ed 367 531 title arizona essential skills for mathematics. … · 2013-11-23 · ed 367 531 title...
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ED 367 531
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INSTITUTIONPUB DATENOTEAVAILABLE FROM
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EDRS PRICEDESCRIPTORS
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SE 053 821
Arizona Essential Skills for Mathematics. ReformattedEdition.Arizona State Dept. of Education, Phoenix.Sep 92235p.; For earlier edition, see ED 325 364.Arizona State Dept. of Education, 1535 W. Jefferson,Phoenix, AZ 85007.Guides Classroom Use Teaching Guides (ForTeacher) (052)
MFOI/PC10 Plus Postage.Algebra; Calculus; *Curriculum Development;*Educational Assessment; Educational Objectives;Elementary Secondary Education; Fractions; Geometry;Mathematical Applications; Mathematical Concepts;Mathematical Logic; *Mathematics Curriculum;Mathematics Education; *Mathematics Skills;Measurement; Numeracy; Probability; Problem Solving;*State Standards; Statistics; *Student Evaluation;Trigonometry
IDENTIFIERS *Arizona; Mathematical Communication; MathematicalThinking; NCTM Curriculum and Evaluation Standards
ABSTRACTThis document ties the essential skills needed in
mathematics to the National Council of Teachers of Mathematics'(NCTM) Curriculum and Evaluation Standards to help facilitate futurecurriculum development. The overall goal for the 21st century is tomake mathematical power a reality for all students. The rationale forthis document can be found in three goals: (1) to restructuremathematics curricula and teaching strategies to change mathematicsfrom a static discipline to a dynamic process; (2) to set highstandards of numeracy for all students; and (3) to integrate studentassessment with the learning process. This document divides essentialmathematical skills into three grade levels: K-3, 4-8, and 9-12. Eachsection presents the essential skills needed at each level in matrixform showing the direct relationship between the content skills andthe NCTM Standards. Each section also supplies a list of studentoutcomes for each content area, as well as examples of indicators. Adirect assessment program which can be used in meeting goal 3 ispresented. Contains 22 references. (RLB)
************************************************************************ Reproductions supplied by EDRS are the best that can be made *
* *from the original document.***********************************************************************
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RM
ISS
ION
TO
RE
PR
OD
UC
E T
HIS
MA
TE
RIA
L H
AS
BE
EN
GR
AN
TE
D B
Y
Linda Edgington
14brarian II
TO
TH
E E
DU
CA
TIO
NA
L R
ES
OU
RC
ES
INF
OR
MA
TIO
N C
EN
TE
R (
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2
Ari
zona
Dep
artm
ent o
f E
duca
tion
C. D
iane
Bis
hop,
Sup
erin
tend
ent
Sept
embe
r 19
92
Ref
orm
atte
d E
ditio
n
BE
ST
CO
PY
AM
IAB
LE
U.I.
DE
PA
RT
ME
NT
OF
ED
UC
AT
ION
Offi
ce o
f Edu
cafio
nai
Res
earc
h an
d im
orov
amen
i
ED
UC
AT
ION
AL
RE
SO
UR
CE
SIN
FO
RM
AT
ION
CE
NT
ER
(E
RIC
)
X0
Thi
s do
cum
ent
has
been
rep
rodu
ced
as
rece
ived
from
!tie
par
son
or o
rgan
izat
ion
orig
inat
ing
it0
Min
or c
hang
esna
ve b
een
mad
e 10
impr
ove
repr
oduc
tion
Qut
hly
Poi
nts
of v
iew
or
opin
ons
stat
ed in
this
doc
u .
mer
it do
not
nec
essa
nIy
repr
esen
t offi
cial
OE
RI p
oliti
c.°
or p
olic
y
3
STA
TE
BO
AR
D O
F E
DU
CA
TIO
N
Cla
udee
n B
ates
Art
hur
Ken
neth
Ben
nett
Hon
orab
le C
. Dia
ne B
isho
p
Dr.
Joh
n H
osm
er
Dr.
Eug
ene
Hug
hes
Dr.
Ray
mon
d K
ellis
Hon
orab
le D
avid
Silv
a
Jam
es U
llman
Dr.
Mor
riso
n W
arre
n
4
The
Ariz
ona
Dep
artm
ent o
f Edu
catio
n is
a a
n eq
ual o
ppor
tuni
tyem
ploy
er a
nd e
duca
tiona
l age
ncy
and
affir
ms
that
it d
oes
not
disc
rimin
ate
on th
e ba
sis
of r
ace,
col
or, n
atio
nal o
rigin
,ag
e, s
ex,
or h
andi
capp
ing
cond
ition
.
0AD
E
Prin
ted
in P
hoen
ix, A
rizon
a by
the
Ariz
ona
Dep
artm
ent o
f Edu
catio
nT
otal
Cop
ies
Prin
ted
3000
Tot
al P
rintin
g C
ost
$629
0.00
Uni
t Prin
ting
Cos
t$2
.10
Dat
e of
Prin
ting
1/93
Con
tent
s
Page
Fore
wor
dB
ackg
roun
d on
the
1987
Doc
umen
tiii
Intr
oduc
tion
1
Goa
ls f
or E
nter
ing
the
21st
Cen
tury
2
Rat
iona
le3
It's
Tim
e to
Cha
nge
3
Let
's C
hang
e w
ith th
e T
imes
3
Mak
ing
the
Com
mitm
ent
5
Mee
ting
the
Cha
lleng
e6
Mak
e M
athe
mat
ical
Pow
er a
Rea
lity
for
All
Stud
ents
7
From
Goa
ls to
Act
ion:
It's
Tim
e8
GO
AL
1.
Res
truc
ture
Mat
hem
atic
s C
urri
cula
and
Tea
chin
g St
rate
gies
to C
hang
e
Mat
hem
atic
s fr
om a
Sta
tic D
isci
plin
e to
a D
ynam
icPr
oces
s9
GO
AL
2.
Set H
igh
Stan
dard
s of
Num
erac
y fo
r A
llSt
uden
ts10
How
Thi
s Se
ctio
n Is
Org
aniz
ed10
K-3
MA
TH
EM
AT
ICS
ESS
EN
TIA
L S
KIL
LS
13
4-8
MA
TH
EM
AT
ICS
ESS
EN
TIA
L S
KIL
LS
33
9-12
MA
TH
EM
AT
ICS
ESS
EN
TIA
L S
KIL
LS
65
GO
AL
3.
Inte
grat
e St
uden
t Ass
essm
entw
ith th
e L
earn
ing
Proc
ess
98
Ari
zona
Stu
dent
Ass
essm
ent P
rogr
am (
ASA
P)98
Dis
tric
t Ass
essm
ent P
lan
(DA
P)10
3
Perf
orm
ance
in th
e C
lass
room
107
Scor
ing
112
Res
ourc
es11
5
9
Fore
wor
d
The
Ariz
ona
Ess
entia
l Ski
lls fo
r M
athe
mat
ics
was
dev
elop
ed in
198
7 by
the
Ess
entia
l Ski
lls C
omm
ittee
app
oint
ed b
y th
e st
ate
Boa
rd o
f Edu
catio
n. T
hedo
cum
ent s
erve
d as
the
fram
ewor
k th
at g
uide
d th
e de
velo
pmen
t of t
hem
athe
mat
ics
port
ion
of th
e A
rizon
a S
tude
nt A
sses
smen
t Pro
gram
. But
the
1987
doc
umen
t was
inte
rpre
ted
by m
any
dist
ricts
as
a lis
t of i
sola
ted
skill
s an
dw
as ta
ught
and
ass
esse
d ac
cord
ingl
y. T
he in
tent
of t
he d
ocum
ent w
as fa
r m
ore
com
preh
ensi
ve. I
n 19
91, a
com
mitt
ee w
as a
ppoi
nted
by
the
stat
e S
uper
inte
n-de
nt to
ref
orm
at th
e A
rizon
a E
ssen
tial S
kills
for
Mat
hem
atic
s in
ord
er to
illus
trat
e th
e in
tent
of t
he o
rigin
al E
ssen
tial S
kills
doc
umen
t.
The
pro
cess
use
d to
dev
elop
the
orig
inal
doc
umen
t is
show
n on
pag
es ii
, iii
and
iv. T
he E
ssen
tial S
kills
incl
uded
in th
e 19
87 d
ocum
ent w
ere
base
d on
the
1986
Cal
iforn
ia F
ram
ewor
k de
velo
ped
hy th
e C
alifo
rnia
Dep
artm
ent o
fE
duca
tion.
The
199
1 co
mm
ittee
ref
orm
atte
d th
e do
cum
ent a
nd a
ligne
d it
toth
e N
atio
nal C
ounc
il of
Tea
cher
s of
Mat
hem
atic
s C
urric
ulum
and
Eva
lua-
tion
Sta
ndar
ds p
r S
choo
l Mat
hem
atic
s. T
he N
atio
nal C
ounc
il of
Tea
cher
s of
Mat
hem
atic
s ha
s gi
ven
the
Ariz
ona
Dep
artm
ent o
f Edu
catio
n pe
rmis
sion
to
repr
int p
ortio
ns o
f the
Cur
ricul
um a
nd E
valu
atio
n S
tand
ards
for
Sch
ool
Mat
hem
atic
s in
the
1992
Ess
entia
l Ski
lls d
ocum
ent.
The
199
1 M
athe
mat
ics
Ess
entia
l Ski
lls C
omm
ittee
was
cha
ired
by L
inda
Jasl
ow, m
athe
mat
ics
spec
ialis
t at t
he A
rizon
a D
epar
tmen
t of E
duca
tion.
Mem
bers
of t
he c
omm
ittee
wer
e
Mic
ki A
ley,
Ariz
ona
Dep
artm
ent o
f Edu
catio
nD
an B
enso
n, P
hoen
ix U
nion
Hig
h S
choo
l Dis
tric
tJi
m C
ochr
ane,
Mar
ana
Uni
fied
Sch
ool D
istr
ict
Nan
cy F
isk,
Bal
sz E
lem
enta
ry S
choo
l Dis
tric
tJe
ri H
amilt
on, P
resc
ott U
nifie
d S
choo
l Dis
tric
tC
hris
tene
Mei
ster
, Tem
pe E
lem
enta
ry S
choo
l Dis
tric
tN
ancy
Ost
ergr
en, L
itchf
ield
Ele
men
tary
Sch
ool D
istr
ict
Col
ette
Poi
nter
-Pul
len,
Alh
ambr
a E
lem
enta
ry S
choo
l Dis
tric
tG
erd
Sha
ft, M
aran
a U
nifie
d S
choo
l Dis
tric
t
Bac
kgro
und
on th
e 19
87 D
ocum
ent
The
sta
te B
oard
of E
duca
tion
appo
inte
d th
e 19
87 a
dvis
ory
com
mitt
ee to
perf
orm
the
initi
al ta
sks
invo
lved
in r
evis
ion
and
revi
ew. O
ne-t
hird
of t
heC
omm
ittee
mem
bers
wer
e ap
poin
ted
as p
rofe
ssio
nal e
duca
tors
. The
rem
aini
ngtw
o-th
irds
of th
e C
omm
ittee
wer
e ci
tizen
mem
bers
, alth
ough
sev
eral
had
pas
tan
d pr
esen
t exp
erie
nce
as e
duca
tors
, man
y as
form
er te
ache
rs o
r m
embe
rs o
fsc
hool
boa
rds
or d
istr
icts
. Not
with
stan
ding
thei
r ex
trem
ely
broa
d an
d di
vers
eba
ckgr
ound
s, C
omm
ittee
mem
bers
sha
red
an in
tens
e in
tere
st in
mat
hem
atic
sed
ucat
ion.
The
edu
cato
r m
embe
rs o
f the
Com
mitt
ee w
ere
from
all
sect
ors
of e
duca
tion,
incl
udin
g el
emen
tary
, sec
onda
ry a
nd u
nive
rsity
teac
hers
and
adm
inis
trat
ors.
Ast
r.ff
faci
litat
or w
as a
ppoi
nted
to th
e C
omm
ittee
by
the
Dep
artm
ent o
f Edu
ca-
ti..)
n at
the
Boa
rd's
req
uest
. Thi
s pe
rson
was
Kay
Dea
n, m
athe
mat
ics
spec
ialis
t.
Oth
er k
ey r
esou
rces
at t
he C
omm
ittee
's d
ispo
sal w
ere
mat
hem
atic
s cu
rric
ula
from
sev
eral
oth
er s
tate
s an
d so
me
of th
e fo
reig
n co
untr
ies
whi
ch s
core
d hi
gher
than
the
Uni
ted
Sta
tes
on th
e re
cent
Inte
rnat
iona
l Mat
hem
atic
s T
est.
Som
e of
the
appr
oach
es fr
om th
e ot
her
stat
es a
nd c
ount
Hes
wer
e fo
und
suita
ble
orad
apta
ble
for
use
in A
rizon
a an
d w
ere
inco
rpor
ated
into
the
Com
mitt
ee's
fina
lre
port
and
the
Ess
entia
l Ski
lls th
emse
lves
.
1 2
The
Mat
hem
atic
s E
ssen
tial S
kills
Com
mitt
ee w
as c
haire
d by
Jam
es N
elso
nfr
om P
hoen
ix. M
embe
rs o
f the
Com
mitt
ee a
nd c
omm
uniti
es th
ey r
epre
sent
edar
e
Cha
mbe
rs:
Els
ie D
uran
Sco
ttsda
le:
Larr
y S
mith
Eag
ar:
Sta
n S
mith
Tem
pe:
Dr.
Ron
Bro
wn
Jan
Bur
khar
dtF
lags
taff:
Cha
rles
Littl
eJo
an P
fuhl
Mes
a:
Pho
enix
:
Bon
nie
Mor
ales
Tuc
son:
Car
ol B
rook
sD
r. S
teve
n W
alte
rsD
ale
Cur
tisIlo
na G
ayD
r. V
irgin
ia H
orak
Mar
tha
Bac
aC
indy
Gilb
ert
Joy
Han
ley
Joan
ne K
imur
aJa
n M
iller
Jam
es N
elso
nA
dria
Ren
keS
usan
Sal
dafia
Dea
n S
ulze
r
Will
iam
s:D
an B
aert
lein
3
The
Pro
cess
The
Com
miu
ee h
ad it
s f
irst
mee
ting
in M
ay 1
986.
Sub
sequ
ent t
o th
is in
itial
mee
ting,
the
Com
mitt
ee m
et o
n a
mon
thly
bas
is th
roug
h D
ecem
ber
1986
.
1 )u
ring
the
cour
se o
f di
scus
sion
s, th
e C
omm
ittee
dis
play
ed a
will
ingn
ess
toex
amin
e in
gre
at d
etai
l all
face
ts o
f m
athe
mat
ics
educ
atio
n. T
he C
omm
ittee
mad
e a
cons
ciou
s de
cisi
on e
arly
in th
e pr
oces
s, h
owev
er, t
o ke
ep th
e "b
igpi
ctur
e" in
foc
us e
ven
whi
le e
xam
inin
g th
e de
tails
nec
essa
ry f
or e
stab
lishi
ngth
e E
ssen
tial S
kills
. Am
ong
the
impo
rtan
t con
side
ratio
ns g
uidi
ng th
e C
omm
it-te
e w
ere
the
follo
win
g:
.T
he o
v er
all o
bjec
tive
of th
ese
Ess
entia
l Ski
lls is
to in
crea
se s
tude
nt m
ath
prof
icie
ncy.
Thi
s ob
ject
ive
can
only
he
met
by
impr
ovin
g th
e qu
ality
of
mat
hem
atic
s in
stru
ctio
n.
The
Ess
entia
l Ski
lls o
utlin
ed h
ere
are
inte
nded
as
a fl
exib
le g
uide
for
the
loca
l dis
ii id
s. T
he E
ssen
tial S
k ill
s sh
ould
be
adap
ted
to lo
cal c
ondi
tions
so th
at th
e ne
eds
of a
ll st
uden
ts in
the
educ
atio
nal c
omm
unity
can
he
met
.
The
mem
bers
of
the
Com
mitt
ee w
ere
unif
orm
in th
eir
belie
f th
at d
istr
icts
and
scho
ols
shou
ld b
e re
spon
sibl
e fo
r te
achi
ng th
ese
skill
s an
d st
rivi
ngto
exc
eed
them
whe
neve
r po
ssib
le.
4.A
lthou
gh th
e E
ssen
tial S
kills
wer
e de
sign
ed p
rim
arily
as
a gu
ide
to lo
cal
dist
rict
s an
d in
divi
dual
sch
ools
in th
eir
deve
lopm
ent o
f cu
rric
ulum
and
nnpl
emen
tatio
n of
that
cur
ricu
lum
, the
y al
so a
re in
tend
ed to
pro
vide
guid
ance
to th
e te
xtbo
ok s
elec
tion
com
mitt
ee.
iv
Dis
tric
t Res
pons
es a
nd I
leai
ngs
The
Com
mitt
ee w
as k
eenl
y aw
are
that
the
succ
ess
or f
ailu
re o
f th
e pr
opos
edre
visi
ons
to th
e E
ssen
tial S
kills
in M
athe
mat
ics
wer
e co
ntin
gent
, in
larg
e pa
il,on
the
acce
ptan
ce a
nd im
plem
enta
tion
of th
ose
skill
s by
loca
l dis
tric
ts a
ndsc
hool
s. T
here
fore
, a d
raft
of
the
prop
osed
rev
isio
ns w
as s
ubm
itted
to s
choo
ldi
stri
cts
stat
ewid
e fo
r di
stri
ct-l
evel
inpu
t.
Supe
rint
ende
nts
wer
e as
ked
to r
espo
nd to
the
revi
sion
or
desi
gnat
e so
meo
neel
se in
thei
r di
stri
ct to
do
so. T
he in
divi
dual
dis
tric
ts w
ere
also
pro
vide
d a
resp
onse
she
et o
n w
hich
they
cou
ld r
epor
t the
ir im
pres
sion
s, a
nd w
ere
info
rmed
of
the
date
s, p
lace
s an
d tim
es o
f th
e th
ree
regi
onal
pub
lic h
eari
ngs.
Res
pons
e sh
eets
wer
e re
ceiv
ed f
rom
man
y di
stri
cts
and
scho
ols
with
in d
isIn
lets
. Som
e di
stri
cts
and
scho
ols
mad
e co
pies
of
the
entir
e do
cum
ent a
ndte
ache
rs w
ere
aske
d to
res
pond
. All
resp
onse
s w
ere
revi
ewed
by
Com
mitt
eem
embe
rs. C
ompl
ete
data
, inc
ludi
ng a
nar
rativ
e of
com
men
ts a
nd s
tatis
tical
repo
rts,
can
be
obta
ined
fro
m th
e D
epar
tmen
t of
Edu
catio
n. S
ome
dist
rict
s.sc
hool
s an
d te
ache
rs w
rote
sep
arat
e le
tters
or
atta
ched
sep
arat
e co
mm
ents
, and
thes
e na
rrat
ive
resp
onse
s ar
e al
so a
vaila
ble
from
the
Dep
artm
ent o
f E
duca
tion.
I le
anin
gs w
ere
sche
dule
d in
thro
e lo
catio
ns a
cros
s th
e st
ate
to p
rovi
de f
or d
irec
tci
ti/en
and
dis
tric
t inp
ut.
1 5
Sche
dule
of
Publ
ic H
eari
ngs
Flag
staf
f, A
rizo
na
Plac
e:D
ate:
Tim
e:
Phoe
nix,
Ari
zona
Plac
e:
Dat
e:T
ime:
Tuc
son,
Ari
zona
Plac
e:
Dat
e:.'i
me:
Flag
staf
f H
igh
Scho
ol M
ini-
Aud
itori
umT
uesd
ay. J
anua
ry 1
3,1987
7 p.
m.
9 p.
m.
Ari
zona
Dep
artm
ent o
f E
duca
tion
1535
Wes
t Jef
fers
on. R
oom
417
Wed
nesd
ay, J
anua
ry14.
1987
5 p.
m.
7 p.
m.
Boa
rd R
oom
, Tuc
son
Uni
fied
Sch
ool
Dis
tric
t No.
110
10 E
ast T
enth
Str
eet
Thu
rsda
y, J
anua
ry 1
5, 1
987
5 p.
m.
7p.
m.
App
roxi
mat
e./ 5
0 pe
ople
atte
nded
the
hear
ings
, and
par
ticip
ants
com
men
tsw
ere
reco
rded
on
note
s an
d ta
pe. A
sum
mar
y of
the
note
s ta
ken
at th
e he
arin
gsis
ava
ilabl
e fr
om th
e D
epar
tmen
t of
Edu
catio
n.
Com
men
ts m
ade
duri
ng th
e he
arin
gs w
ere
posi
tive,
and
spe
cifi
c su
gges
tions
for
impr
ovem
ent w
ere
mad
e. C
once
rns
expr
esse
d at
all
site
s re
late
d to
the
degr
ee o
f au
tono
my
loca
l dis
tric
ts w
ould
hav
e in
impl
emen
ting
the
Ess
entia
lSk
ills
in M
athe
mat
ics.
Fla
gsta
ff p
artic
ipan
ts w
ere
conc
erne
d th
at th
e re
visi
onw
as o
verl
y am
bitio
us a
nd ig
nore
d va
riou
s pr
oble
ms
whi
ch d
istr
icts
in th
eno
rthe
rn p
ortio
n of
the
stat
e fa
ced.
The
se p
robl
ems
incl
uded
bud
get r
estr
ic-
tions
and
hig
h pe
rcen
tage
s of
min
ority
stu
dent
s. F
lags
taff
par
ticip
ants
als
oex
pres
sed
a co
ncer
n th
at th
e st
ate
Boa
rd o
f E
duca
tion,
the
Ari
zona
Dep
artm
ent
of E
duca
tion
and
the
Ari
zona
Leg
isla
ture
wer
e bu
rden
ing
scho
ol d
istr
icts
with
a re
visi
on w
hen
they
had
onl
y re
cent
ly m
anag
ed to
inco
rpor
ate
the
1984
list
into
thei
r cu
rric
ulum
.
Fina
l Rev
isio
n
The
Com
mitt
ee m
et o
n Fe
brua
ry 3
,1987,
to c
onsd
er a
ll re
spon
ses
to th
e dr
aft
docu
men
t. C
omm
ittee
mem
bers
mad
e re
com
men
datio
ns th
at th
e dr
aft b
efu
rthe
r re
vise
d to
acc
omm
odat
e m
any
of th
e re
spon
ses;
ther
efor
e, s
ever
alch
ange
s w
ere
mad
e to
the
draf
t the
dis
tric
ts r
evie
wed
. Tse
rev
isio
ns in
clud
edde
letio
n of
som
e sk
ills
from
the
list i
n re
spon
se to
con
cern
s th
at s
ome
of th
emw
ere
too
soph
istic
ated
for
the
grad
e le
vel.
The
fin
al r
evis
ion,
as
subm
itted
for
appr
oval
by
the
stat
e B
oard
of
Edu
catio
n on
Feb
ruar
y 23
,1987,
refl
ects
the
Com
mitt
ee's
dec
isio
ns r
egar
ding
res
pons
es r
ecei
ved
from
dis
tric
ts, s
choo
lsan
d te
ache
rs a
cros
s th
e st
ate.
17
Intr
oduc
tion
The
l9g7
ver
sion
of t
he A
rizon
a E
ssen
tial S
kills
for
Mat
hem
atic
s ha
s be
en in
terp
rete
d by
som
e as
a c
heck
list o
f iso
late
d sk
ills.
The
inte
ntof
the
docu
men
t was
far
mor
e
com
preh
ensi
ve.
Wel
com
e to
the
I 992
edi
tion
of th
e A
rizon
a E
ssen
tial S
kills
for
Mat
hem
atic
s.T
his
docu
men
t lea
ves
inta
ct th
e A
rizon
a E
ssen
tial S
kills
pub
lishe
d in
198
7. It
cite
s cu
rren
t agr
eem
ent a
mon
g th
ose
with
in th
e m
athe
mat
ics
educ
atio
n co
m-
mun
ity a
s to
how
to p
repa
re s
tude
nts
for
the
21st
cen
tury
.
The
Ariz
ona
Ess
entia
l Ski
lls fo
r M
athe
mat
ics
was
firs
t pub
lishe
d in
Jul
y 19
87by
the
Ariz
ona
Dep
artm
ent o
f Edu
catio
n an
d is
ref
lect
ed in
the
mat
hem
atic
sco
mpo
nent
of t
he A
rizon
a S
tude
nt A
sses
smen
t Pro
gram
(A
SA
P).
The
del
inea
-tio
n of
thes
e m
athe
mat
ics
skill
s he
lped
edu
cato
rs d
efin
e w
hat c
onte
nt a
ndpr
oces
s sk
ills
stud
ents
nee
d. T
his
docu
men
t has
sub
sequ
ently
bee
n im
pact
edby
the
1989
pub
lishi
ng o
f the
Nat
iona
l Cou
ncil
of T
each
ers
of M
athe
mat
ics
( N
CT
M )
Cur
ricul
um a
nd E
valu
atio
n S
tand
ards
lOr
Sch
ool M
athe
mat
ics.
The
irdo
cum
ent,
whi
ch h
as b
ecom
e kn
own
as th
e N
CT
M S
tand
ards
, has
affe
cted
the
dire
ctio
n of
mat
hem
atic
s ed
ucat
ion
in th
is c
ount
ry.
3
Mat
hem
atic
s m
ust b
e fu
lly e
xplo
red
with
in th
e co
ntex
tth
e re
al w
orld
.A
lthou
gh c
onte
nt is
impo
rtan
t, it
is o
ne's
abi
lity
to p
robi
om s
olve
that
ultim
atel
y de
term
ines
the
outc
omes
of o
ne's
enc
ount
ers
with
life
. Pro
blem
solv
ing,
com
mun
icat
ion,
rea
soni
ng a
nd m
akin
g m
athe
mat
ical
con
nect
ions
thes
e as
pect
s of
mat
hem
atic
s th
read
thro
ugh
all g
rade
leve
ls. I
t is
nece
ssar
y to
unde
rsta
nd th
e in
terr
elat
ions
hips
of t
hese
four
pro
cess
es. W
ithou
t thi
s un
der-
stan
ding
, it w
ill n
ot b
e po
ssib
le to
mak
e th
e tr
ansi
tion
smoo
thly
into
the
next
cent
ury.
Thi
s ne
w d
ocum
ent i
s yo
urs,
min
e, o
urs.
Its
form
at ti
es th
e E
ssen
tial S
kills
toth
e N
CT
M S
tand
ards
to h
elp
faci
litat
e fu
ture
cur
ricul
um d
evel
opm
ent.
Fro
mit
you
will
lear
n w
hat c
an b
e do
nein
deed
, wha
t cha
nges
mus
t be
will
ingl
y
mad
eto
pre
pare
our
you
ng p
eopl
e to
take
cha
rge
of th
e w
orld
-to-
be s
o th
atth
ey c
an u
nder
stan
d its
pro
blem
s an
d be
pre
pare
d to
wor
k to
war
d so
lutio
ns.
Whe
re to
beg
in?
Pro
babl
y no
t sur
pris
ingl
y, c
hang
e be
gins
with
in e
ach
of u
s.
1 3
Goa
ls f
or E
nter
ing
the
21st
Cen
tury
As
we
ente
r th
e 2I
st c
entu
ry, w
e m
ust m
ake
mat
hem
atic
al p
ower
a r
ealit
y fo
ral
l stu
dent
s.
