Euclidean m-Space & Linear Equations Systems of Linear Equations.
It’s in the System Systems of Linear Equations and...
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New York City Graphic Organizers for CMP3 It’s in the System Systems of Linear Equations and Inequalities Essential Ideas • A system of linear equations can be used to solve problems when two or more equations that represent constraints on the variables in a situation are identified. • The solution to a system of linear equations can be found graphically or algebraically. Analyzing the equations and the situation can help you to determine which strategy is most appropriate to apply. • The strategies for solving linear equations, linear inequalities and systems of linear equations can be extended to solving systems of linear inequalities. Investigation 1 Linear Equations With Two Variables Problem 1.1 Shirts and Caps: Solving Equations With Two Variables Problem 1.2 Connecting ax + by = c and y = mx + b Problem 1.3 Booster Club Members: Intersecting Lines Investigation 2 Solving Linear Systems Symbolically Problem 2.1 Shirts and Caps Again: Solving Systems With y = mx + b Problem 2.2 Taco Truck Lunch: Solving System by Combining Equations I Problem 2.3 Solving Systems by Combining Equations II Investigation 3 Systems of Functions and Inequalities Problem 3.1 Comparing Security Services: Linear Inequalities Problem 3.2 Solving Linear Inequalities Symbolically Problem 3.3 Operating at a Profit: Systems of Lines and Curves Investigation 4 Systems of Linear Inequalities Problem 4.1 How We Pollute: Inequalities With Two Variables Problem 4.2 What Makes a Car Green: Solving Inequalities by Graphing I Problem 4.3 Feasible Points: Solving Inequalities by Graphing II Problem 4.4 Miles of Emissions: Systems of Linear Inequalities Investigation 1 Linear Equations With Two Variables Problem 1.1 Shirts and Caps: Solving Equations With Two Variables Focus Question What kind of solutions will be found for an equation like 3x + 5y = 13 with two variables? What will the graphs of those solutions look like? Problem 1.2 Connecting ax + by = c and y = mx + b Focus Question How can one change an equation from ax + by = c form to an equivalent y = mx + b form and vice versa? Problem 1.3 Booster Club Members: Intersecting Lines Focus Question What happens when you search for common solutions to two linear equations with two variables? Investigation 2 Solving Linear Systems Symbolically Problem 2.1 Shirts and Caps Again: Solving Systems With y = mx + b Focus Question How can you solve a system of two linear equations with two variables by writing each equation in equivalent y = mx + b form? What are the solution possibilities for such systems and how are they shown by graphs of the solutions? Problem 2.2 Taco Truck Lunch: Solving System by Combining Equations I Focus Question How can you solve a system of linear equations by combining the two equations into one simpler equation by addition or subtraction? Problem 2.3 Solving Systems by Combining Equations II Focus Question How can equations in a system be transformed to equivalent forms that make it easier to solve by combination to eliminate variables? Investigation 3 Systems of Functions and Inequalities Problem 3.1 Comparing Security Services: Linear Inequalities Focus Question How can you use function graphs to find the solutions of an inequality like ax + b < cx + d? How can the solutions be represented on a number line graph? Problem 3.2 Solving Linear Inequalities Symbolically Focus Question How does applying the same operation to both sides of an inequality change the relationship of the two quantities being compared (or not)? How can linear inequalities be solved by strategies that are very similar to strategies for solving linear equations? Problem 3.3 Operating at a Profit: Systems of Lines and Curves Focus Question What are the possible solutions for a system that includes one linear and one quadratic function and how can you find these solutions? Investigation 4 Systems of Linear Inequalities Problem 4.1 How We Pollute: Inequalities With Two Variables Focus Question If a problem involves solving an inequality like ax +by ≤ c, how many solutions would you expect to find and what would a coordinate graph of those solutions look like? Problem 4.2 What Makes a Car Green: Solving Inequalities by Graphing I Focus Question What graph of solutions (in the first quadrant) would you expect for an inequality with the general form ax + by ≤ c? Problem 4.3 Feasible Points: Solving Inequalities by Graphing II Focus Question What graph of solutions would you expect for an inequality with the general form ax + by ≤ c? Problem 4.4 Miles of Emissions: Systems of Linear Inequalities Focus Question What do you look for to solve a system of linear inequalities and what will the graph of a solution look like? 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 8 85
Transcript of It’s in the System Systems of Linear Equations and...
New York City Graphic Organizers for CMP3
It’s in the System Systems of Linear Equations and Inequalities
Essential Ideas
•Asystemoflinearequationscanbeusedtosolveproblemswhentwoormoreequationsthatrepresentconstraintsonthevariablesinasituationareidentified.
•Thesolutiontoasystemoflinearequationscanbefoundgraphicallyoralgebraically.Analyzingtheequationsandthesituationcanhelpyoutodeterminewhichstrategyismostappropriatetoapply.
•Thestrategiesforsolvinglinearequations,linearinequalitiesandsystemsoflinearequationscanbeextendedtosolvingsystemsoflinearinequalities.
Investigation 1LinearEquationsWithTwoVariables
Problem 1.1 ShirtsandCaps:SolvingEquationsWithTwoVariables
Problem 1.2 Connectingax+by=candy=mx+b
Problem 1.3 BoosterClubMembers:IntersectingLines
Investigation 2SolvingLinearSystemsSymbolically
Problem 2.1 ShirtsandCapsAgain:SolvingSystemsWithy=mx+b
Problem 2.2 TacoTruckLunch:SolvingSystembyCombiningEquationsI
Problem 2.3 SolvingSystemsbyCombiningEquationsII
Investigation 3SystemsofFunctionsandInequalities
Problem 3.1 ComparingSecurityServices:LinearInequalities
Problem 3.2 SolvingLinearInequalitiesSymbolically
Problem 3.3 OperatingataProfit:SystemsofLinesandCurves
Investigation 4SystemsofLinearInequalities
Problem 4.1 HowWePollute:InequalitiesWithTwoVariables
Problem 4.2 WhatMakesaCarGreen:SolvingInequalitiesbyGraphingI
Problem 4.3 FeasiblePoints:SolvingInequalitiesbyGraphingII
Problem 4.4 MilesofEmissions:SystemsofLinearInequalities
Investigation 1Linear Equations With Two Variables
Problem 1.1 Shirts and Caps: Solving Equations With Two Variables
Focus Question What kind of solutions will be found for an equation like 3x + 5y = 13 with two variables? What will the graphs of those solutions look like?
Problem 1.2 Connecting ax + by = c and y = mx + b
Focus Question How can one change an equation from ax + by = c form to an equivalent y = mx + b form and vice versa?
Problem 1.3 Booster Club Members: Intersecting Lines
Focus Question What happens when you search for common solutions to two linear equations with two variables?
Investigation 2Solving Linear Systems Symbolically
Problem 2.1 Shirts and Caps Again: Solving Systems With y = mx + b
Focus Question How can you solve a system of two linear equations with two variables by writing each equation in equivalent y = mx + b form? What are the solution possibilities for such systems and how are they shown by graphs of the solutions?
