N S Common Cor e 7 Mathem atics Curr iculum B: Unit Rat Lesson 7 Lessons 8 Lesson 1 ‐Module Ass cs...

G

N

© 2013

Ta

RaMod

Top

Top

MidTopi

Top

Top

1 Eac

7 RADE

ew York S

3 Common Core, Inc. Som

able of C

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

Ma

Module Date:

e rights reserved. commo

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

mmon Cor

athem

1: Ratios a3/26/14

oncore.org

ts1

Propor.....................

nships (7.RP.

nce in Relatio

al Relationsh

ying Proportio

ying Proportio

onstant of Pro

s the Constan

enting Propor

ng Graphs of

d Rubric ........nt 1 day, retu

olving Fractio

s of Fractions

quivalent Rati

Ratio Proble

of Graphs of

wings (7.RP.2b

cale Drawing

Rate as the Sc

g Actual Leng

g Actual Area

se in Creating

xercise in Cha

ay is considered

e

matics

nd Proportional R4

rtiona.....................

2a) ...............

onships as Me

ips ................

onal and Non‐

onal and Non‐

oportionality

nt of Proporti

rtional Relatio

Proportional

.....................urn 1 day, rem

ons (7.RP.1, 7

s and Their U

ios Given the

ms ................

f Proportiona

b, 7.G.1) ........

gs to Ratios an

cale Factor ....

gths from a Sc

as from a Scal

g a Scale Draw

anging Scales .

a 45 minute per

s Curr

elationships

ThisCre

al Rela......................

......................

easuring Rate

......................

‐Proportional

‐Proportional

(7.RP.2b, 7.R

ionality .........

onships with

Relationship

......................mediation or f

.RP.3) ...........

nit Rates .......

Total Quanti

......................

l Relationship

......................

nd Rates ........

......................

cale Drawing

le Drawing ....

wing ...............

......................

riod.

riculu

s work is licensed underative Commons Attribut

ations.....................

.....................

e ....................

.....................

l Relationship

l Relationship

RP.2c, 7.RP.2d

.....................

Equations ....

ps ..................

.....................further applic

.....................

.....................

ity .................

.....................

ps Involving F

.....................

.....................

.....................

....................

.....................

.....................

.....................

um

GR

r a tion‐NonCommercial‐Sh

ships .....................

.....................

.....................

.....................

ps in Tables ...

ps in Graphs ..

d) ..................

.....................

.....................

.....................

.....................cations 2 days

.....................

.....................

.....................

.....................

Fractions .......

.....................

.....................

.....................

.....................

.....................

.....................

.....................

RADE 7 • MODU

hareAlike 3.0 Unported L

......................

......................

......................

......................

......................

......................

......................

......................

......................

......................

......................s)

......................

......................

......................

......................

......................

......................

......................

......................

......................

......................

......................

......................

ULE 1

1

License.

..... 3

..... 8

..... 9

... 17

... 24

... 38

... 57

... 59

... 66

... 84

... 93

. 101

. 103

. 116

. 126

. 132

. 139

. 140

. 154

. 165

. 175

. 185

. 194

1

© 2013

NY

EndTopi


YS COMMON C

‐of‐Module Aics A through

Module Date:


CORE MATHEM

Assessment ah D (assessme

1: Ratios a3/26/14

oncore.org

MATICS CURRIC

and Rubric ....ent 1 day, retu

nd Proportional R4

CULUM

.....................urn 1 day, rem

elationships

ThisCre

......................mediation or f


Mo

.....................further applic


dule Over

.....................cations 2 days


7•rview

......................s)

2

License.

•1

. 209

2

© 2013

NY

Gr

Ra

OVIn M6.RPstudana

In Topropwherelasitua

In Tofor rappinvo

In ththe worof pdrawstud

Late

The AsseThe


YS COMMON C

rade 7 •

atios a

VERVIEWModule 1, studP.3) to formadents examinlysis is applie

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

er in the year,

module is coessments, retMid‐Module

Module Date:


CORE MATHEM

• Modu

and P

W dents build ully define proe situations cd to relations

nts learn thatnd can be useonstant of proatio tables and).

nts extend thtes specified brience in the fns to solve mu

of this moduale drawings e lengths andy, calculate tho‐dimensionag a new scale

, in Module 4

omprised of 2turning the ase Assessment

1: Ratios a3/26/14

oncore.org

MATICS CURRIC

ule 1

Propor

pon their Graoportional relacarefully to deships given in

t the unit rateed to represeoportionality d to graphs a

heir reasoningby rational nufirst two topicultistep ratio

le, students b(7.RP.2b, 7.Gd areas of polyhe actual lengal view of a ro drawing by c

, students wi

2 lessons; 8 dssessments, afollows Topic

nd Proportional R4

CULUM

rtiona

ade 6 reasoniationships anetermine if th tables, graph

e of a collectient proportio(7.RP.2b, 7.Rand interpret

g about ratiosumbers, such cs and their nword proble

bring the sumG.1). Given a ygons (6.G.1,gths and areaoom or buildichanging the

ll extend the

days are reserand remediatc B. The End‐

elationships

ThisCre

al Rela

ng about ratind the constahey are descrihs, and verba

on of equivalnal relationshRP.2c). Studenthe points on

s and proportas a speed o

new understams (7.RP.3).

m of their expescale drawin 6.G.3) as theas of objects ing. The topicscale of an ex

concepts of t

rved for admiing or providi‐of‐Module A


Mo

ations

os, rates, andnt of proportibing a propoal descriptions

lent ratios is chips with equnts relate then the graph w

tional relationf ½ mile per ¼anding of unit

erience with g, students reey identify thin the drawinc culminates xisting drawin

this module t

inistering theing further apAssessment fo


dule Over

ships

d unit rates (6tionality (7.RPortional relatios (7.RP.2a).

called the conuations of thee equation of within the con

nships to com¼ hour (7.RP.t rates for rat

proportional ely on their be scale factorng, and createwith a two‐dng.

to percent pro

e Mid‐ and Enpplications ofollows Topic D


7•rview

6.RP.1, 6.RP.2P.2). In Topiconship. Their

nstant of form y = kx, a proportion

ntext of the

mpute unit rat.1). Studentsios and rates

relationshipsbackground inr as the conste their own scay experienc

oblems.

nd‐of‐Modulef the conceptsD.

3

License.

•1

2, c A, r

nal

tes s

s to n tant cale e of

s.

3

© 2013

NY

Fo

Anapro

Drathe

Fo

Und


YS COMMON C

ocus Stan

alyze propooblems.

7.RP.1 Coee

7.RP.2 R

a

b

c

d

7.RP.3 Usin

aw, construem.

7.G.1 Sle

oundatio

derstand ra

6.RP.1 Ubzre

6.RP.2 Ulao

Module Date:


CORE MATHEM

ndards

ortional rela

Compute unit other quantitieach 1/4 hourequivalently 2

Recognize and

. Decide wequivalengraph is a

b. Identify thand verba

. Represenproportiobetween

d. Explain wthe situat

Use proportioimple interesncrease and d

ct, and des

Solve problemengths and ar

onal Stan

atio concept

Understand thbetween two oo was 2:1, beceived, cand

Understand thanguage in thof flour to 4 cu

1: Ratios a3/26/14

oncore.org

MATICS CURRIC

ationships a

rates associaes measuredr, compute th2 miles per ho

d represent p

hether two qnt ratios in a ta straight line

he constant oal description

t proportionanal to the numthe total cost

what a point (xtion, with spe

onal relationsht, tax, markudecrease, per

cribe geom

ms involving screas from a sc

ndards

ts and use r

he concept ofquantities. Fbecause for evdidate C recei

he concept ofhe context of ups of sugar,

nd Proportional R4

CULUM

and use the

ated with rati in like or diffe unit rate asur.

