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New York City Graphic Organizers for CMP3
Accentuate the Negative Integers and Rational Numbers
Essential Ideas
•Rationalnumberscanbecompared,orderedandlocatedonanumberline.Theycanalsobeusedtoindicateadistanceordifferencebetweenpointsonanumberline.Numberlinesareusefulmodelsforsolvingproblemswithrationalnumbers.
•Modelsfacilitateunderstandingthemeaningofaddition,subtraction,multiplication,anddivisionofpositiveandnegativenumbers,andimproveunderstandingofthestandardalgorithmsfortheseoperations.
•Mathematicalsentences,withorwithoutvariables,canmodelreal-worldproblems.Sometimesrewritingaproblemusingadifferentoperationcanbehelpfulinfindingthesolution.
•Propertiesofoperationsextendtoallrationalnumbersandunderstandingthesepropertiesishelpfulinsolvingproblems.
Investigation 1ExtendingtheNumberSystem
Problem 1.1 PlayingMathFever:UsingPositiveandNegativeNumbers
Problem 1.2 ExtendingtheNumberLine
Problem 1.3 FromSaunatoSnowbank:UsingaNumberLine
Problem 1.4 IntheChips:UsingaChipModel
Investigation 2AddingandSubtractingRationalNumbers
Problem 2.1 ExtendingAdditiontoRationalNumbers
Problem 2.2 ExtendingSubtractiontoRationalNumbers
Problem 2.3 The“+/–”Connection
Problem 2.4 FactFamilies
Investigation 3MultiplyingandDividingRationalNumbers
Problem 3.1 MultiplicationPatternsWithIntegers
Problem 3.2 MultiplicationofRationalNumbers
Problem 3.3 DivisionofRationalNumbers
Problem 3.4 PlayingtheIntegerProductGame:ReasoningAboutMultiplicationandDivisionof Integers
Investigation 4PropertiesofOperations
Problem 4.1 OrderofOperations
Problem 4.2 TheDistributiveProperty
Problem 4.3 WhatOperationsAre Needed?
Investigation 1Extending the Number System
Problem 1.1 Playing Math Fever: Using Positive and Negative Numbers
Focus Question How can you find the total value of a combination of positive and negative integers?
Problem 1.2 Extending the Number Line
Focus Question How can you use a number line to compare two numbers?
Problem 1.3 From Sauna to Snowbank: Using a Number Line
Focus Question How can you write a number sentence to represent a change on a number line, and how can you use a number line to represent a number sentence?
Problem 1.4 In the Chips: Using a Chip Model
Focus Question How can you use a chip model to represent addition and subtraction?
Investigation 2Adding and Subtracting Rational Numbers
Problem 2.1 Extending Addition to Rational Numbers
Focus Question How can you predict whether the result of addition of two numbers will be positive, negative, or zero?
Problem 2.2 Extending Subtraction to Rational Numbers
Focus Question How is a chip model or number line useful in determining an algorithm for subtraction?
Problem 2.3 The “+/–” Connection
Focus Question How are the algorithms for addition and subtraction of integers related?
Problem 2.4 Fact Families
Focus Question What related sentence is equivalent to 4 + n = 43 and makes it easier to find the value of n?
Investigation 3Multiplying and Dividing Rational Numbers
Problem 3.1 Multiplication Patterns With Integers
Focus Question How is multiplication of two integers represented on a number line and a chip board?
Problem 3.2 Multiplication of Rational Numbers
Focus Question What algorithm can you use for multiplying integers?
Problem 3.3 Division of Rational Numbers
Focus Question What algorithm can you use for dividing integers? How are multiplication and division of integers related?
Problem 3.4 Playing the Integer Product Game: Reasoning About Multiplication and Division of Integers
Focus Question What patterns do you notice on the game board for the Integer Product Game that can help you win?
Investigation 4Properties of Operations
Problem 4.1 Order of Operations
Focus Question Does the Order of Operations work for integers? Explain.
Problem 4.2 The Distributive Property
Focus Question How can you use the Distributive Property to expand an expression or factor an expression that involves integers?
Problem 4.3 What Operations Are Needed?
Focus Question What information in a problem is useful to help you decide which operation to use to solve the problem?
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 7 85
Acc
entu
ate
the
Neg
ativ
e In
tege
rs a
nd R
atio
nal N
umbe
rs
Ess
enti
al Id
eas
•Rationa
lnum
bersca
nbeco
mpared
,ordered
and
loca
tedon
anu
mberline
.The
yca
nalso
beus
edtoin
dicateadistanc
eor
differen
cebetwee
npointsonanu
mberline
.Num
berline
sare
usefulm
odelsforso
lvingproblemswithrationa
lnum
bers.
•Modelsfacilitateun
derstan
dingthe
mea
ning
ofad
dition,
subtrac
tion,m
ultiplication,and
divisionofpositive
and
neg
ative
numbers,and
improve
und
erstan
dingofthestan
dardalgorithms
forthes
eoperations
.
•Mathe
maticalsen
tenc
es,w
ithorwitho
utvariables,can
model
real-w
orldproblems.Sometim
esrew
riting
aproblemusing
a
differen
toperationca
nbehe
lpfulinfin
dingthe
solution.
•Properties
ofoperations
exten
dtoallrationa
lnum
bersan
d
understan
dingthe
seproperties
ishelpfulinso
lvingproblems.
Inve
stig
atio
n 1
Exten
dingthe
Num
berSystem
Pro
ble
m 1
.1 P
laying
Math
Feve
r:U
sing
Positive
and
Neg
ativeNum
bers
Pro
ble
m 1
.2 E
xten
dingthe
Num
berLine
Pro
ble
m 1
.3 F
romSau
nato
Snowban
k:U
sing
aN
umberLine
Pro
ble
m 1
.4 IntheChips:
Using
aC
hipM
odel
Inve
stig
atio
n 2
Addingand
Sub
trac
ting
Rationa
lNum
bers
Pro
ble
m 2
.1 E
xten
dingA
ddition
toRationa
lNum
bers
Pro
ble
m 2
.2 E
xten
ding
Subtrac
tiontoRationa
lNum
bers
Pro
ble
m 2
.3 T
he“+/–”Conn
ection
Pro
ble
m 2
.4 F
actFa
milies
Inve
stig
atio
n 3
Multiplyingand
Dividing
Rationa
lNum
bers
Pro
ble
m 3
.1 M
ultiplication
PatternsWithIntegers
Pro
ble
m 3
.2 M
ultiplication
ofRationa
lNum
bers
Pro
ble
m 3
.3 D
ivision
ofRationa
lNum
bers
Pro
ble
m 3
.4 P
laying
the
IntegerProduc
tGam
e:
Rea
soning
AboutM
ultiplication
andD
ivisionof Integers
Inve
stig
atio
n 4
Properties
ofOperations
Pro
ble
m 4
.1 O
rderofOperations
Pro
ble
m 4
.2
TheDistributiveProperty
Pro
ble
m 4
.3 W
hatOperations
Are N
eeded
?
Teacher Implementation Toolkit86
Inve
stig
atio
n 1
Exten
dingthe
Num
berSystem
Pro
ble
m 1
.1 P
laying
Math
Feve
r:U
sing
Positive
and
Neg
ativeNum
bers
Focu
s Q
uest
ion
Howcan
youfin
d
thetotalv
alue
ofaco
mbinationof
positive
and
neg
ativeintegers?
Pro
ble
m 1
.2 E
xten
dingthe
Num
berLine
Focu
s Q
uest
ion
Howcan
youus
eanu
mberline
tocompare
twonum
bers?
Pro
ble
m 1
.3 F
romSau
nato
Snowban
k:U
sing
aN
umberLine
Focu
s Q
uest
ion
Howcan
you
writeanum
bersen
tenc
eto
represe
ntacha
ngeonanu
mber
line,and
howcan
youus
ea
numberline
torep
rese
nta
numbersen
tenc
e?
Pro
ble
m 1
.4 IntheChips:U
sing
a
ChipM
odel
Focu
s Q
uest
ion
Howcan
youus
each
ipm
odeltorep
rese
ntaddition
andsub
trac
tion?
Inve
stig
atio
n 2
Addingand
Sub
trac
ting
Rationa
lNum
bers
Pro
ble
m 2
.1 E
xten
dingA
ddition
toRationa
lNum
bers
Focu
s Q
uest
ion
Howcan
you
predictwhe
therthe
res
ultof
additionoftw
onum
berswillbe
positive
,neg
ative,orze
ro?
