Rational Expressions & Functions - Mathematics Vision Project€¦ · RATIONAL EXPRESSIONS & FUNCTIONS Mathematics Vision Project Licensed under the Creative Commons Attribution CC

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 4

Rational Expressions & Functions

SECONDARY

MATH THREE

An Integrated Approach

Standard Teacher Notes

SECONDARY MATH III // MODULE 4

RATIONAL EXPRESSIONS & FUNCTIONS

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

MODULE 4 - TABLE OF CONTENTS

RATIONAL EXPRESSIONS AND FUNCTIONS

4.1 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 4.1

4.2 Shift and Stretch – A Solidify Understanding Task

Applying transformations to the graph of !(#) = &' . (F.BF.3, F.IF.7d, A.CED.2)


4.3 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)


4.4 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)


4.5 Just Act Rational – A Solidify Understanding Task

Adding, subtracting, multiplying, and dividing rational expressions. (A.APR.7, A.SSE.3)



RATIONAL EXPRESSIONS & FUNCTIONS


4.6 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)


4.7 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)



RATIONAL EXPRESSIONS & FUNCTIONS -4.1


4.1 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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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.

2




9.Whatdoyounoticewhenyoucomparethetwomodelsthatyouhavewritten?

Now,let’sputitalltogethertographtheentirefunction,!(#) = &'.

10.Createatablefor!(#) = &'thatincludesnegativeinputvalues.

11.Howdothevaluesof!(#)intheintervalfrom(−∞, 0)comparetothevaluesof!(#)from(0,∞)?Usethiscomparisontopredictthegraphof!(#) = &

'.

3




12.Graph!(#) = &'.

13.Describethefeaturesof!(#) = &',includingdomain,range,intervalsofincreaseordecrease,#-

and=-intercepts,endbehavior,andanymaximum(s)orminimum(s).

4




4.1 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




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.




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




theequationis!(#) = &'inbothsituations.Emphasizethatthetwodifferentwaysofthinkingabout

thestorycontextarepresentedheresothattheycanseethesimilaritiesintheverticaland

horizontalasymptotes.

Askastudenttopresenthis/herworkonquestion10andthenaskanotherstudenttopresenta

graphoftheentirefunction.AsktheclasswhattherelationshipisbetweenthegraphinQuadrantI

andthegraphinQuadrantIII.Theyshouldrecognizethatitisarotationof180degreesaroundthe

origin,so!(#) = &'isanoddfunction.Finally,discussthefeaturesof!(#) =

&'.Besurethat

studentscanusepropernotationtodescribethedomainandrange,aswellasrecognizingthatthe

functionisdecreasingthroughtheentiredomain.

AlignedReady,Set,Go:RationalExpressionsandFunctions4.1


RATIONAL EXPRESSIONS & FUNCTIONS – 4.1


4.1

Needhelp?Visitwww.rsgsupport.org

READY Topic:Recallingtransformationsonquadraticfunctions

Describethetransformationofeachfunction.Thenwritetheequationinvertexform.1.Description:Equation:

2.Description:Equation:





READY, SET, GO! Name PeriodDate

5




4.1


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?

6




4.1


GO Topic:Reviewingthehorizontalasymptoteinanexponentialfunction

Allexponentialfunctionshaveahorizontalasymptote.Allofthegraphsbelowshowexponentialfunctions.

Matchthefunctionrulewiththecorrectgraph.Thenwritetheequationofthehorizontalasymptote.

11.!(#) = 2) Equationofhorizontalasymptote:

12.+(#) = 2) − 3Equationofhorizontalasymptote:

13.ℎ(#) = 2)/&Equationofhorizontalasymptote:

14.0(#) = −(2)) − 3Equationofhorizontalasymptote:

15.1(#) = 2(/)) + 3Equationofhorizontalasymptote:

16.3(#) = −2(/))Equationofhorizontalasymptote:

a.

b. c.

d.

e. f.

17.Use!(#) = 45()/6) + 7toexplainwhichvaluesaffectthepositionofthehorizontalasymptoteinanexponentialfunction.Beprecise.

