1.1 brutus bites back - utah education network · 2018-07-12 · 1.1 brutus bites back – teacher...
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SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1.1 Brutus Bites Back
A Develop Understanding Task
RememberCarlosandClarita?Acoupleofyearsago,theystartedearningmoneybytakingcareofpetswhiletheirownersareaway.Duetotheiramazingmathematicalanalysisandtheirlovingcareofthecatsanddogsthattheytakein,CarlosandClaritahavemadetheirbusinessverysuccessful.Tokeepthehungrydogsfed,theymustregularlybuyBrutusBites,thefavoritefoodofallthedogs.CarlosandClaritahavebeensearchingforanewdogfoodsupplierandhaveidentifiedtwopossibilities.TheCanineCateringCompany,locatedintheirtown,sells7poundsoffoodfor$5.Carlosthoughtabouthowmuchtheywouldpayforagivenamountoffoodanddrewthisgraph:
1. WritetheequationofthefunctionthatCarlosgraphed.
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SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Claritathoughtabouthowmuchfoodtheycouldbuyforagivenamountofmoneyanddrewthisgraph:
2.WritetheequationofthefunctionthatClaritagraphed.
3. WriteaquestionthatwouldbemosteasilyansweredbyCarlos’graph.WriteaquestionthatwouldbemosteasilyansweredbyClarita’sgraph.Whatisthedifferencebetweenthetwoquestions?
4. Whatistherelationshipbetweenthetwofunctions?Howdoyouknow?
5.Usefunctionnotationtowritetherelationshipbetweenthefunctions.
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SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Lookingonline,Carlosfoundacompanythatwillsell8poundsofBrutusBitesfor$6plusaflat$5shippingchargeforeachorder.Thecompanyadvertisesthattheywillsellanyamountoffoodatthesamepriceperpound.
6. ModeltherelationshipbetweenthepriceandtheamountoffoodusingCarlos’approach.
7. ModeltherelationshipbetweenthepriceandtheamountoffoodusingClarita’sapproach.
8. Whatistherelationshipbetweenthesetwofunctions?Howdoyouknow?
9. Usefunctionnotationtowritetherelationshipbetweenthefunctions.
10. WhichcompanyshouldClaritaandCarlosbuytheirBrutusBitesfrom?Why?
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SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1.1 Brutus Bites Back – Teacher Notes A Develop Understanding Task
Purpose:Thepurposeofthistaskistodevelopandextendtheconceptofinversefunctionsina
linearcontext.Inthetask,studentsusetables,graphs,andequationstorepresentinversefunctions
astwodifferentwaysofmodelingthesamesituation.Therepresentationsexposetheideathatthe
domainofthefunctionistherangeoftheinverse(andviceversa)forsuitablyrestricteddomains.
Studentsmayalsonoticethatthegraphsoftheinversefunctionsarereflectionsoverthe! = $line,butwiththeunderstandingthattheaxesarepartofthereflection.
CoreStandardsFocus:
F.BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.F.BF.4Findinversefunctions.
a. Solveanequationoftheform%($) = (forasimplefunctionfthathasaninverseandwriteanexpressionfortheinverse.
RelatedStandards:
F-BF.4.b,F.BF.4c
StandardsforMathematicalPractice:
SMP1–Makesenseofproblemsandpersevereinsolvingthem
SMP7–Lookforandmakeuseofstructure
Vocabulary:Inversefunction
TheTeachingCycle:
Launch(WholeClass):Beginthetaskbyensuringthatstudentsunderstandthecontext.Ifthey
havenotdonethe“PetSitters”unitfromSecondaryMathematicsI,theymaynotbefamiliarwith
thestoryofCarlosandClarita,twostudentsthatstartedabusinesscaringforpets.Theimportant
partofthecontextissimplythattheyareplanningtobuydogfoodthatispricedbythepound.Ask
studentstobeginbyworkingindividuallyonquestions1,2,and3.Circulatearoundtheroom
lookingathowstudentsareworkingwiththegraphs.Identifystudentstopresenttotheclassthat
haveconstructedatableofvaluestoaidinwritingtheequation.Alsowatchtoseethevariables
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
thatstudentsareusingastheywriteequations.Iftheyareusing$astheindependentvariableforbothequations,thentheyhavenotproperlydefinedthevariable.Ifthisisacommonerror,you
maychoosetodiscussitwiththeclass.
Oncestudentshavewrittentheirequations,askastudenttopresentatablefor#1.Identifythe
differencebetweentermsinthetableandthenhaveastudentpresentanequationthatuses
variableslikep(pounds)andd(dollars)andwritesanequationfordollarsasafunctionofpounds.
Thenhaveastudentpresentatableofvaluesfor#2.Asktheclasstocomparethetablefor#1to
thetablefor#2.Theyshouldnoticethatthepanddcolumnhavebeenswitched,makingdthe
independentvariableandpthedependentvariable.Theyshouldalsonoticethechangeinthe
differencebetweentermsonthetwotables.Askforastudenttopresenttheequationwrittenfor
#2.Then,discussquestion#3.BesurethatstudentsunderstandthatCarlosthinksabouthow
muchitwillcostforacertainnumberofpounds,whileClaritathinksabouthowmuchfoodshecan
buywithacertainamountofmoney.