Mat
hem
atic
al p
ower
is th
e ca
paci
ty to
do
purp
osef
ul a
nd w
orth
whi
le m
ath-
emat
ical
Ivo
rk. T
he c
ritic
al m
anif
esta
tion
of m
athe
mat
ical
pow
er li
es in
the
stud
ent's
abi
lity
to e
mpl
oy
Mat
hem
atic
al th
inki
ngto
use
kno
wle
dge
«Rd
unde
rsta
ndin
g to
anal
yze,
con
ject
ure,
des
ign,
eva
luat
e, f
orm
ulat
e, g
ener
aliz
e, in
ves-
tigat
e, m
odel
, pre
dict
, tra
nsfO
rm o
r ve
rify
Mat
hem
atic
al u
nder
stan
ding
to u
se m
athe
mat
ical
con
cept
s an
dco
nnec
tions
am
ong
conc
epts
bot
h w
ithin
mat
hem
atic
s an
d ac
ross
disc
iplin
esT
ools
and
tech
niqu
esto
eff
icie
ntly
and
eff
ectiv
ely
solv
e m
ath-
emat
ical
pro
blem
s: f
or e
xam
ple,
usi
ng d
iagr
ams
and
tabl
es, c
alcu
-la
tors
and
com
pute
rs, m
anip
ulat
ives
, and
oth
er c
oncr
ete
mat
eria
lsC
omm
unic
atio
n Sk
ills
to c
omm
wlic
ate
resu
lts w
ith v
ario
usau
dien
ces
and
for
vari
ous
purp
oses
.(C
alifo
rnia
Dep
artm
ent o
fE
duca
tion,
199
1, 5
6)
20
2
To
brea
the
life
into
mat
hem
atic
s ed
ucat
ion,
thre
e go
als
mus
t be
achi
eved
:
I.R
estr
uctu
re m
athe
mat
ics
curr
icul
a an
d te
achi
ngst
rate
gies
to c
hang
e m
athe
mat
ics
from
a s
tatic
disc
iplin
e to
a d
ynam
ic p
roce
ss.
2.S
et h
igh
stan
dard
s of
num
erac
y fo
ral
lst
uden
ts.
3.In
tegr
ate
stud
ent a
sses
smen
t with
the
lear
ning
proc
ess.
Rat
iona
le
It's
Tim
e to
Cha
nge
"Oth
er in
dust
rial
ized
cou
ntri
es a
wak
ened
20
year
s ag
o to
the
sign
ific
ance
of
mat
hem
atic
s an
d sc
ienc
e pr
ofic
ienc
y fo
r th
eir
natio
nal w
ell-
bein
g an
d be
gan
effo
rts
to s
tren
gthe
n th
ese
com
pone
nts
of e
duca
tion.
We
in th
e U
nite
d St
ates
have
bee
n to
o sl
ow to
res
pond
. As
a re
sult,
we
face
har
sh r
ealit
ies.
..."
(Mat
h-em
atic
al S
cien
ce E
duca
tion
Boa
rd, 1
991a
, 3)
We
have
bee
n de
alin
g w
ith m
inim
al e
xpec
tatio
ns a
nd h
ave
atta
ined
med
iocr
ity.
We
have
use
d tr
aditi
onal
mat
hem
atic
s cu
rric
ula
to a
chie
ve th
e re
sults
of
the
past
, inc
ludi
ng o
ur c
urre
nt le
vels
of
achi
evem
ent.
At a
poin
t in
time
whe
re w
ear
e ge
nera
ting
mor
e qu
estio
ns th
an a
nsw
ers,
are
we
prep
arin
g ou
r st
uden
ts to
ask
the
righ
t que
stio
ns in
ord
er to
dea
l with
con
stan
t cha
nge,
or
are
we
send
ing
them
into
the
wor
ld w
ith th
e kn
owle
dge
that
ther
e is
onl
y on
e ri
ght a
nsw
er to
ever
y pr
oble
m a
nd th
e im
pres
sion
that
ther
e is
only
one
way
to f
ind
it'?
Arc
we
help
ing
our
stud
ents
to u
nder
stan
d m
athe
mat
ics
and
its b
enef
its e
ffec
tivel
y, s
oth
ey c
an h
e sh
ared
by
all'? .;
2
3
We
mus
t bui
ld f
or a
fut
ure
of c
hang
e. S
ome
scho
ols
and
thei
r co
mm
uniti
esha
ve
alre
ady
star
ted.
Acc
ordi
ng to
Res
hapi
ng S
choo
l Mat
hem
atic
s,"T
he r
uts
of th
eol
d cu
rric
ulum
are
bei
ng e
rode
d by
the
wav
es o
f ch
ange
sw
eepi
ng a
cros
sth
e
land
scap
e of
mat
hem
atic
s ed
ucat
ion.
" (N
atio
nal R
esea
rch
Cou
ncil
1990
, 6)
How
ever
, muc
h m
ore
is n
eede
d if
toda
y's
stud
ents
are
to b
ecom
em
athe
mat
i-ca
lly li
tera
te a
nd u
se th
e la
ngua
ge o
f m
athe
mat
ics
com
fort
ably
and
conf
iden
tly
in th
eir
daily
live
s.
Let
's C
hang
e w
ith th
e T
imes
Our
mis
sion
for
ent
erin
g th
e 21
st c
entu
ry is
cle
ar: M
ake
mat
hem
atic
al p
ower
a re
ality
for
all
stud
ents
. Thr
ee g
oals
have
bee
n id
entif
ied
for
achi
evem
ent.
Inor
der
to g
ain
a be
tter
unde
rsta
ndin
g of
thes
e go
als,
a s
umm
ary
of e
ach
has
been
prep
ared
. Thi
s in
form
atio
n es
tabl
ishe
s w
here
we
are
in m
athe
mat
ics
educ
atio
nan
d w
here
we
mus
t set
our
sig
hts
if o
ur s
tude
nts
are
to a
ttain
mat
hem
atic
al
pow
er. T
he C
omm
ittee
can
not o
vere
mph
asiz
eth
e im
port
ance
of
each
goa
l.
GO
AL
1.
Res
truc
ture
Mat
hem
atic
s C
urric
ula
and
Tea
chin
g S
trat
egie
s to
Cha
nge
Mat
hem
atic
s fr
om a
Sta
tic D
isci
plin
e to
a D
ynam
ic P
roce
ss
Tra
ditio
nal c
urri
cula
tend
s to
bre
ak m
athe
mat
ical
ski
lls in
to m
inut
e pi
eces
,pr
esen
ting
the
skill
s as
isol
ated
fra
gmen
ts o
f kn
owle
dge
with
littl
e ca
rryo
ver
to th
c re
al w
orld
.
Tea
cher
s m
ust t
each
stu
dent
s to
thin
k fo
r th
emse
lves
, as
wel
l as
to w
ork
coop
erat
ivel
y w
ith o
ther
s. C
urri
culu
m a
nd in
stru
ctio
n m
ust h
e ch
ange
d to
enco
urag
e st
uden
ts to
exp
lore
, inv
estig
ate,
rea
son,
con
ject
ure,
dis
cuss
, app
ly,
crea
te a
nd c
omm
unic
ate.
"N
o si
ngle
teac
hing
met
hod
nor
any
sing
le k
ind
ofle
arni
ng e
xper
ienc
e ca
n de
velo
p th
e va
ried
mat
hem
atic
al a
bilit
ies
impl
ied
unde
r th
e de
fini
tion
of m
athe
mat
ical
pow
er."
(N
atio
nal R
esea
rch
Cou
ncil,
1990
h, 3
9)
CO
AL
2.
Set
Hig
h S
tand
ard\
of N
umer
acyP
rS
tude
nts
Few
U. S
. stu
dent
s ac
hiev
e le
vels
of
mat
hem
atic
s ne
cess
ary
to m
eet t
hede
man
ds o
f ou
r so
ciet
y. M
ost s
tude
nts
are
rele
gate
d to
mat
hem
atic
s cl
asse
sth
at h
ave
min
imal
exp
ecta
tions
and
fai
l to
equi
p th
em w
ith s
kills
nec
essa
ry to
prep
are
them
for
the
wor
ld o
f w
ork.
Onl
y in
this
cou
ntry
do
peop
le b
elie
ve th
atle
arni
ng m
athe
mat
ics
is d
epen
dent
on
havi
ng s
peci
al a
bilit
ies.
Thi
s m
yth,
whi
ch h
as d
omin
ated
mat
hem
atic
s ed
ucat
ion,
can
no
long
er b
e to
lera
ted.
Tea
cher
s m
ust p
rom
ote
a co
mm
on c
ore
of m
athe
mat
ics
for
all s
tude
nts
and
am
odel
of
stud
ent-
cent
ered
pra
ctic
e th
roug
h gr
oup
lear
ning
. Rai
sing
per
for-
man
ce le
vels
, kee
ping
up
with
tech
nolo
gica
l adv
ance
s, a
nd in
tegr
atin
g m
ath-
emat
ical
con
cept
s an
d cr
itica
l thi
nkin
g w
ill h
elp
us m
ake
mat
hem
atic
al p
ower
a re
ality
for
all
stud
ents
.
"Eve
ryon
e de
pend
s on
the
succ
ess
of m
athe
mat
ics
educ
atio
n; e
very
one
is h
urt
whe
n it
fails
." (
Nat
iona
l Res
earc
h C
ounc
il, 1
989,
7)
9 4
4
GO
AL
3.
Inte
grat
e S
tude
nt A
sses
smen
t with
the
Lear
ning
Pro
cess
In c
hang
ing
mat
hem
atic
s fr
om a
sta
t:c d
isci
plin
e to
a d
ynam
ic p
roce
ss, i
t is
evid
ent t
hat m
etho
ds o
f as
sess
men
t nee
d to
be
rede
fine
d. T
hey
requ
ire
deve
lopi
ng w
ays
to a
sses
s hi
gher
-ord
er th
inki
ng p
roce
sses
. Set
ting
high
stan
dard
s in
num
erac
y fo
r al
l nec
essi
tate
s fo
cusi
ng o
n w
hat s
tude
nts
know
, as
wel
l as
on w
hat t
hey
need
to k
now
.
Ass
essm
ent m
ust b
e on
goin
g th
roug
hout
the
inst
ruct
iona
l pro
cess
. Onl
y th
enca
n a
stud
ent's
gro
wth
in u
nder
stan
ding
mat
hem
atic
al c
once
pts
and
appl
ying
thos
e co
ncep
ts h
e fa
irly
and
acc
urat
ely
asse
ssed
. As
stud
ents
exp
lore
and
deve
lop
thei
r m
etac
ogni
tive
skill
s (i
.e.,
thin
king
abo
ut th
inki
ng),
and
use
appr
opri
ate
tech
nolo
gy, t
hey
begi
n to
com
mun
icat
e an
d ill
ustr
ate
thei
r th
ink-
ing
proc
esse
s us
ing
both
ora
l and
wri
tten
lang
uage
.
Mak
ing
the
Com
mitm
ent
As
we
mov
e to
mee
t the
goa
ls s
et to
mak
e m
athe
mat
ical
pow
er a
rea
lity
for
all
stud
ents
, we
mis
t con
stan
tly m
onito
r an
d ad
just
beh
avio
ral a
nd c
urri
cula
rou
tcom
es to
mee
t the
nee
ds o
f st
uden
ts a
nd s
ocie
ty. I
t is
not e
noug
h to
str
ive
to m
eet t
he p
rese
nt-d
ay c
halle
nges
with
out g
augi
ng o
ur p
rogr
ess
agai
nst o
urco
ntin
ually
and
rap
idly
cha
ngin
g so
ciet
y. A
s te
ache
rs a
nd a
dmin
istr
ator
s, w
ear
e pi
lotin
g a
plan
e fu
ll of
pot
entia
l; an
d w
e m
ust b
e w
illin
g to
mon
itor
and
tom
ake
corr
ectio
ns to
sta
y on
cou
rse.
Mak
ing
adju
stm
ents
doe
s no
t say
we
have
faile
d; r
athe
r it
indi
cate
s th
at w
e ha
ve le
arne
d an
d ar
e w
illin
g to
use
that
know
ledg
e to
hel
p ou
r st
uden
ts g
row
to h
elp
all o
f us
rea
ch o
ur g
oal.
The
tech
nolo
gica
l nee
ds o
f so
ciet
y an
d w
hat w
e cu
rren
tly a
re te
achi
ng d
iffe
rgr
eatly
. "O
f al
l the
infl
uenc
es th
at s
hape
mat
hem
atic
s ed
ucat
ion,
tech
nolo
gysu
mds
out
as
the
one
with
gre
ates
t pot
entia
l for
rev
olut
iona
ry im
pact
."c
Nat
iona
l Res
earc
h C
ounc
il, 1
990b
, 22)
How
ever
, the
use
of
tech
nolo
gy in
the
5
mat
hem
atic
s cl
assr
oom
has
bee
n lim
ited
to r
einf
orci
ng tr
aditi
onal
cur
ricu
laan
d ha
s ye
t to
appr
oach
its
pote
ntia
l. A
s th
e ag
e of
tech
nolo
gy g
row
s, th
e ga
pis
wid
enin
g be
twee
n w
hat o
ur s
tude
nts
can
do a
nd th
e so
phis
ticat
ion
requ
ired
to u
se th
e te
chno
logy
.
With
the
Info
rmat
ion
Age
upo
n us
, kno
wle
dge
is e
xplo
ding
at a
rat
e w
ith w
hich
we
can
bare
ly k
eep
pace
. Stu
dent
s m
ust l
earn
to u
se c
alcu
lato
rs, c
ompu
ters
and
othe
r te
chno
logi
cal t
ools
to s
olve
pro
blem
s an
d to
go
beyo
nd w
here
we
pres
ently
are
. Mat
hem
atic
s m
akes
tech
nolo
gy p
ossi
ble.
"T
echn
olog
y m
akes
mat
henz
atic
s re
alis
tic.
..."
(Nat
iona
l Res
earc
h C
ounc
il 19
90b,
21)
"If
we
mak
e a
long
-ter
m c
omm
itmen
t to
the
stan
dard
s, ..
. if
we
appr
oach
the
task
with
the
will
to p
erse
vere
, if
we
are
criti
cal o
f th
e st
eps
we
take
, ...
we
will
mak
e pr
ogre
ss to
war
d th
e go
al o
f de
velo
ping
mat
hem
atic
al p
ower
for
all
stud
ents
." (
Mat
hem
atic
al S
cien
ces
Edu
catio
n B
oard
, 199
1a, 2
5) T
he b
enef
itsto
the
indi
vidu
al a
nd to
soc
iety
are
imm
easu
rabl
e.
Mee
ting
the
Cha
lleng
e
In o
rder
to p
repa
re s
tude
nts
for
the
chal
leng
es o
f th
e fu
ture
, we
have
exp
lore
dch
ange
s th
at c
an b
e m
ade
in c
urri
culu
m a
nd m
etho
dolo
gy to
bri
ng th
e te
achi
ngof
mat
hem
atic
s in
line
with
wha
t we
now
kno
w. H
ere
is a
list
of
sugg
estio
ns f
orch
ange
that
can
be
real
ized
at c
omm
unity
, sta
te a
nd n
atio
nal l
evel
s:
Mak
e pr
oble
m s
olvi
ng th
e fo
cus
of m
athe
mat
ics
inst
ruct
ion
and
lear
ning
.H
elp
stud
ents
und
erst
and
that
they
use
mat
hem
atic
s to
sol
ve r
eal-
life
prob
lem
s. "
All
child
ren
lear
n an
d us
e m
athe
mat
ics
bette
r if
it is
der
ived
from
thei
r re
ality
, abs
trac
ted,
pra
ctic
ed in
enj
oyab
le a
nd e
ffec
tive
way
s.an
d ap
pl ie
d to
situ
atio
ns th
at a
re in
tere
stin
g an
d re
al to
them
." (
Will
ough
by,
1990
, 102
)
Em
phas
ize
that
mat
hem
atic
s is
a p
roce
ss, n
ot a
set
of
fact
s. O
ffer
han
ds-
on. e
xper
ient
ial,
risk
-tak
ing
oppo
rtun
ities
that
bui
ld c
onfi
denc
e an
din
tere
st in
that
pro
cess
. Sho
w b
y ex
ampl
e th
at th
ere
are
no "
righ
t/wro
ng,
quic
k an
swer
" ap
proa
ches
to le
arni
ng.
Prov
ide
oppo
rtun
ities
for
usi
ng m
athe
mat
ical
pro
babi
lity
mod
els
and
prob
lem
sol
ving
in a
ll di
scip
lines
. Pro
mpt
stu
dent
s to
pra
ctic
e an
dst
reng
then
thei
r co
mm
unic
atio
n sk
ills
as th
ey m
ake
pred
ictio
ns, r
easo
nan
d so
lve
prob
lem
s.
Use
con
cret
e m
titer
ial s
that
rel
ate
prio
r kn
owle
dge
to n
ew c
once
pts.
The
1987
Ess
entia
l Ski
lls d
ocum
ent e
mph
asiz
es th
at th
e us
e of
con
cret
em
ater
ials
is c
ritic
al to
lear
ning
at a
ll le
vels
.
9 3
6
Use
app
ropr
iate
tech
nolo
gy in
the
clas
sroo
m. "
The
Sta
ndar
dsm
ake
clea
rth
at a
text
book
alo
ne is
not
suf
fici
ent f
or te
achi
ng a
nd le
arni
ng m
athe
mat
-ic
s. C
lass
room
s m
ust b
e eq
uipp
ed w
ith c
alcu
lato
rs, c
ompu
ters
, phy
sica
lm
ater
ials
, and
the
asso
ciat
ed 's
oftw
are'
to s
uppo
rt th
ese.
" (T
raft
on, 1
989,
11-1
2)
Supp
ort t
he te
ache
r as
des
igne
r of
new
lear
ning
env
iron
men
ts a
ndpr
ovid
er o
f in
stru
ctio
n, f
oste
ring
stu
dent
enj
oym
ent a
nd a
ppre
ciat
ion
ofm
athe
mat
ics.
Sele
ct c
ompr
ehen
sive
sta
ndar
ds f
or c
urri
cula
and
ass
essm
ent.
Use
as-
sess
men
ts n
ot f
or c
ompa
ring
stu
dent
s bu
t for
hel
ping
stu
dent
s un
ders
tand
mat
hem
atic
sfo
r pr
ovid
ing
info
rmat
ion
with
whi
ch to
mak
e in
stru
c-tio
nal d
ecis
ions
.
And
per
haps
the
mos
t cri
tical
poi
nt to
be
mad
e: S
how
that
res
truc
turi
ngis
a w
orth
whi
le g
oal.
Dev
elop
and
rew
ard
part
ners
hips
of
scho
ols
(inc
lud-
ing
thei
r bo
ards
), h
ome
and
com
mun
ity w
ith c
omm
itmen
t to
com
mon
goal
s an
d sh
ared
acc
ount
abili
ty f
or m
akin
g ch
ange
hap
pen
in e
duca
tion.
With
out a
ded
icat
ion
to th
is p
artn
ersh
ip, c
hang
e is
impo
ssib
le.
We'
ve lo
oked
at t
he r
oles
of
the
teac
her
and
stud
ent,
of m
athe
mat
ics
educ
atio
nan
d le
arni
ng, a
nd o
f th
e co
mm
unity
in th
e ch
ange
pro
cess
. Wha
t will
be
requ
ired
for
pur
pose
ful r
estr
uctu
ring
with
bui
lt-in
acc
ount
abili
ty?
A c
ritic
alva
riab
le w
ill b
e pe
rsev
eran
ce. I
ts p
oten
tial r
ewar
d? M
athe
mat
ical
pow
er!
9 .
Mak
e M
athe
mat
ical
Pow
er a
Rea
lity
for
AL
L S
tude
nts
Prob
lem
Sol
ving
Com
mun
icat
ion
Rea
soni
ngC
onne
ctio
ns
Mat
hem
atic
sE
ssen
tial S
kills
and
NC
TM
Stan
dard
3.)
DA
P
A Port
folio
3
From
Goa
ls to
Act
ion:
It's
Tim
e
In th
e pa
st. t
he A
rizon
a E
ssen
tial S
kills
/OrM
athe
mat
ics
has
been
inte
rpre
ted
by s
ome
as a
list
of m
inim
al e
xpec
tatio
ns fo
r st
uden
t per
form
ance
, whi
ch h
asge
nera
lly b
een
dem
onst
rate
d by
com
putin
g an
alg
orith
m o
ut o
f con
text
. In
fact
,th
e E
ssen
tial S
kills
are
mul
ti-fa
cete
d. T
here
fore
, the
Com
mitt
ee a
ligne
d th
eA
7izo
na E
ssen
tial S
kills
to th
e :V
CT
M S
tand
ards
in o
rder
to s
how
the
man
yla
cets
of t
he E
ssen
tial S
kills
. By
corr
elat
ing
them
to th
eSt
anda
rds,
grea
ter
:tent
h. m
eani
ng a
nd u
nder
stan
ding
of t
he A
rizon
a E
ssen
tial S
kills
are
illu
s-tr
ated
.
The
rem
aind
er o
f thi
s do
cum
ent i
s di
vide
d in
to th
ree
sect
ions
. Eac
h se
ctio
nad
dres
ses
the
thre
e go
als
for
prep
arin
g A
rizon
as s
tude
nts
for
the
2Ist
cen
tury
.
Goa
l 1:
Res
truc
ture
mat
hem
atic
s cu
rric
ula
and
teac
hing
str
ateg
ies
to c
hang
e m
athe
mat
ics
from
a s
tatic
dis
cipl
ine
to a
dyn
amic
proc
ess.
Goa
l 2:
Set h
igh
stan
dard
s of
num
erac
y fo
r al
l stu
dent
s.
Goa
l 3:
Inte
grat
e st
uden
t ass
essm
ent w
ith th
e le
arni
ng p
roce
ss.
3
All
thre
e go
als
have
bee
n co
rrel
ated
to th
e N
CT
M S
tand
ards
. The
rel
atio
nshi
pbe
twee
n ea
ch g
oal a
nd th
eSt
anda
rds
and
its im
pact
on
the
Ariz
ona
Mat
hem
at-
ics
Ess
entia
l Ski
lls w
ill b
e hi
ghlig
hted
in e
ach
sect
ion.
The
Nat
iona
l Cou
ncil
of T
each
ers
of M
athe
mat
ics
Cur
ricu
lum
and
Eva
luat
ion
Stan
dard
sis
div
ided
into
two
maj
or s
ectio
ns. I
n th
e fir
st s
ectio
n, 1
3-14
stan
dard
s ar
e id
entif
ied
for
each
gra
de le
vel t
o de
scrib
e w
hat a
ll st
uden
ts s
houl
dbe
abl
e to
do
in o
rder
to b
e m
athe
mat
ical
ly e
mpo
wer
ed. T
he fi
rst f
our
of th
ese
stan
dard
s re
late
pre
dom
inan
tly to
the
proc
ess
of te
achi
ng m
athe
mat
ics,
whi
leth
e re
mai
nder
rel
ate
mor
e to
the
cont
ent a
nd c
onte
xt o
f the
con
cept
s th
at s
houl
dhe
taug
ht. T
he s
econ
d se
ctio
n of
the
docu
men
t con
cent
rate
s on
pro
gram
eval
uatio
n an
d st
uden
t ass
essm
ent.
3 3
GO
AL
1. R
estr
uctu
re M
athe
mat
ics
Cur
ricu
la a
nd T
each
ing
Stra
tegi
es to
Cha
nge
Mat
hem
atic
s fr
om a
Sta
tic D
isci
plin
e to
a D
ynam
ic P
roce
ss
The
fir
st f
our
stan
dard
s fo
r al
l gra
de le
vels
. K-1
2. a
re
Mat
hem
atic
s as
Pro
blem
Sol
ving
Mat
hem
atic
s as
Com
mun
icat
ion
Mat
hem
atic
s as
Rea
soni
ngM
athe
mat
ical
Con
nect
ions
The
se f
our
stan
dard
s pe
rmea
te e
very
Ess
entia
l Ski
ll in
this
doc
umen
t. T
hey
are
the
proc
ess,
or
vehi
cle,
by
whi
chth
e E
ssen
tial S
kills
sho
uld
be ta
ught
.L
earn
ing
an E
ssen
tial S
kill
out o
f th
e co
ntex
t of
real
-lif
e ap
plic
atio
n an
dw
ithou
t con
cept
ual u
nder
stan
ding
is a
n ex
erci
se in
fut
ility
. A c
ompr
ehen
sive
unde
rsta
ndin
g of
the
Ess
entia
l Ski
lls c
anno
t occ
ur w
ithou
t int
egra
ting
and
appl
ying
thes
e fo
ur s
tand
ards
thro
ugho
ut in
stru
ctio
n.
"Pro
blem
sol
ving
sho
uld
be th
e ce
ntra
l foc
us o
f th
e m
athe
mat
ics
curr
icul
um.-
(NC
T14
Sta
ndar
ds. 1
989,
23
) It
is th
e ve
hicl
e th
at is
nec
essa
ry to
cre
ate
a fu
ll, r
ich
clas
sroo
m e
nvir
onm
ent.
It g
ives
stu
dent
s th
e op
port
unity
to c
omm
unic
ate
thei
rth
inki
ng, d
evel
op h
ighe
r-or
der
reas
onin
g sk
ills
and
mak
e co
nnec
tions
.
Stud
ents
nee
d to
hav
e m
any
oppo
rtun
ities
to c
omm
unic
ate
to d
iscu
ss,
desc
ribe
and
exp
lain
thei
r th
inki
ng r
egar
ding
mat
hem
atic
al c
once
pts.
Thi
sin
clud
es c
lass
room
and
gro
up d
iscu
ssio
ns, d
iagr
ams
or p
ictu
res
repr
esen
ting
mat
hem
atic
al th
inki
ng, a
nd o
ral o
r w
ritte
n de
scri
ptio
ns th
at c
lari
fy s
tude
nts'
thin
king
abo
ut m
athe
mat
ical
con
cept
s.
Prob
lem
sol
ving
..."p
lace
s cr
itica
l thi
nkin
g at
the
hear
t of
inst
ruct
ion.
- (N
CT
MSt
amla
rds,
198
9. 2
9) I
t inv
ites
stud
ents
to r
easo
nto
thin
k fo
r th
emse
lves
,ap
ply
diff
eren
t str
ateg
ies
and
mak
e th
e co
nnec
tions
nec
essa
ry to
und
erst
and
and
appl
y th
e m
athe
mat
ical
con
cept
s.
Usi
ng a
pro
blem
-sol
ving
app
roac
h to
lear
ning
allo
ws
stud
ents
to m
ake
conn
ec-
tions
. All
stud
ents
K-1
2 m
ust h
e gi
ven
the
oppo
rtun
ity to
exp
lore
new
mat
hem
ati-
9
cal c
once
pts
usin
g co
ncre
te m
ater
ials
. The
y sh
ould
be
assi
sted
thro
ugh
the
disc
over
y pr
oces
s in
ord
er to
hel
p th
em d
evel
op w
ithin
thei
r ow
n m
inds
an
unde
rsta
ndin
g of
the
conc
epts
. The
teac
her
guid
es th
is p
roce
ss b
y he
lpin
g th
est
uden
ts c
lari
fy th
eir
thin
king
and
mak
e co
nnec
tions
bet
wee
n th
e co
ncre
te a
ndth
e ab
stra
ct. B
y cr
eatin
g an
und
erst
andi
ng f
rom
with
in, s
tude
nts
are
mor
e ab
le to
mak
e co
nnec
tions
bet
wee
n di
ffer
ent m
athe
mat
ical
str
ands
and
app
ly th
e co
ncep
tsin
rea
l-lif
e si
tuat
ions
.
The
se s
tand
ards
bri
ng m
athe
mat
ics
to li
fe a
nd e
mpo
wer
stu
dent
s m
athe
mat
i-ca
lly. T
hese
pro
cess
es c
reat
e an
env
iron
men
t tha
t enc
oura
ges
our
stud
ents
tore
ason
, com
mun
icat
e an
d m
ake
conn
ectio
ns in
pro
blem
-sol
ving
situ
atio
ns. B
yin
tegr
atin
g th
ese
four
sta
ndar
ds in
to th
e in
stru
ctio
nal p
roce
ss. m
athe
mat
ics
will
be c
hang
ed f
rom
a s
tatic
dis
cipl
ine
to a
dyn
amic
pro
cess
.
