sequences - mathematicsvisionproject.org · the purpose of this task is to solidify student...

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

SECONDARY MATH 1 // MODULE 1

SEQUENCES

Mathematics Vision Project

mathematicsvisionproject.org

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)


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)


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)


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)



SEQUENCES



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)


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)


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)


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)


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)


1.11 I Know… What Do You Know? – A Practice Understanding Task

Developing fluency with geometric and arithmetic sequences (F.LE.2)


SECONDARY MATH I // MODULE 1

SEQUENCES – 1.1


Licensed under the Creative Commons Attribution CC BY 4.0


1.1 Checkerboard Borders

A Develop Understanding Task

Inpreparationforbacktoschool,theschooladministrationplanstoreplacethetileinthecafeteria.Theywouldliketohaveacheckerboardpatternoftilestworowswideasasurroundforthetablesandservingcarts.Belowisanexampleoftheboarderthattheadministrationisthinkingofusingtosurroundasquare5x5setoftiles.A. Findthenumberofcoloredtilesinthecheckerboardborder.Trackyourthinkingandfinda wayofcalculatingthenumberofcoloredtilesintheborderthatisquickandefficient.Be preparedtoshareyourstrategyandjustifyyourwork.

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SEQUENCES – 1.1




B. Thecontractorthatwashiredtolaythetileinthecafeteriaistryingtogeneralizeawaytocalculatethenumberofcoloredtilesneededforacheckerboardbordersurroundingasquareoftileswithanydimensions.Torepresentthisgeneralsituation,thecontractorstartedsketchingthesquarebelow.

FindanexpressionforthenumberofcoloredbordertilesneededforanyNxNsquare center.

N

N

2


SEQUENCES – 1.1




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


SEQUENCES – 1.1




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


SEQUENCES – 1.1




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


SEQUENCES – 1.1




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

3


SEQUENCES – 1.1




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:

4


SEQUENCES – 1.1




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


SEQUENCES – 1.2

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


1.2 Growing Dots


1. Describethepatternthatyouseeinthesequenceoffiguresabove.

2. Assumingthepatterncontinuesinthesameway,howmanydotsarethereat3minutes?

3. Howmanydotsarethereat100minutes?

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SEQUENCES – 1.2



4. Howmanydotsarethereattminutes?Solvetheproblemsbyyourpreferredmethod.Yoursolutionshouldindicatehowmanydotswillbeinthepatternat3minutes,100minutes,andtminutes.Besuretoshowhowyoursolutionrelatestothepictureandhowyouarrivedatyoursolution.

7


SEQUENCES – 1.2



1.2 Growing Dots– Teacher Notes


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.


SEQUENCES – 1.2



2.Constructlinearandexponentialfunctions,includingarithmeticandgeometric

sequences,givenagraph,adescriptionofarelationship,ortwoinput-outputpairs(include

readingthesefromatable).

Interpretexpressionforfunctionsintermsofthesituationtheymodel.

5.Interprettheparametersinalinearorexponentialfunctionintermsofacontext.

ThistaskalsofollowsthestructuresuggestedintheModelingstandard:


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.


SEQUENCES – 1.2



Explore(SmallGrouporPairs):Askstudentstocompletethetask.Monitorstudentsasthey

work,observingtheirstrategiesforcountingthedotsandthinkingaboutthegrowthofthefigures.

Somestudentsmaythinkaboutthefiguresrecursively,describingthegrowthbysayingthatthe

nextfigureisobtainedbyplacingfourdotsontothepreviousfigureasshown:

Somemaythinkofthefigureasfourarmsoflengtht.withadotinthemiddle.

Othersmayusea“squares”strategy,noticingthatanewsquareisaddedeachminute,asshown:


SEQUENCES – 1.2



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


SEQUENCES – 1.2



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)


SEQUENCES – 1.2



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.