Asktheclasstorespondtoquestion#4.Theyshouldhavesomeexperiencewithinversefunctions
fromSecondaryMathematicsII,sotheyshouldrecognizetheminthiscontext.Modelthenotation
for#5as:)(*) = +,-./0+(0) =),-.Askstudentsforthedomainandrangeofeachfunction.
Explore(SmallGroup):Atthispoint,letstudentsworktogetherontheremainingproblems.As
youmonitorstudentsastheywork,encouragetheuseoftablestosupporttheirwritingof
equations.
Discuss(WholeClass):Proceedwiththediscussionmuchlikethepreviousdiscussionduringthe
launch.Presentthetable,graph,andequationfor#6andthenthesameprocessfor#7.Besure
thatstudentscanrelatetheirequationsbacktothecontext.Forinstance,besurethattheycan
identifywhythe5isaddedorsubtractedintheequations.Helpstudentstoseehowthestructure
oftheequationofthefunctionisreversedintheinversefunctiononestepatatime.Tiethiswork
tothestorycontexttogiveitmeaning.Again,askstudentstocomparethedomainandrangefor
thefunctions.Besurethattheynoticethatthedomainofthefunctionistherangeoftheinverse
andviceversa.
SECONDARY MATH III // MODULE 1
FUNCTIONS AND THEIR INVERSES – 1.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Note:Problem#10isanextensionproblemforstudentsthatmaybeaheadoftheclassin
completingthetask.Itisintendedtogivestudentsanopportunityforadditionalanalysis,
butisnotcriticalforthedevelopmentoftheideasofinverseinthismodule.
AlignedReady,Set,Go:FunctionsandTheirInverses1.1
SECONDARY MATH III // MODULE 1
FUNCTIONS AND INVERSES –
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1.1
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READY Topic:InverseoperationsInverseoperations“undo”eachother.Forinstance,additionandsubtractionareinverseoperations.
Soaremultiplicationanddivision.Inmathematics,itisoftenconvenienttoundoseveraloperations
inordertosolveforavariable.
Solveforxinthefollowingproblems.Thencompletethestatementbyidentifyingthe
operationyouusedto“undo”theequation.
1.24=3x Undomultiplicationby3by_______________________________________________
2. Undodivisionby5by______________________________________________________
3. Undoadd17by_____________________________________________________________
4. Undothesquarerootby___________________________________________________
5. Undothecuberootby_____________________________then___________________
6. Undoraisingxtothe4thpowerby________________________________________
7. Undosquaringby_______________________________then______________________
SET Topic:Linearfunctionsandtheirinverses
CarlosandClaritahaveapetsittingbusiness.Whentheyweretryingtodecidehowmanyeachof
dogsandcatstheycouldfitintotheiryard,theymadeatablebasedonthefollowinginformation.
Catpensrequire6ft2ofspace,whiledogrunsrequire24ft2.CarlosandClaritahaveupto360ft2
availableinthestorageshedforpensandruns,whilestillleavingenoughroomtomovearoundthe
cages.Theymadeatableofallofthecombinationsofcatsanddogstheycouldusetofillthespace.
Theyquicklyrealizedthattheycouldfitin4catsinthesamespaceasonedog.
cats 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60
dogs 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
x5= −2
x +17 = 20
x = 6
x +1( )3 = 2
x4 = 81
x − 9( )2 = 49
READY, SET, GO! Name PeriodDate
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SECONDARY MATH III // MODULE 1
FUNCTIONS AND INVERSES –
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1.1
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8.Usetheinformationinthetabletowrite5orderedpairsthathavecatsastheinputvalue
anddogsastheoutputvalue.
9.Writeanexplicitequationthatshowshowmanydogstheycanaccommodatebasedonhow
manycatstheyhave.(Thenumberofdogs“d”willbeafunctionofthenumberofcats“c”or
! = #(%).)
10.Usetheinformationinthetabletowrite5orderedpairsthathavedogsastheinputvalue
andcatsastheoutputvalue.
11.Writeanexplicitequationthatshowshowmanycatstheycanaccommodatebasedonhow
manydogstheyhave.(Thenumberofcats“c”willbeafunctionofthenumberofdogs“d”
or% = '(!).)
Baseyouranswersin#12and#13onthetableatthetopofthepage.
12.Lookbackatproblem8andproblem10.Describehowtheorderedpairsaredifferent.
13.a)Lookbackattheequationyouwroteinproblem9.Describethedomainfor! = #(%).
b)Describethedomainfortheequation% = '(!)thatyouwroteinproblem11.
c)Whatistherelationshipbetweenthem?
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SECONDARY MATH III // MODULE 1
FUNCTIONS AND INVERSES –
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1.1
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GO Topic:Usingfunctionnotationtoevaluateafunction.
Thefunctions aredefinedbelow.
Calculatetheindicatedfunctionvaluesinthefollowingproblems.Simplifyyouranswers.
14.
15. 16. 17.
18.
19. 20. 20.'() + +)
22.
23. 24. 25.
f x( ), g x( ), and h x( )f x( ) = x g x( ) = 5x −12 h x( ) = x2 + 4x − 7
f 10( ) f −2( ) f a( ) f a + b( )
g 10( ) g −2( ) g a( )
h 10( ) h −2( ) h a( ) h a + b( )
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