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Lesson 1

Lesson 2

Lesson 3

Lesson 4

Lesson 5

Lesson 6

Lesson 7

ic B: Functio

Lesson 8

Lessons 9

Lesson 1

Lesson 1

Lesson 1

Lesson 1

d‐Module Assics A through

ic C: Transfo

Lesson 1

Lesson 1

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Module 3: Date:

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Content

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: Integer Seq

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Lesson 2

Lesson 2

Lesson 2

Lesson 2

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CORE MATHEM

Module 3: Date:

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1: Comparin

2: Modeling

3: Newton’s

4: Piecewise

Assessment ah D (assessme

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Module 3: Date:


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CORE MATHEM

Module 3: Date:


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CORE MATHEM

Lesson 15: Date:


estions (h)–(j) a

n we create th

we create the g

hat make a true

value in the do

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n yields the sdomain values

paired with ex

ymbol and th

tion solved for

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absolute value

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ble equation, w

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number senten

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e, and then plo

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en we talk abo

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unctions and

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esian plane is iden

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mary of the follo

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exceot that ordered

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out the graph o

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ntical to the graph

on as a piecewi

y to write is as a

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owing

ally the same

plane and plot

when we grap

variable equat

ess of creating t

ept we run

d pair. Since e

s reason, we o

of a function o

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CORE MATHEM

Lesson 15: Date:


nutes)

ws students ho

graph of this fu

1

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ll need to deriv

graph of the linea

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as a piecewise fu

defined:

s this graph co

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you use your k

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translated the

where 5 an

nterpret in wo

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w to express a

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f the absolute v

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as a piecewise

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function.

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whole class. B

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real numb

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CORE MATHEM

Lesson 15: Date:


llenge 2 (8 m

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inutes)

two types of st

test integer fu

test integer fun

n defined the f

n mathematics

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real numbers i

d open at the r

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following table to

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nd a third func

eiling function,

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student has a c

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awtooth number o

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lated to them:

ooth function.

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lements of the do

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Mon 15

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: the floor

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ts why the inte

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onal part”

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CORE MATHEM

Lesson 15: Date:


For the floor, c

interval ,

Floor: , Cei

Floor: , , C

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spond to summ

use different e

se functions. T

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s a function from

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name the absolu

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name the floor fu

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name the ceiling

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ceiling, and sawto

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Ceiling: , Sawt

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marize key poin

expressions to

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Given a number

the union of the i

unctions on each

The absolute valu

e value function is

te value function

of a real number

ction such that for

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near function suc

function by sayin

awtooth function

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nts of the lesso

define a funct

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for all real numb

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are the same

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eal numbers .”

eger not greater th

the function is

ers .”

st integer not less

value of the funct

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ch that for each re

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for all real numb

rovide evidenc

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all real numbers

?

a neighbor.

f the domain. T

presented as p

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nction is defined b

ween 0 and on t

al number , the v

han . The floor f

.

than . The ceilin

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eal number , the

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bers then

ce of each stud

Mon 15

ALGEBRA


on the

These are calle

iecewise

ise‐linear

by

the

value of

function

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he right

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Graph A

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Lesson 15: Date:


___________

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ation.

Graph C

our reasoning.

e function who

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ose graph wou

Mon 15

ALGEBRA


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CORE MATHEM

Lesson 15: Date:


mple Solution

ch of these graph

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solution set show

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lain why the sawt

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ns

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n be represented b

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use each in the

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with exactly one

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between the ceilin

Mon 15

ALGEBRA


in the

ntical to

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nal part”

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YS COMMON C


4. The

a.

b.

5. Let

a.

b.

c.

d.

e.

CORE MATHEM

Lesson 15: Date:


following piecew

Graph this func

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Create a graph

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ALGEBRA


4 5 x

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YS COMMON C


6. Grap

a.

b.

c.

CORE MATHEM

Lesson 15: Date:


ph the following p

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piecewise function

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YS COMMON C


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CORE MATHEM

Lesson 15: Date:


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MATICS CURRIC

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YS COMMON C


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CORE MATHEM

Lesson 15: Date:


te a piecewise fun

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YS COMMON C


b.

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CORE MATHEM

Lesson 15: Date:


MATICS CURRIC

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CORE MATHEM

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n opportunity t

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CORE MATHEM

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MATICS CURRIC

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CORE MATHEM

For all 4 To see this, not

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MATICS CURRIC

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Mon 16

ALGEBRA

reater than zer| 9. The last

4 or only solutions

the graphs of

ly one point,

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intersection

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CORE MATHEM

Determine the

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MATICS CURRIC

intersection poin

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Mon 16

ALGEBRA

ghbor. Listen t

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CORE MATHEM

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MATICS CURRIC

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Mon 16

ALGEBRA

10

x

10 x

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ction,

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these exercise

misconception

intersection po

be estimated.

et using graphs

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Mon 16

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CORE MATHEM

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Mon 16

ALGEBRA

he form

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YS COMMON C

-10

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In each pr

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Lesson 16: Date:


CORE MATHEM

-5

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MATICS CURRIC

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Mon 16

ALGEBRA

ion set to

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YS COMMON C

6. The

His n

walk

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b.

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Lesson 16: Date:


CORE MATHEM

graph below show

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MATICS CURRIC

ws Glenn’s distanc

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CULUM

ce from home as

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coordinates of the

verify your answe

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Mon 16

ALGEBRA

street.

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YS COMMON C

c.

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Lesson 18: Date:


CORE MATHEM

A vertical scali

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MATICS CURRIC

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YS COMMON C

2. Let

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MATICS CURRIC

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Mon 18

ALGEBRA

an plane

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YS COMMON C

c.