Problem 2.2 Taco Truck Lunch: Solving System by Combining Equations I
Focus Question How can you solve a system of linear equations by combining the two equations into one simpler equation by addition or subtraction?
Problem 2.3 Solving Systems by Combining Equations II
Focus Question How can equations in a system be transformed to equivalent forms that make it easier to solve by combination to eliminate variables?
Investigation 3Systems of Functions and Inequalities
Problem 3.1 Comparing Security Services: Linear Inequalities
Focus Question How can you use function graphs to find the solutions of an inequality like ax + b < cx + d? How can the solutions be represented on a number line graph?
Problem 3.2 Solving Linear Inequalities Symbolically
Focus Question How does applying the same operation to both sides of an inequality change the relationship of the two quantities being compared (or not)? How can linear inequalities be solved by strategies that are very similar to strategies for solving linear equations?
Problem 3.3 Operating at a Profit: Systems of Lines and Curves
Focus Question What are the possible solutions for a system that includes one linear and one quadratic function and how can you find these solutions?
Investigation 4Systems of Linear Inequalities
Problem 4.1 How We Pollute: Inequalities With Two Variables
Focus Question If a problem involves solving an inequality like ax +by ≤ c, how many solutions would you expect to find and what would a coordinate graph of those solutions look like?
Problem 4.2 What Makes a Car Green: Solving Inequalities by Graphing I
Focus Question What graph of solutions (in the first quadrant) would you expect for an inequality with the general form ax + by ≤ c?
Problem 4.3 Feasible Points: Solving Inequalities by Graphing II
Focus Question What graph of solutions would you expect for an inequality with the general form ax + by ≤ c?
Problem 4.4 Miles of Emissions: Systems of Linear Inequalities
Focus Question What do you look for to solve a system of linear inequalities and what will the graph of a solution look like?
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 8 85
Thin
king
With
Mat
hem
atic
al M
odel
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near
and
Inve
rse
Vari
atio
n
Ess
enti
al Id
eas
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ctionca
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unctions
allo
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werque
stions
ormak
epredictions
aboutarelations
hip.L
inea
rrelations
hips
arefunc
tions
.
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outtwovariablesfromrea
l-worldobse
rvations
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entscan
beco
llected
and
rep
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ingraphs
and
tables.The
serep
rese
ntations
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fulforan
alyzingrelations
hips
among
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clud
ingthe
variabilityofthedata.
•Datamay
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the
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ellthe
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odeling
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Pro
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Pro
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Pro
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oatRen
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rea
Pro
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inding
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lCost
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odeling
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Varia
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umericD
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Pro
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Vitruvian
Man
:Relating
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Pro
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.2
Olderand
Faster:
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ativeCorrelations
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orrelation
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utlie
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dardD
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Inve
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Pro
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.1 W
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Stee
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Que
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Pro
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m 4
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nalyzing
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ayTab
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Pro
ble
m 4
.3 A
fter-Sch
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Jobsan
dH
omew
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WorkingBac
kward:
SettingupaTwo-W
ay
Table
Teacher Implementation Toolkit86
Inve
stig
atio
n 1
Exp
loring
DataPatterns
Pro
ble
m 1
.1 B
ridge
Thickn
essan
dStren
gth
Focu
s Q
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How
wouldyoudes
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relations
hipbetwee
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urexp
erim
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Pro
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m 1
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s Q
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How
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Pro
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m 1
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ustom
Cons
truc
tionParts:
Find
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Focu
s Q
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Howcan
yo
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linea
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stig
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n 2
Line
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Pro
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Howcan
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Pro
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Slope
Focu
s Q
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Howdo
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uationfor
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s Q
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Wha
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Pro
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oatRen
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ines
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rEqua
tions
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s Q
uest
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Wha
tstrategiesdoyoufin
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usefultofind
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?
Pro
ble
m 2
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mus
emen
tPa
rkorMov
ies:
Intersec
tingLinea
rMod
els
Focu
s Q
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Whe
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oftw
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Inve
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Pro
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m 3
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Whe
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Pro
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Spee
d,a
ndTim
eFo
cus
Que
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tex
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ingdistanc
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toano
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Pro
ble
m 3
.3 P
lann
ing
aFieldTrip:F
inding
Individua
lCost
Focu
s Q
uest
ion
How
does
the
costperperso
nch
angeifafix
edtotal
costissplitamong
an
increa
sing
num
berof
individua
lpay
ers?
Pro
ble
m 3
.4 M
odeling
DataPatterns
Focu
s Q
uest
ion
Wha
tpatternin
atab
leor
graphofdatasu
gges
tsan
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hatstrategies
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Inve
stig
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n 4
Varia
bilityand
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inN
umericD
ata
Pro
ble
m 4
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itruvian
Man
:RelatingBody
Mea
suremen
ts
Focu
s Q
uest
ion
Ifyo
uha
vedatarelating
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variab
les,howcan
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Pro
ble
m 4
.2 O
lder
andFaster:
Neg
ativeCorrelations
Focu
s Q
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Fromthe
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re
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knowifthe
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?
Pro
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m 4
.3 C
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Coeffic
ientsan
dO
utlie
rs
Focu
s Q
uest
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Wha
tdoes
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r−1
sugges
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hipbetwee
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ovariables?
Pro
ble
m 4
.4 M
easu
ring
Va
riab
ility:S
tand
ard
Dev
iation
Focu
s Q
uest
ion
How
doyouca
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stan
darddev
iationfora
datadistributionan
dw
hat
does
tha
tstatistictell
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Inve
stig
atio
n 5
Varia
bilityand
Associations
inC
ateg
oric
alD
ata
Pro
ble
m 4
.1 W
oodor
Stee
l?Tha
t’sthe
Que
stion
Focu
s Q
uest
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Wha
tdoes
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ay
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wyou
aboutpreferenc
es
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group
s?
Pro
ble
m 4
.2 P
oliticsof
Girlsand
Boys:A
nalyzing
DatainTwo-W
ayTab
les
Focu
s Q
uest
ion
Su
ppose
youha
ve
reco
rded
the
coun
tsof
differen
tpreferenc
esby
group
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aytab
le.
Howcan
youus
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unts,o
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theco
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ide
iftw
ogroup
sha
vethe
samepreferenc
esorno
t?
Pro
ble
m 4
.3 A
fter-Sch
ool
Jobsan
dH
omew
ork:
WorkingBac
kward:
SettingupaTwo-W
ay
Table
Focu
s Q
uest
ion
Suppose
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uha
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Howcan
youorgan
ize
thedatatocomparean
d
dec
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trelative
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trait?
Graphic Organizers for Grade 8 87
Gro
win
g, G
row
ing,
Gro
win
g Ex
pone
ntia
l Fun
ctio
ns
Ess
enti
al Id
eas
•Situations
tha
tca
nbemodeled
byan
exp
one
ntialfun
ctionsh
owa
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the
tab
leofdata;the
rateofch
angegrows
ordec
aysbyaco
nstantfac
tor.Ta
blesan
dgraphs
can
provide
morein
form
ationab
outafun
ctionan
dhelpsolveproblems.
