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The Mathematics Vision Project
Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius
© 2016 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Off ice of Education
This work is licensed under the Creative Commons Attribution CC BY 4.0
MODULE 1
Sequences
SECONDARY
MATH ONE
An Integrated Approach
Standard Teacher Notes
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MODULE 1 - TABLE OF CONTENTS
SEQUENCES
1.1 Checkerboard Borders – A Develop Understanding Task
Defining quantities and interpreting expressions (N.Q.2, A.SSE.1)
READY, SET, GO Homework: Sequences 1.1
1.2 Growing Dots – A Develop Understanding Task
Representing arithmetic sequences with equations, tables, graphs, and story context (F.LE.1, F.LE.2,
F.LE.5)
READY, SET, GO Homework: Sequences 1.2
1.3 Growing, Growing Dots – A Solidify Understanding Task
Representing geometric sequences with equations, tables, graphs and story context (F.BF.1, F.LE.1a,
F.LE.1c, F.LE.2, F.LE.5)
READY, SET, GO Homework: Sequences 1.3
1.4 Scott’s Workout – A Solidify Understanding Task
Arithmetic Sequences: Constant difference between consecutive terms, initial values (F.BF.1,F.LE.1a,
F.LE.1c, F.lE.2, F.LE.5)
READY, SET, GO Homework: Sequences 1.4
1.5 Don’t Break the Chain – A Solidify Understanding Task
Geometric Sequences: Constant ration between consecutive terms, initial values (F.BF.1, F.LE.1a,
F.LE.1c, F.LE.2, F.LE.5)
READY, SET, GO Homework: Sequences 1.5
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1.6 Something to Chew On – A Solidify Understanding Task
Arithmetic Sequences: Increasing and decreasing at a constant rate (F.BF.1, F.LE.1a, F.LE.1b, F.LE.2,
F.LE.5)
READY, SET, GO Homework: Sequences 1.6
1.7 Chew on This! – A Solidify Understanding Task
Comparing rates of growth in arithmetic and geometric sequences (F.BF.1, F.LE.1, F.LE.2)
READY, SET, GO Homework: Sequences 1.7
1.8 What Comes Next? What Comes Later? – A Practice Understanding Task
Recursive and explicit equations for arithmetic and geometric sequences (F.BF.1, F.LE.1, F.LE.2)
READY, SET, GO Homework: Sequences 1.8
1.9 What Does it Mean? – A Solidify Understanding Task
Using rate of change to find missing terms in a an arithmetic sequence (A.REI.3)
READY, SET, GO Homework: Sequences 1.9
1.10 Geometric Meanies – A Solidify and Practice Understanding Task
Using a constant ratio to find missing terms in a geometric sequence (A.REI.3)
READY, SET, GO Homework: Sequences 1.10
1.11 I Know… What Do You Know? – A Practice Understanding Task
Developing fluency with geometric and arithmetic sequences (F.LE.2)
READY, SET, GO Homework: Sequences 1.11
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1.1 Checkerboard Borders
A Develop Understanding Task
Inpreparationforbacktoschool,theschooladministrationplanstoreplacethetileinthecafeteria.Theywouldliketohaveacheckerboardpatternoftilestworowswideasasurroundforthetablesandservingcarts.Belowisanexampleoftheboarderthattheadministrationisthinkingofusingtosurroundasquare5x5setoftiles.A. Findthenumberofcoloredtilesinthecheckerboardborder.Trackyourthinkingandfinda wayofcalculatingthenumberofcoloredtilesintheborderthatisquickandefficient.Be preparedtoshareyourstrategyandjustifyyourwork.
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B. Thecontractorthatwashiredtolaythetileinthecafeteriaistryingtogeneralizeawaytocalculatethenumberofcoloredtilesneededforacheckerboardbordersurroundingasquareoftileswithanydimensions.Torepresentthisgeneralsituation,thecontractorstartedsketchingthesquarebelow.
FindanexpressionforthenumberofcoloredbordertilesneededforanyNxNsquare center.
N
N
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1 . 1 Checkerboard Borders – Teacher Notes A Develop Understanding Task
Purpose:
Thefocusofthistaskisonthegenerationofmultipleexpressionsthatconnectwiththevisuals
providedforthecheckerboardborders.Theseexpressionswillalsoprovideopportunitytodiscuss
equivalentexpressionsandreviewtheskillsstudentshavepreviouslylearnedaboutsimplifying
expressionsandusingvariables.
CoreStandardsFocus:
N.Q.2Defineappropriatequantitiesforthepurposeofdescriptivemodeling.
A.SSE.1Interpretexpressionsthatrepresentaquantityintermsofitscontext.�
a.Interpretpartsofanexpression,suchasterms,factors,andcoefficients.
b.Interpretcomplicatedexpressionsbyviewingoneormoreoftheirpartsasasingleentity.
RelatedStandards:A.CED.2,A.REI.1
StandardsforMathematicalPracticeofFocusintheTask:
SMP1–Makesenseofproblemsandpersevereinsolvingthem.
SMP7–Lookforandmakeuseofstructure
TheTeachingCycle:
Launch(WholeClass):
Afterreadinganddiscussingthe“CheckerboardBorders”scenario,challengestudentstocomeup
withawaytoquicklycountthenumberofcoloredtilesintheborder.Havethemcreatenumeric
expressionsthatexemplifytheirprocessandrequirestudentstoconnecttheirthinkingtothe
visualrepresentationofthetiles.
Thefirstphaseofworkshouldbedoneindividually,allowingstudentsto“see”theproblemand
patternsinthetilesintheirownway.Thiswillprovideformorerepresentationstobeconsidered
later.AfterstudentsworkindividuallyforafewminutesonpartA,havethemsharewithapartner
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andbegintodevelopadditionalideasasapairorassisteachotheringeneralizingtheirstrategyfor
partB.
Explore(SmallGroup):
Forstudentswhodon’tknowwheretobegin,itmaybeusefultoasksomestarterquestionslike:
“Howmanytilesaretherealongoneside?”,“Howcanyoucountthetilesingroupsratherthanone-
by-one?”
Pressonstudentstoconnecttheirnumericrepresentationstothevisualrepresentation.Youmight
ask,“Howdoesthatfourinyournumbersentenceconnecttothevisualrepresentation?”Encourage
studentstomarkonthevisualortoredrawitsotheycandemonstratehowtheywerethinking
aboutthediagramnumerically.
Watchforstudentswhocalculatethenumberofbordertilesindifferentways.Makenoteoftheir
numericstrategiesandthedifferentgeneralizedexpressionsthatarecreated.Thediffering
strategiesandalgebraicexpressionswillbethefocusofthediscussionattheend,allowingfor
studentstoconnectbacktopriorworkfrompreviousmathematicalexperiencesandbetter
understandequivalencebetweenexpressionsandhowtoproperlysimplifyanalgebraic
expression.Promptstudentstocalculatethenumberoftilesforagivensidelengthusingtheir
expressionandthentodrawthevisualmodelandcheckforaccuracy.Requirestudentstojustify
whytheirexpressionwillworkforanysidelengthNoftheinnersquareregion.Pressthemto
generalizetheirjustificationsratherthanjustrepeattheprocesstheyhavebeenusing.Youmight
ask,“Howdoyouknowyourexpressionwillworkforanysidelength?”,or“Whatisitaboutthe
natureofthepatternthatsuggeststhiswillalwayswork?”,or“Whatwillhappenifwelookataside
lengthofsix?ten?fifty-three?”Considertheseideasbothvisuallyandintermsofthegeneral
expression.
Note:Basedonthestudentworkandthedifficultiestheymayormaynotencounter,a
determinationwillneedtobemadeastowhetheradiscussionofpartAofthetaskshouldbeheld
priortostudentsworkingonpartB.Workingwithaspecificcasemayfacilitateaccesstothe
generalcaseformorestudents.However,ifstudentsarereadyforwholeclassdiscussionoftheir
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generalrepresentations,thenstartingtherewillallowformoretimetobespentonmaking
connectionsbetweenthedifferentexpressions,andextendingthetasktomoregeneral
representations.
Asavailable,selectstudentstopresentwhofounddifferentwaysofgeneralizing.Somepossible
waysstudentsmight“see”thecoloredtilesgroupedareprovidedafterthechallengeactivity.It
wouldbeusefultohaveatleastthreedifferentviewstodiscussandpossiblymore.
Discuss(WholeClass):
Basedonthestudentworkavailable,youwillneedtodeterminetheorderofthestrategiestobe
presented.Alikelyprogressionwouldstartwithastrategythatdoesnotprovidethemost
simplifiedformoftheexpression.Thiswillpromotequestioningandunderstandingfromstudents
thatmayhavedoneitdifferentlyandallowfordiscussionaboutwhateachpieceoftheexpression
represents.Afteracoupleofdifferentstrategieshavebeenshareditmightbeusefultogetthemost
simplifiedformoftheexpressionoutonthetableandthenlookforanexplanationastohowallof
theexpressionscanbeequivalentandrepresentthesamethinginsomanydifferentways.
AlignedReady,Set,Go:GettingReady1.1
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1.1
READY
Topic:RecognizingSolutionstoEquations
Thesolutiontoanequationisthevalueofthevariablethatmakestheequationtrue.Intheequation
9! + 17 = −21, "a”isthevariable.Whena=2,9! + 17 ≠ −19, because 9 2 + 17 = 35. Thus! = 2 is NOT a solution.However,when! = −4, the equation is true 9 −4 + 17 = −19.Therefore,! = −4mustbethesolution.Identifywhichofthe3possiblenumbersisthesolutiontotheequation.
1.3! + 7 = 13 (! = −2; ! = 2; ! = 5) 2.8 − 2! = −2 (! = −3; ! = 0; ! = 5)
3.5 + 4! + 8 = 1 (! = −3;! = −1;! = 2) 4.6! − 5 + 5! = 105 (! = 4; ! = 7; ! = 10)
Someequationshavetwovariables.Youmayrecallseeinganequationwrittenlikethefollowing:
! = 5! + 2.Wecanletxequalanumberandthenworktheproblemwiththisx-valuetodeterminetheassociatedy-value.Asolutiontotheequationmustincludeboththex-valueandthey-value.Oftenthe
answeriswrittenasanorderedpair.Thex-valueisalwaysfirst.Example: !, ! .Theordermatters!
Determinethey-valueofeachorderedpairbasedonthegivenx-value.
5.! = 6! − 15; 8, , −1, , 5, 6.! = −4! + 9; −5, , 2, , 4,
7.! = 2! − 1; −4, , 0, , 7, 8.! = −! + 9; −9, , 1, , 5,
READY, SET, GO! Name PeriodDate
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1.1
SET
Topic:Usingaconstantrateofchangetocompleteatableofvalues
Fillinthetable.Thenwriteasentenceexplaininghowyoufiguredoutthevaluestoputineachcell.9.Yourunabusinessmakingbirdhouses.Youspend$600tostartyourbusiness,anditcostsyou$5.00
tomakeeachbirdhouse.
#ofbirdhouses 1 2 3 4 5 6 7
Totalcosttobuild
Explanation:
10.Youmakea$15paymentonyourloanof$500attheendofeachmonth.
#ofmonths 1 2 3 4 5 6 7
Amountofmoneyowed
Explanation:
11.Youdeposit$10inasavingsaccountattheendofeachweek.
#ofweeks 1 2 3 4 5 6 7
Amountofmoneysaved
Explanation:
12.Youaresavingforabikeandcansave$10perweek.Youhave$25whenyoubeginsaving.
#ofweeks 1 2 3 4 5 6 7
Amountofmoneysaved
Explanation:
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1.1
GO
Topic:GraphLinearEquationsGivenaTableofValues.
Graphtheorderedpairsfromthetablesonthegivengraphs.
13.
! !
0 3
2 7
3 9
5 13
14.
! !
0 14
4 10
7 7
9 5
15.
! !
2 11
4 10
6 9
8 8
16.
! !
1 4
2 7
3 10
4 13
5
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1.2 Growing Dots
A Develop Understanding Task
1. Describethepatternthatyouseeinthesequenceoffiguresabove.
2. Assumingthepatterncontinuesinthesameway,howmanydotsarethereat3minutes?
3. Howmanydotsarethereat100minutes?
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4. Howmanydotsarethereattminutes?Solvetheproblemsbyyourpreferredmethod.Yoursolutionshouldindicatehowmanydotswillbeinthepatternat3minutes,100minutes,andtminutes.Besuretoshowhowyoursolutionrelatestothepictureandhowyouarrivedatyoursolution.
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1.2 Growing Dots– Teacher Notes
A Develop Understanding Task
Purpose:Thepurposeofthistaskistodeveloprepresentationsforarithmeticsequencesthat
studentscandrawuponthroughoutthemodule.Thevisualrepresentationinthetaskshouldevoke
listsofnumbers,tables,graphs,andequations.Variousstudentmethodsforcountingand
consideringthegrowthofthedotswillberepresentedbyequivalentexpressionsthatcanbe
directlyconnectedtothevisualrepresentation.
CoreStandards:
F-BF:Buildafunctionthatmodelsarelationshipbetweentwoquantities.
1:Writeafunctionthatdescribesarelationshipbetweentwoquantities.*
a.Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfroma
context.
F-LE:Linear,Quadratic,andExponentialModels*(SecondaryIfocusonlinearandexponential
only)
Constructandcomparelinear,quadraticandexponentialmodelsandsolveproblems.
1.Distinguishbetweensituationsthatcanbemodeledwithlinearfunctionsandwith
exponentialfunctions.
a.Provethatlinearfunctionsgrowbyequaldifferencesoverequalintervalsand
thatexponentialfunctionsgrowbyequalfactorsoverequalintervals.
b.Recognizesituationsinwhichonequantitychangesataconstantrateperunit
intervalrelativetoanother.
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2.Constructlinearandexponentialfunctions,includingarithmeticandgeometric
sequences,givenagraph,adescriptionofarelationship,ortwoinput-outputpairs(include
readingthesefromatable).
Interpretexpressionforfunctionsintermsofthesituationtheymodel.
5.Interprettheparametersinalinearorexponentialfunctionintermsofacontext.
ThistaskalsofollowsthestructuresuggestedintheModelingstandard:
StandardsforMathematicalPracticeofFocusintheTask:
SMP1:Makesenseofproblemsandpersevereinsolvingthem.
SMP7:Lookforandmakeuseofstructure.
TheTeachingCycle:
Launch(WholeClass):Startthediscussionwiththepatternongrowingdotsdrawnontheboard
orprojectedfortheentireclass.Askstudentstodescribethepatternthattheyseeinthedots
(Question#1).Studentsmaydescribefourdotsbeingaddedeachtimeinvariousways,depending
onhowtheyseethegrowthoccurring.Thiswillbeexploredlaterinthediscussionasstudents
writeequations,sothereshouldnotbeanyemphasisplaceduponaparticularwayofseeingthe
growth.Askstudentsindividuallytoconsideranddrawthefigurethattheywouldseeat3minutes
(Question#2).Then,askonestudenttodrawitontheboardtogiveotherstudentsachanceto
checkthattheyareseeingthepattern.
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Explore(SmallGrouporPairs):Askstudentstocompletethetask.Monitorstudentsasthey
work,observingtheirstrategiesforcountingthedotsandthinkingaboutthegrowthofthefigures.