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Lesson 18: Date:


CORE MATHEM

First, a reflectio

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MATICS CURRIC

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Lesson 19: Date:


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CORE MATHEM

Lesson 19: Date:


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The graph is a

graph of the equa

te a formula (in te

n draw the transfo

A translation

Four Interes3/25/14

oncore.org

MATICS CURRIC

is transla

vertical stretch of

horizontal shrink

vertical stretch of

ation is

rms of ) for the f

ormed graph of th

units left and u

sting Transformati

CULUM

ated right units.

f by a fa

of by a f

f by a fa

provided below.

function that is re

he function on the

units up.

ons of Functions

ThisCre

actor of and refle

factor of .

actor of and tran

For each of the fo

epresented by the

e same set of axes


lected about the

nslated down un

ollowing transform

e transformation o

s as the graph of


Lesso

‐axis.

nits.

mations of the gra

of the graph of

.


Mon 20

ALGEBRA

aph,

.

26

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YS COMMON C

b.

c.

4. Reex

the

prob

with

a.

Lesson 20: Date:


CORE MATHEM

A vertical stret

A horizontal sh

xamine your work

equations

blem, we investiga

h the piecewise‐lin

Write the func

.

Four Interes3/25/14

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MATICS CURRIC

tch by a scale facto

hrink by a scale fac

k on Example 2 an

and

ate whether it is p

near function di

tion in Example

. ,,

,

sting Transformati

CULUM

or of .

ctor of .

nd Exercises 3 and

could be graph

possible to determ

rectly.

e 2 as a piecewise‐

ons of Functions

ThisCre

d 4 from this lesso

hed with the help

mine the graphs of

‐linear function.


n. Parts (b) and (

of the strategic p

f and


Lesso

c) of Example 2 as

points found in (a)

by wo


Mon 20

ALGEBRA

sked how

. In this

rking

26

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YS COMMON C

b.

c.

d.

Lesson 20: Date:


CORE MATHEM

Let

the function

,

Let

the function

Compare the p

defining each p

Function : Ea

domains are th

Function : Ea

the formula for

in the domain o

Four Interes3/25/14

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MATICS CURRIC

. Use the grap

as a piecewise‐lin

,

,

. Use the grap

as a piecewise‐lin

. , ., .

, .

piecewise linear fu

piece change? If s

ach piece of the fo

he same.

ach piece of the fo

r . The length of

of .

sting Transformati

CULUM

ph you sketched in

near function.

ph you sketched in

near function.

..

.

unctions and t

so, how? Did the

ormula for is t

ormula for is fou

f each interval in t

ons of Functions

ThisCre

n Example 2, part

n Example 2, part

to the piecewise l

domains of each p

times the correspo

und by substituting

the domain of is


(b) of

(c) of

inear function .

piece change? If s

onding piece of th

g in for in th

s the length of t


Lesso

to write the form

to write the form

Did the expressio

so how?

e formula for . T

he corresponding p

the corresponding


Mon 20

ALGEBRA

mula for

mula for

ons

The

piece of

g interval

26

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Nam

1.


YS COMMON C

me

Given a. Describe b. Describe c. Sketch t

d. Use you

Explain e. Were yo

why not

Module 3: Date:


CORE MATHEM

| 2| 3

e how to obta

e how to obta

the graphs of

ur graphs to e

how you got

our estimatiot.

Linear and E3/26/14

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MATICS CURRIC

3 and

ain the graph

ain the graph

f and

estimate the s

your answer

ons you made

Exponential Functi

CULUM

| | 4.

h of from th

h of from th

on the sam

solutions to th| 2|

.

in part (d) co

ons

ThisCre

End‐of‐M

he graph of

e graph of

me coordinat

he equation:3 | |

orrect? If yes


Module As

Date

| | usi

| | usin

e plane.

4

s, explain how


sessment

ng transform

ng transform

w you know.


MTask

ALGEBRA I

ations.

ations.

If not explain

30

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2.


YS COMMON C

Let and b

a. Find

b. What is c. What is d. Evaluate e. Compar f. Is there

value ex g. Is there

exists.

Module 3: Date:


CORE MATHEM

be the functio

, 4 , and

the domain o

the range of

e 67

re and contra

a value of , xists.

a value of x

Linear and E3/26/14

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MATICS CURRIC

ons given by

d √3 .

of ?

?

67 .

st and . H

such that

such that

Exponential Functi

CULUM

an

How are they

5

ons

ThisCre

End‐of‐M

nd |

alike? How a

100 ? If s

50 ? If so, fin


Module As

|.

are they diffe

so, find . If n

nd . If not, e


sessment

erent?

not, explain w

explain why n


MTask

ALGEBRA I

why no such

no such value

30

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3.


YS COMMON C

A boy boughtank had doutetras at the since they bo

a. Create aof guppbought

b. How ma c. Create a

guppies d. Use gra

arrived

Module 3: Date:


CORE MATHEM

t6 guppies aubled. His gusame time. Tought the fish

a function ties at the begthe 6 guppie

any guppies w

an equation ts there will be

phs or tables at your estim

, months

, tetras

Linear and E3/26/14

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MATICS CURRIC

t the beginnippy populatioThe table beloh.

to model the ginning of eacs. What is a r

will there be o

that could be e 100 guppies

to approximmate.

08

Exponential Functi

CULUM

ng of the monon continuedow shows the

growth of thech month, anreasonable do

one year afte

solved to dets.

ate a solution

116

ons

ThisCre

End‐of‐M

nth. One mo to grow in the number of t

e boy’s guppyd n is the numomain for i

r he bought t

termine how

n to the equa

224


Module As

onth later the his same mantetras, , afte

y population,mber of montn this situatio

the 6 guppies

many month

tion from par

332


sessment

number of gnner. His sisteer months h

, where ths that haveon?

s?

hs after he bo

rt (c). Explain

2


MTask

ALGEBRA I

uppies in his er bought somhave passed

is the numbee passed since

ought the

n how you

30

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YS COMMON C

e. Create aof tetra

f. Compara compaover tim

g. Use gra

and tetr

Module 3: Date:


CORE MATHEM

a function, , s after mon

re the growtharison of the me.

phs to estimaras will be the

Linear and E3/26/14

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MATICS CURRIC

to model thenths have pas

h of the sisteraverage rate

ate the numbe same.