•Th
ereisoften
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none
way
tow
riteaneq
uation.The
ab
ilitytorew
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lentrelations
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henso
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lvingexp
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ntialfun
ctions
an
drelations
hips.
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erearerulesforworkingw
ithex
pone
ntiale
xpressions
.Th
eseproperties
ofex
pone
ntsareus
efulin
writing
equiva
lent
expressions
and
particu
larlywhe
nworkingw
ithva
lues
written
in
scientificno
tation.
Inve
stig
atio
n 1
Exp
one
ntialG
rowth
Pro
ble
m 1
.1 M
aking
Ballots:Introduc
ing
Exp
one
ntialF
unctions
Pro
ble
m 1
.2 R
eque
sting
aRew
ard:R
eprese
nting
Exp
one
ntialF
unctions
Pro
ble
m 1
.3
Mak
ingaN
ewO
ffer:
GrowthFac
tors
Inve
stig
atio
n 2
Exa
miningG
rowth
Patterns
Pro
ble
m 2
.1 K
illerPlant
Strike
sLa
keVictoria:
y-intercep
tsO
therTha
n1
Pro
ble
m 2
.2
GrowingM
old:
Interpreting
Equa
tions
for
Exp
one
ntialF
unctions
Pro
ble
m 2
.3 S
tudying
Sn
akePopulations
:Interpreting
Graphs
of
Exp
one
ntialF
unctions
Inve
stig
atio
n 3
GrowthFac
torsand
GrowthRates
Pro
ble
m 3
.1
Rep
roduc
ingRab
bits
:Frac
tiona
lGrowthPatterns
Pro
ble
m 3
.2 Inv
esting
for
theFu
ture:G
rowthRates
Pro
ble
m 3
.3
Mak
ingaD
ifferen
ce:
Conn
ecting
GrowthRate
andG
rowthFac
tor
Inve
stig
atio
n 4
Exp
one
ntialD
ecay
Pro
ble
m 4
.1
Mak
ingSmaller
Ballots:Introduc
ing
Exp
one
ntialD
ecay
Pro
ble
m 4
.2 F
ighting
Flea
s:Rep
rese
nting
Exp
one
ntialD
ecay
Pro
ble
m 4
.3 C
oolin
g
Water:M
odeling
Exp
one
ntialD
ecay
Inve
stig
atio
n 5
PatternsWithExp
one
nts
Pro
ble
m 4
.1
Looking
forPatterns
Among
Exp
one
nts
Pro
ble
m 4
.2
Rules
forExp
one
nts
Pro
ble
m 4
.3 E
xten
ding
theRules
ofExp
one
nts
Pro
ble
m 4
.4
Operations
With
ScientificNotation
Pro
ble
m 4
.5 R
evisiting
Exp
one
ntialF
unctions
Teacher Implementation Toolkit88
Inve
stig
atio
n 1
Exp
one
ntialG
rowth
Pro
ble
m 1
.1 M
aking
Ballots:Introduc
ing
Exp
one
ntialF
unctions
Focu
s Q
uest
ion
Wha
taretheva
riab
lesinthis
situationan
dhoware
they
related
?
Pro
ble
m 1
.2 R
eque
sting
aRew
ard:R
eprese
nting
Exp
one
ntialF
unctions
Focu
s Q
uest
ion
Inw
hatway
sarethe
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nted
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ssboardand
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tions
simila
r?D
ifferen
t?
Pro
ble
m 1
.3
Mak
ingaN
ewO
ffer:
GrowthFac
tors
Focu
s Q
uest
ion
Howdoes
the
growth
patternforan
ex
pone
ntialfun
ction
showupin
atab
le,
graph,oreq
uationthat
represe
ntsthefunc
tion
andhowdoes
itcompare
tothe
growthpatternin
a
linea
rfunc
tion?
Inve
stig
atio
n 2
Exa
miningG
rowth
Patterns
Pro
ble
m 2
.1 K
illerPlant
Strike
sLa
keVictoria:
y-intercep
tsO
therTha
n1
Focu
s Q
uest
ion
Wha
tinform
ationdoyou
need
tow
riteaneq
uation
thatrep
rese
ntsan
ex
pone
ntialfun
ction?
Pro
ble
m 2
.2
GrowingM
old:
Interpreting
Equa
tions
for
Exp
one
ntialF
unctions
Focu
s Q
uest
ion
How
isthe
growthfac
toran
d
initialp
opulationforan
ex
pone
ntialfun
ction
represe
nted
inan
equa
tionthatrep
rese
nts
thefunc
tion?
Pro
ble
m 2
.3 S
tudying
Sn
akePopulations
:Interpreting
Graphs
of
Exp
one
ntialF
unctions
Focu
s Q
uest
ion
Howisthe
growthfac
tor
andin
itialp
opulation
foran
exp
one
ntial
func
tionreprese
nted
inagraphthatrep
rese
nts
thefunc
tion?
Inve
stig
atio
n 3
GrowthFac
torsand
GrowthRates
Pro
ble
m 3
.1
Rep
roduc
ingRab
bits
:Frac
tiona
lGrowthPatterns
Focu
s Q
uest
ion
Howis
thegrowthfac
torinthis
Problemsim
ilartotha
tin
theprevious
Problems?
Howisitdifferen
t?
Pro
ble
m 3
.2 Inv
esting
for
theFu
ture:G
rowthRates
Focu
s Q
uest
ion
Howarethe
growth
factoran
dgrowth
rateforan
exp
one
ntial
func
tionrelated?Whe
nmightyouus
eea
ch
inanex
pone
ntial
growthpattern?
Pro
ble
m 3
.3
Mak
ingaD
ifferen
ce:
Conn
ecting
GrowthRate
andG
rowthFac
tor
Focu
s Q
uest
ion
Howdoes
the
initial
populationaffectthe
growthpatternsinan
expone
ntialfun
ction?
Inve
stig
atio
n 4
Exp
one
ntialD
ecay
Pro
ble
m 4
.1
Mak
ingSmaller
Ballots:Introduc
ing
Exp
one
ntialD
ecay
Focu
s Q
uest
ion
How
does
the
patternof
chan
geinthissituation
comparetothe
growth
patternsyo
uha
vestudied
inprevious
Problems?
Howdoes
the
differen
ce
showupin
atab
le,g
raph
andequa
tion?
Pro
ble
m 4
.2 F
ighting
Flea
s:Rep
rese
nting
Exp
one
ntialD
ecay
Focu
s Q
uest
ion
How
canyo
ureco
gnize
an
expone
ntiald
ecay
func
tionfromacontex
tual
setting,tab
le,g
raph,and
eq
uationthatrep
rese
nts
thefunc
tion?
Pro
ble
m 4
.3 C
oolin
g
Water:M
odeling
Exp
one
ntialD
ecay
Focu
s Q
uest
ion
How
canyo
ufin
dthe
initial
populationan
ddec
ay
factorforan
exp
one
ntial
dec
ayrelations
hip?
Inve
stig
atio
n 5
PatternsWithExp
one
nts
Pro
ble
m 4
.1
Looking
forPatterns
Among
Exp
one
nts
Focu
s Q
uest
ion
Wha
tpatternsdidyouobse
rve
inthe
tab
leofpowers?