Pro
ble
m 2
.2 E
xten
ding
Subtrac
tiontoRationa
lNum
bers
Focu
s Q
uest
ion
Howisachip
modelornu
mberline
use
ful
indetermininganalgorithm
forsu
btrac
tion?
Pro
ble
m 2
.3 T
he“+/–”Conn
ection
Focu
s Q
uest
ion
Howarethe
algorithmsforad
ditionan
d
subtrac
tionofintegersrelated?
Pro
ble
m 2
.4 F
actFa
milies
Focu
s Q
uest
ion
Wha
trelated
senten
ceisequiva
lentto
4+n=43an
dm
akes
iteasier
tofind
the
value
of
n?
Inve
stig
atio
n 3
Multiplyingand
Dividing
Rationa
lNum
bers
Pro
ble
m 3
.1 M
ultiplication
PatternsWithIntegers
Focu
s Q
uest
ion
Howis
multiplicationoftw
oin
tegers
represe
nted
onanu
mberline
an
dachipboard?
Pro
ble
m 3
.2 M
ultiplicationof
Rationa
lNum
bers
Focu
s Q
uest
ion
Wha
talgorithm
canyo
uus
eformultiplying
integers?
Pro
ble
m 3
.3 D
ivisionof
Rationa
lNum
bers
Focu
s Q
uest
ion
Wha
talgorithm
canyo
uus
efordividingin
tegers?
Howarem
ultiplicationan
ddivision
ofintegersrelated?
Pro
ble
m 3
.4 P
laying
the
IntegerProduc
tGam
e:
Rea
soning
AboutM
ultiplication
andD
ivisionof Integers
Focu
s Q
uest
ion
Wha
tpatternsdo
youno
tice
onthegam
eboardfor
theIntegerProduc
tGam
ethatcan
he
lpyouwin?
Inve
stig
atio
n 4
Properties
ofOperations
Pro
ble
m 4
.1 O
rderofOperations
Focu
s Q
uest
ion
Does
the
OrderofOperations
workfor
integers?Exp
lain.
Pro
ble
m 4
.2
TheDistributiveProperty
Focu
s Q
uest
ion
Howcan
you
usetheDistributivePropertyto
expan
danex
pressionorfactoran
ex
pressionthatin
volves
integers?
Pro
ble
m 4
.3 W
hatOperations
AreN
eeded
?
Focu
s Q
uest
ion
Wha
tinform
ation
inaproblemisuse
fultohelpyou
dec
idewhich
operationtouse
to
solvetheproblem?
Graphic Organizers for Grade 7 87
Stre
tchi
ng a
nd S
hrin
king
Und
erst
andi
ng S
imila
rity
Ess
enti
al Id
eas
•Simila
rfig
ures
hav
eco
ngruen
tco
rres
pond
ingang
lesan
d
corres
pond
ingsides
leng
thsareinaproportiona
lrelations
hip.
•Th
escalefactorfortw
osim
ilarfig
ures
isestab
lishe
dbyfin
ding
theratioofapairofco
rres
pond
ingsides
.Sca
lefac
torus
edw
ith
othertoolsallo
wsyo
utom
akedrawingsofsimila
rfig
ures
and
to
comparetheperim
etersan
dareasofsimila
rfig
ures
.
•Iftw
ofigures
aresim
ilar,then
youca
nus
eaproportiona
lrelations
hipbetwee
nco
rres
pond
ingsides
tofind
unk
nown
sideleng
ths.
Inve
stig
atio
n 1
Enlargingand
Red
ucingSha
pes
Pro
ble
m 1
.1 S
olvingaM
ystery:
AnIntroduc
tiontoSim
ilarity
Pro
ble
m 1
.2 S
calin
gU
pand
Down:C
orres
pond
ingSides
an
dA
ngles
Inve
stig
atio
n 2
Simila
rFigures
Pro
ble
m 2
.1 D
rawingW
umps:
Mak
ingSim
ilarFigures
Pro
ble
m 2
.2 H
atsOfftothe
Wum
ps:C
hang
ingaFigure’sSize
an
dLoca
tion
Pro
ble
m 2
.3 M
outhing
Offand
Nosing
Aroun
d:S
caleFac
tors
Inve
stig
atio
n 3
ScalingPerim
eterand
Area
Pro
ble
m 3
.1 R
ep-Tile
Qua
drilaterals:F
orm
ingRep
-Tile
sWithSimila
rQua
drilaterals
Pro
ble
m 3
.2 R
ep-Tile
Trian
gles:
Form
ingRep
-Tile
sWith
Simila
rTriang
les
Pro
ble
m 3
.3 D
esigning
Und
er
Cons
traints:Sca
leFac
torsand
Simila
rSh
apes
Pro
ble
m 3
.4 O
utofRea
ch:
Find
ingLen
gthswith
Simila
rTriang
les
Inve
stig
atio
n 4
Simila
rityand
Ratios
Pro
ble
m 4
.1 R
atiosWithin
Simila
rParallelograms
Pro
ble
m 4
.2 R
atiosWithin
Simila
rTriang
les
Pro
ble
m 4
.3 F
indingM
issing
Parts:U
sing
Sim
ilarityto
Find
Mea
suremen
ts
Pro
ble
m 4
.4 U
sing
Sha
dows
toFindH
eights:U
sing
Simila
rTriang
les
Teacher Implementation Toolkit88
Inve
stig
atio
n 1
Enlargingand
Red
ucingSha
pes
Pro
ble
m 1
.1 S
olvingaM
ystery:
AnIntroduc
tiontoSim
ilarity
Focu
s Q
uest
ion
Wha
tdoes
it
mea
nfortw
ofigures
tobesimila
r?
Pro
ble
m 1
.2 S
calin
gU
pand
Down:C
orres
pond
ingSides
an
dA
ngles
Focu
s Q
uest
ion
Whe
nyo
uco
py
afig
ureatacertainsca
lefac
tor
(e.g.1
50%),ho
wdoes
thisva
lue
affectthe
mea
suremen
tsofthe
newfigure?
Inve
stig
atio
n 2
Simila
rFigures
Pro
ble
m 2
.1 D
rawingW
umps:
Mak
ingSim
ilarFigures
Focu
s Q
uest
ion
Howcan
you
determineiftw
osha
pes
are
simila
rbylooking
attherulefor
produc
ingspec
ificco
ordinates
for
theim
age?
Pro
ble
m 2
.2 H
atsOfftothe
Wum
ps:C
hang
ingaFigure’sSize
an
dLoca
tion
Focu
s Q
uest
ion
Wha
ttypes
of
coordinaterulesproduc
esimila
rfig
ures
?Nons
imila
rfig
ures
?
Pro
ble
m 2
.3 M
outhing
Offand
Nosing
Aroun
d:S
caleFac
tors
Focu
s Q
uest
ion
Howcan
you
dec
idewhe
therorno
ttw
osha
pes
aresimila
r?
Inve
stig
atio
n 3
ScalingPerim
eterand
Area
Pro
ble
m 3
.1 R
ep-Tile
Qua
drilaterals:F
orm
ingRep
-Tile
sWithSimila
rQua
drilaterals
Focu
s Q
uest
ion
Wha
ttypes
ofqua
drilateralsarerep
-tile
s?
Howdorep
-tile
ssh
owtha
tthe
scalefactorsand
areasofsimila
rqua
drilateralsarerelated
?
Pro
ble
m 3
.2 R
ep-Tile
Trian
gles:
Form
ingRep
-Tile
sWith
Simila
rTriang
les
Focu
s Q
uest
ion
Which
typ
esof
triang
lesarerep-tile
s?Exp
lain.
Pro
ble
m 3
.3 D
esigning
Und
er
Cons
traints:Sca
leFac
torsand
Simila
rSh
apes
Focu
s Q
uest
ion
Howcan
youus
escalefactorstodrawsim
ilarfig
ures
ortofind
missing
sideleng
thsin
simila
rfig
ures
?
Pro
ble
m 3
.4 O
utofRea
ch:
Find
ingLen
gthswithSimila
rTriang
les
Focu
s Q
uest
ion
Howcan
youus
esimila
rtriang
lestofind
adistanc
ethatisdiffi
culttom
easu
redire
ctly?
Inve
stig
atio
n 4
Simila
rityand
Ratios
Pro
ble
m 4
.1 R
atiosWithin
Simila
rParallelograms
Focu
s Q
uest
ion
Wha
tinform
ation
does
the
ratioofad
jace
ntside
leng
thswithinarectan
gle
giveyo
u?
