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New York City Graphic Organizers for CMP3
Prime Time Factors and Multiples
Essential Ideas
• IfanumberNcanbewrittenasaproductoftwowholenumbers,N=a×b,thenaandbarefactorsofN.Multiplesofacanbefoundusingtheexpressiona×(somewholenumber),suchas2a,3a,4aetc.
•Whenallfactorsofanumberarebrokendownintoprimenumbers,youhaveauniqueprimefactorization.Findingtheprimefactorizationoftwonumberscanbeusefulinfindingtheleastcommonmultipleandgreatestcommonfactorofthenumbers.
•Whencalculatingthevalueofanexpression,theoperationshavetobeperformedinaconventionalorder,theorderofoperations.
•Sometimesanumericalexpressioncanbewrittenindifferentwaysbuttheexpressionsareequivalentbecausethevalueisthesame.
Investigation 1BuildingonFactorsandMultiples
Problem 1.1 PlayingtheFactorGame:FindingProperFactors
Problem 1.2 PlayingtoWin:PrimeandCompositeNumbers
Problem 1.3 TheProductGame:FindingMultiples
Problem 1.4 FindingPatterns:RectanglesandFactorPairs
Investigation 2CommonMultiplesandCommonFactors
Problem 2.1 RidingFerrisWheels:ChoosingCommonMultiplesorCommonFactors
Problem 2.2 LookingatCicadaCycles:ChoosingCommonMultiplesorCommonFactors
Problem 2.3 BaggingSnacks:ChoosingCommonMultiplesorCommonFactors
Investigation 3Factorizations:SearchingforFactorStrings
Problem 3.1 TheProductPuzzle:FactorStrings
Problem 3.2 FindingtheLongestFactorString
Problem 3.3 UsingPrimeFactorizations
Problem 3.4 UnravelingtheLockerProblem:PuttingItAllTogether
Investigation 4LinkingMultiplicationandAddition:TheDistributiveProperty
Problem 4.1 ReasoningwithEvenandOddNumbers
Problem 4.2 UsingtheDistributiveProperty
Problem 4.3 OrderingOperations
Problem 4.4 ChoosinganOperation
Investigation 1Building on Factors and Multiples
Problem 1.1 Playing the Factor Game: Finding Proper Factors
Focus Question How can you find all the factors (or divisors) of a number?
Problem 1.2 Playing to Win: Prime and Composite Numbers
Focus Question What information about a number can you find by looking at its factors?
Problem 1.3 The Product Game: Finding Multiples
Focus Question If you know one factor of a number, how can you find another factor of the number?
Problem 1.4 Finding Patterns: Rectangles and Factor Pairs
Focus Question How do you know when you have found all of the factors of a number?
Investigation 2Common Multiples and Common Factors
Problem 2.1 Riding Ferris Wheels: Choosing Common Multiples or Common Factors
Focus Question How can you decide when finding common multiples is useful in solving a problem?
Problem 2.2 Looking at Cicada Cycles: Choosing Common Multiples or Common Factors
Focus Question How can you find the least common multiple of two or more numbers?
Problem 2.3 Bagging Snacks: Choosing Common Multiples or Common Factors
Focus Question How can you decide when finding common factors is useful in solving a problem? How can you find the greatest common factor of two numbers?
Investigation 3Factorizations: Searching for Factor Strings
Problem 3.1 The Product Puzzle: Factor Strings
Focus Question How can you find the prime factorization of a number?
Problem 3.2 Finding the Longest Factor String
Focus Question How many unique prime factorizations of a number are there?
Problem 3.3 Using Prime Factorizations
Focus Question How can the prime factorization of a number be used to find the LCM and GCF of two or more numbers?
Problem 3.4 Unraveling the Locker Problem: Putting It All Together
Focus Question What characteristics of numbers, such as factors and multiples, did you use to answer the questions? What special numbers, such as prime numbers, composite numbers, and square numbers, did you use?
Investigation 4Linking Multiplication and Addition: The Distributive Property
Problem 4.1 Reasoning with Even and Odd Numbers
Focus Question How do you decide whether a number is even or odd?
Problem 4.2 Using the Distributive Property
Focus Question How is the Distributive Property used to create equivalent expressions? How is finding the area of a rectangle related to the Distributive Property?
Problem 4.3 Ordering Operations
Focus Question How do you decide the order when you work on number sentences with more than one operation?
Problem 4.4 Choosing an Operation
Focus Question How do you decide what operations are needed in a given situation?
The following pages contain a high-level graphic organizer for each Unit in Connected Mathematics 3. The first page of each graphic organizer includes the Essential Ideas of the Unit as well as a list of the Investigations and the Problems. The second page of each graphic organizer provides a full overview of the Unit, including the Focus Questions for each Problem.
Page 1 (example)
Page 2 (example)
Graphic Organizers for Grade 6 77
Prim
e Ti
me
Fact
ors
and
Mul
tiple
s
Ess
enti
al Id
eas
•Ifanu
mberN
can
bewritten
asaproduc
toftw
ow
holenum
bers,
N=a×b,the
naan
dbarefac
torsof
N.M
ultiplesof
aca
nbe
foun
dusing
the
exp
ression
a×(s
omewho
lenum
ber),su
chas2a
,3a
,4aetc.
•Whe
nallfac
torsofanu
mberarebroke
ndownintoprime
numbers,youha
veaunique
primefactorization.Findingthe
prime
factorizationoftw
onum
bersca
nbeus
efulin
find
ingthe
least
commonmultipleand
greates
tco
mmonfactorofthenu
mbers.
•Whe
nca
lculatingthe
value
ofan
exp
ression,the
operations
hav
etobeperform
edin
aconv
entiona
lorder,the
orderofoperations
.
•So
metim
esanum
ericalexp
ressionca
nbewritten
indifferen
tway
sbutthe
exp
ressions
areequiva
lentbec
ause
the
value
isthe
sam
e.
Inve
stig
atio
n 1
BuildingonFa
ctorsand
Multiples
Pro
ble
m 1
.1 P
laying
the
Fac
tor
Gam
e:FindingProperFac
tors
Pro
ble
m 1
.2 P
laying
toW
in:
Primean
dC
ompositeN
umbers
Pro
ble
m 1
.3 T
heProduc
tGam
e:
Find
ingM
ultiples
Pro
ble
m 1
.4 F
indingPatterns:
Rec
tang
lesan
dFac
torPairs
Inve
stig
atio
n 2
CommonMultiples
andC
ommonFa
ctors
Pro
ble
m 2
.1 R
idingFerris
Whe
els:C
hoosing
Common
MultiplesorCommonFa
ctors
Pro
ble
m 2
.2 L
ooking
atCicad
aCyc
les:C
hoosing
Common
MultiplesorCommonFa
ctors
Pro
ble
m 2
.3 B
aggingSna
cks:
Cho
osing
CommonMultiplesor
CommonFa
ctors
Inve
stig
atio
n 3
Factorizations
:Sea
rching
forFa
ctorString
s
Pro
ble
m 3
.1 T
heProduc
tPuz
zle:
FactorString
s
Pro
ble
m 3
.2 F
indingthe
Long
est
FactorString
Pro
ble
m 3
.3 U
sing
Prime
Factorizations
Pro
ble
m 3
.4 U
nrav
elingthe
Lo
ckerProblem:P
utting
It
AllTo
gethe
r
Inve
stig
atio
n 4
Link
ingM
ultiplicationan
d
Addition:The
DistributiveProperty
Pro
ble
m 4
.1 R
easo
ning
withEve
nan
dO
ddN
umbers
Pro
ble
m 4
.2 U
sing
the
DistributiveProperty
Pro
ble
m 4
.3 O
rderingO
perations
Pro
ble
m 4
.4 C
hoosing
an
Operation
Teacher Implementation Toolkit78
Inve
stig
atio
n 1
BuildingonFa
ctorsand
Multiples
Pro
ble
m 1
.1 P
laying
the
Fac
tor
Gam
e:FindingProperFac
tors
Focu
s Q
uest
ion
Howcan
youfin
d
allthe
fac
tors(o
rdivisors)
ofanu
mber?