Pro
ble
m 4
.2 R
ules
forExp
one
nts
Focu
s Q
uest
ion
Wha
tarese
veralrules
for
workingw
ithex
pone
nts
andw
hydothe
ywork?
Pro
ble
m 4
.3 E
xten
ding
theRules
ofExp
one
nts
Focu
s Q
uest
ion
Howarethe
rules
for
integrale
xpone
nts
relatedtorationa
lex
pone
nts?H
oware
therulesforex
pone
nts
usefulin
writing
eq
uiva
lentexp
ressions
withex
pone
nts?
Pro
ble
m 4
.4
Operations
With
ScientificNotation
Focu
s Q
uest
ion
How
does
scien
tific
notation
helptosolveproblems?
Pro
ble
m 4
.5 R
evisiting
Exp
one
ntialF
unctions
Focu
s Q
uest
ion
Wha
taretheeffectsof
aan
dbonthegraphof
y=a
(bx ),b≠0?
Graphic Organizers for Grade 8 89
But
terfl
ies,
Pin
whe
els,
and
Wal
lpap
er S
ymm
etry
and
Tra
nsfo
rmat
ions
Ess
enti
al Id
eas
•Va
rious
trans
form
ations
affec
tdistanc
esand
ang
lesoffig
ures
differen
tly.The
seeffec
tshelpyouco
mparefig
ures
and
determine
thesimila
rityorco
ngruen
cebetwee
nfig
ures
.
•Tw
osha
pes
arecong
ruen
tifasp
ecificse
que
nceofrigid
tran
sform
ations
willtrans
form
one
sha
petothe
other.T
wofigures
aresimila
rifasp
ecificse
que
nceofrigidtrans
form
ations
and
dila
tionwilltrans
form
one
sha
petothe
other.
•Properties
oftran
sform
ations
,cong
ruen
ce,a
ndsim
ilaritycan
be
used
tosolveproblemsab
outsha
pes
and
mea
suremen
ts.
Inve
stig
atio
n 1
Symmetryand
Trans
form
ations
Pro
ble
m 1
.1 B
utterflySy
mmetry:
Line
Refl
ections
Pro
ble
m 1
.2 InaSp
in:R
otations
Pro
ble
m 1
.3
SlidingA
roun
d:T
rans
lations
Pro
ble
m 1
.4
Properties
ofTran
sform
ations
Inve
stig
atio
n 2
Tran
sform
ations
and
Cong
ruen
ce
Pro
ble
m 2
.1 C
onn
ecting
Cong
ruen
tPolygons
Pro
ble
m 2
.2 S
upportingthe
World:C
ong
ruen
tTriang
lesI
Pro
ble
m 2
.3
Minim
umM
easu
remen
t:
Cong
ruen
tTriang
lesII
Inve
stig
atio
n 3
Tran
sform
ingC
oordinates
Pro
ble
m 3
.1 F
lippingin
aG
rid:
CoordinateRules
forRefl
ections
Pro
ble
m 3
.2 S
lidingonaGrid:
CoordinateRules
forTran
slations
Pro
ble
m 3
.3 S
pinning
onaGrid:
CoordinateRules
forRotations
Pro
ble
m 3
.4 A
Spec
ialP
roperty
ofTran
slations
and
Half-Tu
rns
Pro
ble
m 3
.5 P
arallelL
ines
,Tran
sversals,a
ndA
ngleSum
s
Inve
stig
atio
n 4
Dila
tions
and
Sim
ilarFigures
Pro
ble
m 4
.1 F
ocu
sonDila
tions
Pro
ble
m 4
.2 R
eturnofSu
per
Sleu
th:S
imila
rityTrans
form
ations
Pro
ble
m 4
.3 C
heck
ingSim
ilarity
Witho
utTrans
form
ations
Pro
ble
m 4
.4
Using
Sim
ilarTriang
les
Teacher Implementation Toolkit90
Inve
stig
atio
n 1
Symmetryand
Trans
form
ations
Pro
ble
m 1
.1 B
utterflySy
mmetry:
Line
Refl
ections
Focu
s Q
uest
ion
Wha
tdoes
it
mea
ntosay
tha
tafig
ureha
srefle
ctionorflipsym
metry?How
iseac
hpointrelatedtoitsim
age
undertrans
form
ationbyrefle
ction
inaline
?
Pro
ble
m 1
.2 InaSp
in:R
otations
Focu
s Q
uest
ion
Wha
tdoes
it
mea
ntosay
tha
tafig
ureha
srotationorturnsym
metry?How
iseac
hpointrelatedtoitsim
age
undertrans
form
ationbyrotation?
Pro
ble
m 1
.3
SlidingA
roun
d:T
rans
lations
Focu
s Q
uest
ion
Wha
tdoes
it
mea
ntosay
tha
tafig
ureha
stran
slationorslidesymmetry?
Howiseac
hpointrelatedto
itsim
ageun
dertrans
form
ation
bytran
slation?
Pro
ble
m 1
.4
Properties
ofTran
sform
ations
Focu
s Q
uest
ion
How,ifatall,
willthe
sha
pe,size,and
position
ofageo
metricfig
urech
ange
afterea
chofthetran
sform
ations
stud
iedin
thisinve
stigation—
flip,
turn,o
rslide?
Inve
stig
atio
n 2
Tran
sform
ations
and
Cong
ruen
ce
Pro
ble
m 2
.1 C
onn
ecting
Cong
ruen
tPolygons
Focu
s Q
uest
ion
Wha
tdoes
it
mea
ntosay
twogeo
metricsh
apes
areco
ngruen
ttoeac
hother
andhowcouldyoudem
ons
trate
cong
ruen
cew
ithmova
blecopies
ofthefig
ures
?
Pro
ble
m 2
.2 S
upportingthe
World:C
ong
ruen
tTriang
lesI
Focu
s Q
uest
ion
Howm
uch
inform
ationdoyoune
edtodec
ide
thattwotrian
glesareprobab
ly
cong
ruen
torno
tco
ngruen
t?
Howdoyougoaboutplann
ing
tran
sform
ations
tha
t‘m
ove
’one
triang
leontoano
ther?
Pro
ble
m 2
.3
Minim
umM
easu
remen
t:
Cong
ruen
tTriang
lesII
Focu
s Q
uest
ion
Wha
tisthe
sm
allestnum
berofsidean
d
anglem
easu
remen
tstha
twill
allowyoutoconc
ludethattwo
triang
lesareco
ngruen
t?
Inve
stig
atio
n 3
Tran
sform
ingC
oordinates
Pro
ble
m 3
.1 F
lippingin
aG
rid:
CoordinateRules
forRefl
ections
Focu
s Q
uest
ion
Howcan
you
des
cribethe‘m
otion’ofpoints
underrefl
ections
withco
ordinate
rulesinthe
form
(x,y
)→(■
,■)
whe
ntherefle
ctionlin
eis:
(1)the
x-axis?(2
)the
y-axis?
(3)the
line
y=x?
