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1 Eac
7 RADE
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3 Common Core, Inc. Som
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atios adule Overview
ic A: Proport
Lesson 1
Lesson 2
Lessons 3
Lessons 5
ic B: Unit Rat
Lesson 7
Lessons 8
Lesson 1
d‐Module Assics A through
ic C: Ratios a
Lessons 1
Lesson 1
Lesson 1
Lesson 1
ic D: Ratios o
Lesson 1
Lesson 1
Lesson 1
Lesson 1
Lesson 2
Lessons 2
ch lesson is ONE
State Com
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Content
and Pw .................
tional Relatio
: An Experien
: Proportiona
3–4: Identify
5–6: Identify
te and the Co
: Unit Rate a
8–9: Represe
0: Interpretin
sessment andh B (assessme
and Rates Invo
11–12: Ratio
3: Finding Eq
4: Multistep
5: Equations
of Scale Draw
6: Relating S
7: The Unit R
8: Computin
9: Computin
0: An Exercis
21–22: An Ex
day and ONE da
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Propor.....................
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‐of‐Module Aics A through
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Assessment ah D (assessme
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MATICS CURRIC
and Rubric ....ent 1 day, retu
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opic B, studeportionality aere k is the cotionship to raation (7.RP.2
opic C, studeratios and ratly their experolving fraction
he final topic context of scrking with sideroportionalitwings of a twdents creating
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CORE MATHEM
• Modu
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nts extend thtes specified brience in the fns to solve mu
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CORE MATHEM
ndards
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Compute unit other quantitieach 1/4 hourequivalently 2
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CORE MATHEM
15 hamburger
Use ratio and easoning aboequations.
a. Make tabfind missiUse table
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CORE MATHEM
onstant of prepresenting scale drawingspicture.
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Ratio Ta
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CORE MATHEM
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the forest, conser
unted 216 deer in
a.
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CORE MATHEM
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MATICS CURRIC
nit Rate
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he following si
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MATICS CURRIC
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y box with stud
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7: Unit Rat3/26/14
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MATICS CURRIC
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CORE MATHEM
s that in the e
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square mile is
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work with a pa
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MATICS CURRIC
example with
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CORE MATHEM
nch Class Coo
3: French Class Co
nd Margo want to
mount of flour, mi
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What does the
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How many egg
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7: Unit Rat3/26/14
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MATICS CURRIC
oking
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dered pairs that sa
rtionality.
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ch class. A recipe
ve plenty of flour a
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CORE MATHEM
___________
Unit Ra
Susan paid $5
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7: Unit Rat3/26/14
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MATICS CURRIC
____________
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CORE MATHEM
mple Solution
tions indicate a
ame store, Susa
ne the constant
tant of proportion
bottle of water.
ample Soluti
of the following p
anas are $0.59/po
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K = 0.59
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. $
dry cleaning fee f
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so K=6
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6(11) = $66
every $5 that Mic
What is the co
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7: Unit Rat3/26/14
oncore.org
MATICS CURRIC
ns
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what it means in c
ach $1 that he sav
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lesson:
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s context.
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4. Each
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a.
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Lesson 7Date:
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CORE MATHEM
h school year, the
ool paid $1260 for
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Is the price the
zoo?
Explain why or
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Identify the co
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7: Unit Rat3/26/14
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MATICS CURRIC
7th graders who s
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pecial field trip to
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CORE MATHEM
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MATICS CURRIC
te as the Constant 4
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MATICS CURRIC
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build conceptua
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g the line, from
m the table to o
take to pass th
ng, “What is th
resent this situ
ut the amount
amples 1 and
w students to w
dents to persev
dents write an
Example 1
Your mot
indicating
log where
informati
know tha
determine
a.
b.
c.
Lesson 8Date:
e rights reserved. commo
CORE MATHEM
y (10 minutes
al understandi
ssroom and st
m student to st
one person, the
he book from t
he unit rate in t
ation?” “How c
of time it will
2 (33 minute
work with a pa
vere before pro
equation that
1: Do We have En
her has accelerate
g that there is a ha
e she records the m
on in the table be
t if you can determ
e whether or not
Find the constan
Constant of pro
Write equation(
or
Knowing that th
gas station? Exp
No, she will not
tank, she can tr
8: Represe3/26/14
oncore.org
MATICS CURRIC
s)
ng for student
art with a boo
tudent, at a re
en from the ta
he table along
this situation?”
could we use e
take for the bo
es)
artner or small
oviding assistan
models a real
nough Gas to Mak
ed onto the inters
alf a gallon left in
mileage and the n
elow to determine
mine the amount
you can make it t
Ga
nt of proportional
Gallons
oportionality is
(s) that will relate
here is a half gallo
plain why or why
t make it because
ravel 14 miles. Sin
enting Proportiona4
CULUM
s that may ben
k at one end, o
gular pace. Fir
ble along the l
the line to rea
” “What is the
either the unit
ook to reach 10
group to attem
nce.
world situation
ke it to the Gas Sta
state beginning a
the gas tank. The
number of gallons
e whether you wil
of gas that her ca
o the next gas sta
Mother’s Gas Rec
allons Mile
8
10
4
lity and explain w
Miles Driven
. The car tra
the miles driven t
n left in the gas ta
not.
she gets 28 miles
nce the nearest ga
al Relationships wi
ThisCre
nefit from a kin
on a desk. Instr
st, time how lo
line to the thir
ach the tenth p
constant of pr
rate or consta
0 students?”
mpt Example 1
n.
ation?
long road trip and
e nearest gas stati
purchased each t
l make it to the ga
ar consumes in a p
ation.
cord
es Driven
224
280
112
hat it represents i
avels 28 miles for e
to the number of
ank when the ligh
to one gallon. Sin
as station is 26 mi
th Equations
s work is licensed underative Commons Attribut
nesthetic appr
ruct students t
ong it takes for
d person. Ask
person. Relate
roportionality?
ant of proportio
1 before discus
d you notice that t
ion is 26 miles aw
time she fills up th
as station before
particular number
in this situation.
every one gallon o
gallons of gas.
t comes on, will s
ince she has ½ gal
iles away, she will
r a tion‐NonCommercial‐Sh
Less
oach, line 10 s
to pass the boo
r students to p
students to pr
to concepts di
?” “What equa
onality to mak
ssing as a class
the low fuel light
way. Your mother
he tank. Use the
the gas runs out.
r of miles, then yo
of gas.
she make it to the
llon remaining in t
l not have enough
Scaffoldin
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number
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tudents up alo
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pass the book
edict how long
iscussed so far
tion would
e predictions
. Encourage
is on,
keeps a
You
ou can
nearest
the gas
h gas.
ng:
age and use a
entations, suc
r line:
67
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ong
ble
g it
r by
a variety of
ch as a double
7
e
© 2013
NY
Have
= m
mile
stud
3 Common Core, Inc. Som
YS COMMON C
d.
e.
e students writ
miles driven. R
s divided by a
dents that you w
Write eq
In order
depende
Which p
T
Which is
T
Tell stud
Remind
of the de
Ask, whe
variable?
The inde
Tell stud
Lesson 8Date:
e rights reserved. commo
CORE MATHEM
Using the equat
problem in two
Using arithmeti
Using Algebra:
Your mother ca
Using the equat
Using arithmeti
Using algebra:
Proportionality
26.8 (rounded t
te the pairs of
Remind studen
certain numbe
will use the co
quation(s) that
to write the eq
ent variables th
art depends on
The number of
s the dependen
The number of
ents that is u
students the c
ependent varia
en and are
?