Somestudentsmaythinkaboutthefiguresrecursively,describingthegrowthbysayingthatthe
nextfigureisobtainedbyplacingfourdotsontothepreviousfigureasshown:
Somemaythinkofthefigureasfourarmsoflengtht.withadotinthemiddle.
Othersmayusea“squares”strategy,noticingthatanewsquareisaddedeachminute,asshown:
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Asstudentsworktofindthenumberofdotsat100minutes,theymaylookforpatternsinthe
numbers,writingsimply1,5,9,...Ifstudentsareunabletoseeapattern,youmayencouragethem
tomakeatableorgraphtoconnectthenumberofdotswiththetime:
Watchforstudentsthathaveusedagraphtoshowthenumberofdotsatagiventimeandtohelp
writeanequation.Encouragestudentstoconnecttheircountingstrategytotheequationthatthey
write.
Forthediscussion,selectastudentforeachofthethreecountingstrategiesshown,atable,agraph,
arecursiveequation,andatleastoneformofanexplicitequation.
Discuss(WholeGroup):Beginthediscussionbyaskingstudentshowmanydotsthattherewillbe
at100minutes.Theremaybesomedisagreement,typicallybetween100and101.Askastudent
thatsaid101toexplainhowtheygottheiranswer.Ifthereisgeneralagreement,moveontothe
discussionofthenumberofdotsattimet.
Startbyaskingagrouptochartandexplaintheirtable.Askstudentswhatpatternstheyseeinthe
table.Whentheydescribethatthenumberofdotsisgrowingby4eachtime,addadifference
columntothetable,asshown.
Time(Minutes) Numberof
Dots
0 1
1 5
2 9
3 13
t
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Time(Minutes) NumberofDots
0 1
1 5
2 9
3 13
… …
t
Askstudentswheretheyseethedifferenceof4occurringinthefigures.Notethatthedifference
betweentermsisconstanteachtime.
Continuethediscussionbyaskingagrouptoshowtheirgraph.Besurethatitisproperlylabeled,
asshown.
Askstudentshowtheyseetheconstantdifferenceof4onthegraph.Theyshouldrecognizethat
they-valueincreasesby4eachtime,makingalinewithaslopeof4.
Now,movethediscussiontoconsiderthenumberofdotsattimet,asrepresentedbyanequation.
Startwithagroupthatconsideredthegrowthasarecursivepattern,recognizingthatthenextterm
is4plusthepreviousterm.Theymayrepresenttheideaas:! + 4,withXrepresentingthepreviousterm.Thismaycausesomecontroversywithstudentsthatwroteadifferentformula.Ask
thegrouptoexplaintheirworkusingthefigures.Itmaybeusefultorewritetheirformulawith
words,like:
>4
>4
>4
Difference
Numberofdots
Time(Minutes)
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Thenumberofdotsinthecurrentfigure=thenumberofdotsinthepreviousfigure+4
Orsimply,Current=previous+4
Thismaybewritteninfunctionnotationas:! ! = ! ! − 1 + 4.(Althoughstudentshavesomeexposuretofunctionnotationingrade8,theyhavenotseenitusedtowriterecursiveformulas.
Youmaychoosetointroducethisnotationinlaterlessons,simplyfocusingonwritingtherecursive
ideainwordsasshownabove.)
Nextaskagroupthathasusedthe“fourarmsstrategy”towriteandexplaintheirequation.Their
equationshouldbe:! ! = 4! + 1. Askstudentstoconnecttheirequationtothefigure.Theyshouldarticulatethatthereis1dotinthemiddleand4arms,eachwithtdots.The4inthe
equationshows4groupsofsizet.
Next,askagroupthatusedthe“squares”strategytodescribetheirequation.Theymayhave
writtenthesameequationasthe“fourarms”group,butaskthemtorelateeachofthenumbersin
theequationtothefiguresanyway.Inthiswayofthinkingaboutthefigures,therearetgroupsof4
dots,plus1dotinthemiddle.Althoughitisnottypicallywrittenthisway,thiscountingmethod
wouldgeneratetheequation! ! = ! ∙ 4 + 1.
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Nowaskstudentstoconnecttheequationswiththetableandgraphs.Askthemtoshowwhatthe4
andthe1representinthegraph.Askhowtheysee4t+1inthetable.Itmaybeusefultoshowthis
patterntohelpseethepatternbetweenthetimeandthenumberofdots:
Time(Minutes) NumberofDots
0 1 1
1 5 1+4
2 9 1+4+4
3 13 1+4+4+4
… …
t 1+4t
Youmayalsopointoutthatwhenthetableisusedtowritearecursiveequationlike
! ! = ! ! − 1 + 4,youmaysimplylookdownthetablefromoneoutputtothenext.Whenwritinganexplicitformulalike! ! = 4! + 1,itisnecessarytolookacrosstherowsofthetabletoconnecttheinputwiththeoutput.
Finalizethediscussionbyexplainingthatthissetoffigures,equations,table,andgraphrepresent
anarithmeticsequence.Anarithmeticsequencecanbeidentifiedbytheconstantdifference
betweenconsecutiveterms.Tellstudentsthattheywillbeworkingwithothersequencesof
numbersthatmaynotfitthispattern,buttables,graphsandequationswillbeusefultoolsto
representanddiscussthesequences.
AlignedReady,Set,GoHomework:Sequences1.2
Difference
>4
>4
>4
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1.2
READY
Topic:UsingfunctionnotationToevaluateanequationsuchas! = 5! + 1 whengivenaspecificvalueforx,replacethevariablexwiththegivenvalueandworktheproblemtofindthevalueofy.Example:Findywhenx=2.Replacexwith2.! = 5 2 + 1 = 10 + 1 = 11. Therefore,y=11whenx=2.Thepoint 2, 11 isonesolutiontotheequation! = 5! + 1.Insteadofusing! !"# !inanequation,mathematiciansoftenwrite! ! = 5! + 1becauseitcangivemoreinformation.Withthisnotation,thedirectiontofind! 2 ,meanstoreplacethevalueof!with2andworktheproblemtofind! ! .Thepoint !, ! ! isinthesamelocationonthegraphas !, ! ,where!describesthelocationalongthex–axis,and! ! istheheightofthegraph.Giventhat! ! = !" − !and! ! = !" − !",evaluatethefollowingfunctionswiththeindicatedvalues.
1.! 5 = 2.! 5 = 3.! −4 = 4.! −4 =
5.! 0 = 6.! 0 = 7.! 1 = 8.! 1 =
Topic:LookingforpatternsofchangeCompleteeachtablebylookingforthepattern.
9. Term 1st 2nd 3rd 4th 5th 6th 7th 8thValue 2 4 8 16 32
10. Term 1st 2nd 3rd 4th 5th 6th 7th 8thValue 66 50 34 18
11. Term 1st 2nd 3rd 4th 5th 6th 7th 8thValue 160 80 40 20
12. Term 1st 2nd 3rd 4th 5th 6th 7th 8thValue -9 -2 5 12
READY, SET, GO! Name PeriodDate
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1.2
SET
Topic:Usevariablestocreateequationsthatconnectwithvisualpatterns.
Inthepicturesbelow,eachsquarerepresentsonetile.
13.DrawStep4andStep5.
Thestudentsinaclasswereaskedtofindthenumberoftilesinafigurebydescribinghowtheysawthe
patternoftileschangingateachstep.Matcheachstudent’swayofdescribingthepatternwiththe
appropriateequationbelow.Notethat“s”representsthestepnumberand“n”representsthenumberof
tiles.
(a)! = !" − ! + ! − ! (b)! = !" − ! (c)! = ! + ! ! − !
14._____Danexplainedthatthemiddle“tower”isalwaysthesameasthestepnumber.Healsopointed
outthatthe2armsoneachsideofthe“tower”containonelessblockthanthestepnumber.
15._____Sallycountedthenumberoftilesateachstepandmadeatable.Sheexplainedthatthenumber
oftilesineachfigurewasalways3timesthestepnumberminus2.
stepnumber 1 2 3 4 5 6
numberoftiles 1 4 7 10 13 16
16._____Nancyfocusedonthenumberofblocksinthebasecomparedtothenumberofblocksabovethebase.Shesaidthenumberofbaseblocksweretheoddnumbersstartingat1.Andthenumberoftilesabovethebasefollowedthepattern0,1,2,3,4.Sheorganizedherworkinthetableattheright.
Stepnumber #inbase+#ontop
1 1+0
2 3+1
3 5+2
4 7+3
5 9+4
Step 2 Step 3 Step 1 Step 4 Step 5
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1.2
GO
Topic:ThemeaningofanexponentWriteeachexpressionusinganexponent.17.6×6×6×6×6 18.4×4×4 19.15×15×15×15 20.!!×
!!
A)Writeeachexpressioninexpandedform.B)Thencalculatethevalueoftheexpression.21.7!
A)B)
22.3!A)B)
23.5!A)B)
24.10!A)B)
25.7(2)!A)B)
26.10 8! A)B)
27.3 5 !A)B)
28.16 !!!
A)B)
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1.3 Growing, Growing Dots
A Develop Understanding Task
Atthe Atoneminute Attwominutesbeginning
Atthreeminutes Atfourminutes
1. Describeandlabelthepatternofchangeyouseeintheabovesequenceoffigures.
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2. Assumingthesequencecontinuesinthesameway,howmanydotsarethereat5minutes?
3. Writearecursiveformulatodescribehowmanydotstherewillbeaftertminutes.
4. Writeanexplicitformulatodescribehowmanydotstherewillbeaftertminutes.
12
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1.3 Growing, Growing Dots – Teacher Notes
A Develop Understanding Task
Purpose:Thepurposeofthistaskistodeveloprepresentationsforgeometricsequencesthat
studentscandrawuponthroughoutthemodule.Thevisualrepresentationinthetaskshouldevoke
listsofnumbers,tables,graphs,andequations.Variousstudentmethodsforcountingand
consideringthegrowthofthedotswillberepresentedbyequivalentexpressionsthatcanbe
directlyconnectedtothevisualrepresentation.
CoreStandards:
F-BF:Buildafunctionthatmodelsarelationshipbetweentwoquantities.
1:Writeafunctionthatdescribesarelationshipbetweentwoquantities.*
a.Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfroma
context.
F-LE:Linear,Quadratic,andExponentialModels*(SecondaryMathematicsIfocusonlinearand
exponentialonly)
Constructandcomparelinear,quadraticandexponentialmodelsandsolveproblems.
1.Distinguishbetweensituationsthatcanbemodeledwithlinearfunctionsandwith
exponentialfunctions.
a.Provethatlinearfunctionsgrowbyequaldifferencesoverequalintervalsand
thatexponentialfunctionsgrowbyequalfactorsoverequalintervals.
c.Recognizesituationsinwhichonequantitygrowsordecaysbyaconstantpercent
rateperunitintervalrelativetoanother.
2.Constructlinearandexponentialfunctions,includingarithmeticandgeometric
sequences,givenagraph,adescriptionofarelationship,ortwoinput-outputpairs(include
readingthesefromatable).
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Interpretexpressionforfunctionsintermsofthesituationtheymodel.
5.Interprettheparametersinalinearorexponentialfunctionintermsofacontext.
ThistaskalsofollowsthestructuresuggestedintheModelingstandard:
StandardsforMathematicalPracticeofFocusintheTask:
SMP1–Makesenseofproblemsandpersevereinsolvingthem.
SMP6–Attendtoprecision.
TheTeachingCycle:
Launch(WholeClass):Startthediscussionwiththepatternofgrowingdotsdrawnontheboard
orprojectedfortheentireclass.Askstudentstodescribethepatternthattheyseeinthedots
(Question#1).Studentsmaydescribeanincreasingnumberoftrianglesbeingaddedeachtimeor
seeingthreegroupsthateachhaveanincreasingnumberofdotseachtime,dependingonhowthey
seethegrowthoccurring.Thiswillbeexploredlaterinthediscussionasstudentswriteequations,
sothereshouldnotbeanyemphasisplaceduponaparticularwayofseeingthegrowth.Ask
studentsindividuallytoconsideranddrawthefigurethattheywouldseeat5minutes(Question
#2).Then,askonestudenttodrawitontheboardtogiveotherstudentsachancetocheckthat
theyareseeingthepatterncorrectly.Remindstudentsoftheworktheydidyesterdaytowrite
explicitandrecursiveformulas.Thesearenewtermsthatshouldbereinforcedatthebeginningto
clarifytheinstructionsforquestions3and4.
Explore(SmallGrouporPairs):Askstudentstocompletethetask.Monitorstudentsasthey
work,observingtheirstrategiesforcountingthedotsandthinkingaboutthegrowthofthefigures.
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Somestudentsmaythinkaboutthefiguresrecursively,describingthegrowthbysayingthatthe
nextfigureisobtaineddoublingthepreviousfigureasshown:
! = 0 ! = 1
Somemaythinkofthefigureasthreegroupsthatareeachdoubling,asshownbelow.
! = 0 ! = 1 t=2
Asstudentsworktofindtheformulas,theymaylookforpatternsinthenumbers,writingsimply3,
6,12,24,48.Ifstudentsareunabletoseeapattern,youmayencouragethemtomakeatableor
graphtoconnectthenumberofdotswiththetime:
Time(Minutes) NumberofDots
0 3
1 6
2 12
3 24
4 48
Watchforstudentsthathaveusedagraphtoshowthenumberofdotsatagiventimeandtohelp
writeanequation.Encouragestudentstoconnecttheircountingstrategytotheequationthatthey
write.
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Forthediscussion,selectastudentforeachofthecountingstrategiesshown,atable,agraph,a
recursiveequation,andatleastoneformofanexplicitequation.Havetwolargechartsshowingthe
dotfigurespreparedinadvanceforstudentstouseinexplainingtheircountingstrategies.
Discuss(WholeGroup):
Beginthediscussionwiththegroupthatsawthepatternasdoublingthepreviousfigureeachtime.
Askthemtoexplainhowtheythoughtaboutthepatternandhowtheyannotatedthefigures.
3 6 12
Often,studentswhoareusingthisstrategywillthinkofthenumberofdots,withoutthinkingofthe
relationshipbetweenthenumberofdotsandthetime.Iftheydon’tmentionthetimeatthispoint,
becarefultopointouttherelationshipwithtimewhenthenextgrouppresentsastrategythat
connectsthetimeandthenumberofdots.
Askstudentstodescribethepatterntheyseeandrecordtheirwords:
Nextfigure=2×Previousfigure
Supportstudentsinrepresentingthisideaalgebraicallyas:! 0 = 3, ! ! = 2!(! − 1)andhelpthemtounderstandthatthisformulaexpressestheideathatawaytofindatermattimetisto
doublethepreviousterm,startingwith3attime0.
Next,askthegroupthatsawthispatternofgrowthtoexplainthewaytheysawthepatternof
growth.
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! = 0 ! = 1 t=2
Askforatablethatshowstherelationshipbetweentimeandthenumberofdots.Askstudents
whatpatternstheyseeinthetable.Askstudentstoaddadifferencecolumntothetable,likethey
didinGrowingDots.Studentsmaybesurprisedtoseethedifferencebetweentermsrepeatingthe
patterninthenumberofdots.Askstudentsiftheyseeacommondifferencebetweenterms.
Explainthatsincethereisnocommondifference,itisnotanarithmeticsequence.