Exponential Functi

CULUM

e growth of thssed since she

’s tetra popuof change fo

ber of months

ons

ThisCre

End‐of‐M

he sister’s tete bought the t

lation to the r the function

s that will hav


Module As

ra populationtetras.

growth of thens that mode

ve passed whe


sessment

n, where

e guppy popuel each popula

en the popula


MTask

ALGEBRA I

is the numb

ulation. Incluation’s growt

ation of gupp

30

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YS COMMON C

h. Use graeven th

i. Write th

can be iguppies

Module 3: Date:


CORE MATHEM

phs or tables ough there w

he function dentified. Cirs per month.

Linear and E3/26/14

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MATICS CURRIC

to explain wwere more tet

in such arcle or underl

Exponential Functi

CULUM

hy the guppytras to start w

way that theine the expre

ons

ThisCre

End‐of‐M

y population wwith.

e percent incression represe


Module As

will eventually

rease in the nenting percen


sessment

y exceed the

number of gunt increase in


MTask

ALGEBRA I

tetra populat

ppies per mon number of

30

License.

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onth

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4.


YS COMMON C

Regard the scomposed ofon.

a. How ma

(

b. Describedark tria

c. Create a d. Suppose

triangle

the dark

Figure 0

Module 3: Date:


CORE MATHEM

olid dark equf three dark t

any dark trian

(Figure Num

# of dark tria

e in words hoangles in the

a function tha

e the area of in each figur

k triangles in

F

Linear and E3/26/14

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MATICS CURRIC

ilateral triangriangles, the

ngles are in ea

mber)

ngles)

ow, given the next figure.

at models this

the solid darkre form a sequ

the figure

Figure 1

Exponential Functi

CULUM

gle as Figure 0second figure

ach figure? M

number of d

s sequence. W

k triangle in Fuence. Creat

e in the sequ

Figu

ons

ThisCre

End‐of‐M

0. Then, the e is the one co

Make a table t

ark triangles

What is the d

Figure 0 is 1 ste an explicit

ence.

re 2


Module As

first figure inomposed of n

to show this d

in a figure, to

domain of this

quare meter.formula that

Figure 3


sessment

this sequencnine dark tria

data.

o determine t

s function?

. The areas ot gives the are

3


MTask

ALGEBRA I

ce is the one angles, and so

the number o

of one dark ea of just one

Figure 4

30

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e of

09

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YS COMMON C

e. The sum

case. Th

of all thas inc

f. Let figures.

is true f

Module 3: Date:


CORE MATHEM

m of the areas

he sum of the

e dark triangreases?

be the sum There is a re

or each posit

Linear and E3/26/14

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MATICS CURRIC

s of all the da

e areas of all t

les in the

of the perimeal number

tive whole nu

Exponential Functi

CULUM

rk triangles in

the dark trian

figure in the

meters of the aso that:

1

mber . Wha

ons

ThisCre

End‐of‐M

n Figure 0 is 1

ngles in Figure

sequence? I

all dark triang

1

at is the value


Module As

1 m2; there is

e 1 is m2. W

Is this total ar

gles in the

e of ?


sessment

only one tria

What is the su

rea increasing

figure in the


MTask

ALGEBRA I

angle in this

um of the are

g or decreasin

e sequence o

31

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f

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5.


YS COMMON C

The graph of

a. Create a

segmen

b. Sketch t

Module 3: Date:


CORE MATHEM

f a piecewise f

an algebraic rnts.

the graph of

Linear and E3/26/14

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MATICS CURRIC

function is

representatio

2 an

Exponential Functi

CULUM

shown to the

on for . Assu

nd state the d

ons

ThisCre

End‐of‐M

e right. The d

ume that the

domain and ra


Module As

domain of is

graph of is

ange.


sessment

s 3 3

composed of


MTask

ALGEBRA I

3.

f straight line

31

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c. Sketch t

d. How do e. How do

Module 3: Date:


CORE MATHEM

the graph of

oes the range

oes the domai

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MATICS CURRIC

2 an

of

in of

Exponential Functi

CULUM

nd state the d

compare to t

compare to

ons

ThisCre

End‐of‐M

domain and ra

the range of

o the domain


Module As

ange.

, w

of


sessment

where 1?

, where


MTask

ALGEBRA I

1?

31

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A

AsTa

1

2


YS COMMON C

Progression

ssessment ask Item

1 a–b

F‐BF.B.3

c–e

A‐REI.D.1F‐BF.B.3

2 a

F‐IF.A.2

Module 3: Date:


CORE MATHEM

Toward Mas

STEP 1 Missing oincorrectand littleevidencereasoningapplicatiomathemasolve the

3

Student ansmissing or eincorrect.

1 3

Student cresketches thresemble avalue functOR Student is uuse the graestimate thto the equaStudent manot have arcorrect soluthe equatioanother meas trial and

Student procorrect ans

Linear and E3/26/14

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MATICS CURRIC

tery

or answer of g or on of atics to e problem

SMaerams

swer is entirely

Stc

eates hat do not bsolute ions.

unable to phs to he solutions ation. ay or may rrived at utions of on via ethod such error.

Stofinsttscstinwp

ovides no wers.

So

Exponential Functi

CULUM

STEP 2Missing or incanswer but evidence of soreasoning or application ofmathematics solve the prob

Student describeransformations correctly.

Student creates shat resemble thof an absolute vaunction but are naccurate. Studhows evidence ohe intersection he graphs to finolution but is unconfirm his or heolution points; herefore, the con part (e) is incowith the intersecpoints.