Pro
ble
m 1
.2 P
laying
toW
in:
Primean
dC
ompositeN
umbers
Focu
s Q
uest
ion
Wha
tinform
ation
aboutanum
bercan
youfin
dby
looking
atitsfactors?
Pro
ble
m 1
.3 T
heProduc
tGam
e:
Find
ingM
ultiples
Focu
s Q
uest
ion
Ifyo
ukn
owone
factorofanu
mber,h
owcan
you
findano
therfac
torofthenu
mber?
Pro
ble
m 1
.4 F
indingPatterns:
Rec
tang
lesan
dFac
torPairs
Focu
s Q
uest
ion
Howdoyou
knoww
henyo
uha
vefoun
dallof
thefactorsofanu
mber?
Inve
stig
atio
n 2
CommonMultiples
andC
ommon Fa
ctors
Pro
ble
m 2
.1 R
idingFerris
Whe
els:C
hoosing
Common
MultiplesorCommonFa
ctors
Focu
s Q
uest
ion
Howcan
you
dec
idewhe
nfin
dingcommon
multiplesisuse
fulinso
lving
aproblem?
Pro
ble
m 2
.2 L
ooking
atCicad
aCyc
les:C
hoosing
Common
MultiplesorCommonFa
ctors
Focu
s Q
uest
ion
Howcan
youfin
d
theleastco
mmonmultipleoftw
o
ormorenum
bers?
Pro
ble
m 2
.3 B
aggingSna
cks:
Cho
osing
CommonMultiplesor
CommonFa
ctors
Focu
s Q
uest
ion
Howcan
you
dec
idewhe
nfin
dingcommon
factorsisuse
fulinso
lvinga
problem?Howcan
youfin
d
thegreates
tco
mmonfactorof
twonum
bers?
Inve
stig
atio
n 3
Factorizations
:Sea
rching
forFa
ctorString
s
Pro
ble
m 3
.1 T
heProduc
tPuz
zle:
FactorString
s
Focu
s Q
uest
ion
Howcan
you
findthe
primefactorizationof
a nu
mber?
Pro
ble
m 3
.2 F
indingthe
Long
est
FactorString
Focu
s Q
uest
ion
Howm
any
unique
primefactorizations
ofanu
mberarethe
re?
Pro
ble
m 3
.3 U
sing
Prime
Factorizations
Focu
s Q
uest
ion
Howcan
the
primefactorizationofanu
mber
beus
edtofind
the
LCMand
GCF
oftw
oormorenum
bers?
Pro
ble
m 3
.4 U
nrav
elingthe
Lo
ckerProblem:P
utting
It
AllTo
gethe
r
Focu
s Q
uest
ion
Wha
tch
arac
teristicsofnu
mbers,suc
has
factorsand
multiples,didyouus
etoans
werthe
que
stions
?Wha
tsp
ecialn
umbers,suc
hasprime
numbers,compositenum
bers,and
sq
uarenum
bers,didyouus
e?
Inve
stig
atio
n 4
Link
ingM
ultiplicationan
d
Addition:The
DistributiveProperty
Pro
ble
m 4
.1 R
easo
ning
withEve
nan
dO
ddN
umbers
Focu
s Q
uest
ion
Howdoyou
dec
idewhe
theranum
beriseve
norodd?
Pro
ble
m 4
.2 U
sing
the
DistributiveProperty
Focu
s Q
uest
ion
Howisthe
DistributivePropertyuse
dto
crea
teequiva
lentexp
ressions
?Howisfind
ingthe
areaof
arectan
glerelated
tothe
DistributiveProperty?
Pro
ble
m 4
.3 O
rderingO
perations
Focu
s Q
uest
ion
Howdoyou
dec
idetheorderw
henyo
uwork
onnu
mbersen
tenc
esw
ithmore
than
one
operation?
Pro
ble
m 4
.4 C
hoosing
an
Operation
Focu
s Q
uest
ion
Howdoyou
dec
idewha
toperations
are
need
edin
agiven
situa
tion?
Graphic Organizers for Grade 6 79
Com
pari
ng B
its a
nd P
iece
s R
atio
s, R
atio
nal N
umbe
rs, a
nd E
quiv
alen
ce
Ess
enti
al Id
eas
•Rationa
lnum
bersca
nbewritten
infractionordec
imalform
an
d can
bereprese
nted
aspointsordistanc
esonanu
mber-line
.Anum
ber-line
rep
rese
ntationisuse
fulfororderingand
comparing
rationa
lnum
bers.
•Frac
tions
and
dec
imalsca
nberena
med
orrepartitione
dto
findequiva
lentfractions
ordec
imals.Equiva
lenc
eisuse
fulfor
moving
betwee
nfrac
tionan
ddec
imalrep
rese
ntations
and
for so
lving problems.
•Ratiosareco
mparisons
betwee
ntw
onum
bers.Youca
nscaleratios
tom
akeeq
uiva
lentratios.Perce
ntsareratioswhe
re100
parts
represe
ntsthewho
le.
•Arateisaparticu
larkind
ofratio,w
herethe
amoun
tscompared
areindifferen
tun
its.A
unitrateisaratethathasbee
nscaled
tox:1.
Inve
stig
atio
n 1
Mak
ingC
omparisons
Pro
ble
m 1
.1 F
undraising:
ComparingW
ithFrac
tions
an
dRatios
Pro
ble
m 1
.2 F
undraising
Thermometers:In
troduc
ingRatios
Pro
ble
m 1
.3 O
ntheLine
:Equiva
lentFractions
and
the
Num
berLine
Pro
ble
m 1
.4 M
easu
ring
Progress:
Find
ingFractiona
lParts
Pro
ble
m 1
.5 C
omparing
Fund
raisingG
oals:U
sing
Fractions
an
dRatios
Inve
stig
atio
n 2
Conn
ecting
Ratiosan
dRates
Pro
ble
m 2
.1 E
qua
lSha
res:
Introduc
ingU
nitRates
Pro
ble
m 2
.2 U
nequa
lSha
res:
Using
Ratiosan
dFractions
Pro
ble
m 2
.3 M
akingComparisons
WithRateTa
bles
Inve
stig
atio
n 3
Exten
dingthe
Num
berLine
Pro
ble
m 3
.1 E
xten
dingthe
Num
berLine:In
tegersan
d
Mixed
Num
bers
Pro
ble
m 3
.2 E
stim
atingand
OrderingRationa
lNum
bers:
ComparingFractions
toBen
chmarks
Pro
ble
m 3
.3 S
haring
100
Thing
s:
Using
Ten
thsan
dH
undredths
Pro
ble
m 3
.4 D
ecim
alsonthe
Num
berLine
Pro
ble
m 3
.5 E
arthqua
ke
Relief:M
oving
FromFractions
toD
ecim
als
Inve
stig
atio
n 4
WorkingW
ithPerce
nts
Pro
ble
m 4
.1 W
hoIs
the
Bes
t?