Time(Minutes) NumberofDots
0 3
1 6
2 12
3 24
4 48
Atthispoint,itcanbepointedoutthatsinceyougetthenexttermbydoublingthepreviousterm,
thereisacommonratiobetweenterms.Demonstratethat:
!! =
!"! = !"!" = 2
>3
Difference
>24
>12
>6
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Thecommonratiobetweentermsistheidentifyingfeatureofageometricsequence,another
specialtypeofnumbersequence.Continuethediscussionbyaskingagrouptoshowtheirgraph.
Asktheclasswhattheypredictthegraphtolooklike.Whywouldwenotexpectthegraphtobea
line?Besurethegraphisproperlylabeled,asshown.
Now,movethediscussiontoconsiderthenumberofdotsattimet,asrepresentedbyanexplicit
equation.Askagrouptoshowtheirexplicitformulaforthenumberofdotsattimet,whichis:
! ! = 3 ∙ 2!.
Nowaskstudentstoconnecttheequationswiththetableandgraphs.Askthemtoshowwhatthe2
andthe3representinthegraph.Askhowtheysee3 ∙ 2!inthetable.Itmaybeusefultoshowtheconnectiontothetabletohelpdemonstratethepatternbetweenthetimeandthenumberofdots:
Time
(Minutes)
NumberofDots
0 3 3
1 6 3∙2
2 12 3∙2∙2
3 24 3∙2∙2∙2
4 48 3∙2∙2∙2∙2
… …
t 3 ∙ 2!
Difference
>3
>24
>12
>6
Numberofdots
Time(Minutes)
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Youmayalsoremindstudentsthatwhenthetableisusedtowritearecursiveequationsuchas:
! 0 = 3, ! ! = 2! ! − 1 , one maysimplylookdownthetablefromoneoutputtothenext.Whenwritinganexplicitformulasuchas! ! = 3 ∙ 2! ,itisnecessarytolookacrosstherowsofthetabletoconnecttheinputwiththeoutput.
Finalizethediscussionbyexplainingthatthissetoffigures,equations,table,andgraphrepresenta
geometricsequence.Ageometricsequencecanbeidentifiedbytheconstantratiobetween
consecutiveterms.Tellstudentsthattheywillcontinuetoworkwithsequencesofnumbersusing
tables,graphsandequationstoidentifyandrepresentgeometricandarithmeticsequences.
AlignedReady,Set,GoHomework:Sequences1.3
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READY
Topic:Interpretingfunctionnotation
A)Usethegiventabletoidentifytheindicatedvalueforn.B)ThenusingthevaluefornthatyoudeterminedinA,usethetabletofindtheindicatedvalueforB.
! 1 2 3 4 5 6 7 8 9 10! ! -8 -3 2 7 12 17 22 27 32 37
1.!) When ! ! = 12,what is the value of !?
!) What is the value of ! ! − 1 ?
2.!) When ! ! = 17,what is the value of !?
!) What is the value of ! ! − 1 ?
3.!) When ! ! = 32,what is the value of !?
!) What is the value of ! ! + 1 ?
4.!) When ! ! = 2,what is the value of !?
!) What is the value of ! ! + 3 ?
5.!) When ! ! = 27,what is the value of !?
!) What is the value of ! ! − 6 ?
6.!) When ! ! = −8,what is the value of !?
!) What is the value of ! ! + 9 ?
SET Topic:Comparingexplicitandrecursiveequations
Usethegiveninformationtodecidewhichequationwillbetheeasiesttousetofindtheindicatedvalue.Findthevalueandexplainyourchoice.7.Explicitequation:y=3x+7
Recursive:!"# = !"#$%&'( !"#$ + 3
term# 1 2 3 4
value 10 13 16
Findthevalueofthe4thterm._________________
Explanation:
8.Explicitequation:y=3x+7
Recursive:!"# = !"#$%&'( !"#$ + 3
term# 1 2 … 50
value 10 13 …
Findthevalueofthe50thterm.__________________
Explanation:
READY, SET, GO! Name PeriodDate
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9.Thevalueofthe8thtermis78.
Thesequenceisincreasingby10ateachstep.
Explicitequation:y=10x–2
Recursive:!"# = !"#$%&'( !"#$ + 10
Findthe20thterm.__________________________
Explanation:
10.Thevalueofthe8thtermis78.
Thesequenceisincreasingby10ateachstep.
Explicitequation:y=10x–2
Recursive:!"# = !"#$%&'( !"#$ + 10
Findthe9thterm.__________________________
Explanation:
11.Thevalueofthe4thtermis80.
Thesequenceisbeingdoubledateachstep.
Explicitequation:! = 5 2! Recursive: !"# = !"#$%&'( !"#$ ∗ 2
Findthevalueofthe5thterm._______________
Explanation:
12.Thevalueofthe4thtermis80.
Thesequenceisbeingdoubledateachstep.
Explicitequation:! = 5 2! Recursive:: !"# = !"#$%&'( !"#$ ∗ 2
Findthevalueofthe7thterm._______________
Explanation:
GO
Topic:EvaluatingExponentialEquations
Evaluatethefollowingequationswhenx={1,2,3,4,5}.Organizeyourinputsandoutputsintoatableofvaluesforeachequation.Letxbetheinputandybetheoutput.13.y=4x
14y=(-3)x 15.y=-3x 16.y=10x
xinput
youtput
1
2
3
4
5
xinput
youtput
1
2
3
4
5
xinput
youtput
1
2
3
4
5
xinput
youtput
1
2
3
4
5
17.If! ! = 5!,!ℎ!" !" !ℎ! !"#$% !" ! 4 ?
14
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1.4 Scott’s Workout
A Solidify Understanding Task
Scott has decided to add push-ups to his daily exercise routine. He is keeping track of the number of push-ups he completes each day in the bar graph below, with day one showing he completed three push-ups. After four days, Scott is certain he can continue this pattern of increasing the number of push-ups he completes each day.
1234
1. How many push-ups will Scott do on day 10? 2. How many push-ups will Scott do on day n?
CCBYNew
YorkNationa
lGua
rd
https://flic.kr/p/fKtkCW
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3. Model the number of push-ups Scott will complete on any given day. Include both explicit and
recursive equations.
4. Aly is also including push-ups in her workout and says she does more push-ups than Scott
because she does fifteen push-ups every day. Is she correct? Explain.
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1.4 Scott’s Push-Ups – Teacher Notes
A Solidify Understanding Task
Purpose:
Thistaskistosolidifyunderstandingthatarithmeticsequenceshaveaconstantdifferencebetweenconsecutiveterms.Thetaskisdesignedtogeneratetables,graphs,andbothrecursiveandexplicitformulas.Thefocusofthetaskshouldbetoidentifyhowtheconstantdifferenceshowsupineachoftherepresentationsanddefinesthefunctionsasanarithmeticsequence.
StandardsFocus:
F-BF:Buildafunctionthatmodelsarelationshipbetweentwoquantities.1:Writeafunctionthatdescribesarelationshipbetweentwoquantities.*
a.Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfromacontext.
F-LE:Linear,Quadratic,andExponentialModels*(SecondaryIfocusonlinearandexponentialonly)Constructandcomparelinear,quadraticandexponentialmodelsandsolveproblems.
1.Distinguishbetweensituationsthatcanbemodeledwithlinearfunctionsandwithexponentialfunctions.
a.Provethatlinearfunctionsgrowbyequaldifferencesoverequalintervalsandthatexponentialfunctionsgrowbyequalfactorsoverequalintervals.b.Recognizesituationsinwhichonequantitychangesataconstantrateperunitintervalrelativetoanother.
2.Constructlinearandexponentialfunctions,includingarithmeticandgeometricsequences,givenagraph,adescriptionofarelationship,ortwoinput-outputpairs(includereadingthesefromatable).
Interpretexpressionforfunctionsintermsofthesituationtheymodel.
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5.Interprettheparametersinalinearorexponentialfunctionintermsofacontext.
ThistaskalsofollowsthestructuresuggestedintheModelingstandard:
StandardsforMathematicalPracticeofFocusintheTask:
SMP2-Reasonabstractlyandquantitatively.
SMP7–Lookforandmakeuseofstructure.
TheTeachingCycle:
Launch(WholeClass):RemindstudentsoftheworktheyhavedonepreviouslywithGrowingDotsandGrowing,GrowingDots.ReadScott’sWorkoutwiththestudentsandaskastudenttoaddthenumberofpush-upsthatScottwilldoonthefifthdaytothediagramfortheclass.Askstudentswhattheyareobservingaboutthepattern.Allowjustafewresponsessothatyouknowthatstudentsunderstandthetask,butavoidgivingawaytheworkofthetask.
Explore(SmallGroup):Monitorstudentthinkingastheyworkbymovingfromonegrouptoanother.Encouragestudentstousetables,graphs,andrecursiveandexplicitequationsastheyworkonthetask.Listentostudentsandidentifydifferentgroupstopresentandexplaintheirworkononerepresentationeach.Ifstudentsarehavingdifficultywritingtheequation,askthemtobesurethattheyhavetheotherrepresentationsfirst.
Discuss(WholeClass):Whenthevariousgroupsarepreparedtopresent,startthediscussionwithatable.Besurethatthecolumnsofthetablearelabeled.Afterstudentshavepresentedtheir
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table,askstudentstoidentifythedifferencebetweenconsecutivetermsandmarkthetablesothatitlookslikethis:
!Days
!(!)Push-ups
1 32 53 74 95 11… …! 3 + 2(! − 1)
Askifthesequenceisarithmeticorgeometricbaseduponthetable.Studentsshouldbeabletoidentifythatitisarithmeticbecausethereisaconstantdifferencebetweenconsecutiveterms.
Next,askthestudentstopresentthegraph.Thegraphshouldbelabeledandlooklikethis:
Askstudentswheretheyseethedifferencebetweentermsfromthetableonthegraph.Identifythatforeachday,thenumberofpush-upsincreasesby2,soforeachincreaseof1inthexvalue,theyvalueincreasesby2.Thestudentsshouldrecognizethisasaslopeof2.Askwhythepointsinthegrapharenotconnected.Studentsshouldbeabletoanswerthatthepush-upsareassumedtobe
>2
>2
>2
Differencebetweenterms
NumberofPush-ups
NumberofDays
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doneallatonceintheday.Acontinuousgraphwouldsuggestthatthepush-upswerehappeningfortheentiretimeshownonthegraph.
Next,askstudentsfortheirrecursiveequations.Studentsmayhavewrittenanyoftheseequations:
Numberofpush-upstoday=Numberofpush-upsyesterday+2
Or:
Nextterm=Previousterm+2
Askhowtheyseethisequationintheirtableandtheirgraph.Onthetable,theyshouldpointoutthatasyoumovefromonerowtothenext,youadd2tothepreviousterm.Theyshouldbeabletodemonstrateasimilarideaonthegraphasyoumovefromoney-valuetothenext.
Askifanyonehaswrittenarecursiveequationinfunctionform.Ifnoonehaswrittentheequationinfunctionform,explainthatthemoreformalmethodofwritingtheequationis: ! 1 = 3, ! ! = ! ! − 1 + 2.Thisformstilldenotestheideathatthecurrenttermis2morethanthepreviousterm.Askstudentshowtheycanidentifythatthisisanarithmeticsequenceusingtherecursiveequation.Theanswershouldbethattheconstantdifferenceof2betweentermsshowsupintheequationasadding2togetthenextterm.Alsonotethattousearecursiveformulayouhavetoknowthepreviousterm.Thatmeansthatwhenyoumakearecursiveformulaforanarithmeticsequence,youneedtoprovidethefirsttermaspartoftheformula.
Concludethediscussionwiththeexplicitequation,! ! = 3 + 2(! − 1).Althoughthisequationcouldbesimplified,itisusefultoconsideritinthisform.Askstudentshowtheyusedthetabletowritethisequation.Howdoesthisformulashowtheconstantdifferencebetweenterms?Alsoask,“Ifyouarelookingforthe10thterm,whatnumberwillyoumultiplyby2?”Helpstudentstoconnectthatthe(! − 1)intheformulatellsthemthatthenumbertheymultiplyby2isonelessthanthetermtheyarelookingfor.Cantheyexplainthatusingthetableorgraph?
Concludethelessonbyaskingstudentstocomparerecursiveandexplicitformulas.Whatinformationdoyouneedtouseeithertypeofformula?Whataretheadvantagesofeach?Whatideasaboutarithmeticsequencesarehighlightedineach?
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READY
Topic:'Use'function'notation'to'evaluate'equations."Evaluate"the"given"equation"for"the"indicated"function"values.""1.''! ! = 5! + 8''''! 4 ='
''''''''''! −2 =''
2.'''! ! = −2! + 1''''''! 10 ='''''''! −1 =''
3.'''! ! = 6! − 3''''''! −5 =''''''! 0 =''
4.'''! ! = −!'''''''! 9 ='''''''! −11 =''
5.''! ! = 5!'''''''''''! 2 =''''''''''''! 3 =''
6.'''! ! = 3!''''''! 4 =''''''! 1 =''
7.'''! ! = 10!'''''''! 6 ='''''''! 0 =''
8.'''! ! = 2!'''''''! 0 ='''''''! 5 =''
SET
Topic:'Finding'terms'for'a'given'sequence' Find"the"next"3"terms"in"each"sequence.""Identify"the"constant"difference."Write"a"recursive"function"and"an"explicit"function"for"each"sequence."Circle"where"you"see"the"common"difference"in"both"functions.""(The"first"number"is"the"1st"term,"not"the"0th"term).""9.''A)' 3','8','13','18','23','______','______','______','…''''''''' B)'Common'Difference:'____________'
''''''C)' Recursive'Function:'____________________________' D)'Explicit''''''''
''''''Function:_______________________________'''
10.'A)' 11','9','7','5','3','______','______','______','…''''''''' B)'Common'Difference:'____________''
''''''C)' Recursive'Function:'____________________________' D)'Explicit''''''''''Function:_______________________________'''
11.'A)' 3','1.5','0','N1.5','N3','______','______','______','…''''''''' B)'Common'Difference:'____________''
''''''C)' Recursive'Function:'____________________________' D)'Explicit''''''''''Function:_______________________________''
READY, SET, GO! """"""Name" """"""Period"""""""""""""""""""""""Date"
17
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'
GO'
Topic:'Reading'a'graph''Olaf"is"a"mountain"climber.""The"graph"shows"Olaf’s"location"on"the"mountain"beginning"at"noon.""Use"the"information"in"the"graph"to"answer"the"following"questions.""12.'''What'was'Olaf’s'elevation'at'noon?'''
'
13.''What'was'his'elevation'at'2'pm?'''
'
14.'''How'many'feet'had'Olaf'descended''from'noon'until'2'pm?'''
'
15.''Olaf'reached'the'base'camp'at'4'pm.'''What'is'the'elevation'of'the'base'camp?'''
''
16.''During'which'hour'was'Olaf''descending'the'mountain'the'fastest?'Explain'how'you'know.''''
'
''17.''Is'the'value'of'! ! 'the'time'or'the'elevation?''''''
18
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1.5 Don’t Break the Chain
A Solidify Understanding Task
Maybeyou’vereceivedanemaillikethisbefore:
Thesechainemailsrelyoneachpersonthatreceivestheemailtoforwarditon.Haveyouever
wonderedhowmanypeoplemightreceivetheemailifthechainremainsunbroken?Tofigurethis
out,assumethatittakesadayfortheemailtobeopened,forwarded,andthenreceivedbythenext
person.Onday1,BillWeightsstartsbysendingtheemailouttohis8closestfriends.Theyeach
forwarditto10peoplesothatonday2itisreceivedby80people.Thechaincontinuesunbroken.