Whe
n ex
amin
ing
the
rem
aind
er o
f th
e N
CT
M S
tand
ards
, it b
ecom
es a
ppar
ent
that
pro
cess
can
not b
e se
para
ted
from
con
tent
: the
refo
re, a
mor
e th
orou
ghex
plan
atio
n of
eac
h of
the
firs
t fou
r st
anda
rds
is in
clud
ed u
nder
Goa
l 2. S
ince
the
firs
t fou
r st
anda
rds
perm
eate
all
of th
e E
ssen
tial S
kills
, the
for
mat
on
the
mat
rix
foun
d in
Goa
l 2 a
nd th
e ex
plan
atio
n of
the
firs
t fou
r st
anda
rds
vary
slig
htly
fro
mth
e re
mai
nder
of
the
stan
dard
s.
3 5
I)
GO
AL
2.S
et H
igh
Sta
ndar
ds o
f Num
erac
y fo
r A
ll S
tude
nts
All
stud
ents
can
do
mat
hem
atic
s, y
et p
rese
nt-d
ay m
athe
mat
ics
curr
icul
a an
din
stru
ctio
n ac
t as
a si
eve
to w
eed
out a
ll bu
t the
ver
y be
st. M
ost m
athe
mat
ics
clas
ses
teac
h a
trad
ition
al c
urric
ulum
usi
ng tr
aditi
onal
str
ateg
ies
a pr
actic
e th
at
keep
s m
ost s
tude
nts
from
rea
chin
g th
eir
true
pot
entia
l.
Tea
cher
s ne
ed to
re-
exam
ine
wha
t is
bein
g ta
ught
and
how
it is
bei
ng ta
ught
. We
all m
ust b
egin
to lo
ok a
t the
mat
hem
atic
s cu
rric
ulum
diff
eren
tly a
nd e
xam
ine
wha
t is
bein
g do
ne in
the
pres
ent-
day
curr
icul
um th
at a
cts
as a
bar
rier
tom
athe
mat
ics
achi
evem
ent,
As
we
look
at w
hat w
e ca
n ch
ange
to m
ake
that
hap
pen,
we
real
ize
that
we
are
talk
ing
not o
nly
abou
t cur
ricul
um b
ut a
lso
abou
t ins
truc
tion
and
perc
eptio
nsto
war
d m
athe
mat
ics.
In o
rder
to g
ive
ever
y st
uden
t an
oppo
rtun
ity to
ach
ieve
mat
hem
atic
al p
ower
, it
is n
eces
sary
tbr
scho
ols
to
"Res
truc
ture
the
curr
icul
wn
so th
at lo
ng-t
erm
out
com
es e
mph
asiz
e th
ink-
ing.
rat
her
than
mem
oriz
ing
fact
s.R
ethi
nk h
ow in
stru
ctio
n is
del
iver
ed s
o th
at th
e st
ruct
ure
of s
choo
ls fo
cuse
son
suc
cess
, not
failu
re, o
f stu
dent
s.H
elp
teac
hers
impr
ove
thei
r in
stru
ctio
nal s
trat
egie
s so
that
they
can
succ
eed
in te
achi
ng s
tude
nts
with
diff
eren
t lea
rnin
g st
yles
, rat
es, a
nd le
vels
of m
otiv
atio
n.C
reat
e a
scho
ol c
omm
unity
that
add
ress
es th
e m
any
need
s of
chi
ldre
n an
dyo
uth,
nee
ds th
at a
ffect
how
stu
dent
s le
arn
and
prog
ress
." (
Lew
is, 1
989,
140)
How
Thi
s Se
ctio
n Is
Org
aniz
ed
As
in th
e 19
87 A
rizon
a E
ssen
tial S
kills
.for
Mat
hem
atic
s, th
is s
ectio
n of
thc
docu
men
t is
divi
ded
into
thre
e pa
rts:
K-3
, 4-8
and
9-1
2. In
eac
h pa
rt, t
he E
ssen
tial
Ski
lls a
re p
rese
nted
in th
eir
orig
inal
ord
er a
nd a
re c
orre
late
d, in
a m
atrix
form
at,
to th
e S
tand
ards
set
fort
h by
the
Nat
iona
l Cou
ncil
of T
each
ers
of M
athe
mat
ics.
10
The
firs
t fou
r st
anda
rds
perm
eate
all
of th
e E
ssen
tial S
kills
, and
the
boxe
s ha
ve b
een
shad
ed to
illu
stra
te th
eir
rela
tions
hip.
Dire
ct r
elat
ions
hips
bet
wee
n co
nten
t ski
llsar
e in
dica
ted
by u
sing
an
X.
Bot
h th
e A
rizon
a E
ssen
tial S
kills
and
the
NC
TM
Sta
ndar
ds a
re o
pen
to a
wid
era
nge
of in
terp
reta
tion.
If n
o re
latio
nshi
p ha
s be
en d
elin
eate
d, th
is d
oes
not
elim
inat
e th
e po
ssib
ility
of c
orre
latio
n. F
urth
er, i
t is
not t
he C
omm
ittee
's in
tent
to r
estr
ict t
he tr
ansf
er fr
om a
ny o
ne a
rea
to a
noth
er. (
In fa
ct, t
here
wer
e m
embe
rsof
the
Com
mitt
ee w
ho fe
lt th
ey c
ould
not
lim
it th
e po
ssib
ility
of t
rans
fer
and
that
virt
ually
all
area
s sh
ould
be
mar
ked.
)
To
help
the
user
bec
ome
fam
iliar
with
the
mul
tifac
eted
rela
tions
hip
of th
eN
CT
M S
tand
ards
to th
e A
rizon
a E
ssen
tial S
kills
, we
have
illu
stra
ted
a po
rtio
nof
the
K-3
mat
rix (
see
page
11)
. As
you
can
see
on th
e m
atrix
, ski
ll 1-
3,D
emon
stra
te a
n un
ders
tand
ing
of th
e m
eani
ng o
f the
four
bas
ic o
pera
tions
, can
no lo
nger
be
taug
ht a
s an
isol
ated
alg
orith
m. R
athe
r, s
tude
nts
mus
t be
give
n th
eop
port
unity
to a
pply
the
skill
in p
robl
em-s
olvi
ng s
ituat
ions
, com
mun
icat
e th
eir
unde
rsta
ndin
g, d
evel
op lo
gica
l rea
soni
ng s
kills
, and
mak
e m
athe
mat
ical
con
nec-
tions
thro
ugh
real
-wor
ld a
pplic
atio
ns. A
pply
ing
this
ski
ll in
the
cont
ext o
fes
timat
ion,
con
cept
s of
who
le n
umbe
r op
erat
ions
, and
who
le n
umbe
r co
mpu
ta-
tion
enab
les
the
stud
ent t
o de
mon
stra
te a
mor
e co
mpl
ete
unde
rsta
ndin
g of
the
skill
. Fur
ther
, stu
dent
s sh
ould
be
able
to a
pply
thes
e op
erat
ions
thro
ugho
ut th
ere
mai
ning
sta
ndar
ds.
3 7
Ariz
ona
Nla
them
atic
s E
ssen
tial S
kills
Mat
hem
atic
al O
utco
mes
Exp
ecte
d of
Prim
ary
Stu
dent
s
Mat
hem
atic
s In
stru
ctio
n in
(ir
ate.
K-3
shou
ld e
nsur
e th
at s
tude
nts
,om
plet
ing
Gra
deha
s e
espe
nenc
es w
hich
ena
ble
them
to p
erfo
rm th
eto
lloss
111!
1.N
umbe
r
Cou
ntb
. one
s i%
% o
tis
es a
ndte
ns
ardl
nal a
nd o
rdin
al n
umhe
rs to
t:MIT
:Ire
and
ord
er q
uann
tle...
iem
onst
rate
an
um:e
l-st
andi
ng o
t the
mea
ning
of
the
four
haq
cop
erat
ions
.1I l
as.:
ah b
asis
addt
tion.
sub
trac
tion
and
mul
tiplic
atio
nta
ct, i.i
incr
ete
mat
eria
ls o
r nu
idel
s to
dem
onst
rate
an
unde
rsta
ndin
gpl
ace
alue
Sta
ndar
ds
= 71 -g
92 z
5F
g
C
A-
, cE
E.
=
:
z
.X-z
.e.
::::.
-7,
,..f
f2, 7
-, x
'. t.1
.4
VIX
XX
X4
Xx
X
xxxx
XX
XX
xxX
X
XX
XX
X
3
Fol
loin
g th
e m
atric
es a
re c
hart
s in
dica
ting
stud
ent o
utco
mes
and
exa
mpl
es o
fpe
rfor
man
ce in
dica
tors
for
the
rem
aini
ng s
tand
ards
. The
stu
dent
out
com
es li
sted
ith e
ach
stan
dard
are
the
Ess
entia
l Ski
lls th
at h
ave
been
dire
ctly
cor
rela
ted
toth
at s
tand
ard.
The
exa
mpl
es o
f ind
icat
ors
refe
r sp
ecifi
cally
to h
ow s
tude
nts
mig
ht d
emon
stra
te a
n un
ders
tand
ing
of th
e E
ssen
tial S
kills
in li
ght o
f tha
tst
anda
rd o
nly.
Sta
ndar
d 5:
Est
imat
ion
Est
imat
ion
is a
n es
sent
ial c
ompo
nent
of m
athe
mat
ics
sinc
e m
athe
mat
ics
invo
lves
mor
e th
an e
xact
ness
.E
stim
atio
n in
tera
cts
with
num
ber
sens
e an
d sp
atia
l sen
se to
ass
ist s
tude
nts
in d
es e
lopi
ng in
sigh
ts in
toco
ncep
ts a
nd p
roce
dure
s. e
nabl
ing
them
to w
ork
flexi
bly
with
num
ber
and
mea
sure
men
t and
toev
alua
te th
e re
ason
able
ness
of r
esul
ts.
Stu
dent
Out
com
esE
xam
ples
of I
ndic
ator
s
Dem
onst
rate
an
unde
rsta
ndin
g of
the
mea
ning
of
App
ly a
var
iety
of e
stim
atio
n st
rate
gies
.th
e fo
ur b
asic
ope
ratio
ns. (
I-3)
Rec
ogni
ze w
hen
estim
atio
n is
app
ropr
iate
.
mul
tiplic
atio
n fa
cts.
(I-
4)D
eter
min
i_ th
e re
ason
able
ness
of r
esul
ts.
Hav
e fa
cilit
y w
ith b
asic
add
ition
, sub
trac
tion
and
Est
imat
e an
swer
s to
com
puta
tiona
l pro
blem
s.A
pply
est
imat
i.m in
sso
rkin
g w
ith q
uant
ities
.(1
-))
mea
sure
men
t, co
mpu
tatio
n an
d pr
oble
m-
solv
ing.
Add
or
subt
ract
two
thre
e-di
git w
hole
num
bers
.11
-9
Rec
ogni
ze a
nd c
ount
mon
ey. (
1- 1
61
It sh
ould
he
note
d th
at th
e A
rizon
a E
ssen
tial S
kills
are
pre
sent
ed in
a K
-3 a
nd 4
-8
form
at, w
hile
the
NC
TM
Sta
ndar
dsha
ve a
K-4
and
5-8
form
at. S
ince
four
thgr
ade
is a
yea
r of
tran
sitio
n fr
om p
rimar
y to
inte
rmed
iate
leve
l, th
e C
omm
ittee
belie
ves
that
the
non-
para
llel f
ocus
bet
wee
n th
e E
ssen
tial S
kills
and
the
NC
7MSt
anda
rds
is to
our
ben
efit.
The
NC
TM
Sta
ndar
dsad
dres
ses
the
prim
ary
focu
sfo
r fo
urth
gra
de, w
hile
the
Ess
entia
l Ski
lls a
re ti
ed d
irect
ly to
the
5-8
sect
ion
inth
is d
ocum
ent,
setti
ng th
e fo
unda
tion
and
dire
ctio
n fo
r th
e tr
ansi
tion
from
prim
ary
to in
term
edia
te le
vels
of m
athe
mat
ics.
It is
impe
rativ
e th
at fo
urth
gra
decu
rric
ulum
and
inst
ruct
iona
l pra
ctic
es in
terw
eave
the
fund
amen
tals
of b
oth
the
K-3
and
4-8
sec
tions
that
follo
w.
4 0
K-3
MA
TH
EM
AT
ICS
ESS
EN
TIA
L S
KIL
LS
BE
ST C
hP1
AV
AIL
AB
LE
13
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
Mat
hem
atic
al O
utco
mes
Exp
ecte
d of
Pri
mar
y St
uden
ts
Mat
hem
atic
s in
stru
ctio
n in
Gra
des
K-3
sho
uld
ensu
re th
at s
tude
nts
com
plet
ing
Gra
de 3
hav
e ex
peri
ence
s w
hich
ena
ble
them
to p
erfo
rm th
e fo
llow
ing
outc
omes
:
1.N
umbe
r
Cou
nt b
y on
es, t
wo,
fiv
es a
nd te
ns.
1. U
se c
ardi
nal a
nd o
rdin
al n
umbe
rs to
com
pare
and
ord
er q
uant
ities
.
3. D
emon
stra
te a
n un
ders
tand
ing
of th
e m
eani
ng o
f th
e fo
ur b
asic
ope
ratio
ns.
4. H
ave
faci
lity
with
bas
ic a
dditi
on, s
ubtr
actio
n an
d m
ultip
licat
ion
fact
s.
5. U
,c c
oncr
ete
mat
eria
ls o
r m
odel
s to
dem
onst
rate
an
unde
rsta
ndin
g of
pla
ce v
alue
.
6. E
xplo
re th
e co
ncep
ts o
f m
ultip
licat
ion
and
divi
sion
with
con
cret
e m
ater
ials
.
7 R
ead
and
wri
te th
e nu
mbe
r re
pres
ente
d w
hen
obje
cts
are
grou
ped
by h
undr
eds,
tens
and
one
s.
8. E
stim
ate
answ
ers
to c
ompu
tatio
nal p
robl
ems.
9. A
dd o
r su
btra
ct tw
o th
ree-
digi
t who
le n
umbe
rs.
10. C
hoos
e th
e ap
prop
riat
e op
erat
ion
to b
e us
ed in
a g
iven
situ
atio
n.
IIIn
terp
ret w
ord
prob
lem
s by
usi
ng r
ole
play
ing.
pic
ture
s an
d m
odel
s.
12. W
rite
mat
hem
atic
al s
ente
nces
to r
epre
sent
a s
ituat
ion,
15
K-3
Stan
dard
s
01) c > -
8 E -
c ...-; g n § (..1
. E ce'
r..-
.,
' .° t -,' (5 .:1-
g..L
--.
. 17: *z
i
u-
5 v) t E 2 .6
5 f.i k 0 .. § 4E
7g.
I: 0 ' ' -
- cjl a 6 r-
-:
. ..F.3 5 . t..)
- cg Z 5 oc
i
. ci. i
7,5 0. 5 r 6' 0.:
'5,
fE ... i
0...:
..:'
--2-
)Eg
2 0 *2 ° 6 --:
, 4 .5 ci,,
"2 .f* (--,
i
'a.
:5'
.z... ti -0 5 , r-;
X X
XX
XX
X X
XX
X
X X
XX
XX
X
X X
XX
XX
X
X X
XX
XX
X X
XX
X
X X
XX
X X
XX
XX
XX
xxxx
xX
X X
XX
X
XX
XX
X
Xxx
xX
XX
4 3
2 3 4 5 6 7 8 9 10 11 12
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
(co
ntin
ued)
I.N
umbe
r (c
ontin
ued)
13. U
se in
form
ally
the
prop
ertie
s of
com
mut
ativ
ity, a
ssoc
iativ
ity a
nd id
entit
y.
14. U
se c
oncr
ete
mat
eria
ls to
rec
ogni
ze. r
epre
sent
and
com
pare
hal
ves,
thir
ds a
nd f
ourt
hs.
15. R
ecog
nize
fra
ctio
nal e
quiv
alen
ts f
or h
alve
s. f
ourt
hs a
nd te
nths
.
16. R
ecog
nize
and
cou
nt m
oney
.
17. U
se m
oney
to r
epre
sent
and
com
pare
dec
imal
val
ues.
II.
Mea
sure
men
t
I.
Use
non
stan
dard
, met
ric
and
Eng
lish
units
of
mea
sure
to e
stim
ate
and
mea
sure
leng
th. v
olum
e an
d w
eigh
t.
2. U
se d
igita
l and
con
vent
iona
l clo
cks
to te
ll tim
e.
3. R
ead
and
inte
rpre
t Cel
sius
and
Fah
renh
eit t
empe
ratu
res
on th
erm
omet
ers.
4. C
hoos
e an
app
ropr
iate
uni
t of
mea
sure
in a
giv
en s
ituat
ion.
5. l'
se a
arie
tv o
f m
easu
rem
ent i
nstr
umen
ts.
K-3
Stan
dard
s
t f,
. >"
.... .;
67 uE9,
2g_o
u.-R
E--
g, . -
o .t., E . c
c .,..
..,
a . ,.;.-2
, . ,2,), a t., E . ..
g ,. . -2 z -5 3 ,-.. 0 r.,. .
r--
., _ 2 s g. u 2 E .00
c-> .: 5 En
:7j
-,2 .a. .
Cr,
.,, ,..--
a-E
- .C
Z: -
>...
...e;
..,s -" .
-- -
.L._
., c'' , .
("I -
.:.. 5 w -0
. r- -
XX
X
X X
XX
X X
X.X
.X
X X
XX
XX
X X
XX
X X
XX
XX
X
XX
XX
XX
x X
XX
,XX
X
XX
X,
xx
X X
X.X
13 14 15 16 17
1 2 3 4
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
(co
ntin
ued)
III.
Geo
met
ry
I.U
se te
rmin
olog
y ap
prop
riat
e to
the
grad
e le
vel.
2.U
se v
isua
l attr
ibut
es, c
oncr
ete
mat
eria
ls a
nd a
ppro
pria
te v
ocab
ular
y to
iden
tify,
cla
ssif
y an
d de
scri
be c
omm
onge
omet
ric
figu
res
and
mod
els.
3.U
se s
ever
al g
eom
etri
c sh
apes
to m
ake
othe
r ge
omet
ric
shap
es.
4.D
ecid
e w
heth
er f
igur
es a
re c
ongr
uent
and
whe
ther
they
are
sim
ilar.
5.E
xplo
re th
e fi
lling
of
spac
e us
ing
man
ipul
ativ
es.
6.U
sc m
anip
ulat
ive
mat
eria
ls to
dev
elop
the
conc
epts
of
poin
t. lin
e an
d lin
e se
gmen
t.
IV. P
atte
rns
and
Rel
atio
ns
I.Id
entif
y, d
escr
ibe
and
exte
nd a
pat
tern
in a
seq
uenc
e of
obj
ects
.
2.L
ke a
con
cret
e m
odel
to c
reat
e a
patte
rn a
nd r
epre
sent
that
pat
tern
sym
bolic
ally
.
3.D
escr
ibe
the
rela
tions
hip
give
n in
a ta
ble
of n
umbe
rs d
eriv
ed f
rom
a s
eque
nce
of o
bjec
ts.
4.D
eter
min
e a
loca
tion
by u
sing
ord
ered
pai
rs o
f nu
mbe
rs o
n a
rect
angu
lar
grid
.
4 k;
I 7
K-3
Stan
dard
s
.IJ
E.
.,.. t) .7
-6, 0
E E , ri
0 ' m ,z,
,-. 0 w g.,-
--
.-1:
5 '177
..;
tr:.
cr;
tt, .0 g .6
5 .....-
.:
^.1
"*- c z :-.. -' -,,, '6. 8 1.
-:
., 13 C., .-. m 4.
,,.
5 (1) e..,
`a)
E
:....
.
.:-..;
c. ,.
XX
XX
XX
XX
X
XX
XX
XX
XX
X
XX
X,
X
X3(
X X
X
XX
X
XX
1X
XX
XX
XX
X
X X
XX
47
-) 3 4 5 6 2 3 4
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
(co
ntin
ued)
V. D
ata
Ana
lysi
s an
d Pr
obab
ility
I.
Col
lect
. org
aniz
e, r
epre
sent
and
inte
rpre
t dat
a de
rived
from
sur
veys
and
exp
erim
ents
con
duct
ed b
y th
est
uden
ts.
2.C
reat
e an
d in
terp
ret c
oncr
ete,
pic
toria
l and
sym
bolic
gra
phs.
3.P
erfo
rm s
impl
e ac
tiviti
es in
volv
ing
prob
abili
ty.
VI.
Ana
lytic
al R
easo
ning
I.C
lass
ify
and
sort
obj
ects
by
obse
rvin
g re
latio
nshi
ps a
nd m
akin
g ge
nera
lizat
ions
.
2.M
ake
reas
onab
le o
r lo
gica
l con
ject
ures
and
con
clus
ions
abo
ut s
ituat
ions
with
con
cret
e m
ater
ials
, usi
ng s
uch
wor
ds a
s th
ese:
and
, or,
if ..
. the
n, a
ll, s
ome,
non
e, n
ot a
nd o
ut o
f.
18
K-3
Stan
dard
s
bb C : 3_73
E04
,-0 c ,..
,
C E c CA
',.1
c . ,
(,)"
c E - Ow
Zr.
..). 7
2.,(
0.4.
gp'..
F...
.2.- 7, ,r
,
ii-! °'-c
Ec3
.ca-
04.
3 E = ,C
F..
ifi .. ifi- , ,..., ... E = ..,.
....
0...)
7^' Z a u c9 r
"--
,...
...., ,-.
:-:: = a E 0 (..) , 2 .? r- .0 00
, , . '.) cn - .0 7,5. c. C/) -0 - e 5 0 0,
,.1). 2 C
>, - cs . .,
in' .0 u C
S (- 1
:,... a , . . a
7.1 E a 14-, .
)(M
O(
X
XX
Xj(
X
XL
XX
XX
X
X X
XX
XX
XX
XX
4 i)
1 2 3 1 2
Stan
dard
1: M
athe
mat
ics
as P
robl
em S
olvi
ng
Pro
blem
sol
ving
is n
ot a
dis
tinct
topi
c bu
t a p
roce
ss th
at s
houl
d pe
rmea
te th
e en
tire
prog
ram
and
pro
vide
the
cont
ext i
n w
hich
con
cept
s an
d sk
ills
can
be le
arne
d.
Stud
ent O
utco
mes
Exa
mpl
es o
f In
dica
tors
Use
all
of th
e m
athe
mat
ics
Ess
entia
l Ski
lls in
a p
robl
em-s
olvi
ng c
onte
xt.
19
Use
pro
blem
-sol
ving
app
roac
hes
to in
vest
igat
e an
d un
ders
tand
mat
hem
ati-
cal c
once
pt.
For
mul
ate
prob
lem
s fr
om e
very
day
and
mat
hem
atic
al s
ituat
ions
.
Dev
elop
and
app
ly s
trat
egie
s to
sol
ve a
wid
e va
riety
of p
robl
ems.
Ver
ify a
nd in
terp
ret r
esul
ts w
ith r
espe
ct to
the
orig
inal
pro
blem
.
Dem
onst
rate
con
fiden
ce in
usi
ng m
athe
mat
ics
mea
ning
fully
.
5 1
K-3
Stan
dard
2: M
athe
mat
ics
as C
omm
unic
atio
n
Com
mun
icat
ing
mat
hem
atic
al th
inki
ng p
rovi
des
a ve
hicl
e fo
r st
uden
ts to
exp
lore
thei
r id
eas
abou
t mat
hem
atic
al c
once
pts,
cla
rify
and
str
engt
hen
thei
r un
ders
tand
ing,
and
cons
truc
t lin
ks b
etw
een
conc
epts
and
rea
l-lif
e si
tuat
ions
. Stu
dent
s ne
ed to
be
give
n am
ple
time
to e
xplo
re, i
nves
tigat
e, d
escr
ibe
and
disc
uss
thei
r th
inki
ng a
s th
eyw
ork
with
mat
hem
atic
al c
once
pts.
Stud
ent O
utco
mes
Exa
mpl
es o
f In
dica
tors
Com
mun
icat
e an
d ex
plai
n th
inki
ng a
bout
all
of th
e m
athe
mat
ics
Ess
entia
lSk
ills
in a
var
iety
of
cont
exts
.
20
Rel
ate
phys
ical
mat
eria
ls, p
ictu
res
and
diag
ram
s to
mat
hem
atic
al id
eas.
Ref
lect
on
and
clar
ify
thei
r th
inki
ng a
bout
mat
hem
atic
al id
eas
and
situ
a-tio
ns.
Rel
ate
thei
r ev
eryd
ay la
ngua
ge to
mat
hem
atic
al la
ngua
ge a
nd s
ymbo
ls.
Dem
onst
rate
how
rep
rese
ntin
g, d
iscu
ssin
g, r
eadi
ng, w
ritin
g an
d lis
teni
ngto
mat
hem
atic
s ar
e a
vita
l par
t of
lear
ning
and
usi
ng m
athe
mat
ics.
Stan
dard
3: M
athe
mat
ics
as R
easo
ning
Mat
hem
atic
s is
rea
soni
ng. O
ne c
anno
t do
mat
hem
atic
s w
ithou
t rea
soni
ng. C
ritic
al th
inki
ng m
ust b
e pl
aced
at t
he h
eart
of
inst
ruct
ion
by p
rovi
ding
stu
dent
s w
ith m
any
oppo
rtun
ities
to e
xplo
re a
nd e
valu
ate
prob
lem
s, m
ake
conj
ectu
res
and
expl
ain
and
just
ify
thei
r th
inki
ng, a
nd s
olve
pro
blem
s in
a v
arie
ty o
f w
ays.
It i
s as
impo
rtan
t
for
stud
ents
to e
xpla
in h
ow th
ey s
olve
d a
prob
lem
as
the
solu
tion
itsel
f.
Stud
ent O
utco
mes
Use
exp
lora
tion.
dis
cuss
ion
and
conj
ectu
re to
eva
luat
e an
d ju
stif
y th
inki
ngab
out e
ach
of th
e m
athe
mat
ics
Ess
entia
l Ski
lls w
ithin
the
cont
ext o
f a
prob
lem
or in
rea
l-lif
e si
tuat
ions
.
2 1
Exa
mpl
es o
f In
dica
tors
Dra
w lo
gica
l con
clus
ions
abo
ut m
athe
mat
ics.
Use
mod
els,
kno
wn
fact
s, p
rope
rtie
s an
d re
latio
nshi
ps to
exp
lain
thei
rth
inki
ng.
Just
ify
thei
r an
swer
s an
d so
lutio
n pr
oces
ses.
Use
pat
tern
s an
d re
latio
nshi
ps to
ana
lyze
mat
hem
atic
al s
ituat
ions
.
5 5
K-3
Stan
dard
4: M
athe
mat
ical
Con
nect
ions
Mat
hem
atic
al c
onne
ctio
ns a
re th
e th
read
s th
at h
old
the
fabr
ic o
f m
athe
mat
ics
toge
ther
. Stu
dent
s ne
ed to
mak
e m
athe
mat
ical
con
nect
ions
bet
wee
n th
e co
ncre
te a
ndth
e ab
stra
ct, b
etw
een
the
diff
eren
t mat
hem
atic
al s
tran
ds, a
nd b
etw
een
the
clas
sroo
m p
robl
ems
and
the
real
wor
ld.
Stud
ent O
utco
mes
Mak
e co
nnec
tions
bet
wee
n th
e m
athe
mat
ics
Ess
entia
l Ski
lls a
nd th
eir
man
yle
vels
of
appl
icat
ion.
J 0
1 1
Exa
mpl
es o
f In
dica
tors
Lin
k co
ncep
tual
and
pro
cedu
ral k
now
ledg
e.
Rel
ate
vari
ous
repr
esen
tatio
ns o
f co
ncep
ts o
r pr
oced
ures
to o
ne a
noth
er.
Rec
ogni
ze r
elat
ions
hips
am
ong
diff
eren
t top
ics
in m
athe
mat
ics.