Mak
ingSen
seofPerce
nts
Pro
ble
m 4
.2 G
eneticTraits:
Find
ingPerce
nts
Pro
ble
m 4
.3 T
heA
rtof
Comparison:U
sing
Ratios
andPerce
nts
Teacher Implementation Toolkit80
Inve
stig
atio
n 1
Mak
ingC
omparisons
Pro
ble
m 1
.1 F
undraising:
ComparingW
ithFrac
tions
an
dRatios
Focu
s Q
uest
ion
Wha
tare
twow
aystocomparea$5
00
fund
raisinggoaltoa$20
0fund
raisinggoal?
Pro
ble
m 1
.2 F
undraising
Thermometers:In
troduc
ingRatios
Focu
s Q
uest
ion
Howdoes
a
“forev
ery”
statemen
tsh
owa
ratiocomparison?
Pro
ble
m 1
.3 O
ntheLine
:Equiva
lentFractions
and
the
Num
berLine
Focu
s Q
uest
ion
Whe
nyo
ufold
frac
tionstrips,w
hatrelations
hips
doyouse
eem
ergethatsho
whow
thenu
meratoran
dden
ominator
chan
getom
akeeq
uiva
lent
frac
tions
?
Pro
ble
m 1
.4 M
easu
ring
Progress:
Find
ingFractiona
lParts
Focu
s Q
uest
ion
Howcan
fraction
stripshe
lpyoutofind
partof
anu
mber?
Pro
ble
m 1
.5 C
omparing
Fund
raisingG
oals:U
sing
Fractions
an
dRatios
Focu
s Q
uest
ion
Wha
tdoes
it
mea
nfortw
ofractions
tobe
equiva
lent?Wha
tdoes
itm
eanfor
tworatiostobeeq
uiva
lent?
Inve
stig
atio
n 2
Conn
ecting
Ratiosan
dRates
Pro
ble
m 2
.1 E
qua
lSha
res:
Introduc
ingU
nitRates
Focu
s Q
uest
ion
Wha
tdoes
aunit
ratecomparisonstatem
enttellus
?
Pro
ble
m 2
.2 U
nequa
lSha
res:
Using
Ratiosan
dFractions
Focu
s Q
uest
ion
Howarepart-to-
partratiorelations
hipsrelatedto
part-to-w
holefractions
?
Pro
ble
m 2
.3 M
akingComparisons
WithRateTa
bles
Focu
s Q
uest
ion
Howdorate
tableshe
lpusfin
dequiva
lent
ratios?
Inve
stig
atio
n 3
Exten
dingthe
Num
berLine
Pro
ble
m 3
.1 E
xten
dingthe
Num
berLine:In
tegersan
d
Mixed
Num
bers
Focu
s Q
uest
ion
Howcan
the
nu
mberline
helpyouthinkab
out
frac
tions
greatertha
n1an
dle
ss
than
0?
Pro
ble
m 3
.2 E
stim
atingand
OrderingRationa
lNum
bers:
ComparingFractions
toBen
chmarks
Focu
s Q
uest
ion
Whe
nco
mparing
tworationa
lnum
bers,w
hat
areso
meus
efulstrateg
iesfor
dec
idingw
hich
isgreater?
Pro
ble
m 3
.3 S
haring
100
Thing
s:
Using
Ten
thsan
dH
undredths
Focu
s Q
uest
ion
Howdoes
wha
tyo
ukn
owaboutfractions
helpyou
understan
ddec
imals?
Pro
ble
m 3
.4 D
ecim
alsonthe
Num
berLine
Focu
s Q
uest
ion
Howdow
eus
ewha
twekn
owaboutfractions
to
estimatean
dcomparedec
imals?
Pro
ble
m 3
.5 E
arthqua
ke
Relief:M
oving
FromFractions
toD
ecim
als
Focu
s Q
uest
ion
Why
does
itm
akese
nsetodividethe
numeratorofafrac
tionbythe
den
ominatortofind
aneq
uiva
lent
dec
imalrep
rese
ntation?
Inve
stig
atio
n 4
WorkingW
ithPerce
nts
Pro
ble
m 4
.1 W
hoIs
the
Bes
t?
Mak
ingSen
seofPerce
nts
Focu
s Q
uest
ion
Howisaperce
nt
baruse
fulinmak
ingcomparisons
withdec
imals?
Pro
ble
m 4
.2 G
eneticTraits:
Find
ingPerce
nts
Focu
s Q
uest
ion
Howcan
partitioning
beus
edtoexp
ress
one
num
berasaperce
ntof
anothernum
ber?
Pro
ble
m 4
.3 T
heA
rtof
Comparison:U
sing
Ratios
andPerce
nts
Focu
s Q
uest
ion
Inw
hatway
isa
perce
ntlike
aratioand
like
afrac
tion?
Graphic Organizers for Grade 6 81
Let’s
Be
Rat
iona
l U
nder
stan
ding
Fra
ctio
n O
pera
tions
Ess
enti
al Id
eas
•Estim
ationisanim
portan
tpartofreasoning
qua
ntitatively.
It enc
ourag
esm
akingsen
seofasituation,allo
wsyo
uto
reco
gnize
errors,a
ndcomplemen
tsotherproblemsolvingskills.
•Fo
rea
choperation,the
reisaneffic
ient,g
eneralalgorithm
for co
mputingw
ithfrac
tions
tha
tworksinallca
ses.
•Va
riab
lesreprese
ntunk
nownva
lues
.
•So
metim
esrew
riting
aproblemusing
adifferen
toperation
can be he
lpfulinfin
dingthe
solution.
Inve
stig
atio
n 1
Exten
dingA
dditionan
d
Subtrac
tionofFrac
tions
Pro
ble
m 1
.1 G
etting
Close
:Estim
atingSum
s
Pro
ble
m 1
.2 E
stim
atingSum
san
dD
ifferen
ces
Pro
ble
m 1
.3 L
andSec
tions
:Addingand
Sub
trac
ting
Fractions
Pro
ble
m 1
.4 V
isitingthe
Spice
Shop:A
ddingand
Sub
trac
ting
Mixed
Num
bers
Inve
stig
atio
n 2
BuildingonMultiplication
WithFrac
tions
Pro
ble
m 2
.1 H
owM
uchofthe
Pan
Hav
eWeSo
ld?Find
ingParts
ofParts
Pro
ble
m 2
.2 M
odeling
MultiplicationSituations
Pro
ble
m 2
.3 C
hang
ing
Form
s:M
ultiplicationWith
Mixed
Num
bers
Inve
stig
atio
n 3
DividingW
ithFrac
tions
Pro
ble
m 3
.1 P
reparingFood:
DividingaFractionbyaFrac
tion
Pro
ble
m 3
.2 IntoPiece
s:W
hole
Num
bersorMixed
Num
bers
Divided
byFrac
tions
Pro
ble
m 3
.3 S
haring
aPrize
:DividingaFractionbya
Who
leN
umber
Pro
ble
m 3
.4 E
xamining
AlgorithmsforDividingFractions
Inve
stig
atio
n 4
WrappingU
pthe
Operations
Pro
ble
m 4
.1 J
usttheFa
cts:
FactFam
ilies
forAddition
andSub
trac
tion
Pro
ble
m 4
.2 M
ultiplicationan
d
DivisionFa
ctFam
ilies
Pro
ble
m 4
.3 B
ecomingan
Operations
Sleuth
Teacher Implementation Toolkit82
Inve
stig
atio
n 1
Exten
dingA
dditionan
d
Subtrac
tionofFrac
tions
Pro
ble
m 1
.1 G
etting
Close
:Estim
atingSum
s
Focu
s Q
uest
ion
Wha
tareso
me
strategiesfores
timatingthe
sum
soffrac
tions
?