roportional r

uantities are table or graphthrough the

of proportionns of proportio

al relationshipmber, n, of itt and the num

x,y) on the graecial attention

hips to solve ps and markdcent error.

metrical figu

cale drawingscale drawing

ratio reason

f a ratio and uFor example, “very 2 wings tived nearly th

f a unit rate aa ratio relatioso there is 3/

elationships

ThisCre

em to solve

os of fractionferent units. s the complex

elationships b

in a proportihing on a cooorigin.

ality (unit ratonal relations

ps by equatioems purchasember of items

aph of a propn to the point

multistep ratdowns, gratui

res and des

s of geometriand reproduc

ning to solv

use ratio lang“The ratio of there was 1 bhree votes.”

a/b associatedonship. For e/4 cup of flour


Mo

real‐world

ns, including rFor example,x fraction ½ /

between qua

onal relationrdinate plane

te) in tables, gships.

ons. For examed at a constacan be expre

portional relats (0,0) and (1

tio and percenities and com

scribe the r

ic figures, inccing a scale d

ve problem

guage to descwings to beakbeak.” “For e

d with a ratio example, “Thisr for each cup


dule Over

d and mathe

ratios of lengt, if a person w¼ miles per h

antities.

ship, e.g., by e and observi

graphs, equat

mple, if total cant price, p, tessed at t = pn

tionship mea1,r), where r i

nt problems. mmissions, fee

relationship

luding compudrawing at a d

ms.

ribe a ratio reks in the bird very vote can

a:b with b ≠ s recipe has ap of sugar.” “


7•rview

ematical

ths, areas andwalks 1/2 milehour,

testing for ng whether t

tions, diagram

cost, t, is the relationshn.

ans in terms os the unit rat

Examples: es, percent

ps between

uting actual different scale

elationship house at the

ndidate A

0, and use raa ratio of 3 cu“We paid $75

4

License.

•1

d e in

the

ms,

hip

of te.

n

e.

e

ate ups 5 for

4

© 2013

NY

Solv

Fo

2 Exp


YS COMMON C

1

6.RP.3 Ure

a

b

c

d

ve real‐wor

6.G.1 Fcte

6.G.3 Dficp

ocus StanMP.1 M

mrrm(1fi

MP.2 Rtan

pectations for un

Module Date:


CORE MATHEM

15 hamburger

Use ratio and easoning aboequations.

a. Make tabfind missiUse table

b. Solve unitexample, mowed in

. Find a perthe quant

d. Use ratio appropria

rld and mat

Find the area omposing intechniques in

Draw polygonind the lengthoordinate. Aproblems.

ndards fMake sense omultistep ratioepresentatioepresentatiomeaning of co1,r) on graphigures on a sc

Reason abstraables to detenumbers will b

nit rates in this g

1: Ratios a3/26/14

oncore.org

MATICS CURRIC

rs, which is a

rate reasoninout tables of e

bles of equivaing values in ts to compare

t rate problemif it took 7 hon 35 hours? A

rcent of a quatity); solve pr

reasoning to ately when m

thematical

of right triangto rectangles the context o

s in the coordh of a side joiApply these te

or Mathof problems ao problems, ins, such as rans to each otonstant propos and the impcale drawing.

actly and quarmine if the dbe analyzed t

grade are limited

nd Proportional R4

CULUM

rate of $5 pe

ng to solve reequivalent rat

lent ratios rethe tables, ane ratios.

ms including tours to mow 4At what rate w

antity as a ratoblems involv

convert meaultiplying or d

problems in

gles, other trior decomposof solving rea

dinate plane gning points wechniques in t

hematicaand perseverencluding caseatio tables, thher and to thortionality in plications of h

antitatively. Sdata represento create and

d to non‐comple

elationships

ThisCre

r hamburger.

al‐world and tios, tape dia

lating quantitnd plot the pa

those involvin4 lawns, thenwere lawns be

te per 100 (e.ving finding t

asurement undividing quan

nvolving ar

iangles, specising into trianl‐world and m

given coordinwith the samethe context o

al Practie in solving thes involving pe coordinate e context of tproportional how scale fact

Students comnts a proportiimplement e

x fractions.


Mo

.”2

mathematicagrams, doubl

ties with whoairs of values

ng unit pricinn at that rate,eing mowed?

.g., 30% of a qthe whole, giv

nits; manipulantities.

rea, surface

ial quadrilatengles and othmathematical

nates for the e first coordinof solving real

ce hem. Studenpairs of ration plane, and ethe problem.relationshipstors magnify

mpute unit rational relationequations incl


dule Over

al problems, le number lin

ole‐number mon the coord

g and consta how many la?

quantity meaven a part and

ate and transf

e area, and v

rals, and polyer shapes; apl problems.

vertices; use nate or the sa‐world and m

nts make sensnal number enequations, and. Students des, the importaor shrink act

tes for pairednship. Use of luding, y = kx


7•rview

e.g., by ne diagrams, o

measurementdinate plane.

nt speed. Foawns could be

ans 30/100 timd the percent

form units

volume.

ygons by pply these

coordinates tme second

mathematical

se of and solvntries; they usd relate theseepict the ance of (0,0) aual lengths of

d data given inconcrete

x, where k is t

5

License.

•1

or

s,

r e

mes t.

to

ve se e

and f

n

he

5

© 2013

NY

Te

New

Fam

3 The4 The


YS COMMON C

crsp

MP.4 Mmpr

erminolo

w or Recent

Proportquantitycorrespo

Proportmeasuretype.)

Constanthat satproporti2/3 x.)

One‐to‐correspowith onpoint Q

Scale Drdrawingunder thand Q o| | =

miliar Terms

Ratio

Rate

Unit Rat

Equivale

ese terms will be

ese are terms an

Module Date:


CORE MATHEM

onstant of prepresenting scale drawingspicture.

Model with mmodel real woproportionalitatios and rate

ogy

tly Introduc

tional To (Mey if there is a onding meas

tional Relatioes of quantiti

nt of Proportiisfies the equionality.; e.g.

‐ One Correspondence if the and only onin S.)

rawing and Sg of S with scahe pairing of of S is related = r | |.)

s and Symb

te

ent Ratio

e formally define

nd symbols stude

1: Ratios a3/26/14

oncore.org

MATICS CURRIC

roportionalitysymbolically ts, scale factor

mathematics. orld situationsty in problemes to describe

ced Terms

easures of onenumber k > 0ure y of a qua

onship (A oneies of the first

ionality (If a puation y = kx, , If the ratio o

pondence (Twhere is a pairinne point P' in

cale Factor3 ale factor r if this one‐to‐oto the distan

bols4

ed in Grade 8. A

ents have seen p

nd Proportional R4

CULUM

y. Students dthe quantitiesrs will be cha

Students uses. They apply scenarios. The real world p

e type of qua0 so that for eantity of the s

‐to‐one matct type are pro

proportional where k is a pof y to x is 2 t

wo figures in tng between tS' and likewis

(For two figuthere exists a

one corresponce | | bet

A description is p

previously.

elationships

ThisCre

econtextualizs involved witnged to creat

e equations oand interprehroughout thphenomena.

ntity are propevery measursecond type i

ching betweeoportional to

relationship ipositive consto 3, then the

the plane, S ahe points in Sse, each poin

res in the plaa one‐to‐one ndence, the dtween corresp

provided in Grad


Mo

ze a given conth the formulte additional

of the form yet the meanine module, th

portional to mre x of a quans given by kx,

n two types othe measure

is described btant, then k iconstant of p

and S', are saiS and S', so tht Q' in S' is pa

ane, S and S', corresponde

distance | |ponding poin

de 7.


dule Over

nstant speed la, distance =scale drawing

= kx, graphs, ng of constantere is an emp

measures of atity of the firs, i.e., y = kx.)

of quantities ses of quantitie

by the set of os called the cproportionali

id to be in onhat, each poinaired with on

S' is said to bence betweenbetween any

nts P and Q of


7•rview

situation, = rate x time. gs of a given

and diagramts of phasis on usin

a second typest type the

such that thees of the seco

ordered pairsconstant of ty is 2/3 and

e‐to‐one nt P of S is pae and only on

be a scale n S and S' so ty two points Pf S by

6

License.

