmh)lp(lno)
.”
Stopthereandtellstudentsthattheywillbeusingthesamestrategythroughoutthetask.They
shoulddecidewhatwasdonewiththenumbers,labelit,andthendothesamethingwiththe
rationalexpressions.Tellstudentsthattheanswerstotheproblemsareprovidedattheendsothat
theycanmakesurethattheirprocessiscorrect.Iftheygettotheendoftheproblemandtheir
workdoesn’tleadtotheanswergiven,theyneedtogobackandfindtheirmistake.
Explore(Individual,FollowedbySmallGroup):
Monitorstudentsastheywork,helpingthemtoseeandlabeleachstepoftheoperations.Expect
manymistakesasstudentsworkwiththerationalexpressions.Watchforstudentsthatnoticetheir
mistakesandfindwaystocorrectthemsothattheycandescribetheirprocessfortheclass.Some
ofthecommonerrorstowatchforsothattheycanbeaddressedduringthediscussion:
• Improperlyreducingterms,notfactors
• Notusingparenthesesandmakingmistakesindistributing(usuallywhenchangingto
commondenominator).
• Signerrorsinsubtraction.
Discuss(WholeGroup):
Beginthediscussionwithastudentsharinghis/herworkonRationalExpressionBin
multiplication.Askthestudenttoexplainhowtheanswerwasreduced,andthenasktheclassifit
makesadifferenceinwhichstepthefractionisreduced.Makesurethediscussionincludestheidea
thatreducingisdividingby1,soitcanhappenineitherstepsincemultiplicationanddivisionareat
thesamelevelinorderofoperations.DiscusswhyitiscorrecttodivideC − 2fromthenumerator
anddenominatorinthiscasetosimplifythefraction.
Next,askastudenttoexplainhowhis/herworkonproblem3,RationalExpressionBindivision.
Askthestudenttoexplaintheprocessfordivision,andthenasktheclasswhythedenominatorof
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
thefractionisgone.Again,emphasizethatthedenominatoris1(not0)becausetheresultof
dividingthecommonfactorsisone.
Followwithastudentthatpresentshis/herworkonproblem7,RationalExpressionBinaddition.
Ideally,findastudentthatdidnotproperlydistributeinthefirstattemptandthenfoundawayto
correcttheproblem.Helpstudentstoseewhentoputparenthesesaroundthenumeratorto
remindthemthattheymustdistributeeachtermwhentheymultiply.
Endthediscussionwithproblem9,RationalExpressionAinsubtraction.Again,itwouldbeidealif
astudentsharesthatinitiallymademistakesanddescribeshowtheycorrectedthem.Thetwo
potentialerrorswithsubtractionarefailingtodistributeproperlywhengettingacommon
denominatorandmakingsignerrorswhensubtracting.Askstudentsforstrategiestohelpthem
avoidtheseerrorsandrecordtheerrors.
Closethediscussionbygoingbacktoeachoftheorganizertablesandrecordingthestepsforthe
operation.Tellstudentsthattheorganizersaretohelpthemremembertheprocedureforeach
operation.
AlignedReady,Set,Go:RationalExpressionsandFunctions5.5
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RATIONAL EXPRESSIONS & FUNCTIONS – 5.5
5.5
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READY Topic:Recallingtrigonometricfunctions
Usethegiventriangletowritethevaluesof!"#$, &'!$, (#)+(#$and!"#,, &'!,, (#)+(#,.
1.
sinA=cosA=tanA=sinB=cosB=tanB=
2.
sinA=cosA=tanA=
√2cmsinB=cosB=tanB=
3.
sinA=cosA=tanA=sinB=cosB=tanB=
4.
sinA=cosA=tanA=sinB=cosB=tanB=
READY, SET, GO! Name PeriodDate
7√2m
34
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RATIONAL EXPRESSIONS & FUNCTIONS – 5.5
5.5
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SET Topic:Adding,subtracting,multiplying,anddividingrationalfunctions
5.Angelasimplifiedthefollowingrationalexpressions.Onlyoneofthethreeproblemsiscorrect.Determinewhichonesheansweredcorrectly.ThenidentifyAngela’serrorsinthetwothatareincorrectandcorrectthem.
a. KL(LNO)
+ R(LNS)
b. L(LTO)
− V(LTO)(LNS)
c.(LTS)(LNR)(LTR)
× (LTK)(LNR)(LTR)
5X(X − 1)(X − 3)(X − 1)
+2(X − 3)
(X − 3)(X − 1)
X1−
4(X − 1)
(X + 1)(X − 2)(X + 5)(X + 2)(X − 2)(X + 2)
5XR − X + 2X − 3(X − 3)(X − 1)
X(X − 1)(X − 1)
−4
(X − 1) (X + 1)(X + 5)
(X + 2)(X + 2)
5XR + X − 3(X − 3)(X − 1)
XR − X − 4(X − 1)
XR + 6X + 5XR + 4X + 4
Simplifyeachexpression.Reducewhenpossible.