‐axis
ependent varia
‐ axis
ents that any v
8: Represe3/26/14
oncore.org
MATICS CURRIC
ion found in part
ways.
ic:
an travel 504 miles
ion found in part
ic: .
or algebraically,
.
to the nearest ten
numbers in th
nts to think of t
er of gallons. T
nstant of prop
will relate the
quation to rep
hat are implied
n the other for
f miles driven d
nt variable – ga
f miles is the de
usually known
onstant of pro
able to the inde
graphed on a
ble?
variable may b
enting Proportiona4
CULUM
b, determine how
– Use sub
– This is t
s on 18 gallons of
b, determine how
– Use sub
– This equ
use the multiplica
th) gallons would
e chart as orde
the constant o
This constant is
portionality (or
miles driven t
resent this situ
d in this proble
r its outcome?
depends on the
allons of gas or
ependent varia
as the indepen
oportionality ca
ependent varia
coordinate gri
be used to repr
al Relationships wi
ThisCre
w far your mother
bstitution to repla
the same as multip
gas.
w many gallons of
bstitution to repla
uation demonstra
ative inverse (mak
d be needed to driv
ered pairs. Exp
f proportionali
s the same as t
unit rate) as a
to the number
uation, direct s
m.
number of ga
r miles driven?
able while the n
ndent variable
an also be expr
able.
d, which axis w
resent the situa
th Equations
s work is licensed underative Commons Attribut
r can travel on 18 g
ace the g (gallons o
plying by the cons
f gas would be nee
ace the m (miles d
ates the same as d
king one) to solve
ve 750 miles.
plain that in th
ity as . T
the unit rate o
a multiplier in y
of gallons of g
students to thi
llons of gas tha
?
number of gall
e, and is know
ressed as fro
would show th
ation as long a
r a tion‐NonCommercial‐Sh
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gallons of gas. So
of gas) with 18.
stant of proportio
eded to travel 750
driven) with 750.
dividing by the Con
the equation.
his example =
The ratio is a ce
f miles per gal
your equation.
gas.
nk of the indep
at are in the ga
lons is the inde
wn as the depe
m an ordered
e values of the
as it is known t
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onality.
0 miles.
nstant of
= gallons and
ertain number
lon. Remind
.
pendent and
as tank.
ependent variab
endent variable
pair. It is the r
e dependent
hat in showing
68
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of
ble.
e.
ratio
g a
8
© 2013
NY
3 Common Core, Inc. Som
YS COMMON C
proporti
variable.
equation
Tell stud
variable
the equa
Ask students
Example 2
Andrea is
People ha
shows the
needed to
a.
b.
c.
Tell stud
equation
Lesson 8Date:
e rights reserved. commo
CORE MATHEM
onal relationsh
. In this proble
n or
ents that this e
when the othe
ation, then stud
to try working
2: Andrea’s Portr
a street artist in N
ave their portrait d
e relationship bet
o draw the portra
Write several or
in the context of
(4, 6) means tha
(6, 9) means tha
(2, 3) means tha
(1, 1 ½) means t
Write several e
spent drawing t
Determine the c
in this situation.
The constant of
draw portrait
ents these ord
n for this situat
8: Represe3/26/14
oncore.org
MATICS CURRIC
hip in an equat
em, students ca
.
equation mode
er is known. If
dents can use
on Example 2
aits
New Orleans. She
drawn and then c
ween the number
its.
rdered pairs from
f this graph.
at in 4 hours she c
at in 6 hours she c
at in 2 hours she c
that in 1 hour she
quations that wou
the portraits.
constant of propo
.
f proportionality is
ts in hours or ca
dered pairs can
tion. Rememb
enting Proportiona4
CULUM
tion that the co
an write 2
els the situatio
f the equation
algebra to solv
independently
e draws caricature
ome back later to
r of portraits she d
the graph and ex
can draw 6 portra
can draw 9 portra
can draw 3 portra
e can draw 1 ½ por
uld relate the num
rtionality and exp
s / which mean
n complete ½ p
n be used to ge
ber that the con
al Relationships wi
ThisCre
onstant of prop
28 or 28
on and will prov
is written so th
ve the equatio
y.
es (cartoon‐like po
o pick it up from h
draws and the am
plain what each c
its
its
its
rtraits
mber of portraits d
plain what it mean
ns that Andrea can
portrait in 1 hour.
enerate the con
nstant of propo
th Equations
s work is licensed underative Commons Attribut
portionality is
8 . We are sub
vide us with a
hat the known
n.
ortraits of tourists
her. The graph bel
mount of time in h
coordinate pair me
drawn to the time
ns
n
nstant of prop
ortionality
r a tion‐NonCommercial‐Sh
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multiplied by t
bstituting the 2
way to determ
information is
s).
low
hours
eans
e
ortionality and
.
Scaffo
Rea
show
to a
Use
soft
Use
opp
inst
whi
inde
If st
que
sign
such
1. “H
pr
in
2. “L
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te
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the independe
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id comprehe
graphing cal
ware.
this exercise
ortunity to of
ruction to a s
le others wor
ependently.
udents are re
stions to intr
nificance of th
h as:
How is the co
roportionality
the graph?”
Locate the po
xplain what th
erms of this si
69
License.
•1
ent
nto
m aloud and
of a caricature
nsion.
culators or
e as an
ffer targeted
small group
rk
eady, pose
oduce the
he point (1,r),
nstant of
y represented
oint (1, 1.5).
his means in
ituation.”
9
e
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© 2013
NY
Clos
Ask s
unde
Exit
Ensu
unde
3 Common Core, Inc. Som
YS COMMON C
sing (2 minut
students to res
erstanding.
How can
T
s
t
t
Ticket (5 min
ure that studen
erstanding of t
Lesson
If a pro
a posit
multip
Lesson 8Date:
e rights reserved. commo
CORE MATHEM
es)
spond to this q
n unit rate be u
The unit rate is
substituted in p
the independen
the constant of
nutes)
nts complete th
the lesson.
n Summary:
oportional relation
tive constant, then
plicative relationsh
8: Represe3/26/14
oncore.org
MATICS CURRIC
question to a p
used to write a
s the constant o
place of in th
nt variable to f
f proportionali
he Exit Ticket i
nship is described
n is called the co
hip between each
enting Proportiona4
CULUM
partner or in wr
n equation rela
of proportiona
he equation
find the depend
ty to find the d
ndependently,
d by the set of ord
onstant of propor
‐value and its co
al Relationships wi
ThisCre
riting. Use this
ating two varia
ality, . After c
. The con
dent variable, a
dependent vari
, in order to pr
ered pairs that sa
rtionality. The con
orresponding ‐va
th Equations
s work is licensed underative Commons Attribut
s as an opportu
ables that are p
computing the
nstant of propo
and the depen
iables.
rovide evidenc
atisfies the equati
nstant of proporti
alue.
r a tion‐NonCommercial‐Sh
Less
unity to inform
proportional?
value for , it
ortionality can
ndent variable c
ce of each stud
on , where
ionality expresses
hareAlike 3.0 Unported L
7•son 8
mally assess
may be
be multiplied b
can be divided
ent’s
e is
s the
70
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•1
by
d by
0
© 2013
NY
Nam
Le
Eq
Exit
John
3 Common Core, Inc. Som
YS COMMON C
me __________
sson 8:
quations
t Ticket
n and Amber w
a. Determin
why not.
b. Determin
explain w
J
Time
2
3
4
Lesson 8Date:
e rights reserved. commo
CORE MATHEM
_____________
Represe
work at an ice c
e whether Joh
e whether Am
why not.
John’s wages
e (h) Wage
2 18
3 27
4 36
8: Represe3/26/14
oncore.org
MATICS CURRIC
____________
enting P
cream shop. Th
n’s wages are
ber’s wages ar
es ($)
8
7
6
enting Proportiona4
CULUM
_____________
roportio
he hours worke
proportional t
re proportiona
al Relationships wi
ThisCre
_______
onal Rela
ed and wages
o time. If they
al to time. If th
th Equations
s work is licensed underative Commons Attribut
Date____
ationshi
earned are giv
y are, determin
hey are, determ
r a tion‐NonCommercial‐Sh
Less
____________
ips with
ven for each pe
ne the unit rate
mine the unit r
hareAlike 3.0 Unported L
7•son 8
______
erson.
e. If not, expla
ate. If not,
7
License.
•1
ain
1
© 2013
NY
3 Common Core, Inc. Som
YS COMMON C
c. Write an e
for each.
d. How muc
e. How long
Lesson 8Date:
e rights reserved. commo
CORE MATHEM
equation to m
Explain what i
h would each w
will it take eac
8: Represe3/26/14
oncore.org
MATICS CURRIC
odel the relatio
it means in the
worker make a
ch worker to e
enting Proportiona4
CULUM
onship betwee
e context of the
after working 1
earn $50?
al Relationships wi
ThisCre
en each person
e situation.