•1

In

s to

ng

e of

e ond

s

y =

ired ne

hat, P

6

© 2013

NY

Su

As

As

MAs

EnAs


YS COMMON C

Ratio Ta

uggested Ratio Ta

Coordin

Equatio

ssessmen

ssessment T

id‐Module ssessment Ta

nd‐of‐Modulessessment Ta

Module Date:


CORE MATHEM

able

d Tools aable (See exam

nate Plane (Se

ns of the form

nt Summ

Type Admi

sk After T

e sk

After T

FlouSugar

2

4

6

3

6

9

Ratio Table

1: Ratios a3/26/14

oncore.org

MATICS CURRIC

and Reprmple below)

ee example be

m y = kx

mary

inistered

Topic B C

Topic D C

r

e

nd Proportional R4

CULUM

resentat

elow)

Constructed r

Constructed r

elationships

ThisCre

tions

Format

esponse with

esponse with


Mo

h rubric

h rubric

Coordinate


dule Over

Standa

7.RP.2

7.RP.1, 77.G.1

e Plane


7•rview

ards Address

7.RP.2, 7.RP.3

7

License.

•1

sed

3,

7

© 2013

G

N

To

U

7.R

Focu

Instr

In Toof econsproprelagrap(1, r

1 Les


7 RADE

ew York S

opic B:

nit Ra

RP.2b, 7.RP

us Standard:

ructional Days

Lesso

Lessons 8

Lesson

opic B, studequivalent ratstant of propportionality frtionship to a phical undersr), where r is t

son Structure Ke

Topic B: Date:


State Com

Ma

ate an

.2c, 7.RP.2d

7.RP.2b

7.RP.2c

7.RP.2d

s: 4

on 7: Unit Rat

8–9: Represe

n 10: Interpre

nts learn to idios. They reportionality (7rom the descvariety of reptandings of pthe unit rate,

ey: P‐Problem S

Unit Rat3/26/14

oncore.org

mmon Cor

athem

nd Con

d,

Recogn

b. Ide

eq

rel

c. Re

cos

pri

ca

d. Ex

me

an

te as the Const

enting Proport

eting Graphs o

dentify the copresent this re.RP.2, 7.RP.2ription of a representationsproportional rin terms of t

Set Lesson, M‐M

te and Constant of4

e

matics

nstant

ize and repres

entify the cons

uations, diagr

lationships.

epresent propo

st t is proporti

ice p, the relat

n be expressed

plain what a p

eans in terms o

d (1, r) where

tant of Propor

ional Relations

f Proportional

onstant of proelationship w2c). In Lessoneal‐world cons (7.RP.2b). Trelationships he situation o

odeling Cycle Le

f Proportionality

ThisCre

s Curr

t of Pr

sent proportion

stant of propo

ams, and verb

ortional relatio

onal to the nu

tionship betwe

d as t = pn.

oint (x, y) on t

of the situatio

r is the unit ra

rtionality (P)1

ships with Equ

Relationships

oportionality with equationss 8 and 9, stuntext and relaTopic B concluas they interpor context of

esson, E‐Explora


riculu

roport

nal relationshi

rtionality (unit

bal descriptions

onships by equ

mber n of item

een the total co

the graph of a

n, with special

ate.

uations (P)

s (P)

by finding ths of the form udents deriveate the equatudes with stupret the meaa given prob

tion Lesson, S‐S


um

GR

tional

ips between q

t rate) in table

s of proportion

uations. For ex

ms purchased a

ost and the num

proportional r

l attention to t

e unit rate iny = kx, where

e the constanttion representudents consolnings of the plem (7.RP.2d

ocratic Lesson


RADE 7 • MODU

lity

uantities.

s, graphs,

nal

xample, if total

at a constant

mber of items

relationship

the points (0, 0

n the collectioe k is the t of ting the lidating their points (0,0) an).

57

License.

ULE 1

l

0)

on

nd

7

© 2013

NY


YS COMMON C

Topic B: Date:


CORE MATHEM

Unit Rat3/26/14

oncore.org

MATICS CURRIC

tes and the Consta4

CULUM

ant of Proportiona

ThisCre

ality



To


7•opic B

58

License.

•1

8

© 2013

NY

Stud

Less

In th

A ke

the u

intro

to m

as th

ratio

one

The

Clas

Exa

Begi

conc

scien

part

cons

cons

Ask s

Exa

Wi

Fo

of

co


YS COMMON C

Lesso

dent Outcom

Students

relations

situation

Students

son Notes

his lesson, stud

ey indicator of t

unit rate, a con

oductory exam

make sense of t

he unit rate of

o and rate lang

square mile.”

problem set m

sswork

ample 1 (20 m

n this lesson b

cerned that the

ntists found th

of the forest,

servationist co

servationists ne

students to att

ample 1: Nationa

ildlife conservatio

rest. The scientis

the forest, conser

unted 216 deer in

a.

b.

Lesson 7Date:


CORE MATHEM

on 7: Un

mes

s identify the s

ship as the con

n.

s find and inter

dents develop a

this understan

ncept already i

mple about dee

this problem. In

deer per squa

guage when na

may be used as

minutes): Nat

by presenting t

e deer populat

at there were

conservationis

unted 216 dee

eed to be worr

tempt Example

al Forest Deer Pop

onists are concern

ts found that ther

rvationists counte

n a 24 square mile

Why does it m

national forest

Have students g

overpopulation,

What is the po

See chart below

7: Unit Rat3/26/14

oncore.org

MATICS CURRIC

nit Rate

ame value rela

nstant of propo

rpret the const

an understand

ding is that stu

ntroduced. Th

r population; s

n it, students d

re mile. While

ming the unit

homework or

tional Forest

he following si

tion might not

144 deer in a 1

sts counted 11

er in a 24 squar

ried?

e 1 with a part

pulation in Danger

ed that the deer p

re were 144 deer

ed 117 deer in a 13

plot of the forest

atter if the deer p

t?

generate as many

, damage to land,

opulation density o

w.

te as the Constant 4

CULUM

as the C

ating the meas

ortionality and

tant of proport

ing of a “const

udents come to

e concept of “

significant time

develop a sense

the unit rate is

rate. For exam

supplementar

Deer Popula

ituation: Wildl

be constant ac

16 square mile

7 deer in a 13

re mile plot of

ner.

r?

population might

in a 16 square mil

3 square mile area

t. Do conservation

population is not c

y theories as possi

, etc.).

of deer per square

of Proportionality

ThisCre

Constant

sures of x and t

recognize it as

tionality within

tant of proport

o see the const

constant of pr

e should be tak

e of what the c

s the numerica

mple, “The unit

ry exercises for

tion in Dange

life conservatio

cross the Natio

e area of the fo

square mile ar

the forest. Do

not be constant a

le area of the fore

a. Yet a third con

nists need to be w

constant in a certa

ible (e.g., food sup

e mile?

y


t of Prop

the measures o

s the unit rate

n the contexts

tionality” and i

tant of propor

roportionality”

ken to ensure a

constant of pro

al part of the ra

rate is 9, beca

r targeted inst

er?

onists are

onal Forest. Th

orest. In anoth

rea. Yet a third

o

across the Nationa

est. In another pa

nservationist

worried?

ain area of the

pply,


Less

portiona

of y in a propo

in the context

of problems.

interpret its m

rtionality as syn

is developed t

all students ha

oportionality is

ate, encourage

ause there are

ruction.

he

her

d

al

art

Scaffol

Showgenercomp(reproprovilesso

Ask stwhat their


7•son 7

ality

rtional

of a given

eaning in cont

nonymous wit

through an

ave an opportu

s, and recogniz

e students to u

9 deer for eve

lding:

w, or ask studenrate, a visual toprehension of toducible cutouded at the endn for students

tudents to parthe problem isown words to

59

License.