1. Howmanypeoplewillreceivetheemailonday7?
CCBYTh
omasKoh
ler
https://flic.kr/p/iV
6hUJ
Hi! My name is Bill Weights, founder of Super Scooper Ice Cream. I am offering you a gift certificate for our signature “Super Bowl” (a $4.95 value) if you forward this letter to 10 people. When you have finished sending this letter to 10 people, a screen will come up. It will be your Super Bowl gift certificate. Print that screen out and bring it to your local Super Scooper Ice Cream store. The server will bring you the most wonderful ice cream creation in the world—a Super Bowl with three yummy ice cream flavors and three toppings! This is a sales promotion to get our name out to young people around the country. We believe this project can be a success, but only with your help. Thank you for your support. Sincerely, Bill Weights Founder of Super Scooper Ice Cream
19
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2. Howmanypeoplewithreceivetheemailondayn?Explainyouranswerwithasmany
representationsaspossible.
3. IfBillgivesawayaSuperBowlthatcosts$4.95toeverypersonthatreceivestheemail
duringthefirstweek,howmuchwillhehavespent?
20
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1.5 Don’t Break the Chain– Teacher Notes
A Solidify Understanding Task
Purpose:
Thistaskistosolidifyunderstandingthatgeometricsequenceshaveaconstantratiobetween
consecutiveterms.Thetaskisdesignedtogeneratetables,graphs,andbothrecursiveandexplicit
formulas.Thefocusofthetaskshouldbetoidentifyhowtheconstantratioshowsupineachofthe
representations.
CoreStandards:
F-BF:Buildafunctionthatmodelsarelationshipbetweentwoquantities.
1:Writeafunctionthatdescribesarelationshipbetweentwoquantities.*
a.Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfroma
context.
F-LE:Linear,Quadratic,andExponentialModels*(SecondaryIfocusonlinearandexponential
only)
Constructandcomparelinear,quadraticandexponentialmodelsandsolveproblems.
1.Distinguishbetweensituationsthatcanbemodeledwithlinearfunctionsandwith
exponentialfunctions.
a.Provethatlinearfunctionsgrowbyequaldifferencesoverequalintervalsand
thatexponentialfunctionsgrowbyequalfactorsoverequalintervals.
c.Recognizesituationsinwhichonequantitygrowsordecaysbyaconstantpercent
rateperunitintervalrelativetoanother.
2.Constructlinearandexponentialfunctions,includingarithmeticandgeometric
sequences,givenagraph,adescriptionofarelationship,ortwoinput-outputpairs(include
readingthesefromatable).
Interpretexpressionforfunctionsintermsofthesituationtheymodel.
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5.Interprettheparametersinalinearorexponentialfunctionintermsofacontext.
ThistaskalsofollowsthestructuresuggestedintheModelingstandard:
StandardsforMathematicalPracticeofFocusintheTask:
SMP8–Lookforandexpressregularityinrepeatedreasoning.
SMP5–Useappropriatetoolsstrategically.
TheTeachingCycle:
Launch(WholeClass):RemindstudentsoftheworktheyhavedonepreviouslywithGrowingDots
andGrowing,GrowingDots.Withouthandingthetasktostudents,read“Don’tBreaktheChain”
pasttheendofthelettertothepointwhereitsaysthatBillstartsbysendingtheletterto8people.
Askstudentshowmanypeoplewillreceivetheemailthenextdayifthechainisunbroken.Ask
studentswhattheyareobservingaboutthepattern.Youmaywishtodrawadiagramthatshows
thefirst8andtheneachofthemsendingtheletterto10moresothatstudentsseethatthisisnota
situationwheretheyaresimplyadding10eachtime.Besuretolabelthediagramtoshowthat
therewere8peopleonday1and80onday2toavoidconfusionaboutthetime.Havestudents
workonthetaskinsmallgroups(2-4studentspergroup).Accesstographingcalculatorswill
facilitategraphingthefunction.
Explore(SmallGroup):Monitorstudentthinkingastheyworkbymovingfromonegroupto
another.Encouragestudentstousetables,graphs,andrecursiveandexplicitequationsasthey
workonthetask.Listentostudentsandidentifydifferentgroupstopresentandexplaintheirwork
ononerepresentationeach.Ifstudentsarehavingdifficultywritingtheequations,askthemtobe
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surethattheyhavetheotherrepresentationsfirst.Thefocusofthetaskisquestions1and2.The
thirdquestionrequiresstudentstosumthefirst7termsofthesequence.Whileitgivesan
interestingresult,itisanextensionforthosewhofinishquickly.Whentheclassisfinishedwiththe
firsttwoquestions,calltheclassbackforawholegroupdiscussion.
Discuss(WholeClass):Startthediscussionwithatable.Besurethatthecolumnsofthetableare
labeled.Afterstudentshavepresentedtheirtable,askstudentstoidentifythedifferencebetween
consecutivetermsandmarkthetablesothatitlookslikethis:
!Days
!(!)#ofemails
1 8
2 80
3 800
4 8000
5 80,000
… …
Askifthesequenceisarithmetic,baseduponthetable.Studentsshouldbeabletoidentifythatitis
notarithmeticbecausethereisnoconstantdifferencebetweenconsecutiveterms.Ask,“Whatother
patternsdoyouseeinthetable?”Studentsshouldnoticethatthenexttermisobtainedby
multiplyingby10.Askstudentstotestifthereisaconstantratiobetweenconsecutiveterms.They
shouldrecognizetheratioof10.Confirmthattheconstantratiobetweenconsecutivetermsmeans
thatthisisageometricsequence.
Next,askthestudentstopresentthegraph.Thegraphshouldbelabeledandlooklikethegraph
below,butwithonlythepointsinthetable.(Thecurvehasbeenshownhereascontinuousonlyto
seeanaccurateplacementofthepoints.):
>720>7,200
>72
Difference
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Askstudentswhatwasdifficultaboutcreatingthegraph.Theywillprobablypointoutthatthey-
valueswereincreasingsoquicklythatitwasdifficulttoscalethegraph.Askifthepointsonthe
graphmakealine.Helpthemtoseethatthepointsmakeacurve;thegraphislinearonlywhen
thereisaconstantdifferencebetweenterms.Identifythatforeachday,thenumberofemails
multipliesby10,soforeachincreaseof1inthexvalue,theyvalueis10timesbigger.Askstudents
whichgraphthismostresembles:GrowingDotsorGrowing,GrowingDots.Helpthemtoidentify
thecharacteristicshapeofanincreasingexponentialfunction.
Next,askstudentsfortheirrecursiveequations.Studentsmayhavewrittenanyoftheseequations:
Numberofemailstoday=10*Numberofemailsyesterday
Or: Nextterm=10*Previousterm
Askhowtheyseethisequationintheirtableandtheirgraph.Onthetable,theyshouldpointout
thatasyoumovefromonerowtothenext,youmultiplytheprevioustermby10.Theyshouldbe
abletodemonstrateasimilarideaonthegraphasyoumovefromoney-valuetothenext.
Askifanyonehaswrittentheirrecursiveequationinfunctionform.Ifnoonehaswrittenthe
equationinfunctionform,explainthatthemoreformalmethodofwritingtheequationis:
! 1 = 8, ! ! = 10! ! − 1 .AskstudentshowthisformulaisdifferentthantherecursiveformulaforScott’sworkout,anarithmeticsequence.Askstudentshowtheycanidentifythatthisis
Numberofemails
NumberofDays
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ageometricsequenceusingtherecursiveequation.Theanswershouldbethattheconstantratioof
10betweentermsshowsupintheequationasmultiplyingby10togetthenextterm.Alsonote
thattousearecursiveformulayouhavetoknowthepreviousterm.Thatmeansthatwhenyou
havearecursiveformulaforanarithmeticsequence,youneedtoprovidethefirsttermaspartof
theformula.
Concludethediscussionwiththeexplicitformula,! ! = 8(10!!!).Manystudentsmayfinditdifficulttowritethepatternthattheyseeinthisform.First,itmaybehardtoseethattheyneedto
useanexponent,andthenitmaybehardtoseehowtheexponentrelatestothetermnumber.It
maybehelpfultoreferbacktothetable,addingthethirdcolumnshownhere.
!Days
!(!)Numberofemails
!(!)Numberofemails
!(!)Numberofemails
1 8 8 8(10!)2 80 8 ∙ 10 8(10!)3 800 8 ∙ 10 ∙ 10 8(10!)4 8,000 8 ∙ 10 ∙ 10 ∙ 10 8(10!)5 80,000 8 ∙ 10 ∙ 10 ∙ 10 ∙ 10 8(10!)… … …
n 8(10!!!)
Askstudentstonoticehowmanytimes10isusedasapowerandcompareittothenumberofdays.
Remindthemthattowriteanexplicitformulatheyneedtoreadacrossthetableandrelatethen
valuetof(n).Youmaywishtoaskstudentstohelpyoucompleteafourthcolumnthatshowsthe
powersof10.Helpstudentstoconnectthatthe(! − 1)intheformulatellsthemthenumberoftimestheymultiplyby10isonelessthanthetermtheyarelookingfor.Askstudentstorelatethe
formulatograph.
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Concludethelessonbyaskingstudentstocomparetherecursiveandexplicitformulasfor
arithmeticandgeometricsequences.Howaretherecursiveformulasforgeometricandarithmetic
sequencesalike?Howaretheydifferent?Howaretheexplicitformulasforgeometricand
arithmeticsequencealike?Howaretheydifferent?
AlignedReady,Set,GoHomework:Sequences1.5
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READY Topic:'Rates'of'change'in'a'table'and'a'graph''The$same$sequence$is$shown$in$both$a$table$and$a$graph.$$Indicate$on$the$table$where$you$see$the$rate$of$change$of$the$sequence.$Then$draw$on$the$graph$where$you$see$the$rate$of$change.$$1.''''''
!!' ! ! '1' 2'2' 5'3' 8'4' 11'5' 14'
''''''''
2.''''''''''''''''''''''''!' ! ! '1' 13'2' 11'3' 9'4' 7'5' 5'
'''''''
'3.'''
!' ! ! '1' 16'2' 11'3' 6'4' 1'5' !!!!!!!−4'
'
'4.''''''''''''''''''''''''''''''''''''
!' ! ! '1' 0'2' 4'3' 8'4' 12'5' 16'
'
'$$$
READY, SET, GO! $$$$$$Name$ $$$$$$Period$$$$$$$$$$$$$$$$$$$$$$$Date$
21
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SET Topic:'Recursive'and'explicit'functions'of'geometric'sequences''Below$you$are$given$various$types$of$information.$$Write$the$recursive$and$explicit$functions$for$each$geometric$sequence.$Finally,$graph$each$sequence,$making$sure$you$clearly$label$your$axes.$$'''5.'''''2','4','8','16','…'
'
'
'
'
'
6.''' Time'
(days)'Number'of'cells'
1' 3'
2' 6'
3' 12'
4' 24'
'
Recursive:_____________________________________'
'
Explicit:________________________________________'
'
Recursive:_____________________________________'
'
Explicit:________________________________________'
'
7.''Claire'has'$300'in'an'account.'She'decides'she'is'going'to'take'out'half'of'what’s'left'in'there'at'the'end'of'each'month.'''
'
'
'
'
'
8.''Tania'creates'a'chain'letter'and'sends'it'to'four'friends.'Each'day'each'friend'is'then'instructed'to'send'it'to'four'friends'and'so'forth.'
'
'
Recursive:_____________________________________'
'
Explicit:________________________________________'
'
Recursive:_____________________________________'
'
Explicit:________________________________________'
'
'
'
22
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9.''
'
'
'
Recursive:_____________________________________'
'
Explicit:________________________________________'
'''
GO Topic:'Recursive'and'explicit'functions'of'arithmetic'sequences''Below$you$are$given$various$types$of$information.$$Write$the$recursive$and$explicit$functions$for$each$arithmetic$sequence.$$Finally,$graph$each$sequence,$making$sure$you$clearly$label$your$axes.$$10.'''''2','4','6','8','…'
'
'
'
'
'
'
11.''' Time'
(days)'Number'of'cells'
1' 3'
2' 6'
3' 9'
4' 12'
'
Recursive:_____________________________________'
Explicit:________________________________________'
'
Recursive:_____________________________________'
Explicit:________________________________________'
Day 3Day 2Day 1
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12.''Claire'has'$300'in'an'account.'She'decides'she'is'going'to'take'out'$25'each'month.'''
'
'
'
'
'
'
13.''Each'day'Tania'decides'to'do'something'nice'for'2'strangers.'What'is'the'relationship'between'the'number'people'helped'and'days?'
'
Recursive:_____________________________________'
'
Explicit:________________________________________'
Recursive:_____________________________________'
'
Explicit:________________________________________'
'
14.''
'
'
'
Recursive:_____________________________________'
'
Explicit:________________________________________'
'
'$
Day 3Day 2Day 1
24
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1.6 Something to Chew On
A Solidify Understanding Task
The Food-Mart grocery store has a candy machine like the one
pictured here. Each time a child inserts a quarter, 7 candies come
out of the machine. The machine holds 15 pounds of candy. Each
pound of candy contains about 180 individual candies.
1. Represent the number of candies in the machine for any given number of customers. About
how many customers will there be before the machine is empty?
2. Represent the amount of money in the machine for any given number of customers.
CCBYlilCystar
https://flic.kr/p/7kW
qZf
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3. To avoid theft, the store owners don’t want to let too much money collect in the machine, so
they take all the money out when they think the machine has about $25 in it. The tricky
part is that the store owners can’t tell how much money is actually in the machine without
opening it up, so they choose when to remove the money by judging how many candies are
left in the machine. About how full should the machine look when they take the money out?
How do you know?
26
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1.6 Something to Chew On – Teacher Notes
A Solidify Understanding Task
Purpose:
Thistaskintroducesadecreasingarithmeticsequencetofurthersolidifytheideathatarithmetic
sequenceshaveaconstantdifferencebetweenconsecutiveterms.Again,connectionsshouldbe
madeamongallrepresentations:table,graph,recursiveandexplicitformulas.Theemphasis
shouldbeoncomparingincreasinganddecreasingarithmeticsequencesthroughthevarious
representations.
CoreStandards:
F-BF:Buildafunctionthatmodelsarelationshipbetweentwoquantities.
1:Writeafunctionthatdescribesarelationshipbetweentwoquantities.*
a.Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfroma
context.
F-LE:Linear,Quadratic,andExponentialModels*(SecondaryIfocusinlinearandexponentialonly)
Constructandcomparelinear,quadraticandexponentialmodelsandsolveproblems.
1.Distinguishbetweensituationsthatcanbemodeledwithlinearfunctionsandwith
exponentialfunctions.
a.Provethatlinearfunctionsgrowbyequaldifferencesoverequalintervalsand
thatexponentialfunctionsgrowbyequalfactorsoverequalintervals.
b.Recognizesituationsinwhichonequantitychangesataconstantrateperunit
intervalrelativetoanother.
2.Constructlinearandexponentialfunctions,includingarithmeticandgeometric
sequences,givenagraph,adescriptionofarelationship,ortwoinput-outputpairs(include
readingthesefromatable).
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Interpretexpressionsforfunctionsintermsofthesituationtheymodel.
5.Interprettheparametersinalinearorexponentialfunctionintermsofacontext.