18.Whydoesanexponentialfunctionhaveahorizontalasymptote?

7




4.2 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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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:

9





- = 0andahorizontalasymptoteat! = 0.Itcontainsthepoints(1,2),(-1,-2),(2,1),(-2,1),

31 26 , 48and3−1 26 ,−48.

Equation:

7.

Equation:

8.

Equation:


- = −6andahorizontalasymptoteat! = −3.Itcrossesthex-axisat−6 #1.Itcontainsthepoints(-5,-4)and(-7,-2).

Equation:

10




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:

11




4.2 Shift and Stretch – Teacher Notes A Solidify Understanding Task

Purpose:

Thepurposeofthistaskistoextendstudents’understandingofrationalfunctionsbyapplying

transformationstothegraphofD(-) = #$.Thetaskhelpsstudentstofocusontheverticaland

horizontalasymptotesandhowtheycanbeusedtoeasilygraphsimplerationalfunctions.This

taskpreparesstudentsformorecomplicatedrationalfunctionsinupcomingtasks.

CoreStandardsFocus:


simplecasesandusingtechnologyformorecomplicatedcases.


showingendbehavior.

F.BF.B.3Identifytheeffectonthegraphofreplacingf(x)byf(x)+k,kf(x),f(kx),andf(x+k)for

specificvaluesofk(bothpositiveandnegative);findthevalueofkgiventhegraphs.Experiment

withcasesandillustrateanexplanationoftheeffectsonthegraphusingtechnology.Include

recognizingevenandoddfunctionsfromtheirgraphsandalgebraicexpressionsforthem.





SMP5–Useappropriatetoolsstrategically

SMP8–Lookforandexpressregularityinrepeatedreasoning




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.




Afterdiscussingtheexamples,helpstudentstogeneralizebyworkingproblem#10andbeingsure

thateachstudentknowstheverticalandhorizontalshifts,alongwiththeverticalstretches.Then,

askapreviouslyselectedstudenttodemonstratehowhe/shegraphedeitheroftheproblemson

#11.





4.2


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:


12




4.2


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.

13




4.2


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.Look

backattherootsandtheconstanttermineachproblem.Makeastatementaboutanythingyounotice.

14




4.3 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




Intoday’stask,youaregoingtolookforpatternsintheformssothatyoucancompletethefollowingchart: Howtofindthe

verticalasymptote:Howtofindthehorizontalasymptote:

Howtofindtheintercepts:

Degreeofthenumerator<Degreeofthedenominator

Degreeofthenumerator=Degreeofthedenominator

Youaregivenseveraldifferentrationalfunctions.Startbyidentifyingthedegreeofthenumeratoranddenominatorandusingtechnologytographthefunction.Asyouareworking,lookforpatternsthatwillhelpyoucompletethetable.Youneedtofindaquickwaytoidentifythehorizontalandverticalasymptoteswhenyouseetheequationofarationalfunction,aswellasnoticingotherpatternsthatwillhelpyouanalyzeandgraphthefunctionquickly.Thelasttwographsaretheresoyoucanexperimentwithyourownrationalfunctionsandtestyourtheories.

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1.0 = '1-'2/

2.0 = '('1/)('2-)

Deg.ofNum._____Deg.ofDenom._____HorizontalAsymptote___________VerticalAsymptote___________Intercepts___________________________________


3.0 = &('13)4

4.0 = -'2&'13



16




5.0 = '42-'('2/)('1&)('1-)

6.0 = /'41-(/'15)(/'25)



7.YourOwnRationalFunction:

8.YourOwnRationalFunction:



9.Nowthatyouhavetriedsomeexamples,it’stimetodrawsomeconclusionsandcompletethetworowsofthetableonthefirstpage.Bepreparedtodiscussyourconclusionsandwhyyouthinkthattheyarecorrect.

17




4.3 Rational Thinking – Teacher Notes A Solidify Understanding Task Purpose:Thepurposeofthistaskistointroducetheterm“rationalfunction”andtouseseveral

examplesofrationalfunctionstodiscovertherelationshipbetweenthedegreeofthenumerator

anddenominatorandthehorizontalasymptotes.Studentswillalsofindverticalasymptotesand

interceptsforrationalfunctionsandusethisinformationtocompleteagraphicorganizer.Graphing

technologyisusedasanimportanttoolforexploringrationalfunctionsinthistask.