•1

text.

h

unity

ze it

use

ry

nts to o aid in this situation uts are d of the to handle):

raphrase s asking in a partner.

9

© 2013

NY

Enco

cons

cons

equa

Allow

Base

sure

MP.1


YS COMMON C

ourage student

stant of propor

stant of propor

ation as the co

What is t

T

When w

know if t

T

We call th

So deer p

the case:

w for commen

We could

column?

between

through m

What is th

Constant

Meaning

(Could be

c.

d.

ed upon the dis

e to support yo

Lesson 7Date:


CORE MATHEM

ts to make a ch

rtionality. Stud

rtionality. Rem

onstant.

the unit rate in

The unit rate is

e look at the re

the relationshi

The square mil

Table:

Square M

16

13

24

his constant (o

er square mile

that the unit r

ts or observat

add the unit r

(Add this to ta

unit rate and c

more examples

he unit rate? How

of Proportionality

of Constant of Pro

e completed later

Use the unit ra

Use the unit ra

scussion of the

ur answer with

7: Unit Rat3/26/14

oncore.org

MATICS CURRIC

hart to organiz

dents have alre

member that th

n this example?

s 9 because the

elationship bet

p is proportion

es is always m

Miles Number

14

11

21

r same) value t

e is 9, and the c

ate and the co

ions but leave

rate to the tabl

able for studen

constant of pro

s.

w do you know?

y: k = 9

oportionality in th

after formalizing t

ate of deer per squ

ate to determine t

e questions abo

h mathematica


CULUM

e the data from

eady found uni

e constant of p

? How do you

ere are 9 deer p

tween square

nal?

ultiplied by the

of Deer

44

17

16

the “constant o

constant of pro

onstant of prop

a lingering que

le so that we h

nts to see). Do

oportionality?

The unit rate is 9,

his problem: Ther

this concept in a f

uare mile to deter

the number of squ

ove, answer th

al reasoning ab

of Proportionality

ThisCre

m the problem

t rate in earlie

proportionality

know?

per square mile

miles and num

e same value, 9

of proportiona

oportionality is

portionality are

estion for now

have 1 square m

es that help to

We will see if

9, because there ar

re are 9 deer for e

few more example

rmine how many d

uare miles in whic

he question: D

bout rate and u

y


m and then exp

r lessons but h

y is also like a s

e.

mber of deer in

9 in this case.

ality”.

s 9. Is that a co

e the same?

w.

mile in the first

o guide your de

your hypothes

re 9 deer for every

very 1 square mile

es)

deer are there for

ch you would find

Do conservation

unit rate.


Less

plicitly model fi

have not identi

scalar and will

n the table belo

oincidence or w

t column and 9

ecision about t

sis holds true a

y one square mile

e of forest.

r every 207 square

486 deer?

nists need to b

Scaffol

As anto deareasnumbcauseworryreaso


7•son 7

nding the

ified it as the

be used in an

ow, how do we

will that always

9 in the second

the relationship

as we move

e.

e miles.

be worried? Be

lding:

n extension, asktermine exams and accompabers of deer the conservationy, and explain toning.

60

License.

•1

e

s be

d

p

e

k students ples of nying at would ists to their

0

© 2013

NY

Revi

exam

cons

To inword


YS COMMON C

ew vocabulary

mple, “In the d

stant of propor

Vocab

A cons

A varia

If a pro

is a p

multip

that m

Note:

quanti

where

for the

ncrease accessd “variable.”

Lesson 7Date:


CORE MATHEM

y box with stud

eer example, t

rtionality is?”

bulary:

stant specifies a u

able is a letter tha

oportional relatio

positive constant,

plicative relationsh

make the equation

In a given situatio

ities. For example

e is the positive c

e given situation.

ibility, use Fray

7: Unit Rat3/26/14

oncore.org

MATICS CURRIC

dents. When po

the constant of

nique number.

at represents a nu

onship is described

, then is called t

hip between two

n true.

on, it would be rea

e, a set of ordered

constant, or the c

yer diagrams t


CULUM

ossible, use the

f proportionali

umber.

d by the set of ord

the constant of pr

quantities, and

asonable to assign

pairs , would

onstant of propor

o introduce an

of Proportionality

ThisCre

e first example

ity was 9. How

dered pairs that sa

oportionality. It i

. The , pai

n any variable as a

d be all the points

rtionality. This val

nd record new

y


e from this less

w would you ex

atisfies the equati

is the value that d

rs represent all th

a placeholder for

s that satisfy the e

lue for specifies

vocabulary wo


Less

son to illustrat

xplain to someo

ion , wher

describes the

he pairs of values

the given

equation s a unique numbe

ords. This is an


7•son 7

e the terms. Fo

one what a

e

, r

n example for t

6

License.

•1

or

the

1

© 2013

NY

Rem

the

rate

of E

Disc

Allow

stud

with

Exa


YS COMMON C

mind students

number of sq

e in deer per s

xample 1 is:

cussion

How are

T

w students to w

dents to work in

h Example 2.

ample 2 (18 m

Example 2

Brandon c

level. He

school on

Encourag

a.

b.

Lesson 7Date:


CORE MATHEM

s that in the e

quare miles co

square mile is

There are 9 d

the constant o

They are the sa

work with a pa

ndependently

minutes): You

2: You Need WHA

came home from

needed 3 cookies

n the very next day

e students to mak

Is the number

a table that sho

cookies needed

What is the un

The unit rate is

The constant o

Meaning of Co

It took 2 hours

they finish bak

96 students (3

If it takes 2 hou

8:00 pm.

# of cookie

sheets

2

4

10

16

7: Unit Rat3/26/14

oncore.org

MATICS CURRIC

example with

ould be defin

s , or 9. Th

deer for every

of proportiona

ame. They both

artner on Exam

on Example 3

u Need WHAT

AT???

school and inform

s for each of the 9

y! Brandon and h

ke a chart to orga

of cookies propor

ows data for the n

d.

it rate? How do y

s 18 because they

of proportionality

nstant of Proport

s to bake 8 sheets

king the cookies?

cookies each) = 28

= 16 shee

urs to bake 8 shee

# of cookies

baked

36

72

180

288


CULUM

the deer pop

ed as x, and t

he constant o

y 1 square mi

ality and the un

h represent the

mple 2; circulat

while offering

T???

med his mother th

96 students in 7th g

his mother determ

nize the data from

rtional to the num

number of sheets

ou know?

can bake 18 cook

is 18.

tionality in this pro

of cookies. If Bra

88 cookies

ets of cookies

ets, it will take 4 h

of Proportionality

ThisCre

pulation, we a

the number o

of proportiona

le of forest.

nit rate alike?

e same number

e to informally

targeted smal

hat he had volunte

grade. Unfortuna

mined that they ca

m the problem.

mber of sheets use

needed for the to

kies for every one

oblem: There are

andon and his mot

hours to bake 16 s

y


are looking fo

of deer could

ality, , is 9. T

r that is the va

y assess progre

ll group instruc

eered to make coo

ately, he needed t

an fit 36 cookies o

ed in baking? Crea

otal number of

cookie sheet.

e 18 cookies per 1

ther begin baking

sheets of cookies.


Less

or deer per sq

be defined a

The meaning

alue of the ratio

ess. Consider t

ction to studen

okies for his entir

he cookies for an

n two cookie shee

ate

sheet.

g at 4:00 pm, when

They will finish ba

Scaffold

Considexplain

For stuchallenproblerate is


7•son 7

quare mile, so

s , so the un

g in the contex

o of y to x.

hen instructing

nts that strugg

e grade

event at

ets.

n will

aking at

ding:

er using a visun the situation,

udents who neenge, have themm in which thea fraction.