Student providesone correct answ

ons

ThisCre

End‐of‐M

correct

ome

f to blem

STEA cowitof rappmasolvor aanssubevidreaappmasolv

es the partially

Studtrancorrbe somivoca

sketches e graph alue

dent of using point of d the nable to er

onclusion onsistent ction

StudthatmorerroevidintefindequconcconsestimmayStudis clemorvocadeta

s only wer.

Studcorr


Module As

EP 3orrect answeth some evidereasoning or plication of thematics tove the problean incorrect swer with bstantial dence of soliasoning or plication of thematics tove the proble

dent describes tnsformations rectly, but there some minor misussion of appropabulary.

dent creates sket are accurate wre than one minoor; the student sdence of using thrsection points t the solutions toation. The clusion in part (esistent with the mated solutionsy have one errordent communicaear but could inre appropriate uabulary or moreail.

dent provides twrect answers.


sessment

er ence

o em

d

o em

STEP 4A corrsupposubstevidereasoapplicmathesolve

he

may use or riate

Studentransfoand coapprop

etches with no or hows he to o the

e) is

s but r. ation clude use of e

Studenthat arsolutiomatch of the ipoints.explanreflectsundersprocessolving

The wo(e) supconclusestimanot solincludeexplanapprop

wo Studenansweritems.


MTask

ALGEBRA I

4 rect answer orted by antial nce of solid

oning or cation of ematics to the problem

nt describes the ormations clearlyrrectly AND usepriate vocabulary

nt creates sketche accurate and ns in part (d) the ‐coordinatintersection The student’s ation for part (ds an standing that thes is analogous tog the system

and ork shown in parpports his or her sion that tes were or werutions and es supporting ation using priate vocabulary

nt provides corrers for all three

31

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y s y.

hes

tes

d)

e o

. rt

re

y.

ect

13

© 2014

NY

3


YS COMMON C

b–c

F‐IF.A.1

d

F‐IF.A.2

e

F‐IF.A.1 F‐IF.A.2 F‐IF.C.7a

f

F‐IF.A.1 F‐IF.A.2

g

F‐IF.A.1 F‐IF.A.2

3 a

A‐CED.A.F‐BF.A.1aF‐IF.B.5

Module 3: Date:


CORE MATHEM

Neither domrange is cor

Student mamajor erroromission inthe expressdoesn’t subinto or )

a

Student mano attemptcompare thfunctions.

Student proincorrect coOR Student mano attempt

Student proincorrect coOR Student mano attempt

1 a

Student doprovide an exponentiaOR Student proexponentiathat does nthe data, andomain is inomitted.

Linear and E3/26/14

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MATICS CURRIC

main nor rrect.

Oidttrfodfo

akes a r or n evaluating sion (e.g., bstitute ‐67 )

Smt

akes little or t to he two

Sdsfidponbfeinta

ovides an onclusion.

akes little or t to answer.

Stpse

ovides an onclusion.

akes little or t to answer.

S

pse

es not

l function.

ovides an l function

not model nd the ncorrect or

ScfisOSetdidt

Exponential Functi

CULUM

One of the two isdentified correcthe student has rhe ideas, giving ange of when or domain of , domain of wheor the range of

Student makes omore errors in evhe expression.

Student comparidoes not note thimilarity of the tunctions yieldindentical outputspositive inputs aopposite outputsnegative inputs; be limited to supeatures, such asnvolves squaringhe other containabsolute value.

Student identifiehere is no solutiprovides little or upporting work explanation.

Student identifie5 is a solutio

provides little or upporting work explanation.

Student providescorrect exponentunction, but thes incorrect or omOR Student providesexponential funchat does not modata but correctldentifies the domhis situation.

ons

ThisCre

End‐of‐M

s tly, or reversed the asked and the en asked .

Bothare mayerro

one or valuating

Studexprworanswtherpres

son e two g s for nd s for it may perficial s one g and ns an

Studthe equposimayarticfuncwhe

es that ion but no or

Studtherprovbut limitinco

es that on but no or

Stud

provbut limitinco

s a tial e domain mitted.

s an ction odel the y main in

Studminan ethata dositua


Module As

h domain and racorrect but notay contain minor ors.

dent evaluates tression correctlyrk to support thewer is limited, ore is one minor esent.

dent recognizes two functions aal for 0 anditive ‐values buy not clearly culate that the tctions are opposen is negative.

dent identifies thre is no solution vides an explanathe explanationted or contains onsistencies or e

dent identifies th5 is a solution avides an explanathe explanationted or contains onsistencies/ err

dent has made oor errors in provexponential funct models the datomain that fits thation.


sessment

ange ation

Both dare corapprop

he y but e r error

Studenexpresshows suppor

that re d ut

two sites

Studenthat thyield idfor posfor an iand opnegativ

hat and ation, n is minor errors.

Studenthere isprovideand/orsuppor

hat and ation, n is minor rors.

Studen5 i

provideand/orsuppor

only viding ction ta and he

Studencorrectfunctiothe domsituatio


MTask

ALGEBRA I

omain and rangrrect and use priate notation.

nt evaluates the sion correctly anthe work to rt the answer.

nt clearly describe two functions dentical outputs sitive inputs andinput of 0, pposite outputs fve inputs.

nt identifies thats no solution anes an explanatior work that clearrts valid reasonin

nt identifies thatis a solution andes an explanatior work that clearrts valid reasonin

nt provides a t exponential on and identifiesmain to fit the on.

31

License.

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bes

for

t d on rly ng.

t d on rly ng.

s

14

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YS COMMON C

b

F‐IF.A.2

c

F‐BF.A.1a

d

F‐IF.A.2

e

F‐BF.A.1a

f

A‐CED.A.2F‐IF.B.6 F‐LE.A.3

g

A‐REI.D.1F‐IF.A.2 F‐IF.C.9

Module 3: Date:


CORE MATHEM

Student givincorrect anno supportcalculations

a

Student proequation orequation thnot demonunderstandis required the problemdescribed.