Pro
ble
m 1
.2 E
stim
atingSum
san
dD
ifferen
ces
Focu
s Q
uest
ion
Howdoyou
knowifyourestim
ateisan
underes
timateoran
ove
restim
ate?
Wha
tinform
ationdoes
an
underes
timateorove
restim
ate
tellyo
u?
Pro
ble
m 1
.3 L
andSec
tions
:Addingand
Sub
trac
ting
Fractions
Focu
s Q
uest
ion
Wha
tare
somestrategiesforad
dingand
su
btrac
ting
fractions
?
Pro
ble
m 1
.4 V
isitingthe
Spice
Shop:A
ddingand
Sub
trac
ting
Mixed
Num
bers
Focu
s Q
uest
ion
Wha
tare
somestrategiesforad
dingand
su
btrac
ting
mixed
num
bers?
Inve
stig
atio
n 2
BuildingonMultiplication
WithFrac
tions
Pro
ble
m 2
.1 H
owM
uchofthe
Pan
Hav
eWeSo
ld?Find
ingParts
ofParts
Focu
s Q
uest
ion
Howdoes
an
area
modelrelatetom
ultiplying
frac
tions
?
Pro
ble
m 2
.2 M
odeling
MultiplicationSituations
Focu
s Q
uest
ion
Wha
tstrategies
canyo
uus
etom
ultiplyall
combinations
offactorsin
clud
ing
who
lenum
bers,fractions
,and
mixed
num
bers?
Pro
ble
m 2
.3 C
hang
ing
Form
s:M
ultiplicationWith
Mixed
Num
bers
Focu
s Q
uest
ion
Howcan
you
usenu
mberproperties
and
eq
uiva
lentfractions
tom
ultiply
rationa
lnum
bers?
Inve
stig
atio
n 3
DividingW
ithFrac
tions
Pro
ble
m 3
.1 P
reparingFood:
DividingaFractionbyaFrac
tion
Focu
s Q
uest
ion
Wha
tdoes
it
mea
ntodivideafrac
tionbya
frac
tion?
Wha
tstrategieshe
lpyou
divideafrac
tionbyafrac
tion?
Pro
ble
m 3
.2 IntoPiece
s:W
hole
Num
bersorMixed
Num
bers
Divided
byFrac
tions
Focu
s Q
uest
ion
Wha
tdoes
it
mea
ntodivideawho
lenum
beror
mixed
num
berbyafrac
tion?
Wha
tstrategieshe
lpyoudivideawho
le
numberormixed
num
berby
afrac
tion?
Pro
ble
m 3
.3 S
haring
aPrize
:DividingaFractionbya
Who
leN
umber
Focu
s Q
uest
ion
Wha
tdoes
it
mea
ntodivideafrac
tionbya
who
lenum
ber?Wha
tstrategies
helpyoudivideafrac
tionbya
who
lenum
ber?
Pro
ble
m 3
.4 E
xamining
AlgorithmsforDividingFractions
Focu
s Q
uest
ion
Wha
tisan
effic
ientalgorithmfordivision
problemsinvo
lvingfractions
and
mixed
num
bers?
Inve
stig
atio
n 4
WrappingU
pthe
Operations
Pro
ble
m 4
.1 J
usttheFa
cts:
FactFam
ilies
forAddition
andSub
trac
tion
Focu
s Q
uest
ion
Howdofac
tfamilies
helpyouso
lveeq
uations
su
chas
?
Pro
ble
m 4
.2 M
ultiplicationan
d
DivisionFa
ctFam
ilies
Focu
s Q
uest
ion
Howdofac
tfamilies
helpyouso
lveeq
uations
su
chas?
Pro
ble
m 4
.3 B
ecomingan
Operations
Sleuth
Focu
s Q
uest
ion
Howdoyou
knoww
henaparticu
laroperation
iscalledfortosolveaproblem?
Howdoyoureprese
ntthe
problemw
ithanu
mbersen
tenc
e?
4 5–
N=3 8
2 9÷N
=2 3
Graphic Organizers for Grade 6 83
Cove
ring
and
Sur
roun
ding
Tw
o-D
imen
sion
al M
easu
rem
ent
Ess
enti
al Id
eas
•Polygons
and
irregularfigures
can
bedec
ompose
din
totrian
gles
andrec
tang
lestofind
the
areaofthefig
ures
.
•Afixe
dnum
berofarea
unitscan
been
close
dbyman
ydifferen
tperim
eters,and
afixe
dnum
berofperim
eterunitscan
enc
lose
man
ydifferen
tarea
s.
•Fo
rmulasforthearea
and
perim
eterofarectan
glecan
help
you so
lveproblemsbyreasoning
aboutthe
relations
hip
betwee
n va
lues
.
•Th
evo
lumeofaprism
can
bethoug
htofasm
ultiplyinga
base laye
rofun
itcub
esbythenu
mberoflaye
rsnee
ded
to
fill the
prism
.
•Su
rfac
earea
softhree-dim
ensiona
lsolid
sca
nbefoun
d
by ad
ding the
areasoftheface
s.
Inve
stig
atio
n 1
Des
igning
Bum
perC
ars:
Exten
dingand
Buildingon
Area an
dPerim
eter
Pro
ble
m 1
.1 D
esigning
Bum
per-
CarRides
:Areaan
dPerim
eter
Pro
ble
m 1
.2 B
uildingStorm
Sh
elters:C
ons
tantA
rea,
Cha
ngingPerim
eter
Pro
ble
m 1
.3 F
encing
in
Spac
es:C
ons
tantPerim
eter,
Cha
ngingA
rea
Inve
stig
atio
n 2
Mea
suring
Trian
gles
Pro
ble
m 2
.1 T
rian
glesonGrids:
Find
ingA
reaan
dPerim
eter
ofTriang
les
Pro
ble
m 2
.2 M
oreTrian
gles:
Iden
tifyingBasean
dH
eight
Pro
ble
m 2
.3 M
akingFam
ilies
of
Triang
les:M
aintaining
the
Base
andthe
Height
Pro
ble
m 2
.4 D
esigning
Trian
gles
Und
erC
ons
traints
Inve
stig
atio
n 3
Mea
suring
Parallelograms
Pro
ble
m 3
.1 P
arallelograms
andTrian
gles:FindingA
reaan
d
Perim
eterofParallelograms
Pro
ble
m 3
.2 M
akingFam
ilies
of
Parallelograms:M
aintaining
the
Basean
dthe
Height
Pro
ble
m 3
.3 D
esigning
ParallelogramsUnd
erC
ons
traints
Pro
ble
m 3
.4 P
olygons
on
CoordinateGrids
Inve
stig
atio
n 4
Mea
suring
Surface
Area
andVolume
Pro
ble
m 4
.1 M
aking
Rec
tang
ularBoxe
s
Pro
ble
m 4
.2 F
illingthe
Boxe
s:
Find
ingVolume
Pro
ble
m 4
.3 D
esigning
Gift
Boxe
s:FindingSurface
Area
Teacher Implementation Toolkit84
Inve
stig
atio
n 1
Des
igning
Bum
perC
ars:
Exten
dingand
BuildingonArea
andPerim
eter
Pro
ble
m 1
.1 D
esigning
Bum
per-
CarRides
:Areaan
dPerim
eter
Focu
s Q
uest
ion
Wha
tarethe
form
ulasforfin
dingthe
areaan
d
perim
eterofarectan
gle?Exp
lain
why
the
ywork.