6. RLT](LTS)
− V(LTS)
7. RLLTR
+ LNSLNK
8. L_T]LT`L_NKLTV
∙ L_TOLNVL_TVLTV
9. VLT`KLNRc
÷ L_NOLNScL_NVL
10. RL
(L_NV)+ V
(LTR) 11.
LNScLNV
− LTRVNL
35
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RATIONAL EXPRESSIONS & FUNCTIONS – 5.5
5.5
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Topic:Comparingrationalnumbersandrationalexpressions
12. Rationalnumbersandrationalexpressionsarecomparablebecausetheyhavesimilarfeatures.Completethetablebelowbywritingthecomparablesituationforeachstatementwritten.
GO Topic:Findingvaluesofxthataffectthedomainofarationalexpression
Identifythevaluesofxforwhichtheexpressionisundefined,ifany.
13.ScLNV
14.RRL
15.LNeLTSK
16.RLK
RationalNumbers RationalExpressions
Wholenumbersarerationalnumberswith
adenominatorofone.
a)
b)
Rationalexpressionsareundefinedwhen
thedenominatorisequaltozero.
Whenyouadd,subtract,multiplyordivide
tworationalnumbers,theresultisalsoa
rationalnumber.
c)
Rationalfractionsareclassifiedasproper
fractionswhenthenumericvalueofthe
numeratorissmallerthanthedenominator.
d)
36
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.6
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5.6 Sign on the Dotted Line
A Practice Understanding Task
JosueandFranciaareworkingongraphingallkindsofrational
functionswhentheyhavethislittledialogue:
Josue:It’seasytofigureoutwheretheasymptotesandinterceptsareonarationalfunction.
Francia:Yes,andit’salmostliketheasymptotessplitthegraphintosections.Allyouneedtoknow
iswhatthegraphisdoingineachsection.
Josue:Itseemsalmosteasierthanthat.It’slikeallyouneedtoknowiswhetherit’sgoingupor
downeithersideoftheverticalasymptoteandthenuselogictofigureitoutfromthere.
Francia:Don’toverlooktheintercepts.Theygivesomeprettyimportantclues.
Josue:Yeah,yeah.Iwonderifwecanfigureoutaneasywaytodeterminethebehaviornearthe
asymptotes.
Francia:Seemseasyenoughtojustpluginnumbersandseewhattheoutputsare,butmaybeyou
don’tevenneedexactvalues.Hmmm.Weneedtothinkaboutthis.
JosueandFranciaaredefinitelyontosomething.Everyonewantstofindawaytobeabletopredict
andsketchgraphseasily.Inthistask,you’regoingtoworkonjustthat.Startbyfinding
asymptotesandintercepts,thenfigureoutastrategythatyoucanuseeverytimetoquickly
sketchthegraph.Afterusingyourstrategytographthefunction,usetechnologytocheck
yourworkandrefineyourstrategy.
Theexamplesyouneedtodevelopyourstrategyareonthefollowingpages.Someofthefunctions
givenneedtobecombinedand/orsimplifiedtomakeonerationalfunction.Ifthisisthecase,write
thesimplifiedfunctioninthespacenexttothegraph.
CC
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RATIONAL EXPRESSIONS & FUNCTIONS -5.6
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1.
" = (% − 5)(% + 1)(% + 2)(% − 2)
VerticalAsymptote(s)______________________
HorizontalorSlantAsymptote_________________
Intercepts_______________________________________
Graph:
2.
" = (% − 3)(% + 1) ∙
%(% − 4)
VerticalAsymptote(s)______________________
HorizontalorSlantAsymptote_________________
Intercepts_______________________________________
Graph:
38
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.6
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3.
" = %/ − 6% + 2(% − 2)
VerticalAsymptote(s)______________________
HorizontalorSlantAsymptote_________________
Intercepts_______________________________________
Graph:
4.