CoreStandardsFocus:


simplecasesandusingtechnologyformorecomplicatedcases.


showingendbehavior.




F.IF.5Relatethedomainofafunctiontoitsgraphand,whereapplicable,tothequantitative

relationshipitdescribes.Forexample,ifthefunctionh(n)givesthenumberofpersonhoursittakesto

assemblenenginesinafactory,thenthepositiveintegerswouldbeanappropriatedomainforthe

function.

Related:F.IF.6,A.CED.3


SMP3–Constructviableargumentsandcritiquethereasoningofothers

SMP7–Lookforandmakeuseofstructure

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.


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.





4.3


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

Fillinthespecifiedfeaturesofeachrationalfunction.Sketchtheasymptotesonthegraphandmarkthelocationoftheintercepts.


18




4.3


10./ = 12(14()(16*)

Degreeofnum.__________Degreeofdenom.__________

Equationofhorizontalasymptote:

Equationofverticalasymptote(s):

y-intercept:(writeasapoint)

x-intercept(s):(writeaspoints)

11./ = (16()14&






12./ = "'1(14&)2






13./ = (14")(14))(16!)






19




4.3


GO Topic:Reducingfractions

Reducethefollowingfractionstolowestform.Thenexplainthemathematicsthatmakesitpossibletorewritethefractioninitsnewform.(Improperfractionsshouldnotbewrittenasmixednumbers.)Ifafractioncan’tbereduced,explainwhy.14.

")"!Explanation:

15.)(""Explanation:

17.!""#Explanation:

18.("&Explanation:

19.""*)# Explanation:

20.6"*,!)."*,!). Explanation:

20




4.4 Are You Rational?


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?

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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




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




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




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




4.4 Are You Rational? – Teacher Notes A Solidify Understanding Task Purpose:

Inthistask,rationalfunctionsareformallydefinedandconnectedtorationalnumbers.Students

willusetheseconnectionstoreducerationalfunctionsandtoidentifyrationalfunctionsthatare

improperandwritetheminanequivalentform.Studentswillgraphrationalfunctionsthatcanbe

reduced,seeingthatthegraphisequivalenttothereducedfunctionexceptthatthereisaholein

thegraph,producedbythefactorthatisreduced.Theywillalsographimproperrationalfunctions

andidentifytheslantasymptote.

CoreStandardsFocus:




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.





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.




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.




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.





4.4


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


26




4.4


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




4.4


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




4.5 Just Act Rational


In4.4AreYouRational?,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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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




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.

31




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




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




4.5 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.


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,




l(lmh)(lno)

∙ jlp.Iwillmultiplythenumeratorstogetherandthedenominatortogetherandmake

asinglefraction:jl(lmh)lp(lno)

.”

Stopthereandtellstudentsthattheywillbeusingthesamestrategythroughoutthetask.They

shoulddecidewhatwasdonewiththenumbers,labelit,andthendothesamethingwiththe

rationalexpressions.Tellstudentsthattheanswerstotheproblemsareprovidedattheendsothat

theycanmakesurethattheirprocessiscorrect.Iftheygettotheendoftheproblemandtheir

workdoesn’tleadtotheanswergiven,theyneedtogobackandfindtheirmistake.


Monitorstudentsastheywork,helpingthemtoseeandlabeleachstepoftheoperations.Expect

manymistakesasstudentsworkwiththerationalexpressions.Watchforstudentsthatnoticetheir

mistakesandfindwaystocorrectthemsothattheycandescribetheirprocessfortheclass.Some

ofthecommonerrorstowatchforsothattheycanbeaddressedduringthediscussion:

• Improperlyreducingterms,notfactors

• Notusingparenthesesandmakingmistakesindistributing(usuallywhenchangingto

commondenominator).

• Signerrorsinsubtraction.