Student proequation ordoes not recorrect dataOR Student faiprovide an graph.

a

Student doprovide a lifunction.

2

Student dodemonstratto recognizdistinguish linear and egrowth or tgrowth rateaverage ratof functions

11

Student doprovide corof the functunable to panswer thaon reasonin

Linear and E3/26/14

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MATICS CURRIC

ves an nswer with ing s.

Sinatf

ovides no r gives an hat does strate ding of what to solve m

Sindurp

ovides an r graph that eflect the a.

ls to equation or

Scbqg

es not near

Sfd

es not te an ability e and between exponential to compare es or te of change s.

Sccrinaa

es not rrect graphs tions and is provide an t is based ng.

SgaaOSinbas

Exponential Functi

CULUM

Student gives an ncorrect answeranswer is supporhe student’s funrom part (a).

Student sets up ancorrect equatiodemonstrates limunderstanding ofequired to solveproblem describe

Student providescorrect graph or but the answer tquestion is eithegiven or incorrec

Student providesunction that is lidoes not reflect d

Student makes acorrect but incomcomparison of grates that does nnclude or incorrapplies the conceaverage rate of c

Student providesgraphs but is unaarrive at a correcanswer from theOR Student’s graphsncomplete or inbut the student aat an answer basound reasoning

ons

ThisCre

End‐of‐M

r, but the rted with nction

StudcalcarrivStudsupp

an on that mited f what is e the ed.

StudcorrsimpincoOR Studerrogivesubsundrequprob

s a table, o the r not ct.

Studcorrbut monexplmar4th m

s a inear but data.

Studcorrbut simpdoenota

partially mplete rowth not ectly ept of change.

StudcomrateanalchanHowcomminof m

s correct able to ct graphs.

s are correct, arrives sed on .

Studthatimptheranswsuppgrap


Module As

dent has a minoulation error in ving at the answdent provides porting work.

dent provides a rect answer but plifies it into an orrect equation.

dent has a minoor in the equatioen but demonstrstantial erstanding of wuired to solve thblem.

dent provides a rect table or grathe answer is 4 nths with an lanation that therk occurs during month.

dent provides a rect linear functithe function is eplified incorrects not use the ation, .

dent makes a comparison of growes that includes alysis of the rate nge of each funcwever, student’smmunication conor errors or mismathematical ter

dent provides grt contain minor recisions and refore arrives atwer that is portable by the phs but incorrec


sessment

r

wer.

Studencorrectproper

then

r on rates

what is he

Studencorrectdemonundersrequireproble

ph,

e 100 the

StudencorrectAND thcorrecta valid

ion, either ly or

Studencorrectusing t

orrect wth an of ction. s ntains use rms.

Studenthe Guwill incrate anexplanan anachange

raphs

t an

ct.

Studengraphsanswersupporgraphs


MTask

ALGEBRA I

nt provides a t answer with r supporting wor

nt provides a t equation that nstrates standing of whated to solve the m.

nt provides a t table or graph,he answer is t (5 months) witexplanation.

nt provides a t linear function he notation,

nt identifies thatppies’ populatiocrease at a fastend provides a vaation that includlysis of the rate e of each functio

nt provides corre and arrives at ar that is rtable by the and correct.

31

License.

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t is

h

.

t on r lid des of on.

ect an

15

© 2014

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4


YS COMMON C

h

F‐IF.B.6 F‐LE.A.1F‐LE.A.3

i

A‐SSE.B.3

4 a–c

F‐BF.A.1aF‐IF.A.3 F‐LE.A.1F‐LE.A.2

d

F‐BF.A.1aF‐LE.A.1F‐LE.A.2

e

F‐BF.A.1aF‐LE.A.1F‐LE.A.2

f

F‐BF.A.1a

Module 3: Date:


CORE MATHEM

Student doprovide tabgraphs thataccurate ensupport an and shows reasoning iexplanation

3c

Student doprovide an exponentiathat shows increase.

a

Student dothe table codoes not derelationshipStudent doprovide an exponentia

a

Student faiprovide an exponentia

a

Student faiprovide an exponentia

a

Student proor no evideunderstand

Linear and E3/26/14

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MATICS CURRIC

es not bles or t are nough to answer little n an n.

Soclie

es not

l function percent

Setvfa2

es not fill in orrectly and escribe the p correctly. es not

l function.

StdcinShd

ls to explicit l formula.

Seeinwrrp

ls to explicit l formula.

Seeinwrrp

ovides little ence of ding how to

Sinn

Exponential Functi

CULUM

Student providesor graphs that arcorrect but proviimited or incorreexplanation of re

Student writes aexponential funchat uses an incoversion of the groactor, such as 0.20%, or 0.20.

Student completable correctly andescribes the seqcorrectly but givencorrect functioStudent may or mhave given a corrdomain.

Student providesexplicit formula texponential but ncorrect; suppowork is missing oeflects limited easoning about problem.

Student providesexplicit formula texponential but ncorrect; suppowork is missing oeflects limited easoning about problem.

Student identifiencorrect value onot provided, bu

ons

ThisCre

End‐of‐M

s tables re ides ect esults.