10 hours? Who
th Equations
s work is licensed underative Commons Attribut
n’s wages. Ide
o will earn mo
r a tion‐NonCommercial‐Sh
Less
ntify constant
re money?
hareAlike 3.0 Unported L
7•son 8
of proportiona
72
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•1
ality
2
© 2013
NY
Exit
The
3 Common Core, Inc. Som
YS COMMON C
t Ticket Sam
following solut
John and
a.
b.
c.
d.
e.
Time
2
3
4
Lesson 8Date:
e rights reserved. commo
CORE MATHEM
mple Solution
tions indicate a
Amber work at an
Determine whet
why not.
Yes, the unit rat
Determine whet
explain why not
Yes, the unit rat
Write an equatio
proportionality
John: ;
Amber:
How much wou
After hours J
multiplied by
earn $ becau
the same amou
How long will it
To determine ho
proportionality
; .
out how long it
John’s wages
e (h) Wages
2 18
3 27
4 36
8: Represe3/26/14
oncore.org
MATICS CURRIC
ns
an understand
n ice cream shop.
ther John’s wages
te is 9. The collec
ther Amber’s wag
t.
te is 8. The collec
on to model the r
for each. Explain w
constant of propo
; constant of pro
ld each worker m
John will earn $90
the constant of p
use her equation is
unt of time.
take each worker
ow long it will tak
. Algebraically, th
(round to t
will take Amber t
; ; .
s ($)
8 18/2
7 27/3
6 36/4
enting Proportiona4
CULUM
ding of the obje
The hours worke
s are proportional
tion of ratios is eq
ges are proportion
tion of ratios is eq
relationship betwe
what it means in t
ortionality is ; Jo
oportionality is ;
ake after working
0 because 10 hour
proportionality.
s ;
r to earn $50?
ke John to earn $
his can be shown a
the nearest hundre
to earn $ divide
. It will ta
= 9
= 9
= 9
al Relationships wi
ThisCre
ectives of this l
ed and wages earn
to time. If they a
quivalent.
nal to time. If they
quivalent.
een each person’s
the context of the
ohn earns $ for e
Amber earns $ f
g 10 hours? Who w
rs is the value of t
;
; . Jo
, the dependent
as a one‐step equ
edth). It will take
e by , her consta
ake Amber . h
th Equations
s work is licensed underative Commons Attribut
lesson:
ned are given for
are, determine the
y are, determine t
s wages. Identify
e situation.
very hour he work
for every hour she
will earn more mo
the independent v
; . After
ohn will earn more
t value will be div
uation: ;
e John nearly ho
ant of proportiona
hours to earn $
r a tion‐NonCommercial‐Sh
Less
each person.
e unit rate. If not
the unit rate. If n
constant of
ks.
e works.
oney?
variable which sho
r hours Amber
e money than Am
vided by 9, the con
ours to earn $ .
ality. ;.
hareAlike 3.0 Unported L
7•son 8
t, explain
not,
ould be
r will
mber in
nstant of
;
To find
73
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•1
3
© 2013
NY
Pro
3 Common Core, Inc. Som
YS COMMON C
blem Set Sa
Write an e
1. Ther
a.
b.
2. In 25
a.
b.
3. Jenn
a.
b.
4. It co
a.
b.
Lesson 8Date:
e rights reserved. commo
CORE MATHEM
ample Soluti
equation that wil
re are 3 cans that
Find the consta
Write an equat
5 minutes Li can r
Find the consta
Write an equat
nifer is shopping w
Find the consta
$
Write an equat
ost $5 to send 6 pa
Find the consta
$
Write an equat
8: Represe3/26/14
oncore.org
MATICS CURRIC
ons
l model the propo
store 9 tennis bal
ant of proportiona
tion to represent
un 10 laps around
ant of proportiona
tion to represent
with her mother.
ant of proportiona
tion to represent
ackages through a
ant of proportiona
tion to represent
enting Proportiona4
CULUM
ortional relationsh
lls. Consider the n
ality for this situat
the relationship.
d the track. Consi
ality in this situati
the relationship.
They pay $2 per p
ality in this situati
the relationship.
a certain shipping
ality for this situat
the relationship.
al Relationships wi
ThisCre
hip given in each r
number of balls p
tion.
der the number o
ion.
pound for tomato
ion.
company. Consid
tion.
th Equations
s work is licensed underative Commons Attribut
real world situatio
er can.
of laps she can run
es at the vegetab
der the number of
r a tion‐NonCommercial‐Sh
Less
on.
n per minute.
le stand.
f packages per dol
hareAlike 3.0 Unported L
7•son 8
llar.
74
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•1
4
© 2013
NY
3 Common Core, Inc. Som
YS COMMON C
5. On a
be d
prom
mon
a.
b.
c.
6. Allis
Nico
char
num
S
# Shi
1
2
5
7
1
Lesson 8Date:
e rights reserved. commo
CORE MATHEM
average, Susan do
downloaded on to
motions. Susan w
nth offered by ano
Find the consta
.
Write an equat
.
Use your equat
evidence and a
Compare the f
(number of so
songs. If she m
fee of $58 per
son’s middle schoo
ole have voluntee
rges a set‐up fee a
mber of shirts. Val
S = # of songs
purchased
40
20
12
5
Print‐o‐Ram
irts (S) Total
(C
10 95
25
50 37
75
100
8: Represe3/26/14
oncore.org
MATICS CURRIC
ownloads 60 songs
o personal digital d
wants to know if sh
other company. W
ant of proportiona
tion to represent
tion to find the an
a written explanat
flat fee of $58 per
ngs), then the res
maintains the sam
month.
ol team has desig
red to call local st
as well as a fixed a
lue T’s and More c
C = total
cost
36
18
10.80
4.50 .
ma
Cost
)
5
5
enting Proportiona4
CULUM
s per month. An o
devices. The grap
he should buy her
Which is the bette
ality for this situat
the relationship.
nswer to Susan’s q
tion.
r month for songs
sult is .me number of song
ned t‐shirts conta
tores to get an est
amount for each s
charges $8 per shi
Constant of
proportionality
.
.
..
..
.
. N
al Relationships wi
ThisCre
online music vend
ph below shows th
r music from this c
er buy?
tion.
question above. J
to $.90 per song.
. She wou
gs, the $0.90 cost
aining their team n
timate on the tota
shirt ordered. The
irt. Which compa
Not Proportional
th Equations
s work is licensed underative Commons Attribut
dor sells package p
he package prices
company or pay a
Justify your answe
. If . and
uld spend $54 on s
per song would b
name and color. A
al cost of purchasi
e total cost is show
any should they us
r a tion‐NonCommercial‐Sh
Less
prices for songs th
for the most pop
a flat fee of $58.00
er with mathemat
d we substitute 60
songs when she bo
be cheaper than th
Allison and her fri
ing t‐shirts. Print‐
wn below for the
se?
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hat can
ular
0 for the
tical
0 for
ought 60
he flat
iend
o‐Rama
given
7
License.
•1
5
© 2013
NY
3 Common Core, Inc. Som
YS COMMON C
a.
b.
c.
d.
e.
Lesson 8Date:
e rights reserved. commo
CORE MATHEM
Does either pri
Explain.
The unit rate f
ratios are equ
Write an equat
for Va
What is the co
8; the cost of o
How much is P
Guess and Che
A
A
Write a propos
choice. Determ
Since we plan
Print‐o‐Rama:
Value T’s and
8: Represe3/26/14
oncore.org
MATICS CURRIC
icing model repre
for Print‐o‐Rama i
ivalent (8) and the
tion relating cost
alue T’s and More
nstant of proporti
one shirt is $8.
Print‐o‐Rama’s set
eck: C= price of a
95 =
Attempt #1
Attempt #2
Set up fee
sal to your teache
mine the number o
on a purchase of
: ;
More: ;
enting Proportiona4
CULUM
sent a proportion
is not constant. Th
e graph shows a s
and shirts for Valu
ionality of Value T
t up fee?
shirt (# of shirts) +
= ______ 10 + ____
= $25
r indicating which
of shirts that you n
90 shirts, we shou
;
;
al Relationships wi
ThisCre
nal relationship be
he graph for Value
straight line throu
ue T’s and More.