SEQUENCES – 1.2



Nowaskstudentstoconnecttheequationswiththetableandgraphs.Askthemtoshowwhatthe4

andthe1representinthegraph.Askhowtheysee4t+1inthetable.Itmaybeusefultoshowthis

patterntohelpseethepatternbetweenthetimeandthenumberofdots:


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


SEQUENCES – 1.2




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


8


SEQUENCES – 1.2




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

9


SEQUENCES – 1.2




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)

10


SEQUENCES – 1.3

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

1.3 Growing, Growing Dots


Atthe Atoneminute Attwominutesbeginning

Atthreeminutes Atfourminutes

1. Describeandlabelthepatternofchangeyouseeintheabovesequenceoffigures.

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SEQUENCES – 1.3


2. Assumingthesequencecontinuesinthesameway,howmanydotsarethereat5minutes?

3. Writearecursiveformulatodescribehowmanydotstherewillbeaftertminutes.

4. Writeanexplicitformulatodescribehowmanydotstherewillbeaftertminutes.

12


SEQUENCES – 1.3


1.3 Growing, Growing Dots – Teacher Notes


Purpose:Thepurposeofthistaskistodeveloprepresentationsforgeometricsequencesthat

studentscandrawuponthroughoutthemodule.Thevisualrepresentationinthetaskshouldevoke

listsofnumbers,tables,graphs,andequations.Variousstudentmethodsforcountingand

consideringthegrowthofthedotswillberepresentedbyequivalentexpressionsthatcanbe

directlyconnectedtothevisualrepresentation.

CoreStandards:




context.

F-LE:Linear,Quadratic,andExponentialModels*(SecondaryMathematicsIfocusonlinearand

exponentialonly)






c.Recognizesituationsinwhichonequantitygrowsordecaysbyaconstantpercent

rateperunitintervalrelativetoanother.





SEQUENCES – 1.3






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.


SEQUENCES – 1.3


Somestudentsmaythinkaboutthefiguresrecursively,describingthegrowthbysayingthatthe

nextfigureisobtaineddoublingthepreviousfigureasshown:

! = 0 ! = 1

Somemaythinkofthefigureasthreegroupsthatareeachdoubling,asshownbelow.

! = 0 ! = 1 t=2

Asstudentsworktofindtheformulas,theymaylookforpatternsinthenumbers,writingsimply3,

6,12,24,48.Ifstudentsareunabletoseeapattern,youmayencouragethemtomakeatableor

graphtoconnectthenumberofdotswiththetime:


0 3

1 6

2 12

3 24

4 48

Watchforstudentsthathaveusedagraphtoshowthenumberofdotsatagiventimeandtohelp

writeanequation.Encouragestudentstoconnecttheircountingstrategytotheequationthatthey

write.


SEQUENCES – 1.3


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.


SEQUENCES – 1.3


! = 0 ! = 1 t=2

Askforatablethatshowstherelationshipbetweentimeandthenumberofdots.Askstudents

whatpatternstheyseeinthetable.Askstudentstoaddadifferencecolumntothetable,likethey

didinGrowingDots.Studentsmaybesurprisedtoseethedifferencebetweentermsrepeatingthe

patterninthenumberofdots.Askstudentsiftheyseeacommondifferencebetweenterms.

Explainthatsincethereisnocommondifference,itisnotanarithmeticsequence.


0 3

1 6

2 12

3 24

4 48

Atthispoint,itcanbepointedoutthatsinceyougetthenexttermbydoublingthepreviousterm,

thereisacommonratiobetweenterms.Demonstratethat:

!! =

!"! = !"!" = 2

>3

Difference

>24

>12

>6


SEQUENCES – 1.3


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)


SEQUENCES – 1.3


Youmayalsoremindstudentsthatwhenthetableisusedtowritearecursiveequationsuchas:

! 0 = 3, ! ! = 2! ! − 1 , one maysimplylookdownthetablefromoneoutputtothenext.Whenwritinganexplicitformulasuchas! ! = 3 ∙ 2! ,itisnecessarytolookacrosstherowsofthetabletoconnecttheinputwiththeoutput.

Finalizethediscussionbyexplainingthatthissetoffigures,equations,table,andgraphrepresenta

geometricsequence.Ageometricsequencecanbeidentifiedbytheconstantratiobetween

consecutiveterms.Tellstudentsthattheywillcontinuetoworkwithsequencesofnumbersusing

tables,graphsandequationstoidentifyandrepresentgeometricandarithmeticsequences.



SEQUENCES – 1.3

1.3


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 ?




!) What is the value of ! ! + 1 ?





6.!) When ! ! = −8,what is the value of !?


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:


13


SEQUENCES – 1.3

1.3


9.Thevalueofthe8thtermis78.

Thesequenceisincreasingby10ateachstep.

Explicitequation:y=10x–2

Recursive:!"# = !"#$%&'( !"#$ + 10

Findthe20thterm.__________________________

Explanation:


Thesequenceisincreasingby10ateachstep.