" = 4(% + 1) +
(% − 5)(% − 3)
VerticalAsymptote(s)______________________
HorizontalorSlantAsymptote_________________
Intercepts_______________________________________
Graph:
39
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.6
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5.
" = 3%(%/ + 2% + 1) ÷
% − 3% + 1
VerticalAsymptote(s)______________________
HorizontalorSlantAsymptote_________________
Intercepts_______________________________________
Graph:
6.
" = % + 5% + 4 −
% + 2% − 1
VerticalAsymptote(s)______________________
HorizontalorSlantAsymptote_________________
Intercepts_______________________________________
Graph:
40
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.6
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
7.
" = 2%/ + % − 15%/ + 4% + 3
VerticalAsymptote(s)______________________
HorizontalorSlantAsymptote_________________
Intercepts_______________________________________
Graph:
8.Summarizeyourstrategyforgraphingrationalfunctionswithastep-by-stepprocess.
41
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.6
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5.6 Sign on the Dotted Line – Teacher Notes A Practice Understanding Task Purpose:Thepurposeofthistaskistohavestudentsconnectallthattheyhavelearnedabout
rationalfunctionssofartosketchthegraphofrationalfunctions.Studentsareaskedtodevelopa
strategyfordeterminingthebehaviorneartheasymptotesaspartofanoverallstrategyfor
graphingrationalfunctions.Someofthefunctionsgiveninthetaskrequirestudentstocombine
tworationalexpressionstomakethefunctionmorepredictabletograph.
CoreStandardsFocus:
F.IF.4Forafunctionthatmodelsarelationshipbetweentwoquantities,interpretkeyfeaturesof
graphsandtablesintermsofthequantities,andsketchgraphsshowingkeyfeaturesgivenaverbal
descriptionoftherelationship.Keyfeaturesinclude:intercepts,intervalswherethefunctionis
increasing/decreasing,positiveornegative;relativemaximumsandminimums;symmetries;end
behavior;andperiodicity.*
F.IF.7dGraphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandin
simplecasesandusingtechnologyformorecomplicatedcases.*
d.Graphrationalfunctions,identifyingzeroswhensuitablefactorizationsareavailable,and
showingendbehavior.
Related:F.IF.5,F.IF.9,A.CED.1
StandardsforMathematicalPractice:
SMP3–Constructviableargumentsandcritiquethereasoningofothers
SMP7–Lookforandmakeuseofstructure
TheTeachingCycle:
Launch(WholeGroup):
StartthetaskbyreadingthedialoguebetweenJosueandFranciaatthebeginningofthetask.Ask
theclasswhatinformationtheycandetermineaboutanyrationalfunctioniftheyjustlookatthe
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.6
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equation.Theyshouldbeabletodescribehowtodetermineifitwillbeatransformationof
" = 1/%andhowtofindtheasymptotesandintercepts.AskwhyitwouldbehelpfultofindthebehaviorneartheasymptotesandwhyFranciamightthinkthatshedoesn’tneedexactvalues.
Ideally,astudentwillsaythatifoneknowsthefunctionispositivenearaverticalasymptote,then
oneknowsthatthegraphisapproachinginfinityonthatsideoftheasymptote.
Describetheexpectationsforthetask,specifically,thatstudentsfindaquickwaytodeterminethe
behaviorneartheasymptotesanduseittodevelopastrategyforgraphinganyrationalfunction.
Tellstudentsthatsomeofthefunctionsarenotgiveninsimplifiedform,sotheymayneedto
combinerationalexpressionsbeforegraphingthefunction.Stressthattheyshouldnotuse
technologytographafunctionuntiltheyarereadytochecktheirwork.Inthisway,theycantrya
techniqueandthenmodifyitifitisnotproducingacorrectgraph.
Explore(Individual,FollowedbySmallGroup):
Listenforstudentswhohaveideasabouthowtodeterminebehaviorneartheasymptote.Ifa
studentisstuck,askthemtotryavaluethatis.1unitsoneithersideofaverticalasymptote.This
willgiveanumberthatiscloseenoughtorepresentthebehaviorbutiscomplicatedenoughthatit
mightleadthemtolookforaslightlysimplerstrategy.Aftertheyhavefoundavalue,askthemhow
theymightthinkaboutjustthesignoftheoutputvalueneartheasymptoteandhowtheycoulduse
thatalongwiththerestoftheinformationaboutthefunction.