Pro
ble
m 4
.2 R
atiosWithin
Simila
rTriang
les
Focu
s Q
uest
ion
Forapairof
triang
les,ifthe
mea
suresof
corres
pond
ingang
lesareeq
ual,
howcan
youus
eratiosofside
leng
thstodeterminewhe
theror
notthetriang
lesaresimila
r?
Pro
ble
m 4
.3 F
indingM
issing
Parts:U
sing
Sim
ilarityto
Find
Mea
suremen
ts
Focu
s Q
uest
ion
Iftw
osha
pes
aresimila
r,ho
wcan
youus
einform
ationab
outthe
sha
pes
tofind
unk
nownsideleng
ths,
perim
eters,and
areas?
Pro
ble
m 4
.4 U
sing
Sha
dows
toFindH
eights:U
sing
Simila
rTriang
les
Focu
s Q
uest
ion
Howcan
youus
esimila
rtriang
lestoestim
atethe
heightsoftallobjects?
Graphic Organizers for Grade 7 89
Com
pari
ng a
nd S
calin
g R
atio
s, R
ates
, Per
cent
s, a
nd P
ropo
rtio
ns
Ess
enti
al Id
eas
•Ratiosmak
eco
mparisons
betwee
ntw
opartsofthewho
le
orbetwee
none
partan
dthe
who
le.R
ates
,unitrates,
andperce
ntsarealltyp
esofratios.
•Being
abletocha
ngetheform
ofaratioisause
ful
problem-solvingstrateg
y.
•Aproportiona
lrelations
hiphasparticu
larch
arac
teristics
whe
nreprese
nted
inatab
le,g
raphoreq
uation.
•Kno
wingthe
des
iredratiobetwee
ntw
ovariablesallowsyo
u
tosca
lethe
ratioorfin
dam
issing
partofaratio.
Inve
stig
atio
n 1
Way
sofComparing:R
atiosan
dProportions
Pro
ble
m 1
.1 S
urve
ying
Opinions
:Ana
lyzing
ComparisonStatem
ents
Pro
ble
m 1
.2 M
ixingJuice
:ComparingRatios
Pro
ble
m 1
.3 T
imetoC
onc
entrate:
ScalingRatios
Pro
ble
m 1
.4 K
eepingThing
sinProportion:
ScalingtoSolveProportions
Inve
stig
atio
n 2
Comparingand
Sca
lingRates
Pro
ble
m 2
.1 S
haring
Pizza:
ComparisonStrategies
Pro
ble
m 2
.2 C
omparingPizzaPrice
s:
ScalingRates
Pro
ble
m 2
.3 F
indingC
osts:U
nitRate
andC
ons
tantofProportiona
lity
Inve
stig
atio
n 3
Marku
ps,M
arkd
owns
,and
Mea
sures:
Using
Ratios,Perce
nts,and
Proportions
Pro
ble
m 3
.1 C
ommissions
,Marku
ps,and
Disco
unts:P
roportions
WithPerce
nts
Pro
ble
m 3
.2 M
easu
ring
tothe
Unit:
Mea
suremen
tConv
ersions
Pro
ble
m 3
.3 M
ixingitU
p:C
onn
ecting
Ratios,
Rates
,Perce
ntsan
dProportions
Teacher Implementation Toolkit90
Inve
stig
atio
n 1
Way
sofComparing:R
atiosan
dProportions
Pro
ble
m 1
.1 S
urve
ying
Opinions
:Ana
lyzing
ComparisonStatem
ents
Focu
s Q
uest
ion
Wha
tdodifferen
tco
mparisons
ofqua
ntitiestellyo
uab
outthe
irrelations
hip?
Pro
ble
m 1
.2 M
ixingJuice
:ComparingRatios
Focu
s Q
uest
ion
Wha
tstrategiesdoyouus
eto
determinewhich
mixisthe
mostorang
ey?
Pro
ble
m 1
.3 T
imetoC
onc
entrate:
ScalingRatios
Focu
s Q
uest
ion
Whe
nyo
uscaleup
arec
ipe
andcha
ngetheun
its,like
fromcup
stooun
ces,
wha
tareso
meoftheissu
esyouha
veto
dea
lwith?
Pro
ble
m 1
.4 K
eepingThing
sinProportion:
ScalingtoSolveProportions
Focu
s Q
uest
ion
Wha
tstrategiesca
nyo
uus
etofind
am
issing
value
inaproportion?
Wha
tis
yourpreferred
strateg
yan
dw
hy?
Inve
stig
atio
n 2
Comparingand
Sca
lingRates
Pro
ble
m 2
.1 S
haring
Pizza:
ComparisonStrategies
Focu
s Q
uest
ion
Howcan
youdetermine
whe
thertworatiosareeq
uiva
lentorfin
dw
hich
oftw
oratiosism
orefav
orable?
Pro
ble
m 2
.2 C
omparingPizzaPrice
s:
ScalingRates
Focu
s Q
uest
ion
Howcan
youus
eratetab
lesto
findm
issing
value
s?H
owareratetablessimila
rtosca
lingqua
ntitiesan
dsolvingproportions
?
Pro
ble
m 2
.3 F
indingC
osts:U
nitRate
andC
ons
tantofProportiona
lity
Focu
s Q
uest
ion
Howcan
youfin
daunitratein
ades
cription,aneq
uation,atab
le,o
ragraph?
Inve
stig
atio
n 3
Marku
ps,M
arkd
owns
,and
Mea
sures:
Using
Ratios,Perce
nts,and
Proportions
Pro
ble
m 3
.1 C
ommissions
,Marku
ps,and
Disco
unts:P
roportions
WithPerce
nts
Focu
s Q
uest
ion
Howcan
youus
eproportions
an
dperce
nttab
lestofind
various
perce
ntag
es
ofava
luewhe
nyo
ukn
owacertainperce
ntag
eofthesameva
lue?
Pro
ble
m 3
.2 M
easu
ring
tothe
Unit:
Mea
suremen
tConv
ersions
Focu
s Q
uest
ion
Howcan
youus
eun
itrates
,proportions
,equa
tions
,and
ratetablestosca
le
ava
rietyofun
its?
Pro
ble
m 3
.3 M
ixingitU
p:C
onn
ecting
Ratios,
Rates
,Perce
ntsan
dProportions
Focu
s Q
uest
ion
Howcan
youus
escale
factors,ratetables,proportions
,equa
tions
,orgraphs
tofind
amoun
tsofamixture,g
iven
theproportions
?
Graphic Organizers for Grade 7 91
Mov
ing
Stra
ight
Ahe
ad L
inea
r R
elat
ions
hips
Ess
enti
al Id
eas
•Tw
ovariablesareinaline
arrelations
hipifone
variableischa
nging
byaco
nstantamoun
twhe
ntheothervariablecha
nges
by
increm
entsof1un
it.
•Th
erateofch
angeinaline
arrelations
hipisrep
rese
nted
bythe
slopeofthelin
ereprese
ntingthe
relations
hip.
•Th
eeq
uation
y=m
xisaparticu
larkind
oflin
earrelations
hipw
here
xan
dyareproportiona
ltoeac
hother.
•So
lutions
forlin
eareq
uations
oftheform
y=m
x+barepairsof
values
(x,y
)which
mak
etheeq
uationtrue
.Graphica
lly,s
olution
pairsarepointsonthegraphofthelin
e.
•Properties
ofeq
ualityca
nbeus
edtom
aintaineq
uiva
lent
expressions
onea
chsideoftheeq
uationwhe
nfin
dingasolution.