62

License.

•1

o

nit

xt

g

led

al to , such as:

ed more m create a e constant

2

© 2013

NY

Exa

Clos

Ask s

Exit

Ensu

unde


YS COMMON C

ample 3: Fren

Example 3

Suzette a

certain am

determine

a.

b.

c.

sing Question

students to res

informally ass

What is a

C

How is th

T

Ticket (5 min

ure that studen

erstanding of t

Lesso

If a pro

a posit

Lesson 7Date:


CORE MATHEM

nch Class Coo

3: French Class Co

nd Margo want to

mount of flour, mi

e the number of e

Considering th

What does the

One egg is nee

How many egg

;

n (2 minutes)

spond to these

sess student un

another name

Constant of pro

he constant of

They are the sa

nutes)

nts complete th

the lesson.

on Summary

oportional relatio

tive constant, the

7: Unit Rat3/26/14

oncore.org

MATICS CURRIC

oking

ooking

o prepare crepes f

ilk, and 2 eggs. Th

eggs needed to ma

e amount of eggs

; The con

e constant or prop

eded to make c

gs will be needed f

Five eggs will be n

e questions, eit

nderstanding o

for the consta

oportionality

f proportionalit

ame.

he Exit Ticket i

nship is described

n is called the c


CULUM

for all of the stude

he girls know that

ake 50 crepes bec

necessary to mak

nstant of proportio

portionality mean

crepes.

for 50 crepes?

needed to make

ther by speakin

of the lesson.

ant that relates

ty related to th

ndependently,

d by the set of ord

onstant of propor

of Proportionality

ThisCre

ents in their Frenc

t they already hav

cause they are not

ke the crepes, wha

onality is .

in the context of

crepes.

ng to a neighbo

s the measures

he unit rate?

, in order to pr

dered pairs that sa

rtionality.

y


ch class. A recipe

ve plenty of flour a

t sure they have e

at is the constant

this problem?

or or in writing

s of two quant

rovide evidenc

atisfies the equati


Less

makes 20 crepes

and milk but need

enough eggs for th

of proportionalit

g. Use this as a

ities?

ce of each stud

ion , where

Scaffo

Ask stindep

Offer instrustrugg


7•son 7

with a

d to

he recipe.

ty?

an opportunity

ent’s

e is

olding:

tudents to worpendently on Extargeted smal

uction to studegled with Exam

63

License.

•1

y to

rk xample 3. l group nts that

mple 2.

3

© 2013

NY

Nam

Le

Exit

At th

the c


YS COMMON C

me _________

sson 7:

t Ticket

he same store,

constant of pro

Lesson 7Date:


CORE MATHEM

___________

Unit Ra

Susan paid $5

oportionality a

7: Unit Rat3/26/14

oncore.org

MATICS CURRIC

____________

te as the

5.10 for six bot

and explain its


CULUM

___________

e Consta

tles of water, w

meaning in thi

of Proportionality

ThisCre

____________

ant of Pr

while John paid

is context.

y


__

roportio

d $7.65 for nin


Less

Date_______

onality

ne bottles of w


7•son 7

___________

water. Determin

64

License.

•1

____

ne

4

© 2013

NY

Exit

The

Pro


YS COMMON C

t Ticket Sam

following solut

At the sa

Determi

The const

for each b

blem Set Sa

For each o

1. Bana

a.

b.

2. The

a.

b.

3. For

a.

b.

Lesson 7Date:


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

What is the co

K = 0.59

How much do

. $

dry cleaning fee f

What is the co

so K=6

How much will

6(11) = $66

every $5 that Mic

What is the co

so k=2.

If Micah saves

2(150) = $300

7: Unit Rat3/26/14

oncore.org

MATICS CURRIC

ns

an understand

n paid $5.10 for

of proportionalit

nality is 0.85. This

ons

roblems, define t

ound.

nstant of proporti

25 pounds of ban

.

for 3 pairs of pant

nstant of proporti

. The cost is $6 for

l the dry cleaner c

cah saves, his pare

nstant of proporti

. Micah’s parents

$150, how much


CULUM

ding of the obje

r six bottles of w

ty and explain it

represents the un

he constant of pro

ionality, k?

anas cost?

ts is $18.

ionality? Explain w

r every pair of pan

charge for 11 pairs

ents give him $10

ionality? Explain w

give him $2 for ea

money will his pa

of Proportionality

ThisCre

ectives of this l

water while John

s meaning in this

nit rate; that is, the

oportionality to a

what it means in c

nts that is dry clea

s of pants?

0.

what it means in c

ach $1 that he sav

arents give him?

y


lesson:

n paid $7.65 for

s context.

e unit rate is 0.85

answer the follow‐

context.

aned.

context.

ves.


Less

nine bottles of

5 because the cost

‐up question.


7•son 7

water.

t is $0.85

6

License.

•1

5

© 2013

NY


YS COMMON C

4. Each

scho

In 20

a.

b.

c.

d.

e.

Lesson 7Date:


CORE MATHEM

h school year, the

ool paid $1260 for

012, the school pa

Is the price the

zoo?

Explain why or

Because the ra

,

Identify the co

K = 15. This is

What would th

120(15) = $1,80

How many stud

,

#

7: Unit Rat3/26/14

oncore.org

MATICS CURRIC

7th graders who s

r 84 students to en

aid $1395 for 93 st

e school pays each

r why not.

atio of the entranc

nstant of proport

the price per stud

he school pay if 12

00

dents would ente

of students

84

70

93


CULUM

study Life Science

nter the zoo. In 20

tudents to enter t

h year in entrance

ce fee paid per stu

ionality and expla

dent; the price is $

20 students entere

er the zoo if the sc

Price

1,260

1,050

1,395

of Proportionality

ThisCre

participate in a sp

011, the school pa

the zoo.

fees proportiona

udent was the sam

ain what it means

$15 for each stude

ed the zoo?

hool paid $1,425?

,

,

,

y


pecial field trip to

aid $1050 for 70 st

al to the number o

me

s in the context of

ent.

?

YES


Less

o the city zoo. In 2

tudents to enter t

of students enteri

f this situation.


7•son 7

2010, the

the zoo.

ng the

66

License.

•1

6

© 2013

NY


YS COMMON C

Lesson 7Date:


CORE MATHEM

7: Unit Rat3/26/14

oncore.org

MATICS CURRIC


CULUM

of Proportionality

ThisCre

y



Less


7•son 7

67

License.

•1

7

© 2013

NY

Stud

Less

In thbe uto grpropdiagcont

The instr

Clas

Disc

Disc

Enco

relat

Stud


YS COMMON C

Lesso

Equa

dent Outcom

Students

contexts

son Notes

his lesson, studused to represeraphs. Throughportional relatirams. If studentext of the grap

problem set mruction.

sswork

cussion (5 min

Points to remem

Proportiona

The constan

Discussion Note

How could we u

uss important

ourage student

tionship by usi

If we know

given set

could we

constant o

rewrite th

dents should no

(may need

Lesson 8Date:


CORE MATHEM

on 8: Re

ations

mes

s use the const

s as they relate

dents continue ent propoprtiohout the lessononships in munts are ready, iph.

may be used as

nutes)

mber:

al relationships ha

nt ratio, or unit rat

es

use what we know

facts.

ts to begin to t

ng an equation

w that the con

of ordered pai

write this equa

of proportiona

his equation to

ote the followi

d to add this se

8: Represe3/26/14

oncore.org

MATICS CURRIC

epresent

tant of proport

e the equations

to develop thenal relationshin, look for oppltiple ways, incinclude discuss

homework or

ave a constant rat

te, can also be cal

w about the consta

think about ho

n by framing w

stant of propo

irs, and , th

ation? What if

ality, but do no

o solve for ?