T’s and More? Wh
+ set up fee
__ or 375 = _____
h company the tea
need for your tea
uld choose Print‐o
th Equations
s work is licensed underative Commons Attribut
etween quantity o
e T’s and More is p
ugh the origin.
hat does it repres
___ 50 + ______
am should use. B
m.
o‐Rama.
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proportional since
sent?
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al cost?
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76
License.
•1
6
© 2013
NY
Stud
Less
In th
grap
corre
num
be e
The exam
Clas
Stud
cons
conn
Ask s
scaff
Exa
Defi
birdh
MP.2
3 Common Core, Inc. Som
YS COMMON C
Lesso
Equa
dent Outcom
Students
contexts a
son Notes
his lesson, stud
phs and equatio
esponding gra
mber lines and t
xposed to this
problem set mmple, consider
sswork (35 m
dents will begin
stant of propor
nected to the r
students to att
folded questio
ample 1: Jack
Example 1
Jackson a
offered to
money fo
requests f
will allow
birdhouse
a. Writ
num
ne the variable
houses)
Lesson 9Date:
e rights reserved. commo
CORE MATHEM
on 9: Re
ations
mes
use the consta
as they relate t
dents gain furth
ons. If possible
phs. In additio
tables. While t
idea in this les
may be used as using targeted
minutes)
n to write equa
rtionality and w
real‐world situa
tempt to solve
ns, once stude
kson’s Birdho
1: Jackson’s Birdh
nd his grandfathe
o buy the birdhou
or his summer cam
for birdhouses. If
w Jackson to calcul
es.
te an equation tha
mber of birdhouse
/ B
es. = # of bir
9: Represe3/26/14
oncore.org
MATICS CURRIC
epresent
ant of proporti
the equations
her exposure t
e, students may
n to these, stu
he significance
sson, as sugges
homework or d instruction w
ations in two v
write the equa
ations they de
Example 1 in
ents have had a
ouses
houses
er constructed a m
ses. Jackson deci
mp, but he is not s
f Jackson can build
ate the time it wi
at you could use t
s.
dhouses and
enting Proportiona4
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ting Pro
onality to repr
to a correspon
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y use graphing
udents have the
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tion with two
scribe.
mixed‐ability g
a chance to ma
model for a birdho
ded that building
ure how long it w
d 7 birdhouses in
ll take him to buil
to find out how lo
= number of
al Relationships wi
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portiona
resent proporti
nding ratio tab
real‐world prop
g calculators to
e opportunity
(1,r) is not form
affolding in Exa
ry exercises forhat struggled o
y will analyze d
variables. The
groups. Assess
ake progress in
ouse. Many of the
birdhouses could
will take him to fill
5 hours, write an
ld any given numb
ng it will take him
hours (time co
th Equations
s work is licensed underative Commons Attribut
al Relati
ional relations
le
portional conte
o connect equa
to use other re
mally introduce
ample 2.
r targeted inston the previou
data that will he
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student progr
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all of the
equation that
ber of
m to build any
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epresentations
ed until Lesson
ruction. Durings day’s Exit Tic
elp them unde
esentations wi
ress and discus
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allow st
cooper
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situatio
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to write
differen
such as
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n 10, students
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ill continuously
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ng:
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tudents to wo
ative groups.
impler, altern
on, such as, “I
ild 2 birdhous
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extension ask
e the equatio
nt equivalent
s / 5/7
76
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•1
rld
y
ble
may
y be
the
loud and
ork in
native
If Jackson
ses in 4
students
on in a
t form,
.
6
© 2013
NY
This
num
varia
equa
Use
Rem
to bu
answ
who
cont
we d
3 Common Core, Inc. Som
YS COMMON C
Ask the st
No
If it is imp
proportio
# of
What doe
It mbirdh
If it is imp
proportio
# of hous
part of the pro
mber of birdhou
able. Our const
ation 5/7
the equation a
b. How
If ques
Jack
c. How
If take
d. How
If
.bird
mind students t
uy a partially b
wers (e.g., part
ole numbers (e
text. We must
determine if an
Lesson 9Date:
e rights reserved. commo
CORE MATHEM
tudents: Does
portant to dete
onality?
f birdhouses/ho
es that mean in
means that Jack
house in one h
portant to dete
onality?
hours / # of bi
se or 5/7 60
oblem asks you
uses, so we wa
tant of proport
7B.
above to deter
w many birdhouse
/ and
stion asks for the
/
/ /
kson can build
w long will it take J
/ and
e Jackson hour
w long will it take
/ and (rounded to the
dhouses.
hat while you
built birdhouse
ts of an hour th
.g., the numbe
consider the r
nd how we rou
9: Represe3/26/14
oncore.org
MATICS CURRIC
it matter whic
ermine the num
ours is 7/5 or 1
n the context o
kson can build
our.
ermine the num
irdhouses is 5/0 43 minu
u to write an e
ant to know .
tionality will be
rmine the follo
es can Jackson bui
, then sub
number or birdho
/ .
birdhouses in
Jackson to build 3
then substi
rs to build bird
to build birdh
, then subs
e nearest tenth).
may work for a
e. Tell students
hat can be writ
er of birdhouse
real‐life contex
und.
enting Proportiona4
CULUM
ch of these var
mber of birdho
1.4.
of this situation
1.4 birdhouses
mber of hours
/7 or 0.71, whiutes to build on
equation that w
This forces H
e dependent/i
wing:
ld in 40 hours?
bstitute 40 in the e
ouses.
hours.
35 birdhouses? Us
itute 35 into the e
dhouses.
houses? Use the eq
stitute for in
It will take Jeff
a fractional pa
that some num
tten as minute
es completed o
xt, and we mus
al Relationships wi
ThisCre
iables is indep
ouses that can
n?
s in one hour o
it takes to buil
ch means that
ne bird house.
will let him dete
to be the depe
ndependent w
equation for and
se the equation fr
equation for ;
quation from part
nto the equation;
hours and m
rt of an hour, a
mbers can be l
s), but others m
or sold), all of t
st consider the
th Equations
s work is licensed underative Commons Attribut
endent or dep
be built in one
or one entire ho
d one birdhou
t it takes him 5
ermine how lo
endent variabl
which is / o
d solve for since
rom part a to solv
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t a to solve the pr
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a customer wil
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him to build a
the independe
7/5 so use th
Scaffolding:
Use, and e
students t
variety of
represent
such as a
H
5
10
15
20
25
30
Provide g
calculator
students t
them to s
graph, the
and interp
unit rate o
graph, (1,
77
License.
•1
of
ird
ny
ent
e
:
encourage
to use, a
f
tations,
table:
B
7
14
21
28
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ketch the
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on the
5/7).
7
© 2013
NY
Exa
Simi
stud
Circu
follo
Let s
Barb
depe
divid
varia
3 Common Core, Inc. Som
YS COMMON C
ample 2: Al’s
lar to above, u
dents an oppor
ulate to assess
ow.
students select
bara’s Produce
endent/indepe
de the cost (de
able). Lead the
Which ma
Costcorn
Based on
num
Which wo
Cost
Have stud
proportio
’
How do y
Write the
Write the
Exmaple 2
Al’s Produ
Write two
corn sold
A
B
Lesson 9Date:
e rights reserved. commo
CORE MATHEM
Produce Stan
use Example 2 a
tunity to tackle
progress infor
t any two pairs
stand to calcu
endent). In ord
ependent varia
em through th
akes more sens
t per ear of cor
n.
the previous q
mber of ears of
ould be the dep
t, because the c
dents voluntee
onality and com
0.21 and
ou write an eq
e equation for A
0.21
e equation for B
0.22
2: Al’s Produce St
uce Stand sells eo equations, one f
and the cost. The
Al’s Produce Stand
Barbara’s Produce
Ears
Cost $1
9: Represe3/26/14
oncore.org
MATICS CURRIC
nd
as an opportun
e the problem
rmally before d
s of numbers fr
ulate the const
der to determi
ble) by the num
e following qu
se: to use a un
rn makes more
question, which
corn
pendent variab
cost depends o
er to share the
mpare the value
’
quation for a pr
Al’s Produce St
Barbara’s Prod
tand
ears of corn for $for each produce s
en use each equat
d: . wh
e Stand: .