Inve
stig
atio
n 1
Walking
Rates
Pro
ble
m 1
.1 W
alking
Maratho
ns:
Find
ingand
Using
Rates
Pro
ble
m 1
.2 W
alking
Rates
an
dLinea
rRelations
hips:Tab
les,
Graphs
,and
Equa
tions
Pro
ble
m 1
.3 R
aising
Mone
y:
Using
Linea
rRelations
hips
Pro
ble
m 1
.4 U
sing
the
Walka
thonMone
y:Rec
ognizing
Line
arRelationh
sips
Inve
stig
atio
n 2
Exp
loring
Linea
rRelations
hips
WithGraphs
and
Tab
les
Pro
ble
m 2
.1 W
alking
toW
in:
Find
ingthe
PointofIntersec
tion
Pro
ble
m 2
.2 C
rossingthe
Line
:Using
Tab
les,G
raphs
,and
Equa
tions
Pro
ble
m 2
.3 C
omparingC
osts:
ComparingRelations
hips
Pro
ble
m 2
.4 C
onn
ecting
Tab
les,
Graphs
,and
Equa
tions
Inve
stig
atio
n 3
SolvingEqua
tions
Pro
ble
m 3
.1 S
olvingEqua
tions
Using
Tab
lesan
dG
raphs
Pro
ble
m 3
.2 M
ysteryPouc
hes
inthe
KingdomofMontarek
:Exp
loring
Equa
lity
Pro
ble
m 3
.3 F
romPouc
hesto
Variab
les:W
riting
Equa
tions
Pro
ble
m 3
.4 S
olving
Line
ar Equa
tions
Pro
ble
m 3
.5 F
indingthe
Point
of Intersec
tion
Inve
stig
atio
n 4
Exp
loring
Slope:C
onn
ecting
Rates
and
Ratios
Pro
ble
m 4
.1 C
limbingStairs:
Using
Risean
dRun
Pro
ble
m 4
.2 F
indingthe
Slope
of aLine
Pro
ble
m 4
.3 E
xploring
Patterns
WithLine
s
Pro
ble
m 4
.4 P
ullin
gItA
llTo
gethe
r:W
riting
Equa
tions
for Line
arRelations
hips
Teacher Implementation Toolkit92
Inve
stig
atio
n 1
Walking
Rates
Pro
ble
m 1
.1 W
alking
Maratho
ns:
Find
ingand
Using
Rates
Focu
s Q
uest
ion
Wha
teq
uation
represe
ntsyo
urw
alking
rateas
arelations
hipbetwee
ndistanc
ean
dtim
e?
Pro
ble
m 1
.2 W
alking
Rates
an
dLinea
rRelations
hips:Tab
les,
Graphs
,and
Equa
tions
Focu
s Q
uest
ion
Howcan
you
predictwhe
therarelations
hip
isline
arfromatab
le,a
graph,
oran
equa
tionthatrep
rese
nts
therelations
hip?
Pro
ble
m 1
.3 R
aising
Mone
y:
Using
Linea
rRelations
hips
Focu
s Q
uest
ion
Wha
tis
thepatternofch
angeina
linea
rrelations
hip?
Pro
ble
m 1
.4 U
sing
the
Walka
thonMone
y:Rec
ognizing
Line
arRelationh
sips
Focu
s Q
uest
ion
Howcan
you
determineifalin
earrelations
hip
isin
crea
sing
ordec
reasing?
Inve
stig
atio
n 2
Exp
loring
Linea
rRelations
hips
WithGraphs
and
Tab
les
Pro
ble
m 2
.1 W
alking
toW
in:
Find
ingthe
PointofIntersec
tion
Focu
s Q
uest
ion
Whe
nisituse
ful
touse
agraphoratabletosolve
aproblem?
Pro
ble
m 2
.2 C
rossingthe
Line
:Using
Tab
les,G
raphs
,an
dEqua
tions
Focu
s Q
uest
ion
Howdoes
the
patternofch
angeforalin
ear
relations
hipappea
rinatab
le,
agraph,oran
equa
tion?
Pro
ble
m 2
.3 C
omparingC
osts:
ComparingRelations
hips
Focu
s Q
uest
ion
Howcan
you
dec
ideifatableoran
equa
tion
represe
ntsalin
earrelations
hip?
Pro
ble
m 2
.4 C
onn
ecting
Tab
les,
Graphs
,and
Equa
tions
Focu
s Q
uest
ion
Howare
solutions
ofan
equa
tionofthe
form
y=b+m
xrelatedto
thegraphan
dthe
tab
lefor
theeq
uation?
Inve
stig
atio
n 3
SolvingEqua
tions
Pro
ble
m 3
.1 S
olvingEqua
tions
Using
Tab
lesan
dG
raphs
Focu
s Q
uest
ion
Howarethe
co
ordinates
ofapointonalin
eor
inatab
lerelated
tothe
equa
tion
ofthelin
e?
Pro
ble
m 3
.2 M
ysteryPouc
hes
inthe
KingdomofMontarek
:Exp
loring
Equa
lity
Focu
s Q
uest
ion
Wha
tdoes
eq
ualitymea
n?
Pro
ble
m 3
.3 F
romPouc
hesto
Variab
les:W
riting
Equa
tions
Focu
s Q
uest
ion
Wha
tare
somestrategiesforso
lving
linea
req
uations
?
Pro
ble
m 3
.4 S
olving
Line
arEqua
tions
Focu
s Q
uest
ion
Howcan
the
properties
ofeq
ualitybeus
edto
solvelin
eareq
uations
?
Pro
ble
m 3
.5 F
indingthe
Point
ofIntersec
tion
Focu
s Q
uest
ion
Howcan
youfin
d
whe
ntw
oexp
ressions
areequa
lorwhe
none
exp
ressionisgreater
orlesstha
ntheother?
Inve
stig
atio
n 4
Exp
loring
Slope:C
onn
ecting
Rates
and
Ratios
Pro
ble
m 4
.1 C
limbingStairs:
Using
Risean
dRun
Focu
s Q
uest
ion
Howisthe
stee
pne
ssofase
tofstairsrelated
toastraight-line
graph?
Pro
ble
m 4
.2 F
indingthe
Slope
ofaLine
Focu
s Q
uest
ion
Howcan
youfin
d
the
y-intercep
tan
dthe
slopeof
alin
efromdatainatab
le,g
raph,
oreq
uation?
Pro
ble
m 4
.3 E
xploring
Patterns
WithLine
s
Focu
s Q
uest
ion
Howcan
you
predictwhe
thertwoline
sare
parallelo
rperpen
dicularfrom
theireq
uations
?
Pro
ble
m 4
.4 P
ullin
gItA
llTo
gethe
r:W
riting
Equa
tions
for Line
arRelations
hips
Focu
s Q
uest
ion
Wha
tinform
ation
doyoune
edtow
riteaneq
uation
foralin
earrelations
hip?Isthe
ex
pressionforthedep
enden
tva
riab
lealw
aysthesame?
Graphic Organizers for Grade 7 93
Wha
t Do
You
Expe
ct?
Prob
abili
ty a
nd E
xpec
ted
Valu
e
Ess
enti
al Id
eas
•Probab
ilities
areratios.Probab
ilitycan
beus
edtopredict
outco
mes
inrea
lworldeve
ntsoran
alyzegam
esforfairne
ss.
•Th
eoretica
lprobab
ilityisdetermined
byreasoning
aboutthe
lik
elihoodofasp
ecificoutco
mebased
onallp
ossibleoutco
mes
of
aneve
nt.Lists,treediagrams,orarea
modelsca
nsh
owallofthe
possibleoutco
mes
and
determinethetheo
retica
lprobab
ilityofa
compoun
deve
nt.
•Th
eex
perim
entalp
robab
ilityofan
eve
ntcan
befoun
dby
gathe
ring
datafromexp
erim
entsorobse
rvations
,coun
ting
the
nu
mberoftimes
the
spec
ified
outco
meocc
urred,a
ndcomparing
thattothe
num
beroftrials.L
ong
run
relativefreq
uenc
ies
colle
cted
fromexp
erim
entsm
akegoodapproximations
of
theo
retica
lprobab
ilities
.