ThistaskalsofollowsthestructuresuggestedintheModelingstandard:
StandardsforMathematicalPracticeofFocusintheTask:
SMP4-Modelwithmathematics.
SMP2–Reasonabstractlyandquantitatively.
TheTeachingCycle:
Launch(Wholeclass):Beforehandingoutthetask,askstudentstodefineanarithmeticsequence.
Laterwewillsaythatitisalinearfunctionwiththedomainofpositiveintegers.Rightnow,expect
studentstoidentifytheconstantrateofchangeorconstantdifferencebetweenconsecutiveterms.
Askstudentstogiveafewexamplesofarithmeticsequences.Sincetheonlysequencestheyhave
seenuptothispointhavebeenincreasing,expectthemtoaddanumbertogettothenextterm.
Then,wonderoutloudwhetherornotitwouldbeanarithmeticsequenceifanumberissubtracted
togetthenextterm.Don’tanswerthequestionorsolicitresponses.
Readtheopeningparagraphofthetaskandbesurethatallstudentsunderstandhowthecandy
machineswork;whenaquarterisinserted,7candiescomeout.Readthefirstpromptinthetask
anddiscusswhatitmeansto“Representthenumberofcandiesinthemachineforagivennumber
ofcustomers.”Explainthattheirrepresentationsshouldincludetables,graphs,andequations.As
soonasstudentsunderstandthetask,setthemtowork.
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Explore(Smallgroups):Tobeginthetask,studentswillneedtodecidehowtosetuptheirtables
orgraphs.Tablesshouldbesetwithcustomersastheindependentvariableandthenumberof
candiesasthedependentvariable.Onewaytorepresentcandiesversuscustomersistocalculate
howmanycandiesinthemachinewhenitisfullandthenbuildtablesandgraphsbysubtracting7
foreachcustomer.Itwouldbeappropriatetohavegraphingcalculatorsavailableforthistask.
Thesecondprompt(#2)issimilartootherincreasinggeometricsequencesthatstudentshave
previouslymodeledinScott’sWorkoutandGrowingDots.Again,encourageasmany
representationsaspossible.Whenyoufindthatmoststudentsarefinishedwithnumber1and2it
probablyagoodtimetostartthediscussion.Number3isanextensionprovidedfordifferentiation,
butnotthefocusofthetaskformoststudents.
Monitorthegroupworkwithparticularfocusontheworkin#1.Haveonegrouppreparedto
presentthetableandonegrouppresentthegraph.Anothergroupcanpresentbothformsofthe
equationsfrom#1.Youmaychoosetohavethepresentersbegintodrawtheirtablesandgraphs
whiletheothergroupsfinishtheirwork.Alsoselectjustonegrouptodoallofproblem#2onthe
boardforcomparison.
Discuss(WholeGroup):Startthediscussionbyrepeatingthequestionthatwasstatedinthe
launch:Canyouformanarithmeticsequencebysubtractinganumberfromeachtermtogetthe
nextterm?Askthegrouptopresentthetablethattheymadefor#1,whichshouldlooksomething
likethis,althoughstudentsmaynothaveincludedthefirstdifferenceattheside.Ifnot,additinas
partofthediscussion.
# of Customers # of Candies 0 2700 1 2693 2 2686 3 2679 4 2672 5 2665 6 2658 7 2651 8 2644 9 2637
10 2632 …. …. ! 2700 − 7!
>-7>-7>-7>-7>-7
Difference
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Askstudentswhattheynoticeaboutthetable.Doesthetableshowaconstantdifferencebetween
terms?Whatistheconstantdifference?Doesthistablerepresentanarithmeticsequence?
Next,discussthegraph,whichshouldbeproperlylabeledandlooksomethinglikethis:
Theconstantdifferencebetweentermshasbeendemonstratedinthetable.Nowaskstudentshow
thisgraphofanarithmeticsequenceisliketheotherarithmeticsequencesthatwehavestudiedin
Scott’sWorkoutandGrowingDots.Theyshouldidentifythatthepointsformalineandthegraphis
notcontinuous.(Thereisnoneedtoemphasizethediscretenatureofthesequencesinceitisa
focusinthenextmodule.)Howisthisgraphdifferent?Itisdecreasingataconstantrate,rather
thanincreasingataconstantrate.
Askstudentstocomparetherecursiveformulasforboth#1and#2.Encouragethemtouse
functionnotationonly,sothattheirformulaslooklike:
1. ! 0 = 2700, ! ! = ! ! − 1 − 7 2.! 1 = .25, ! ! = ! ! − 1 + .25
Askstudents,“Basedontherecursiveformula,is#2anarithmeticsequence?Whyorwhynot?”
Expectstudentstoanswerthat+.25showsthateachtermisincreasingbyaconstantamount.
Numberofcandies
Numberofcustomers
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Nowaskstudentstocomparetheexplicitformulasforboth#1and#2.Infunctionnotation,they
shouldbelike:
1. ! ! = 2700 − 7! 2.! ! = .25!
Askstudentshowtheycanidentifyanarithmeticsequencefromanexplicitequation.Re-
emphasizethedefinitionofanarithmeticsequenceasasequencethathasacommondifference
betweenconsecutiveterms.Thecommondifferencecanbeeitherpositiveornegative.Youmay
wishtoendthediscussionbyworkingwithstudentstocompletethechartgivenintheIntervention
Activity.
AlignedReady,Set,GoHomework:Sequences1.6
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READY !Topic:!Finding!the!common!difference!!Find%the%missing%terms%for%each%arithmetic%sequence%and%state%the%common%difference.%%1.!!!5,!11,!_______!,!23,!29,!_______...!!!!Common!Difference!=!__________!!
2.!!!7,!3,!>1,!_______!,!_______!,!>13…!!!!Common!Difference!=!__________!!
3.!!!8,!_______!,!_______!,!47,!60…!!!Common!Difference!=!__________!!
4.!!0,!_______!,!_______!,!2,!!!!!!…!!
!Common!Difference!=!_________!!
5.!!5,!_______!,!_______!,!_______!,!25…!!!Common!Difference!=!__________!!
6.!!!3,!________!,!________!,!_________!,!>13!…!!Common!Difference!=!__________!!
SET !Topic:!Writing!the!recursive!function!!Two%consecutive%terms%in%an%arithmetic%sequence%are%given.%%Find%the%recursive%function.%%7.!If!!! 3 = 5!!"#!! 4 = 8!…!!!!!!!f"(5)!=!_______.!!f!(6)!=!_______.!!Recursive!Function:!_______________________________________________!
!!!8.!!If!!! 2 = 20!!"#!! 3 = 12!…!!!!!!!f"(4)!=!_______.!!f!(5)!=!_______.!!Recursive!Function:!_______________________________________________!
!!!9.!!If!! 5 = 3.7!!"#!! 6 = 8.7!…!!!!!!!!f"(7)!=!_______.!!f!(8)!=!_______.!!Recursive!Function:!_______________________________________________!
! !!!
READY, SET, GO! %%%%%%Name% %%%%%%Period%%%%%%%%%%%%%%%%%%%%%%%Date%
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!Two%consecutive%terms%in%a%geometric%sequence%are%given.%%Find%the%recursive%function.%!10.!If!!! 3 = 5!!"#!! 4 = 10!…!!!!!!!f"(5)!=!_______.!!f!(6)!=!_______.!!Recursive!Function:!_______________________________________________!
!!!11.!!If!!! 2 = 20!!"#!! 3 = 10!…!!!!!!!f"(4)!=!_______.!!f!(5)!=!_______.!!Recursive!Function:!_______________________________________________!
!!!12.!!If!! 5 = 20.58!!"#!! 6 = 2.94!…!!!!!!!!f"(7)!=!_______.!!f!(8)!=!_______.!!Recursive!Function:!_______________________________________________!
! !!
GO Topic:!Evaluating!using!function!notation!!Find%the%indicated%values%of%! ! .%%13.!!!!! ! = !2!!!!!!!!!!!!!!!!!!!!!!!!!!!Find!! 5 !!"#!! 0 .! !!!14.! !! ! = !5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Find!! 4 !!"#!! 1 .!!!15.! !! ! = (−2)!!!!!!!!!!!!!!!!!!!!!!!!Find!! 3 !!"#!! 0 .!!!16.! !!!! ! = −2!!!!!!!!!!!!!!!!!!!!!!!!!!Find!! 3 !!"#!! 0 .!!!17.! In!what!way!are!the!problems!in!#15!and!#16!different?!!!!18.! !!!! ! = 3 + 4 ! − 1 !!!!!!!!!!!!Find!! 5 !and!! 0 .!!!19.! !! ! = 2 ! − 1 + 6!!!!!!!!!!!!!!Find!! 1 !and!! 6 .!
!
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1.7 Chew On This
A Solidify Understanding Task
Mr.andMrs.Gloopwanttheirson,Augustus,todohishomeworkeveryday.Augustuslovestoeat
candy,sohisparentshavedecidedtomotivatehimtodohishomeworkbygivinghimcandiesfor
eachdaythatthehomeworkiscomplete.Mr.GloopsaysthatonthefirstdaythatAugustusturnsin
hishomework,hewillgivehim10candies.Ontheseconddayhepromisestogive20candies,on
thethirddayhewillgive30candies,andsoon.
1. Writebotharecursiveandanexplicitformulathatshowsthenumberofcandiesthat
Augustusearnsonanygivendaywithhisfather’splan.
2. UseaformulatofindhowmanycandiesAugustuswillgetonday30inthisplan.
Augustuslooksinthemirroranddecidesthatheisgainingweight.Heisafraidthatallthatcandy
willjustmakeitworse,sohetellshisparentsthatitwouldbeokiftheyjustgivehim1candyonthe
firstday,2onthesecondday,continuingtodoubletheamounteachdayashecompleteshis
homework.Mr.andMrs.GlooplikeAugustus’planandagreetoit.
CCBYFrankJacobi
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3. ModeltheamountofcandythatAugustuswouldgeteachdayhereacheshisgoalswiththe
newplan.
4. UseyourmodeltopredictthenumberofcandiesthatAugustuswouldearnonthe30thday
withthisplan.
5. Writebotharecursiveandanexplicitformulathatshowsthenumberofcandiesthat
Augustusearnsonanygivendaywiththisplan.
Augustus is generally selfish and somewhat unpopular at school. He decides that he could improve
his image by sharing his candy with everyone at school. When he has a pile of 100,000 candies, he
generously plans to give away 60% of the candies that are in the pile each day. Although Augustus
30
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may be earning more candies for doing his homework, he is only giving away candies from the pile
that started with 100,000. (He’s not that generous.)
6. How many pieces of candy will be left on day 4? On day 8?
7. Model the amount of candy that would be left in the pile each day.
8. How many days will it take for the candy to be gone?
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1.7 Chew On This – Teacher Notes
A Solidify Understanding Task
Purpose:
Thepurposeofthistaskistosolidifyandextendtheideathatgeometricsequenceshaveaconstant
ratiobetweenconsecutivetermstoincludesequencesthataredecreasing(0<r<1).Thecommon
ratioinonegeometricsequenceisawholenumberandintheothersequenceitisapercent.This
taskcontainsanopportunitytocomparethegrowthofarithmeticandgeometricsequences.This
taskalsoprovidespracticeinwritingandusingformulasforarithmeticsequences.
CoreStandards:
F.BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.*
a.Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfroma
context.
F.LE.1Distinguishbetweensituationsthatcanbemodeledwithlinearfunctionsandwith
exponentialfunctions.
a.Provethatlinearfunctionsgrowbyequaldifferencesoverequalintervalsand
thatexponentialfunctionsgrowbyequalfactorsoverequalintervals.
c.Recognizesituationsinwhichonequantitygrowsordecaysbyaconstantpercent
rateperunitintervalrelativetoanother.
F.LE.2Constructlinearandexponentialfunctions,includingarithmeticandgeometricsequences,
givenagraph,adescriptionofarelationship,ortwoinput-outputpairs(includereadingthesefrom
atable).
F.IF.5Interprettheparametersinalinearorexponentialfunctionintermsofacontext.
ThistaskalsofollowsthestructuresuggestedintheModelingstandard:
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StandardsforMathematicalPracticeofFocusintheTask:
SMP7–Lookforandmakeuseofstructure.
SMP6–Attendtoprecision.
TheTeachingCycle:
Launch(WholeGroup):
Beginbyreadingthefirstpartofthetask,leadingtoquestions1and2.Besurethatstudents
understandtheproblemsituationandthenaskwhattypeofsequencewouldmodelthissituation
andwhy?Studentsshouldbeabletoidentifythatthisisanarithmeticsequencebecausethe
numberofcandiesincreasesby10eachdaythatAugustusmeetshisgoals.Askstudentstoworkon
#1individuallyandthencallonstudentstosharetheirresponses.Theformulasshouldbe:
! ! = 10! and ! 1 = 10, ! ! = ! ! − 1 + 10
Read#2togetherandaskstudentswhichformulawillbeeasiertousetofigureouthowmany
candiestherewillbeonday30.Confirmthattheexplicitformulawillbebetterforthispurpose
sincewedon’tknowthe29thtermsothatitcanbeusedintherecursiveformula.
Readtheinformationprovidedtoanswerquestions3,4,and5.Aftercheckingtoseethatstudents
understandtheproblemsituation,askthemtoanswer#3and#4individually.Thiswillbeagood
assessmentopportunitytoseewhatstudentsunderstandabouttheirpreviouswork,sincethisis
verysimilartothegeometricsequencesthattheyhavealreadymodeled.Aftertheyhavehadsome
time,theyshouldbeabletoproduceproperlylabeledtables,graphsandequationslike:
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Commonratio,r=2
Explicitformula:! ! = 2!!!
Recursiveformula:! 1 = 1, ! ! = 2!(! − 1)
Whenstudentshavesharedtheirwork,askthemtopredictwhichplanwillgiveAugustusmore
candiesonday30andwhy?Thenhavestudentuseaformulatofindthenumberofcandiesonday
30andcompareittowhattheyfoundin#2.Theywillprobablybesurprisedtofindthatinthe
secondplan,Augustuswillhave536,870,912.Askstudents,“Whythehugedifferenceinresults?Whatisitaboutageometricsequencelikethisthatmakesitgrowsoquickly?”
Explore(SmallGroup):
Beforeallowingstudentstoworkintheirgroupsonthelastpartoftheproblem,checktobesure
theyunderstandtheproblemsituation.Thepileofcandystartswith100,000.When60%ofthe
candyisremovedonday1thereare40,000candies.Studentsneedtobeclearthattheyare
keepingtrackofthenumberofcandiesthatremaininthepile,notthenumberofcandiesthatare
removed.Thereareseveraldifferentstrategiesforfiguringthisout,butthatshouldbepartofthe
conversationinthesmallgroups.Asyoumonitorstudentworkatthebeginningofthetask,make
surethattheyarefindingsomewaytocalculatethenumberofcandiesleftinthepile.Letstudents
workintheirgroupstodecidehowtoorganizeandanalyzethisinformation.Studentshave
experienceincreatingtablesandgraphsinthesetypesofsituations,althoughyoushouldcheckto
#ofDays #ofCandies
1 1
2 2
3 4
4 8
5 16
… …
! 2!
NumberofCandies
NumberofDays
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besurethattheyhavescaledtheirgraphsappropriately.Watchforgroupsthatrecognizeareable
towriteformulasandrecognizethatthisisageometricsequence.