Beginthediscussionwithastudentsharinghis/herworkonRationalExpressionBin

multiplication.Askthestudenttoexplainhowtheanswerwasreduced,andthenasktheclassifit

makesadifferenceinwhichstepthefractionisreduced.Makesurethediscussionincludestheidea

thatreducingisdividingby1,soitcanhappenineitherstepsincemultiplicationanddivisionareat

thesamelevelinorderofoperations.DiscusswhyitiscorrecttodivideC − 2fromthenumerator

anddenominatorinthiscasetosimplifythefraction.

Next,askastudenttoexplainhowhis/herworkonproblem3,RationalExpressionBindivision.

Askthestudenttoexplaintheprocessfordivision,andthenasktheclasswhythedenominatorof




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.





4.5


READY Topic:Recallingtrigonometricfunctions

Usethegiventriangletowritethevaluesof!"#$, &'!$, (#)+(#$and!"#,, &'!,, (#)+(#,.

1.

sinA=cosA=tanA=sinB=cosB=tanB=

2.

sinA=cosA=tanA=

√2cmsinB=cosB=tanB=

3.


4.



7√2m

34




4.5


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




4.5


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




4.6 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.

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1.

" = (% − 5)(% + 1)(% + 2)(% − 2)

VerticalAsymptote(s)______________________

HorizontalorSlantAsymptote_________________

Intercepts_______________________________________

Graph:

2.

" = (% − 3)(% + 1) ∙

%(% − 4)



Intercepts_______________________________________

Graph:

38




3.

" = %/ − 6% + 2(% − 2)



Intercepts_______________________________________

Graph:

4.

" = 4(% + 1) +

(% − 5)(% − 3)



Intercepts_______________________________________

Graph:

39




5.

" = 3%(%/ + 2% + 1) ÷

% − 3% + 1



Intercepts_______________________________________

Graph:

6.

" = % + 5% + 4 −

% + 2% − 1



Intercepts_______________________________________

Graph:

40




7.

" = 2%/ + % − 15%/ + 4% + 3



Intercepts_______________________________________

Graph:

8.Summarizeyourstrategyforgraphingrationalfunctionswithastep-by-stepprocess.

41




4.6 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.*




showingendbehavior.

Related:F.IF.5,F.IF.9,A.CED.1


SMP3–Constructviableargumentsandcritiquethereasoningofothers

SMP7–Lookforandmakeuseofstructure

TheTeachingCycle:

Launch(WholeGroup):

StartthetaskbyreadingthedialoguebetweenJosueandFranciaatthebeginningofthetask.Ask

theclasswhatinformationtheycandetermineaboutanyrationalfunctioniftheyjustlookatthe




equation.Theyshouldbeabletodescribehowtodetermineifitwillbeatransformationof

" = 1/%andhowtofindtheasymptotesandintercepts.AskwhyitwouldbehelpfultofindthebehaviorneartheasymptotesandwhyFranciamightthinkthatshedoesn’tneedexactvalues.

Ideally,astudentwillsaythatifoneknowsthefunctionispositivenearaverticalasymptote,then

oneknowsthatthegraphisapproachinginfinityonthatsideoftheasymptote.

Describetheexpectationsforthetask,specifically,thatstudentsfindaquickwaytodeterminethe

behaviorneartheasymptotesanduseittodevelopastrategyforgraphinganyrationalfunction.

Tellstudentsthatsomeofthefunctionsarenotgiveninsimplifiedform,sotheymayneedto

combinerationalexpressionsbeforegraphingthefunction.Stressthattheyshouldnotuse

technologytographafunctionuntiltheyarereadytochecktheirwork.Inthisway,theycantrya

techniqueandthenmodifyitifitisnotproducingacorrectgraph.


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.




Encouragestudentstocheckthattheyhaveperformedtheoperationscorrectlybyusingtechnology

tochecktheirwork.


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.”




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.





HowtoGraphaRationalFunctionSteps: Example: Remember

this:

" = % + 3(% − 1)(% + 2)




4.6


READY Topic:Identifyingextraneoussolutions1.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


42




4.6


SET Topic:Predictingandsketchingrationalfunctions

Findtheverticalasymptote(s),horizontalorslantasymptote,andintercepts.Thensketchthegraph.(Donotusetechnologytogetthegraph.Themaxandminsdonotneedtobeaccurate.)5.