Inve
stig
atio
n 1
AFirstLookatC
hanc
e
Pro
ble
m 1
.1 C
hoosing
Cerea
l:To
ssingC
oinsto
Find
Probab
liliie
s
Pro
ble
m 1
.2 Tossing
Pap
erC
ups:Finding
MoreProbab
ilities
Pro
ble
m 1
.3 O
neM
ore
Try:FindingExp
erim
ental
Probab
ilities
Pro
ble
m 1
.4 A
nalyzing
Eve
nts:U
nderstan
ding
Equa
llyLikely
Inve
stig
atio
n 2
Exp
erim
entala
nd
Theo
retica
lProbab
ility
Pro
ble
m 2
.1 P
redicting
toW
in:F
inding
Theo
retica
lProbab
ilities
Pro
ble
m 2
.2 C
hoosing
Marbles:D
eveloping
Probab
ilityM
odels
Pro
ble
m 2
.3 D
esigning
aFa
irG
ame:Pond
ering
Possibleand
Probab
le
Pro
ble
m 2
.4 W
inning
theBonu
sPrize
:Using
StrategiestoFind
Theo
retica
lProbab
ilities
Inve
stig
atio
n 3
Mak
ingD
ecisions
With
Probab
ility
Pro
ble
m 3
.1 D
esigning
aSp
inne
rtoFind
Probab
ilities
Pro
ble
m 3
.2
Mak
ingD
ecisions
:Ana
lyzing
Fairnes
s
Pro
ble
m 3
.3 R
olle
rDerby:A
nalyzing
aG
ame
Pro
ble
m 3
.4 S
cratch
ing
Spots:D
esigning
and
Using
aSim
ulation
Inve
stig
atio
n 4
Ana
lyzing
Compoun
d
Eve
ntsUsing
an
Area Model
Pro
ble
m 4
.1 D
rawing
AreaModelstoFindthe
Sa
mpleSpac
e
Pro
ble
m 4
.2 M
aking
Purple:A
reaModels
andProbab
ility
Pro
ble
m 4
.3 O
ne-
and-O
neFree-Th
rows:
SimulatingaProbab
ility
Situation
Pro
ble
m 4
.4 S
coring
Points:F
inding
Exp
ected Value
Inve
stig
atio
n 5
BinomialO
utco
mes
Pro
ble
m 5
.1 G
uessing
Ans
wers:FindingM
ore
Exp
ectedValue
s
Pro
ble
m 5
.2 O
rtonv
ille:
BinomialP
robab
ility
Pro
ble
m 5
.3 A
Baseb
all
Series
:Exp
anding
BinomialP
robab
ility
Teacher Implementation Toolkit94
Inve
stig
atio
n 1
AFirstLookatC
hanc
e
Pro
ble
m 1
.1 C
hoosing
Cerea
l:To
ssingC
oinsto
Find
Probab
liliie
s
Focu
s Q
uest
ion
Wha
tisagoodw
aytocho
ose
betwee
ntw
oormore
even
tssotha
tea
cheve
nt
hasthesamech
ance
of
being
selec
ted?
Pro
ble
m 1
.2 Tossing
Pap
erC
ups:Finding
MoreProbab
ilities
Focu
s Q
uest
ion
Which
resu
ltw
ouldyouco
nsider
tobemorelike
lyto
occur—
one
based
on
threetrialsorone
based
on25
trials?W
hy?
Pro
ble
m 1
.3 O
neM
ore
Try:FindingExp
erim
ental
Probab
ilities
Focu
s Q
uest
ion
Ifyo
uca
nnotlistallo
fthe
possibleoutco
mes
of
aneve
nt,h
owcan
you
mak
eadec
isionab
out
thelik
elihoodofan
ev
entoccurring
?
Pro
ble
m 1
.4 A
nalyzing
Eve
nts:U
nderstan
ding
Equa
llyLikely
Focu
s Q
uest
ion
How
canyo
udetermine
whe
therthe
outco
mes
ofaprobab
ilityeve
nt
arealle
qua
llylike
ly
andw
hyw
ouldthis
inform
ationmatter?
Inve
stig
atio
n 2
Exp
erim
entala
nd
Theo
retica
lProbab
ility
Pro
ble
m 2
.1 P
redicting
toW
in:F
inding
Theo
retica
lProbab
ilities
Focu
s Q
uest
ion
Ifyo
uca
nnotlookintoa
containe
r,ho
wcould
youco
llectthe
data
youne
edtopredictthe
proportionofblueblock
sinacontaine
rwithblue,
yello
w,a
ndred
block
s?
Pro
ble
m 2
.2 C
hoosing
Marbles:D
eveloping
Probab
ilityM
odels
Focu
s Q
uest
ion
Wha
tis
thedifferen
cebetwee
nex
perim
entala
nd
theo
retica
lprobab
ility?
Pro
ble
m 2
.3 D
esigning
aFa
irG
ame:Pond
ering
Possibleand
Probab
le
Focu
s Q
uest
ion
How
canyo
udec
idewhe
thera
gam
eisfairorno
t?
Pro
ble
m 2
.4 W
inning
theBonu
sPrize
:Using
StrategiestoFind
Theo
retica
lProbab
ilities
Focu
s Q
uest
ion
Inw
hat
kind
ofsituationwould
youus
eex
perim
ental
probab
ilityand
in
wha
tkind
ofsituation
wouldyouus
e
theo
retica
lprobab
ility?
Inve
stig
atio
n 3
Mak
ingD
ecisions
With
Probab
ility
Pro
ble
m 3
.1 D
esigning
aSp
inne
rtoFind
Probab
ilities
Focu
s Q
uest
ion
Wha
tdoes
itm
eanforasp
inne
rtobe“fair”or“b
aise
d”?
Pro
ble
m 3
.2
Mak
ingD
ecisions
:Ana
lyzing
Fairnes
s
Focu
s Q
uest
ion
Wha
tsh
ouldyouco
nsiderw
hen
youan
alyzeagam
eto
determinewhe
theritis
fairorno
t?
Pro
ble
m 3
.3 R
olle
rDerby:A
nalyzing
aG
ame
Focu
s Q
uest
ion
Whe
nplaying
Rolle
rDerby(or
othergam
es),why
is
strategyim
portan
t?
Pro
ble
m 3
.4 S
cratch
ing
Spots:D
esigning
and
Using
aSim
ulation
Focu
s Q
uest
ion
Ifyo
uca
nnotdire
ctlycompute
theprobab
ilityofan
ev
entocc
urring
,how
canyo
ugathe
rdatathat
wouldhelpyoupredict
theprobab
ilityofthe
even
tocc
urring
?
Inve
stig
atio
n 4
Ana
lyzing
Compoun
d
Eve
ntsUsing
an
Area Model
Pro
ble
m 4
.1 D
rawing
AreaModelstoFindthe
Sa
mpleSpac
e
Focu
s Q
uest
ion
In
wha
tsituations
isan
area
modelforan
alyzing
probab
ilities
helpful?
Pro
ble
m 4
.2 M
aking
Purple:A
reaModels
andProbab
ility
Focu
s Q
uest
ion
Inw
hat
situations
isithelpfulto
useasimulationtofind
aprobab
ility?
Pro
ble
m 4
.3 O
ne-
and-O
neFree-Th
rows:
SimulatingaProbab
ility
Situation
Focu
s Q
uest
ion
Iftw
o
baske
tballp
laye
rstak
eone
sho
tea
chand
the
firstplaye
rmisse
san
d
these
cond
playe
rmak
es
thesh
ot,w
hoisthe
betterplaye
r?Exp
lain
yourans
wer.
Pro
ble
m 4
.4 S
coring
Points:F
inding
Exp
ectedValue
Focu
s Q
uest
ion
Wha
tdoes
exp
ectedvalue
mea
ninaprobab
ility
situation?
Inve
stig
atio
n 5
BinomialO
utco
mes
Pro
ble
m 4
.1 G
uessing
Ans
wers:FindingM
ore
Exp
ectedValue
s
Focu
s Q
uest
ion
Ifyo
udo
notkn
owthe
ans
wersto
atrue
/false
tes
t,w
hyisit
usefultoran
domlygue
ss?
Pro
ble
m 4
.2 O
rtonv
ille:
BinomialP
robab
ility
Focu
s Q
uest
ion
Wha
tdoes
itm
eanfor
asituationtobea
“binomial”probab
ility
situation?
Pro
ble
m 4
.3 A
Baseb
all
Series
:Exp
anding
BinomialP
robab
ility
Focu
s Q
uest
ion
Iftw
obaseb
alltea
msare
equa
llym
atch
edand
one
team
hasw
onthefirst
twogam
esofase
ven
gam
ese
ries
,isitlike
lythe
othertea
mw
illw
initall?
Exp
lainyourans
wer.
Focu
s Q
uest
ion
Wha
tisagoodw
aytocho
ose
betwee
ntw
oormoreeve
ntsso
tha
tea
cheve
nthasthe
sam
ech
ance
ofbeing
selec
ted?
Focu
s Q
uest
ion
Which
res
ultwouldyouco
nsidertobemorelike
lytoocc
ur—
one
based
onthreetrialsorone
based
on25
trials?W
hy?
Focu
s Q
uest
ion
Ifyo
uca
nnotlistallo
fthepossibleoutco
mes
ofan
eve
nt,h
owcan
youmak
eadec
isionab
outthe
like
lihoodofan
eve
ntoccurring
?
Focu
s Q
uest
ion
Howcan
youdeterminewhe
therthe
outco
mes
ofaprobab
ilityeve
ntarealleq
uallylike
lyand
why
wouldthisinform
ationmatter?