Discuss(WholeClass):
Startthediscussionwithstudentspresentingatable.Askstudentsiftheycheckforacommon
differenceorcommonratiobetweenterms.Notethatthecommonratiotothesideofthetable.
Askstudentswhatkindofsequencethismustbe,givenacommonratiobetweenterms.Ask,“How
isthisgeometricsequencedifferentthanotherswehaveseen?”
Thefollowingtableandgraphshowsomeofthemodelsthatstudentsshouldproduce.
Number
ofDays
Numberof
Candies
Left
1 40000
2 16000
3 6400
4 2560
5 1024
6 410
7 164
8 66
Continuethediscussionwiththegraphofthesequence.
Howisthegraphdifferentthanothergeometricsequenceswehaveseen?Whatisitaboutthe
sequencethatcausesthebehaviorofthegraph?
Commonratio,! =0.4
Numberofcandiesleft
Numberofdays
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Next,askfortherecursiveformula.Ifstudentshaveidentifiedthecommonratio,theyshouldbe
abletowrite:! 1 = 40,000, ! ! = .4!(! − 1).Studentsthatcalculatedeachtermofthesequencebysubtracting60%oftheprevioustermmayhavestruggledwiththisformula.
Emphasizetheusefulnessofthetableinfindingthecommonratioforwritingformulas.
Theexplicitformula,! ! = 100,000(0. 4!)willprobablybedifficultforstudentstoproduce.Helpthemtorecallthatinpastgeometricsequencesthatthenumberintheformulathatisraisedtoa
poweristhecommonratio.Thentheyneedtothinkabouttheexponent—isitnor(n-1)?Discuss
waysthattheycantesttheexponent.Finally,theyneedtoconsiderthemultiplier,inthiscase
100,000.Wheredoesitcomefromintheproblem,bothinthiscaseandinpreviousgeometric
sequenceproblemslikeGrowing,GrowingDots?Helpstudentstogeneralizetomaketheirworkin
writingexplicitformulaseasier.
Youmaychoosetowrapupthediscussionbycompletingtheinterventionactivitywiththeclass.
AlignedReady,Set,GoHomework:Sequences1.7
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READY !Topic:!Distinguishing!between!arithmetic!and!geometric!sequences!!Find%the%missing%values%for%each%arithmetic%or%geometric%sequence.%%Underline%whether%it%has%a%constant%difference%or%a%constant%ratio.%%State%the%value%of%the%constant%difference%or%ratio.%Indicate%if%the%sequence%is%arithmetic%or%geometric%by%circling%the%correct%answer.%%1.!!!5,!10,!!15,!____!,!25,!30,!____...!!!!Common!difference!or!ratio?!!Common!Difference/ratio!=!__________!!Arithmetic!or!geometric?!!
2.!!!20,!10,!____!,!2.5,!____,…!!!!Common!difference!or!ratio?!!Common!Difference/ratio!=!__________!!Arithmetic!or!geometric?!!
!3.!!!2,!5,!8,!____!,!14,!____,!…!!!Common!difference!or!ratio?!!Common!Difference/ratio!=!__________!!Arithmetic!or!geometric?!
!4.!!30,!24!,!____!,!12,!6,!…!!!Common!difference!or!ratio?!!Common!Difference/ratio!=!__________!!Arithmetic!or!geometric?!!
SET !Topic:!Recursive!and!explicit!equations!!Determine%whether%the%given%information%represents%an%arithmetic%or%geometric%sequence.%Then%write%the%recursive%and%the%explicit%equation%for%each.%%5.!!!2,!4,!6,!8,!…!!Arithmetic!or!geometric?!!Recursive:!________________________________________!!Explicit:!__________________________________________%
6.!!!2,!4,!8,!16,!…!!Arithmetic!or!geometric?!!Recursive:!________________________________________!!Explicit:!__________________________________________!%
%
READY, SET, GO! %%%%%%Name% %%%%%%Period%%%%%%%%%%%%%%%%%%%%%%%Date%
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7.!!!!Time!(in!days)!
Number!of!dots!
1! 3!2! 7!3! 11!4! 15!
!Arithmetic!or!geometric?!!Recursive:!________________________________________!!Explicit:!__________________________________________!
8.!!!!Time!(in!days)!
Number!of!cells!
1! 5!2! 8!3! 12.8!4! 20.48!!Arithmetic!or!geometric?!!Recursive:!________________________________________!!Explicit:!__________________________________________!!
!9.!!!Michelle!likes!chocolate!but!it!causes!acne.!!She!chooses!to!limit!herself!to!three!chocolate!bars!every!5!days.!(So,!she!eats!part!of!a!bar!each!day.)!!!Arithmetic!or!geometric?!!Recursive:!________________________________________!!Explicit:!__________________________________________!
!10.!!!Scott!decides!to!add!running!to!his!exercise!routine!and!runs!a!total!of!one!mile!his!first!week.!!He!plans!to!double!the!number!of!miles!he!runs!each!week.!!Arithmetic!or!geometric?!!Recursive:!________________________________________!!Explicit:!__________________________________________!!
!11.!!!Vanessa!has!$60!to!spend!on!rides!at!the!state!fair.!!Each!ride!costs!$4.!!Arithmetic!or!geometric?!!Recursive:!________________________________________!!Explicit:!__________________________________________!
!12.!!!Cami!invested!$6,000!into!an!account!that!earns!10%!interest!each!year.!(Hint:!Make!a!table!of!values!to!help!yourself.)!Arithmetic!or!geometric?!!Recursive:!________________________________________!!Explicit:!__________________________________________!
33
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GO Topic:!Graphing!and!counting!slope!between!two!points.!!For%the%following%problems%two%points%and%a%slope%are%given.%%Plot%and%label%the%2%points%on%the%graph.%%Draw%the%line%segment%between%them.%%Then%sketch%on%the%graph%how%you%count%the%slope%of%the%line%by%moving%up%or%down%and%then%sideways%from%one%point%to%the%other.%%%13.!!A(2!,!\1)!and!B(4!,!2)! 14.!!H(\2!,!1)!and!K(2!,!5)! 15.!P(0!,!0)!and!Q(3!,!6)!
! ! !
Slope:!!! = ! !!! Slope:!!! = 1!!"! !!! Slope:!!! = 2!!"! !!!
!For%the%following%problems,%two%points%are%given.%%Plot%and%label%these%points%on%the%graph.%%Then!count%the%slope.%!
16.!!C(\3!,!0)!and!D(0!,!5)! 17.!!E(\2!,!\1)!and!N(\4!,!4)! 18.!!S(0!,!3)!and!W(1!,!6)!
! ! !
!!!!!!!!!!Slope:!!! =!! !!!!!!!!!!Slope:!!!! =! !!!!!!!!!!Slope:!!!! =!
!
34
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1.8 What Comes Next? What
Comes Later?
A Practice Understanding Task
Foreachofthefollowingtables,
• describehowtofindthenextterminthesequence,
• writearecursiveruleforthefunction,
• describehowthefeaturesidentifiedintherecursiverulecanbeusedtowriteanexplicit
ruleforthefunction,and
• writeanexplicitruleforthefunction.
• identifyifthefunctionisarithmetic,geometricorneither
Example:
• Tofindthenextterm:add3tothepreviousterm
• Recursiverule:! 0 = 5, ! ! = ! ! − 1 + 3• Tofindthenthterm:startwith5andadd3ntimes
• Explicitrule:! ! = 5 + 3!• Arithmetic,geometric,orneither?Arithmetic
FunctionA
1. Howtofindthenextterm:_______________________________________
2. Recursiverule:_____________________________________________
3. Tofindthenthterm:_________________________________________
4. Explicitrule:_______________________________________________
5. Arithmetic,geometric,orneither?_____________________________
x y0 5
1 8
2 11
3 14
4 ?
… …
n ?
CCBYHiro
akiM
aeda
https://flic.kr/p/6R8
oDk
x y1 5
2 10
3 20
4 40
5 ?
… …
n ?
35
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FunctionB
6. Howtofindthenextterm:_________________________________________
7. Recursiverule:_______________________________________________
8. Tofindthenthterm:___________________________________________
9. Explicitrule:_________________________________________________
10. Arithmetic,geometric,orneither?_______________________________
FunctionC
11. Tofindthenextterm:______________________________________________
12. Recursiverule:____________________________________________________
13. Tofindthenthterm:________________________________________________
14. Explicitrule:______________________________________________________
15. Arithmetic,geometric,orneither?____________________________________
FunctionD
16. Tofindthenextterm:________________________________________
17. Recursiverule:______________________________________________
18. Tofindthenthterm:__________________________________________
19. Explicitrule:________________________________________________
20. Arithmetic,geometric,orneither?______________________________
x y1 -8
2 -17
3 -26
4 -35
5 -44
6 -53
… …
n
x y1 2
2 6
3 18
4 54
5 162
6 486
… …
n
x y1 3
2 15
3 27
4 39
5 51
6 ?
… …
n ?
36
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FunctionE
21. Tofindthenextterm:________________________________________
22. Recursiverule:______________________________________________
23. Tofindthenthterm:__________________________________________
24. Explicitrule:________________________________________________
25. Arithmetic,geometric,orneither?______________________________
FunctionF
26. Tofindthenextterm:________________________________________
27. Recursiverule:______________________________________________
28. Tofindthenthterm:__________________________________________
29. Explicitrule:________________________________________________
30. Arithmetic,geometric,orneither?______________________________
FunctionG
31. Tofindthenextterm:_______________________________________
32. Recursiverule:_____________________________________________
33. Tofindthenthterm:_________________________________________
34. Explicitrule:_______________________________________________
35. Arithmetic,geometric,orneither?______________________________
x y0 1
11 35
2 2 153 2 454
3 255 4
… …
n
x y0 3
1 4
2 7
3 12
4 19
5 ?
… …
n ?
x y1 10
2 2
3 25
4 225
5 2125
6 2625
… …
n
37
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FunctionH
36. Tofindthenextterm:________________________________________
37. Recursiverule:______________________________________________
38. Tofindthenthterm:__________________________________________
39. Explicitrule:________________________________________________
40. Arithmetic,geometric,orneither?______________________________
x y1 -1
2 0.2
3 -0.04
4 0.008
5 -0.0016
6 0.00032
… …
n
38
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1.8 What Comes Next? What Comes Later? – Teacher Notes
A Practice Understanding Task
Purpose:
Thepurposeofthistaskistopracticewritingrecursiveandexplicitformulasforarithmeticand
geometricsequencesfromatable.Thistaskalsoprovidespracticeinusingtablestoidentifywhen
asequenceisarithmetic,geometric,orneither.Thetaskextendsstudents’experienceswith
sequencestoincludegeometricsequenceswithalternatingsigns,andmoreworkwithfractionsand
decimalnumbersinthesequences.
CoreStandards:
F.BF.1:Writeafunctionthatdescribesarelationshipbetweentwoquantities.*
a.Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfroma
context.
F.LE.1.Distinguishbetweensituationsthatcanbemodeledwithlinearfunctionsandwith
exponentialfunctions.
F.LE.2.Constructlinearandexponentialfunctions,includingarithmeticandgeometricsequences,
givenagraph,adescriptionofarelationship,ortwoinput-outputpairs(includereadingthesefrom
atable).
StandardsforMathematicalPracticeofFocusintheTask:
SMP7–Lookforandmakeuseofstructure.
SMP8–Lookforandexpressregularityinrepeatedreasoning.
TheTeachingCycle:
Launch(WholeClass):Remindstudentsoftheworkdoneinprevioustasksinthismodule.Ask,
“Whatstrategiesareyouworkingonforwritingformulasforarithmeticandgeometricsequences?”
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Studentsmayoffersomeideasthatwillbehelpfulinthistask.Handoutthetaskandworkthrough
theexamplewiththeclass.Itwillbeimportanttoclarifythetwopromptsthataskstudentsto
“describe”whattheyareseeinginthetable.Writingaverbaldescriptionbeforewritingtheformula
canpromotedeeperunderstanding.
Explore(SmallGroup):Monitorstudentworkonthetask.FunctionsAandBshouldbefamiliar
andaccessibleformoststudents.FunctionFismorechallenging;althoughstudentsmayrecognize
apattern,itisneitherarithmeticnorgeometric.Itisnotnecessaryforstudentstowritethe
formulas,sincethisisanunfamiliarfunctiontype.Ifyoufindthatagroupisentirelystuckonthe
problem,youmightaskthemtomoveontosomeoftheothersandgobacktofinishFunctionF.
WatchtheirworkonFunctionHtobesurethattheyconsiderthealternatingsignintheirformula.
Iftheyhaven’tincludeditintheformula,askthemhowtheirformulaswillproducethenegative
signonsomeoftheterms.Allowthemtoworktofindawaytoincludethenegativesignintheir
formula.Watchformisconceptionsthatmightmakeforaproductivediscussionandlistenfor
studentsgeneralizingstrategiesforwritingequations.Identifygroupstopresenttheirworkfor
FunctionsBandEthathaveshownthefirstdifferenceontheirtablesandcanarticulatehowthey
haveusedthecommondifferenceorcommonratiotowritetheequations.
Discuss(WholeClass):Startthediscussionwiththestudentpresentationoftheirworkon
FunctionD.Theirworkwillprobablylooksomethinglike:
x y
1 -8
2 -17
3 -26
4 -35
5 -44
6 -53
… …
n
FunctionB
Tofindthenextterm:Add-9tothepreviousterm
Recursiverule:!(1) = −8, !(!) = !(! − 1) − 9Tofindthenthterm:Startwith-8andadd-9n-1times
Explicitrule:!(!) = −8− 9(! − 1)Arithmetic,geometric,orneither?Arithmetic
Difference
>-9>-9>-9
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Askstudentshowtheywereabletousethecommondifferenceof-9intheformulas.Askhowthe
firstterm,-8,showsupineachformula.
Next,askagrouptoshowtheirworkwithFunctionC.Itshouldlooksomethinglikethis:
x y
1 2
2 6
3 18
4 54
5 162
6 486
Askstudentshowtheywereabletousethecommonratioof3intheformulas.Then,askhowthe
firstterm,2,appearsintheformulas.
Iftimeallows,youmaywishtohavesomeoftheotherspresented.Ifnot,movetoaskingstudents
togeneralizetheirthinkingaboutwritingformulasforarithmeticandgeometricfunctionswiththe
followingchart:
Howdoyouuse:
Commonratioordifference? Firstterm?
ArithmeticSequence–Recursive
Formula
ArithmeticSequence–Explicit
Formula
GeometricSequence–Recursive
Formula
GeometricSequence–Explicit
Formula
Commonratio3
FunctionC
Tofindthenextterm:Multiplytheprevioustermby3
Recursiverule:!(1) = 2, !(!) = 3!(! − 1)Tofindthenthterm:startwith2andmultiplyby3(n-1)times
Explicitrule:!(!) = 2(3)!!!Arithmetic,geometric,orneither?Geometric
>4
>36>12
Difference
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Askifthereareanyotherstrategiesthattheyhavefoundforwritingformulasandrecordthem
beneaththechart.Leavethechartinviewforstudentstouseinupcomingtasks.
AlignedReady,Set,GoHomework:Sequences1.8
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READY !Topic:!Common!Ratios!!Find%the%common%ratio%for%each%geometric%sequence.%%1.!!!2,!4,!8,!16…!!
2.!!!!!!,!1,!2,!4,!8…!!!
3.!!!85,!10,!820,!40…!!
4.!!!10,!5,!2.5,!1.25…!!