Explicitequation:y=10x–2

Recursive:!"# = !"#$%&'( !"#$ + 10

Findthe9thterm.__________________________

Explanation:


Thesequenceisbeingdoubledateachstep.

Explicitequation:! = 5 2! Recursive: !"# = !"#$%&'( !"#$ ∗ 2

Findthevalueofthe5thterm._______________

Explanation:


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


SEQUENCES – 1.4


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

15


SEQUENCES – 1.4


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.

16


SEQUENCES – 1.4


1.4 Scott’s Push-Ups – Teacher Notes


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).



SEQUENCES – 1.4





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


SEQUENCES – 1.4


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


SEQUENCES – 1.4


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?


SEQUENCES – 1.4

1.4


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


SEQUENCES – 1.4

1.4


'

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


SEQUENCES – 1.5


1.5 Don’t Break the Chain


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


SEQUENCES – 1.5


2. Howmanypeoplewithreceivetheemailondayn?Explainyouranswerwithasmany

representationsaspossible.

3. IfBillgivesawayaSuperBowlthatcosts$4.95toeverypersonthatreceivestheemail

duringthefirstweek,howmuchwillhehavespent?

20


SEQUENCES – 1.5


1.5 Don’t Break the Chain– Teacher Notes


Purpose:

Thistaskistosolidifyunderstandingthatgeometricsequenceshaveaconstantratiobetween

consecutiveterms.Thetaskisdesignedtogeneratetables,graphs,andbothrecursiveandexplicit

formulas.Thefocusofthetaskshouldbetoidentifyhowtheconstantratioshowsupineachofthe

representations.

CoreStandards:




context.

F-LE:Linear,Quadratic,andExponentialModels*(SecondaryIfocusonlinearandexponential

only)













SEQUENCES – 1.5





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


SEQUENCES – 1.5


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


SEQUENCES – 1.5


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


SEQUENCES – 1.5


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.


SEQUENCES – 1.5


Concludethelessonbyaskingstudentstocomparetherecursiveandexplicitformulasfor

arithmeticandgeometricsequences.Howaretherecursiveformulasforgeometricandarithmetic

sequencesalike?Howaretheydifferent?Howaretheexplicitformulasforgeometricand

arithmeticsequencealike?Howaretheydifferent?



SEQUENCES – 1.5

1.5


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


SEQUENCES – 1.5

1.5


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


SEQUENCES – 1.5

1.5


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

23


SEQUENCES – 1.5

1.5


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


SEQUENCES – 1.6


1.6 Something to Chew On


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

25


SEQUENCES – 1.6


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


SEQUENCES – 1.6


1.6 Something to Chew On – Teacher Notes


Purpose:

Thistaskintroducesadecreasingarithmeticsequencetofurthersolidifytheideathatarithmetic

sequenceshaveaconstantdifferencebetweenconsecutiveterms.Again,connectionsshouldbe

madeamongallrepresentations:table,graph,recursiveandexplicitformulas.Theemphasis

shouldbeoncomparingincreasinganddecreasingarithmeticsequencesthroughthevarious

representations.

CoreStandards:




context.

F-LE:Linear,Quadratic,andExponentialModels*(SecondaryIfocusinlinearandexponentialonly)






b.Recognizesituationsinwhichonequantitychangesataconstantrateperunit

intervalrelativetoanother.





SEQUENCES – 1.6


Interpretexpressionsforfunctionsintermsofthesituationtheymodel.




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.


SEQUENCES – 1.6


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


SEQUENCES – 1.6


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


SEQUENCES – 1.6


Nowaskstudentstocomparetheexplicitformulasforboth#1and#2.Infunctionnotation,they

shouldbelike:

1. ! ! = 2700 − 7! 2.! ! = .25!

Askstudentshowtheycanidentifyanarithmeticsequencefromanexplicitequation.Re-

emphasizethedefinitionofanarithmeticsequenceasasequencethathasacommondifference

betweenconsecutiveterms.Thecommondifferencecanbeeitherpositiveornegative.Youmay

wishtoendthediscussionbyworkingwithstudentstocompletethechartgivenintheIntervention

Activity.



SEQUENCES – 1.6

1.6


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%

27


SEQUENCES – 1.6

1.6


!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 .!

!

28


SEQUENCES – 1.7


1.7 Chew On This


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

https://flic.kr/p/4DE

69H

29


SEQUENCES – 1.7


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


SEQUENCES – 1.7


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?