Pro
ble
m 1
.2 B
uildingStorm
Sh
elters:C
ons
tantA
rea,
Cha
ngingPerim
eter
Focu
s Q
uest
ion
Forafix
edarea,
wha
tarethesh
apean
dperim
eter
oftherectan
gleswiththegreates
tan
dle
astperim
eters?
Pro
ble
m 1
.3 F
encing
in
Spac
es:C
ons
tantPerim
eter,
Cha
ngingA
rea
Focu
s Q
uest
ion
Forafix
ed
perim
eter,w
hatarethesh
apean
d
area
oftherectan
gleswiththe
greates
tan
dle
astarea
?
Inve
stig
atio
n 2
Mea
suring
Trian
gles
Pro
ble
m 2
.1 T
rian
glesonGrids:
Find
ingA
reaan
dPerim
eter
ofTriang
les
Focu
s Q
uest
ion
Wha
tisaform
ula
forfin
dingthe
areaofatriang
le?
Pro
ble
m 2
.2 M
oreTrian
gles:
Iden
tifyingBasean
dH
eight
Focu
s Q
uest
ion
Does
itm
akean
ydifferen
cew
hich
sideisuse
das
thebasewhe
nfin
dingthe
areaof
atriang
le?
Pro
ble
m 2
.3 M
akingFam
ilies
ofTriang
les:M
aintaining
the
Base
andthe
Height
Focu
s Q
uest
ion
Wha
tca
nyo
usay
istruean
dw
hatca
nyo
usayisnot
true
abouttrian
glesthathav
ethe
samebasean
dheight?
Pro
ble
m 2
.4 D
esigning
Trian
gles
Und
erC
ons
traints
Focu
s Q
uest
ion
Wha
tco
nditions
foratriang
leproduc
etriang
les
thathav
ethesamearea
?Dothe
yha
vethe
sam
esh
ape?
Exp
lain.
Inve
stig
atio
n 3
Mea
suring
Parallelograms
Pro
ble
m 3
.1 P
arallelograms
andTrian
gles:FindingA
reaan
d
Perim
eterofParallelograms
Focu
s Q
uest
ion
Wha
tisa
strategyforfin
dingthe
areaofa
parallelogram?Exp
lainw
hythe
strategyworks.
Pro
ble
m 3
.2 M
akingFam
ilies
of
Parallelograms:M
aintaining
the
Basean
dH
eight
Focu
s Q
uest
ion
Wha
tca
nyo
usayab
outtwoparallelogramsthat
have
the
sam
ebasean
dheight?
Pro
ble
m 3
.3 D
esigning
ParallelogramsUnd
erC
ons
traints
Focu
s Q
uest
ion
Und
erw
hat
cond
itions
willtwoormore
parallelogramsha
vethe
sam
earea
?Dothe
separallelograms
have
the
sam
esh
ape?
Exp
lain.
Pro
ble
m 3
.4 P
olygons
on
CoordinateGrids
Focu
s Q
uest
ion
Howcan
youfin
d
thearea
ofapolygondrawnona
coordinategraph?
Ongridpap
er?
Inve
stig
atio
n 4
Mea
suring
Surface
Area
andVolume
Pro
ble
m 4
.1 M
aking
Rec
tang
ularBoxe
s
Focu
s Q
uest
ion
Wha
tisa
strategyforfin
dingthe
surface
area
ofarectan
gularprism
?Exp
lainw
hythe
strateg
yworks.
Pro
ble
m 4
.2 F
illingthe
Boxe
s:
Find
ingVolume
Focu
s Q
uest
ion
Wha
tisa
strategyforfin
dingthe
volumeof
arectan
gularprism
?Exp
lainw
hy
thestrategyworks.
Pro
ble
m 4
.3 D
esigning
Gift
Boxe
s:FindingSurface
Area
Focu
s Q
uest
ion
Wha
tisa
strategyforfin
dingthe
surface
area
ofathree-dim
ensiona
lobject?Exp
lainw
hythe
strategyworks.
Graphic Organizers for Grade 6 85
Dec
imal
Ops
Com
putin
g W
ith D
ecim
als
and
Perc
ents
Ess
enti
al Id
eas
•Estim
ationisanim
portan
tpartofreasoning
qua
ntitatively.Ithelps
youmak
ese
nseofasituation,allo
wsyo
utorec
ognize
errors,a
nd
complemen
tsotherproblemsolvingskills.
•Th
estan
dardalgorithmfordividingdec
imalsissup
ported
bythe
conn
ections
betwee
nfrac
tionan
ddec
imaloperations
.
•Flue
ncywithdec
imaloperations
allo
wyoutosolveava
riety
of problemsinvo
lvingratiosan
dperce
nts.
•Inve
rseoperations
can
beus
edtoisolateavariablew
hen
solvingequa
tions
.
Inve
stig
atio
n 1
Dec
imalO
perations
an
dEstim
ation
Pro
ble
m 1
.1 O
uttoLun
ch:
Match
ingO
perations
an
dQ
uestions
Pro
ble
m 1
.2 G
etting
Close
:Estim
atingD
ecim
alC
alcu
lations
Pro
ble
m 1
.3 Tak
eaHike:
Conn
ecting
Ratios,Rates
,an
dD
ecim
als
Inve
stig
atio
n 2
Addingand
Sub
trac
ting
Dec
imals
Pro
ble
m 2
.1 G
etting
Thing
sin
theRightPlace
:AddingD
ecim
als
Pro
ble
m 2
.2 W
hat’s
the
Differen
ce?:Sub
trac
ting
Dec
imals
Pro
ble
m 2
.3 C
onn
ecting
Operations
:Fac
tFa
milies
Inve
stig
atio
n 3
Multiplyingand
DividingD
ecim
als
Pro
ble
m 3
.1 It’s
Dec
imalTim
e(s):
MultiplyingD
ecim
alsI
Pro
ble
m 3
.2 ItWorksEve
ryTim
e:
MultiplyingD
ecim
alsII
Pro
ble
m 3
.3 H
owM
anyTimes
?DividingD
ecim
alsI
Pro
ble
m 3
.4 G
oingthe
Long
Way
:DividingD
ecim
alsII
Pro
ble
m 3
.5 C
halle
ngingC
ases
:DividingD
ecim
alsIII
Inve
stig
atio
n 4
Using
Perce
nts
Pro
ble
m 4
.1 W
hat’s
the
Tax
on
ThisItem
?
Pro
ble
m 4
.2 C
omputingTips
Pro
ble
m 4
.3 P
erce
ntD
isco
unts
Pro
ble
m 4
.4 P
utting
Operations
To
gethe
r
Teacher Implementation Toolkit86
Inve
stig
atio
n 1
Dec
imalO
perations
an
dEstim
ation
Pro
ble
m 1
.1 O
uttoLun
ch:
Match
ingO
perations
an
dQ
uestions
Focu
s Q
uest
ion
Wha
tsigna
lsin
a
real-w
orldproblemtelly
ouwhich
operationtouse
?