Pro
ble
m 3
.2 S
lidingonaGrid:
CoordinateRules
forTran
slations
Focu
s Q
uest
ion
Wha
tkind
of
coordinaterule(x
,y)→
(■,■
)tellshowto‘m
ove
’any
pointtoits
imag
eun
deratrans
lation?
Pro
ble
m 3
.3 S
pinning
onaGrid:
CoordinateRules
forRotations
Focu
s Q
uest
ion
Wha
tarethe
coordinaterulesthatdes
cribe
‘motion’ofpointsonagridund
er
turnsof90
°an
d180
°?
Pro
ble
m 3
.4 A
Spec
ialP
roperty
ofTran
slations
and
Half-Tu
rns
Focu
s Q
uest
ion
Howare
lines
and
the
irim
ages
und
er
tran
slationan
dhalf-turnrelated
toeac
hother?
Pro
ble
m 3
.5 P
arallelL
ines
,Tran
sversals,a
ndA
ngleSum
s
Focu
s Q
uest
ion
Whe
ntw
o
parallellines
arecutbya
tran
sversal,wha
tca
nbesaidabout
thean
glesform
ed?W
hatisalw
ays
true
abou
tthean
glem
easu
resin
atriang
le?H
owdoyo
ukn
owtha
tyo
urans
wersareco
rrec
t?
Inve
stig
atio
n 4
Dila
tions
and
Sim
ilarFigures
Pro
ble
m 4
.1 F
ocu
sonDila
tions
Focu
s Q
uest
ion
Wha
tco
ordinate
rulesmodeldila
tions
and
ho
wdodila
tions
cha
ngeor
prese
rvech
arac
teristicsofthe
originalfigure?
Pro
ble
m 4
.2 R
eturnofSu
per
Sleu
th:S
imila
rityTrans
form
ations
Focu
s Q
uest
ion
Howcan
youus
etran
sform
ations
toche
ckw
hether
twofigures
aresim
ilarorno
t?
Pro
ble
m 4
.3 C
heck
ingSim
ilarity
Witho
utTrans
form
ations
Focu
s Q
uest
ion
Wha
tinform
ation
aboutthe
sides
and
ang
lesoftw
o
triang
leswillgua
rantee
tha
tthey
aresimila
r?
Pro
ble
m 4
.4
Using
Sim
ilarTriang
les
Focu
s Q
uest
ion
Wha
tfacts
aboutsim
ilartriang
lesallowyou
tofind
leng
thsinverylargefig
ures
ev
enw
henthey
can
’tbereac
hed
tom
easu
re?
Graphic Organizers for Grade 8 91
Say
It W
ith S
ymbo
ls M
akin
g Se
nse
of S
ymbo
ls
Ess
enti
al Id
eas
•Algeb
raicequa
tions
and
exp
ressions
can
beus
edto
solveproblems.
•Equiva
lenc
eisuse
fulw
henso
lvingequa
tions
and
problems.
Equiva
lentexp
ressions
can
begen
erated
using
properties
of
operations
.Exa
miningequiva
lentform
sofan
exp
ressionca
nreve
al
newin
form
ationab
outthe
contex
tofaproblem.
•Equiva
lentexp
ressions
can
beus
edtodev
elopand
relateform
ulas
forgeo
metricsh
apes
includ
ingvolumes
ofco
nes,sphe
res,
andcylinders.
•Equa
tions
can
hav
eone
solution,nosolution,oran
infin
itenu
mber
ofso
lutions
which
can
beiden
tifie
dbyex
aminingthe
equa
tionor
itsgraph.
•Th
eun
derlyingpatternofch
angeinarelations
hiporfunc
tion
canbereprese
nted
byasymbolicrep
rese
ntationoreq
uation.
Differen
ttypes
offunc
tions
,suc
hasline
ar,inv
erse
,exp
one
ntial,
orqua
dratic,hav
esp
ecificch
arac
teristicsinthe
irsym
bolic
represe
ntations
.
Inve
stig
atio
n 1
Equiva
lentExp
ressions
Pro
ble
m 1
.1
Tilin
gPools:W
riting
Equiva
lentExp
ressions
Pro
ble
m 1
.2 T
hink
ing
inD
ifferen
tWay
s:
DeterminingEquiva
lenc
e
Pro
ble
m 1
.3
TheSc
hoolP
oolP
roblem:
Interpreting
Exp
ressions
Pro
ble
m 1
.4
DivingIn
:Rev
isitingthe
DistributiveProperty
Inve
stig
atio
n 2
Gen
eratingExp
ressions
Pro
ble
m 2
.1 W
alking
To
gethe
r:A
dding
Exp
ressions
Pro
ble
m 2
.2
Predicting
Profit:
SubstitutingExp
ressions
Pro
ble
m 2
.3
Mak
ingC
andles:Volume
ofCylinders,C
one
s,
andSphe
res
Pro
ble
m 2
.4
Selling
IceCream
:Solving
VolumeProblems
Inve
stig
atio
n 3
SolvingEqua
tions
Pro
ble
m 3
.1 S
ellin
g
GreetingC
ards:Solving
Line
arEqua
tions
Pro
ble
m 3
.2 C
omparing
Costs:M
oreSolving
Line
arEqua
tions
Pro
ble
m 3
.3 F
actoring
Qua
draticExp
ressions
Pro
ble
m 3
.4 S
olving
Qua
draticEqua
tions
Inve
stig
atio
n 4
Looking
Bac
katFun
ctions
Pro
ble
m 4
.1 P
umping
Water:L
ooking
at
PatternsofCha
nge
Pro
ble
m 4
.2
Areaan
dProfit:W
hat’s
theConn
ection?
Pro
ble
m 4
.3 C
reating
Patterns:Linea
r,Exp
one
ntial,Qua
dratic
Pro
ble
m 4
.4
Wha
t’sthe
Fun
ction?
Inve
stig
atio
n 5
Rea
soning
WithSy
mbols
Pro
ble
m 4
.1 U
sing
Algeb
ratoSolveaPuz
zle
Pro
ble
m 4
.2 O
ddand
Eve
nRev
isited
Pro
ble
m 4
.3 S
qua
ring
OddN
umbers
Teacher Implementation Toolkit92
Inve
stig
atio
n 1
Equiva
lentExp
ressions
Pro
ble
m 1
.1
Tilin
gPools:W
riting
Equiva
lentExp
ressions
Focu
s Q
uest
ion
Wha
tex
pression(s)rep
rese
nts
thenu
mberofborder
tilesne
eded
tosurroun
d
asq
uarepoolw
ith
sideleng
th3
s?
Pro
ble
m 1
.2 T
hink
ing
inD
ifferen
tWay
s:
DeterminingEquiva
lenc
e
Focu
s Q
uest
ion
Howcan
youdetermineif
twoormoreexp
ressions
areeq
uiva
lent?
Pro
ble
m 1
.3
TheSc
hoolP
oolP
roblem:
Interpreting
Exp
ressions
Focu
s Q
uest
ion
Wha
tinform
ationdoes
an
expressionreprese
ntin
a
given
contex
t?
