Download Growing, Growing Dots

11
SECONDARY MATH 1 // MODULE 1
SEQUENCES – 1.3
CCBYms.neauxneauxnope
1.3 Growing, Growing Dots
A Develop Understanding Task
Atthe
beginning
Atoneminute Attwominutes
Atthreeminutes Atfourminutes
1. Describeandlabelthepatternofchangeyouseeintheabovesequenceoffigures.
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
https://flic.kr/p/7UMF7v
12
SECONDARY MATH 1 // MODULE 1
SEQUENCES – 1.3
2. Assumingthesequencecontinuesinthesameway,howmanydotsarethereat5minutes?
3. Writearecursiveformulatodescribehowmanydotstherewillbeaftertminutes.
4. Writeanexplicitformulatodescribehowmanydotstherewillbeaftertminutes.
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
SECONDARY MATH 1 // MODULE 1
SEQUENCES – 1.3
1.3 Growing, Growing Dots – Teacher Notes
A Develop Understanding Task
Purpose:Thepurposeofthistaskistodeveloprepresentationsforgeometricsequencesthat
studentscandrawuponthroughoutthemodule.Thevisualrepresentationinthetaskshouldevoke
listsofnumbers,tables,graphs,andequations.Variousstudentmethodsforcountingand
consideringthegrowthofthedotswillberepresentedbyequivalentexpressionsthatcanbe
directlyconnectedtothevisualrepresentation.
CoreStandards:
F-BF:Buildafunctionthatmodelsarelationshipbetweentwoquantities.
1:Writeafunctionthatdescribesarelationshipbetweentwoquantities.*
a.Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfroma
context.
F-LE:Linear,Quadratic,andExponentialModels*(SecondaryMathematicsIfocusonlinearand
exponentialonly)
Constructandcomparelinear,quadraticandexponentialmodelsandsolveproblems.
1.Distinguishbetweensituationsthatcanbemodeledwithlinearfunctionsandwith
exponentialfunctions.
a.Provethatlinearfunctionsgrowbyequaldifferencesoverequalintervalsand
thatexponentialfunctionsgrowbyequalfactorsoverequalintervals.
c.Recognizesituationsinwhichonequantitygrowsordecaysbyaconstantpercent
rateperunitintervalrelativetoanother.
2.Constructlinearandexponentialfunctions,includingarithmeticandgeometric
sequences,givenagraph,adescriptionofarelationship,ortwoinput-outputpairs(include
readingthesefromatable).
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
SECONDARY MATH 1 // MODULE 1
SEQUENCES – 1.3
Interpretexpressionforfunctionsintermsofthesituationtheymodel.
5.Interprettheparametersinalinearorexponentialfunctionintermsofacontext.
ThistaskalsofollowsthestructuresuggestedintheModelingstandard:
StandardsforMathematicalPracticeofFocusintheTask:
SMP1–Makesenseofproblemsandpersevereinsolvingthem.
SMP6–Attendtoprecision.
TheTeachingCycle:
Launch(WholeClass):Startthediscussionwiththepatternofgrowingdotsdrawnontheboard
orprojectedfortheentireclass.Askstudentstodescribethepatternthattheyseeinthedots
(Question#1).Studentsmaydescribeanincreasingnumberoftrianglesbeingaddedeachtimeor
seeingthreegroupsthateachhaveanincreasingnumberofdotseachtime,dependingonhowthey
seethegrowthoccurring.Thiswillbeexploredlaterinthediscussionasstudentswriteequations,
sothereshouldnotbeanyemphasisplaceduponaparticularwayofseeingthegrowth.Ask
studentsindividuallytoconsideranddrawthefigurethattheywouldseeat5minutes(Question
#2).Then,askonestudenttodrawitontheboardtogiveotherstudentsachancetocheckthat
theyareseeingthepatterncorrectly.Remindstudentsoftheworktheydidyesterdaytowrite
explicitandrecursiveformulas.Thesearenewtermsthatshouldbereinforcedatthebeginningto
clarifytheinstructionsforquestions3and4.