Pro
ble
m 1
.2 G
etting
Close
:Estim
atingD
ecim
alC
alcu
lations
Focu
s Q
uest
ion
Whe
nyo
uwork
withdec
imalcomputations
,wha
tstrategiesca
nyo
uus
etoestim
ate
theresu
lts?
Pro
ble
m 1
.3 Tak
eaHike:
Conn
ecting
Ratios,Rates
,an
dD
ecim
als
Focu
s Q
uest
ion
Howcan
you
expressaunitrateasadec
imal
anduse
ittosolveproblems?
Inve
stig
atio
n 2
Addingand
Sub
trac
ting
Dec
imals
Pro
ble
m 2
.1 G
etting
Thing
sin
theRightPlace
:AddingD
ecim
als
Focu
s Q
uest
ion
Howdoyou
useplace
value
toaddtwogiven
dec
imalnum
bers?
Pro
ble
m 2
.2 W
hat’s
the
Differen
ce?:Sub
trac
ting
Dec
imals
Focu
s Q
uest
ion
Howdoyou
subtrac
tone
dec
imalnum
ber
fromano
ther?
Pro
ble
m 2
.3 C
onn
ecting
Operations
:Fac
tFa
milies
Focu
s Q
uest
ion
Dofac
tfamilies
ap
plytooperations
with
dec
imalnum
bers?
Inve
stig
atio
n 3
Multiplyingand
DividingD
ecim
als
Pro
ble
m 3
.1 It’s
Dec
imalTim
e(s):
MultiplyingD
ecim
alsI
Focu
s Q
uest
ion
Howdoyou
findthe
produc
tofan
ytw
o
dec
imalnum
bers?
Pro
ble
m 3
.2 ItWorksEve
ryTim
e:
MultiplyingD
ecim
alsII
Focu
s Q
uest
ion
Wha
talgorithm
canbeus
edtofind
any
dec
imalproduc
t?
Pro
ble
m 3
.3 H
owM
anyTimes
?DividingD
ecim
alsI
Focu
s Q
uest
ion
Howcan
a
dec
imaldivisionproblembe
written
inequiva
lentfractionan
d
who
lenum
berform
?
Pro
ble
m 3
.4 G
oingthe
Long
Way
:DividingD
ecim
alsII
Focu
s Q
uest
ion
Howcan
you
carryoutadec
imaldivisionus
inga
metho
dsim
ilartolo
ngdivisionof
who
lenum
bers?
Pro
ble
m 3
.5 C
halle
ngingC
ases
:DividingD
ecim
alsIII
Focu
s Q
uest
ion
Howcan
you
completealo
ngdivisionproblem
thatdoes
n’tgiveawho
lenum
ber
quo
tien
t?Tha
tis,h
owdo
youex
pressrem
aind
ersin
dec
imalform
?
Inve
stig
atio
n 4
Using
Perce
nts
Pro
ble
m 4
.1 W
hat’s
the
Tax
on
ThisItem
?
Focu
s Q
uest
ion
Howdoyou
findthe
tax
and
the
totalc
ostof
anitem
fromagiven
sellin
gprice
an
dtax
rate?
Howdoyoufin
dthe
baseprice
fromagiven
tax
rate
andamoun
t?
Pro
ble
m 4
.2 C
omputingTips
Focu
s Q
uest
ion
Howdoyou
findthe
tipand
the
totalc
ostofa
restau
rantm
ealfromagiven
mea
lprice
and
tiprate?
Howdoyou
findthe
mea
lprice
fromagiven
tipperce
ntand
amoun
t?
Pro
ble
m 4
.3 P
erce
ntD
isco
unts
Focu
s Q
uest
ion
Howdoyoufin
d
thedisco
untan
dthe
totalc
ost
ofan
item
fromagiven
sellin
g
price
and
disco
untrate?How
doyoufin
dthe
baseprice
fromagiven
disco
untrateand
am
oun
t?H
owcan
youex
press
ach
angeinagiven
amoun
tas
aperce
ntcha
nge?
Pro
ble
m 4
.4 P
utting
Operations
To
gethe
r
Focu
s Q
uest
ion
Howdoyou
dec
idewhich
operations
to
perform
whe
naproblemin
volves
dec
imalsan
dperce
nts?
Graphic Organizers for Grade 6 87
Vari
able
s an
d Pa
ttern
s Fo
cus
on A
lgeb
ra
Ess
enti
al Id
eas
•Inm
anyreal-w
orldsitua
tions
,one
variablequa
ntitydep
endson
another.T
ables,graphs
,and
equa
tions
arevarious
rep
rese
ntations
thatcan
beus
edtobetterun
derstan
dthe
patternofch
ange
betwee
nva
riab
lequa
ntities.
•Notallrelations
hipsarelin
ear.Line
arrelations
hipsha
veacons
tant
rateofch
angebetwee
nva
riab
lesan
darew
ritten
inthe
form
y=m
x,y=b+x,a
ndy=b+m
x.
•Th
ereism
oretha
none
way
tow
riteanex
pressiontom
odelarea
l-worldsitua
tion.Properties
ofoperations
allo
wyoutogen
erate
equiva
lentexp
ressions
and
che
ckequiva
lenc
e.
•So
lutions
foreq
uations
and
ineq
ualitiesca
nbefoun
dby
exam
iningthe
tab
leorgraphoftheeq
uationorbyrewriting
itas
arelatedequa
tion.
Inve
stig
atio
n 1
Variab
les,Tab
les,and
Graphs
Pro
ble
m 1
.1 G
etting
Rea
dyto
Ride:D
ata,Tab
les,and
Graphs
Pro
ble
m 1
.2 F
romA
tlan
ticCityto
Lewes
:Tim
e,Rate,and
Distanc
e
Pro
ble
m 1
.3 F
romLew
esto
Chinc
oteag
ueIsland
:Stories
,Ta
bles,and
Graphs
Pro
ble
m 1
.4 F
romC
hinc
oteag
ue
toC
olonialW
illiamsb
urg:
Ave
rageSp
eed
Inve
stig
atio
n 2
Ana
lyzing
Relations
hips
Among
Variables
Pro
ble
m 2
.1 R
enting
Bicyc
les:
Indep
enden
tan
dD
epen
den
tVa
riab
les
Pro
ble
m 2
.2 F
indingC
ustomers:
Line
arand
Non-Line
arPatterns
Pro
ble
m 2
.3 P
redicting
Profits:
FourQ
uadrantG
raphing
Pro
ble
m 2
.4 W
hat’s
the
Story?:
Interpreting
Graphs
Inve
stig
atio
n 3
RelatingVariablesWithEqua
tions
Pro
ble
m 3
.1 V
isittoW
ild
World:F
unctionRules
With
One
Operation
Pro
ble
m 3
.2 M
oving
,Tex
ting
,an
dM
easu
ring
:Using
Rates
and
RateTa
bles
Pro
ble
m 3
.3 G
roup
Disco
unts
andaBonu
sCard:F
unctions
With
TwoO
perations
Pro
ble
m 3
.4 G
etthe
Calcu
lations
Right:E
xpressions
and
Order
ofOperations
Inve
stig
atio
n 4
Exp
ressions
,Equa
tions
,and
Ineq
ualities
Pro
ble
m 4
.1 Tak
ingthe
Plung
e:
Equiva
lentExp
ressions
I
Pro
ble
m 4
.2 E
quiva
lent
Exp
ressions
II
Pro
ble
m 4
.3 P
utting
Itall
Togethe
r:Equiva
lentExp
ressions
III
Pro
ble
m 4
.4 F
indingthe
Unk
nown
Value:SolvingEqua
tions
Pro
ble
m 4
.5 S
olvingIn
equa
lities
Teacher Implementation Toolkit88
Inve
stig
atio
n 1
Variab
les,Tab
les,and
Graphs
Pro
ble
m 1
.1 G
etting
Rea
dyto
Ride:D
ata,Tab
les,and
Graphs
Focu
s Q
uest
ion
Given
atab
le
ofdatash
owinghowaqua
ntity
chan
ges
ove
rtime,howcan
you
cons
truc
tagraphofthatdata?
