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SECONDARY MATH I // MODULE 1
1.2 Growing Dots
A Develop Understanding Task
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SEQUENCES – 1.2
1. Describethepatternthatyouseeinthesequenceoffiguresabove.
2. Assumingthepatterncontinuesinthesameway,howmanydotsarethereat3minutes?
3. Howmanydotsarethereat100minutes?
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SECONDARY MATH I // MODULE 1
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4. Howmanydotsarethereattminutes?Solvetheproblemsbyyourpreferredmethod.Your
solutionshouldindicatehowmanydotswillbeinthepatternat3minutes,100minutes,
andtminutes.Besuretoshowhowyoursolutionrelatestothepictureandhowyou
arrivedatyoursolution.
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1.2 Growing Dots– Teacher Notes
A Develop Understanding Task
Purpose:Thepurposeofthistaskistodeveloprepresentationsforarithmeticsequencesthat
studentscandrawuponthroughoutthemodule.Thevisualrepresentationinthetaskshouldevoke
listsofnumbers,tables,graphs,andequations.Variousstudentmethodsforcountingand
consideringthegrowthofthedotswillberepresentedbyequivalentexpressionsthatcanbe
directlyconnectedtothevisualrepresentation.
CoreStandards:
F-BF:Buildafunctionthatmodelsarelationshipbetweentwoquantities.
1:Writeafunctionthatdescribesarelationshipbetweentwoquantities.*
a.Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfroma
context.
F-LE:Linear,Quadratic,andExponentialModels*(SecondaryIfocusonlinearandexponential
only)
Constructandcomparelinear,quadraticandexponentialmodelsandsolveproblems.
1.Distinguishbetweensituationsthatcanbemodeledwithlinearfunctionsandwith
exponentialfunctions.
a.Provethatlinearfunctionsgrowbyequaldifferencesoverequalintervalsand
thatexponentialfunctionsgrowbyequalfactorsoverequalintervals.
b.Recognizesituationsinwhichonequantitychangesataconstantrateperunit
intervalrelativetoanother.
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2.Constructlinearandexponentialfunctions,includingarithmeticandgeometric
sequences,givenagraph,adescriptionofarelationship,ortwoinput-outputpairs(include
readingthesefromatable).
Interpretexpressionforfunctionsintermsofthesituationtheymodel.
5.Interprettheparametersinalinearorexponentialfunctionintermsofacontext.
ThistaskalsofollowsthestructuresuggestedintheModelingstandard:
StandardsforMathematicalPracticeofFocusintheTask:
SMP1:Makesenseofproblemsandpersevereinsolvingthem.
SMP7:Lookforandmakeuseofstructure.
TheTeachingCycle:
Launch(WholeClass):Startthediscussionwiththepatternongrowingdotsdrawnontheboard
orprojectedfortheentireclass.Askstudentstodescribethepatternthattheyseeinthedots
(Question#1).Studentsmaydescribefourdotsbeingaddedeachtimeinvariousways,depending
onhowtheyseethegrowthoccurring.Thiswillbeexploredlaterinthediscussionasstudents
writeequations,sothereshouldnotbeanyemphasisplaceduponaparticularwayofseeingthe
growth.Askstudentsindividuallytoconsideranddrawthefigurethattheywouldseeat3minutes
(Question#2).Then,askonestudenttodrawitontheboardtogiveotherstudentsachanceto
checkthattheyareseeingthepattern.
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Explore(SmallGrouporPairs):Askstudentstocompletethetask.Monitorstudentsasthey
work,observingtheirstrategiesforcountingthedotsandthinkingaboutthegrowthofthefigures.
Somestudentsmaythinkaboutthefiguresrecursively,describingthegrowthbysayingthatthe
nextfigureisobtainedbyplacingfourdotsontothepreviousfigureasshown:
Somemaythinkofthefigureasfourarmsoflengtht.withadotinthemiddle.
Othersmayusea“squares”strategy,noticingthatanewsquareisaddedeachminute,asshown:
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Asstudentsworktofindthenumberofdotsat100minutes,theymaylookforpatternsinthe
numbers,writingsimply1,5,9,...Ifstudentsareunabletoseeapattern,youmayencouragethem
tomakeatableorgraphtoconnectthenumberofdotswiththetime:
Time(Minutes)
Numberof
Dots
0
1
1
5
2
9
3
13
t
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.
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Difference
Time(Minutes)
NumberofDots
0
1
1
5
2
9
3
13
…
…
t
>4
>4
>4
Askstudentswheretheyseethedifferenceof4occurringinthefigures.Notethatthedifference
betweentermsisconstanteachtime.
Continuethediscussionbyaskingagrouptoshowtheirgraph.Besurethatitisproperlylabeled,
asshown.
Numberofdots
Time(Minutes)
Askstudentshowtheyseetheconstantdifferenceof4onthegraph.Theyshouldrecognizethat
they-valueincreasesby4eachtime,makingalinewithaslopeof4.
Now,movethediscussiontoconsiderthenumberofdotsattimet,asrepresentedbyanequation.
Startwithagroupthatconsideredthegrowthasarecursivepattern,recognizingthatthenextterm
is4plusthepreviousterm.Theymayrepresenttheideaas:! + 4,withXrepresentingthe
previousterm.Thismaycausesomecontroversywithstudentsthatwroteadifferentformula.Ask
thegrouptoexplaintheirworkusingthefigures.Itmaybeusefultorewritetheirformulawith
words,like:
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Thenumberofdotsinthecurrentfigure=thenumberofdotsinthepreviousfigure+4
Orsimply,Current=previous+4
Thismaybewritteninfunctionnotationas:! ! = ! ! − 1 + 4.(Althoughstudentshavesome
exposuretofunctionnotationingrade8,theyhavenotseenitusedtowriterecursiveformulas.