ng in their ma

cond equation

enting Proportiona4

CULUM

ting Pro

tionality to rep

s to a correspo

eir understandps. Additionallortunities to hcluding tables, sion of the sign

supplementar

io, or unit rate.

lled the constant o

ant of proportiona

w we can mod

with the followi

ortionality, , t

en we can writ

f we know the

ot know the ‐v

terials: /n after Example

al Relationships wi

ThisCre

portiona

present propor

onding ratio tab

ding of how eqly, students arehave students rgraphs, equatnificance of (1,

ry exercises for

of proportionality

ality to write an e

del a proportio

ing probing qu

o be equal to

te / . H

‐values, and

values. Could

/ and eventu

e 1).

th Equations


al Relati

rtional relation

ble and/or gra

uations can e exposed represent ions, and r) in the

r targeted

y.

equation?

nal

estions:

/ for a

How else

the

we

ually


Less

ionships

nships by equat

phical represe

Scaffolding

Frame th

concrete

present

below an

problem

Give stu

these qu

understa

of

1. “What

propo

examp

2. “If re

square

the nu

we rep

using a

3. “Is the

possib


7•son 8

s with

tions in real wo

ntation.

g:

his discussion

e example. Fo

students with

nd recall the

m” from the la

dents time to

uestions to de

anding of the

/ and

t was the con

rtionality in t

ple?”

epresents the

e miles and

umber of dee

present this r

an equation?

ere more than

ble equation?

66

License.

•1

orld

n around a

or example,

h the table

“deer

ast lesson:

o think about

evelop their

e equivalence

:

nstant of

he deer

e number of

represents

r, how could

relationship

? Explain.”

n one

Explain.”

6

© 2013

NY

Opt

To b

one

alon

from

will t

askin

repr

abou

Exa

Allow

stud

Stud

MP.4


YS COMMON C

ional Activity

build conceptua

wall of the cla

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:


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


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


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

Encoura

represe

number


7•son 8

tudents up alo

ok from the tab

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

License.

•1

ong

ble

g it

r by

a variety of

ch as a double

7

e

© 2013

NY

Have

= m

mile

stud


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:


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


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


Less

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


7•son 8

lve the

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

License.

•1

of

ble.

e.

ratio

g a

8

© 2013

NY


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:


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


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


Less

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

Ex

te


7•son 8

the independe

28 for in the

mine either

s substituted in

d write the

olding:

d the problem

w an image o

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

d

© 2013

NY

Clos

Ask s

unde

Exit

Ensu

unde


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:


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 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.


Less

unity to inform

proportional?

value for , it

ortionality can

ndent variable c

ce of each stud

on , where

ionality expresses


7•son 8

mally assess

may be

be multiplied b

can be divided

ent’s

e is

s the

70

License.

•1

by

d by

0

© 2013

NY

Nam

Le

Eq

Exit

John


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:


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


Date____

ationshi

earned are giv

y are, determin

hey are, determ


Less

____________

ips with

ven for each pe

ne the unit rate

mine the unit r


7•son 8

______

erson.

e. If not, expla

ate. If not,

7

License.

•1

ain

1

© 2013

NY


YS COMMON C

c. Write an e

for each.

d. How muc

e. How long

Lesson 8Date:


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


n’s wages. Ide

o will earn mo


Less

ntify constant

re money?


7•son 8

of proportiona

72

License.

•1

ality

2

© 2013

NY

Exit

The


YS COMMON C

t Ticket Sam

following solut

John and

a.

b.

c.

d.

e.

Time

2

3

4

Lesson 8Date:


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


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 $


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. ;.


7•son 8

t, explain

not,

ould be

r will

mber in

nstant of

;

To find

73

License.

•1

3

© 2013

NY

Pro


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:


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


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


real world situatio

er can.

of laps she can run

es at the vegetab

der the number of


Less

on.

n per minute.

le stand.

f packages per dol


7•son 8

llar.

74

License.

•1

4

© 2013

NY


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:


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


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


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


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?


7•son 8

hat can

ular

0 for the

tical

0 for

ought 60

he flat

iend

o‐Rama

given

7

License.

•1

5

© 2013

NY


YS COMMON C

a.

b.

c.

d.

e.

Lesson 8Date:


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


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.


Less

of t‐shirts and tota

proportional since

sent?

e sure to support


7•son 8

al cost?

e the

your

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


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:


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

CULUM

ting Pro

onality to repr

to a correspon

o connecting r

y use graphing

udents have the

e of the point (

sted by the sca

supplementarwith students th

ariables. They

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

ThisCre

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


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

e abstract repre

student progr

n their groups.

eir neighbors

d help him earn

all of the

equation that

ber of

m to build any

onstructing


Less

ionships

ships by equati

exts with repre

ations and tabl

epresentations

ed until Lesson

ruction. Durings day’s Exit Tic

elp them unde

esentations wi

ress and discus

Scaffoldin

Read th

allow st

cooper

Use a s

situatio

can bui

hours…

As an e

to write

differen

such as


7•son 9

s with

ons in real wo

esentations by

es to

s, such as doub

n 10, students

g the opening cket.

erstand the

ill continuously

ss, along with t

ng:

he problem a

tudents to wo

ative groups.

impler, altern

on, such as, “I

ild 2 birdhous

…”

extension ask

e the equatio

nt equivalent

s / 5/7

76

License.

•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


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:


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


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

/ );

t a to solve the pr

/ minutes . t

a customer wil

left as non‐inte

must be round

his depends on

e real‐life situat


Less

pendent?

e hour, what is

ouse and part o

use, what is the

5/7of an hour

ong it will take

le and B to be

r / which is

e the

ve the problem.

. It will

roblem.

; to build

ll not want

eger

ded to

n the

tion before


7•son 9

the constant o

of a second

e constant of

to build one bi

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

35

42

raphing

rs for

to use. Ask

ketch the

en locate

pret the

on the

5/7).

7

© 2013

NY

Exa

Simi

stud

Circu

follo

Let s

Barb

depe

divid

varia


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:


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


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

14 21

$3.08 $4.62


Less

ollars/cents pe

r of corn not pa

heir unit rate o

for $ . . mber of ears of

esents cost

d y represents cost

d

227

2 $50.00

Scaffoldin

Use an

more re

such as,

more se

per doll

cents pe


7•son 9

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


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:


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

enting Proportiona4

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

ThisCre

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


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

alue is the dep

eled using this t

ependent varia

rovide evidenc


Less

e of the constant o

nd values are alw

the unknown usin

of and ,ho

as an opportu

, and what do

. You n

e of equation a

n unchanging

pendent)?

pendent variab

type of equatio

able because tim

ce of each stud


7•son 9

of

ways

g

ow would you

unity to inform

o you need to

need to know t

and explain wh

speed (constan

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


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:


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

enting Proportiona4

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


Date____

ationshi

oportional rela

th 93 miles. If

miles to kilomet

1.61 represent

aid that0.62re


Less

____________

ips with

tionship betwe

= number of

ters? Explain w

ts miles per km

epresents mile


7•son 9

______

een distance in

f kilometers an

why.

m.

es per km.

80

License.

•1

n

nd

0

© 2013

NY

Exit

The

Pro


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:


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

enting Proportiona4

CULUM

ding of the obje

at they felt repres

y in the table paire

ct equation that w

, and he said that

as her equation,

o be . by divid

he variables were

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.

nt variable and wh

he dependent varia

u have in your tan

al Relationships wi

ThisCre

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


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


Less

ip between distan

mber of kilomete

d why?

per km.

miles per km.

ichael used repres

know which value

mple, Maria shoul

the moon to weig

th.

?

you can drive dep


7•son 9

nce in

rs and

sents km

es are

ld have

gh

ends on

8

License.

•1

1

© 2013

NY


YS COMMON C

b.

c.

d.

e.

f.

g.

h.

i.

3. Sup

a.

b.