Wha
tca
nyo
utellfromthe
pattern
ofpointsin
the
graph?
Pro
ble
m 1
.2 F
romA
tlan
ticCityto
Lewes
:Tim
e,Rate,and
Distanc
e
Focu
s Q
uest
ion
Wha
tarethe
adva
ntag
esand
disad
vantag
esof
tablesan
dgraphs
indisco
vering
an
ddes
cribingthe
patternof
chan
geinavariableove
rtime?
Pro
ble
m 1
.3 F
romLew
esto
Chinc
oteag
ueIsland
:Stories
,Ta
bles,and
Graphs
Focu
s Q
uest
ion
Which
prese
ntationofdata—
table,
graph,orwritten
notes—
seem
sto
bettersh
owpatternsofch
angein
distanc
eove
rtimean
dw
hy?
Pro
ble
m 1
.4 F
romC
hinc
oteag
ue
toC
olonialW
illiamsb
urg:
Ave
rageSp
eed
Focu
s Q
uest
ion
Howdoyou
calculateav
erag
esp
eedforatrip?
Howdoatab
leand
agraphof
(tim
e,distanc
e)datash
owspee
d?
Inve
stig
atio
n 2
Ana
lyzing
Relations
hips
Among
Variables
Pro
ble
m 2
.1 R
enting
Bicyc
les:
Indep
enden
tan
dD
epen
den
tVa
riab
les
Focu
s Q
uest
ion
Whe
ntw
o
variab
lesinasitua
tionarerelated,
howdoyoudec
idewhich
tocall
theindep
enden
tva
riab
leand
which
the
dep
enden
tva
riab
le?
Pro
ble
m 2
.2 F
indingC
ustomers:
Line
arand
Non-Line
arPatterns
Focu
s Q
uest
ion
Howare
therelations
hipsbetwee
nindep
enden
tan
ddep
enden
tva
riab
lesinthisProblemdifferen
tfromtho
sein
Problem2.1?How
arethedifferen
cessh
ownintab
les
andgraphs
ofdata?
Pro
ble
m 2
.3 P
redicting
Profits:
FourQ
uadrantG
raphing
Focu
s Q
uest
ion
Howarethe
va
riab
lestourin
comean
dtour
profitrelated
toeac
hother?How
doyouplotdatapointsw
ithone
orbothcoordinates
neg
ative?
Pro
ble
m 2
.4 W
hat’s
the
Story?:
Interpreting
Graphs
Focu
s Q
uest
ion
Whe
nthe
relations
hipbetwee
ndep
enden
tan
din
dep
enden
tva
riab
lesis
displaye
din
agraph,w
hatca
nyo
ulearnab
outthe
relations
hipfrom
arising
graph,ale
velg
raph,and
a
falling
graph?
Inve
stig
atio
n 3
RelatingVariablesWithEqua
tions
Pro
ble
m 3
.1 V
isittoW
ild
World:F
unctionRules
With
One
Operation
Focu
s Q
uest
ion
Inw
hatkind
sof
situations
willthe
equa
tionforthe
relations
hipbetwee
ndep
enden
tan
din
dep
enden
tva
riab
lesbein
theform
y=x+k?
y=x–k?
y =
kx?
y =
x/k?
Pro
ble
m 3
.2 M
oving
,Tex
ting
,an
dM
easu
ring
:Using
Rates
and
RateTa
bles
Focu
s Q
uest
ion
Wha
tca
nyo
utellab
outarelations
hipbetwee
ndep
enden
tan
din
dep
enden
tva
riab
leswhe
ngiven
aneq
uation
inthe
form
y=m
x?H
owistha
trelations
hipsho
wninatab
leand
a
graphofsample(x
,y)v
alue
s?W
hy
isthe
point(1,m
)onev
erygraph?
Pro
ble
m 3
.3 G
roup
Disco
unts
andaBonu
sCard:F
unctions
With
TwoO
perations
Focu
s Q
uest
ion
Howdoyou
calculateva
lues
of
yfroman
equa
tionlik
ey=3
x+5w
hen
values
of
xaregiven
?Howabout
y=5+3
x?W
hendoyoune
ed
such
equa
tions
tha
tinvo
lve
twooperations
?
Pro
ble
m 3
.4 G
etthe
Calcu
lations
Right:E
xpressions
and
Order
ofOperations
Focu
s Q
uest
ion
Whe
nan
equa
tion
relatin
gtwova
riablesinvo
lves
two
orm
oreop
erations
,how
doyou
usetheeq
uatio
ntofind
value
sof
the dep
enden
tva
riablefromgiven
va
lues
ofthe
indep
enden
tva
riable?
Inve
stig
atio
n 4
Exp
ressions
,Equa
tions
,an
dIn
equa
lities
Pro
ble
m 4
.1 Tak
ingthe
Plung
e:
Equiva
lentExp
ressions
I
Focu
s Q
uest
ion
Isitpossibleto
have
twodifferen
t,butequiva
lent,
expressions
foragiven
situation?
Exp
lain.
Pro
ble
m 4
.2 E
quiva
lent
Exp
ressions
II
Focu
s Q
uest
ion
Wha
tdoes
it
mea
ntosay
tha
ttw
oalgeb
raic
expressions
areequiva
lent?
Pro
ble
m 4
.3 P
utting
Itall
Togethe
r:Equiva
lentExp
ressions
III
Focu
s Q
uest
ion
Howcan
ex
pressions
suc
has3
x+7
xor
3(x +2)b
ewritten
inequiva
lent
form
?
Pro
ble
m 4
.4 F
indingthe
Unk
nown
Value:SolvingEqua
tions
IFo
cus
Que
stio
n Wha
tstrategies
willalw
aysso
lveeq
uations
inthe
form
sx+a=b,x
–a=b,a
x=b,
andx÷a=b(a
≠0)?
Pro
ble
m 4
.5 S
olvingIn
equa
lities
Focu
s Q
uest
ion
Wha
tstrategies
willalw
aysso
lveineq
ualitiesinthe
form
sx+a
< b,x
–a<b,a
x<b,
andx÷a<b(w
ith>,≤
,or≥as
theca
sem
aybe)?
Graphic Organizers for Grade 6 89
Dat
a A
bout
Us
Stat
istic
s an
d D
ata
Ana
lysi
s
Ess
enti
al Id
eas
•Th
ean
swerstoastatisticalque
stionareca
lleddata.D
ataca
nbe
eithernum
ericalorca
tegorica
l.