Studor gcorrexplpredbut errothe

n ction orrect owth .02, 2%,

Studversusinexprexprnoteto 2minor innot

tes the nd quence es an n. may not rect

StudtabldesccorrerrofuncfuncexpogrowDescmayerro

s an that is

rting or

the

StudcorrexpoNotaworerro

s an that is

rting or

the

StudcorrexpoNotaworerro

es the of or is t

Studsigntow


Module As

dent provides tagraphs that are rect and gives anlanation that is dominantly corrcontains minor ors or omissions explanation.

dent creates a cosion of the functng a growth factoressed as 200% ressed as 2 withe that 2 is equiva00%. Student hor error in notatn the domain or specify the dom

dent completes e correctly and cribes the sequerectly but has a mor in either his oction or domain.ction provided isonential with a wth factor of 3. cription or notaty contain minor ors.

dent provides a rect explicit onential formulaation or supportrk may contain mors.

dent provides a rect explicit onential formulaation or supportrk may contain mors.

dent solution shnificant progress wards identifying


sessment

ables

n

ect

in

Studenor grapcorrectcomplethat usvocabu

orrect tion or or h a alent has a tion does

main.

Studenversionusing aexpresexpresnote thto 200%specifiecorrect

the

ence minor r her . The s

tion

Studentable cthe seqand proexponeincludiof the duses prand pro(eithersubscrithe fun

a. ting minor

Studencorrectexponeusing fsubscriformulwork a

a. ting minor

Studencorrectexponeusing fsubscriformulwork a

ows that

Studencorrectenough


MTask

ALGEBRA I

nt provides tablephs that are t and gives a ete explanation ses mathematicaulary correctly.

nt creates a corren of the functiona growth factor sed as 200% or sed as 2 with a hat 2 is equivale%. Student es the domain tly.

nt completes theorrectly, describquence correctlyovides a correctential function ng the declaratidomain. Studenrecise language oper notation function or ipt notation) fornction.

nt provides a t explicit ential formula unction or ipt notation; a and supportinre free of errors

nt provides a t explicit ential formula unction or ipt notation; a and supportinre free of errors

nt identifies the t value of withh supporting

31

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M3

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al

ect n

nt

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on nt

r

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g s.

h

16

© 2014

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5


YS COMMON C

F‐LE.A.1

5 a

F‐BF.A.1a

b–c

F‐BF.B.3F‐IF.A.1

d–e

F‐BF.B.3F‐IF.A.1

Module 3: Date:


CORE MATHEM

determine tperimeter otriangles norecognize tcommon fabetween twsuccessive fperimeter.

a

Student doprovide a pdefinition ofunction anthan two exin the answincorrect.

3

Student prographs thatmajor errorand range aor are inconwith the gra

3

Both explansolutions aror have maconceptual (e.g., confudomain and

Linear and E3/26/14

oncore.org

MATICS CURRIC

the of the dark or how to he actor wo figures’

sudo

es not iecewise of the nd/or more xpressions wer are

Spweteo

ovides t contain rs; domain are missing nsistent aphs.

SfocvOSpMt

nations and re incorrect jor errors sing d range).

SmeOSoc

Exponential Functi

CULUM

olution shows sunderstanding ofdetermine the peof the dark triang

Student providespiecewise functiowhich at least onexpressions is cohe solution mayerrors with the inor notation.

Student providesor (b) that wouldcorrect for (c) anversa. OR Student answerspart (b) or (c) corMinor errors mayhe domain and

Student answersmore than one merror. OR Student answersone of part (d) ancorrectly.

ons

ThisCre

End‐of‐M

ome f how to erimeter gles.

is

errocomOR Studincoa miothea waeiththatis a geomby aprob

s a on in ne of the orrect; y contain ntervals

Studpieccorrthe minintefuncOR Oneincoand nota

s a graph d be d vice

s either rrectly. y exist in range.

Studthaterrorangwith

s contain minor

s only nd (e)

Studexpldomit merro


Module As

but contains m

ors or is not mplete.

dent computes aorrect value duinor error but erwise demonstay to determineer by recognizint the given equarecursive form ometric sequenceapproaching the blem algebraica

dent provides a cewise function wrect expressionsanswer may conor errors with thrvals or use of ction notation.

e expression is orrect, but intervuse of function ation is correct.

dent provides grt contain one mior. The domain age are consistenh the graphs.

dent answer onllains how the main/range chanay contain one mor.


sessment

minor

an ue to

rates e ng tion of a e or

lly.

evidenthinkingraph, diagramthat shshe arrsolutio

with , but ntain he

vals

Studencorrectpiecewcorrect

raphs inor and nt

Studengraphsand (c)domaineach thwith st

y

nges; minor

Studenexplaindomainbut alsknowinfindingrange.


MTask

ALGEBRA I

ce of student ng (correct table,marking on m, or calculationhows how he or rived at the n.

nt provides a tly defined wise function witt intervals.

nt provides corre for both parts ( and provides a n and range for hat are consistentudent graphs.

nt answer not onns how the n/range changeso explains how ng 1 aids in g the new domai

31

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M3

I

,

ns)

th

ect b)

nt

nly

s

in/

17

© 2014

NY

Nam

1.


YS COMMON C

me

Given a. Describe b. Describe c. Sketch t

d. Use you

Explain

e. Were yo

not.

To obthis g

To obunits

Solution: The solutio

Let Is |2YesYes

Module 3: Date:


CORE MATHEM

| 2| 3

e how to obta

e how to obta

the graphs of

ur graphs to e

how you got

our estimatio

btain the ggraph up 4

btain the gs down.

x ≈ 2.5 orons are the

x = 2.5 2.5 + 2|-3, 4.5 – 3 , the estim

Linear and E3/26/14

oncore.org

MATICS CURRIC

3 and

ain the graph

ain the graph

f and

estimate the s

your answer

ons you made

graph of g,4 units.

graph of h,

r x ≈-4.5 e x-coordin

3 = -|2.5| + = -2.5 + 4

mates are c

Exponential Functi

CULUM

| | 4.

h of from th

h of from th

on the sam

solutions to th| 2|

.

in part (d) co

, reflect th

, translate

nates of th

+ 4 true? 4 is true. orrect. Th

ons

ThisCre

End‐of‐M

he graph of

e graph of

me coordinat

he equation:3 | | 4

orrect? If yes

e graph of

the graph

he intersect

Let Is |- Yes,

hey each m


Module As

Date

| | usi

| | usin

e plane.