Afterstudentshavefiguredoutawaytousethesignsratherthanexactvalue,theywillneedtofind
awaytoorganizeandkeeptrackoftheirfindings.Somepeopleusetables,somepeopleuseasign
chartthatlookslikeanumberlinedividedbytheverticalasymptotes,somepeoplemarkthe
asymptotesontheirgraphandthenrecordtheappropriatesignnearthetoporbottomofthe
asymptotes.Lookforstudentsthathavefoundaneffectiveorganizationalstrategysothattheycan
shareduringthediscussion.
Beonthealertforproblemswithcombiningexpressions.Thistaskisdesignedtogivestudents
anotheropportunitytoperformoperationsonrationalexpressionswiththesupportofateacher.
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.6
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Encouragestudentstocheckthattheyhaveperformedtheoperationscorrectlybyusingtechnology
tochecktheirwork.
Discuss(WholeGroup):
Beginthediscussionwithaskingpreviously-selectedstudentstosharequestion#1.Sincethereare
severalstepsineachproblem,youmaywanttohavedifferentstudentsshareeachstep.The
examplebelowshowsonepossiblewayforstudentstotalkaboutthegraph.
Step1:“Istartedbyfindingtheverticalasymptotesbysettingthedenominatorequaltozero.Igotthehorizontalasymptotebyseeingthatthedegreeofthenumeratoranddenominatorarethesame,soItooktheratiooftheleadterms.Then,Igottheinterceptsbysubstituting% = 0and" = 0.”
Step2:“Imarkedtheasymptotesonthegraphwithdottedlines.ThenIplottedtheintercepts.”
Step3:“Iputin% = −20toseewhattheleftendbehaviorwouldbe.
" = (4/546)(4/578)(4/57/)(4/54/).That
makes4/6(48:)48;(4//)whichis
greaterthan1,soIknowthatthegraphisabovetheasymptote.ThenIdidasimilarprocesstoseethatitwasbelowtheasymptoteontherightside.”
VerticalAsymptote(s)% = 2, % = −2HorizontalorSlantAsymptote" = 1Intercepts(5,0),(-1,0),(0, 6=)
Step4:“Icheckedx=-2.1.Ididn’tfindexactvalues,onlylookedtoseeifitmadethefactorspositiveornegative.
" = (−)(−)(−)(−) = +
Ididasimilarprocessontheothersideoftheasymptoteandontheasymptoteatx=2.ThenImarkeditonthegraphlikethis:
Step5:“IusedthepointsthatIhadandwhatIknewabouttheasymptotestosketchthegraph.”
Step6:“Icheckeditandfeltprettygoodaboutmyself.”
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.6
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Proceedinasimilarfashionwithstudentssharingquestions#3,6,and7.Theyallfollowasimilar
process,buteachoftheproblemshasaspecialfeaturetodiscuss:
• Question#3hasaslantasymptote,sostudentsmustdividethenumeratorbythedenominatortofindtheendbehavior.
• Question#6requiresstudentstosubtractthetworationalexpressions.Thegraphcrossesthehorizontalasymptote.
• Question#7hasaholeinthegraphbecausetherationalexpressionhasacommonfactorinthenumeratoranddenominator.
Finalizethelessonbyrecordingaprocessforgraphingrationalfunctionssuchasthis:
• Simplifythefunction.
• Findtheverticalasymptotesbyfindingthezerosofthedenominator.
• Findthehorizontalorslantasymptote.
O Ifdegreeofnumerator<degreeofdenominator," = 0O Ifdegreeofnumerator=degreeofdenominator," =ratioofleadtermsO Ifdegreeofnumerator>degreeofdenominator,dividenumeratorby
denominatortogetslantasymptote
• Findinterceptsbysubstituting% = 0and" = 0.• Testthebehaviorneartheverticalasymptotesbyfindingifthevaluesarepositiveor
negative.Keeptrackoftheresultsusingasignchartofsometype.
• Sketchthefunction.
AlignedReady,Set,Go:RationalExpressionsandFunctions4.6
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.6
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HowtoGraphaRationalFunctionSteps: Example: Remember
this:
" = % + 3(% − 1)(% + 2)
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS – 5.6
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READY Topic:Identifyingextraneoussolutions
1.Belowistheworkdonetosolvearationalequation.Theproblemhasbeenworkedcorrectly.Explainwhytheequationhasonlyonesolution.