Use
mat
hem
atic
s in
oth
er c
urri
culu
m a
reas
.
Use
mat
hem
atic
s in
rea
l-lif
e ac
tiviti
es.
5 7
Stan
dard
5: E
stim
atio
n
Est
imat
ion
is a
n es
sent
ial c
ompo
nent
of
mat
hem
atic
s si
nce
mat
hem
atic
s in
volv
es m
ore
than
exa
ctne
ss. E
stim
atio
n in
tera
cts
with
num
ber
sens
e an
d sp
atia
l sen
se to
assi
st s
tude
nts
in d
evel
opin
g in
sigh
ts in
to c
once
pts
and
proc
edur
es, e
nabl
ing
them
to w
ork
flex
ibly
with
num
ber
and
mea
sure
men
t and
to e
valu
ate
the
reas
onab
lene
ssof
res
ults
.
Stud
ent O
utco
mes
Dem
onst
rate
an
unde
rsta
ndin
g of
the
mea
ning
of
the
four
bas
ic o
pera
tions
.(1
-3)
Hav
e fa
cilit
y w
ith b
asic
add
ition
, sub
trac
tion
and
mul
tiplic
atio
n fa
cts.
(1-
4)
Est
imat
e an
swer
s to
com
puta
tiona
l pro
blem
s. (
1-8)
Add
or
subt
ract
two
thre
e-di
git w
hole
num
bers
. (1-
9)
Rec
ogni
ze a
nd c
ount
mon
ey. (
1- 1
6)
Use
non
stan
dard
, met
ric
and
Eng
lish
units
of
mea
sure
to e
stim
ate
and
mea
sure
leng
th, v
olum
e an
d w
eigh
t. (I
I- I
)
Use
dig
ital a
nd c
onve
ntio
nal c
lock
s to
tell
time.
(II
-2)
Rea
d an
d in
terp
ret C
elsi
us a
nd F
ahre
nhei
t tem
pera
ture
s on
ther
mom
eter
s.(I
I-3)
Perf
orm
sim
ple
activ
ities
invo
lvin
g pr
obab
ility
. (V
-3)
Mak
e re
ason
able
or
logi
cal c
onje
ctur
es a
nd c
ontk
lusi
ons
abou
t situ
atio
nsw
ith c
oncr
ete
mat
eria
ls, u
sing
suc
h w
ords
as
thes
e:an
d, o
r. if
...th
en. a
ll,so
me,
non
e,no
f an
d m
a 4
(v1-
2)
r fS
23
Exa
mpl
es o
f In
dica
tors
App
ly a
var
iety
of
estim
atio
n st
rate
gies
.
Rec
ogni
ze w
hen
estim
atio
n is
app
ropr
iate
.
Det
erm
ine
the
reas
onab
lene
ss o
f re
sults
.
App
ly e
stim
atio
n in
wor
king
with
qua
ntiti
es, m
easu
rem
ent,
com
puta
tion
and
prob
lem
sol
ving
.
5;4
K-3
Stan
dard
6: N
umbe
r Se
nse
and
Num
erat
ion
Num
ber
sens
e an
d nu
mer
atio
n ar
e th
e de
velo
pmen
t of
who
le n
unth
er c
once
pts
and
skill
s th
roug
h th
e us
e of
con
cret
e m
ater
ial a
nd p
hysi
cal m
anip
ulat
ions
. Thi
s en
able
sch
ildre
n to
bui
ld a
nd e
xten
d nu
mbe
r re
latio
nshi
ps a
nd h
elps
them
to d
evel
op a
link
bet
wee
n th
eir
wor
ld a
nd th
e w
orld
of
mat
hem
atic
s.
Stud
ent O
utco
mes
Cou
nt b
y on
es. t
w o
s. f
ives
and
tens
. ( I
-1 )
Use
car
dina
l and
ord
inal
num
bers
to c
ompa
re a
nd o
rder
qua
ntiti
es. (
1-2)
Use
con
cret
e m
ater
ials
or
mod
els
to d
emon
stra
te a
n un
ders
tand
ing
of p
lace
valu
e. (
I-5
)
Rea
d an
d w
rite
the
num
ber
repr
esen
ted
whe
n ob
ject
s ar
e gr
oupe
d by
hun
dred
s,te
ns a
nd o
nes.
( 1
-7 )
Est
imat
e an
swer
s to
com
puta
tiona
l pro
blem
s. (
1-8)
Use
non
stan
dard
, met
ric
and
Eng
lish
units
of
mea
sure
to e
stim
ate
and
mea
sure
leng
th. v
olum
e an
d w
eigh
t. (I
I-1
)
Rea
d an
d in
tel p
ret C
elsi
us a
nd F
ahre
nhei
t tem
pera
ture
s on
ther
mom
eter
s.(
11-3
)
Cho
ose
an a
ppro
pria
te u
nit o
f m
easu
re in
a g
iven
situ
atio
n. (
24
Exa
mpl
es o
f In
dica
tors
Use
con
cret
e m
ater
ials
to c
onst
ruct
and
exp
lain
num
ber
mea
ning
s an
dop
erat
ions
.
Inte
rpre
t the
mul
tiple
use
s of
num
bers
enc
ount
ered
in th
e re
al w
orld
.
Rel
ate
coun
ting,
gro
upin
g an
d pl
ace
valu
e co
ncep
ts w
ithin
the
base
10
num
erat
ion
syst
em.
6 1
Stan
dard
7: C
once
pts
of W
hole
Num
ber
Ope
ratio
ns
Und
erst
andi
ng th
e fu
ndam
enta
l ope
ratio
ns o
f ad
ditio
n. s
ubtr
actio
n, m
ultip
licat
ion
and
divi
sion
is c
entr
al to
kno
win
g m
athe
mat
ics.
It i
s im
port
ant f
or c
hild
ren
to b
uild
an a
war
enes
s of
mod
els
and
prop
ertie
s of
thes
e op
erat
ions
, see
ing
the
rela
tions
hips
am
ong
them
.
Stud
ent O
utco
mes
Dem
onst
rate
an
unde
rsta
ndin
g of
the
mea
ning
of
the
four
bas
ic o
pera
tions
.(1
-3)
Hav
e fa
cilit
y w
ith b
asic
add
ition
, sub
trac
tion
and
mul
tiplic
atio
n fa
cts.
(I-
4)
Exp
lore
the
conc
epts
of
mul
tiplic
atio
n an
d di
visi
on w
ith c
oncr
ete
mat
eria
ls.
(1-6
)
Est
imat
e an
swer
s to
com
puta
tiona
l pro
blem
s. (
1-8)
Cho
ose
the
appr
opri
ate
oper
atio
n to
be
used
in a
giv
en s
ituat
ion.
(1-
10)
Inte
rpre
t wor
d pr
oble
ms
by u
sing
rol
e pl
ayin
g, p
ictu
res
and
mod
els.
(1-
11)
Wri
te m
athe
mat
ical
sen
tenc
es to
rep
rese
nt a
situ
atio
n. (
1-12
)
Lke
info
rmal
ly th
e pr
oper
ties
of c
omm
utat
ivity
, ass
ocia
tivity
and
iden
tity.
(1-1
3)
6 2
25
Exa
mpl
es o
f In
dica
tors
Mod
el, u
sing
con
cret
e m
ater
ials
, the
mea
ning
for
the
oper
atio
ns.
Rel
ate
the
mat
hem
atic
al la
ngua
ge a
nd s
ymbo
lism
of
oper
atio
ns to
pro
blem
situ
atio
ns a
nd in
form
al la
ngua
ge.
Dem
onst
rate
ope
ratio
n se
nse
by r
ole
play
ing
or m
odel
ing
a w
ide
vari
ety
ofpr
oble
m s
ituat
ions
.
6 3
K-3
Stan
dard
8: W
hole
Num
ber
Com
puta
tion
Who
le n
umbe
r co
mpu
tatio
n en
com
pass
es a
var
iety
of
way
s fo
r ch
ildre
n to
sol
ve p
robl
ems,
fro
m d
evel
opm
enta
l lev
els
usin
g co
ncre
te m
ater
ials
to s
olvi
ng p
robl
ems
by u
sing
abs
trac
t alg
orith
ms.
Thi
s st
reng
then
s a
child
's r
easo
ning
ski
lls, m
athe
mat
ical
insi
ght a
nd c
onfi
denc
e to
do
mat
hem
atic
s.
Stud
ent O
utco
mes
Dem
onst
rate
an
unde
rsta
ndin
g of
the
mea
ning
of
the
four
bas
ic o
pera
tions
.(1
-3)
Hav
e fa
cilit
y w
ith b
asic
add
ition
, sub
trac
tion
and
mul
tiplic
atio
n fa
cts.
(1-
4)
Est
imat
e an
swer
s to
com
puta
tiona
l pro
blem
s. (
I-8)
Add
or
subt
ract
two
thre
e-di
git w
hole
num
bers
. (1-
9)
Cho
ose
the
appr
opri
ate
oper
atio
n to
be
used
in a
giv
en s
ituat
ion.
(1-
10)
Wri
te m
athe
mat
ical
sen
tenc
es to
rep
rese
nt a
situ
atio
n. (
1-12
)
ti 4
26
Exa
mpl
es o
f In
dica
tors
Use
con
cret
e m
ater
ials
to d
emon
stra
te u
nder
stan
ding
of
mat
hem
atic
alpr
oced
ures
.
Dem
onst
rate
rea
sona
ble
prof
icie
ncy
with
bas
ic f
acts
and
alg
orith
ms.
Use
a v
arie
ty o
f m
enta
l com
puta
tion
and
estim
atio
n te
chni
ques
.
Use
cal
cula
tors
in a
ppro
pria
te c
ompu
tatio
nal s
ituat
ions
.
Sele
ct a
nd u
se c
ompu
tatio
n te
chni
ques
app
ropr
iate
to s
peci
fic
prob
lem
s,an
d de
term
ine
whe
ther
the
resu
lts a
re r
easo
nabl
e.
6,i
Stan
dard
9: G
eom
etry
and
Spa
tial S
ense
Geo
met
ry a
nd s
patia
l sen
se h
elp
a st
uden
t to
repr
esen
t and
des
crib
e in
an
orde
rly m
anne
r ou
r in
here
ntly
geo
met
ric w
orld
. Und
erst
andi
ng s
houl
d gr
ow n
atur
ally
from
expl
orat
ion
and
expe
rienc
e w
ith tw
o- a
nd th
ree-
dim
ensi
onal
figu
res
in p
robl
em-s
olvi
ng s
ituat
ions
.
Stud
ent O
utco
mes
Use
non
stan
dard
, met
ric a
nd E
nglis
h un
its o
f mea
sure
to e
stim
ate
and
mea
sure
leng
th. v
olum
e an
d w
eigh
t. (I
I-1
)
Use
term
inol
ogy
appr
opria
te to
the
grad
e le
vel.
(Ill-
I )
Use
vis
ual a
ttrib
utes
, con
cret
e m
ater
ials
and
app
ropr
iate
voc
abul
ary
to id
en-
tify,
cla
ssify
and
des
crib
e co
mm
on g
eom
etric
figu
res
and
mod
els.
(III
-2)
Use
sev
eral
geo
met
ric s
hape
s to
mak
e ot
her
geom
etric
sha
pes.
(III
-3)
Dec
ide
whe
ther
figu
res
are
cong
ruen
t and
whe
ther
they
are
sim
ilar.
(11
1-4)
Exp
lore
the
fillin
g of
spa
ce u
sing
man
ipul
ativ
es. (
III-5
)
Use
man
ipul
ativ
e m
ater
ials
to d
evel
op th
e co
ncep
ts o
f poi
nt, l
ine
and
line
segm
ent.
(III-
6)
Cla
ssify
and
sor
t obj
ects
by
obse
rvin
g re
latio
nshi
ps a
nd m
akin
g ge
nera
liza-
tions
. (V
I-I )
Mak
e re
ason
able
or
logi
cal c
onje
ctur
es a
nd c
oncl
usio
ns a
bout
situ
atio
ns w
ithco
ncre
te m
ater
ials
, usi
ng s
uch
wor
ds a
s th
ese:
and
, or,
if ..
. the
n, a
ll, s
ome,
nolie
, /lo
t and
out
of V
I-21
6)
)
27
Exa
mpl
es o
f In
dica
tors
Des
crib
e, m
odel
, dra
w a
nd c
lass
ify s
hape
s.
Inve
stig
ate
and
pred
ict t
he r
esul
ts o
f com
bini
ng, s
ubdi
vidi
ng a
nd c
hang
ing
shap
es.
Rel
ate
geom
etric
idea
s to
num
ber
and
mea
sure
men
t.
Rec
ogni
ze g
eom
etry
in th
e re
al w
orld
.
,,
K-3
Stan
dard
10:
Mea
sure
men
t
Mea
sure
men
t is
com
pari
son.
A u
nit a
ppro
pria
te f
or th
e at
trib
ute
to h
e m
easu
red
is s
elec
ted,
the
com
pari
son
mad
e. a
nd th
e re
sults
rep
orte
d. S
tude
nts
mus
t dev
elop
an in
form
al u
nder
stan
ding
of
the
proc
ess
of m
easu
rem
ent w
ith n
onst
anda
rd u
nits
bef
ore
stan
dard
inst
rum
ents
are
intr
oduc
ed.
Stud
ent O
utco
mes
Exa
mpl
es o
f In
dica
tors
Use
non
stan
dard
, met
ric
and
Eng
lish
units
of
mea
sure
to e
stim
ate
and
mea
sure
leng
th, v
olum
e an
d w
eigh
t. 01
-1)
Use
dig
ital a
nd c
onve
ntio
nal c
lock
s to
tell
time.
(11
-2)
Rea
d an
d in
terp
ret C
elsi
us a
nd F
ahre
nhei
t tem
pera
ture
s on
ther
mom
eter
s.(
11-3
Cho
ose
an a
ppro
pria
te u
nit o
f m
easu
re in
a g
iven
situ
atio
n. (
11-4
)
Use
a v
arie
ty o
f m
easu
rem
ent i
nstr
umen
ts. (
11-5
)
6 3
28
Use
a v
arie
ty o
f m
easu
rem
ent i
nstr
umen
ts to
det
erm
ine
leng
th, c
apac
ity,
wei
ght,
area
, vol
ume,
tim
e. te
mpe
ratu
re a
nd a
ngle
.
Dev
elop
an
unde
rsta
ndin
g of
the
proc
ess
of m
easu
ring
.
Use
for
mal
and
info
rmal
mea
sure
men
ts in
pro
blem
sol
ving
and
eve
ryda
ysi
tuat
ions
.
6 9
Stan
dard
11:
Sta
tistic
s an
d Pr
obab
ility
Stat
istic
s an
d pr
obab
ility
are
tool
s fo
r pr
oble
m s
olvi
ng. B
y co
llect
ing,
org
aniz
ing,
dis
play
ing
and
thin
king
abo
ut d
ata
in m
any
way
s, w
e le
arn
to u
se n
umbe
rs to
desc
ribe
and
inte
rpre
t the
wor
ld a
roun
d us
.
Stud
ent O
utco
mes
Col
lect
. org
aniz
e, r
epre
sent
and
inte
rpre
t dat
a de
rive
d fr
om s
urve
ys a
ndex
peri
men
ts c
ondu
cted
by
the
stud
ents
. (V
-1 )
Cre
ate
and
inte
rpre
t con
cret
e, p
icto
rial
and
sym
bolic
gra
phs.
(V
-2)
Perf
orm
sim
ple
activ
ities
invo
lvin
g pr
obab
ility
. (V
-3)
19
Exa
mpl
es o
f In
dica
tors
Col
lect
, org
aniz
e an
d de
scri
be d
ata.
Con
stru
ct, r
ead
and
inte
rpre
t dis
play
s of
dat
a.
Form
ulat
e an
d so
lve
prob
lem
s th
at in
volv
e co
llect
ing
and
anal
yzin
g da
ta.
Exp
lore
con
cept
s of
cha
nce
and
mak
e ap
prop
riat
e co
njec
ture
s.
K-3
Stan
dard
12:
Fra
ctio
ns a
nd D
ecim
als
Frac
tions
and
dec
imal
s ar
e ne
cess
ary
to d
escr
ibe
the
real
wor
ld. S
tude
nts
need
opp
ortu
nitie
s to
exp
lore
fra
ctio
nal r
elat
ions
hips
and
bui
ld a
n un
ders
tand
ingo
f or
der
and
equi
vale
nce.
Stu
dent
s bu
ild u
nder
stan
ding
thro
ugh
the
use
of c
oncr
ete
mat
eria
ls, d
iagr
ams
and
real
-wor
ld s
ituat
ions
.
Stud
ent O
utco
mes
Exa
mpl
es o
f In
dica
tors
U,,e
con
cret
e m
ater
ials
or
mod
els
to d
emon
stra
te a
n un
ders
tand
ing
of p
lace
valu
e. (
1-5)
Wri
te m
athe
mat
ical
sen
tenc
es to
rep
rese
nt a
situ
atio
n. (
1-12
)
Use
con
cret
e m
ater
ials
to r
ecog
nize
, rep
rese
nt a
nd c
ompa
re h
alve
s, th
irds
and
four
ths.
(1-
14 )
Rec
ogni
ze f
ract
iona
l equ
ival
ents
for
hal
ves,
fou
rths
and
tent
hs. (
1-15
)
Rec
ogni
ze a
nd c
ount
mon
ey. (
1-16
)
Use
mon
ey to
rep
rese
nt a
nd c
ompa
re d
ecim
al v
alue
s. (
1-17
)
Use
non
stan
dard
, met
ric
and
Eng
lish
units
of
mea
sure
to e
stim
ate
and
mea
sure
leng
th, v
olum
e an
d w
eigh
t. (1
1-1)
Use
dig
ital a
nd c
onve
ntio
nal c
lock
s to
tell
time.
(11
-2)
Use
con
cept
s of
fra
ctio
ns, m
ixed
num
bers
and
dec
imal
s to
sol
ve p
robl
emsi
tuat
ions
.
Use
num
ber
sens
e in
ord
er to
det
erm
ine
reas
onab
lene
ss o
f an
swer
s.
Use
mod
els
to e
xpla
in th
e re
latio
nshi
p be
twee
n fr
actio
ns a
nd d
ecim
als
and
to f
ind
equi
vale
nt f
ract
ions
.
Use
mod
els
to e
xpla
in o
pera
tions
for
fra
ctio
ns a
nd d
ecim
als.
App
ly f
ract
ions
and
dec
imal
s in
rea
l-lif
e si
tuat
ions
. 73
Stan
dard
13:
Pat
tern
s an
d R
elat
ions
hips
Patte
rns
and
rela
tions
hips
ena
ble
stud
ents
to c
lass
ify
and
onza
nize
info
rmat
ion
mat
hem
atic
ally
. Stu
dent
s sh
ould
he
give
n op
port
uniti
es to
foc
us o
n th
e re
gula
ritie
sof
eve
nts,
sha
pes.
des
igns
and
set
s of
num
bers
in o
rder
to id
entif
N p
atte
rns.
Stud
ent O
utco
mes
Cou
nt b
y on
es, t
wos
. fiv
es a
nd te
ns. (
Iden
tif\ ,
desc
ribe
and
ext
end
a pa
ttern
in a
seq
uenc
e of
obi
ects
. (
Exa
mpl
es o
f In
dica
tors
Rec
ogni
ze, d
escr
ibe,
ext
end
and
crea
te a
wid
e va
riet
y of
pat
tern
s us
ing
conc
rete
and
pic
tori
al r
epre
sent
atio
ns.
Rep
rese
nt a
nd d
escr
ibe
mat
hem
atic
al r
elat
ions
hips
.U
se a
con
cret
e m
odel
to c
reat
e a
patte
rn a
nd r
epre
sent
that
pat
tern
sym
boli-
cally
. (IV
-2)
Use
var
iabl
es a
nd o
pen
sent
ence
s to
exp
ress
rel
atio
nshi
ps.
Des
crib
e th
e re
latio
nshi
p gi
ven
in a
tabl
e of
num
bers
der
ived
fro
m a
seq
uenc
eC
lass
ify
and
orga
nize
info
rmat
ion
base
d on
pat
tern
s id
entif
ied.
of o
bjec
ts. (
IV
-3)
Det
erm
ine
a lo
catio
n by
usi
ng o
rder
ed p
airs
of
num
bers
on
a re
ctan
gula
r gr
id.
( IV
-4)
Perf
orm
sim
ple
activ
ities
invo
lvin
g pr
obab
ility
. ( V
-3)
I -4
4
11
Stan
dard
13:
Pat
tern
s an
d R
elat
ions
hips
Patte
rns
and
rela
tions
hips
ena
ble
stud
ents
to c
lass
ify
and
orga
nize
info
rmat
ion
mat
hem
atic
ally
. Stu
dent
s sh
ould
be
give
n op
port
uniti
es to
foc
us o
n th
e re
gula
ritie
sof
eve
nts.
sha
pes,
des
igns
and
set
s of
num
bers
in o
rder
to id
entit
, pat
tern
s.
Stud
ent O
utco
mes
Cou
nt b
y on
es, t
wos
, fiv
es a
nd te
ns. (
I-I)
Iden
tif.v
.de
scri
be a
nd e
xten
d a
patte
rn in
a s
eque
nce
of o
bjec
ts. (
IV-1
)
Use
a c
oncr
ete
mod
el to
cre
ate
a pa
ttern
and
rep
rese
nt th
at p
atte
rn s
ymbo
li-ca
ll,. (
lV-2
)
Des
crib
e th
e re
latio
nshi
p gi
ven
in a
tabl
e of
num
bers
der
ived
fro
m a
seq
uenc
eof
obj
ects
. ( I
V-3
)
Det
erm
ine
a lo
catio
n by
usi
ng o
rder
ed p
airs
of
num
bers
on
a re
ctan
gula
r gr
id.
(IV
-4)
Perf
orm
sim
ple
activ
ities
invo
lvin
g pr
obab
ility
. (V
-3)
3 1
Exa
mpl
es o
f In
dica
tors
Rec
ogni
ze, d
escr
ibe,
ext
end
and
crea
te a
wid
e va
riet
y of
pat
tern
s us
ing
conc
rete
and
pic
tori
al r
epre
sent
atio
ns.
Rep
rese
nt a
nd d
escr
ibe
mat
hem
atic
al r
elat
ions
hips
.
Use
var
iabl
es a
nd o
pen
sent
ence
s to
exp
ress
rel
atio
nshi
ps.
Cla
ssif
y an
d or
gani
ze in
form
atio
n ba
sed
on p
atte
rns
iden
tifie
d.
4-8
MA
TH
EM
AT
ICS
ESS
EN
TIA
L S
KIL
LS
78
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
Mat
hem
atic
al O
utco
mes
Exp
ecte
d of
Mid
dle
Scho
ol S
tude
nts
Mat
hem
atic
s in
stru
ctio
n in
Gra
des
4-8
shou
ld e
nsur
e th
at s
tude
nts
com
plet
ing
Gra
de 8
hav
e ex
perie
nces
whi
ch e
nabl
e th
emto
per
form
the
follo
win
g O
utco
mes
:
I.N
umbe
r
I.U
se c
oncr
ete
mat
eria
ls a
nd il
lust
rativ
e m
odel
s to
rep
rese
nt p
lace
val
ue in
who
le a
nd d
ecim
al n
umbe
rs, f
ract
ions
and
thei
r eq
uial
ents
and
per
cent
s.
2.R
ead
and
writ
e w
hole
num
bers
, int
eger
s, c
omm
on fr
actio
ns a
nd d
ecim
al fr
actio
ns.
3.U
se a
ppro
pria
te s
ymbo
ls a
nd o
rder
rel
atio
ns w
hen
com
parin
g w
hole
num
bers
, int
eger
s, c
omm
on fr
actio
ns a
ndde
cim
al fr
actio
ns.
4.Id
entit
iv a
nd u
se n
umbe
r pr
oper
ties
to s
impl
ify e
xpre
ssio
ns a
nd c
alcu
latio
ns. i
.e.,
com
mut
ativ
e, a
ssoc
iativ
e,di
strib
utiv
e. id
entit
y of
one
and
zer
o. r
ule
of o
rder
of o
pera
tions
.
5.C
onve
rt n
umbe
rs in
bas
e 10
not
atio
n to
and
from
sci
entif
ic n
otat
ion.
6.C
onve
rt n
umbe
rs in
bas
e I(
) no
tatio
n to
and
from
exp
ande
d no
tatio
n.
7.U
nder
stan
d th
e sq
uare
roo
t of a
num
ber
and
the
met
hods
of f
indi
ng it
or
its a
ppro
xim
atio
n.
8.Li
st th
e pr
ime
fact
oriz
atio
n of
a w
hole
num
ber
and
ultim
atel
y w
rite
it as
a n
umer
ical
exp
ress
ion
invo
lvin
g po
sitiv
e
expo
nent
s.
9.U
nder
stan
d gr
eate
st c
omm
on fa
ctor
and
leas
t com
mon
mul
tiple
and
thei
r ap
plic
atio
ns.
10.
Fin
d eq
uiva
lent
frac
tions
.
35
4-8
Stan
dard
s
E . .f . ,- ,
-5 a:.
E c E ( :.
i .e-
1
c CC
..ir
i
t4.
c-.
, A . = r-) c C.; .
--1-
C -
1 cd t., .c Z = .w
-.
b 0 C., I .1 a n E c Z . --
.4 g' Z .so
, F_ g : :: 4; ,-- -5 5 ,-1
,E,,
( : ;
.r-
--
. . .
r. E .;.,..
-
Lt.
.. t = al_ . x
.c ...:.-
f',
7
. s,
, . .
.17.
'..T
5
c. -
.. .c-
,
= rz -F.,
a ..- .
ns,
,f,-
;
(.7 .
c-I
5 E.:j 51
.r-
,
XX
XX
XX
X X
X
XX
XX
X
xxxx
XX
XX
XX
X
XX
XX
X
XX
X X
Xy
X
XX
XX
XX
XX
XX
XX
XX
X X
XX
X
1 3 4 5 6 7 8 9 10
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
(co
ntin
ued)
I.N
umbe
r (c
ontin
ued)
I1
.R
ecal
l add
ition
, sub
trac
tion,
mul
tiplic
atio
n an
d di
visi
on fa
cts
with
rea
sona
ble
spee
d.
12.
I. ;s
e ba
sic
oper
atio
ns w
ith w
hole
num
bers
. int
eger
s, c
omm
on fr
actio
ns a
nd d
ecim
al fr
actio
ns.
13.
Use
var
ious
tech
niqu
es o
f men
tal a
rithm
etic
.
14.
Fst
imat
e to
pre
dict
ans
wer
s to
com
puta
tiona
l pro
blem
s an
d to
che
ck fo
r re
ason
able
ness
of a
nsw
ers.
15.
Ilse
a ca
lcul
ator
in a
ppro
pria
te s
ituat
ions
.
16.
l'se
the
addi
tive
and
mul
tiplic
ativ
e la
ws
of e
xpon
ents
.
17.
Use
rat
ios
and
prop
ortio
nsto
solv
e pr
oble
ms.
18.
Con
vert
am
ong
perc
ent,
frac
tiona
l and
dec
imal
equ
ival
ents
.
19.
Sol
ve fo
r th
e un
know
n in
a p
erce
nt p
robl
em.
10.
Inte
rpre
t pro
blem
s, tr
ansl
ate
to n
umer
ical
exp
ress
ions
,ch
oose
an
appr
opria
te m
etho
d of
cal
cula
tion
and
solv
e.
21.
Sol
ve p
robl
ems
nivo
lvin
g m
oney
.
11.
Rou
nd o
ff nu
mbe
rs to
a s
peci
fied
plac
e va
lue.
821(
)
4-8
' 7 .5. E I, .7
,4 :
Stan
dard
s
.?.." -,i
..... z E 2 r i
-2 0 fi ri
Ir. c t; .1-,
--i
ii' I- Z t, .c 2 tr;
5' 6-E
.5. g = ... ;,-, " .2 g -.