Al’s Produce Sta
7 14
1.50 $2.94 $
enting Proportiona4
CULUM
nity to build pe
first, independ
discussing the
rom either Al’s
ant of proport
ne the unit pri
mber of ears o
estions to orga
it rate of “ears
sense because
h would be the
ble and why?
on the number
pair of numbe
es for Al’s and
0.22
roportional rel
tand:
duce Stand:
. . Barbara’s Pstand that models
ion to help compl
here x represents
where x repr
and
21 238
$4.41 $50.00
al Relationships wi
ThisCre
erseverance; g
dently or with
scaffolded que
s Produce stand
ionality ( = ce, students n
of corn (indepe
anize their thin
s of corn per do
e corn is sold a
e independent
of ears of corn
ers they used to
for Barbara’s.
lationship?
Produce stand sells
s the relationship
lete the tables bel
number of ears of
resents number of
Barbar
Ears 13
Cost $2.85
th Equations
s work is licensed underative Commons Attribut
ive
a partner.
estions that
d or
eed to
endent
nking.
ollar” or of “do
as an entire ear
variable?
n purchased
o determine th
s 13 ears of corn f
between the num
low.
of corn and y repre
f ears of corn and
ra’s Produce Stand
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r a tion‐NonCommercial‐Sh
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r of corn not pa
heir unit rate o
for $ . . mber of ears of
esents cost
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Use an
more re
such as,
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er ear of corn”?
art of an ear of
or constant of
t
ng:
example that
elevant to stu
, “Which mak
ense, ‘gallons
lar’ or ‘dollars
er gallon of m
Scaffoldin
Provide
calcula
studen
Ask stu
locate t
on each
0.21) a
and exp
(1,r) wi
show t
78
License.
•1
?
f
t may be
udents,
kes
s of milk
s and
milk’?”
ng:
e graphing
tors for
ts to use.
udents to
the unit rate
h graph, (1,
nd (1, 0.22),
plain why
ill always
he unit rate.
8
© 2013
NY
Clos
Ask s
asse
Exit
Ensu
unde
3 Common Core, Inc. Som
YS COMMON C
Lesson
How d
How d
propor
What i
left as
algebra
If you use
write the
sing Question
students to res
ess student pro
What type
know to
A prcons
Give an ex
Distathen
How do y
The
Give an ex
why.
The mult
Ticket (5 min
ure that studen
erstanding of t
Lesson 9Date:
e rights reserved. commo
CORE MATHEM
n Summary:
o you find the con
o you write an eq
rtionality in place
is the structure of
variables, and wh
a.
ed numbe
equations?
0.21 and
ns (5 minutes
spond to these
ogress.
e of relationsh
write an equa
roportional rela
stant of propor
xample of a re
ance equals ra
n the distance w
ou determine
value that is d
xample of a re
distance is the
tiplied by the r
nutes)
nts complete th
the lesson.
9: Represe3/26/14
oncore.org
MATICS CURRIC
nstant of proporti
quation for a prop
of .
f proportional rela
hen one of them is
er of ears of co
0.22
s)
e questions, eit
hip can be mod
tion in this for
ationship can b
rtionality, whic
al‐world relati
ate multiplied b
will depend on
which value is
determined by m
al‐world relati
e dependent va
rate.
he Exit Ticket i
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CULUM
ionality? Divide to
ortional relations
ationship equation
s known, they are
rn and
ther with a nei
deled using an e
m?
be modeled usi
ch is represente
onship that ca
by time. If the r
time elapsed.
(independen
multiplying a c
onship that ca
ariable, and the
ndependently,
al Relationships wi
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o find the unit rate
hip? ,subs
ns, and how do w
e substituted into
for the var
ighbor or in wr
equation in the
ing an equation
ed by in the e
n be modeled
rate of a vehicl
nt) and which v
constant to a va
nnot be mode
e time the inde
, in order to pr
th Equations
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e, / .
stituting the value
we use them? an to find t
riables instead
riting. Use this
e form
n in the formequation.
using this type
le is going at a
value is (dep
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of
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ow would you
unity to inform
o you need to
need to know t
and explain wh
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ble.
on and explain
me is being
ent’s
79
License.
•1
ally
the
hy.
nt),
n
9
© 2013
NY
Nam
Le
Eq
Exit
Osca
kilom
3 Common Core, Inc. Som
YS COMMON C
me __________
sson 9:
quations
t Ticket
ar and Maria e
meters and dis
number of m
a. Oscar wr
b. Maria w
Lesson 9Date:
e rights reserved. commo
CORE MATHEM
_____________
Represe
ach wrote an e
tance in miles.
iles, who wrot
rote the equat
rote the equat
9: Represe3/26/14
oncore.org
MATICS CURRIC
____________
enting P
equation that t
. One entry in
e the correct e
ion 1.61
tion 0.62
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CULUM
_____________
roportio
they felt repres
the table paire
equation that w
, and he said
as her equat
al Relationships wi
ThisCre
_______
onal Rela
sented the pro
ed 150 km wit
would relate m
that the rate 1
ion, and she sa
th Equations
s work is licensed underative Commons Attribut
Date____
ationshi
oportional rela
th 93 miles. If
miles to kilomet
1.61 represent
aid that0.62re
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ters? Explain w
ts miles per km
epresents mile
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______
een distance in
f kilometers an
why.
m.
es per km.
80
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•1
n
nd
0
© 2013
NY
Exit
The
Pro
3 Common Core, Inc. Som
YS COMMON C
t Ticket Sam
following solut
Oscar and
kilometer
num
a. Osca
b. Mar
Maria is c
per mile.
independ
stated tha
blem Set Sa
1. A pe
a.
b.
c.
d.
2. Use
a.
Lesson 9Date:
e rights reserved. commo
CORE MATHEM
mple Solution
tions indicate a
d Maria each wrot
rs and distance in
mber of miles, who
ar wrote the equa
ria wrote the equa
correct. Maria fou
However, it shoul
ent or dependent,
at represents nu
ample Soluti
erson who weighs
Which variable
Weight on the
themselves firs
What is an equ
. /
How much wo
. lb.
How much wo
lb.
this table to answ
Which variable
The number of
the number of
9: Represe3/26/14
oncore.org
MATICS CURRIC
ns
an understand
te an equation tha
miles. One entry
o wrote the correc
ation . ,
ation .
und the unit rate to
ld be noted that th
, each equation sh
umber of km and m
ons
s pounds on E
e is the independe
earth is the indep
st. The weight on
uation that relates
uld a 185 pound a
uld a man that we
wer the following
e is the dependen
f miles driven is th
gallons of gas you
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ding of the obje
at they felt repres
y in the table paire
ct equation that w
, and he said that
as her equation,
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hould include a de
m represents num
Earth weighs .
ent variable? Exp
pendent variable b
the moon depend
s weight on Earth
astronaut weigh o
eighed 50 pounds
questions.
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he dependent varia
u have in your tan
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ectives of this l
sented the propor
ed km with 9
would relate miles
the rate . /
and she said that
ding miles by km.
e not well‐defined.
efinition of each v
mber of miles.
lb. on the moon
plain why.
because most peo
ds on a person’s w
to weight on the
on the moon?
on the moon wei
hy?
able because the n
nk.
th Equations
s work is licensed underative Commons Attribut
lesson:
rtional relationshi
3 miles. If = nu
s to kilometers an
represents miles
. represents
The rate that Mi
. Since we do not
variable. For exam
n.
ople do not fly to
weight on the eart
moon?
igh back on Earth
number of miles y
r a tion‐NonCommercial‐Sh
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ip between distan
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per km.
miles per km.
ichael used repres
know which value
mple, Maria shoul
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th.
?
you can drive dep
hareAlike 3.0 Unported L
7•son 9
nce in
rs and
sents km
es are
ld have
gh
ends on
8
License.
•1
1
© 2013
NY
3 Common Core, Inc. Som
YS COMMON C
b.
c.
d.
e.
f.
g.
h.
i.
3. Sup
a.
b.
Lesson 9Date:
e rights reserved. commo
CORE MATHEM
Is miles driven
gallons?
Yes, miles drive
to get every co
In any ratio rel
Yes, miles
If the number o
Yes, multiply th
If the number o
Explain how th
proportionality
How many mil
558 miles
How many gall
/ of a gal
How many mil
. mil
How many gall
/ of a gall
ppose that the cos
If the = cost a
is the depend
use it. .