Solve: !
"!#!"− &
"#!= &
!"("#!)
− (")&(")("#!)
= & Writeusingacommondenominator.
! − "(")(" − !)
= & Subtract.
(")(" − !)! − "
(")(" − !)= &(")(" − !) Multiplybothsidesbythecommon
denominator.
! − " = "! − !" Simplify.
"! − " − ! = * Writeaquadraticequationinstandardform.
(" − !)(" + &) = * Factor
" = !,-" = −& ApplytheZero-ProductPropertyandsolveforz.
Substitute2and-1intotheoriginalequationtoseeifthenumbersaresolutions.
Substitutethegivennumbersintothegivenequation.Identifywhichareactualsolutionsandwhich,ifany,areextraneous.2..: − 1.12 3
4
. −3
2. + 1= 2
3.d:0and3
3224 − 2
−1
2 − 1= 1
4.7: 1
174 − 7
−1
7 − 1= 0
Solve5and6.Watchforextraneoussolutions.
5.
9:;#:
− 9:#9
= 94
6.2< + =
:>4= 1
READY, SET, GO! Name PeriodDate
42
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RATIONAL EXPRESSIONS & FUNCTIONS – 5.6
5.6
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SET Topic:Predictingandsketchingrationalfunctions
Findtheverticalasymptote(s),horizontalorslantasymptote,andintercepts.Thensketchthegraph.(Donotusetechnologytogetthegraph.Themaxandminsdonotneedtobeaccurate.)5.
? =(< + 4)(−2< − 6)
VerticalAsymptote(s)______________________
HorizontalorSlantAsymptote_________________
Intercepts_______________________________________
Graph:
6.
? =3<
(< − 3)∙(< − 4)(< + 1)
VerticalAsymptote(s)______________________
HorizontalorSlantAsymptote_________________
Intercepts_______________________________________
Graph:
43
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS – 5.6
5.6
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7.
? =(<4 − 4<)(4< − 8)
÷(< + 2)< + 4
VerticalAsymptote(s)______________________
HorizontalorSlantAsymptote_________________
Intercepts_______________________________________
Graph:
8.
? =(< − 6)(< − 3)
+(< + 3)
<4 − 6< + 9
VerticalAsymptote(s)______________________
HorizontalorSlantAsymptote_________________
Intercepts_______________________________________
Graph:
44
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS – 5.6
5.6
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GO Topic:Exploringlinearequations
9.WhatvalueofkintheequationF< + 10 = 6?wouldgivealinewithslope-3?10.WhatvalueofkintheequationF< − 12 = −15?wouldgivealinewithslope4
3?
11.ThestandardformofalinearequationisH< + I? = J.Rewritethisequationin
slope–interceptform.Whatistheslope?Whatisthey–intercept?12.Ifbisthey–interceptofalinearfunctionwhosegraphhasslopem,then? = 7< + Ldescribes
theline.Belowisanincompletejustificationofthisstatement.Fillinthemissinginformation.
Statements Reasons
1.7 = M;#MN:;#:N
1.slopeformula
2.7 = M#O:#P
2.Bydefinition,ifbisthey–intercept,then
Q ,LSisapointontheline.(<, ?)isany
otherpointontheline.
3.7 = M#O:
3.?
4.7 ̇ = ? − L
4.MultiplicationPropertyofEquality
(Multiplybothsidesoftheequationbyx.)
5.7< + L = ?, UV? = 7< + L
5.?
45
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.7
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
5.7 We All Scream for Ice Cream A Practice Understanding Task
TheGlacierBowlisanenormousicecreamtreatsoldattheneighborhoodicecreamparlor.Itissolargethatanyperson
whocaneatitwithin30minutesgetsat-shirtandhispicturepostedonawall.BecausetheGlacierBowlissobig,itcosts$60andmostpeoplesplitthetreatwithagroup.
Ameraandsomeofherfriendsareplanningtogettogethertosharethebowloficecream.Theyplantosplitthecostbetweenthemequally.
1.Whatisanalgebraicexpressionfortheamountthateachpersoninthegroupwillpay?
2.Atthelastminute,oneofthefriendscouldn’tgowiththegroup.Writeanexpressionthat
representstheamountthateachpersoninthegroupnowpays.