Explore(SmallGrouporPairs):Askstudentstocompletethetask.Monitorstudentsasthey
work,observingtheirstrategiesforcountingthedotsandthinkingaboutthegrowthofthefigures.
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
SECONDARY MATH 1 // MODULE 1
SEQUENCES – 1.3
Somestudentsmaythinkaboutthefiguresrecursively,describingthegrowthbysayingthatthe
nextfigureisobtaineddoublingthepreviousfigureasshown:
! = 0 ! = 1
Somemaythinkofthefigureasthreegroupsthatareeachdoubling,asshownbelow.
! = 0 ! = 1 t=2
Asstudentsworktofindtheformulas,theymaylookforpatternsinthenumbers,writingsimply3,
6,12,24,48.Ifstudentsareunabletoseeapattern,youmayencouragethemtomakeatableor
graphtoconnectthenumberofdotswiththetime:
Time(Minutes)
NumberofDots
0
3
1
6
2
12
3
24
4
48
Watchforstudentsthathaveusedagraphtoshowthenumberofdotsatagiventimeandtohelp
writeanequation.Encouragestudentstoconnecttheircountingstrategytotheequationthatthey
write.
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
SECONDARY MATH 1 // MODULE 1
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)andhelp
themtounderstandthatthisformulaexpressestheideathatawaytofindatermattimetisto
doublethepreviousterm,startingwith3attime0.
Next,askthegroupthatsawthispatternofgrowthtoexplainthewaytheysawthepatternof
growth.
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
SECONDARY MATH 1 // MODULE 1
SEQUENCES – 1.3
! = 0 ! = 1
t=2
Askforatablethatshowstherelationshipbetweentimeandthenumberofdots.Askstudents
whatpatternstheyseeinthetable.Askstudentstoaddadifferencecolumntothetable,likethey
didinGrowingDots.Studentsmaybesurprisedtoseethedifferencebetweentermsrepeatingthe
patterninthenumberofdots.Askstudentsiftheyseeacommondifferencebetweenterms.
Explainthatsincethereisnocommondifference,itisnotanarithmeticsequence.
Difference
Time(Minutes)
NumberofDots
0
3
1
6
2
12
3
24
4
48
>3
>6
>12
>24
Atthispoint,itcanbepointedoutthatsinceyougetthenexttermbydoublingthepreviousterm,
thereisacommonratiobetweenterms.Demonstratethat:
!
!
=
!"
!
=
!"
!"
= 2
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
SECONDARY MATH 1 // MODULE 1
SEQUENCES – 1.3
Thecommonratiobetweentermsistheidentifyingfeatureofageometricsequence,another
specialtypeofnumbersequence.Continuethediscussionbyaskingagrouptoshowtheirgraph.
Asktheclasswhattheypredictthegraphtolooklike.Whywouldwenotexpectthegraphtobea
line?Besurethegraphisproperlylabeled,asshown.
Numberofdots
Time(Minutes)
Now,movethediscussiontoconsiderthenumberofdotsattimet,asrepresentedbyanexplicit
equation.Askagrouptoshowtheirexplicitformulaforthenumberofdotsattimet,whichis:
! ! = 3 ∙ 2! .
Nowaskstudentstoconnecttheequationswiththetableandgraphs.Askthemtoshowwhatthe2
andthe3representinthegraph.Askhowtheysee3 ∙ 2! inthetable.Itmaybeusefultoshowthe
connectiontothetabletohelpdemonstratethepatternbetweenthetimeandthenumberofdots:
Time
NumberofDots
Difference
0
3
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! (Minutes)
>6
>12
>24
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
SECONDARY MATH 1 // MODULE 1
SEQUENCES – 1.3
Youmayalsoremindstudentsthatwhenthetableisusedtowritearecursiveequationsuchas:
! 0 = 3, ! ! = 2! ! − 1 , one maysimplylookdownthetablefromoneoutputtothenext.When
writinganexplicitformulasuchas! ! = 3 ∙ 2! ,itisnecessarytolookacrosstherowsofthetable
toconnecttheinputwiththeoutput.