What would be
$
9: Represe3/26/14
oncore.org
MATICS CURRIC
proportionally re
en is proportional
orresponding mea
ating gallons and
of gallons is know
he constant of pro
of miles driven is
his value would be
y (31 mpg).
es could be driven
lons are used whe
llon
es have been driv
les
lons have been us
lon
st of renting a sno
and = hours, wh
dent variable beca
e the cost of renti
enting Proportiona4
CULUM
elated to gallons?
lly related to gallo
sure of miles drive
miles driven, will
wn, can you find th
oportionality (31 m
known, can you f
e calculated? Yes,
n with 18 gallons o
en the car has bee
ven when ½ of a ga
sed when the car h
wmobile is $ .
hich variable is the
ause the cost of us
ng 2 snow mobile
al Relationships wi
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If so, what is the
ons because every
en.
l one of the values
he miles driven? E
mpg) by the numb
ind the number o
divide the numbe
of gas?
en driven 18 miles
allon is used?
has been driven ½
for 5 hours.
e dependent varia
sing the snowmob
es for 5 hours each
th Equations
s work is licensed underative Commons Attribut
equation that rel
y measure of gallo
s always be larger
Explain how this v
ber of gallons.
of gallons consume
er of miles driven
s?
½ mile?
able? Explain why
bile depends on th
h?
r a tion‐NonCommercial‐Sh
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lates miles driven
ons can be multipl
r, if so, which one
value would be ca
ed?
by constant of
y.
he number of hou
hareAlike 3.0 Unported L
7•son 9
to
lied by 31
e?
alculated?
urs you
82
License.
•1
2
© 2013
NY
3 Common Core, Inc. Som
YS COMMON C
4. In m
a.
b.
c.
d.
e.
Lesson 9Date:
e rights reserved. commo
CORE MATHEM
mom’s car, the num
Write the equa
What is the co
How many mil
miles
If your family t
/ gallons
If you drive
you use?
gallons
9: Represe3/26/14
oncore.org
MATICS CURRIC
mber of miles driv
ation that will rela
nstant of proporti
es could you go if
takes a trip of 600
s
miles during on
enting Proportiona4
CULUM
ven is proportiona
Gallons
0
4
6
8
10
ate the number of
ionality?
f you filled your 22
0 miles, how many
ne week of commu
al Relationships wi
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al to the number o
Miles
driven
0
112
168
224
280
f miles driven to t
2‐gallon tank?
y gallons of gas wo
uting to school an
th Equations
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of gallons of gas u
he gallons of gas.
ould be needed to
nd work, how man
r a tion‐NonCommercial‐Sh
Less
sed.
o make the trip?
ny gallons of gas w
hareAlike 3.0 Unported L
7•son 9
would
83
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•1
3
© 2013
NY
Stud
Less
In th
intro
been
alwa
emp
The exam
Clas
Exa
Exam
to th
Have
Exa
Gr
coo
Tab
flou
MP.1 &
MP.2
3 Common Core, Inc. Som
YS COMMON C
Lesso
Relat
dent Outcom
Students
interpret
the prob
Students
relations
son Notes
his lesson, focu
oduced to the s
n exposed to th
ays, on the gra
phasized as we
problem set mmple, consider
sswork
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mple 1 is a revi
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andma’s Special C
okies. Using this i
ble – Create a char
ur used to the am
Lesson 1Date:
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CORE MATHEM
on 10: I
tionship
mes
s consolidate t
t what points o
blem, including
s are able to id
ship where is
us turns to the
significance of
his in Lessons 8
ph of a propor
ll.
may be used as using targeted
5 minutes)
ew of previous
complete the
rk individually
Chocolate‐Chip Co
nformation, comp
rt comparing the a
ount of cookies.
10: Interpre3/26/14
oncore.org
MATICS CURRIC
nterpret
ps
heir understan
on the graph o
g the point 0, 0
entify and inte
s the unit rate.
graphical repre
the point (1,r)
8 and 9. In par
rtional relation
homework or d instruction w
sly taught conc
table.
and then comp
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plete the chart:
amount of
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pro
flou
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ting Gra
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erpret in conte
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ship, indicate t
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cepts, but the
pare and critiq
h yields 4 dozen c
ble – Is the numbe
oportional to the a
ur used? Explain.
s, because there e
nstant = / or at each measure o
flour multiplied b
nstant gives the
rresponding meas
okies
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aphs of P
ions represent
al relationship
ext the point 1
a proportional
of a proportio
understanding
the unit rate. A
ry exercises forhat struggled o
lesson will be
que each other
cookies, calls for 3
er of cookies
amount of
exists a
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ra
in
1
1
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Proporti
ting proportion
p mean in term
1, on the gra
relationship. S
onal relationshi
developed in t
Additionally, th
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built upon this
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3 cups of flour to m
nit Rate – What is
te, and what is th
the context of th
1 1/3
1 1/3 dozen cookie
cookies) for every
flour
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partner.
make 4 dozen
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1 cup of
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ion or context
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se the challeng
Scaffolding:
Use and en
students to
representa
a double n
Use a more
task, such a
this proble
different w
how both m
the situatio
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of
ve
will
ge
ncourage
o use a variety o
aitons, including
umber line:
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as, “Represent
m in two
ways and explain
models relate to
on.”
4
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© 2013
NY
3 Common Core, Inc. Som
YS COMMON C
Graph –
Example 2
Below is a
Record th
Grandma
above.
1
How man
do we ca
Lesson 1Date:
e rights reserved. commo
CORE MATHEM
Model the relatio
2
a graph modeling
he coordinates of
, ;, O cups of su
, ; 2 cups of su
, ; 4 cups of su
, ); 8 cups of s
, ; 12 cups
, ; 16 cups
a has 1 remaining
1.5 dozen cookies
ny dozen cookies c
ll this point?
, ; 0 cup of su
10: Interpre3/26/14
oncore.org
MATICS CURRIC
onship on a graph
the amount of su
flour of the point
ugar will give 0 do
ugar yields 3 doze
ugar yields 6 doze
sugar yields 12 do
of sugar yields 18
of sugar yields 24
cup of sugar, how
can grandma mak
ugar = 0 dozen coo
eting Graphs of Pro4
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.
Does the
quantitie
each oth
The poin
that pas
(0,0).
gar required to m
ts from the graph
ozen cookies
n cookies
n cookies
ozen cookies
8 dozen cookies
4 dozen cookies
w many dozen coo
ke if she has no su
okies, point is calle
oportional Relatio
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e graph show the
es being proportio
her? Explain.
nts lie on a straigh
sses through the o
make Grandma’s C
in a table. What
okies will she be a
ugar? Can you gra
ed the origin
nships
s work is licensed underative Commons Attribut
two
onal to
ht line
origin
Equatio
that ca
the rela
Or
.
D = Nu
F = Num
Chocolate‐Chip Coo
do these ordered
ble to make? Plo
aph this on the gri
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on – Write an equ
an be used to repr
ationship.
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mber of Dozen Co
mber of cups of Fl
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ookies
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Scaffolding:
Provide gra
calculators
students to
fluidly betw
equations,
and graphs
8
License.
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aphing
s to allow
o move
ween
tables,
s.