3.Itturnsoutthateachpersoninthegrouphadtopay$2morethantheywouldhaveifeveryone
intheoriginalgrouphadsharedtheicecream.Howmanypeoplewereintheoriginalgroup?
CC
BY
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s://f
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46
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.7
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
4.Explainwhyyouranswer(s)makessenseinthissituation.
Thisstoryandtheproblemitrepresentsprovidesanopportunitytomodelasituationthatrequiresarationalequation.Rationalequationscantakemanyforms,buttheyaresolvedusingprincipleswehaveworkedwithbefore.Tryapplyingsomeofthestrategiesforworkingwithrationalexpressionsthatwehaveusedinthismoduletosolvetheseequations.5. !
"#$ −&" =
!(" 6. !")("#& =
"#*")!
7.+ + !-")$ =
."")$ − 2 8. ""#( −
$")! =
)."0"0#")*
47
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.7
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
5.7 We All Scream for Ice Cream – Teacher Notes A Practice Understanding Task Purpose:Thepurposeofthistaskistwo-fold:tomodelasituationusingrationalfunctions,andtosolveequationsthatcontainrationalexpressions.Studentswillwritearationalequationtoanswer
aquestionaboutcombinedrates.Theywilldevelopstrategiestosolverationalequationsusing
theirpreviousworkwithrationalexpressions.
CoreStandardsFocus:A.REI.A.2Solvesimplerationalandradicalequationsinonevariable,andgiveexamplesshowinghowextraneoussolutionsmayarise.A.SSE.3Chooseandproduceanequivalentformofanexpressiontorevealandexplainpropertiesofthequantityrepresentedbytheexpression.
StandardsforMathematicalPractice:
SMP1–Makesenseofproblemsandpersevereinsolvingthem
SMP4–Modelwithmathematics
Vocabulary:Extraneoussolution
TheTeachingCycle:
Launch(WholeGroup):
Beginthetaskbyreadingthescenarioandensuringstudentsunderstandthesituation.Askstudentstocompletequestions#1and#2.Thenaskstudentswhichexpressionisgreater:*-" or
*-")&?Sharejustificationsthatrelatetotheproblemsituation,suchas,weknowthateach
personwillpaymoreiffewerpeoplesharethecost.Pressstudentstojustifytheiranswerfromastrictlynumericperspectivealso.Thiswillhelppreparethemtowritetheequationtheywillneed
fortheproblem.
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.7
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Explore(Individual,FollowedbySmallGroup):
Monitorstudentsastheyareworkingtoseethattheyareusingtheanswerstothefirsttwoquestionstohelpwriteanequation.Theymayneedsupporttothinkabouthowtousethe
additional$2intheequation.Thereareseveralpossibleapproaches.Somestudentsmaysubtractthetwoalgebraicexpressionsandsaythedifferenceis$2.Somemaysaythattheoriginalamount
plus$2isequaltotheamountpaidwithfewerpeopleinthegroup.Takenoteofthepossible
approachesforwritingtheequationandforsolvingtheequationsothattheycanbesharedduringthediscussion.
Oncestudentshaveanequation,watchforvariousmethodsforsolvingtheequation.Moststudents
willthinkaboutfindingacommondenominatorandcombiningthetwoexpressions.Findastudent
tosharethatcanarticulatethisstrategyfortheclass.Itislikelythatthistaskwillproduceanumberofmisconceptionsthatshouldbeanalyzedbytheclass.Ifyourclassisasafeenvironment
forsharingmistakes,thenaskastudenttoshare.Otherwise,bepreparedtosharecommonmisconceptionswithoutidentifyingthestudentssothatthemisconceptionscanbeaddressed.
Discuss(WholeGroup):
Beginbyaskingastudenttoexplaintheequationhe/shewrote.Mostofthepossibleequationslead
tosimilarsteps,anditwillbeusefulfortheclasstoseethis.Besurethestudentthatsharesexplainshowhe/shegoteachrationalexpressionandcombinedthemtoformtheequation.Next,
askastudenttosharethathasfoundacommondenominatorandcombinedtermsbeforetaking
thereciprocalofbothsides(orusingcross-products).Askstudentstoexplainwhyitis“legal”totakethereciprocalofbothsidesoftheequationortousecross-products.Besuretohavestrong
mathematicaljustificationofthisstep.Atthispoint,askastudenttosharewhotookthereciprocalofeachtermwithoutcombiningterms(oranyothercommonmisconceptionthatmayhave
occurred).(Ifyouarenotcomfortablewithstudentssharingerrors,thentelltheclassthatthiswas
astrategyyouhaveseenandpresentityourself.)Sharetheentireprocessandasktheclasswhytheanswerdoesn’tmakesense.Askwhythisstrategyforsolvingtheequationdoesnotworkandis
notthesameascombiningtermsandthentakingthereciprocalofbothsidesorusingcrossproducts.