Finalizethediscussionbyexplainingthatthissetoffigures,equations,table,andgraphrepresenta
geometricsequence.Ageometricsequencecanbeidentifiedbytheconstantratiobetween
consecutiveterms.Tellstudentsthattheywillcontinuetoworkwithsequencesofnumbersusing
tables,graphsandequationstoidentifyandrepresentgeometricandarithmeticsequences.
AlignedReady,Set,GoHomework:Sequences1.3
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
13
SECONDARY MATH I // MODULE 1
1.3
SEQUENCES – 1.3
READY, SET, GO!
Name
PeriodDate
READY
Topic:Interpretingfunctionnotation
A)Usethegiventabletoidentifytheindicatedvalueforn.B)Thenusingthevaluefornthatyou
determinedinA,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 !? 4.!) When ! ! = 2, what is the value of !?
!) What is the value of ! ! − 1 ?
!) What is the value of ! ! + 3 ?
2.!) When ! ! = 17, what is the value of !? 5.!) When ! ! = 27, what is the value of !?
!) What is the value of ! ! − 1 ?
!) What is the value of ! ! − 6 ?
3.!) When ! ! = 32, what is the value of !?
6.!) When ! ! = −8, what is the value of !?
!) What is the value of ! ! + 1 ?
!) What is the value of ! ! + 9 ?
SET
Topic:Comparingexplicitandrecursiveequations
Usethegiveninformationtodecidewhichequationwillbetheeasiesttousetofindtheindicated
value.Findthevalueandexplainyourchoice.
7.Explicitequation:y=3x+7
8.Explicitequation:y=3x+7
Recursive:!"# = !"#$%&'( !"#$ + 3
Recursive:!"# = !"#$%&'( !"#$ + 3
term#
1
2
3
4
term#
1
2
…
50
value
10
13
16
value
10
13
…
th
Findthevalueofthe4 term._________________
Findthevalueofthe50thterm.__________________
Explanation:
Explanation:
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
14
SECONDARY MATH I // MODULE 1
1.3
SEQUENCES – 1.3
9.Thevalueofthe8thtermis78.
Thesequenceisincreasingby10ateachstep.
Explicitequation:y=10x–2
Recursive:!"# = !"#$%&'( !"#$ + 10 Findthe20thterm.__________________________
Explanation:
10.Thevalueofthe8thtermis78.
Thesequenceisincreasingby10ateachstep.
Explicitequation:y=10x–2
Recursive:!"# = !"#$%&'( !"#$ + 10
Findthe9thterm.__________________________
Explanation:
11.Thevalueofthe4thtermis80.
Thesequenceisbeingdoubledateachstep.
Explicitequation:! = 5 2 ! Recursive: !"# = !"#$%&'( !"#$ ∗ 2
Findthevalueofthe5thterm._______________
Explanation:
12.Thevalueofthe4thtermis80.
Thesequenceisbeingdoubledateachstep.
Explicitequation:! = 5 2 ! Recursive:: !"# = !"#$%&'( !"#$ ∗ 2
Findthevalueofthe7thterm._______________
Explanation:
GO
Topic:EvaluatingExponentialEquations
Evaluatethefollowingequationswhenx={1,2,3,4,5}.Organizeyourinputsandoutputsintoa
tableofvaluesforeachequation.Letxbetheinputandybetheoutput.
13.y=4x
14y=(-3)x 15.y=-3x
16.y=10x
x
y
x
y
x
y
x
y
input output
input output
input output
input output
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
17.If! ! = 5! , !ℎ!" !" !ℎ! !"#$% !" ! 4 ?
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org