Youmaychoosetointroducethisnotationinlaterlessons,simplyfocusingonwritingtherecursive
ideainwordsasshownabove.)
Nextaskagroupthathasusedthe“fourarmsstrategy”towriteandexplaintheirequation.Their
equationshouldbe:! ! = 4! + 1. Askstudentstoconnecttheirequationtothefigure.They
shouldarticulatethatthereis1dotinthemiddleand4arms,eachwithtdots.The4inthe
equationshows4groupsofsizet.
Next,askagroupthatusedthe“squares”strategytodescribetheirequation.Theymayhave
writtenthesameequationasthe“fourarms”group,butaskthemtorelateeachofthenumbersin
theequationtothefiguresanyway.Inthiswayofthinkingaboutthefigures,therearetgroupsof4
dots,plus1dotinthemiddle.Althoughitisnottypicallywrittenthisway,thiscountingmethod
wouldgeneratetheequation! ! = ! ∙ 4 + 1. Mathematics Vision Project
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Nowaskstudentstoconnecttheequationswiththetableandgraphs.Askthemtoshowwhatthe4
andthe1representinthegraph.Askhowtheysee4t+1inthetable.Itmaybeusefultoshowthis
patterntohelpseethepatternbetweenthetimeandthenumberofdots:
Time(Minutes)
NumberofDots
0
1
1
1
5
1+4
2
9
1+4+4
3
13
1+4+4+4
…
…
t
1+4t
Difference
>4
>4
>4
Youmayalsopointoutthatwhenthetableisusedtowritearecursiveequationlike
! ! = ! ! − 1 + 4,youmaysimplylookdownthetablefromoneoutputtothenext.When
writinganexplicitformulalike! ! = 4! + 1,itisnecessarytolookacrosstherowsofthetableto
connecttheinputwiththeoutput.
Finalizethediscussionbyexplainingthatthissetoffigures,equations,table,andgraphrepresent
anarithmeticsequence.Anarithmeticsequencecanbeidentifiedbytheconstantdifference
betweenconsecutiveterms.Tellstudentsthattheywillbeworkingwithothersequencesof
numbersthatmaynotfitthispattern,buttables,graphsandequationswillbeusefultoolsto
representanddiscussthesequences.
AlignedReady,Set,GoHomework:Sequences1.2
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1.2
SEQUENCES – 1.2
READY, SET, GO!
Name
PeriodDate
READY
Topic:Usingfunctionnotation
Toevaluateanequationsuchas! = 5! + 1 whengivenaspecificvalueforx,replacethevariablex
withthegivenvalueandworktheproblemtofindthevalueofy.
Example:Findywhenx=2.Replacexwith2.! = 5 2 + 1 = 10 + 1 = 11. Therefore,y=11whenx=2.Thepoint 2, 11 isonesolutiontotheequation! = 5! + 1.Insteadof
using! !"# !inanequation,mathematiciansoftenwrite! ! = 5! + 1becauseitcangivemore
information.Withthisnotation,thedirectiontofind! 2 ,meanstoreplacethevalueof!with2and
worktheproblemtofind! ! .Thepoint !, ! ! isinthesamelocationonthegraphas !, ! ,where
!describesthelocationalongthex–axis,and! ! istheheightofthegraph.
Giventhat! ! = !" − !and! ! = !" − !",evaluatethefollowingfunctionswiththeindicated
values.
1.! 5 =
2.! 5 =
3.! −4 =
4.! −4 =
5.! 0 =
6.! 0 =
7.! 1 =
8.! 1 =
Topic:Lookingforpatternsofchange
Completeeachtablebylookingforthepattern.
9.
Term
Value
1st
2
2nd
4
3rd
8
4th
16
5th
32
6th
7th
8th
Term
Value
1st
66
2nd
50
3rd
34
4th
18
5th
6th
7th
8th
Term
Value
1st
160
2nd
80
3rd
40
4th
20
5th
6th
7th
8th
Term
Value
1st
-9
2nd
-2
3rd
5
4th
12
5th
6th
7th
8th
10.
11.
12.
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SECONDARY MATH I // MODULE 1
1.2
SEQUENCES – 1.2
SET
Topic:Usevariablestocreateequationsthatconnectwithvisualpatterns.
Inthepicturesbelow,eachsquarerepresentsonetile.
Step 1
Step 2
Step 4
Step 3
Step 5
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._____Nancyfocusedonthenumberofblocks
inthebasecomparedtothenumberofblocks
abovethebase.Shesaidthenumberofbase
blocksweretheoddnumbersstartingat1.And
thenumberoftilesabovethebasefollowedthe
pattern0,1,2,3,4.Sheorganizedherworkin
thetableattheright.
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Stepnumber
#inbase+#ontop
1
1+0
2
3+1
3
5+2
4
7+3
5
9+4
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SECONDARY MATH I // MODULE 1
1.2
SEQUENCES – 1.2
GO
Topic:Themeaningofanexponent
Writeeachexpressionusinganexponent.
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)
25.7(2)! A)
B)
23.5! A)
B)
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A)
B)
A)
B)
26.10 8! 24.10! 27.3 5 ! A)
B)
28.16
A)
B)
! !
!