Focu
s Q
uest
ion
Ifyo
uca
nnotlookintoacontaine
r,ho
wcouldyouco
llectthe
datayo
une
edtopredicttheproportionofblueblock
sinacontaine
rwithblue,yellow,a
ndred
block
s?
Focu
s Q
uest
ion
Wha
tisthe
differen
cebetwee
nex
perim
entala
ndthe
oretica
lprobab
ility?
Focu
s Q
uest
ion
Howcan
youdec
idewhe
theragam
eisfairorno
t?
Focu
s Q
uest
ion
Inw
hatkind
ofsituationwouldyouus
eex
perim
entalp
robab
ilityand
inw
hatkind
ofsituationwouldyouus
etheo
retica
lprobab
ility?
Focu
s Q
uest
ion
Wha
tdoes
itm
eanforasp
inne
rtobe“fair”or“b
aise
d”?
Focu
s Q
uest
ion
Wha
tsh
ouldyouco
nsiderw
henyo
uan
alyzeagam
etodeterminewhe
theritisfairorno
t?
Focu
s Q
uest
ion
Whe
nplaying
Rolle
rDerby(orothergam
es),why
isstrateg
yim
portan
t?
Focu
s Q
uest
ion
Ifyo
uca
nnotdire
ctlycomputetheprobab
ilityofan
eve
ntocc
urring
,howcan
yougathe
rdatathatw
ouldhelpyoupredicttheprobab
ilityoftheev
entoccurring
?
Focu
s Q
uest
ion
Inw
hatsituations
isanarea
modelforan
alyzingprobab
ilities
helpful?
Focu
s Q
uest
ion
Inw
hatsituations
isithelpfultouse
asim
ulationtofind
aprobab
ility?
Focu
s Q
uest
ion
Iftw
obaske
tballp
laye
rstak
eone
sho
tea
chand
the
firstplaye
rmisse
san
dthe
sec
ond
playe
rmak
esthe
sho
t,w
hoisthe
betterplaye
r?E
xplainyourans
wer.
Focu
s Q
uest
ion
Wha
tdoes
exp
ectedvalue
mea
ninaprobab
ilitysitua
tion?
Focu
s Q
uest
ion
Ifyo
udonotkn
owthe
ans
werstoatrue/falsetest,w
hyisituse
fultoran
domlygue
ss?
Focu
s Q
uest
ion
Wha
tdoes
itm
eanforasituationtobea“b
inomial”probab
ilitysitua
tion?
Focu
s Q
uest
ion
Iftw
obaseb
alltea
msareeq
uallym
atch
edand
one
tea
mhasw
onthefirsttwogam
esofase
vengam
ese
ries
,isitlike
lythe
othertea
mw
illw
initall?E
xplainyourans
wer.
Graphic Organizers for Grade 7 95
Filli
ng a
nd W
rapp
ing
Thre
e-D
imen
sion
al M
easu
rem
ent
Ess
enti
al Id
eas
•Prism
sarena
med
fortheirbases
.The
nam
eofaprism
indicates
thenu
mberofve
rtices
,edges
,and
fac
esthe
prism
has.
•Slicingprism
sve
rtically,h
orizo
ntally,o
ronaslan
tca
nex
pose
differen
tsh
apes
ofcross-sec
tions
,dep
endingonwhich
ofthe
originaledges
arein
tersec
ted.
•Comparing,rea
soning
about,a
ndexten
dingw
hatyo
ukn
ow
aboutareaan
dvolumelead
stoanun
derstan
dingoftheform
ulas
forfin
dingthe
surface
areaan
dvolumeofprism
s,cone
s,
andpyram
ids.
•Proportiona
lcha
nges
todim
ensions
ofthesides
ofaprism
lead
stopredictablecha
nges
inthe
surface
areaan
dthe
volume.
•Approximations
oftheratioofthecircum
ferenc
eofacircleto
thecircle’sdiameterle
adstoexa
ctform
ulasforthearea
and
circum
ferenc
eofacircle.
Inve
stig
atio
n 1
BuildingSmartBoxe
s:
Rec
tang
ularPrism
s
Pro
ble
m 1
.1 H
owBigA
reTho
se
Boxe
s?FindingVolume
Pro
ble
m 1
.2 O
ptimalC
ontaine
rsI:
Find
ingSurface
Area
Pro
ble
m 1
.3 O
ptimalC
ontaine
rsII:
Find
ingthe
Lea
stSurface
Area
Pro
ble
m 1
.4 C
ompost
Containe
rs:S
calin
gU
pPrism
s
Inve
stig
atio
n 2
Polygona
lPrism
s
Pro
ble
m 2
.1 S
urface
Areaan
d
VolumeofPrism
s
Pro
ble
m 2
.2 C
alcu
lating
Volume
ofPrism
s
Pro
ble
m 2
.3 S
licingPrism
san
d Pyram
ids
Inve
stig
atio
n 3
Areaan
dC
ircum
ferenc
eofCirc
les
Pro
ble
m 3
.1 G
oingA
roun
d
inC
ircles:C
ircum
ferenc
e
Pro
ble
m 3
.2 P
ricing
Pizza:
Conn
ecting
Area,D
iameter,
andRad
ius
Pro
ble
m 3
.3 S
qua
ring
aC
ircleto
Find
ItsArea
Pro
ble
m 3
.4 C
onn
ecting
Circ
umferenc
ean
dA
rea
Inve
stig
atio
n 4
Cylinders,C
one
s,and
Sphe
res
Pro
ble
m 4
.1 S
urface
Area
of Cylinders
Pro
ble
m 4
.2 V
olumeofCylinders
Pro
ble
m 4
.3 C
omparing
Juice Containe
rs:
ComparingVolumes
Pro
ble
m 4
.4 F
illingC
one
san
d Pyram
ids
Pro
ble
m 4
.5 C
omparingVolumes
ofSp
heres,C
ylinders,and
Cone
s
Teacher Implementation Toolkit96
Inve
stig
atio
n 1
BuildingSmartBoxe
s:
Rec
tang
ularPrism
s
Pro
ble
m 1
.1 H
owBigA
reTho
se
Boxe
s?FindingVolume
Focu
s Q
uest
ion
Wha
tdothe
su
rfac
earea
and
volumetellab
out
thesize
ofarectan
gularprism
?Wha
tmea
suremen
tsdoyou
need
tocalcu
latesurface
area
andvolume?
Pro
ble
m 1
.2 O
ptimalC
ontaine
rsI:
Find
ingSurface
Area
Focu
s Q
uest
ion
Ifyo
udes
igna boxinthe
sha
peofa
rectan
gular prism
withavo
lume
of24
cm
3 ,des
cribethesh
apean
d
dim
ensions
oftheprism
tha
tha
sminim
umsurface
area.
Pro
ble
m 1
.3 O
ptimalC
ontaine
rsII:
Find
ingthe
Lea
stSurface
Area
Focu
s Q
uest
ion
Ifyo
udes
ign
arectan
gularprism
withgiven
vo
lume,w
hatarethedim
ensions
oftherectan
gularprism
tha
tha
stheleastsu
rfac
earea
?
Pro
ble
m 1
.4 C
ompostC
ontaine
rs:
ScalingU
pPrism
s
Focu
s Q
uest
ion
Asyo
uch
ange
dim
ensions
byace
rtainscale
factor,ho
wdothe
surface
area
andvolumeoftheprism
cha
nge?
Inve
stig
atio
n 2
Polygona
lPrism
s
Pro
ble
m 2
.1 S
urface
Areaan
d
VolumeofPrism
s
Focu
s Q
uest
ion
Howdothe
vo
lumean
dsurface
areaofaprism
ch
angeasthe
num
berofsides
increa
sebutthe
heightand
area
ofsides
rem
ainthesame?
Pro
ble
m 2
.2 C
alcu
lating
Volume
ofPrism
s
Focu
s Q
uest
ion
Wha
tgen
eral
strategyca
nbeus
edtofind
the
vo
lumeofan
yprism
—triang
ular,
rectan
gular,p
entagona
l,an
d
so on?
Pro
ble
m 2
.3 S
licingPrism
s
andPyram
ids
Focu
s Q
uest
ion
Wha
tsu
rfac
esh
apes
and
three
dim
ensiona
lsh
apes
can
becrea
tedbyslicing
arectan
gularprism
withcu
tsin
va
rious
dire
ctions
?