6
C 2 F 5L
I-1 ,,c .c.. '-' n.
E.:'
5'5.
g r.:
,' ..- L.1.
.,
-. c .,..
.1.,
ri ?:.-
-17.
,
-.:-
.5 -
;54. 7: -9 -9 -: -
... ?, '. 2 -
'a' u E ?-
'- ,. 53 -
xxxx
x,x
.xx
XX
xxxx
xxki
cxx
xx
xx,1
4._
x
.2U
(XX
X
xxxx
xxxx
xxxx
xxxx
xxx
x&xx
xxxx
xxxx
xxxx
xxx
11 12 13 14 15 16 17 18 19 20 21
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
(co
ntin
ued)
II.
Mea
sure
men
t
I.K
now
that
all
mea
sure
men
t is
appr
oxim
ate.
Sele
ct th
e ap
prop
riat
e m
easu
ring
inst
rum
ent.
3.A
dd a
nd s
ubtr
act m
easu
res
of ti
me.
4.C
ompa
re te
mpe
ratu
res
and
dete
rmin
e th
e am
ount
of
chan
ge.
5M
easu
re le
ngth
. vol
ume
and
wei
ght i
n bo
th E
nglis
h an
d m
etri
c un
its.
6.C
onve
rt E
nglis
h an
d m
etri
c m
easu
rem
ent u
nits
to e
quiv
alen
t uni
ts w
ithin
the
gix
en s
yste
m.
7.R
e aw
are
of a
ppro
xim
ate
equi
vale
nt m
easu
rem
ents
bet
wee
n E
nglis
h an
d m
etri
c un
its.
8.I.
'se
an a
ppro
pria
te f
orm
ula
to c
alcu
late
the
peri
met
er a
nd a
rea
of p
olyg
ons
and
the
circ
umfe
renc
e of
aci
rcle
.
9.U
se a
n ap
prop
riat
e fo
rmul
a to
cal
cula
te th
e vo
lum
e an
d su
rfac
e ar
ea o
f th
e fo
llow
ing
solid
s:sp
here
s. p
rism
s.fl
y ra
tnid
s, c
y tin
ders
and
con
es.
10.
I. 's
e it
prot
ract
or to
mea
sure
and
dra
w a
ngle
s.
11.
te
an a
ppro
pria
te f
orm
ula
to d
eter
min
e L
. mea
sure
men
t whe
n a
dire
ct m
easu
rem
ent t
ool i
s un
avai
labl
e.
12.
I ..,st
.mat
e le
ngth
s. a
reas
.(f
lum
es a
nd %
teig
ht,,
and
chec
k by
mea
suri
ng in
Eng
lish
and
met
ric
units
.
IA
.So
k e
prob
lem
sith
mea
sure
men
ts in
clud
ing
dist
ance
. wei
ght,
time.
are
a. c
apac
ity a
nd te
mpe
ratu
re.
17
4-8
tana
aras
c ....
- --
.7 t: -
.- 'E' r
L''
,--:
.':: .
..; -3:
1g v-:
1-,
-= i-- ,_ ti g sa
- - 74 P N:
1-.; . rf'-'
. x:
I
..c al'
z:
.'-' 5. -7,
,;:i
r_--
-.:
-.' :; -S .._.
t...5
' ri
F g 2 r-:
XX
XX
XX
XX
,X
XX
XX
X,
XX
_XX
XX
XX
X
x A
xx
xx
x x.
,x x
. I-x
x x
x,,x
'xx
xx
xxxx
xx
xx
x x,
x,x
xx
x xr
x x,
X1
xl1
x x
x xx_
x x)
(xx
xxx
xx
3 4 5 6 7 8 9 10 11 12 13
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
(co
ntin
ued)
III.
Geo
met
ry
Use
term
inol
ogy
appr
opria
te to
the
grad
e le
vel.
2.R
ecog
nize
two-
and
thre
e-di
men
sion
alfi
gure
s by
thei
r sh
apes
and
iden
tify
thei
r co
mpo
nent
par
ts a
nd a
ssoc
iate
dpr
oper
ties.
3.T
hrou
gh o
bser
vatio
n. m
easu
rem
ent,
draw
ing
and
mod
elin
g, id
entif
y ge
omet
ric
prop
ertie
s su
ch a
s sy
mm
etry
,co
ngru
ence
, sim
ilari
ty, p
aral
lelis
m a
nd p
erpe
ndic
ular
ity.
4.V
isua
lize,
dra
w a
nd c
onst
ruct
two-
and
thre
e-di
men
sion
al f
igur
es th
roug
h co
ncre
te e
xper
ienc
es.
5.t '
tiliz
e ge
omet
ric
prin
cipl
es a
nd id
eas
to s
olve
pro
blem
s.
ftL
xper
imen
t with
, dis
cove
r an
d ap
ply
the
know
n fo
rmul
as o
f ge
omet
ry.
7.lf
ilder
stan
d pi
and
its
impo
rtan
ce in
the
geom
etry
of
the
circ
le.
S.R
ecog
nize
and
cla
ssif
!, a
ngle
s as
wel
l as
iden
tify
rela
tions
hips
bet
wee
n an
gles
.
9.D
raw
and
/or
cons
truc
t a v
arie
t, of
sha
pes
havi
ng th
e sa
me
area
.
IV,
Patte
rns
and
Rel
atio
ns
Org
aniz
e da
ta to
sho
w ic
latio
nshi
ps in
tabl
es a
nd g
saph
s.
1,4
4-8
Stan
dard
s
c .._ : L., 5
:-E
., 0 a.
u ,.- U
*g - ,cc
r4
a ;6; = U
, ce ..., 5 0 l - Z
. . -E5 ,.. .2 P 5 z .2i ;-,.
1- E o Z
, . .° ;-= .E ,...
E.1
.1
- z c a z' ,..
...., U
,
.2 t. c c E ° 14 a.
22 0 <.., V
,cn
>.
:2 .2 _ `: DL
z p L ,
5 4.' - r 3
-.4
x x"
Xx
xjcx
x
xxxx
xxx
4 X
X,,X
xx
X X
Xx
xx
xx
AX
IS1
x x
XX
xx
xx
x
xx
xx
x
XX
XX
x i
xx
x X
XX
xx
X X
XX
xx
xx
3 4 5 6 7 8 9
Ari
zona
Nla
them
atic
s E
ssen
tial S
kills
(co
ntin
ued)
IV. P
atte
rns
and
Rel
atio
ns (
cont
inue
d)
2.In
terp
ret d
ata
from
tabl
es a
nd c
hart
s to
det
erm
ine
rela
tions
hips
.
3.D
eter
min
e an
exp
ress
ion
or im
equ
atio
n fo
r a
rela
tions
hip
and
then
eva
luat
e or
sol
ve it
for
a gi
ven
valu
e of
the
varia
ble.
4.R
ecog
niic
or
find
a sp
ecifi
c pa
ttern
occ
urrin
g in
a s
eque
nce
of n
umbe
rs.
5.E
xten
d pa
ttern
s an
d cr
eate
new
one
s.
6.ira
ph o
rder
ed p
airs
to s
hox%
a p
atte
rn o
r re
latio
nshi
p.
7.t n
ders
tand
the
rela
tions
hip
netw
een
angl
e. th
at a
re c
ompl
emen
tary
, sup
plem
enta
ry a
nd v
ertic
al.
Iden
tify
and
unde
rsta
nd th
e re
latio
nshi
ps o
f ang
les
form
ed b
y a
tran
sver
,,a1
of tv
.o p
aral
lel l
ines
.
9.l'n
ders
tand
the
rela
tions
hips
occ
urrin
g he
Mee
n si
mila
r fig
ures
.
V.
Dat
a A
naly
sis
and
Prob
abili
ty
I.C
olle
ct a
nd o
rgan
ile d
ata
u,in
g lis
ts. t
ahle
,, an
d gr
aphs
.
sdu
et p
ret.
anal
\ it:
and
dra
v. c
oncl
uion
s fr
om g
rapk
.
.1,ik
e po
ll,' a
nd s
ure
,.
4-8
Stan
dard
s
c e- :a --:
F, ri
E ,,-:
',5 -1:
-,-.
: P t ;:-:
4,-.
, ,.. _. ... E
. t P.:
..6
. _ --;.! Fl ,. = 6 c r-:
. c a ... 2-.)
, x
= to
" :.* --:
z .-:
t 6 .5 (-;
d 73 t., r-:
XX
XX
Xxx
xxX
XX
XX
X X
X X
,X X
XX
X X
X X
X X
,X
XX
XX
XX
X
XX
XX
X
X X
_X X
x,
X X
X X
,x
X
ji_X
.42L
____
_
XrX XX
XX
X A
X X
X
_l__
LIX,X
X X
X4, I, i
3 4 5 6 7 8 9 3
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
(co
ntin
ued)
V.
Dat
a A
naly
sis
and
Prob
abili
ty (
cont
inue
d)
4.Pr
edic
t and
rec
ord
the
prob
abili
ty o
f ev
ents
fro
m s
impl
e ex
peri
men
ts.
5.E
xpre
ss p
roba
bilit
y as
a r
atio
or
frac
tion.
6.Fi
nd th
e m
ean.
med
ian,
mod
e an
d ra
nge
for
a se
t of
data
.
7.D
eter
min
e th
e pr
obab
ility
of
sim
ple
even
ts a
nd d
raw
con
clus
ions
or
mak
e in
terp
reta
tions
.
8.Fi
nd th
e em
piri
cal p
roba
bilit
y of
an
even
t fro
m a
sam
ple
of o
bser
ved
outc
omes
.
9.Fi
nd th
e pr
obab
ility
of
com
plem
enta
ry e
vent
s an
d of
mut
ually
exc
lusi
ve e
vent
s.
In.
Gen
erat
e a
freq
uenc
y di
stri
butio
n fo
r a
give
n lis
t of
data
.
II.
L's
e a
list o
r tr
ee d
iagr
am to
cou
nt p
erm
utat
ions
or
com
bina
tions
.
Vl.
Ana
lytic
al R
easo
ning
Rec
ogni
re a
nd id
entif
y nu
mer
ical
and
geo
met
ric
patte
rns
and
sequ
ence
s.
Cla
ssif
y an
d ca
tego
ri/e
set
s of
num
bers
or
grou
ps o
f ge
omet
ric
shap
es.
(4)1
3.t's
ing
a gi
Nen
attr
ibut
e. li
nd s
imila
ritie
s an
d di
ffer
ence
s am
ong
geom
etri
c sh
apes
or
desi
gns.
4-8
Stan
dard
s
,...
7: i :''. z ZL
i--..... ,1 ''..: 'S ,-
. i
E ,-,:.
A 7 ''.1' = 4
7.: . u.. .. . . Z
. _, .:2 ,,-,
J .. , .., ......
...
= , .'..
,-. ..., , 2 E ...
c_...
.
, ....
,-..,
... .4 ,g, E. " ',.
.)-'
r--
. ._ = Li.. J '
. 6 rz.
X
. . 3 .i. a
-9,
...1
T) e
= 4
x,x
x x
xX
X
xxxx
X1
X
xx
x
xxxx
,X
xX
X
xxxx
xX
X
xX
.,Xx
xX
X
xxxx
XX
Xk_
XX
x
..1I.
ixx
XX
IA
iE
llx
x x
4 5 6 7 8 9 10 II 2
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
(co
ntin
ued)
VI.
Ana
l)tic
al R
easo
ning
(co
ntin
ued)
4.A
ppro
pria
tely
use
term
s su
ch a
s an
d. o
r. n
ot. o
nly
and
if ...
then
, in
a m
athe
mat
ical
sen
se.
5.P
ro id
e sp
ecifi
c ex
ampl
es o
f a g
iven
num
eric
al o
r ge
omet
ric p
rinci
ple.
6.P
rovi
de a
cou
nter
exam
ple
of a
con
ditio
n th
at is
not
alw
ays
true
.
7.P
erfo
rm ta
sks
invo
lvin
g in
duct
ive
and
dedu
ctiv
e re
ason
ing.
VII
. Alg
ebra
Lnd
erst
and
the
mea
ning
of "
varia
ble-
and
"co
nsta
nt.-
1.R
e fa
mili
ar a
nd p
rofic
ient
with
the
stan
dard
ord
er o
f ope
ratio
ns.
3.R
epre
sent
mat
hem
atic
al r
elat
ions
hips
usi
ng a
riahl
es.
4.S
impl
if po
l.nom
ial e
xpre
ssio
ns.
ralu
ate
sim
ple
alge
brai
c ex
pres
sion
s h
subs
titut
ing
xalu
es fo
r th
e va
riabl
e'
6.S
ol \
c tw
0-st
epeq
uatio
ns a
nd in
equa
litie
s us
ing
inte
gers
, fra
ctio
ns a
nd d
ecim
als.
41
4-8
Stan
dard
s
7) _-c-
'
---:
.g 0 r- i
,,, ,--.
t; t, : I-
,l t- f--,- t _c. - vs,
' :3 :. _ -F... -- 5 Z t sc sd
.. -,1
.7.: -,1
'-'-
-= - 5 v.l.
:1-, r
, = 2 t5 - 41 .-:- v 36
Ei z v
-.2
3, := -3.
1- Z; - rz.
c i
.2 t, ,- = x ,--:
xXxx
xxxx
xxxx
_XX
XX
XX
XX
x X
XX
x X
XX
XX
XX
XX
XX
XX
XX
XX
x
XX
XX
XX
XX
XX
XX
XX
XX
XX
XX
X
XX
XX
xX
I
5 7 3 5
4-8
Stan
dard
1: M
athe
mat
ics
as P
robl
em S
olvi
ng
Prob
lem
sol
ving
is th
e pr
oces
s by
whi
ch s
tude
nts
expe
rien
ce th
e po
wer
of
mat
hem
atic
s an
d its
use
fuln
ess
in th
e w
orld
aro
und
them
. Thr
ough
dif
fere
nt p
robl
em-s
olvi
ngac
tiviti
es, s
tude
nts
can
use
a va
riet
y of
str
ateg
ies
and
mat
hem
atic
al to
ols
to s
olve
rea
l-w
orld
pro
blem
s. C
hild
ren
need
to h
e gi
ven
man
y op
port
uniti
es to
wor
k w
ithot
her
stud
ents
in a
naly
zing
, org
aniz
ing
and
solv
ing
prob
lem
s, u
sing
"m
essy
dat
a" f
rom
the
real
wor
ld.
Stud
ent O
utco
mes
Use
a v
arie
ty o
f pr
oble
m-s
olvi
ng s
trat
egie
s to
sol
ve r
eal-
wor
ld p
robl
ems
invo
lvin
g th
e m
athe
mat
ics
Ess
entia
l Ski
lls.
42
Exa
mpl
es o
f In
dica
tors
Use
pro
blem
-sol
ving
app
roac
hes
to in
vest
igat
e an
d un
ders
tand
mat
hem
ati-
cal c
onte
nt.
Form
ulat
e pr
oble
ms
from
situ
atio
ns w
ithin
and
out
side
mat
hem
atic
s.
Dev
elop
and
app
ly a
var
iety
of
stra
tegi
es to
sol
ve p
robl
ems.
with
em
phas
ison
mul
tiste
p an
d no
nrou
tine
prob
lem
s.
Ver
ify
and
inte
rpre
t res
ults
with
res
pect
to th
e or
igin
al p
robl
em s
ituat
ion.
Gen
eral
ize
solu
tions
and
str
ateg
ies
to n
ew p
robl
em s
ituat
ions
.
4-8
Stan
dard
2: M
athe
mat
ics
as C
omm
unic
atio
n
In g
rade
s 4-
8. c
omm
unic
atio
n be
com
es e
ven
mor
e im
port
ant.
Stud
ents
nee
d to
dis
cuss
, exp
lain
, con
ject
ure,
just
ify
and
valid
ate
thei
r th
inki
ng in
ord
er to
deep
en th
eir
unde
rsta
ndin
g of
mat
hem
atic
al c
once
pts.
Stu
dent
s ne
ed to
use
that
und
erst
andi
ng to
ana
lyze
and
rel
ate
mat
hem
atic
s to
the
real
wor
ld.
Stud
ent O
utco
mes
Com
mun
icat
e m
athe
mat
ical
ly th
eir
unde
rsta
ndin
g of
the
mat
hem
atic
s E
ssen
-tia
l Ski
lls a
nd r
elat
e th
at u
nder
stan
ding
to r
eal-
wor
ld a
pplic
atio
ns.
43
Exa
mpl
es o
f In
dica
tors
Mod
el s
ituat
ions
usi
ng o
ral,
wri
tten,
con
cret
e, p
icto
rial
, gra
phic
al a
ndal
gebr
aic
met
hods
.
Ref
lect
on
and
clar
ify
thin
king
abo
ut m
athe
mat
ical
idea
s an
d si
tuat
ions
.
Dev
elop
com
mon
und
erst
andi
ng o
f m
athe
mat
ical
idea
s, in
clud
ing
the
role
of d
efin
ition
s.
Use
the
skill
s of
rea
ding
, lis
teni
ng a
nd v
iew
ing
to in
terp
ret a
nd e
valu
ate
mat
hem
atic
al id
eas.
Dis
cuss
mat
hem
atic
al id
eas
and
mak
e co
njec
ture
s an
d co
nvin
cing
arg
u-m
ents
.
9 7
4-8
Stan
dard
3: M
athe
mat
ics
as R
easo
ning
Rea
soni
ng is
fun
dam
enta
l to
know
ing
and
doin
g m
athe
mat
ics.
Stu
dent
s ne
ed to
be
give
n am
ple
time
to e
xplo
rean
d an
alyz
e m
athe
mat
ical
con
cept
s an
d pr
oble
m-
solv
ing
situ
atio
ns, a
nd to
mak
e co
njec
ture
s an
d co
nstr
uct v
alid
arg
umen
ts p
erta
inin
g to
thos
e pr
oble
ms.
Stud
ent O
utco
mes
1..1
w m
athe
mat
ical
rea
soni
ng to
gai
n an
und
erst
andi
ng o
f th
e m
athe
mat
ics
Fsse
ntia
l Ski
lls a
nd u
se th
is u
nder
stan
ding
to s
olve
a v
arie
ty o
f pr
oble
ms.
44
Exa
mpl
es o
f In
dica
tors
Rec
ogni
ze a
nd a
pply
ded
uctiv
e an
d in
duct
ive
reas
onin
g.
Und
erst
and
and
appl
y re
ason
ing
proc
esse
s, w
ith s
peci
al a
ttent
ion
to s
patia
lre
ason
ing
and
reas
onin
g w
ith p
ropo
rtio
ns a
nd g
raph
s.
Mak
e an
d ev
alua
te m
athe
mat
ical
con
ject
ures
and
arg
umen
ts.
Val
idat
e th
inki
ng.
Use
per
suas
ion
and
the
pow
er o
f re
ason
ing
as a
par
t of
mat
hem
atic
s.
Stan
dard
4: M
athe
mat
ical
Con
nect
ions
Mat
hem
atic
al c
onne
ctio
ns f
ocus
es o
n m
athe
mat
ics
as a
n in
tegr
ated
who
le a
s op
pose
d to
a li
st o
f is
olat
ed f
acts
that
need
to b
e m
emor
ized
. Stu
dent
s ne
ed to
be
give
n
ampl
e op
port
uniti
es to
exp
lore
the
inte
rcon
nect
ions
bet
wee
n m
athe
mat
ical
con
cept
s. a
sw
ell t
heir
rel
atio
nshi
ps to
the
real
wor
ld. I
f st
uden
ts a
re to
be
empo
wer
ed
mat
hem
atic
ally
, the
y m
ust b
e ab
le to
app
ly m
athe
mat
ics
usin
g a
vari
ety
of s
trat
egie
sin
dif
fere
nt p
robl
em s
ituat
ions
.
Stud
ent O
utco
mes
Exa
mpl
es o
f In
dica
tors
Mak
e m
athe
mat
ical
con
nect
ions
bet
wee
n m
athe
mat
ics
Ess
entia
l Ski
lls a
ndpr
oble
m s
ituat
ions
and
dem
onst
rate
thei
r un
ders
tand
ing
that
mat
hem
atic
sis
ln
inte
grat
ed w
hole
.
45
Exp
lore
pro
blem
s an
d de
scri
be r
esul
ts u
sing
gra
phic
al,
num
eric
al, p
hysi
-ca
l, al
gebr
aic
and
verb
al m
athe
mat
ical
mod
els
or r
epre
sent
atio
ns.
Use
a m
athe
mat
ical
idea
to f
urth
er u
nder
stan
ding
of
othe
r m
athe
mat
ical
idea
s.
App
ly m
athe
mat
ical
thin
king
and
mod
elin
g to
sol
ve p
robl
ems
that
ari
se in
othe
r di
scip
lines
.
App
ly m
athe
mat
ical
thin
king
and
mod
elin
g to
sol
ve p
robl
ems
refl
ectiv
e of
our
cultu
re a
nd s
ocie
ty.
4-8
Stan
dard
5: N
umbe
r an
d N
umbe
r R
elat
ions
hips
Num
ber
and
num
ber
rela
tions
hips
are
the
unde
rsta
ndin
g of
mul
tiple
rep
rese
ntat
ions
of
who
le n
umbe
rs, f
ract
ions
,de
cim
als,
inte
gers
and
rat
iona
l num
bers
thro
ugh
conc
rete
exp
erie
nces
in th
e co
ntex
t of
real
-wor
ld s
ituat
ions
.
Stud
ent O
utco
mes
Use
con
cret
e m
ater
ials
and
illu
stra
tive
mod
els
to r
epre
sent
pla
ce v
alue
inw
hole
and
dec
imal
num
bers
, fra
ctio
ns a
nd th
eir
equi
vale
nts
and
perc
ents
. (I-
1 )
Rea
d an
d w
rite
who
le n
umbe
rs, i
nteg
ers,
com
mon
fra
ctio
ns a
nd d
ecim
alfr
actio
ns. (
1-2)
Use
app
ropr
iate
sym
bols
and
ord
er r
elat
ions
whe
n co
mpa
ring
who
le n
umbe
rs,
inte
gers
, com
mon
fra
ctio
ns a
nd d
ecim
al f
ract
ions
. (1-
3)
Con
vert
num
bers
in b
ase
10 n
otat
ion
to a
nd f
rom
sci
entif
ic n
otat
ion.
(1-
5)
Con
vert
num
bers
in b
ase
10 n
otat
ion
to a
nd f
rom
exp
ande
d no
tatio
n. (
1-6)
Und
erst
and
the
squa
re r
oot o
f a
num
ber
and
the
met
hods
of
find
ing
it or
its
appr
oxim
atio
n. (
1-7)
Lis
t the
pri
me
fact
oriz
atio
n of
a w
hole
num
ber
and
ultim
atel
y w
rite
it a
s a
num
eric
al e
xpre
ssio
n in
volv
ing
posi
tive
expo
nent
s. (
I-8)
Und
erst
and
grea
test
com
mon
fac
tor
and
leas
t com
mon
mul
tiple
and
thei
rap
plic
atio
ns. (
1-9)
Find
equ
ival
ent f
ract
ions
. (I-
10)
Use
the
addi
tive
and
mul
tiplic
ativ
e la
ws
of e
xpon
ents
. (1-
16)
Use
rat
ios
and
prop
ortio
ns to
sol
ve p
robl
ems.
(1-
17)
1 i
46
Exa
mpl
es o
f In
dica
tors
Rep
rese
nt a
nd u
se n
umbe
rs in
a v
arie
ty o
f eq
uiva
lent
for
ms
(int
eger
,fr
actio
n, d
ecim
al, p
erce
nt, e
xpon
entia
l and
sci
entif
ic n
otat
ion)
in r
eal-
wor
ld a
nd m
athe
mat
ical
pro
blem
situ
atio
ns.
App
ly r
atio
s, p
ropo
rtio
ns a
nd p
erce
nts
in a
wid
e va
riet
y of
situ
atio
ns.
Des
ign,
inve
stig
ate
and
dem
onst
rate
rel
atio
nshi
ps a
mon
g fr
actio
ns, d
eci-
mal
s an
d pe
rcen
ts, a
nd a
pply
thes
e in
terr
elat
ions
hips
in r
eal-
wor
ld s
itua-
tions
.
Rep
rese
nt n
umer
ical
rel
atio
nshi
ps in
one
- an
d tw
o-di
men
sion
al g
raph
s.
Util
ize
num
ber
lines
, are
a m
odel
s, g
raph
s, a
nd c
alcu
lato
r- a
nd c
ompu
ter-
gene
rate
d nu
mbe
rs to
rep
rese
nt a
giv
en q
uant
ity.
Stan
dard
5: N
umbe
r an
d N
umbe
r R
elat
ions
hips
(co
ntin
ued)
Stud
ent O
utco
mes
Con
vert
am
ong
perc
ent,
frac
tiona
l and
dec
imal
equ
ival
ents
. (l -
18)
Rou
nd o
ft' n
umbe
rs to
a s
peci
fied
pla
ce v
alue
. (I-
22)
Org
aniz
e da
ta to
sho
w r
elat
ions
hips
in ta
bles
and
gra
phs.
(IV
- I
1
Inte
rpre
t dat
a fr
om ta
bles
and
cha
rts
to d
eter
min
e re
latio
nshi
ps. (
IV-2
)
Rec
ogni
ze o
r fi
nd a
spe
cifi
c pa
ttern
occ
urri
ng in
a s
eque
nce
of n
umbe
rs.
(IV
-4)
Ext
end
patte
rns
and
crea
te n
ew o
nes.
(IV
-5)
Gra
ph o
rder
ed p
airs
to s
how
a p
atte
rn o
r re
latio
nshi
p. (
IV-(
i)
Rec
ogni
zean
d id
entif
y nu
mer
ical
and
geo
met
ric
patte
rns
and
sequ
ence
s.(V
)
Cla
ssif
y an
d ca
tego
rize
set
s of
num
bers
or
grou
ps o
f ge
omet
ric
shap
es. (
VI-
2)
App
ropr
iate
ly u
se te
rms
such
as
and,
or,
not
, onl
y an
d if
...tlw
n,in
am
athe
mat
ical
sen
se. (
VI-
41
Pert
'orm
task
s in
volv
ing
indu
ctiv
e an
d de
duct
ive
reas
onin
g. (
VI-
7)
1 1
44-
Exa
mpl
es o
f In
dica
tors 5
4-8
Stan
dard
6: N
umbe
r Sy
stem
s an
d N
umbe
r T
heor
y
Num
ber
syst
ems
and
num
ber
theo
ry a
re th
e fo
unda
tion
of th
e st
ruct
ure
of m
athe
mat
ics.
The
y sh
ow h
ow li
mite
d el
emen
ts in
tegr
ate
to f
orm
a w
orki
ng s
yste
m.
Stud
ent O
utco
mes
LIs
e co
ncre
te m
ater
ials
and
illu
stra
tive
mod
els
to r
epre
sent
pla
ce v
alue
inIA
hol
e au
d de
cim
al n
umbe
rs, f
ract
ions
and
thei
r eq
uiva
lent
s an
d pe
rcen
ts.
( 1-
1)
t Ise
app
ropr
iate
sym
bols
and
ord
er r
elat
ions
whe
n co
mpa
ring
who
le n
umbe
rs.
inte
gers
, com
mon
fra
ctio
ns a
nd d
ecim
al &
actio
ns. (
I-3)
Iden
tify
and
use
num
ber
prop
ertie
s to
sim
plif
y ex
pres
sion
s an
d ca
lcul
atio
ns,
i.e.,
com
mut
ativ
e, a
ssoc
iativ
e, d
istr
ibut
ive,
iden
tity
of o
ne a
nd z
ero.
rul
e of
orde
r of
ope
ratio
ns. (
14)
Con
vert
num
bers
in b
ase
10 n
otat
ion
to a
nd f
rom
sci
entif
ic n
otat
ion.