•Th
erearese
veralw
aystotrytosay
wha
tistyp
icalofase
tof
data;in
eac
hca
seasinglenum
ber,c
alledam
easu
reofce
nter,
summarizes
the
data.Bec
ause
various
mea
suresofce
nterare
calculated
differen
tly,the
yresp
ond
differen
tlytocha
nges
inthe
dataortounu
sualdatava
lues
.
•Th
eva
riab
ilityofase
tofdataca
nbemea
sured,interpreted,a
nd
compared
withtheva
riab
ilityofotherdatase
ts.M
easu
resof
variab
ilitytelly
ouho
wspread
outthe
dataareinrelationtoeac
hotherortothe
cen
ter.
•Find
ingm
easu
resofce
nterorva
riab
ilityand
graphing
dataare
usefulforsu
mmarizingthe
inform
ationinavariabledatase
t.
Visua
lrep
rese
ntations
ofadatase
tca
nhe
lpyoutoin
terpretthe
mea
suresofce
nterand
variability,a
ndrelatethes
etothe
ove
rall
shap
eofthereprese
ntation.
Inve
stig
atio
n 1
Wha
t’sin
aN
ame?
Organ
izing,
Rep
rese
nting,a
ndD
escribingD
ata
Pro
ble
m 1
.1 H
owM
anyLe
tters
Arein
aN
ame?
Pro
ble
m 1
.2 D
escribingN
ame
Leng
ths:W
hatArethe
Sha
pe,
Mode,and
Ran
ge?
Pro
ble
m 1
.3 D
escribingN
ame
Leng
ths:W
hatIsthe
Med
ian?
Inve
stig
atio
n 2
Who
’sin
YourH
ous
ehold?
Using
the
Mea
n
Pro
ble
m 2
.1 W
hat’s
aM
ean
Hous
eholdSize?
Pro
ble
m 2
.2 C
omparing
Distributions
WiththeSa
meMea
n
Pro
ble
m 2
.3 E
xperim
enting
With
theMea
n
Pro
ble
m 2
.4 W
hoElseIsin
Yo
urH
ous
ehold?
Inve
stig
atio
n 3
Wha
tIsYourFav
oriteC
erea
l?
Mea
suring
Variability
Pro
ble
m 3
.1 E
stim
atingC
erea
lPortionSize
s:U
sing
IQR
Pro
ble
m 3
.2 C
onn
ecting
Cerea
lSh
elfLo
cationan
dSug
arC
onten
t:
Des
cribingVariabilityU
sing
the IQ
R
Pro
ble
m 3
.3 W
aiting
inLineto
Buy
Cerea
l:Using
the
MAD
Inve
stig
atio
n 4
Wha
tNum
bersDes
cribeUs?
Using
Graphs
toG
roup
Data
Pro
ble
m 4
.1 T
rave
lingtoSch
ool:
Mak
ingH
istograms
Pro
ble
m 4
.2 M
easu
ring
Perform
ance
Whe
nJu
mping
Rope:M
akingBox-an
d-W
hisker
Plots
Pro
ble
m 4
.3 H
owM
uchTa
ller
IsaSixth-G
radeStud
entTh
ana
Seco
nd-G
radeStud
ent?
Teacher Implementation Toolkit90
Inve
stig
atio
n 1
Wha
t’sin
aN
ame?
Organ
izing,
Rep
rese
nting,a
ndD
escribingD
ata
Pro
ble
m 1
.1 H
owM
anyLe
tters
Arein
aN
ame?
Focu
s Q
uest
ion
Wha
taredata?
Wha
tareobse
rvations
?Howdo
youreprese
ntdataus
ingaline
plotorafreq
uenc
ytable?How
canyo
uco
mparetw
odistributions
ofdata?
Pro
ble
m 1
.2 D
escribingN
ame
Leng
ths:W
hatArethe
Sha
pe,
Mode,and
Ran
ge?
Focu
s Q
uest
ion
Wha
tare
mea
suresofce
ntralten
den
cy
andspread
orva
riab
ility?How
doyouco
mputean
duse
mode
andran
ge?
Pro
ble
m 1
.3 D
escribingN
ame
Leng
ths:W
hatIsthe
Med
ian?
Focu
s Q
uest
ion
Howdoyou
iden
tifyand
use
the
med
ian?
How
doyouco
mparetw
odistributions
ofdataus
ingthe
med
ians
fordata
fromProblems1.1an
d1.2?
Inve
stig
atio
n 2
Who
’sin
YourH
ous
ehold?Using
theMea
n
Pro
ble
m 2
.1 W
hat’s
aM
ean
Hous
eholdSize?
Focu
s Q
uest
ion
Howm
ightyou
goaboutfind
inganum
bertha
tis
agoodestim
ateofho
useh
oldsize
forallsixthgradestud
entsbased
onthes
edata?
Pro
ble
m 2
.2 C
omparing
Distributions
WiththeSa
meMea
n
Focu
s Q
uest
ion
Howdothe
med
ianan
dm
eanresp
ond
tothe
datainadistribution?
Pro
ble
m 2
.3 E
xperim
enting
With
theMea
n
Focu
s Q
uest
ion
Howdoyou
interpret,computean
duse
themea
n?
Pro
ble
m 2
.4 W
hoElseIsin
Yo
urH
ous
ehold?
Focu
s Q
uest
ion
Howdoyou
disting
uish
differen
ttypes
of
data?
Wha
tstatisticsareuse
dw
ith
differen
ttypes
ofdata?
Inve
stig
atio
n 3
Wha
tIsYourFav
oriteC
erea
l?
Mea
suring
Variability
Pro
ble
m 3
.1 E
stim
atingC
erea
lPortionSize
s:U
sing
IQR
Focu
s Q
uest
ion
Wha
tinform
ation
does
the
interqua
rtile
ran
ge
provideab
outhowdatava
ryin
adistribution?
Pro
ble
m 3
.2 C
onn
ecting
Cerea
lSh
elfLo
cationan
dSug
arC
onten
t:
Des
cribingVariabilityU
sing
theIQ
R
Focu
s Q
uest
ion
Howis
interqua
rtile
ran
geus
edtom
ake
comparisons
among
distributions
?
Pro
ble
m 3
.3 W
aiting
inLineto
Buy
Cerea
l:Using
the
MAD
Focu
s Q
uest
ion
Wha
tinform
ation
does
the
mea
nab
solutedev
iation
provideab
outhowdatava
ryin
adistribution?
Inve
stig
atio
n 4
Wha
tNum
bersDes
cribeUs?
Using
Graphs
toG
roup
Data
Pro
ble
m 4
.1 T
rave
lingtoSch
ool:
Mak
ingH
istograms
Focu
s Q
uest
ion
Howdoyou
interpretdatareprese
nted
using
ahistogram?
Pro
ble
m 4
.2 M
easu
ring
Perform
ance
Whe
nJu
mping
Rope:M
akingBox-an
d-W
hisker
Plots
Focu
s Q
uest
ion
Howdoyou
interpretdatareprese
nted
using
a
box-an
d-w
hiskerplot?
Pro
ble
m 4
.3 H
owM
uchTa
ller
IsaSixth-G
radeStud
entTh
ana
Seco
nd-G
radeStud
ent?
Focu
s Q
uest
ion
Howdoyou
comparean
dcontrastdata
represe
nted
using
dotplots,
histograms,and
boxplots?
Graphic Organizers for Grade 6 91