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS -5.7
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Iftheclasshasotherproductivestrategiesforsolvingtheequation,suchasmultiplyingeverythingintheequationbythecommondenominatortogetridofthedenominators,youmaychooseto
sharethemtoo.Becarefulofthisstrategybecauseitoftenleadstotheideathatmultiplyingbyacommondenominatorjustmakesthedenominatorsgoawayinanyexpression.Askforstrong
justificationofwhythisworksinanequationbutisnotappropriatewhensimplyaddingtwoterms.
Useanumericexampletoreinforcetheidea.
Next,askstudentstosharetheirworkontheremainingequations.Theyareeachalittledifferent,soitwillbeusefultoshareasmanyastimeallows.Besurethateither#7or#8isdiscussed,sothat
extraneoussolutionscanbeconsidered.
AlignedReady,Set,Go:RationalExpressionsandFunctions5.7
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS – 5.7
5.7
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Needhelp?Visitwww.rsgsupport.org
READY Topic:Calculatingvolume,surfaceareaandsolvingrighttriangles
Findtheindicatedvaluesforthegeometricfiguresbelow.1.Volume:SurfaceArea:
rectangularprism
2.Volume:! = #
$%ℎ
SurfaceArea:'( = )*(, + *)/ℎ0*0,234ℎ0,540*5,ℎ026ℎ4.
rightcone
3.Solvetherighttriangle.∠9 =
:;<<<< =
9;<<<< =
4.Solvetherighttriangle.∠9 =
∠: =
:;<<<< =
5.Volume:! = =
$)*$
Surfacearea:'( = 4)*?* ≈ 3959D2,03sphereThisistheradiusoftheearth.
6.Volume:! = 5?E
$
Surfacearea:'( = 5? + 25GHI
=+ ℎ?
ℎ = 147D5 = 230DrightsquarepyramidThesearethedimensionsofthegreatpyramidofGiza.
READY, SET, GO! Name PeriodDate
2
0
m
48
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS – 5.7
5.7
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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SET Topic:Solvingrationalequations
Solveeachequation.Identifyextraneoussolutions.7.M + ?
N= 3 8.N
?−
#
$N=
#
P 9.2M + $
NQ?= 1
10. ?
NIR?N−
#
NR?= 1 11.3M − #
?NR#= 4 12. ?N
NIQ$N−
?
NQ$=
?
N
Topic:Usingworkandraterelationshipstosolveproblems
13.ChanningtakestwiceaslongasDakotatocompleteaschoolproject.Ittakesthem15hourstocompletetheprojecttogether.Howlongwouldittakeeachstudenttocompletetheprojectifheworksalone?
14.AprintshopcanprinttheMVPmathbookin24minutesifbothoftheirprintmachinesare
workingtogethertodothejob.Ifaprintmachineisworkingalone,thejobtakeslonger.MachineAcanprintthebook20minutesfasterthanmachineB.Howlongdoesittakeeachmachinetoprintthebook?
49
ALGEBRA II // MODULE 5
RATIONAL EXPRESSIONS & FUNCTIONS – 5.7
5.7
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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15.Theproblemin#14generatesanextraneoussolution,eventhoughneithersolutionmakesadenominatorequalzero.Whatisanotherreasonforhavinganextraneoussolution?
GO
Topic:Simplifyingrationalexpressions(10–11)Reducetosimplestform.(12–15)Performtheindicatedoperations.Reduceeachofyouranswerstoitssimplestform.(Assumealldenominators≠0)16. 17. 18.
M? + 8M + 12
M? + 3M − 18
M? − 3M − 40
M? − 11M + 24
M? + 8M + 12
M? + 3M − 18+M? − 3M − 40
M? − 11M + 24
19.
20. 21.
M? + 5M − 36
(M − 4) ∙ 3(M + 2)
M + 9
4
M? − 4−
1
M − 2 M? − 2M − 3
M + 1÷M − 3
5
50