(I-
5)
Con
vert
num
bers
in b
ase
10 n
otat
ion
to a
nd f
rom
exp
ande
d no
tatio
n. (
1-61
Inde
rsta
nd th
e sq
uare
roo
t of
a nu
mbe
r an
d th
e m
etho
ds o
f fi
ndin
g it
or it
sap
prox
imat
ion.
(1-
7)
Lis
t the
pri
me
fact
oriz
atio
n of
a w
hole
num
ber
and
ultim
atel
y w
rite
it a
s a
num
eric
al e
xpre
ssio
n in
volv
ing
posi
tive
expo
nent
s. (
I-8)
Und
erst
and
grea
test
com
mon
fac
tor
and
leas
t com
mon
mul
tiple
and
thei
rap
plic
atio
ns. (
1-9)
Find
equ
ival
ent f
ract
ions
. ( I
-10)
Use
bas
ic o
pera
tions
with
who
le n
umbe
rs, i
nteg
ers,
com
mon
fra
ctio
ns a
ndde
cim
al f
ract
ions
. (1-
12)
48
Exa
mpk
s of
Ind
icat
ors
Dem
onst
rate
an
unde
rsta
ndin
g of
the
need
for
num
bers
bey
ond
the
who
lenu
mbe
rs.
Dev
elop
and
use
ord
er r
elat
ions
for
who
le-n
umbe
r fr
actio
ns a
nd d
ecim
als.
Iden
tify
the
rela
tions
hip
of w
hole
num
bers
as
inte
gers
and
fra
ctio
ns a
sra
tiona
l num
bers
.
Show
how
the
basi
c ar
ithm
etic
ope
ratio
ns a
re r
elat
ed to
One
ano
ther
.
Dev
elop
and
vpp
ly n
Um
ber
theo
ry c
once
pts
(e.g
., pr
imes
, fac
tors
and
mul
tiple
s) in
rea
l-w
orld
and
mat
hem
atic
al p
robl
em s
ituat
ions
.
Inve
stig
ate
the
arith
met
ic o
f fr
actio
ns, d
ecim
als,
inte
gers
and
rat
iona
lnu
mbe
rs th
roug
h th
e un
ity o
f co
mm
on id
eas.
Dev
elop
mat
hem
atic
al c
once
pts
by e
xplo
ring
num
ber
theo
ry. 1
7
Stan
dard
6: N
umbe
r Sy
stem
s an
d N
umbe
r T
heor
y (c
ontin
ued)
Stud
ent O
utco
mes
Use
var
ious
tech
niqu
es o
f m
enta
l ari
thm
etic
. (1-
13)
Est
imat
e to
pre
dict
ans
wer
s to
com
puta
tiona
l pro
blem
s an
d to
che
ck f
orre
ason
able
ness
of
answ
ers.
(1-
14)
Use
the
addi
tive
and
mul
tiplic
ativ
e la
ws
of e
xpon
ents
. (I-
16)
Use
rat
ios
and
prop
ortio
ns to
sol
ve p
robl
ems.
(I-
17)
Con
vert
am
ong
perc
ent,
frac
tiona
l and
dec
imal
equ
ival
ents
. (1-
18)
Solv
e fo
r th
e un
know
n in
a p
erce
nt p
robl
em. (
1-19
)
Inte
rpre
t pro
blem
s, tr
ansl
ate
to n
umer
ical
exp
ress
ions
,ch
oose
an
appr
opri
ate
met
hod
of c
alcu
latio
n an
d so
lve.
(I-
20)
Solv
e pr
oble
ms
invo
lvin
g m
oney
. (I-
21)
Rou
nd o
ff n
umbe
rs to
a s
peci
fied
pla
ce v
alue
. (1-
22)
Ext
end
patte
rns
and
crea
te n
ew o
nes.
(IV
-5)
App
ropr
iate
ly u
se te
rms
such
as
and,
or,
not
, onl
yan
d if
...th
en, i
n a
mat
hem
atic
al s
ense
. (V
I-4)
Prov
ide
spec
ific
exa
mpl
es o
f a
give
n nu
mer
ical
or
geom
etri
c pr
inci
ple.
(V
I-5)
Perf
orm
task
s in
volv
ing
indu
ctiv
e an
d de
duct
ive
reas
onin
g. (
VI-
7)
Be
fam
iliar
and
pro
fici
ent w
ith th
e st
anda
rd o
rder
of
oper
atio
ns. (
VII
-2)
49
Exa
mpl
es o
f In
dica
tors
4-8
Stan
dard
7: C
ompu
tatio
n an
d E
stim
atio
n
Com
puta
tion
and
estim
atio
n ar
e w
orki
ng s
kills
dev
elop
ed th
roug
h co
ncep
tual
exp
erie
nces
. The
abi
lity
to w
ork
flex
ibly
with
who
le n
umbe
rs, f
ract
ions
, dec
imal
s,in
tege
rs a
nd r
atio
nal n
umbe
rs w
ithin
a c
onte
xt o
f re
ason
able
ness
(es
timat
ion)
pro
vide
s th
e to
ols
nece
ssar
y fo
r al
l pro
blem
sol
ving
.
Stud
ent O
utco
mes
Iden
tify
and
use
num
ber
prop
ertie
s to
sim
plif
y ex
pres
sion
s an
d ca
lcul
atio
ns,
i.e.,
com
mut
ativ
e, a
ssoc
iativ
e, d
istr
ibut
ive,
iden
tity
of o
ne a
nd z
ero,
rul
e of
orde
r of
ope
ratio
ns. (
1-4)
Con
vert
num
bers
in b
ase
10 n
otat
ion
to a
nd f
rom
sci
entif
ic n
otat
ion.
(I-
5)
Con
vert
num
bers
in b
ase
10 n
otat
ion
to a
nd f
rom
exp
ande
d no
tatio
n. (
1-6)
Und
erst
and
the
squa
re r
oot o
f a
num
ber
and
the
met
hods
of
find
ing
it or
its
appr
oxim
atio
n. (
1-7)
Lis
t the
pri
me
fact
oriz
atio
n of
a w
hole
num
ber
and
ultim
atel
y w
rite
it a
s a
num
eric
al e
xpre
ssio
n in
volv
ing
posi
tive
expo
nent
s. (
1-8)
Und
erst
and
grea
test
com
mon
fac
tor
and
leas
t com
mon
mul
tiple
and
thei
rap
plic
atio
ns. (
I-9)
Find
equ
ival
ent f
ract
ions
. (1-
10)
Rec
all a
dditi
on, s
ubtr
actio
n, m
ultip
licat
ion
and
divi
sion
fac
ts w
ith r
easo
nabl
esp
eed.
(I-
1)
Use
bas
ic o
pera
tions
with
who
le n
umbe
rs, i
nteg
ers,
com
mon
fra
ctio
ns a
ndde
cim
al f
ract
ions
. ( 1
- 1
2 )
Use
var
ious
tech
niqu
es o
f m
enta
l ari
thm
etic
. (1-
13)
Est
imat
e to
pre
dict
ans
wer
s to
com
puta
tiona
l pro
blem
s an
d to
che
ck f
orre
ason
able
ness
of
answ
ers.
(I-
14)
50
Exa
mpl
es o
f In
dica
tors
Com
pute
with
who
le n
umbe
rs, f
ract
ions
, dec
imal
s, in
tege
rs a
nd r
atio
nal
num
bers
.
Dev
elop
, ana
lyze
and
exp
lain
pro
cedu
res
for
com
puta
tion
and
tech
niqu
esfo
r es
timat
ion.
Dev
elop
, ana
lyze
and
exp
lain
met
hods
for
sol
ving
pro
port
ions
.
Use
the
appr
opri
ate
met
hod
of c
ompu
tal i
on: m
enta
l ari
thm
etic
, pap
er-a
nd-
penc
il, c
alcu
lato
r or
com
pute
r.
Use
com
puta
tion,
est
imat
ion
and
prop
ortio
ns to
sol
ve p
robl
ems.
Use
est
imat
ion
to c
heck
the
reas
onab
lene
ss o
f re
sults
.
Dev
elop
ski
lls n
eces
sary
to u
se a
ppro
pria
te te
chno
logy
.
Inte
rpre
t and
rel
ate
info
rmat
ion
gain
ed th
roug
h th
e us
e of
tech
nolo
gy.
Stan
dard
7: C
ompu
tatio
n an
d E
stim
atio
n (c
ontin
ued)
Stud
ent O
utco
mes
Use
a c
alcu
lato
r in
app
ropr
iate
situ
atio
ns. (
I-15
)
Use
the
addi
tive
and
mul
tiplic
ativ
e la
ws
of e
xpon
ents
. (1-
16)
Use
rat
ios
and
prop
ortio
ns to
sol
ve p
robl
ems.
(1-
17)
Con
vert
am
ong
perc
ent,
frac
tiona
l and
dec
imal
equ
ival
ents
. (I-
18)
Sol
ve fo
r th
e un
know
n in
a p
erce
nt p
robl
em. (
I-19
)
inte
rpre
t pro
blem
s, tr
ansl
ate
to n
umer
ical
exp
ress
ions
, cho
ose
an a
ppro
pria
tem
etho
d of
cal
cula
tion
and
solv
e. (
I-20
)
Sol
ve p
robl
ems
invo
lvin
g m
oney
. (1-
21)
Rou
nd o
ff nu
mbe
rs to
a s
peci
fied
plac
e va
lue.
(1-
22)
Add
and
sub
trac
t mea
sure
s of
tim
e. (
11-3
)
Com
pare
tem
pera
ture
s an
d de
term
ine
the
amou
nt o
f cha
nge.
(1I
-4)
Con
vert
Eng
lish
and
met
ric m
easu
rem
ent u
nits
to e
quiv
alen
t uni
ts w
ithin
the
give
n sy
stem
. (11
-6)
Use
an
appr
opria
te fo
rmul
a to
cal
cula
te th
e pe
rimet
er a
nd a
rea
of p
olyg
ons
and
the
circ
umfe
renc
e of
a c
ircle
. (I1
-8)
Use
an
appr
opria
te fo
rmul
a to
cal
cula
te th
e vo
lum
e an
d su
rfac
e ar
ea o
f the
follo
win
g so
lids:
sph
eres
, pris
ms,
pyr
amid
s, c
ylin
ders
and
con
es. (
II-9)
Use
an
appr
opria
te fo
rmul
a to
det
erm
ine
a m
easu
rem
ent w
hen
a di
rect
mea
sure
men
t too
l is
unav
aila
ble.
(II-
11)
112
51
Exa
mpl
es o
f In
dica
tors
113
Stan
dard
7: C
ompu
tatio
n an
d E
stim
atio
n (c
ontin
ued)
Stud
ent O
utco
mes
Solv
e pr
oble
ms
with
mea
sure
men
ts in
clud
ing
dist
ance
, wei
ght,
time,
are
a,ca
paci
ty a
nd te
mpe
ratu
re. (
II-
1 3)
Util
ite g
eom
etri
c pr
inci
ples
and
idea
s to
sol
ve p
robl
ems.
(11
1-5)
Exp
erim
ent w
ith, d
isco
ver
and
appl
y th
e kn
own
form
ulas
of
geom
etry
. (11
1-6)
Und
erst
and
pi a
nd it
s im
port
ance
in th
e ge
omet
ry o
f th
e ci
rcle
. (11
1-7)
Ext
end
patte
rns
and
crea
te n
ew o
nes.
(1V
-5)
Exp
ress
pro
babi
lity
as a
rat
io o
r fr
actio
n. (
V-5
)
Find
the
mea
n, m
edia
n, m
ode
and
rang
e fo
r a
set o
f da
ta. (
V-6
)
App
ropr
iate
ly u
se te
rms
such
as
and,
or,
not
, onl
y an
d if
...
then
. in
am
athe
mat
ical
sen
se. (
V1-
4)
Prov
ide
spec
ific
exa
mpl
es o
f a
give
n nu
mer
ical
or
geom
etri
c pr
inci
ple.
(V
1-5)
Perf
orm
task
s in
volv
ing
indu
ctiv
e an
d de
duct
ive
reas
onin
g. (
V1-
7)
Be
fam
iliar
and
pro
fici
ent w
ith th
e st
anda
rd o
rder
of
oper
atio
ns. (
V11
-2)
Eva
luat
e si
mpl
e al
gebr
aic
expr
essi
ons
by s
ubst
itutin
g va
lues
for
the
vari
able
s.(V
1l-5
)
Solv
e tw
o-st
ep e
quat
ions
and
ineq
ualit
ies
usin
g in
tege
rs, f
ract
ions
and
dec
i-m
als.
(V
11-6
)
p 4
52
Exa
mpl
es o
f In
dica
tors
Stan
dard
8: P
atte
rns
and
Func
tions
Patte
rns
and
func
tions
are
the
perc
eptu
al li
nks
to c
onne
ctin
g an
d un
ders
tand
ing
the
wor
ld a
roun
d us
. Exp
lori
ng p
atte
rns
requ
ires
obs
erva
tion
skill
s th
at id
entif
y,an
alyz
e, c
onne
ct a
nd e
xten
d re
latio
nshi
ps.
Stud
ent O
utco
mes
Rec
ogni
ze tw
o- a
nd th
ree-
dim
ensi
onal
fig
ures
by
tiir
sha
pes
and
iden
tify
thei
r co
mpo
nent
par
ts a
nd a
ssoc
iate
d pr
oper
ties.
(11
1-2)
Thr
ough
obs
erva
ti.)n
. mea
sure
men
t, dr
awin
g an
d m
odel
ing,
iden
tify
geom
et-
ric
prop
ertie
s su
ch a
s sy
mm
etry
. con
grue
nce,
sim
ilari
ty, p
aral
lelis
m a
ndpe
rpen
dicu
lari
ty. (
111-
3)
Exp
erim
ent w
ith, d
isco
ver
and
appl
y th
e kn
own
form
ulas
of
geom
etry
. (II
I-6)
Und
erst
and
pi a
nd it
s im
port
ance
in th
e ge
omet
ry o
f th
e ci
rcle
. (11
1-7)
Rec
ogni
ze a
nd c
lass
ify
angl
es a
s w
ell a
s id
entif
y re
latio
nshi
ps b
etw
een
angl
es.
(III
-8)
Org
aniz
e da
ta to
sho
w r
elat
ions
hips
in ta
bles
and
gra
phs.
(IV
-)
Inte
rpre
t dat
a fr
om ta
bles
and
cha
rts
to d
eter
min
e re
latio
nshi
ps. (
IV-2
)
Det
erm
ine-
an e
xpre
ssio
n or
an
equa
tion
for
a re
latio
nshi
p an
d th
en e
valu
ate
orso
lve
it fo
r a
give
n va
lue
of th
e va
riab
le. (
IV-3
)
Rec
ogni
ze o
r fi
nd a
spe
cifi
c pa
ttern
occ
urri
ng in
a s
eque
nce
of n
umbe
rs.
(IV
-4)
Ext
end
patte
rns
and
crea
te n
ew o
nes.
(IV
-5)
(ira
ph o
rder
ed p
airs
to s
how
a p
atte
rn o
r re
latio
nshi
p. (
IV-6
)
53
Exa
mpl
es o
f In
dica
tors
Des
crib
e, e
xten
d, a
naly
ze a
nd c
reat
e a
wid
e va
riet
y of
pat
tern
s.
Des
crib
e an
d re
pres
ent r
elat
ions
hips
with
tabl
es, g
raph
s an
d nu
mbe
rm
anip
ulat
ions
.
Ana
lyze
fun
ctio
nal r
elat
ions
hips
to e
xpla
in h
ow a
cha
nge
in o
ne q
uant
ityre
sults
in a
cha
nge
in a
noth
er.
Ilse
pat
tern
s an
d fu
nctio
ns to
rep
rese
nt a
nd s
olve
pro
blem
s.
117
Stan
dard
8: P
atte
rns
and
Func
tions
(co
ntin
ued)
Stud
ent O
utco
mes
Und
erst
and
the
rela
tions
hips
occ
urri
ng b
etw
een
sim
ilar
figu
res.
(IV
-9)
Col
lect
and
org
aniz
e da
ta u
sing
list
s, ta
bles
and
gra
phs.
(V
-)
Inte
rpre
t, an
alyz
e an
d dr
aw c
oncl
usio
ns f
rom
gra
phs.
(V
-2)
Pred
ict a
nd r
ecor
d th
e pr
obab
ility
of
even
ts f
rom
sim
ple
expe
rim
ents
. (V
-4)
Det
erm
ine
the
prob
abili
ty o
f si
mpl
e ev
ents
and
dra
w c
oncl
usio
ns o
r m
ake
inte
rpre
tatio
ns. (
V-7
)
Find
the
empi
rica
l pro
babi
lity
of a
n ev
ent f
rom
a s
ampl
e of
obs
erve
d ou
tcom
es.
(V-8
)
Find
the
prob
abili
ty o
f co
mpl
emen
tary
eve
nts
and
of im
itual
ly e
xclu
sive
even
ts. (
V-9
)
Rec
ogni
ze a
nd id
ent4
num
eric
al a
nd g
eom
etri
c pa
ttern
s an
d se
quen
ces.
(V1-
l
Cla
ssif
y an
d ca
tego
rize
set
s of
num
bers
or
grou
ps o
f ge
omet
ric
shap
es. (
V1-
2)
App
ropr
iate
ly u
se te
rms
such
as
and,
or,
/lot,
Onl
yan
df
..th
en,
in a
mat
hem
atic
al s
ense
. (V
1-4)
Perf
orm
task
s in
volv
ing
indu
ctiv
e an
d de
duct
ive
reas
onin
g. (
VI-
7)
Rep
rese
nt m
athe
mat
ical
rel
atio
nshi
ps u
sing
var
iabl
es. (
V 1
1-3)
54
Exa
mpl
es o
f In
dica
tors
Stan
dard
9: A
lgeb
ra
Alg
ebra
is th
e la
ngua
ge th
roug
h w
hich
mos
t mat
hem
atic
s is
com
mun
icat
ed.
Stud
ent O
utco
mes
Use
an
appr
opri
ate
form
ula
to c
alcu
late
the
peri
met
er a
nd a
rea
of p
olyg
ons
and
the
circ
umfe
renc
e of
a c
ircl
e. (
11-8
)
Use
an
appr
opri
ate
form
ula
to c
alcu
late
the
volu
me
and
surf
ace
urea
of
the
follo
win
g so
lids:
sph
eres
, pri
sms,
pyr
amid
s, c
ylin
ders
and
con
es. (
11-9
)
Use
an
appr
opri
ate
form
ula
to d
eter
min
e a
mea
sure
men
t whe
n a
dire
ctm
easu
rem
ent t
ool i
s un
avai
labl
e. (
II-1
)
Util
ize
geom
etri
c pr
inci
ples
and
idea
s to
sol
ve p
robl
ems.
(11
1-5)
Exp
erim
ent w
ith, d
isco
ver
and
appl
y th
e kn
own
form
ulas
of
geom
etry
. (11
1-6)
Und
erst
and
pi a
nd it
s im
port
ance
in th
e ge
omet
ry o
f th
e ci
rcle
. (II
I-7)
Org
aniz
e da
ta to
sho
w r
elat
ions
hips
in ta
bles
and
gra
phs.
(IV
- 1
)
Inte
rpre
t dat
a fr
om ta
bles
and
cha
rts
to d
eter
min
e re
latio
nshi
ps. (
1V-2
)
Det
erm
ine
an e
xpre
ssio
n or
an
equa
tion
for
a re
latio
nshi
p an
d th
en e
valu
ate
orso
lve
it fo
r a
give
n va
lue
of th
e va
riab
le. (
IV-3
)
Rec
ogni
ze o
r fi
nd a
spe
cifi
c pa
ttern
occ
urri
ng in
a s
eque
nce
of n
umbe
rs.
( IV
-4)
Ext
end
patte
rns
and
crea
te n
ew o
nes.
(IV
-5)
Gra
ph o
rder
ed p
airs
to s
how
a p
atte
rn o
r re
latio
nshi
p. (
IV-6
)
1 20
55
4-8
Exa
mpl
es o
f In
dica
tors
Def
ine
and
use
vari
able
s w
ithin
the
cont
ext o
f ex
pres
sion
s an
d eq
uatio
ns.
Rep
rese
nt s
ituat
ions
and
num
ber
patte
rns
with
tabl
es, g
raph
s, v
erba
l rul
esan
d eq
uatio
ns a
nd e
xplo
re th
eir
inte
rrel
atio
nshi
ps.
Iden
tify
prop
ertie
s an
d re
latio
nshi
ps b
y us
ing
tabl
es a
nd g
raph
s.
Solv
e lin
ear
equa
tions
usi
ng c
oncr
ete,
info
rmal
and
for
mal
met
hods
.
Solv
e in
equa
litie
s an
d no
nlin
ear
equa
tions
info
rmal
ly.
App
ly a
lgeb
raic
met
hods
to s
olve
a v
arie
ty o
f re
al-w
orld
and
mat
hem
atic
alpr
oble
ms.
121
Stan
dard
9: A
lgeb
ra (
cont
inue
d)
Stud
ent O
utco
mes
Col
lect
and
org
aniz
e da
ta u
sing
list
s, ta
bles
and
gra
phs.
(V
-1)
Inte
rpre
t, an
alyz
e an
d dr
aw c
oncl
usio
ns f
rom
gra
phs.
(V
-2)
Rec
ogni
ze a
nd id
entif
y nu
mer
ical
and
geo
met
ric
patte
rns
and
sequ
ence
s.(V
I-1)
App
ropr
iate
ly u
se te
rms
such
as
and,
or,
not
, onl
yan
dif
...th
en,
in a
mat
hem
atic
al s
ense
. (V
I-4)
Prov
ide
spec
ific
exa
mpl
es o
f a
give
n nu
mer
ical
or
geom
etri
c pr
inci
ple.
(V
I-5)
Perf
orm
task
s in
volv
ing
indu
ctiv
e an
d de
duct
ive
reas
onin
g. (
VI-
7)
Und
erst
and
the
mea
ning
of
"var
iabl
e" a
nd "
cons
tant
." (
VII
-1)
Be
fam
iliar
and
pro
fici
ent w
ith th
e st
anda
rd o
rder
of
oper
atio
ns. (
VII
-2)
Rep
rese
nt m
athe
mat
ical
rel
atio
nshi
ps u
sing
var
iabl
es. (
VII
-3)
Sim
plif
y po
lyno
mia
l exp
ress
ions
. (V
II-4
)
Eva
luat
e si
mpl
e al
gebr
aic
expr
essi
ons
by s
ubst
itutin
g va
lues
for
the
vari
able
s.(V
II-5
)
Solv
e tw
o-st
ep e
quat
ions
and
ineq
ualit
ies
usin
g in
tege
rs, f
ract
ions
and
dec
i-m
als.
(V
II-6
)
1 2
2
56
Exa
mpl
es o
f In
dica
tors
4-8
Stan
dard
10:
Sta
tistic
s
Col
lect
ing,
rep
rese
ntin
g an
d pr
oces
sing
dat
a ar
e ac
tiviti
es o
f m
ajor
impo
rtan
ce to
con
tem
pora
ry s
ocie
ty. S
tude
nts
shou
ldre
cogn
ize
that
sta
tistic
s pl
ay a
n im
port
ant
role
bet
wee
n th
e ex
actn
ess
of m
athe
mat
ical
stu
dies
and
the
natu
re o
f a
wor
ld d
epen
dent
larg
ely
on in
divi
dual
opi
nion
.
Stud
ent O
utco
mes
Org
aniz
e da
ta to
sho
w r
elat
ions
hips
in ta
bles
and
gra
phs.
(IV
- )
Inte
rpre
t dat
a fr
om ta
bles
and
cha
rts
to d
eter
min
e re
latio
nshi
ps. (
IV-2
)
Det
erm
ine
an e
xpre
ssio
n or
an
equa
tion
for
a re
latio
nshi
p an
d th
en e
valu
ate
orso
lve
it fo
r a
give
n va
lue
of th
e va
riab
le. (
IV-3
)
Rec
ogni
ze o
r fi
nd a
spe
cifi
c pa
ttern
occ
urri
ng in
a s
eque
nce
of n
umbe
rs.
(IV
-4)
Col
lect
and
org
aniz
e da
ta u
sing
list
s, ta
bles
and
gra
phs.
(V
-1)
Inte
rpre
t, an
alyz
e an
d dr
aw c
oncl
usio
ns f
rom
gra
phs.
(V
-2)
Tak
e po
lls a
nd s
urve
ys. (
V-3
)
Pred
ict a
nd r
ecor
d th
e pr
obab
ility
of
even
ts f
rom
sim
ple
expe
rim
ents
. (V
-4)
Find
the
mea
n, m
edia
n, m
ode
and
rang
e fo
r a
set o
f da
ta. (
V-6
)
Det
erm
ine
the
prob
abili
ty o
f si
mpl
e ev
ents
and
dra
w c
oncl
usio
ns o
r m
ake
inte
rpre
tatio
ns. (
V-7
)
Find
the
empi
rica
l pro
babi
lity
of a
n ev
ent f
rom
a s
ampl
e of
obs
erve
d ou
tcom
es.
(V-8
)
Find
the
prob
abili
ty o
f co
mpl
emen
tary
eve
nts
and
of m
utua
lly e
xclu
sive
even
ts. (
V-9
)
124
57
Exa
mpl
es o
f In
dica
tors
Syst
emat
ical
ly c
olle
ct, o
rgan
ize
and
desc
ribe
dat
a.
Con
stru
ct, r
ead
and
inte
rpre
t tab
les,
cha
rts
and
grap
hs.
Mak
e in
fere
nces
and
con
vinc
ing
argu
men
ts th
at a
re b
ased
on
data
ana
lysi
s.
Eva
luat
e ar
gum
ents
that
are
bas
ed o
n da
ta a
naly
sis.
Form
ulat
e qu
estio
ns; c
olle
ct a
nd o
rgan
ize
data
bas
ed o
n th
ose
ques
tions
;re
pres
ent t
he d
ata
usin
g gr
aphs
, tab
les,
and
fre
quen
cy d
istr
ibut
ions
; ana
lyze
data
; mak
e co
njec
ture
s; a
nd c
omm
unic
ate
info
rmat
ion
in a
mea
ning
ful
way
.
Use
sta
tistic
al m
etho
ds a
s a
mea
ns f
or d
ecis
ion
mak
ing.
125
Stan
dard
10:
Sta
tistic
s (c
ontin
ued)
Stud
ent O
utco
mes
Gen
erat
e a
freq
uenc
y di
stri
butio
n fo
r a
give
n lis
t of
data
. (V
-10)
Use
a li
st o
r tr
ee d
iagr
am to
cou
nt p
erm
utat
ions
or
com
bina
tions
. (V
-11)
App
ropr
iate
ly u
se te
rms
such
as
and,
or,
not
, onl
y an
d if
... t
hen,
in a
mat
hem
atic
al s
ense
. (V
I-4)
Perf
orm
task
s in
volv
ing
indu
ctiv
e an
d de
duct
ive
reas
onin
g. (
VI-
7)
1 2
b
58
Exa
mpl
es o
f In
dica
tors
4-8
Stan
dard
11:
Pro
babi
lity
Prob
abili
ty is
the
mea
sure
of
the
likel
ihoo
d of
an
even
t tha
t can
be
dete
rmin
ed th
eore
tical
ly o
rexp
erim
enta
lly. S
tude
nts
mus
t not
onl
y un
ders
tand
the
rela
tions
hip
betw
een
the
num
eric
al e
xpre
ssio
n an
d th
e pr
obab
ility
of
the
even
ts b
ut r
ealiz
e th
at th
e m
easu
reof
cer
tain
ty o
r un
cert
aint
y va
ries
as
mor
e da
ta a
re c
olle
cted
.
Stud
ent O
utco
mes
Inte
rpre
t dat
a fr
om ta
bles
and
cha
rts
to d
eter
min
e re
latio
nshi
ps. (
IV-2
)
Inte
rpre
t, an
alyz
e an
d dr
aw c
oncl
usio
ns f
rom
gra
phs.