31


SEQUENCES – 1.7


1.7 Chew On This – Teacher Notes


Purpose:

Thepurposeofthistaskistosolidifyandextendtheideathatgeometricsequenceshaveaconstant

ratiobetweenconsecutivetermstoincludesequencesthataredecreasing(0<r<1).Thecommon

ratioinonegeometricsequenceisawholenumberandintheothersequenceitisapercent.This

taskcontainsanopportunitytocomparethegrowthofarithmeticandgeometricsequences.This

taskalsoprovidespracticeinwritingandusingformulasforarithmeticsequences.

CoreStandards:

F.BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.*


context.

F.LE.1Distinguishbetweensituationsthatcanbemodeledwithlinearfunctionsandwith






F.LE.2Constructlinearandexponentialfunctions,includingarithmeticandgeometricsequences,

givenagraph,adescriptionofarelationship,ortwoinput-outputpairs(includereadingthesefrom

atable).

F.IF.5Interprettheparametersinalinearorexponentialfunctionintermsofacontext.



SEQUENCES – 1.7





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:


SEQUENCES – 1.7


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


SEQUENCES – 1.7


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


SEQUENCES – 1.7


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.



SEQUENCES – 1.7

1.7


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:!__________________________________________!%

%


32


SEQUENCES – 1.7

1.7


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


SEQUENCES – 1.7

1.7


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


SEQUENCES – 1.8


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


SEQUENCES – 1.8


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


SEQUENCES – 1.8


FunctionE


22. Recursiverule:______________________________________________


24. Explicitrule:________________________________________________


FunctionF


27. Recursiverule:______________________________________________


29. Explicitrule:________________________________________________


FunctionG

31. Tofindthenextterm:_______________________________________

32. Recursiverule:_____________________________________________

33. Tofindthenthterm:_________________________________________

34. Explicitrule:_______________________________________________


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


SEQUENCES – 1.8


FunctionH


37. Recursiverule:______________________________________________


39. Explicitrule:________________________________________________


x y1 -1

2 0.2

3 -0.04

4 0.008

5 -0.0016

6 0.00032

… …

n

38


SEQUENCES – 1.8


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.*


context.

F.LE.1.Distinguishbetweensituationsthatcanbemodeledwithlinearfunctionsandwith


F.LE.2.Constructlinearandexponentialfunctions,includingarithmeticandgeometricsequences,


atable).



SMP8–Lookforandexpressregularityinrepeatedreasoning.

TheTeachingCycle:

Launch(WholeClass):Remindstudentsoftheworkdoneinprevioustasksinthismodule.Ask,

“Whatstrategiesareyouworkingonforwritingformulasforarithmeticandgeometricsequences?”


SEQUENCES – 1.8


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


SEQUENCES – 1.8


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


SEQUENCES – 1.8


Askifthereareanyotherstrategiesthattheyhavefoundforwritingformulasandrecordthem

beneaththechart.Leavethechartinviewforstudentstouseinupcomingtasks.



SEQUENCES – 1.8 1.8


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! !


39




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


SEQUENCES – 1.9


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


SEQUENCES – 1.9


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


SEQUENCES – 1.9


1.9 What Does It Mean? – Teacher Notes


Purpose:

Thepurposeofthistaskistosolidifystudentunderstandingofarithmeticsequencesbyfinding

missingtermsinthesequence.Studentswilldrawupontheirpreviousworkinusingtablesand

writingexplicitformulasforarithmeticsequences.Thetaskwillalsoreinforcetheirfluencyin

solvingequationsinonevariable.

CoreStandards:

A.REI.3Solvelinearequationsandinequalitiesinonevariableincludingequationswithcoefficients

representedbyletters.

ClusterswithInstructionalNotes:Solveequationsandinequalitiesinonevariable.

Extendearlierworkwithsolvinglinearequationstosolvinglinearinequalitiesinonevariableand

tosolvingliteralequationsthatarelinearinthevariablebeingsolvedfor.Includesimple

exponentialequationsthatrelyonlyonapplicationofthelawsofexponents,

suchas5x=125or2x=1/16.



TheTeachingCycle:

Launch(WholeClass):Referstudentstothepreviousworkwitharithmeticsequences.Inthe

past,wehavebeengiventermsandaskedtowriteformulas.Inthistask,wewillbeusingwhatwe

knowaboutsequencestofindmissingterms.Abriefdiscussionofthechartthatwascreatedas

partofthe“WhatComeNext,WhatComesLater”taskmaybeusefulindrawingstudents’attention

totheroleofthefirsttermandthecommondifferenceinwritingexplicitformulas.