4

s, how do you

f a about th

h of a 2 un

tion points

x =-4.5 -4.5 + 2| -, 2.5 – 3 =

make the eq


sessment

ng transform

ng transform

u know? If no

he x-axis a

its to the l

s of the gra

- 3 = - |4.5= -4.5 + 4quation tru


MTask

ALGEBRA I

ations.

ations.

ot explain why

and transla

left and 3

aphs of g a

5| + 4 true4 is true. ue.

31

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y

ate

and h.

e?

18

© 2014

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2.


YS COMMON C

Let and b

a. Find

b. What is c. What is d. Evaluate e. Compar f. Is there

value ex g. Is there

exists.

f(1/3

D:

R: a

(-6

Whef givopp

No,posno

Yes

Module 3: Date:


CORE MATHEM

be the functio

, 4 , and

the domain o

the range of

e 67

re and contra

a value of , xists.

a value of x

3) = 1/9,

all real num

all real num

7)2 + -67|

en x is posves the alw

posite of wh

, f and g aritive sum, way to get

, if x = 5,

Linear and E3/26/14

oncore.org

MATICS CURRIC

ons given by

d √3 .

of ?

?

67 .

st and . H

such that

such that

g(4) = 1

mbers.

mbers.

-67| = 0.

itive, both ways positivhat f gives.

re either bor the oppt a negativ

f(x) = g(x)

Exponential Functi

CULUM

an

How are they

5

16, g(-√

functions gve value of

both zero, gposite of eave sum.

= 25, thus

ons

ThisCre

End‐of‐M

nd |

alike? How a

100 ? If s

50 ? If so, fin

√3) = -3

give the saf x2, where

giving a sumach other, g

s f(x) + g(x


Module As

|.

are they diffe

so, find . If n

nd . If not, e

ame value. as g gives a

m of zero, giving a su

x) = 50.


sessment

erent?

not, explain w

explain why n

But whena value tha

both positum of zero.


MTask

ALGEBRA I

why no such

no such value

n x is negatat is the

tive, giving So, ther

31

License.

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e

tive,

g a re is

19

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3.


YS COMMON C

A boy boughtank had doutetras at the since they bo

a. Create aof guppbought

b. How ma c. Create a

guppies d. Use gra

arrived

g(n)

g(12

100

Module 3: Date:


CORE MATHEM

t 6 guppies aubled. His gusame time. Tought the fish

a function ties at the begthe 6 guppie

any guppies w

an equation ts there will be

phs or tables at your estim

, months

, tetras

= 6•2n D

2) = 6•212 =

= 6•2n

Linear and E3/26/14

oncore.org

MATICS CURRIC

t the beginnippy populatioThe table beloh.

to model the ginning of eacs. What is a r

will there be o

that could be e 100guppies

to approximmate.

0

8

Domain: n

= 24,576

Exponential Functi

CULUM

ng of the moon continuedow shows the

growth of thech month andreasonable do

one year afte

solved to dets.

ate a solution

1

16

is a whole

guppies

ons

ThisCre

End‐of‐M

nth. One mo to grow in the number of t

e boy’s guppyd is the numomain for i

r he bought t

termine how

n to the equa

2

24

number.


Module As

onth later the his same mantetras, , afte

y population,mber of montn this situatio

the 6 guppies

many month

tion from par

3

32


sessment

number of gnner. His sisteer months h

, where ths that have on?

s?

hs after he bo

rt (c). Explain

2


MTask

ALGEBRA I

uppies in his er bought somhave passed

is the numbepassed since

ought the

n how you

32

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er he

20

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YS COMMON C

e. Create aof tetra

f. Compara compaover tim

g. Use gra

and tetr

The monThe gupp

t(nOr,

Module 3: Date:


CORE MATHEM

a function, t, s after mon

re the growtharison of the me.

phs to estimaras will be the

guppies’ pnth, the nu rate of chpies is alwa

n) = 8(n+1), t(n) = 8n

Linear and E3/26/14

oncore.org

MATICS CURRIC

to model thenths have pas

h of the sisteraverage rate

ate the numbe same.

population umber of ghange for tays increas

), n is a wn + 8, n is

Exponential Functi

CULUM

growth of thssed since she

’s tetra popuof change fo

ber of months

is increasinguppies douhe tetras issing.

whole numb a whole n

ons

ThisCre

End‐of‐M

he sister’s tetre bought the t

lation to the r the function

s that will hav

ng faster thubles, whileis constant

ber. number.


Module As

ra populationtetras.

growth of thens that mode

ve passed whe

han the tee the numbt, but the r


sessment

n, where

e guppy popuel each popula

en the popula

etras’ popuber of tetrarate of cha


MTask

ALGEBRA I

is the numb

ulation. Incluation’s growt

ation of gupp

lation. Eaca’s increasenge for the

32

License.

M3

I

er

ude h

pies

ch es by 8. e

21

© 2014

NY


YS COMMON C

h. Use graeven th

i. Write th

can be iguppies

n g(n) t(n)

The gugrowtguppieshows

g(n) =

Module 3: Date:


CORE MATHEM

phs or tables ough there w

he function dentified. Cirs per month.

0 1 6 12 8 16

uppy poputh is linear.es eventual that by th

= 6(200%)n

Linear and E3/26/14

oncore.org

MATICS CURRIC

to explain wwere more tet

in such arcle or underl

2 24 4 24 3

lation’s gro. The graplly overtakhe end of t

n

Exponential Functi

CULUM

hy the guppytras to start w

way that theine the expre

3 4 48 96 32 40

owth is expph in part kes the popthe 3rd mo

ons

ThisCre

End‐of‐M

y population wwith.

e percent incression represe

5 192 The48 The

ponential, a(g) shows hulation of nth, there


Module As

will eventually

rease in the nenting percen

e average re rate of ch

and the tehow the pothe tetras. are more


sessment

y exceed the

number of gunt increase in

rate of chahange is co

etra populaopulation o. The table guppies th


MTask

ALGEBRA I

tetra populat

ppies per mon number of

ange is douonstant.

ations’ of the e below han tetras.