Inve
stig
atio
n 3
Areaan
dC
ircum
ferenc
eofCirc
les
Pro
ble
m 3
.1 G
oingA
roun
d
inC
ircles:C
ircum
ferenc
e
Focu
s Q
uest
ion
Ifyo
ukn
owthe
diameter(o
rradius)ofacircle,
howcan
youca
lculate
itscircum
ferenc
e?
Pro
ble
m 3
.2 P
ricing
Pizza:
Conn
ecting
Area,D
iameter,
andRad
ius
Focu
s Q
uest
ion
Howdoes
the
area
ofacirclein
crea
seasthe
radiusan
ddiameterin
crea
se?
Pro
ble
m 3
.3 S
qua
ring
aC
ircle
toFindItsArea
Focu
s Q
uest
ion
Howcan
youfin
d
thearea
ofacircleifyoukn
ow
theradius?H
owcan
youfin
dthe
radiusofacircleifyoukn
owthe
area
?Wha
tev
iden
cedoyouha
ve
thattho
sestrateg
ieswillw
orkfor
anycircle?
Pro
ble
m 3
.4 C
onn
ecting
Circ
umferenc
ean
dA
rea
Focu
s Q
uest
ion
Howdothe
diagramsatthe
startofthe
problemgiveev
iden
ceconn
ecting
circum
ferenc
ean
dareaofacircle
tothe
rad
iusan
dthe
num
ber p
?
Inve
stig
atio
n 4
Cylinders,C
one
s,and
Sphe
res
Pro
ble
m 4
.1 S
urface
Area
of Cylinders
Focu
s Q
uest
ion
Howcan
you
calculatethesu
rfac
earea
of
acy
linderand
why
does
tha
tstrategywork?
Pro
ble
m 4
.2 V
olumeofCylinders
Focu
s Q
uest
ion
Howcan
you
calculatethevo
lumeofacy
linder
andhowisthe
proce
duresimila
rtotha
tforprism
s?
Pro
ble
m 4
.3 C
omparing
JuiceContaine
rs:
ComparingVolumes
Focu
s Q
uest
ion
Howdocylinders
comparewithrectan
gularprism
sinpac
kagingagiven
volumefor
minim
umcost?
Pro
ble
m 4
.4 F
illingC
one
san
d Pyram
ids
Focu
s Q
uest
ion
Howdoes
the
vo
lumeofasp
hereand
cone
co
mparetotha
tofacy
linderw
ith
adiameterand
heightequa
lto
thatofthesp
hereand
cone
?
Wha
tform
ulaforvo
lumeofa
sphe
reand
cone
isim
plie
dby
thes
erelations
hips?
Pro
ble
m 4
.5 C
omparingVolumes
ofSp
heres,C
ylinders,and
Cone
s
Focu
s Q
uest
ion
Which
co
ntaine
r—aco
neoracu
p—
seem
stoholdthe
mostic
ecrea
m
andhowdoyoukn
ow?
Graphic Organizers for Grade 7 97
Sam
ples
and
Pop
ulat
ions
Mak
ing
Com
pari
sons
and
Pre
dict
ions
Ess
enti
al Id
eas
•Asurve
yallowsyo
utogathe
rdataus
ingasam
pleofapopulation
anduse
tha
tdatatorep
rese
ntthe
population.Tab
lesan
dgraphs
,asw
ella
smea
suresofce
nterand
variabilityena
bleyouto
comparedatafromdifferen
tsamplesan
ddrawconc
lusions
about
thesamplesan
dthe
populations
.
•Ran
domsam
plesarewitho
utbias,and
the
reforeareuse
ful
forpredicting
populationch
arac
teristics.Probab
ilitym
odels
allowyoutoselec
tarand
omsam
plefromapopulation.Ran
dom
samples,eve
nofthesamesize
,varyfromeac
hotherand
from
theun
derlyingpopulation.
•Yo
uca
nco
mparetw
osam
pleswithap
proximatelythesame
mea
sureofva
riab
ilitybyus
ingtha
tmea
suretodeterminethe
distanc
ebetwee
nthece
ntersofthesamples.
Inve
stig
atio
n 1
Mak
ingSen
seofSa
mples
Pro
ble
m 1
.1 C
omparingPerform
ance
s:
Looking
atCen
teran
dSpread
Pro
ble
m 1
.2 W
hich
Tea
misM
ostSuc
cessful?
Using
the
MADtoC
ompareSa
mples
Pro
ble
m 1
.3 C
astYo
urVote:D
isting
uish
ing
Categ
orica
land
Num
ericalD
ata
Pro
ble
m 1
.4 A
reSteelC
oastersFaster
than
WoodC
oasters?Using
the
IQR
toC
ompareSa
mples
Inve
stig
atio
n 2
Cho
osing
Sam
plesFromPopulations
Pro
ble
m 2
.1 A
skingA
boutH
one
sty:
Using
aSam
pletoM
akePredictions
Pro
ble
m 2
.2 S
elec
ting
aSam
ple:
Differen
tKindsofSa
mples
Pro
ble
m 2
.3 C
hoosing
Ran
domSam
ples:
ComparingSam
plesUsing
Cen
teran
dSpread
Pro
ble
m 2
.4 G
rowingSam
ples:
Wha
tSize
Sam
pletoU
se?
Inve
stig
atio
n 3
Using
Sam
plestoM
akePredictions
Pro
ble
m 3
.1 S
olvinganArche
ologicalM
ystery:
ComparingSam
plesUsing
BoxPlots
Pro
ble
m 3
.2 F
iveCho
colateC
hipsinE
very
Cookie:U
sing
Sam
plin
gin
aSim
ulation
Pro
ble
m 3
.3 E
stim
atingthe
Dee
rPopulation:
Using
Sam
plestoEstim
atetheSize
of
aPopulation
Pro
ble
m 3
.4 C
omparingN
BABaske
tball
Playe
rsand
WNBABaske
tballP
laye
rs:
Using
MADsan
dM
eans
Teacher Implementation Toolkit98
Inve
stig
atio
n 1
Mak
ingSen
seofSa
mples
Pro
ble
m 1
.1 C
omparingPerform
ance
s:Looking
atC
enteran
dSpread
Focu
s Q
uest
ion
Ifyo
uus
ethesamplesco
res
provided
,howm
ightyouan
swerthisque
stion:
Who
perform
sbetteronmathtests,John
orMary?
Pro
ble
m 1
.2 W
hich
Tea
misM
ostSuc
cessful?
Using
the
MADtoC
ompareSa
mples
Focu
s Q
uest
ion
Wha
tareso
medifferen
tstrategiesyo
umightuse
toans
werthisque
stion:
Which
tea
misthe
mostsuc
cessfula
ndsho
uld
winthe
prize
?
Pro
ble
m 1
.3 C
astYo
urVote:D
isting
uish
ing
Categ
orica
land
Num
ericalD
ata
Focu
s Q
uest
ion
Howm
ightw
eco
mparethes
eresu
ltstosee
ifeac
hgroup
res
pond
edtothe
su
rvey
inasim
ilarway
?Howm
ightw
eus
eperce
ntag
estohelpusmak
eco
mparisons
?
Pro
ble
m 1
.4 A
reSteelC
oastersFaster
than
WoodC
oasters?Using
the
IQR
toC
ompareSa
mples
Focu
s Q
uest
ion
Howm
ightyoudec
idewhich
arefaster:s
teelcoastersorwoodcoasters?
Inve
stig
atio
n 2
Cho
osing
Sam
plesFromPopulations
Pro
ble
m 2
.1 A
skingA
boutH
one
sty:
Using
aSam
pletoM
akePredictions
Focu
s Q
uest
ion
Wha
tisapopulation?
Wha
tisa
samplin
gplan?
Wha
tmak
esagoodsam
ple?
Pro
ble
m 2
.2 S
elec
ting
aSam
ple:
Differen
tKindsofSa
mples
Focu
s Q
uest
ion
Howcouldyouse
lectasam
ple
ofyo
ursch
oolp
opulationtosurve
y?
Pro
ble
m 2
.3 C
hoosing
Ran
domSam
ples:
ComparingSam
plesUsing
Cen
teran
dSpread
Focu
s Q
uest
ion
Howcan
youus
estatistics
aboutaran
domsam
pleofthes
edatatom
ake
predictions
aboutthe
entire
populationof10
07thgradestud
entsin
the
sch
ool?
Pro
ble
m 2
.4 G
rowingSam
ples:
Wha
tSize
Sam
pletoU
se?
Focu
s Q
uest
ion
Areyouab
letom
ake
goodstatisticalestim
ates
withlessw
orkby
selectingsmallersamples?H
owdoes
the
size
ofthesamplerelatetothe
acc
urac
yof
statistica
lestim
ates
?