SEQUENCES – 1.9


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


SEQUENCES – 1.9


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)


SEQUENCES – 1.9


Pointoutthatthisstrategyiscloselyrelatedtothepreviousstrategy.Noticehowthetwoformulas

arereallyjustthesameformularearranged?

Formulafromsecondstrategy:! = ! ! !!(!)!!!

Formulafromthirdstrategy:! ! = ! 1 + !(! − 1).

Eitherwaytheythinkaboutit,studentscangetthecommondifferenceandaddittothetermthey

knowtogetthenextterm.Askstudentsifthestrategyofwritinganequationworkswiththethree

problemsonside1andgivethemachancetoconvincethemselvesthatitwill.

Finally,askstudentstoamend,edit,oraddtotheirstrategyandexplanationonthelastquestion.

Concludethediscussionbyhavingafewstudentssharewhattheyhavewrittenwiththeclass.





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!


43




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:!__________________________________!

!

!

44


SEQUENCES – 1.10


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


CCBYJDHan

cock

https://flic.kr/p/eod

TxP

45


SEQUENCES – 1.10


Table4

x 1 2 3 4 5 6y 4 972


A. Describeyourmethodforfindingthegeometricmeans.

B. Howcanyoutelliftherewillbemorethanonesolutionforthegeometricmeans?

46


SEQUENCES – 1.10


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.



TheTeachingCycle:

Launch(WholeClass):Explaintostudentsthattoday’spuzzlesinvolvefindingmissingterms

betweentwonumbersinageometricsequence.Thesenumbersarecalled“geometricmeans”.Ask

themtorecalltheworkthattheyhavedonepreviouslywitharithmeticmeans.Ask,“What

informationdoyouthinkwillbeusefulforfindinggeometricmeans?Somemayrememberthatthe

firsttermandthecommondifferencewereimportantforarithmeticmeans;similarlythefirstterm

andthecommonratiomaybeimportantforgeometricmeans.


SEQUENCES – 1.10


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


SEQUENCES – 1.10


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:


SEQUENCES – 1.10


• Dividethelasttermbythefirstterm

• Takethen-1rootoftheresulttogetthecommonratio(Studentswillprobablynotusethis

terminology)

Ifthisoccursinclass,itmaybeproductivetocomparehowto“undo”anarithmeticsequenceanda

geometricsequence.Thedifferencesresultfromthedifferentnatureofthetwosequencetypes;

arithmeticsequencesareadditiveandgeometricsequencesaremultiplicative.





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$

47




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!

!!!!!!

!

48




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! !

49


SEQUENCES – 1.11


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

50


SEQUENCES – 1.11


3. Iknowthat:theexplicitformulaforthesequenceis! ! = −10(3)!Whatdoyouknow?

4. Iknowthat:Thefirst4termsofthesequenceare2,3,4.5,6.75...

Whatdoyouknow?

5. Iknowthat:thesequenceisarithmeticand! 3 = 10 and ! 7 = 26Whatdoyouknow?

51


SEQUENCES – 1.11


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

52


SEQUENCES – 1.11


9. Iknowthat:thefirsttermofthesequenceis-2,andthefifthtermis-!!.

Whatdoyouknow?

10. Iknowthat:agraphofthesequenceis:

Whatdoyouknow?

53


SEQUENCES – 1.11


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,


atable).




TheTeachingCycle:

Launch(WholeClass):Thetaskgivesstudentssomeinformationaboutasequenceandasksthem

tocompletetheremaininginformation.Askstudentswhataresomethingsthatwecanknowabout

asequence.Thisquestionshouldgeneratealistlike:

• Atablewith4termsofthesequence• Agraphofthesequence• Anexplicitformulaforthesequence• Arecursiveformulaforthesequence• Thecommonratioorcommondifferencebetweenterms• Whetherthesequenceisgeometric,arithmeticorneither


SEQUENCES – 1.11


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.



SEQUENCES – 1.11

1.11


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*

54


SEQUENCES – 1.11

1.11


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*!

!!

!

55


SEQUENCES – 1.11

1.11


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

sequences - mathematicsvisionproject.org · the purpose of this task is to solidify student...

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