? =(< + 4)(−2< − 6)



Intercepts_______________________________________

Graph:

6.

? =3<

(< − 3)∙(< − 4)(< + 1)



Intercepts_______________________________________

Graph:

43




4.6


7.

? =(<4 − 4<)(4< − 8)

÷(< + 2)< + 4



Intercepts_______________________________________

Graph:

8.

? =(< − 6)(< − 3)

+(< + 3)

<4 − 6< + 9



Intercepts_______________________________________

Graph:

44




4.6


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




4.7 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?

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4.Explainwhyyouranswer(s)makessenseinthissituation.

Thisstoryandtheproblemitrepresentsprovidesanopportunitytomodelasituationthatrequiresarationalequation.Rationalequationscantakemanyforms,buttheyaresolvedusingprincipleswehaveworkedwithbefore.Tryapplyingsomeofthestrategiesforworkingwithrationalexpressionsthatwehaveusedinthismoduletosolvetheseequations.5. !

"#$ −&" =

!(" 6. !")("#& =

"#*")!

7.+ + !-")$ =

."")$ − 2 8. ""#( −

$")! =

)."0"0#")*

47




4.7 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.


SMP1–Makesenseofproblemsandpersevereinsolvingthem

SMP4–ModelwithmathematicsVocabulary:Extraneoussolution

TheTeachingCycle:

Launch(WholeGroup):

Beginthetaskbyreadingthescenarioandensuringstudentsunderstandthesituation.Askstudentstocompletequestions#1and#2.Thenaskstudentswhichexpressionisgreater:*-" or

*-")&?Sharejustificationsthatrelatetotheproblemsituation,suchas,weknowthateach

personwillpaymoreiffewerpeoplesharethecost.Pressstudentstojustifytheiranswerfromastrictlynumericperspectivealso.Thiswillhelppreparethemtowritetheequationtheywillneed

fortheproblem.





Monitorstudentsastheyareworkingtoseethattheyareusingtheanswerstothefirsttwoquestionstohelpwriteanequation.Theymayneedsupporttothinkabouthowtousethe

additional$2intheequation.Thereareseveralpossibleapproaches.Somestudentsmaysubtractthetwoalgebraicexpressionsandsaythedifferenceis$2.Somemaysaythattheoriginalamount

plus$2isequaltotheamountpaidwithfewerpeopleinthegroup.Takenoteofthepossible

approachesforwritingtheequationandforsolvingtheequationsothattheycanbesharedduringthediscussion.

Oncestudentshaveanequation,watchforvariousmethodsforsolvingtheequation.Moststudents

willthinkaboutfindingacommondenominatorandcombiningthetwoexpressions.Findastudent

tosharethatcanarticulatethisstrategyfortheclass.Itislikelythatthistaskwillproduceanumberofmisconceptionsthatshouldbeanalyzedbytheclass.Ifyourclassisasafeenvironment

forsharingmistakes,thenaskastudenttoshare.Otherwise,bepreparedtosharecommonmisconceptionswithoutidentifyingthestudentssothatthemisconceptionscanbeaddressed.


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.




Iftheclasshasotherproductivestrategiesforsolvingtheequation,suchasmultiplyingeverythingintheequationbythecommondenominatortogetridofthedenominators,youmaychooseto

sharethemtoo.Becarefulofthisstrategybecauseitoftenleadstotheideathatmultiplyingbyacommondenominatorjustmakesthedenominatorsgoawayinanyexpression.Askforstrong

justificationofwhythisworksinanequationbutisnotappropriatewhensimplyaddingtwoterms.

Useanumericexampletoreinforcetheidea.

Next,askstudentstosharetheirworkontheremainingequations.Theyareeachalittledifferent,soitwillbeusefultoshareasmanyastimeallows.Besurethateither#7or#8isdiscussed,sothat

extraneoussolutionscanbeconsidered.





4.7


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.


2

0

m

48




4.7


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




4.7


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

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