Lesson 9Date:


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

ThisCre

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


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?


Less

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


7•son 9

to

lied by 31

e?

alculated?

urs you

82

License.

•1

2

© 2013

NY


YS COMMON C

4. In m

a.

b.

c.

d.

e.

Lesson 9Date:


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

ThisCre

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


of gallons of gas u

he gallons of gas.

ould be needed to

nd work, how man


Less

sed.

o make the trip?

ny gallons of gas w


7•son 9

would

83

License.

•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


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

ample 1–2 (15

mple 1 is a revi

he students to

e students wor

ample 1

andma’s Special C

okies. Using this i

ble – Create a char

ur used to the am

Lesson 1Date:


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

ookie recipe, whic

plete the chart:

amount of

Tab

pro

flou

Yes

con

tha

of f

con

cor

coo

eting Graphs of Pro4

CULUM

ting Gra

nding of equati

f a proportion

0 .

erpret in conte

esentation of a

) on the graph

ticular, a key u

ship, indicate t

supplementarwith students th

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

oportional Relatio

ThisCre

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

/ such

of the cups

by the

sure of

Un

ra

in

1

1

c

fl

nships


Proporti

ting proportion

p mean in term

1, on the gra

relationship. S

onal relationshi

developed in t

Additionally, th

r targeted inston the previou

built upon this

r’s work with a

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


Lesso

onal

nal relationship

ms of the situati

aph of a propo

Students are fo

ip, though stud

this lesson is w

he importance

ruction. Durings day’s Exit Tic

s example. Pos

partner.

make 4 dozen

s the unit

he meaning

e problem?

ies (16

1 cup of


7•on 10

ps as they

ion or context

ortional

ormally

dents may hav

why this point w

e of (0,0) is

g the opening cket.

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

84

License.

•1

of

ve

will

ge

ncourage

o use a variety o

aitons, including

umber line:

e open‐ended

as, “Represent

m in two

ways and explain

models relate to

on.”

4

f

© 2013

NY


YS COMMON C

Graph –

Example 2

Below is a

Record th

Grandma

above.

1

How man

do we ca

Lesson 1Date:


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


CULUM

.

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

ThisCre

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


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


Lesso

on – Write an equ

an be used to repr

ationship.

/

. F

mber of Dozen Co

mber of cups of Fl

okies.

d pairs (values) rep

ot the point on the

d provided above


7•on 10

uation

resent

ookies

lour

present?

e graph

e? What

Scaffolding:

Provide gra

calculators

students to

fluidly betw

equations,

and graphs

8

License.

•1

aphing

s to allow

o move

ween

tables,

s.

5

© 2013

NY

Gene

and

Exe

Ask s


YS COMMON C

erate class disc

what it means

How is th

T

c

What qua

s

When you

T

What qua

T

When you

T

What is th

1

Starting a

to reach t

1

Why are w

T

n

Continue

1

Does this

quantities

T

t

Graphs of

proportio

T

v

ercises (15 mi

students to work

Exercises

1. The

Lesson 1Date:


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


CULUM

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

ThisCre

ad to the conc

ordinate of the

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


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


Lesso

nt 1, must

graph, which h

ould you have t

axis or value

d equal the unit

tically do you m

s be the case, w

nce will always

int must be on

ate describes th

f water.


7•on 10

be on the grap

has an ‐

to move vertica

e represents th

t rate of 1.5 un

move?

whenever two

s be the equal

n every graph o

he change in th

86

License.

•1

ph

ally

e

nits.

to

of a

he

6

© 2013

NY


YS COMMON C

a.

b.

c.

d.

e.

f.

2. You

Amu

a.

b.

c.

Lesson 1Date:


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


CULUM

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

ThisCre

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


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?


Lesso

portional to the n

ne that passes thro

er?

m.

.

on

ed

nt‐

ent Park? Explain

eople multiplied b

Scaffold

Consgrouinstrrepreprop

Exte1. Is it p

labe2. Can

the m

3. How

prob


7•on 10

number of

ough the

n why or

by the

ding:

sider selecting up for targetedruction on the esentation of aportional relatinsion questionpossible to swils on the andthe gallons de

minutes?

w would this ch

blem?

87

License.

•1

a small graphical a onship. ns: itch the d axis? pend on

ange the

7

© 2013

NY

Clos

Ask s

infor


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:


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

e form s through the o

proportional.

the meaning o

her?

on a graph t

to the ‐axis o


CULUM

relationship.

raph of the line if

this point in the c

he cost would be

ost would be $50

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

oportional Relatio

ThisCre

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

uation, or graph

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


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


Lesso

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

represents a n

e ‐axis or vert


7•on 10

each

ation?

total of

opportunity t

antities that a

‐value, or you

tion, you can us

e point 1, a

/ is equivalenional. In an

graph of a strai

nit rate, then th

hat are

number

tical unit.

88

License.

•1

o

re

u can

se

and

nt

ight

he

8

© 2013

NY

Exit

Ensu

unde


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:


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


CULUM

e unit rate, will al

nts the amount of

e is zero amount o

en as part of the se

line that passes t

ndependently,

oportional Relatio

ThisCre

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


ine representing t

e for every horizon

there will also be z

ven real‐world or

data points.

rovide evidenc


Lesso

two quantities th

ntal increase of 1

zero amount of th

mathematical

ce of each stud


7•on 10

at are

unit

he

ent’s

89

License.

•1

9

© 2013

NY

Nam

Le

Exit

Grea

the g


YS COMMON C

me _________

sson 10:

t Ticket

at Rapids Whit

graph of the re

Lesson 1Date:


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


CULUM

___________

Graphs o

afts for $125 p

points mean in

oportional Relatio

ThisCre

____________

of Propor

per hour. Expla

the context o

nships


__

rtional R

ain why the po

f the problem.


Lesso

Date_______

Relation

oint 0,0 and .


7•on 10

___________

ships

1,125 are on

90

License.

•1

____

n

0

© 2013

NY

Exit

The

Pro

The

grap

grap


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:


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


CULUM

ding of the obje

ts rafts for $125 pe

oints mean in the

must include the po

et the correspondi

it change on the h

would cost $0, an

ull understandi

ven the informa

and other point

nce (ft.) run by a J

resent in the cont

0 feet.

esent in the conte

conds.

proportional to th

hat passes throug

he distance ran by

r unit rate, is 58 a

h.

aph for the quant

represent in t

ou spent a total of

oportional Relatio

ThisCre

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


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?


Lesso

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

her?


7•on 10

on the

ph

ecause it

units. The

ost $125.

es, equations a

le, equation or

ace of k.

9

License.

•1

and

r

1

© 2013

NY


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:


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


CULUM

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

ThisCre

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


f

he

an

el

s. The table below

o make the mix.

uts to be

s that passes

eanuts

f the amount of ca


Lesso

w shows how man

5

12.5

andy multiplied by

Packages

of M&M’s

Ca

Pe

0

1

2

3

4


7•on 10

ny

y the

ans of

anuts

0

2

4

6

8

92

License.

•1

2

© 2013

NY

Nam

1.

2.


YS COMMON C

me

Josiah and Tiyear, Josiah wevery year wJosiah’s pay a

A recent studsent or receivby boys propusing a graph

Module Date:


CORE MATHEM

llery have newill be paid $with the compand Tillery’s p

dy claimed thved 7 text meportional to thh on the coor

1: Ratios a3/26/14

oncore.org

MATICS CURRIC

w jobs at Yum14 per hour aany, they wilpay rate prop

at in any giveessages. Is thhe number ofdinate plane.

nd Proportional R4

CULUM

mYum’s Ice Cand Tillery wil each be giveportional? Exp

en month, forhe relationshif text messag.

elationships

ThisCre

Mid‐M

ream Parlor. ll be paid $7 en a raise of $plain your rea

r every 5 text p between nues sent or rec


Module As

Date

Josiah is Tilleper hour. Th$2 per hour. asoning using

messages a bumber of textceived by girl


ssessment

ery’s manageey have beenIs the relatiog a table.

boy sent or ret messages ses? Explain yo


7• Task

r. In their firsn told that aftnship betwee

eceived, a girent or receiveour reasoning

93

License.