32

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tion

onth

bling.

22

© 2014

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4.


YS COMMON C

Regard the scomposed ofon.

a. How ma

(

b. Describedark tria

c. Create a d. Suppose

triangle

the dark

Figure 0

Thpre

T(n

Figun 0 1 2 3

Module 3: Date:


CORE MATHEM

olid dark equf three dark t

any dark trian

(Figure Num

# of dark tria

e in words hoangles in the

a function tha

e the area of in each figur

k triangles in

F

e number evious figur

n) = 3n, D:

ure, Area oftriangle

1 1/4 1/16 1/64

Linear and E3/26/14

oncore.org

MATICS CURRIC

ilateral triangriangles, the

ngles are in ea

mber)

ngles)

ow, given the next figure.

at models this

the solid darkre form a sequ

the figure

Figure 1

of trianglere.

: n is a wh

f one dark e, A(n)

Exponential Functi

CULUM

gle as Figure 0second figure

ach figure? M

number of d

s sequence. W

k triangle in Fuence. Creat

e in the sequ

Figu

s in a figur

0

1

hole numbe

A(n)

ons

ThisCre

End‐of‐M

0. Then, the e is the one co

Make a table t

ark triangles

What is the d

Figure 0 is 1 ste an explicit

ence.

re 2

re is 3 tim

1

3

r.

=


Module As

first figure inomposed of n

to show this d

in a figure, to

domain of this

quare meter.formula that

Figure 3

mes the num

2

9


sessment

this sequencnine dark tria

data.

o determine t

s function?

. The areas ot gives the are

3

mber of tri

3

27


MTask

ALGEBRA I

ce is the one angles, and so

the number o

of one dark ea of just one

Figure 4

iangles in t

4

81

32

License.

M3

I

o

of

e of

the

23

© 2014

NY


YS COMMON C

e. The sumcase. Th

of all thas n inc

f. Let figures.

is true f

Fig0 1 2 3

Let of o

P is term

So, f

For n

K = 3

Module 3: Date:


CORE MATHEM

m of the areashe sum of the

e dark triangreases?

be the sum There is a re

or each posit

gure Area 1 3/4 9/1627/6

x representne side of th

a geometricm, so k = P(

for example,

n = 1,

3/2

Linear and E3/26/14

oncore.org

MATICS CURRIC

s of all the dae areas of all t

les in the

of the perimeal number

tive whole nu

in m2

6 64

t the numbehe triangle

c sequence a(n+1)/P(n).

, for n = 0,

Exponential Functi

CULUM

rk triangles inthe dark trian

figure in the

meters of the aso that:

1

mber . Wha

T(n) = The total

er of metersin Figure 0.

and k is the

ons

ThisCre

End‐of‐M

n Figure 0 is 1ngles in Figure

sequence? I

all dark triang

1

at is the value

area is de

s long .

ratio betwe


Module As

1 m2; there is e 1 is ¾ m2. W

Is this total ar

gles in the

e of ?

creasing as

Figure P(n)

0 3

1 3

2 92

een any ter


sessment

only one triaWhat is the su

rea increasing

figure in the

s n increas

)

32

92

94

274

rm and the


MTask

ALGEBRA I

angle in this um of the are

g or decreasin

e sequence o

ses.

previous

32

License.

M3

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eas

ng

f

24

© 2014

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5.


YS COMMON C

The graph of

a. Create a

segmen

b. Sketch t

or

DR

Module 3: Date:


CORE MATHEM

f a piecewise –

an algebraic rnts.

the graph of

, 31,1,03,

, 1, |

Domain: -Range: -6

Linear and E3/26/14

oncore.org

MATICS CURRIC

–defined fun

representatio

2 an

11 00 22 3

3 12| 1,0

3 ≤ x ≤ 3 6 ≤ y ≤ 2

Exponential Functi

CULUM

ction is sho

on for . Assu

nd state the d

103

ons

ThisCre

End‐of‐M

own to the rig

ume that the

domain and ra


Module As

ght. The dom

graph of is

ange.


sessment

main of is 3

composed of


MTask

ALGEBRA I

3 3.

f straight line

32

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YS COMMON C

c. Sketch t

d. How do e. How do

Everycan rrang

Everyreprey = f(

DomaRang

Module 3: Date:


CORE MATHEM

the graph of

oes the range

oes the domai

y value in represent te of y = f(x

y value in tesent this bf(x) by 1/k,

ain: -1.5 ge: -3 ≤ y

Linear and E3/26/14

oncore.org

MATICS CURRIC

2 an

of

in of

the range this by mux) by k, giv

the domainby multiply, giving -3

≤ x ≤ 1.5 y ≤ 1

Exponential Functi

CULUM

nd state the d

compare to t

compare to

of y = f(x) ltiplying thving -3k ≤

n of y = f(xying the co3/k ≤ x ≤ 3

ons

ThisCre

End‐of‐M

domain and ra

the range of

o the domain

would be he compou y ≤ k.

x) would beompound in3/k.


Module As

ange.

, w

of

multiplied nd inequal

e divided bnequality t


sessment

where 1?

, where

by k. Sinclity that gi

by k. Sincethat gives t


MTask

ALGEBRA I

?

1?

ce k>1 we ives the

e k>1 we cthe domain

32

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N S Common Core Mathem atics Curr iculum - Achieve · students ap ewise, and st el situations f an express s of function Assessment ATICS CURRIC Linear and E 3/25/14 ncore.org ule

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