(V
-2)
Pred
ict a
nd r
ecor
d th
e pr
obab
ility
of
even
ts f
rom
sim
ple
expe
rim
ents
. (V
-4)
Exp
ress
pro
babi
lity
as a
rat
io o
r fr
actio
n. (
V-5
)
Det
erm
ine
the
prob
abili
ty o
f si
mpl
e ev
ents
and
dra
w c
oncl
usio
ns o
r m
ake
inte
rpre
tatio
ns. (
V-7
)
Find
the
empi
rica
l pro
babi
lity
of a
n ev
ent f
rom
a s
ampl
e of
obs
erve
d ou
tcom
es.
(V-8
)
Find
the
prob
abili
ty o
f co
mpl
emen
tary
eve
nts
and
of m
utua
lly e
xclu
sive
even
ts. (
V-9
)
Gen
erat
e a
freq
uenc
y di
stri
butio
n fo
r a
give
n lis
t of
data
. (V
- 10
)
Use
a li
st o
r tr
ee d
iagr
am to
cou
nt p
erm
utat
ions
or
com
bina
tions
.(V
- I
I )
App
ropr
iate
ly u
se te
rms
such
as
and,
or,
not
, onl
yan
dif
...th
en,
in a
mat
hem
atic
al s
ense
. (V
I-4)
Prov
ide
a co
unte
rexa
mpl
e of
a c
ondi
tion
that
is n
ot a
lway
s tr
ue.
(VI-
6)
Perf
orm
task
s in
volv
ing
indu
ctiv
e an
d de
duct
ive
reas
onin
g.(V
I-7)
128
59
Exa
mpl
es o
f In
dica
tors
Exp
lore
situ
atio
ns b
y ex
peri
men
ting
and
sim
ulat
ing
prob
abili
ty m
odel
s.
Con
stru
ct a
sam
ple
to d
eter
min
e pr
obab
ility
.
Com
pare
exp
erim
enta
l res
ults
with
mat
hem
atic
al e
xpec
tatio
ns.
Mak
e pr
edic
tions
that
are
bas
ed o
n ex
peri
men
tal o
r th
eore
tical
pro
babi
li-tie
s.
Use
cha
rts,
gra
phs
and
plot
s to
mak
e pr
edic
tions
.
Mak
e hy
poth
eses
, tes
t con
ject
ures
and
ref
ine
theo
ries
bas
ed o
n ne
win
form
atio
n us
ing
expe
rim
enta
tion
and
sim
ulat
ion.
Exp
lore
eve
nts
and
situ
atio
ns r
elev
ant t
o th
eir
daily
live
s.
129
Stan
dard
12:
Geo
met
ry
4-8
Geo
met
ry is
the
stud
y of
spa
tial r
elat
ions
hips
that
def
ine
and
mak
e se
nse
of th
e w
orld
in w
hich
we
live.
Stud
ent O
utco
mes
Use
an
appr
opri
ate
form
ula
to c
alcu
late
the
peri
met
er a
nd a
rea
of p
olyg
ons
and
the
circ
umfe
renc
e of
a c
ircl
e. (
II-8
)
Use
an
appr
opri
ate
form
ula
to c
alcu
late
the
volu
me
and
surf
ace
area
of
the
follo
win
g so
lids:
sph
eres
, pri
sms,
pyr
amid
s, c
ylin
ders
and
con
es. (
II-9
)
Use
a p
rotr
acto
r to
mea
sure
and
dra
w a
ngle
s. (
II-1
0)
Use
an
appr
opri
ate
form
ula
to d
eter
min
e a
mea
sure
men
t whe
n a
dire
ctm
easu
rem
ent t
ool i
s un
avai
labl
e. (
II-
1 I
)
Est
imat
e le
ngth
s, a
reas
, vol
umes
and
wei
ghts
, and
che
ck b
y m
easu
ring
inE
nglis
h an
d m
etri
c un
its. (
II-1
2)
Solv
e pr
oble
ms
with
mea
sure
men
ts in
clud
ing
dist
ance
, wei
ght,
time,
are
a,ca
paci
ty a
nd te
mpe
ratu
re. (
11-1
3)
Use
term
inol
ogy
appr
opri
ate
to th
e gr
ade
leve
l. (I
II-I
)
Rec
ogni
ze tw
o- a
nd th
ree-
dim
ensi
onal
fig
ures
by
thei
r sh
apes
, and
iden
tify
thei
r co
mpo
nent
par
ts a
nd a
ssoc
iate
d pr
oper
ties.
(II
I-2)
Thr
ough
obs
erva
tion,
mea
sure
men
t, dr
awin
g an
d m
odel
ing,
iden
tify
geom
et-
ric
prop
ertie
s su
ch a
s sy
mm
etry
, con
grue
nce,
sim
ilari
ty, p
aral
lelis
m a
ndpe
rpen
dicu
lari
ty. (
III-
3)
Vis
ualiz
e, d
raw
and
con
stru
ct tw
o- a
nd th
ree-
dim
ensi
onal
fig
ures
thro
ugh
conc
rete
exp
erie
nces
. (II
I-4)
Util
ize
geom
etri
c pr
inci
ples
and
idea
s to
sol
ve p
robl
ems.
(11
1-5)
1 31
1
60
Exa
mpl
es o
f In
dica
tors
Iden
tify,
des
crib
e, c
ompa
re a
nd c
lass
ify
figu
res.
Use
geo
met
ric
term
s to
des
crib
e an
d id
entif
y re
latio
nshi
ps.
Def
ine
two-
and
thre
e-di
men
sion
al f
igur
es th
roug
h ex
peri
ence
s in
con
-st
ruct
ing,
dra
win
g, m
easu
ring
, and
com
pari
ng a
nd c
ontr
astin
g th
em.
Dra
w in
fere
nces
and
mak
e co
njec
ture
s fr
om g
eom
etri
c pr
oble
m s
ituat
ions
.
Ded
uce
the
Pyth
agor
ean
theo
rem
thro
ugh
expl
orat
ions
.
Rep
rese
nt a
nd s
olve
pro
blem
s us
ing
geom
etri
c m
odel
s.
App
ly g
eom
etri
c pr
inci
ples
to s
olve
rea
l-w
orld
pro
blem
s.
Use
con
nect
ions
of
geom
etry
and
the
wor
ld th
roug
h ar
t, na
ture
, con
stru
c-tio
n an
d an
atom
y.
131
Stan
dard
12:
Geo
met
ry (
cont
inue
d)
Stud
ent O
utco
mes
Exp
erim
ent w
ith, d
isco
ver
and
appl
y th
e kn
own
form
ulas
of
geom
etry
. (II
I-6)
Und
erst
and
pi a
nd it
s im
port
ance
in th
e ge
omet
ry o
f th
e ci
rcle
. (II
I-7)
Rec
ogni
ze a
nd c
lass
ify
angl
es a
s w
ell a
s id
entif
y re
latio
nshi
ps b
etw
een
angl
es.
( II
I-8)
Dra
w a
nd/o
r co
nstr
uct a
var
iety
of
shap
es h
avin
g th
e sa
me
area
. (11
1-9)
Und
erst
and
the
rela
tions
hip
betw
een
angl
es th
at a
re c
ompl
emen
tary
, sup
ple-
men
tary
and
ver
tical
. (IV
-7)
Iden
tify
and
unde
rsta
nd th
e re
latio
nshi
ps o
f an
gles
for
med
by
a tr
ansv
ersa
l of
two
para
llel l
ines
. (IV
-8)
Und
erst
and
the
rela
tions
hips
occ
urri
ng b
etw
een
sim
ilar
figu
res.
(IV
-9)
Rec
ogni
ze a
nd id
entif
y nu
mer
ical
and
geo
met
ric
patte
rns
and
sequ
ence
s.(V
1- I
)
Cla
sszf
y an
d ca
tego
rize
set
s of
num
bers
or
grou
ps o
f ge
omet
ric
shap
es. (
VI-
2)
Usi
ng a
giv
en a
ttrib
ute,
fin
d si
mila
ritie
s an
d di
ffer
ence
s am
ong
geom
etri
csh
apes
or
desi
gns.
(V
I-3)
App
ropr
iate
ly u
se te
rms
such
as
and,
or,
not
, onl
yan
d if
...
then
,in
am
athe
mat
ical
sen
se. (
V1-
4)
Prov
ide
spec
ific
exa
mpl
es o
f a
give
n nu
mer
ical
or
geom
etri
c pr
inci
ple.
(V
l-5)
Prov
ide
a co
unte
rexa
mpl
e of
a c
ondi
tion
that
is n
ot a
lway
s tr
ue. (
VI-
6)
Perf
orm
task
s in
volv
ing
indu
ctiv
e an
d de
duct
ive
reas
onin
g. (
VI-
7)
132
61
Exa
mpl
es o
f In
dica
tors
4-8
Stan
dard
13:
Mea
sure
men
t
In a
dditi
on to
siz
e, d
ista
nce
and
leng
th o
f tim
e, m
easu
rem
ent i
nclu
des
the
mor
e im
port
ant c
once
pt o
f un
ders
tand
ing
dim
ensi
onal
rel
atio
nshi
ps r
egar
dles
s of
the
stan
dard
of
mea
sure
.
Stud
ent O
utco
mes
Kno
w th
at a
ll m
easu
rem
ent i
s ap
prox
imat
e. (
II-1
)
Sele
ct th
e ap
prop
riat
e m
easu
ring
inst
rum
ent.
(11-
2)
Add
and
sub
trac
t mea
sure
s of
tim
e. (
11-3
)
Com
pare
tem
pera
ture
s an
d de
term
ine
the
amou
nt o
f ch
ange
. (11
-4)
Mea
sure
leng
th. v
olum
e an
d w
eigh
t in
both
Eng
lish
and
met
ric
units
. (11
-5)
Con
vert
Eng
lish
and
met
ric
mea
sure
men
t uni
ts to
equ
ival
ent u
nits
with
in th
egi
ven
syst
em. (
11-6
)
13e
awar
e of
app
roxi
mat
e eq
uiva
lent
mea
sure
men
ts b
etw
een
Eng
lish
and
met
ric
units
. (11
-7)
Use
an
appr
opri
ate
form
ula
to c
alcu
late
the
peri
met
er a
nd a
rea
of p
olyg
ons
and
the
circ
umfe
renc
e of
a c
ircl
e. (
11-8
)
Use
an
appr
opri
ate
form
ula
to c
alcu
late
the
volu
me
and
surf
ace
area
of
the
follo
win
g so
lids:
sph
eres
, pri
sms,
pyr
amid
s, c
ylin
ders
and
con
es. (
11-9
)
Use
a p
rotr
acto
r to
mea
sure
and
dra
w a
ngle
s. (
11-1
0)
Use
an
appr
opri
ate
form
ula
to d
eter
min
e a
mea
sure
men
t whe
n a
dire
ctm
easu
rem
ent t
ool i
s un
avai
labl
e. (
11-
11
)
Est
imat
e le
ngth
s, a
reas
, vol
umes
and
wei
ghts
, and
che
ck b
y m
easu
ring
inE
nglis
h an
d m
etri
c un
its. (
11-
1 2)
1 3
62
Exa
mpl
es o
f In
dica
tors
Use
app
ropr
iate
tool
s to
mea
sure
obj
ects
,
Est
imat
e, m
ake
and
use
mea
sure
men
ts to
des
crib
e an
d co
mpa
re o
bjec
ts.
Sele
ct a
ppro
prilt
e un
its a
nd to
ols
to m
easu
re th
e de
gree
of
accu
racy
requ
ired
in a
par
tiz.'u
lar
situ
atio
n.
Und
erst
and
the
stru
ctur
e an
d us
e of
sys
tem
s of
mea
sure
men
t.
Thr
ough
exp
lora
tion.
dev
elop
pro
cedu
res
and
form
ulas
for
det
erm
inin
gpe
rim
eter
, are
a, v
olum
e, a
ngle
, mea
sure
, cap
acity
, and
wei
ght a
nd m
ass.
Dev
elop
con
cept
s of
for
mal
and
info
rmal
sys
tem
s of
mea
sure
men
ts.
Dev
elop
for
mul
as a
nd p
roce
dure
s fo
r de
term
inin
g m
easu
rem
ents
whe
nso
lvin
g pr
oble
ms.
Con
stru
ct s
cale
mod
els.
135
Stan
dard
13:
Mea
sure
men
t (co
ntin
ued)
Stud
ent O
utco
mes
Solv
e pr
oble
ms
with
mea
sure
men
ts in
clud
ing
dist
ance
, wei
ght,
time,
are
a,ca
paci
ty a
nd te
mpe
ratu
re. (
II-1
3)
Thr
ough
obs
erva
tion,
mea
sure
men
t, dr
awin
g an
d m
odel
ing,
iden
tify
geom
et-
ric
prop
ertie
s su
ch a
s sy
mm
etry
, con
grue
nce,
sim
ilari
ty, p
aral
lelis
m a
ndpe
rpen
dicu
lari
ty. (
III-
3)
Vis
ualiz
e, d
raw
and
con
stru
ct tw
o- a
nd th
ree-
dim
ensi
onal
fig
ures
thro
ugh
conc
rete
exp
erie
nces
. (II
I-4)
Util
ize
geom
etri
c pr
inci
ples
and
idea
s to
sol
ve p
robl
ems.
(II
I-5)
Exp
erim
ent w
ith, d
isco
ver
and
appl
y th
e kn
own
form
ulas
of
geom
etry
. (II
I-6)
Und
erst
and
pi a
nd it
s im
port
ance
in th
e ge
omet
ry o
f th
e ci
rcle
. (II
I-7)
Rec
ogni
ze a
nd c
lass
ify
angl
es a
s w
ell a
s id
entif
y re
latio
nshi
ps b
etw
een
angl
es.
(III
-8)
Dra
w a
nd/o
r co
nstr
uct a
var
iety
of
shap
es h
avin
g th
e sa
me
area
. (II
I-9)
136
63
Exa
mpl
es o
f In
dica
tors
138
9-12
MA
TH
EM
AT
ICS
ESS
EN
TIA
L S
KIL
LS
pri i
ffi
r?7
1F
65
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
Mat
hem
atic
al O
utco
mes
Exp
ecte
d of
All
Hig
h Sc
hool
Gra
duat
es
Hig
h sc
hool
mat
hem
atic
s in
stru
ctio
n sh
ould
ens
ure
that
eac
h st
uden
t gra
duat
ing
from
hig
h sc
hool
per
form
s th
efo
llow
ing
outc
omes
:
I.N
umbe
r
1.U
se th
e fo
ur a
rith
met
ic o
pera
tions
on
ratio
nal n
umbe
rs (
expr
esse
d as
com
mon
fra
ctio
ns, d
ecim
al f
ract
ions
or
perc
ents
) w
ith a
ccur
acy
and
reas
onab
le s
peed
. usi
ng p
enci
l and
pap
er o
r ca
lcul
ator
.
Perf
orm
men
tal c
alcu
latio
ns o
f th
e fo
ur o
pera
tions
with
app
ropr
iate
leve
ls o
f pr
ecis
ion.
3. U
nder
stan
d th
e co
ncep
t of
orde
ring
the
real
num
bers
by
loca
ting
thei
r co
rres
pond
ing
poin
ts o
n th
e nu
mbe
r lin
e.
4. W
rite
mat
hem
atic
al e
xpre
ssio
ns a
nd s
ente
nces
to r
epla
ce v
erba
l exp
ress
ions
, usi
ng a
ppro
pria
te o
pera
tion,
num
ber
and
rela
tion
sym
bols
.
5.U
se s
tand
ard
rule
s fo
r or
der
of o
pera
tions
, tog
ethe
r w
ith s
ymbo
ls o
f in
clus
ion
to c
orre
ctly
eva
luat
e nu
mer
ical
expr
essi
ons
and
form
ulas
in a
var
iety
of
prob
lem
set
tings
.
6.U
nder
stan
d an
d us
e pr
oper
ties
of e
qual
ity a
nd in
equa
lity.
II.
Mea
sure
men
t
I.U
nder
stan
d th
e ap
prox
imat
e na
ture
of
mea
sure
men
t and
app
ly th
e pr
ecis
ion
of m
easu
rem
ents
to th
e re
sulti
ngac
cura
cy o
f ca
lcul
atio
ns.
Mak
e un
it co
nver
sion
s w
ith a
ppro
pria
te le
vels
of
accu
racy
bet
wee
n an
d w
ithin
sta
ndar
d sy
stem
s of
mea
sure
s.
3. U
se a
ppro
pria
te s
tand
ard
units
of
mea
sure
men
t to
mak
e re
ason
able
est
imat
es o
f lin
ear,
are
a, v
olum
e an
dw
eigh
t mea
sure
s of
obj
ects
com
mon
ly e
ncou
nter
ed in
dai
ly li
fe.
9-12
Stan
dard
s
(1) E 1) :-
E .
.
.E = E
.r-
4
g) .2 o .r
r ,
- ....
C; .
. kt
.k
r .
cS...
.k
. c
--... - .r-
-
.,..., tt, c E ...
.
I-. E .
oc
0 ._,. .
0 ,
.A . _
.c-
,
.:3 0 2 .
, a. ...., .
r- I
6 r, (....
),..
_ 0 ...., c E.:. ,, - (2
.--
,
_E
.,-
3-
XX
XX
X
XX
X
X X
XX
X
X X
XX
X
X X
XX
X
X X
XX
X
X X
XX
X
X X
XX
X
XX
XX
XX
140
67
141
1 2 3 4 5 1 2
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
(con
tinue
d)
II.
Mea
sure
men
t (co
ntin
ued)
4.Se
lect
and
use
app
ropr
iate
for
mul
as o
rpr
oced
ures
to d
eter
min
e in
dire
ctly
a m
easu
re w
hen
a di
rect
mea
sure
men
t is
not a
vaila
ble
or f
easi
ble.
III.
Geo
met
ry
I.Id
entif
y an
d di
stin
guis
h be
twee
n ge
omet
ric
elem
ents
, suc
h as
line
s,an
gles
, pla
nes,
pol
ygon
s, c
ircl
es a
ndre
gula
r so
lids.
2.U
nder
stan
d an
d ap
ply
basi
c ge
omet
ric
rela
tions
hips
suc
has
par
alle
l, in
ters
ectin
g, p
erpe
ndic
ular
, sim
ilari
ty,
prop
ortio
nalit
y an
d co
ngru
ence
.
3.C
alcu
late
are
as o
f re
gula
r po
lygo
ns a
nd c
ircl
es a
nd v
olum
es o
fre
ctan
gula
r so
lids,
cyl
inde
rs a
nd s
pher
es.
4.U
se c
ompa
ss, p
rotr
acto
r an
d ru
ler
to p
erfo
rm s
tand
ard
geom
etri
c co
nstr
uctio
ns.
5. A
pply
the
Pyth
agor
ean
theo
rem
in s
olvi
ngpr
oble
ms.
IV. D
ata
Ana
lysi
s an
d Pr
obab
ility
I.Se
lect
and
use
app
ropr
iate
pri
ncip
les
of c
ount
ing
colle
ctio
ns a
ndar
rang
emen
ts o
f ob
ject
s or
out
com
es o
fse
quen
tial p
roce
dure
s.
2.Se
lect
and
use
app
ropr
iate
sta
tistic
al m
easu
res
to d
escr
ibe
sets
of
data
.14
'4.
6 8
9-12
Stan
dard
s
.0 . .5 -a v) = .65 2 .
= -2 (1 - c L) = 14 2 .
t^ 4
I:, 4 c
(24 = 2 .
en
,,, = E u .= 7,1; 2 .
.1-
iY.
-,9 D'U
Tt .
,r,
0*,
-.1" = LE.
v::+
.. r., = >,
cn ,. 15 E ° 61.
t---
.. .8 (It a c t.l t E ° 'd
.00
V, g 0 &Q i- .
ON
.2 .- Ii14 ei., . 0 ..-.
:2 .2 0 ct.
.--.
(1 d .
C`I .--.
6 d 0 : . '- c 5".0 c .F c c u .en .-
.
1.) 2 . ,t .-
-.
XX
XX
X
XX
XX
XX
XX
XX
XX
.X
X
XX
X
XX
XX
XX
XX
XX
XX
4 5
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
(co
ntin
ued)
IV.
Dat
a A
naly
sis
and
Prob
abili
ty (
cont
inue
d)
3.E
xpla
in v
aria
bilit
y of
sta
tistic
al m
easu
res
in te
rms
of s
ampl
ing
proc
ess
or p
opul
atio
n di
ffer
ence
s.
4.U
se th
e co
ncep
t of
mat
hem
atic
al e
xpec
tatio
n to
est
imat
e ou
tcom
es o
fran
dom
pro
cess
es.
5.Id
entif
y an
d ex
plai
n m
isus
es o
f st
atis
tics.
6. U
se e
xper
imen
tal o
bser
vatio
ns to
est
imat
e em
piri
cal p
roba
bilit
ies
and
popu
latio
n pa
ram
eter
s.
7.D
istin
guis
h be
twee
n in
depe
nden
t and
dep
ende
nt e
vent
san
d us
e co
nditi
onal
pro
babi
litie
s.
V.
Alg
ebra
I.A
dd. s
ubtr
act,
mul
tiply
and
div
ide
with
mon
omia
ls a
nd b
inom
ials
.
/. Si
mpl
ify
and
eval
uate
alg
ebra
ic e
xpre
ssio
ns w
hich
invo
lve
inte
gral
exp
onen
tsan
d sq
uare
roo
ts.
3.Si
mpl
ify
ratio
nal a
lgeb
raic
exp
ress
ions
hav
ing
mon
omia
l den
omin
ator
s.
4. S
olve
line
ar e
quat
ions
and
ineq
ualit
ies
in o
ne v
aria
ble,
and
appl
y th
ese
to s
olve
pro
blem
s.
5. S
olve
,,ys
tem
s of
line
ar e
quat
ions
and
ineq
ualit
ies
in tw
o va
riab
les,
gra
phic
ally
and
ana
lytic
ally
, and
form
ulat
e su
ch s
yste
ms
to s
olve
pro
blem
s.
6. S
olve
pro
blem
s us
ing
ratio
and
pro
port
ion,
per
cent
ord
irec
t and
inve
rse
vari
atio
n.
144
69
9-12
Stan
dard
s
.0 c -; v ) 'A :15
= .2 .a ("t)
Vs
n ,15
0 u *F',
El
-E ..sp
c 2 .r.; a
.9. J.;
.= >,
'41 E 2
0 .,-1 .0 %), < c 't5 E 2
2 o t0 .2 t.-3 .-.
..--- ,..1
-6
m u ,.. o .0 c -c , g-,
-, .
XX
XX
XX
XX
X,X
X
XX
XX
X
XX
rXX
X
Xxx
xX
Xxx
xxx_
XX
XX
XX
XX
X
XX
XX
X
XX
XX
X
Xli_
xX
X
145
3 4 5 6 7 1 4 5 6
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
(co
ntin
ued)
VI.
Patte
rns
and
Rel
atio
ns
1.Id
entif
y an
d di
stin
guis
h be
twee
n ar
ithm
etic
and
geo
met
ric
prog
ress
ions
and
sta
te a
rul
e or
for
mul
a fo
r th
ege
nera
l ter
m o
f su
ch a
pro
gres
sion
.
I.W
rite
the
linea
r re
latio
nshi
p be
twee
n tw
o va
riab
les
by in
spec
tion
of it
s gr
aph
or o
f th
e se
t of
its o
rder
ed p
airs
.
3. G
raph
the
inve
rse
of a
fun
ctio
n by
insp
ectio
n of
the
grap
h of
the
func
tion.
4.Id
entif
y an
d ex
plai
n gr
aphi
c m
isre
pres
enta
tion
or d
isto
rtio
ns o
f se
ts o
f da
ta.
5.Id
entif
y, in
terp
ret a
nd c
onst
ruct
gra
phs
of n
onlin
ear
rela
tions
suc
h as
par
abol
as a
nd c
ircl
es.
VII
. Ana
lytic
Rea
soni
ng
I.D
istin
guis
h be
twee
n in
duct
ive
and
dedu
ctiv
e re
ason
ing
and
expl
ain
oow
eac
h is
mos
t app
ropr
iate
ly u
sed.
2.R
ecog
nize
whe
n co
nditi
ons
of d
efin
ition
s ar
e sa
tisfi
ed.
3. U
se d
educ
tive
reas
onin
g to
gen
erat
e co
nclu
sion
s.
4. U
se s
tatis
tical
rea
soni
ng to
test
hyp
othe
ses
info
rmal
ly.
5.Id
entif
y va
lid a
nd in
valid
arg
umen
ts.
14G
70
9-12
Stan
dard
s
elo
.... = 174
17;
CI
,-,-
;
d,,,
1,7i 4
..... f) 10 vi
g .... ti c .6
0 r.,
"S .6 e 8 1.--
:
u 6. JD - . E 8 cc
0 . 0 .L4) ci
0 ....: .4 B c;
.,..-
., - 2 0 :'
eq
ci To U ,.... o a C = a (19- g c.
,-i
'Fd. 4
xxX
xx.
....'
X
XX
, X x
X
Xx x
x,
kx
xx
.. ..
x..
x,
.0,..
..
x-
.
X.
..
'. X
* x
XX
2 3 4 5 2 3 4 5 147
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
(co
ntin
ued)
Add
ition
al C
ours
es a
nd S
kills
Req
uire
d fo
r U
nive
rsity
Adm
issi
on
Upo
n co
mpl
etin
g co
urse
s in
Alg
ebra
I, A
lgeb
ra I
I an
d G
eom
etry
, a h
igh
scho
ol s
tude
nt s
houl
d ha
ve m
aste
red
the
follo
win
g ou
tcom
es:
I.A
lgeb
ra I
I.
Sim
plif
y al
gebr
aic
expr
essi
ons.
2.E
valu
ate
alge
brai
c ex
pres
sion
s.
3. S
olve
line
ar e
quat
ions
.
4. U
se th
e la
ws
of e
xpon
ents
to s
impl
ify
expr
essi
ons
invo
lvin
g in
tegr
al e
xpon
ents
.
5. A
dd, s
ubtr
act,
mul
tiply
and
div
ide
poly
nom
ial e
xpre
ssio
ns.
6. F
acto
r po
lyno
mia
ls.
7. S
olve
line
ar in
equa
litie
s an
d gr
aph
them
on
a nu
mbe
r lin
e.
8.Fi
nd th
e sl
ope
of a
line
and
wri
te e
quat
ions
of
lines
in s
tand
ard
and
slop
e in
terc
ept f
orm
.
9. S
olve
sys
tem
s of
line
ar e
quat
ions
in tw
o va
riab
les
by s
ubst
itutio
n, a
dditi
on m
etho
d an
d gr
aphi
cally
.
10. A
dd, s
ubtr
act,
mul
tiply
and
div
ide
alge
brai
c fr
actio
ns a
nd c
ompl
ex f
ract
ions
.
148
7 1
9-12
Stan
dard
s--
--1
t4 1
...9 'Z ce) u 2 v: MI
_a --;
g ri .....- = E C . :,3 .5 r-i
011 e c 0.1
VI
CI = e-;
= ,,
V = 4
- -c,) trl
c '5 szi
Q f.) = .... 5, cn co E -. C
IJ E r-:
.Q ....,3 L.
..0 U WI < m E - 0) ,g.
oci
0 C 0 ci
.. c
-0 2 :ZU
.' i7 CA
. v: tI)
C E c a . e-:
1.1
=a
v) .. *1:
X'
Jx CI,
'
'3
'
Xr ,
.!.
X' X
XiX
XX
. XX
XJC
.iti
t:X
149
2 3 4 5 6 7 8 9 10
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
(co
ntin
ued)
Alg
ebra
I (
cont
inue
d)
11. S
impl
ify,
add
, sub
trac
t and
mul
tiply
rad
ical
s, a
nd r
atio
naliz
e de
nom
inat
ors.
12. S
olve
dir
ect a
nd in
vers
e va
riat
ion
prob
lem
s.
13. S
olve
ele
men
tary
wor
d pr
oble
ms.
14. U
se f
orm
ulas
.