Pro
ble
m 1
.4
DivingIn
:Rev
isitingthe
DistributiveProperty
Focu
s Q
uest
ion
Howcan
yo
uus
etheDistributive
andC
ommutative
Properties
tosho
w
thattwoexp
ressions
areeq
uiva
lent?
Inve
stig
atio
n 2
Gen
eratingExp
ressions
Pro
ble
m 2
.1 W
alking
To
gethe
r:A
dding
Exp
ressions
Focu
s Q
uest
ion
Wha
tarethead
vantag
esand
disad
vantag
esofus
ing
one
equa
tionrathertha
ntw
oormoreequa
tions
to
represe
ntasitua
tion?
Pro
ble
m 2
.2
Predicting
Profit:
SubstitutingExp
ressions
Focu
s Q
uest
ion
Wha
tareso
meway
sthatyou
canco
mbineone
ormore
expressions
(oreq
uations
)tocreateane
w
expression(oreq
uation)?
Pro
ble
m 2
.3
Mak
ingC
andles:Volume
ofCylinders,C
one
s,
andSphe
res
Focu
s Q
uest
ion
Wha
teq
uationreprese
ntsthe
relations
hipamong
the
vo
lumes
ofcy
linders,
cone
s,and
sphe
res?
Pro
ble
m 2
.4
Selling
IceCream
:Solving
VolumeProblems
Focu
s Q
uest
ion
Wha
tform
ulasareuse
ful
insolvingproblems
invo
lvingvolumeof
cylin
ders,cone
s,
andsphe
res?
Inve
stig
atio
n 3
SolvingEqua
tions
Pro
ble
m 3
.1 S
ellin
g
GreetingC
ards:Solving
Line
arEqua
tions
Focu
s Q
uest
ion
Wha
tstrategiesca
nyo
uus
etosolveeq
uations
tha
tco
ntainparen
thes
es?
Pro
ble
m 3
.2 C
omparing
Costs:M
oreSolving
Line
arEqua
tions
Focu
s Q
uest
ion
Wha
tarestrategiesforfin
ding
aso
lutionthatiscommon
totwoline
arequa
tions
?
Pro
ble
m 3
.3 F
actoring
Qua
draticExp
ressions
Focu
s Q
uest
ion
Wha
tareso
me
strategiesforfactoring
a
qua
draticex
pression?
Pro
ble
m 3
.4 S
olving
Qua
draticEqua
tions
Focu
s Q
uest
ion
Wha
tareso
me
strategiesforso
lving
qua
draticeq
uations
?
Inve
stig
atio
n 4
Looking
Bac
katFun
ctions
Pro
ble
m 4
.1 P
umping
Water:L
ooking
at
PatternsofCha
nge
Focu
s Q
uest
ion
Howcan
yo
uus
ean
equa
tionto
solveparticu
larque
stions
ab
outthe
fun
ctionan
d
contex
titrep
rese
nts?
Pro
ble
m 4
.2
Areaan
dProfit:W
hat’s
theConn
ection?
Focu
s Q
uest
ion
Des
cribeho
wtwo
differen
tco
ntex
tscan
bereprese
nted
bythe
sameeq
uation.
Pro
ble
m 4
.3 C
reating
Patterns:Linea
r,Exp
one
ntial,Qua
dratic
Focu
s Q
uest
ion
How
canyo
udeterminethe
patternsofch
angeofa
func
tionfromatab
leof
dataforthefunc
tion?
Pro
ble
m 4
.4
Wha
t’sthe
Fun
ction?
Focu
s Q
uest
ion
How
canyo
udeterminewhich
func
tiontouse
tosolveor
represe
ntaproblem?
Inve
stig
atio
n 5
Rea
soning
WithSy
mbols
Pro
ble
m 4
.1 U
sing
Algeb
ratoSolveaPuz
zle
Focu
s Q
uest
ion
Howcan
yo
uus
ealgeb
ratosolvea
numbertrick
?
Pro
ble
m 4
.2 O
ddand
Eve
nRev
isited
Focu
s Q
uest
ion
Howcan
youus
e
algeb
ratorep
rese
nt
andprove
aconjec
ture
aboutnum
bers?
Pro
ble
m 4
.3 S
qua
ring
OddN
umbers
Focu
s Q
uest
ion
Des
cribeso
mestrategies
formak
ingand
proving
aco
njec
ture.
Graphic Organizers for Grade 8 93
It’s
in th
e Sy
stem
Sys
tem
s of
Lin
ear
Equa
tions
and
Ineq
ualit
ies
Ess
enti
al Id
eas
•Asystemoflin
eareq
uations
can
beus
edtosolveproblemswhe
ntw
oormoreequa
tions
tha
treprese
ntcons
traintsontheva
riab
les
inasitua
tionareiden
tifie
d.
•Th
eso
lutiontoasystemoflin
eareq
uations
can
befoun
d
graphica
llyoralgeb
raically.A
nalyzing
the
equa
tions
and
the
situationca
nhe
lpyoutodeterminewhich
strateg
yism
ost
appropriatetoapply.
•Th
estrategiesforso
lvingline
arequa
tions
,linea
rineq
ualitiesan
d
system
soflin
eareq
uations
can
beex
tend
edtosolvingsystemsof
linea
rineq
ualities.
Inve
stig
atio
n 1
Line
arEqua
tions
With
TwoVariables
Pro
ble
m 1
.1 S
hirtsan
d
Cap
s:SolvingEqua
tions
With
TwoVariables
Pro
ble
m 1
.2 C
onn
ecting
ax
+b
y=cand
y=m
x+b
Pro
ble
m 1
.3 B
oosterC
lub
Mem
bers:In
tersec
ting
Lines
Inve
stig
atio
n 2
SolvingLinea
r
System
sSy
mbolically
Pro
ble
m 2
.1 S
hirtsan
dC
aps
Again:SolvingSystemsWith
y=m
x+b
Pro
ble
m 2
.2 Tac
oTruck
Lu
nch:SolvingSystemby
CombiningEqua
tions
I
Pro
ble
m 2
.3 S
olvingSystemsby
CombiningEqua
tions
II
Inve
stig
atio
n 3
System
sofFu
nctions
an
dIn
equa
lities
Pro
ble
m 3
.1 C
omparingSec
urity
Services
:Linea
rIneq
ualities
Pro
ble
m 3
.2 S
olvingLinea
rIneq
ualitiesSy
mbolically
Pro
ble
m 3
.3 O
perating
ataProfit:S
ystemsofLine
s
andC
urve
s
Inve
stig
atio
n 4
System
sofLine
arIn
equa
lities
Pro
ble
m 4
.1 H
owW
ePollu
te:
Ineq
ualitiesWithTw
oVariables
Pro
ble
m 4
.2 W
hatMak
esa
CarG
reen
:SolvingIn
equa
lities
byGraphing
I
Pro
ble
m 4
.3 F
easiblePoints:
SolvingIn
equa
litiesbyGraphing
II
Pro
ble
m 4
.4 M
ilesofEmissions
:Sy
stem
sofLine
arIn
equa
lities
Teacher Implementation Toolkit94
Inve
stig
atio
n 1
Line
arEqua
tions
With
TwoVariables
Pro
ble
m 1
.1 S
hirtsan
dC
aps:
SolvingEqua
tions
With
TwoVariables
Focu
s Q
uest
ion
Wha
tkind
ofso
lutions
willbefoun
dfor
anequa
tionlik
e3x
+5
y=13
withtw
ovariables?W
hatwill
thegraphs
ofthose
solutions
looklik
e?