!SET !Topic:!Recursive!and!explicit!equations!!Fill!in!the!blanks!for!each!table;!then!write!the!recursive!and!explicit!equation!for!each!sequence.!!5.%%Table%1%
x" 1! 2! 3! 4! 5!y" 5! 7! 9! ! !
!Recursive:!_________________________________________!!!!!!!Explicit:!__________________________________________!!!6.%%Table%2% % % % 7.%%Table%3% % % % 8.%%Table%4%
! ! !%% % ! !!!!
!!Recursive:! ! !!!!!!!! ! Recursive:! ! ! ! Recursive:!!Explicit:! ! ! ! Explicit!! ! ! ! Explicit:!!
x! y%1! 82!2! 84!3! 86!4! !5! !
x! y%1! 3!2! 9!3! 27!4! !5! !
x! y%1! 27!2! 9!3! 3!4! !5! !
READY, SET, GO! %%%%%%Name% %%%%%%Period%%%%%%%%%%%%%%%%%%%%%%%Date%
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GO Topic:!Writing!equations!of!lines!given!a!graph.!!Write%each%equation%of%the%line%in%! = !" + !%form.%%Name%the%value%of%m%and%b.%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Recall%that%m%is%the%slope%or%rate%of%change%and%b!is%the%yGintercept.%%9.!!!!!!!!!!!!!!!!!
! = !!!!!!!!!!!!!!! =!!!!!!!!!!!!!!!Equation:!!
10.!!!!!!!!!!!!!!!!!!!!!! = !!!!!!!!!!!!!!! =!!!!!!!!!!!!!!!Equation:!!
!!!11.!!!!!!!!!!!!!!!!!!! = !!!!!!!!!!!!!!! =!!!!!!!!!!!!!!!Equation:!
!!!12.!!!!!!!!!!!!!!!!! = !!!!!!!!!!!!!!! =!!!!!!!!!!!!!!!Equation:!!!
!
40
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1.9 What Does It Mean?
A Solidify Understanding Task Eachofthetablesbelowrepresentsanarithmeticsequence.
Findthemissingtermsinthesequence,showingyour
method.
1.
x 1 2 3y 5 11
2.
x 1 2 3 4 5y 18 -10
3.
x 1 2 3 4 5 6 7y 12 -6
4.Describeyourmethodforfindingthemissingterms.Willthemethodalwayswork?Howdoyou
know?
CCBYTimEvanson
https://flic.kr/p/c6A
iWu
41
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Hereareafewmorearithmeticsequenceswithmissingterms.Completeeachtable,eitherusing
themethodyoudevelopedpreviouslyorbyfindinganewmethod.
5.
x 1 2 3 4y 50 86
6.
x 1 2 3 4 5 6y 40 10
7.
x 1 2 3 4 5 6 7 8y -23 5
8.Themissingtermsinanarithmeticsequencearecalled“arithmeticmeans”.Forexample,inthe
problemabove,youmightsay,“Findthe6arithmeticmeansbetween-23and5”.Describea
methodthatwillworktofindarithmeticmeansandexplainwhythismethodworks.
42
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1.9 What Does It Mean? – Teacher Notes
A Solidify Understanding Task
Purpose:
Thepurposeofthistaskistosolidifystudentunderstandingofarithmeticsequencesbyfinding
missingtermsinthesequence.Studentswilldrawupontheirpreviousworkinusingtablesand
writingexplicitformulasforarithmeticsequences.Thetaskwillalsoreinforcetheirfluencyin
solvingequationsinonevariable.
CoreStandards:
A.REI.3Solvelinearequationsandinequalitiesinonevariableincludingequationswithcoefficients
representedbyletters.
ClusterswithInstructionalNotes:Solveequationsandinequalitiesinonevariable.
Extendearlierworkwithsolvinglinearequationstosolvinglinearinequalitiesinonevariableand
tosolvingliteralequationsthatarelinearinthevariablebeingsolvedfor.Includesimple
exponentialequationsthatrelyonlyonapplicationofthelawsofexponents,
suchas5x=125or2x=1/16.
StandardsforMathematicalPracticeofFocusintheTask:
SMP7–Lookforandmakeuseofstructure.
TheTeachingCycle:
Launch(WholeClass):Referstudentstothepreviousworkwitharithmeticsequences.Inthe
past,wehavebeengiventermsandaskedtowriteformulas.Inthistask,wewillbeusingwhatwe
knowaboutsequencestofindmissingterms.Abriefdiscussionofthechartthatwascreatedas
partofthe“WhatComeNext,WhatComesLater”taskmaybeusefulindrawingstudents’attention
totheroleofthefirsttermandthecommondifferenceinwritingexplicitformulas.
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ExplorePart1(SmallGroup):Givestudentsthetaskandbesurethattheyunderstandthe
instructions.Tellthemthattheyarelookingforastrategyforsolvingtheseproblemsthatworksall
thetime,sotheyshouldbepayingattentiontotheirmethods.Thetaskbuildsincomplexityfrom
theproblemsonfirstpagetotheproblemsonsecondpage.Becausetheproblemsonthefirstside
areallaskingstudentstofindanoddnumberofterms,moststudentswillprobablyaveragethe
firstandlasttermstofindthemiddleterm,andthenaveragethenexttwo,andsoonuntilthetable
iscompleted.Somestudentsmaychoosetowriteanequationanduseittofindthemissingterms.
Othersmayuseguessandchecktofindthecommondifferencetogettoeachterm.Watchforthe
variousstrategiesandnotethestudentsthatyouwillselectforthediscussionofthefirstpart.Let
studentsworkonthetaskuntiltheycompletethefirstpage,andthencallthembackforashort
discussion.
Discuss(WholeClass):Beginthediscussionbyaskingastudentthatusedguessandcheckto
explainhis/herstrategy.Asktheclasswhatunderstandingofarithmeticsequencesisnecessaryto
usethisstrategy?Theyshouldanswerthatitreliesonthecommondifferenceandthatitcanbe
addedtoatermtogetthenextterm.Thisisusingthereasoningthatweusetowriterecursive
formulas.Next,askthestudentthatusedanaveragingmethodtoexplainhis/hermethod.Askthe
classwhythismethodworks.Nowasktheclasswhattheadvantagesaretoeachofthesemethods.
Willtheybothworkeverytime?Ifstudentshaven’tnoticedthatalloftheseexamplesareworking
withanoddnumberofmissingterms,askthemwhatwouldhappenifyouneededtouseoneof
thesestrategiestofindanevennumberofmissingterms.Don’tdiscusstheanswerasaclass,just
useitasateaserforthenextsection.
Explore(SmallGroup):Havestudentsworkonside2ofthetask,withthesameinstructions
aboutkeepingtrackoftheirstrategies.Monitorstudentsastheyworknotingthestrategiesthat
theyaretrying.Somewillcontinuetotryguessandcheck,althoughitbecomesmoredifficult.You
mightencouragethemtolookforamorereliablemethod,justincasethenumbersaretoomessyto
guessat.Themethodoftakingasimpleaveragewillnotworkwellfortheseproblemseither,so
studentsmaybecomestuck.Youmightaskthemwhattheywouldneedtoknowandhowthey
woulduseittofillinthemissingterms.Whentheyknowthattheyneedthedifferencebetween
terms,thenaskiftheycanthinkofawaytofinditwithoutguessing.Thismayhelpthemtothinkof
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writinganequationtofindthedifference.Listentostudentsastheyexplaintheirthinkingabout
theformulaandbepreparedtoaskstudentstopresenttheirvariousstrategies.
Discuss(WholeGroup):Sincemoststudentswillusethesamemethodforallthreeproblems,
startthediscussionbytalkingaboutthesecondproblem.Acoupleofusefulideasthatcould
emergefromthediscussionfollowinthesectionbelow:
x 1 2 3 4 5 6
y 40 10
Onewaytothinkaboutsettingupanequationtofindthedifferenceistosay:IknowthatIneedto
startat40andaddthedifference(d),5timestogetto10.So,Iwrotetheequation:
40 + 5! = 10
Asecondcommonapproachistosubtractthefirsttermfromthelasttermanddividebythe
numberof“jumps”togetthecommondifference.Thisisreallyjustfindingtheslopebyfindingthe
changeinydividedbythechangeinx,althoughstudentsprobablywon’trecognizeitassuch.This
canbewritten:
! = ! ! − !(1)! − 1
Thisisworthpointingoutifitarises,sincestudentsoftenfindananalogouspatterningeometric
sequencesinthenexttask.
Athirdstrategyistothinkmoreabouttheexplicitformulasthatwehavewritteninthepastasa
generalformula:! ! = ! 1 + !(! − 1).Studentsprobablywon’tusethisnotation,butmightsaysomethinglike,“Welearnedinthelasttaskthatwhenyouwriteanexplicitformulatogetanyterm,
youmultiplythecommondifferenceby1lessthanthetermyou’relookingforandaddthefirst
term.”Iftheyapplytheidealikeageneralformulatheywouldwrite:
10 = 40 + !(6 − 1)
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Pointoutthatthisstrategyiscloselyrelatedtothepreviousstrategy.Noticehowthetwoformulas
arereallyjustthesameformularearranged?
Formulafromsecondstrategy:! = ! ! !!(!)!!!
Formulafromthirdstrategy:! ! = ! 1 + !(! − 1).
Eitherwaytheythinkaboutit,studentscangetthecommondifferenceandaddittothetermthey
knowtogetthenextterm.Askstudentsifthestrategyofwritinganequationworkswiththethree
problemsonside1andgivethemachancetoconvincethemselvesthatitwill.
Finally,askstudentstoamend,edit,oraddtotheirstrategyandexplanationonthelastquestion.
Concludethediscussionbyhavingafewstudentssharewhattheyhavewrittenwiththeclass.
AlignedReady,Set,GoHomework:Sequences1.9
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READY !
Topic:!Comparing!arithmetic!and!geometric!sequences!
!
1.!!How!are!arithmetic!and!geometric!sequences!similar?!!
!
!
!
!
!
2.!!How!are!they!different?!
SET Topic:!Finding!missing!terms!in!an!Arithmetic!sequence!
!
Each%of%the%tables%below%represents%an%arithmetic%sequence.%%Find%the%missing%terms%in%the%
sequence,%showing%your%method.%
%3.%%Table%1%
%
x" 1! 2! 3!
y" 3! ! 12!
!
4.%%Table%2% % % %%%%%%%5.%%Table%3% % % % 6.%%Table%4%
! ! ! !!!!!!
!
x! y%1! 24!
2! !
3! 6!
4! !
x! y%1! 16!
2! !
3! !
4! 4!
5! !
x! y%
1! 2!
2! !
3! !
4! 26!
READY, SET, GO! %%%%%%Name% %%%%%%Period%%%%%%%%%%%%%%%%%%%%%%%Date%
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GO
Topic:!Sequences!
Determine%the%recursive%and%explicit%equations%for%each.%(if%the%sequence%is%not%arithmetic%or%
geometric,%identify%it%as%neither%and%don’t%write%the%equations).%
%
7.!!5,!9,!13,!17,…!!!!!!!!!!!This!sequence!is:!!Arithmetic!,!Geometric!,!Neither!
!
Recursive!Equation:!______________________________!!!Explicit!Equation:!_________________________________!
!
!
!
8.!!60,!30,!0,!R30!,…!!!!!!!This!sequence!is:!!Arithmetic!,!Geometric!,!Neither!
!
Recursive!Equation:!_____________________________!!!Explicit!Equation:!__________________________________!
!
!
!
9.!!60,!30,!15,!!"! ,!,…!!!!!!!This!sequence!is:!!Arithmetic!,!Geometric!,!Neither!
!
Recursive!Equation:!____________________________!!!!Explicit!Equation:!__________________________________!
!
!
!
10.!!!
!
!
!
(The!number!of!black!tiles!above)!!!!!!!!!!!This!sequence!is:!!Arithmetic!,!Geometric!,!Neither!
!
Recursive!Equation:!____________________________!!!!Explicit!Equation:!__________________________________!
!
!
!
!
11..!!!4,!7,!12,!19,!,…!!!!!!!!!!!This!sequence!is:!!Arithmetic!,!Geometric!,!Neither!
!
Recursive!Equation:!_____________________________!!!Explicit!Equation:!__________________________________!
!
!
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1.10 Geometric Meanies
A Practice Understanding Task Eachofthetablesbelowrepresentsageometric
sequence.Findthemissingtermsinthesequence,
showingyourmethod.
Table1
x 1 2 3y 3 12
Isthemissingtermthatyouidentifiedtheonlyanswer?Whyorwhynot?
Table2
x 1 2 3 4y 7 875
Arethemissingtermsthatyouidentifiedtheonlyanswers?Whyorwhynot?
Table3
x 1 2 3 4 5y 6 96
Arethemissingtermsthatyouidentifiedtheonlyanswers?Whyorwhynot?
CCBYJDHan
cock
https://flic.kr/p/eod
TxP
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Table4
x 1 2 3 4 5 6y 4 972
Arethemissingtermsthatyouidentifiedtheonlyanswers?Whyorwhynot?
A. Describeyourmethodforfindingthegeometricmeans.
B. Howcanyoutelliftherewillbemorethanonesolutionforthegeometricmeans?
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1.10 Geometric Meanies – Teacher Notes
A Practice Understanding Task Purpose:
Thepurposeofthistaskistosolidifystudentunderstandingofgeometricsequencestofindmissing
termsinthesequence.Studentswilldrawupontheirpreviousworkinusingtablesandwriting
explicitformulasforgeometricsequences.
CoreStandards:
A.REI.3Solvelinearequationsandinequalitiesinonevariableincludingequationswithcoefficients
representedbyletters.
ClusterswithInstructionalNotes:Solveequationsandinequalitiesinonevariable.
Extendearlierworkwithsolvinglinearequationstosolvinglinearinequalitiesinonevariableand
tosolvingliteralequationsthatarelinearinthevariablebeingsolvedfor.Includesimple
exponentialequationsthatrelyonlyonapplicationofthelawsofexponents,
suchas5x=125or2x=1/16.
StandardsforMathematicalPracticeofFocusintheTask:
SMP7–Lookforandmakeuseofstructure.
TheTeachingCycle:
Launch(WholeClass):Explaintostudentsthattoday’spuzzlesinvolvefindingmissingterms
betweentwonumbersinageometricsequence.Thesenumbersarecalled“geometricmeans”.Ask
themtorecalltheworkthattheyhavedonepreviouslywitharithmeticmeans.Ask,“What
informationdoyouthinkwillbeusefulforfindinggeometricmeans?Somemayrememberthatthe
firsttermandthecommondifferencewereimportantforarithmeticmeans;similarlythefirstterm
andthecommonratiomaybeimportantforgeometricmeans.
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Ask,“Howdoyoupredictthemethodsforfindingarithmeticandgeometricmeanstobesimilar?”
Ideally,studentswouldthinktousethecommonratiotomultiplytogetthenextterminthesame
waythattheyaddedthecommondifferencetogetthenextterm.Theymayalsothinkabout
writingtheequationusingthenumberof“jumps”thatittakestogetfromthefirsttermtothenext
termthattheyknow.
Ask,“Howdoyouthinkthemethodswillbedifferent?”Ideally,studentswillrecognizethatthey
havetomultiplyratherthansimplyadd.Theymayalsothinkthattheequationshaveexponentsin
thembecausetheformulasthattheyhavewrittenforgeometricsequenceshaveexponentsinthem.