•1

st ter en

l ed g

3

© 2013

NY

3.


YS COMMON C

When a songrest. The grabrought in by

a. Identifybrought

b. Write ansong, x.

Module Date:


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

t of proportioof the song.

elating dollars

nd Proportional R4

CULUM

c store, the stch money a pc store from

onality betwe

s earned by th

elationships

ThisCre

Mid‐M

tore takes sompop singer masales of the s

en dollars ea

he pop singer


Module As

me of the moakes given thesong.

rned by the p

r, y, to dollars


ssessment

oney and the e total amoun

pop singer an

s brought in b


7• Task

singer gets thnt of money

d dollars

by sales of the

94

License.

•1

he

e

4

© 2013

NY


YS COMMON C

c. Accordifirst we

d. Describethe grap

e. Which pbrought

Module Date:


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

oint (0, 0) on

graph represes of the song

nd Proportional R4

CULUM

ationship, how$800 that we

the graph rep

ents the amouby the store?

elationships

ThisCre

Mid‐M

w much moneek?

presents in te

unt of money?


Module As

ney did the so

erms of the si

y the pop sing


ssessment

ong bring in fr

ituation being

ger gets for $


7• Task

rom sales in t

g described b

1 in money

9

License.

•1

he

by

5

© 2013

NY

A

AsTa

1

2


YS COMMON C

Progression

ssessment ask Item

1

7.RP.2a

2 7.RP.2a

Module Date:


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

Student answincorrectly. unable to givcomplete grawas unable tproportionalto the graph

1: Ratios a3/26/14

oncore.org

MATICS CURRIC

tery

r incorrect nd little of or n of tics to problem.

wered Student was mplete at rrect pairs of e table. unable to eason out r.

wered Student was ve a aph AND/OR to relate the l relationship .

nd Proportional R4

CULUM

STEP 2Missing or inanswer but evidence of sreasoning orapplication omathematicssolve the pro

Student may orhave answered relationship waproportional. Swas able to comleast two correcvalues in the taStudent providelimited expressireasoning.

Student may orhave answered relationship waproportional. Sprovided a grapmistakes (unlabincorrect pointsStudent providelimited expressireasoning.

elationships

ThisCre

Mid‐M

ncorrect

some r of s to oblem.

STA cwiof apmasooransuevreaapmaso

r may not that the s not Student mplete at ct pairs of ble. ed a ion of

Stuansrelprowaat entreacon

r may not that the s Student ph with beled axis, s). ed a ion of

Stuansrelprolabplomiexpinc


Module As

TEP 3correct answith some evidreasoning or

pplication of athematics toolve the problr an incorrectnswer with ubstantial vidence of solasoning or pplication of athematics toolve the probl

udent correctly swered that theationship was noportional. The as correctly set uleast two correctries. Student’s asoning may havntained a minor

udent correctly swered that theationship was oportional. Studbeled the axis ANotted points withnor error. Studeplanation was slcomplete.


ssessment

wer dence r

o lem, t

lid

o lem.

STEPA cosuppsubsevidereasoapplmathsolve

e ot table up with ct

ve r error.

Studeanswerelatiopropoprovidand vtwo oentriereasodemowas npropothe copropofor ea

e

dent ND h ent ightly

StudeanswerelatiopropocorrecAND pon theStudethe prrelatioconfirthat tstraigthrou


7• Task

P 4 rrect answerported by stantial ence of solid oning or ication of hematics to e the problem

ent correctly ered that the onship was not ortional. Studended correct set‐alues of table wor more correct es. Student ned AND onstrated that thno constant of ortionality or thaonstant of ortionality changach pair of value

ent correctly ered that the onship was ortional. Studenctly labeled the plotted the grape coordinate plaent explained tharoportional onship was rmed by the facthe graph was a ht line going gh the origin.

96

License.

•1

m.

nt up

with

here

at

ges s.

nt axis ph ane. at

t

6

© 2013

NY

3


YS COMMON C

3

a

7.RP.2b

b

7.RP.2c

c

7.RP.2d

d

7.RP.2d

e

7.RP.2d

Module Date:


CORE MATHEM

Student was

answer

work was sh

Student was write an equwrote an equwas not in th

or evfor any value

Student answincorrectly aor little undeanalyzing gra

Student was describe thecorrectly.

Student was identify eitheor y‐coordinapoint.

1: Ratios a3/26/14

oncore.org

MATICS CURRIC

unable to

AND no

own.

unable to uation or uation that he form ven e k.

wered nd shows no erstanding of aphs.

unable to situation

unable to er of the x‐ ate of the

nd Proportional R4

CULUM

Student was un

answer . C

of constant of proportionality incorrectly.

Student wrote aincorrect equat

as 5 , or

AND/OR used aincorrect valuepart (a) to writeequation in the

.

Student answerincorrectly, but some understananalyzing graphAND/OR solvingequations.

Student was abexplain that thethe dollar amoueither the singeearnings or salewas unable to dthe relationship

Student answerone of the ordevalues correctly

elationships

ThisCre

Mid‐M

able to

Concept

was used

Stu

ans

prosup

an ion, such

,

n of k from e the form

Stueqconprowrthesomeq

red shows nding of hs g

Stu$4,stuincstudetakeqbuterr

le to e zero was unt to ers’ es, but describe p.

Stusituwit

rs only ered pair y.

Stuidecoo

y‐c

doord


Module As

udent correctly

swered , b

ovided no work pport answer.

udent created anuation using thenstant of oportionality, buote the equatioe form 5 ome other equivauation.

udent answered,000 in sales, buudent’s work wacomplete OR theudent correctly monstrated the ken to solve the uation from part with a computror.

udent describes uation correctlyth minor error.

udent correctly entifies the x‐ordinate as 1 an

coordinate as ,

es not put it in adered pair form.


ssessment

but

to

Stude

answe

Studefree wanswe

n e

ut n in or alent

Stude

answe

ut as e

steps

rt (b), tational

Stude$4,00no errtakenanswe

the , but

Studeexplaireprethat zzero esinger

nd the

but

an .

Stude

answe


7• Task

ent correctly

ered .

ent provided errowork to supporter.

ent correctly

ered .

ent answered 00 in sales AND hrors in the steps to arrive at theer.

ent correctly ins that (0,0) sents the situatizero sales leads tearnings for the r.

ent correctly

ered (1, ).

97

License.

•1

or‐

had s

ion to

7

© 2013

NY


YS COMMON C

Module Date:


CORE MATHEM

1: Ratios a3/26/14

oncore.org

MATICS CURRIC

nd Proportional R4

CULUM

elationships

ThisCre

Mid‐M


Module As


ssessment


7• Task

98

License.

•1

8

© 2013

NY


YS COMMON C

Module Date:


CORE MATHEM

1: Ratios a3/26/14

oncore.org

MATICS CURRIC

nd Proportional R4

CULUM

elationships

ThisCre

Mid‐M


Module As


ssessment


7• Task

99

License.

•1

9

© 2013

NY


YS COMMON C

Module Date:


CORE MATHEM

1: Ratios a3/26/14

oncore.org

MATICS CURRIC

nd Proportional R4

CULUM

elationships

ThisCre

Mid‐M


Module As


ssessment


7• Task

10

License.

•1

00

N S Common Cor e 7 Mathem atics Curr iculum B: Unit Rat Lesson 7 Lessons 8 Lesson 1 ‐Module Ass cs...

Documents

Transcript of N S Common Cor e 7 Mathem atics Curr iculum B: Unit Rat Lesson 7 Lessons 8 Lesson 1 ‐Module Ass cs...