Pro
ble
m 1
.2 C
onn
ecting
ax
+b
y=cand
y=m
x+b
Focu
s Q
uest
ion
Howcan
one
cha
ngean
equa
tionfrom
ax+b
y=cform
toaneq
uiva
lent
y=m
x+bform
and
viceve
rsa?
Pro
ble
m 1
.3 B
oosterC
lub
Mem
bers:In
tersec
ting
Lines
Focu
s Q
uest
ion
Wha
tha
ppen
swhe
nyo
use
arch
forco
mmon
solutions
totwoline
arequa
tions
withtw
ovariables?
Inve
stig
atio
n 2
SolvingLinea
r
System
sSy
mbolically
Pro
ble
m 2
.1 S
hirtsan
dC
aps
Again:SolvingSystemsWith
y=m
x+b
Focu
s Q
uest
ion
Howcan
you
solveasystem
oftw
oline
ar
equa
tions
withtw
ovariables
bywriting
eac
heq
uationin
equiva
lenty=m
x+bform
?W
hat
aretheso
lutionpossibilities
for
such
systemsan
dhowarethe
ysh
ownbygraphs
oftheso
lutions
?
Pro
ble
m 2
.2 Tac
oTruck
Lu
nch:SolvingSystemby
CombiningEqua
tions
I
Focu
s Q
uest
ion
Howcan
you
solveasystem
oflin
eareq
uations
byco
mbiningthe
twoequa
tions
intoone
sim
plereq
uationby
additionorsu
btrac
tion?
Pro
ble
m 2
.3 S
olvingSystemsby
CombiningEqua
tions
II
Focu
s Q
uest
ion
Howcan
eq
uations
inasystembe
tran
sform
edtoequiva
lent
form
sthatm
akeiteasierto
solvebyco
mbinationto
elim
inateva
riab
les?
Inve
stig
atio
n 3
System
sofFu
nctions
an
dIn
equa
lities
Pro
ble
m 3
.1 C
omparingSec
urity
Services
:Linea
rIneq
ualities
Focu
s Q
uest
ion
Howcan
you
usefunc
tiongraphs
tofind
the
so
lutions
ofan
ineq
ualitylik
e
ax+b<c
x+d?H
owcan
the
so
lutions
bereprese
nted
ona
numberline
graph?
Pro
ble
m 3
.2 S
olvingLinea
rIneq
ualitiesSy
mbolically
Focu
s Q
uest
ion
Howdoes
ap
plyingthe
sam
eoperation
tobothsides
ofan
ineq
uality
chan
getherelations
hipofthetw
o
qua
ntitiesbeing
compared
(or
not)?H
owcan
line
arin
equa
lities
beso
lved
bystrategiesthat
areve
rysim
ilartostrateg
iesfor
solvingline
arequa
tions
?
Pro
ble
m 3
.3 O
perating
ataProfit:S
ystemsofLine
s
andC
urve
s
Focu
s Q
uest
ion
Wha
tarethe
possiblesolutions
forasystem
thatin
clud
esone
line
arand
one
qua
draticfunc
tionan
dhowcan
yo
ufin
dthe
sesolutions
?
Inve
stig
atio
n 4
System
sofLine
arIn
equa
lities
Pro
ble
m 4
.1 H
owW
ePollu
te:
Ineq
ualitiesWithTw
oVariables
Focu
s Q
uest
ion
Ifaproblem
invo
lves
solvinganineq
ualitylik
eax
+b
y≤
c,howm
anyso
lutions
wouldyouex
pec
ttofind
and
wha
twouldacoordinategraphofthose
so
lutions
looklik
e?
Pro
ble
m 4
.2 W
hatMak
esa
CarG
reen
:SolvingIn
equa
lities
byGraphing
I
Focu
s Q
uest
ion
Wha
tgraphof
solutions
(inthefirstqua
drant)
wouldyouex
pec
tforan
ineq
uality
withthegen
eralform
ax+b
y≤
c?
Pro
ble
m 4
.3 F
easiblePoints:
SolvingIn
equa
litiesbyGraphing
II
Focu
s Q
uest
ion
Wha
tgraphof
solutions
wouldyouex
pec
tforan
ineq
ualitywiththegen
eralform
ax
+b
y≤
c?
Pro
ble
m 4
.4 M
ilesofEmissions
:Sy
stem
sofLine
arIn
equa
lities
Focu
s Q
uest
ion
Wha
tdoyou
lookfortosolveasystem
oflin
ear
ineq
ualitiesan
dw
hatwillthe
graphofaso
lutionlooklik
e?
Graphic Organizers for Grade 8 95
Look
ing
for
Pyth
agor
as T
he P
ytha
gore
an T
heor
em
Ess
enti
al Id
eas
•Th
erelations
hipbetwee
nanu
mberand
itssq
uarerootisthe
sam
easthe
relations
hipbetwee
nthearea
ofasq
uareand
the
leng
th
ofitsside.The
relations
hipbetwee
nanu
mberand
itscu
beroot
isthe
sam
easthe
relations
hipbetwee
nthevo
lumeofacu
bean
d
theleng
thofone
ofitsed
ges
.
•Th
ePytha
gorean
The
oremrelates
the
areasofthesq
uareson
thesides
ofarighttrian
gletothe
areaofthesq
uareonthe
hypotenu
se.A
saresu
lt,the
Pytha
gorean
The
oremisuse
fulfor
findingthe
leng
thofan
unk
nownsideofarighttrian
glegiven
theleng
thoftheothertwosides
,find
ingthe
leng
thofase
gmen
tjoiningany
twopointsonaco
ordinategrid,a
ndforwriting
the
eq
uationofacirclecen
teredattheorigin.
•Th
eco
nverse
ofthePytha
gorean
The
oremcan
beus
edto
determinewhe
theratrian
gleisarighttrian
gle.
•Th
ese
tofrealnum
bersiscomprise
dofthese
tofrationa
lnu
mbersan
dthe
setofirrationa
lnum
bers.D
ecim
alsthatneither
repea
tno
rterm
inateareca
lledirrationa
lnum
bers.Youca
nloca
te
irrationa
lnum
bersonanu
mberline
,and
workw
iththem
inthe
sameway
aswithrationa
lnum
bers.