Inve
stig
atio
n 3
Using
Sam
plestoM
akePredictions
Pro
ble
m 3
.1 S
olvinganArche
ologicalM
ystery:
ComparingSam
plesUsing
BoxPlots
Focu
s Q
uest
ion
Howm
ightyouan
alyzethes
edatatohelpyoupredictthese
ttlemen
tperiods
ofthearrowhe
addatafromnew
sites
?
Pro
ble
m 3
.2 F
iveCho
colateC
hipsinE
very
Cookie:U
sing
Sam
plin
gin
aSim
ulation
Focu
s Q
uest
ion
Wha
tisthe
typ
icalnum
berof
chipsne
eded
tohav
eatle
ast5ch
ipsineve
ry
cookie?
Wha
tad
vice
wouldyougivetoJeff
andTed
tohelpthe
msolvetheirqua
lity-co
ntrol
problem?
Pro
ble
m 3
.3 E
stim
atingthe
Dee
rPopulation:
Using
Sam
plestoEstim
atetheSize
of
aPopulation
Focu
s Q
uest
ion
Howisitpossibletoestim
ate
thedee
rpopulationofastate,orev
enofasm
all
partofastate?
Pro
ble
m 3
.4 C
omparingN
BABaske
tball
Playe
rsand
WNBABaske
tballP
laye
rs:
Using
MADsan
dM
eans
Focu
s Q
uest
ion
AreN
BAbaske
tballp
laye
rs
talle
rthan
WNBAbaske
tballp
laye
rs?Exp
lain.
Graphic Organizers for Grade 7 99
Shap
es a
nd D
esig
ns T
wo-
Dim
ensi
onal
Geo
met
ry
Ess
enti
al Id
eas
•Th
esu
moftheinterioran
glesofapolygonrelatestothe
num
ber
oftriang
lesthatareform
edbydrawingdiagona
lsfromone
vertex.
•Triang
lesha
ve3sides
,butnotev
eryco
mbinationof3sideleng
ths
willm
akeatriang
le.
•Aswithtriang
les,spec
ificco
mbinations
ofsideleng
ths
ofapolygonca
nproduc
eco
ngruen
tco
piesofthepolygon.
•Ang
lesca
nbeclassifie
dbytheirsize
,the
irlo
cationinrelation
toeac
hotherin
afigureordes
ign,and
the
ircombined
ang
le
mea
sure.A
ngleclassifica
tionbyloca
tionorco
mbined
ang
le
mea
surecan
helpyouwriteequa
tions
tofind
unk
nown
anglem
easu
res.
Inve
stig
atio
n 1
TheFa
milyofPolygons
Pro
ble
m 1
.1 S
ortingand
Ske
tching
Polygons
Pro
ble
m 1
.2 InaSp
in:A
nglesan
dRotations
Pro
ble
m 1
.3 E
stim
atingM
easu
res
ofRotations
and
Ang
les
Pro
ble
m 1
.4 M
easu
ring
Ang
les
Pro
ble
m 1
.5 D
esignCha
lleng
eI:
DrawingW
ithTo
ols—
Rulerand
Protrac
tor
Inve
stig
atio
n 2
Des
igning
Polygons
:The
Ang
leC
onn
ection
Pro
ble
m 2
.1 A
ngleSum
sofReg
ularPolygons
Pro
ble
m 2
.2 A
ngleSum
sofAny
Polygon
Pro
ble
m 2
.3 T
heBee
sDoIt:P
olygons
inN
ature
Pro
ble
m 2
.4 T
heIn
san
dO
utsofPolygons
Inve
stig
atio
n 3
Des
igning
Trian
glesan
dQ
uadrilaterals
Pro
ble
m 3
.1 B
uildingTrian
gles
Pro
ble
m 3
.2 D
esignCha
lleng
eII:
DrawingTrian
gles
Pro
ble
m 3
.3 B
uildingQ
uadrilaterals
Pro
ble
m 3
.4 P
arallelL
ines
and
Trans
versals
Pro
ble
m 3
.5 D
esignCha
lleng
eIII:
TheQua
drilateralG
ame
Teacher Implementation Toolkit100
Inve
stig
atio
n 1
TheFa
milyofPolygons
Pro
ble
m 1
.1 S
ortingand
Ske
tching
Polygons
Focu
s Q
uest
ion
Wha
tproperties
doall
polygons
sha
re?Wha
tproperties
dosome
sub-group
sofpolygons
sha
re?
Pro
ble
m 1
.2 InaSp
in:A
nglesan
dRotations
Focu
s Q
uest
ion
Wha
tareso
meco
mmon
ben
chmarkan
gles?W
hatpartofafullturn
iseac
han
gleequa
lto?
Pro
ble
m 1
.3 E
stim
atingM
easu
res
ofRotations
and
Ang
les
Focu
s Q
uest
ion
Whe
nadrawingsho
wstw
o
raysw
ithaco
mmonen
dpoint,howm
any
rotationan
glesarethere?
Howw
ouldyou
estimatethemea
sureofea
chang
le?
Pro
ble
m 1
.4 M
easu
ring
Ang
les
Focu
s Q
uest
ion
Howdoyoumea
sureanan
gle
withan
ang
lerulerand
aprotrac
tor?
Pro
ble
m 1
.5 D
esignCha
lleng
eI:
DrawingW
ithTo
ols—
Rulerand
Protrac
tor
Focu
s Q
uest
ion
Inatrian
gle,w
hatmea
suresof
sides
and
ang
lesgivejusteno
ughinform
ationto
drawafigurethatisunique
lydetermined
?
Inve
stig
atio
n 2
Des
igning
Polygons
:The
Ang
leC
onn
ection
Pro
ble
m 2
.1 A
ngleSum
sofReg
ularPolygons
Focu
s Q
uest
ion
Wha
tisthe
sizeofea
chang
le
andthe
sum
ofalla
nglesinareg
ularpolygon
with
nsides
?
Pro
ble
m 2
.2 A
ngleSum
sofAny
Polygon
Focu
s Q
uest
ion
Wha
tisthe
ang
lesum
ofan
ypolygonwith
nsides
?Howdoyoukn
owtha
tyo
urform
ulaiscorrec
t?
Pro
ble
m 2
.3 T
heBee
sDoIt:P
olygons
inN
ature
Focu
s Q
uest
ion
Which
reg
ularpolygons
can
be
used
totile
asurface
witho
utove
rlap
sorgap
s,
andhowdoyoukn
owtha
tyo
urans
wer
iscorrec
t?
Pro
ble
m 2
.4 T
heIn
san
dO
utsofPolygons
Focu
s Q
uest
ion
Wha
tisanex
terioran
gleof
apolygon,and
wha
tdoyoukn
owaboutthe
mea
suresofexterioran
gles?
Inve
stig
atio
n 3
Des
igning
Trian
glesan
dQ
uadrilaterals
Pro
ble
m 3
.1 B
uildingTrian
gles
Focu
s Q
uest
ion
Wha
tco
mbinations
ofthree
sideleng
thsca
nbeus
edtom
akeatriang
le?
Howm
anydifferen
tsh
apes
arepossibleforsu
ch
aco
mbinationofsideleng
ths?
Pro
ble
m 3
.2 D
esignCha
lleng
eII:
DrawingTrian
gles
Focu
s Q
uest
ion
Wha
tisthe
smallestnum
berof
sidean
dang
lem
easu
remen
tstha
twilltelly
ou
howtodrawanex
actco
pyofa
nygiven
trian
gle?
Pro
ble
m 3
.3 B
uildingQ
uadrilaterals
Focu
s Q
uest
ion
Wha
tco
mbinations
ofside
leng
thsca
nbeus
edtom
akeaqua
drilateral?
Howm
anydifferen
tsh
apes
arepossibleforan
ysu
chcombinationofsideleng
ths?
Pro
ble
m 3
.4 P
arallelL
ines
and
Trans
versals
Focu
s Q
uest
ion
Whe
ntw
oparallellines
are
cutbyatran
sversal,wha
tca
nbesaidaboutthe
eightang
lesthatareform
ed?
Pro
ble
m 3
.5 D
esignCha
lleng
eIII:
TheQua
drilateralG
ame
Focu
s Q
uest
ion
Howaresqua
res,rho
mbus
es,
rectan
gles,and
trapez
oidssimila
r?H
oware
they
differen
t?
Graphic Organizers for Grade 7 101