5
© 2013
NY
Gene
and
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Ask s
3 Common Core, Inc. Som
YS COMMON C
erate class disc
what it means
How is th
T
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What qua
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What qua
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When you
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What is th
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to reach t
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quantities
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t
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proportio
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v
ercises (15 mi
students to work
Exercises
1. The
Lesson 1Date:
e rights reserved. commo
CORE MATHEM
cussion using t
s.
e unit rate rela
The unit rate m
coordinate of o
antity is measu
sugar
u plot the orde
The amount of
antity is measu
The amount of
u plot the poin
The total amou
he unit rate for
1.5
t the origin, if
the line you gra
1.5 units
we always mov
The unit rate is
number of cook
moving one un
1.5 units
number look f
s that are prop
The vertical dis
the unit rate w
f different prop
onal relationshi
The point 1,vertical distanc
nutes)
k independently
graph below show
10: Interpre3/26/14
oncore.org
MATICS CURRIC
the following q
ated to the gra
must be the val
one.
red along the
ered pair ,
f sugar in cups
red along the
f cookies (dozen
t , , what
unt of cookies
r this proportio
you move one
aphed?
ving 1.5 units v
s 1.5 dozen coo
kies. Since the
nit at a time al
familiar? Is it t
portional are gr
stance is the sa
when moving on
portional relat
ip? Explain wh
or unit rate mce for every 1 u
y on these exerci
ws the amount of
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questions to lea
aph?
ue of the ‐coo
horizontal axis
,what does
vertical axis?
ns)
does repres
onal relationsh
e unit along the
vertically?
okies for every
e unit rate is 1.5
ong the horizo
the unit rate?
raphed?
ame as the unit
ne unit horizon
ionship have d
hy.
must be on eve
unit change in t
ises.
f time a person ca
oportional Relatio
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ad to the conc
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s?
represent?
sent?
hip?
e horizontal ax
1 cup of sugar
5, every vertica
ontal axis. Wha
Do you think t
t rate. Yes, the
ntally on the ax
different points
ery graph becau
the horizontal
n shower with a c
nships
s work is licensed underative Commons Attribut
lusion the poin
e point on the g
xis, how far wo
r. The vertical
al move would
at distance vert
this will always
e vertical dista
xis.
s, but what poi
use the unit ra
axis.
certain amount of
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nt 1, must
graph, which h
ould you have t
axis or value
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be on the grap
has an ‐
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t rate of 1.5 un
move?
whenever two
s be the equal
n every graph o
he change in th
86
License.
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ph
ally
e
nits.
to
of a
he
6
© 2013
NY
3 Common Core, Inc. Som
YS COMMON C
a.
b.
c.
d.
e.
f.
2. You
Amu
a.
b.
c.
Lesson 1Date:
e rights reserved. commo
CORE MATHEM
Can you determ
gallons of wate
Yes, the quanti
origin , .
How long can a
minutes,
What are the c
, If the
Can you use th
Since the graph
the graph, such
Plot the unit ra
Yes, the unit ra
Write the equa
and the length
whe
r friend uses the e
usement Park.
Create a table an
sized groups of p
Is the cost of ad
why not.
Yes, because the
constant gives t
What is the uni
50, 1 person cos
10: Interpre3/26/14
oncore.org
MATICS CURRIC
mine by looking at
er? Explain how y
ities are proportio
a person shower w
minutes
coordinates of poi
ere are 30 gallons
he graph to identif
h is a line that pas
h as , and g
ate on the graph.
ate is a point on th
ation to represent
of a shower.
re is minutes a
equation
nd record the cost
people.
dmission proportio
ere exists a consta
he corresponding
t rate, and what d
sts $50
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t the graph wheth
you know.
onal to each other
with 15 gallons of
int A? Describe po
s of water, then a
fy the unit rate?
sses through ,
get .
Is the point on th
he graph of the rel
t the relationship
nd is gallons
to find the total
of entering the a
onal to the amoun
ant = 50 such that
measures of cost
does it represent i
oportional Relatio
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her the length of t
r since all points li
water and with 6
oint A in the conte
person can showe
and , ), you c
he line of this relat
lationship.
between the num
cost of P people e
musement park fo
nt of people enter
t each measure of
t
in the context of t
nships
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the shower is prop
ie on a straight lin
60 gallons of wate
ext of the problem
er for 10 minutes.
can take a point o
tionship?
mber of gallons use
entering the local
or several differen
ring the Amuseme
f the amount of pe
the situation?
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minutes?
w would this ch
blem?
87
License.
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a small graphical a onship. ns: itch the d axis? pend on
ange the
7
© 2013
NY
Clos
Ask s
infor
3 Common Core, Inc. Som
YS COMMON C
d.
e.
f.
sing (5 minut
students to res
rmally assess u
What poin
T
How can y
proportio
I
d
t
How can y
F
f
e
l
r
How do y
proportio
A
c
Lesson 1Date:
e rights reserved. commo
CORE MATHEM
Sketch a graph
What point(s) M
other? Explain w
, , , .
If 0 people ente
If 1 person ente
For every 1‐unit
Would the poin
Yes the point (5
$250 for 5 peop
tes)
spond to these
understanding
nts are always
The points 0,0
you use the un
onal to each oth
In a table you c
divide every v
the equation
0,0 must be o
you identify th
From a table, y
for the entire ta
equation in the
line that passes
relationship is p
you determine
onal to each oth
Any point ,corresponding
10: Interpre3/26/14
oncore.org
MATICS CURRIC
to represent this
MUST be on the gr
why and describe
er the park, then th
ers the park, the co
t increase along th
nt , be on t
5,250) would be on
ple to enter an Am
e questions, eit
of the lesson.
on the graph o
0 and 1, , w
nit rate to creat
her?
can multiply ea
value by the un
and rep
on the straight
he unit rate fro
you can divide e
able, then the
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proportional.
the meaning o
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relationship.
raph of the line if
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he cost would be
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he horizontal axis,
the graph? What
n the graph becau
musement Park.
ther by speakin
of two quantit
where is the u
te a table, equ
ach value by
nit rate to obta
place the k wit
t line of the pro
m a table, equ
each value b
ratio / is th
,the unit rateorigin and cont
of any point on
that represents
or horizontal un
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the two quantitie
context of the pro
$0.
s, the change in th
does this point re
use 5(50) = 250. T
ng to a partner
ies that are pro
unit rate, are a
uation, or graph
the unit rate to
ain the corresp
th the value of
oportional rela
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by the correspo
e unit rate, an
e is the number
tains the point
n a graph that
s a proportiona
nit, and corre
nships
s work is licensed underative Commons Attribut
es represented are
oblem.
he vertical distanc
epresent in the co
The meaning is tha
r or in writing.
oportional to e
always on the g
h of a relations
o obtain the co
ponding ‐valu
f the unit rate.
ationship.
h?
onding x value.
d the relations
r represented b
t 1, , repre
represents two
al relationship
responds to the
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e proportional to
ce is 50 units.
ontext of the situa
at it would cost a
Use this as an
each other?
graph.
ship of two qu
orresponding
e. In an equat
In a graph, th
If the ratio /ship is proporti
by the . If a gesenting the un
o quantities th
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each
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total of
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‐value, or you
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graph of a strai
nit rate, then th
hat are
number
tical unit.
88
License.
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o
re
u can
se
and
nt
ight
he
8
© 2013
NY
Exit
Ensu
unde
3 Common Core, Inc. Som
YS COMMON C
Lesson
The po
propor
The un
on the
The po
second
These
situatio
Ticket (5 min
ure that studen
erstanding of t
Lesson 1Date:
e rights reserved. commo
CORE MATHEM
n Summary:
oints , and
rtional to each oth
nit rate r in the po
graph.
oint , indicated quantity.
two points may n
on, but they will a
nutes)
nts complete th
the lesson.
10: Interpre3/26/14
oncore.org
MATICS CURRIC
, , where r is th
her.
int , represe
es that when ther
ot always be give
always fall on the
he Exit Ticket i
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e unit rate, will al
nts the amount of
e is zero amount o
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line that passes t
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oportional Relatio
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lways fall on the l
f vertical increase
of one quantity, t
et of data for a giv
hrough the given
, in order to pr
nships
s work is licensed underative Commons Attribut
ine representing t
e for every horizon
there will also be z
ven real‐world or
data points.
rovide evidenc
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two quantities th
ntal increase of 1
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mathematical
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Nam
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Exit
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the g
3 Common Core, Inc. Som
YS COMMON C
me _________
sson 10:
t Ticket
at Rapids Whit
graph of the re
Lesson 1Date:
e rights reserved. commo
CORE MATHEM
___________
: Interp
e Watering Co
elationship, an
10: Interpre3/26/14
oncore.org
MATICS CURRIC
____________
reting G
mpany rents r
d what these p
eting Graphs of Pro4
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___________
Graphs o
afts for $125 p
points mean in
oportional Relatio
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____________
of Propor
per hour. Expla
the context o
nships
s work is licensed underative Commons Attribut
__
rtional R
ain why the po
f the problem.
r a tion‐NonCommercial‐Sh
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Date_______
Relation
oint 0,0 and .