Inve
stig
atio
n 1
CoordinateGrids
Pro
ble
m 1
.1
Driving
Aroun
dEuc
lid:
Loca
ting
Pointsand
Find
ingD
istanc
es
Pro
ble
m 1
.2 P
lann
ing
Parks:S
hapes
ona
CoordinateGrid
Pro
ble
m 1
.3
Find
ingA
reas
Inve
stig
atio
n 2
Squa
ring
Off
Pro
ble
m 2
.1 L
ooking
forSq
uares
Pro
ble
m 2
.2
Squa
reRoots
Pro
ble
m 2
.3 U
sing
Sq
uarestoFindLen
gths
Pro
ble
m 2
.4 C
ubeRoots
Inve
stig
atio
n 3
ThePy
thag
orea
nTh
eorem
Pro
ble
m 3
.1 D
isco
vering
thePytha
gorean
The
orem
Pro
ble
m 3
.2 A
Proofof
thePytha
gorean
The
orem
Pro
ble
m 3
.3
Find
ingD
istanc
es
Pro
ble
m 3
.4 M
easu
ring
theEgyp
tian
Way
:Le
ngthsTh
atForm
aRightTrian
gle
Inve
stig
atio
n 4
Using
the
Pytha
gorean
Th
eorem:U
nderstan
ding
Rea
lNum
bers
Pro
ble
m 4
.1 A
nalyzing
theWhe
elofTh
eodorus:
Squa
reRootsona
Num
berLine
Pro
ble
m 4
.2
Rep
rese
ntingFractions
asD
ecim
als
Pro
ble
m 4
.3
Rep
rese
ntingD
ecim
als
asFractions
Pro
ble
m 4
.4 G
etting
Rea
l:Irrationa
lNum
bers
Inve
stig
atio
n 5
Using
the
Pytha
gorean
Th
eorem:A
nalyzing
Triang
lesan
dC
ircles
Pro
ble
m 4
.1 S
topping
Snea
kySally:F
inding
Unk
nownSideLe
ngths
Pro
ble
m 4
.2
Ana
lyzing
Trian
gles
Pro
ble
m 4
.3
Ana
lyzing
Circ
les
Teacher Implementation Toolkit96
Inve
stig
atio
n 1
CoordinateGrids
Pro
ble
m 1
.1
Driving
Aroun
dEuc
lid:
Loca
ting
Pointsand
Find
ingD
istanc
es
Focu
s Q
uest
ion
Howdodriving
distanc
ean
dflying
distanc
ebetwee
ntw
ocoordinates
relatetoeac
hother?
Pro
ble
m 1
.2
Plann
ingParks:S
hapes
on
aCoordinateGrid
Focu
s Q
uest
ion
Howdothe
coordinates
ofen
dpointsofa
segmen
the
lpdraw
otherline
s,w
hich
are
parallelo
rperpen
dicular
tothe
seg
men
t?
Pro
ble
m 1
.3
Find
ingA
reas
Focu
s Q
uest
ion
How
does
kno
winghow
tocalcu
lateareasof
rectan
glesan
dtrian
gles
helpin
the
calcu
lationof
irregularareas?
Inve
stig
atio
n 2
Squa
ring
Off
Pro
ble
m 2
.1 L
ooking
forSq
uares
Focu
s Q
uest
ion
Howm
anydifferen
tsq
uareareasarepossible
todrawusing
the
dots
onadotgridasve
rtices
?Why
aresomesq
uare
area
sno
tpossible?
Pro
ble
m 2
.2
Squa
reRoots
Focu
s Q
uest
ion
Wha
tdoes
√xmea
n?H
ow
does
itrelatetox
2 ?
Pro
ble
m 2
.3 U
sing
Sq
uarestoFindLen
gths
Focu
s Q
uest
ion
How
canyo
ufin
dthe
distanc
ebetwee
nan
ytw
opoints
onagrid?
Pro
ble
m 2
.4 C
ubeRoots
Focu
s Q
uest
ion
Wha
tdoes
itm
eantotak
ethe
cuberootofanu
mber?
Inve
stig
atio
n 3
ThePy
thag
orea
nTh
eorem
Pro
ble
m 3
.1 D
isco
vering
thePytha
gorean
The
orem
Focu
s Q
uest
ion
You
know
the
sum
ofthe
two
shortestsideleng
ths
ofatria
nglem
ustbe
greatertha
nthethird
side
leng
th.Isthereasimila
rrelatio
nshipamon
gthe
sq
uareson
the
sidesofa
triang
le?Isthe
relationship
thesameforalltria
ngles?
Pro
ble
m 3
.2 A
Proofof
thePytha
gorean
The
orem
Focu
s Q
uest
ion
How
canyo
uprove
tha
tthe
relations
hipobse
rved
in
Problem3.1w
illw
orkfor
allrighttrian
gles?
Pro
ble
m 3
.3
Find
ingD
istanc
esFo
cus
Que
stio
n How
canyo
ufin
dthe
distanc
ebetwee
nan
ytw
opoints
onaplane
?
Pro
ble
m 3
.4 M
easu
ring
theEgyp
tian
Way
:Len
gths
ThatForm
aRightTrian
gle
Focu
s Q
uest
ion
Ifa
triang
lew
ithsideleng
ths
a,b,a
ndcsatisfie
sthe
relatio
nshipa
2 +b
2 =c
2 ,is
thetriang
learighttria
ngle?
Inve
stig
atio
n 4
Using
the
Pytha
gorean
Th
eorem:U
nderstan
ding
Rea
lNum
bers
Pro
ble
m 4
.1 A
nalyzing
theWhe
elofTh
eodorus:
Squa
reRootsona
Num
berLine
Focu
s Q
uest
ion
Can
you
finddistanc
estha
tare
exac
tsq
uarerootsofall
who
lenum
bers?C
anyou
ordersqua
rerootsona
numberline
?
Pro
ble
m 4
.2
Rep
rese
ntingFractions
asD
ecim
als
Focu
s Q
uest
ion
Why
ca
nyo
ureprese
nteve
ry
frac
tionasarep
eating
or
term
inatingdec
imal?How
canyo
upredictwhich
represe
ntations
willrep
eat
andw
hich
willterminate?
Pro
ble
m 4
.3
Rep
rese
ntingD
ecim
als
asFractions
Focu
s Q
uest
ion
Can
you
represe
nteve
ryrep
eating
orterm
inatingdec
imalas
afrac
tion?
Pro
ble
m 4
.4 G
etting
Rea
l:Irrationa
lNum
bers
Focu
s Q
uest
ion
Can
youiden
tifyeve
ry
numberaseitherrationa
lorirrationa
l?
Inve
stig
atio
n 5
Using
the
Pytha
gorean
Th
eorem:A
nalyzing
Triang
lesan
dC
ircles
Pro
ble
m 4
.1 S
topping
Snea
kySally:F
inding
Unk
nownSideLe
ngths
Focu
s Q
uest
ion
Howcan
yo
uus
ethePytha
gorean
Th
eoremtofind
distanc
es
inageo
metricsh
ape?
Pro
ble
m 4
.2
Ana
lyzing
Trian
gles
Focu
s Q
uest
ion
Howdo
theleng
thsofthesides
of
a30
-60-90
trian
glerelate
toeac
hother?
Pro
ble
m 4
.3
Ana
lyzing
Circ
les
Focu
s Q
uest
ion
Wha
tis
therelations
hipbetwee
ntheco
ordinates
ofapoint
(x,y
)onacirclew
itha
centerattheorigin?
Graphic Organizers for Grade 8 97