Explore(SmallGroup):Theproblemsinthistaskgetlargerandrequirestudentstosolve
equationsofincreasinglyhigherorder.Somestudentswillstartwithaguessandcheckstrategy,
whichmayworkformanyofthese.Evenifitisworking,youmaychoosetoaskthemtoworkona
strategythatismoreconsistentandwillworknomatterwhatthenumbersare.Studentsarelikely
totrythesamestrategiesastheyusedforarithmeticsequences.Thesestrategieswillbesuccessful
iftheythinktomultiplybythecommonratio,ratherthanaddthecommondifference.Iftheyare
writingequations,theymayhavesimilarthinkingtothis:
Table2
x 1 2 3 4
y 7 875
“Ineedtostartat7andmultiplybyanumber,r,togettothesecondterm,thenmultiplybyragain
togettothethirdterm,andbyragaintoendupat875.So,Iwillwritetheequation:
7 ∙ ! ∙ ! ∙ ! = 875 or 7!! = 875
Somestudentsmayhavedifficultysolvingtheequationthattheywrite.Itmaybehelpfultoget
themtodivideby7andthenthinkaboutthenumberthatcanberaisedtothethirdpowertoget
125.Calculatorswillalsobehelpfulifstudentsunderstandhowtotakerootsofnumbers.Allofthe
numbersusedinthistaskarestraightforward,makingthemaccessibleformoststudentstothink
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aboutwithoutacalculator.Watchforstudentsthatcanexplainthestrategyshownabove,along
withotherproductivestrategiessothatyoucanhighlighttheirworkinthediscussion.
Alongwithcompletingthetable,studentsareaskedifthesolutiontheyfoundistheonlysolution.
Thisisareminderforthemtothinkofboththepositiveandnegativesolutioniftheexponentis
even.Asyouaremonitoringstudentwork,watchtoseeiftheyareansweringthisprompt
appropriately.Ifnot,youmaywanttoaskaquestionlike,“Iseethatyouhave!! = 4, ! = 2.Isthereanothernumberthatyoucanmultiplybyitselftoget4?”
Discuss(WholeClass):StartthediscussionwithTable1.Selectastudentthatguessedatthe
commonratioandaskhowhe/shefigureditoutandthenhowthecommonratiowasusedtofind
themissingterm.Sincemostguessersonlygetthepositiveratio(2),youwillprobablyhave
studentsthatwanttoaddthenegativesolution(-2)also.Great!Askhowtheyfiguredthesolution
outandtoverifyfortheclassthatacommonratioof-2alsoproducesatermthatworksinthe
sequence.Somestudentsmayhavesimplyreasonedthat-2wouldworkasacommonratio.Be
suretoselectastudentthathaswrittenanequationlike:3!! = 12.Usethisasanopportunityforstudentstoseethatthealgebraicsolutiontothisequationis! = ±2.
Next,askastudenttowriteandexplainanequationforTable2,asshownabove.Astheysolvethe
equation,7!! = 875,!! = 125,askstudentsifthereismorethanonenumberthatcanbecubedtoobtain125?Theyshoulddecidethat5isasolution,but-5isnot.Haveastudentdemonstrate
usingthecommonratiothatwasfoundtocompletethetable.
Nowthatstudentshaveworkedacoupleofproblemsasaclass,theyhavehadachancetocheck
theirwork,askthemtoreconsiderthegeneralizationsattheendofthetask.Givestudentsafew
minutestomodifytheirresponsestoquestionsAandB.Then,askafewstudentstosharetheir
procedureforfindinggeometricmeansandknowingthenumberofsolutionswiththeentireclass.
Studentsmaynoticethatoneofthestrategiestheyareusingisanalogoustotheprocesstheywere
usingtofindthecommondifferenceinanarithmeticsequence.Withthearithmeticsequence,the
processledtotheslopeformula.Withageometricsequence,theprocessbecomes:
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• Dividethelasttermbythefirstterm
• Takethen-1rootoftheresulttogetthecommonratio(Studentswillprobablynotusethis
terminology)
Ifthisoccursinclass,itmaybeproductivetocomparehowto“undo”anarithmeticsequenceanda
geometricsequence.Thedifferencesresultfromthedifferentnatureofthetwosequencetypes;
arithmeticsequencesareadditiveandgeometricsequencesaremultiplicative.
AlignedReady,Set,GoHomework:Sequences1.10
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READY !
Topic:!Arithmetic!and!geometric!sequences!!For$each$set$of$sequences,$find$the$first$five$terms.$Then$compare$the$growth$of$the$arithmetic$sequence$and$the$geometric$sequence.$Which$grows$faster?$$ $When?$$1.!! Arithmetic!sequence:!! 1 = 2, common!difference,! = 3!!!!!!!!! Geometric!sequence:!! 1 = 2, common!ratio, ! = 3!
Arithmetic! Geometric!! 1 =! ! 1 =!! 2 =! ! 2 =!! 3 =! ! 3 =!! 4 =! ! 4 =!! 5 =! ! 5 =!!a)! Which!value!do!you!think!will!be!more,!! 100 !or!!(100)?!! b)!!!Why?!!
2.!! Arithmetic!sequence:!! 1 = 2, common!difference,! = 10!!!!!!!!!! Geometric!sequence:!! 1 = 128, common!ratio, ! = !
!!
Arithmetic! Geometric!
! 1 =! ! 1 =!! 2 =! ! 2 =!! 3 =! ! 3 =!! 4 =! ! 4 =!! 5 =! ! 5 =!!a)! Which!value!do!you!think!will!be!more,!! 100 !or!!(100)?!! b)!!!Why?!!
3.!! Arithmetic!sequence:!! 1 = 20,! = 10!Geometric!sequence:!! 1 = 2, ! = 2!
Arithmetic! Geometric!! 1 =! ! 1 =!! 2 =! ! 2 =!! 3 =! ! 3 =!! 4 =! ! 4 =!! 5 =! ! 5 =!!a)! Which!value!do!you!think!will!be!more,!! 100 !or!!(100)?!! b)!!!Why?!!
!
READY, SET, GO! $$$$$$Name$ $$$$$$Period$$$$$$$$$$$$$$$$$$$$$$$Date$
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4.!! Arithmetic!sequence:!! 1 = 50, common!difference,! = −10!Geometric!sequence:!! 1 = 1, common!ratio, ! = 2!
Arithmetic! Geometric!! 1 =! ! 1 =!! 2 =! ! 2 =!! 3 =! ! 3 =!! 4 =! ! 4 =!! 5 =! ! 5 =!!a)! Which!value!do!you!think!will!be!more,!! 100 !or!!(100)?!! b)!!!Why?!
!!
5.!! Arithmetic!sequence:!! 1 = 64, common!difference,! = −2!Geometric!sequence:!! 1 = 64, common!ratio, ! = !
!!Arithmetic! Geometric!! 1 =! ! 1 =!! 2 =! ! 2 =!! 3 =! ! 3 =!! 4 =! ! 4 =!! 5 =! ! 5 =!!a)! Which!value!do!you!think!will!be!more,!! 100 !or!!(100)?!! b)!!!Why?!
!!
!6.!! Considering!arithmetic!and!geometric!sequences,!would!there!ever!be!a!time!that!a!! geometric!sequence!does!not!out!grow!an!arithmetic!sequence!in!the!long!run!as!the!! number!of!terms!of!the!sequences!becomes!really!large?!!!!!!!!!!!!!!!!!!!!!!!!!Explain.!
SET
Topic:!Finding!missing!terms!in!a!geometric!sequence!!Each$of$the$tables$below$represents$a$geometric$sequence.$$Find$the$missing$terms$in$the$sequence.$Show$your$method.$$7.!!Table!1!
x" 1! 2! 3!y" 3! ! 12!
!!!!!!
!
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8.!!Table!2! ! ! ! 9.!!Table!3! ! ! ! 10.!!Table!4!! ! ! !!!!!
!
GO !Topic:!Writing!the!explicit!equations!of!a!geometric!sequence!$Given$the$following$information,$determine$the$explicit$equation$for$each$geometric$sequence.$$11.!!! 1 = 8, !"##"$!!"#$%!! = 2!!!!!12.!!! 1 = 4, ! ! = 3!(! − 1)!!!!!13.!!! ! = 4! ! − 1 ; !! 1 = !
!!!!!!!14.!!Which!geometric!sequence!above!has!the!greatest!value!at!!! 100 !?! ! ! ! !
!
x) y$1! 4!2! !3! !4! !5! 324!
x) y$1! 2!2! !3! !4! 54!
x) y$1! 5!2! !3! 20!4! !
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1.11 I Know … What Do You Know?
A Practice Understanding Task IneachoftheproblemsbelowIsharesomeoftheinformationthatIknowaboutasequence.Your
jobistoaddallthethingsthatyouknowaboutthesequencefromtheinformationthatIhavegiven.
Dependingonthesequence,someofthethingsyoumaybeabletofigureoutforthesequenceare:
• atable;
• agraph;
• anexplicitequation;
• arecursiveformula;
• theconstantratioorconstantdifferencebetweenconsecutiveterms;
• anytermsthataremissing;
• thetypeofsequence;
• astorycontext.
Trytofindasmanyasyoucanforeachsequence,butyoumusthaveatleast4thingsforeach.
1. Iknowthat:therecursiveformulaforthesequenceis! 1 = −12, ! ! = ! ! − 1 + 4Whatdoyouknow?
2. Iknowthat:thefirst5termsofthesequenceare0,-6,-12,-18,-24...
Whatdoyouknow?
CCBYJim
Larrison
https://flic.kr/p/9mP2
c9
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3. Iknowthat:theexplicitformulaforthesequenceis! ! = −10(3)!Whatdoyouknow?
4. Iknowthat:Thefirst4termsofthesequenceare2,3,4.5,6.75...
Whatdoyouknow?
5. Iknowthat:thesequenceisarithmeticand! 3 = 10 and ! 7 = 26Whatdoyouknow?
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6. Iknowthat:thesequenceisamodelfortheperimeterofthefollowingfigures:
Figure1 Figure2 Figure3
Whatdoyouknow?
7. Iknowthat:itisasequencewhere! 1 = 5 andtheconstantratiobetweentermsis-2.Whatdoyouknow?
8. Iknowthat:thesequencemodelsthevalueofacarthatoriginallycost$26,500,butloses
10%ofitsvalueeachyear.
Whatdoyouknow?
Lengthofeachside=1
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9. Iknowthat:thefirsttermofthesequenceis-2,andthefifthtermis-!!.
Whatdoyouknow?
10. Iknowthat:agraphofthesequenceis:
Whatdoyouknow?
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1.11 I Know… What Do You Know? – Teacher Notes
A Practice Understanding Task Purpose:
Thepurposeofthistaskistopracticeworkingwithgeometricandarithmeticsequences.Thisis
thefinaltaskinthemoduleandisintendedtohelpstudentsdevelopfluencyinusingvarious
representationsforsequences.Thistaskcouldbeusedasaperformanceassessment.
CoreStandards:
F-LE.2:Constructlinearandexponentialfunctions,includingarithmeticandgeometricsequences,
givenagraph,adescriptionofarelationship,ortwoinput-outputpairs(includereadingthesefrom
atable).
StandardsforMathematicalPracticeofFocusintheTask:
SMP6–Attendtoprecision.
SMP7–Lookforandmakeuseofstructure.
TheTeachingCycle:
Launch(WholeClass):Thetaskgivesstudentssomeinformationaboutasequenceandasksthem
tocompletetheremaininginformation.Askstudentswhataresomethingsthatwecanknowabout
asequence.Thisquestionshouldgeneratealistlike:
• Atablewith4termsofthesequence• Agraphofthesequence• Anexplicitformulaforthesequence• Arecursiveformulaforthesequence• Thecommonratioorcommondifferencebetweenterms• Whetherthesequenceisgeometric,arithmeticorneither
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Iftheclassdoesn’tidentifyalloftheserepresentations,addthemissingonestothelist.Handout
thetaskandtellstudentsthattheywillbetryingtofigureoutallthethingsthatcanbeknownfrom
theinformationgiven.
Explore(SmallGroup):Monitorstudentsastheywork.Aftermoststudentshavecompleteda
substantialportionofthetask,beginassigningonegroupperproblemtomakealargechartand
presentittotheclass.Eachchartshouldhavealltheinformationlistedaboveaboutthesequence
thattheyhavebeenassigned.(Ineachcase,someoftheinformationwasgiven.)Askstudentstobe
preparedtodiscusshowtheyfoundeachpieceofinformation.
Discuss(WholeGroup):Askeachgrouptopresenttheirworkwiththesequenceoneatatime.
Encouragethegroupsthatarenotpresentingtobecheckingtheirworkandaskingquestionstothe
presenters.Intheinterestoftime,youmaychoosejustafewgroupstopresent,possibly#6and
#8,andthenhavetheothergroupsposttheirwork.Youcouldthenorganizea“gallerystroll”in
whichstudentswentfromonecharttothenext,comparingtheirworktothechartandformulating
questionsforthegroups.Whentheyhavehadachancetogotoeachchart,facilitateadiscussion
wherestudentscanasktheirquestionstothegroupthatdevelopedthechart.
AlignedReady,Set,GoHomework:Sequences1.11
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READY !Topic:!Comparing!linear!equations!and!arithmetic!sequences!!1.!!Describe!the!similarities!and!differences!between!linear!equations!and!arithmetic!sequences.!
Similarities* Differences*!!
!!!!!!
!
SET Topic:!Representations!of!arithmetic!sequences!!Use*the*given*information*to*complete*the*other*representations*for*each*arithmetic*sequence.*
!2.!!!!!!Recursive*Equation:************************************************************************Graph!
!!!!!!!!!!!!!!!!!!!!!!!!!!!Explicit!Equation:!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
*Table*
Days! Cost!1!2!3!4!!
8!16!24!32!
!!!
!
Create*a*context*!
!!!
!
READY, SET, GO! ******Name* ******Period***********************Date*
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3.!!!!!!Recursive*Equation:*! 1 = 4,!!!!! ! = ! ! − 1 + 3**************!!!!!!!!!!!!!!!!!!!!!!!!!!!Explicit!Equation:!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Graph!
*Table*
! !!!!!!
!!
!!!!
!
Create*a*context*!
!!!!!!
!
!!!!
4.!!!!!!Recursive*Equation:******************************************************************************Graph!!!!!!!!!!!!!!!!!!!!!!Explicit!Equation:!!!! ! = 4 + 5(! − 1)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
*Table*
! !!!!!!
!!
!!
!
Create*a*context*!
!!
!
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5.!!!!!!Recursive*Equation:*********************************************************************Graph!!!!!!!!!!!!!!!!!!!!!!Explicit!Equation:!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
*Table*
! !!!!!!
!!
!!
!
Create*a*context*Janet!wants!to!know!how!many!seats!are!in!each!row!of!the!theater.!Jamal!lets!her!know!that!each!row!has!2!seats!more!than!the!row!in!front!of!it.!The!first!row!has!14!seats.!
!!
!GO
!Topic:!Writing!explicit!equations!!Given*the*recursive*equation*for*each*arithmetic*sequence,*write*the*explicit*equation.*!6.!!!!! ! = ! ! − 1 − 2; ! 1 = 8!!!!!7.!!! ! = 5 + ! ! − 1 ; ! 1 = 0!!!!!8.!!! ! = ! ! − 1 + 1; !! 1 = !
!!!
!!!
56