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ships
1,125 are on
90
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____
n
0
© 2013
NY
Exit
The
Pro
The
grap
grap
3 Common Core, Inc. Som
YS COMMON C
t Ticket Sam
following solut
Great Rap
graph of t
Every gra
because 0
is the unit
point ,
blem Set Sa
problem set re
phs. Within ea
ph and student
1. The
a.
b.
c.
d.
2. Cham
a.
b.
c.
Lesson 1Date:
e rights reserved. commo
CORE MATHEM
mple Solution
tions indicate a
pids White Wateri
the relationship, a
ph of a proportion
0 can be multiplied
t rate. On the gra
means 0 hours
ample Soluti
equires studen
ch problem, st
ts have to conn
graph to the right
What does the p
the situation?
In 5 seconds, a j
What does the p
situation?
A jaguar can run
Is the distance r
Explain why or w
Yes, because it i
origin (0,0)
Write an equatio
The constant of p
mpionship T‐shirt
What point(s) M
, , ,
What does the o
5 T‐shirts would
How many T‐shi
; /
10: Interpre3/26/14
oncore.org
MATICS CURRIC
ns
an understand
ing Company rent
and what these po
nal relationship m
d constant to ge
aph for every 1 uni
of renting a raft w
ons
nts to have a fu
tudents are giv
nect unit rate a
t shows the distan
point (5, 280) repr
aguar can run 280
point (3,174) repre
n 174 feet in 3 sec
un by the Jaguar p
why not.
is a straight line th
on to represent th
proportionality, o
ts sell for $22 each
MUST be on the gr
ordered pair ,
d cost $110
irts were sold if yo
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ding of the obje
ts rafts for $125 pe
oints mean in the
must include the po
et the correspondi
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ull understandi
ven the informa
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nce (ft.) run by a J
resent in the cont
0 feet.
esent in the conte
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proportional to th
hat passes throug
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r unit rate, is 58 a
h.
aph for the quant
represent in t
ou spent a total of
oportional Relatio
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ectives of this l
er hour. Explain w
context of the pro
oints , and
ing value of 0. Th
horizontal axis, th
d , means
ng of proportio
ation in a diffe
ts to the equat
aguar.
ext of
ext of the
he time?
gh the
y the Jaguar. Expl
and can be substit
tities to be propor
the context of thi
f $88?
nships
s work is licensed underative Commons Attribut
lesson:
why the point (0,0
oblem.
, .The point (0he point ,he vertical axis wil
s 1 hour of renting
onal relationsh
erent format, s
tion and graph
lain or model you
tuted into the equ
rtional to each oth
s problem?
r a tion‐NonCommercial‐Sh
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0) and (1,125) are
0,0) is on the grap
is on the graph b
ll change by 125 u
g the raft would co
hips, their tabl
ometimes tabl
h.
r reasoning.
uation y = kx in pla
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on the
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ecause it
units. The
ost $125.
es, equations a
le, equation or
ace of k.
9
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and
r
1
© 2013
NY
3 Common Core, Inc. Som
YS COMMON C
3. The
rent
car i
amo
a.
b.
4. Jack
pack
a.
b.
c.
5. The
Am
a.
b.
c.
d.
e.
Lesson 1Date:
e rights reserved. commo
CORE MATHEM
following graph r
ting a car is a fixed
is driven. It does
ount per day.
What does the o
It would cost $2
What would be
your reasoning.
Since the unit ra
.
kie is making a sna
kages of M&M’s s
What points MU
proportional to
(0,0) (1,2), All gr
through the orig
Write an equatio
Describe the ord
In the mixture y
following table sh
mount of Candy (pounds)
Cost (Dollars)
Is the cost of can
Yes, because the
constant gives t
Write an equatio
the amount of c
.
Using the equat
pounds of candy
. $
What is the max
with $60?
/ .
Graph the relati
10: Interpre3/26/14
oncore.org
MATICS CURRIC
represents the tot
d amount each da
not matter how m
ordered pair (4,25
50 to rent a car fo
the cost to rent th
ate is 62.5, the cos
.
ack mix for a party
he needs to how
UST be on the grap
the packages of M
raphs of proportio
gin (0,0) and the u
on to represent th
dered pair (12,24)
ou will need 12 pa
hows the amount
ndy proportional t
ere exists a consta
he corresponding
on to illustrate th
candy and the cost
ion, predict how m
y?
ximum amount of
onship
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tal cost of renting
y, regardless of h
many miles you dr
50) represent?
or 4 days.
he car for a week?
st for a week woul
y. She is using M&
many cans of pea
ph for the numbe
M&M’s? Explain w
onal relationships
unit rate (1,r).
his relationship.
in the context of
ackages of M&M’
of candy and pric
2
5
to the amount of
ant = 2.5 such that
measure of cost
e relationship bet
t.
much it will cost f
f candy you can bu
oportional Relatio
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a car. The cost of
ow many miles th
rive; you just pay a
? Explain or mode
ld be
&M’s and peanuts
anuts she needs to
r of cans of peanu
why.
are straight lines
the problem.
’s to 24 cans of pe
ce paid.
3
7.5
candy?
t each measure of
tween
for 12
uy
nships
s work is licensed underative Commons Attribut
f
he
an
el
s. The table below
o make the mix.
uts to be
s that passes
eanuts
f the amount of ca
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Josiah and Tiyear, Josiah wevery year wJosiah’s pay a
A recent studsent or receivby boys propusing a graph
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CORE MATHEM
llery have newill be paid $with the compand Tillery’s p
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1: Ratios a3/26/14
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MATICS CURRIC
w jobs at Yum14 per hour aany, they wilpay rate prop
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Josiah is Tilleper hour. Th$2 per hour. asoning using
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3.
3 Common Core, Inc. Som
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When a songrest. The grabrought in by
a. Identifybrought
b. Write ansong, x.
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e rights reserved. commo
CORE MATHEM
g is sold by anaph below shoy one popula
y the constantt in by sales o
n equation re
1: Ratios a3/26/14
oncore.org
MATICS CURRIC
n online musicows how mucr online musi
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3 Common Core, Inc. Som
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c. Accordifirst we
d. Describethe grap
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CORE MATHEM
ng to the proek, if the pop
e what the poph.
point on the gt in from sale
1: Ratios a3/26/14
oncore.org
MATICS CURRIC
oportional relap star earned
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ationship, how$800 that we
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rom sales in t
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3 Common Core, Inc. Som
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Progression
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1
7.RP.2a
2 7.RP.2a
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CORE MATHEM
Toward Mas
STEP 1 Missing oranswer anevidence oreasoning applicatiomathematsolve the p
Student answincorrectly. unable to coleast two corvalues in theStudent was respond or rtheir answer
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1: Ratios a3/26/14
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MATICS CURRIC
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STEP 2Missing or inanswer but evidence of sreasoning orapplication omathematicssolve the pro
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TEP 3correct answith some evidreasoning or
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udent correctly swered that theationship was noportional. The as correctly set uleast two correctries. Student’s asoning may havntained a minor
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3
a
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c
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d
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e
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e rights reserved. commo
CORE MATHEM
Student was
answer
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or evfor any value
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1: Ratios a3/26/14
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MATICS CURRIC
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3 Common Core, Inc. Som
YS COMMON C
Module Date:
e rights reserved. commo
CORE MATHEM
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oncore.org
MATICS CURRIC
nd Proportional R4
CULUM
elationships
ThisCre
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s work is licensed underative Commons Attribut
Module As
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ssessment
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7• Task
98
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3 Common Core, Inc. Som
YS COMMON C
Module Date:
e rights reserved. commo
CORE MATHEM
1: Ratios a3/26/14
oncore.org
MATICS CURRIC
nd Proportional R4
CULUM
elationships
ThisCre
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s work is licensed underative Commons Attribut
Module As
r a tion‐NonCommercial‐Sh
ssessment
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7• Task
99
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3 Common Core, Inc. Som
YS COMMON C
Module Date:
e rights reserved. commo
CORE MATHEM
1: Ratios a3/26/14
oncore.org
MATICS CURRIC
nd Proportional R4
CULUM
elationships
ThisCre
Mid‐M
s work is licensed underative Commons Attribut
Module As
r a tion‐NonCommercial‐Sh
ssessment
hareAlike 3.0 Unported L
7• Task
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