15. S
olve
and
gra
ph a
bsol
ute
valu
e pr
oble
ms.
16. S
olve
qua
drat
ic e
quat
ions
.
17. U
nder
stan
d th
e co
ncep
t of
func
tions
and
thei
r gr
aphs
.
H.
Alg
ebra
II
1.E
valu
ate
alge
brai
c ex
pres
sion
s.
2. U
se la
ws
of e
xpon
ents
to c
hang
e th
e fo
rms
of e
xpre
ssio
ns in
volv
ing
inte
gral
exp
onen
tsan
d ra
tiona
l exp
onen
ts.
3.Si
mpl
ify
radi
cals
.
4. C
hang
e fr
om f
ract
iona
l exp
onen
ts to
rad
ical
for
m o
r ra
dica
l for
m to
frac
tiona
l exp
onen
ts w
hene
ver
nece
ssar
y.
5. S
olve
for
mul
as f
or s
peci
fied
sym
bols
.1
5
7 2
9-12
Stan
dard
s
c 5 -6 -8 ct Ki = c
ii .
= ..9 Ei = g u Q ..= ;4
.04
= &) ts = 'al .
01
U .c rt i .
'I'
V,
-8 b.° :i .
1/1
0 ..,-7
; c LE.
\ 0
' .V
- .
.5 . ,,:t
cn cl 0) E g o .0.
-
° 419
m 0) E 0 6 .00
°G' 0 . E-.
iiict
5U.
as
re:
;.:4 .
CD .-.
E 0' .--
....
.
.`" .
(-SI ,--,
6 61 , ) ctd) - 2 .....
,,,`,
'-
a 74 g .en ...
.
g ..r,i . 't .....
.-.
,.
XX
-,
Ix.
x.,
X: '
,, '''X
,O. "
XX
4,..
?
X
.' X.'
X..
.
4:.
X
xkr
.)t
x
xxxx
x
11 12 13 14 15 16 17
2 3 4 5 51
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
(co
ntin
ued)
11.
Alg
ebra
II (
cont
inue
d)
6.S
olve
equ
atio
ns a
nd in
equa
litie
s in
volv
ing
abso
lute
valu
e.
7. P
erfo
rm o
pera
tions
with
pol
ynom
ials
incl
udin
g sy
nthe
ticdi
visi
on.
S.
Fac
tor
vario
us ty
pes
of p
olyn
omia
ls.
9.S
olve
qua
drat
ic e
quat
ions
by
fact
orin
g an
d us
ing
the
quad
ratic
form
ula.
10.
Sol
ve q
uadr
atic
ineq
ualit
ies.
11.
Per
form
ope
ratio
ns w
ith r
atio
nal e
xpre
ssio
ns.
12.
Sol
ve e
quat
ions
and
ineq
ualit
ies
invo
lvin
g ra
tiona
l exp
ress
ions
.
13.
Per
form
ope
ratio
ns w
ith c
ompl
ex n
umbe
rs.
14. W
ork
NA
,ith
form
s of
the
equa
tion
of a
line
and
the
conc
ept
of th
e sl
ope
of a
line
.
15.
Sol
ve li
near
sys
tem
s of
equ
atio
ns in
thre
eva
riabl
es.
16. W
ork
with
the
func
tion
conc
ept a
nd fu
nctio
nno
tatio
n.
17. U
se th
e m
etho
d of
com
plet
ing
the
squa
re to
find
the
vert
ex o
f a p
arab
ola
and
then
gra
ph.
18.
Sol
ve e
xpon
entia
l and
loga
rithm
ic e
quat
ions
.
152
73
Stan
dard
s.
-;.- -5 - C.-
.. -:
r-.. E - C."
;
r-i
1
riI 1,
'e:
1
c.,-
; Li:';.:; -
7r.
:1--
.
tr:
E E,..
....
,-.
e.,
L.%
..:::
r-
;.= E ,- .:-. 3 x:
',"-
!--
C7.
:
,-,-
.7:
c.-.
..'
11
i14 I ; I
in.
..e.
a. i-
EIC
-.-:
L. 1
s , - 'E ...
'21
1 ,-
-:
-, .!'
XX
XX
X
)(X
XV
(.
xxxx
x
X3X
xxX
XX
xx-
xxX
xxxx
xxx
i
xxxx
xxxx
xxxx
XX
xxx
,
xxxx
xxxX
xxx
xxxx
x
153
6 7 8 9 10 11 11 13 14 15 16 17 18
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
(co
ntin
ued)
II.
Alg
ebra
II
(con
tinue
d)
19. W
ork
with
ari
thm
etic
and
geo
met
ric
prog
ress
ions
.
20. U
se a
cal
cula
tor
to o
btai
n tw
o-de
cim
al-p
lace
app
roxi
mat
ions
to th
e an
swer
for
rad
ical
, exp
onen
tial o
rlo
gari
thm
ic f
orm
s.
21. W
ork
with
con
ic s
ectio
ns. a
nd u
se a
lgeb
ra to
sol
ve p
robl
ems
invo
lvin
g co
ordi
nate
geo
met
ry.
Solv
e eq
uatio
ns a
nd p
robl
ems
whi
ch r
ely
on p
rope
rtie
s of
loga
rith
mic
and
exp
onen
tial
func
tions
.
23. U
nder
stan
d de
gree
and
rad
ian
mea
sure
of
angl
es.
24. W
ork
with
trig
onom
etri
c fu
nctio
ns a
s ra
tios
of s
ides
of
righ
t tri
angl
es a
nd a
sci
rcul
ar f
unct
ions
.
25. G
raph
trig
onom
etri
c fu
nctio
ns.
26. U
se c
alcu
lato
rs a
nd ta
bles
to a
ppro
xim
ate
valu
es o
f tr
igon
omet
ric
func
tions
.
27. D
eriv
e ex
act v
alue
s of
trig
onom
etri
c fu
nctio
ns a
ssoc
iate
d w
ith a
ngle
s of
30,
45,
60 a
nd 9
0 de
gree
s.
28. D
escr
ibe
and
grap
h on
e-to
-one
fun
ctio
ns a
nd th
eir
inve
rses
.
29. W
ork
with
inve
rse
trig
onom
etri
c fu
nctio
ns.
30. V
erif
y tr
igon
omet
ric
iden
titie
s an
d so
lve
trig
onom
etri
c eq
uatio
ns.
1 5
431
. Sol
ve r
ight
tria
ngle
s an
d us
e th
e la
w o
f si
nes
and
the
law
of c
osin
es.
74
9-12
Stan
dard
s
to .= v.) v: c c
1
.c z,-4 :g . v; m ..c liv
-i
c-i
co c &) v: c .c .a.
eri
,,, a s ° U .c v, 4
:1 I5 2) tri
. , . 9 ti = 43
15 .= Cr) as E e '' - >,
b. cE) v r-
-:
.0 ID , ca E P. ; - , g v 06
'iE) 0 go
.r.:
.c a;
ci"
Vy
c-.^
..:
z-,
: ::, 2 0 : c
,i --.
c.;
e2) o c .E. c 74 2'
, -, . g r-
7.
v C,
I . -
1
..c ,,, 4
. X, XX
XX
XX
XX
X
. XX
XX
X..
X
XX
X.X
X
X.
XX
X X
X)t
,..x
xx
xx
I Iii;
".
xx
x
,xxe
xx
x...
,..
IJL
Icic
xx
jc4r
xxx
xii-
xic
xx
___t
19 20 21 22 23 24 25 26 27 28 29 30
r-o
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
(co
ntin
ued)
III.
Geo
met
ry
1.U
se d
efin
ition
s, p
ostu
late
s an
d th
eore
ms
to d
evel
op p
roof
s.
2. S
olve
wor
d pr
oble
ms.
3. P
erfo
rm a
lgeb
raic
man
ipul
atio
ns.
4. U
se f
orm
al, l
ogic
al r
easo
ning
to s
olve
abs
trac
t, th
eore
tical
and
pra
ctic
al p
robl
ems.
5U
nder
stan
d an
gle
and
line
rela
tions
hips
, e.g
.. ve
rtic
al a
ngle
s, p
aral
lel o
r pe
rpen
dicu
lar
lines
, and
sup
plem
en-
tary
and
com
plem
enta
ry a
ngle
s.
6.t s
e co
ncep
ts o
f co
nizr
uenc
e an
d si
mila
rity
.
7.E
xplo
re p
olyg
onal
rel
atio
nshi
ps to
incl
ude
and
iden
tify
com
mon
geo
met
ric
shap
es a
nd d
escr
ibe
thei
r an
gle
mea
sure
s.
8.E
xplo
re c
ircl
es a
nd a
rc c
hara
cter
istic
s to
det
erm
ine
the
mea
sure
of
arcs
and
thei
r re
late
d an
gles
and
cho
rds,
tang
ents
and
sec
ants
.
9. K
now
and
use
the
Pyth
agor
ean
theo
rem
.
10.
Find
the
peri
met
er a
nd a
rea
of a
pla
ne f
igur
e.
11.
Find
vol
ume
and
surf
ace
area
of
a so
lid.
1_5
G7
5
9-12
Stan
dard
s
G - t 2oua
= .,-; 2
ti) . 2 1,i; 2 rei
, , , c,-,
) 0 1,--
, 2 zr:
v tt.,
.;- .r:
13' g u..
sc:
CA co P. ,... z E 6 r-:
U a co ,... z E a c6
t ,9 ,,, cr:
. `=
1
.74 ci
>, 2!..-
,,,-.
0 --:
il cc
u tc 73 1-4
c,s,
t; .1:
XX
X,X
X )
00(
..
XX
XA
X X
XX
X
)(X
XX
XxX
XX
X
xxxx
X
XX
XX
XX
XX
XX
XX
XX
_X
X X
XX
X
157
1 2 3 4 5 8 9 10 11
Ari
zona
Mat
hem
atic
s E
ssen
tial S
kills
(co
ntin
ued)
ill.
Geo
met
ry (
cont
inue
d)
P. M
ake
and
appl
y ge
omet
ric c
onst
ruct
ion.
1 3.
Sol
ve p
robl
ems
appl
ying
the
met
hods
of c
oord
inat
e ge
omet
ry,e
.g.,
the
dist
ance
form
ula,
the
mid
poin
t for
mul
a
and
findi
ng th
e eq
uatio
n of
a li
ne.
14. U
se r
ight
tria
ngle
trig
onom
etry
to s
olve
pro
blem
s.
158
76
9-12
Stan
dard
s
to .E -6 uo _ 2 0.. = ...-1 2 --:
c c 143; = = 7. c") .= .E.-
4 2 r.i
WI c .E i5 c4 .= - ,-
,-;
, C 0 F, .= 75 2 -A:
-,9 ti, :Tt
,-;
c 0 .c-;
" = LE sc;
II _a C >-,
En :.4 ,.-. , <
1.) - 6 r--:
' 6- .0 0 - < = cs ,.... p, Z 0 6' 06
.65 E ,-.
..b.-
P
O
v, C:
--.
2 c ei...
.--:
--.
2 .,-.°
), b ...-
ci u "E to c..... C .E'
......
-. g ?f, u .
....
., a- ..= 1-.5 2 -
X.XX.X
x
Xxx
xx
XX
XX
xx
15,)
12 13 14
9-12
Stan
dard
1: M
athe
mat
ics
as P
robl
em S
olvi
ng
Prob
lem
sol
ving
is m
athe
mat
ics
in it
s br
oade
st s
ense
. In
the
prim
ary
grad
es it
is s
omet
imes
use
ful t
o di
ffer
entia
te a
mon
g co
ncep
tual
, pro
cedu
ral a
nd p
robl
em-
solv
ing
goal
s. I
n gr
ades
9-1
2 th
e di
stin
ctio
ns b
etw
een
thes
e go
als
blur
, and
they
are
inte
grat
ed a
nd in
tern
aliz
ed. T
his
give
s th
e st
uden
t a b
road
fou
ndat
ion
for
appr
oach
ing
and
doin
g m
athe
mat
ics,
reg
ardl
ess
of to
pic
or s
ubje
ct. P
robl
em s
olvi
ng is
mor
e th
an a
pply
ing
a sp
ecif
ic r
ule
to a
spe
cifi
c pr
oble
m. P
robl
em s
olvi
ngis
the
fabr
ic th
at h
olds
the
stat
e E
ssen
tial S
kills
.NC
TM
Sta
ndar
dsan
d al
l of
mat
hem
atic
s to
geth
er.
Stud
ent O
utco
mes
App
ly th
e m
athe
mat
ics
Ess
entia
l Ski
lls in
a v
arie
ty o
f pr
oble
m-s
olvi
ngco
ntex
ts a
nd le
arn
the
mor
e co
mpl
ex E
ssen
tial S
kills
with
in th
e co
ntex
t of
real
-lif
e pr
oble
ms
or s
ituat
ions
.
1 6)
77
Exa
mpl
es o
f In
dica
tors
Use
, with
incr
easi
ng c
onfi
denc
e, p
robl
em-s
olvi
ng a
ppro
ache
s to
inve
sti-
gate
and
und
erst
and
mat
hem
atic
al c
onte
nt.
App
ly in
tegr
ated
mat
hem
atic
al p
robl
em-s
olvi
ng s
trat
egie
s to
sol
ve p
rob-
lem
s fr
om w
ithin
and
out
side
mat
hem
atic
s.
Rec
ogni
ze a
nd f
orm
ulat
e pr
oble
ms
from
situ
atio
ns w
ithin
and
out
side
mat
hem
atic
s.
App
ly th
e pr
oces
s of
mat
hem
atic
al m
odel
ing
to r
eal-
wor
ld p
robl
em s
itua-
tions
.
161
9-12
Stan
dard
2: M
athe
mat
ics
as C
omm
unic
atio
ns
All
stud
ents
mus
t lis
ten
to, r
ead
abou
t, w
rite
abo
ut, s
peak
abo
ut, r
efle
ct o
n an
d de
mon
stra
te m
athe
mat
ical
idea
s. S
tude
nts
invo
lved
in in
divi
dual
and
sm
all-
grou
pac
tiviti
es w
ill h
ave
man
y op
port
uniti
es to
dis
cuss
, que
stio
n, li
sten
and
sum
mar
ize.
The
se a
ctiv
ities
will
dir
ect i
nstr
uctio
n aw
ay f
rom
rec
all o
f te
rmin
olog
y an
d ro
utin
em
anip
ulat
ion
to d
eepe
r co
ncep
tual
und
erst
andi
ng o
f m
athe
mat
ics.
It i
s no
t eno
ugh
for
stud
ents
to f
ind
the
answ
er. I
t is
just
as
impo
rtan
t for
stu
dent
s to
desc
ribe
how
they
get
thei
r an
swer
and
the
diff
icul
ties
they
enc
ount
er in
tryi
ng to
fin
d it.
It h
as b
ecom
e m
ore
appa
rent
, with
the
deve
lopm
ent o
f th
e A
rizo
naSt
uden
t Ass
essm
ent
Prog
ram
, tha
t mat
hem
atic
s is
not
just
cal
cula
tbns
and
man
ipul
atio
ns. D
iscu
ssio
ns a
bout
peo
ple,
issu
es a
nd c
ultu
ral i
mpl
icat
ions
of
mat
hem
atic
s w
ill a
dd to
the
unde
rsta
ndin
g of
the
conn
ectio
n be
twee
n m
athe
i tat
ics
and
soci
ety.
Stud
ent O
utco
mes
Com
mun
icat
e th
eir
expe
rien
ces
with
, and
an
unde
rsta
ndin
g of
, the
mat
hem
at-
ics
Ess
entia
l Ski
lls.
1 6
78
Exa
mpl
es o
f In
dica
tors
Ref
lect
upo
n an
d cl
arif
y th
inki
ng a
bout
mat
hem
atic
al id
eas
and
rela
tion-
ship
s.
Form
ulat
e m
athe
mat
ical
def
initi
ons
and
expr
ess
gene
raliz
atio
ns d
isco
v-er
ed th
roug
h in
vest
igat
ions
.
Exp
ress
mat
hem
atic
al id
eas
oral
ly a
nd in
wri
ting.
Dem
onst
rate
und
erst
andi
ng o
f w
ritte
n pr
esen
tatio
ns o
f m
athe
mat
ics.
Ask
cla
rify
ing
and
exte
ndin
g qu
estio
ns r
elat
ed to
mat
hem
atic
s th
at a
pply
toth
e re
al w
orld
.
163
Stan
dard
3: M
athe
mat
ics
as R
easo
ning
Ded
uctiv
e an
d in
duct
ive
reas
onin
g ar
e us
ed b
y th
emse
lves
in a
ll ar
eas
of m
athe
mat
ics.
Whi
le d
oing
mat
hem
atic
s, s
tude
nts
mak
e uo
njec
ture
s fr
om p
atte
rns
mad
e in
part
icul
ar c
ases
(in
duct
ive
reas
onin
g) a
nd te
st th
ese
conj
ectu
res
by lo
gica
l ver
ific
atio
n, o
r a
coun
tere
xam
ple
(ded
uctiv
e re
ason
ing)
. The
se m
etho
ds a
re v
ery
usef
ulin
mat
hem
atic
s an
d in
oth
er f
ield
s. A
ll st
uden
ts n
eed
to k
now
that
ded
uctiv
e re
ason
ing
is th
e m
etho
d us
ed to
est
ablis
h th
e va
lidity
of
a m
athe
mat
ical
asse
rtio
n.R
easo
ning
and
pro
ofs
need
to b
e us
ed in
all
mat
hem
atic
s co
urse
s.
Stud
ent O
utco
mes
Use
mat
hem
atic
al r
easo
ning
to d
emon
stra
te c
once
ptua
l und
erst
andi
ng o
f th
em
athe
mat
ics
Ess
entia
l Ski
lls a
nd th
eir
abili
ty to
use
them
in a
var
iety
of
prob
lem
situ
atio
ns. 16
4
79
Exa
mpl
es o
f In
dica
tors
Mak
e an
d te
st c
onje
ctur
es.
Form
ulat
e co
unte
rexa
mpl
es.
Judg
e th
e va
lidity
of
argu
men
ts.
Con
stru
ct s
impl
e va
lid a
rgum
ents
.
And
so
that
, in
addi
tion,
col
lege
-int
endi
ng s
tude
nts-
Con
stru
ct p
roof
s fo
r m
athe
mat
ical
ass
ertio
ns, i
nclu
ding
indi
rect
pro
ofs
and
proo
fs b
y m
athe
mat
ical
indu
ctio
n. 165
9-12
Stan
dard
4: M
athe
mat
ical
Con
nect
ions
Con
nect
ions
are
the
thre
ads
that
take
isol
ated
ski
lls a
nd w
eave
them
into
use
ful t
ools
. Con
nect
ions
nee
d to
be
culti
vate
dbe
twee
n m
athe
mat
ical
topi
cs, a
s w
ell a
s w
ith
othe
r di
scip
lines
.
Dev
elop
ing
mat
hem
atic
s as
an
inte
grat
ed w
hole
incr
ease
s th
e po
tent
ial f
or r
eten
tion
and
tran
sfer
of
mat
hem
atic
al c
once
pts.
Con
nect
ing
mat
hem
atic
s w
ith o
ther
disc
iplin
es a
nd w
ith d
aily
aff
airs
und
ersc
ores
the
utili
ty o
f th
e su
bjec
t.
Stud
ent O
utco
mes
Stud
ents
can
mak
e co
nnec
tions
bet
wee
n th
edi
ffer
ent m
athe
mat
ics
Ess
entia
lSk
ills,
as
wel
l as
with
rea
l-lif
e si
tuat
ions
.
16t;
Exa
mpl
es o
f In
dica
tors
Rec
ogni
ze e
quiv
alen
t rep
rese
ntat
ions
of
the
sam
e co
ncep
t.
Rel
ate
proc
edur
es in
one
rep
rese
ntat
ion
to p
roce
dure
sin
an
equi
vale
nt
repr
esen
tatio
n.
Use
the
conn
ectio
ns a
mon
g m
athe
mat
ical
topi
cs.
Use
the
conn
ectio
ns b
etw
een
mat
hem
atic
s an
d ot
herd
isci
plin
es a
nd th
e re
al
wor
ld.
Art
: the
use
of
sym
met
ry, p
ersp
ectiv
e, s
patia
lre
pres
enta
tions
, and
pat
-
tern
s (i
nclu
ding
fra
ctal
s) to
cre
ate
orig
inal
art
istic
wor
ks
Bio
logy
: the
use
cf
scal
ing
to id
entif
y lim
iting
fac
tors
on
the
orga
nism
s
Bus
ines
s: th
e op
timiz
atio
n of
a c
omm
unic
atio
nne
twor
k
Indu
stri
al A
rts:
the
use
of m
athe
mat
ics-
base
dco
mpu
ter-
aide
d de
sign
in
prod
ucin
g sc
ale
draw
ings
or
mod
els
of th
ree-
dim
ensi
onal
obje
cts
such
as
hous
es
Med
icin
e: m
odel
ing
an in
ocul
atio
n pl
an to
elim
inat
e an
infe
ctio
us d
isea
se
Phys
ics:
the
use
of v
ecto
rs to
add
ress
pro
blem
sin
volv
ing
forc
es
Soci
al S
cien
ce: t
he u
se o
f st
atis
tical
tech
niqu
esin
pre
dict
ing
and
anal
yz-
ing
elec
tion
resu
lts
167
go
Stan
dard
5: A
lgeb
ra
Alg
ebra
is th
e la
ngua
ge th
roug
h w
hich
mos
t of
mat
hem
atic
s is
com
mun
icat
ed. T
he in
crea
sing
use
of
quan
titat
ive
met
hods
, bot
h in
nat
ural
sci
ence
s an
d in
dis
cipl
ines
such
as
econ
omic
s, p
sych
olog
y an
d so
ciol
ogy,
has
mad
e al
gebr
a an
impo
rtan
t too
l for
app
lied
mat
hem
atic
s.
Stud
ent O
utco
mes
Wri
te m
athe
mat
ical
exp
ress
ions
and
sen
tenc
es to
rep
lace
ver
bal e
xpre
ssio
ns.
usin
g ap
prop
riat
e op
erat
ion,
num
ber
and
rela
tion
sym
bols
. (1-
4)
Use
app
ropr
iate
sta
ndar
d un
its o
f m
easu
rem
ent t
o m
ake
reas
onab
le e
stim
ates
of li
near
, are
a, v
olum
e an
d w
eigh
t mea
sure
s of
obj
ects
com
mon
ly e
ncou
nter
edin
dai
ly li
fe. (
I1-
3)
Sele
ct a
nd u
se a
ppro
pria
te f
orm
ulas
or
proc
edur
es to
det
erm
ine
indi
rect
ly a
mea
sure
whe
n a
dire
ct m
easu
rem
ent i
s no
t ava
ilabl
e or
fea
sibl
e. (
11-4
)
Add
, sub
trac
t, m
ultip
ly a
nd d
ivid
e w
ith m
onom
ials
and
bin
omia
ls. (
V-
)
Sim
plif
y an
d ev
alua
te a
lgeb
raic
exp
ress
ions
whi
ch in
volv
e in
tegr
al e
xpon
ents
and
squa
re r
oots
. (V
-2)
Sim
plif
y ra
tiona
l alg
ebra
ic e
xpre
ssio
ns h
avin
g m
onom
ial d
enom
inat
ors.
(V
-3)
Solv
e lin
ear
equa
tions
and
ineq
ualit
ies
in o
ne v
aria
ble,
and
app
ly th
ese
to s
olve
prob
lem
s. (
V-4
)
Solv
e sy
stem
s of
line
ar e
quat
ions
and
ineq
ualit
ies
in tw
o va
riab
les,
gra
phic
ally
and
anal
ytic
ally
. and
for
mul
ate
such
sys
tem
s to
sol
ve p
robl
ems.
(V
-5)
Solv
e pr
oble
ms
usin
g ra
tio a
nd p
ropo
rtio
n. p
erce
nt o
r di
rect
and
inve
rse
vari
atio
n. (
V-6
)
Iden
tify
and
dist
ingu
ish
hem
cel
l ari
thm
etic
and
geo
met
ric
prog
ress
ions
and
stat
e a
rule
or
form
ula
for
the
gene
ral t
erm
of
such
a p
rogr
essi
on. (
VI-
1)
168
Exa
mpl
es o
f In
dica
tors
Rep
rese
nt s
ituat
ions
that
invo
lve
vari
able
qua
ntiti
es w
ith e
xpre
ssio
ns,
equa
tions
, ine
qual
ities
and
mat
rice
s.
Use
tabl
es a
nd g
raph
s as
tool
s to
inte
rpre
t exp
ress
ions
, equ
atio
ns a
ndin
equa
l itie
s.
Ope
rate
on
expr
essi
ons
and
mat
rice
s, a
nd s
olve
equ
atio
ns a
nd in
equa
litie
s.
Tra
nsfe
r th
e al
gebr
aic
proc
ess
to p
robl
em s
olvi
ng in
oth
er d
isci
plin
es.
169
Stan
dard
5: A
lgeb
ra (
cont
inue
d)
Stud
ent O
utco
mes
Wri
te th
e lin
ear
rela
tions
hip
betw
een
two
vari
able
s by
insp
ectio
n of
its
grap
hor
of
the
set o
f its
ord
ered
pai
rs. (
VI-
2)
Gra
ph th
e in
vers
e of
a f
unct
ion
by.in
spec
tion
of th
e gr
aph
of th
e fu
nctio
n.(V
1-3)
Iden
tify
and
expl
ain
grap
hic
mis
repr
esen
tatio
n or
dis
tort
ions
of
sets
of
data
.(V
I-4)
Iden
tify,
inte
rpre
t and
con
stru
ct g
raph
s of
non
linea
r re
latio
ns s
uch
as p
arab
olas
and
circ
les.
( V
I-5)
Rec
ogni
ze w
hen
cond
ition
s of
def
initi
ons
are
satis
fied
. (V
II-2
)
Add
ition
al C
ours
es a
nd S
kills
Req
uire
d fo
r U
nive
rsity
Adm
issi
on
Upo
n co
mpl
etin
g co
urse
s in
Alg
ebra
I. A
lgeb
ra I
I an
d G
eom
etry
, a h
igh
scho
olst
uden
t sho
uld
have
mas
tere
d th
e fo
llow
ing
outc
omes
:
Sim
plif
y al
gebr
aic
expr
essi
ons.
(1-
1)
Eva
luat
e al
gebr
aic
expr
essi
ons.
(1-
2)
Solv
e lin
ear
equa
tions
. (1-
3)
Use
the
law
s of
exp
onen
ts to
sim
plif
y ex
pres
sion
s in
volv
ing
inte
gral
exp
o-ne
nts.
(1-
4)-;
82
Exa
mpl
es o
f In
dica
tors
And
so
that
, in
addi
tion,
col
lege
-int
endi
ng s
tude
nts
can
Use
mat
rice
s to
sol
ve li
near
sys
tem
s.
Dem
onst
rate
tech
nica
l fac
ility
with
alg
ebra
ic tr
ansf
orm
atio
ns, i
nclu
ding
tech
niqu
es b
ased
on
the
theo
ry o
f eq
uatio
ns.
171
Stan
dard
5: A
lgeb
ra (
cont
inue
d)
Stud
ent O
utco
mes
Add
, sub
trac
t, m
ultip
ly a
nd d
ivid
e po
lyno
mia
l exp
ress
ions
. (1-
5)
Fact
or p
olyn
omia
ls. (
1-6)
Solv
e lin
ear
ineq
ualit
ies
and
grap
h th
em o
n a
num
ber
line.
(1-
7)
Find
the
slop
e of
a li
ne a
nd w
rite
equ
atio
ns o
f lin
es in
sta
ndar
d an
d sl
ope
inte
rcep
t for
m. (
1-8)
Solv
e sy
stem
s of
line
ar e
quat
ions
in tw
o va
riab
les
by s
ubst
itutio
n, a
dditi
onm
etho
d an
d gr
aphi
cally
. (1-
9)
Add
, sub
trac
t, m
ultip
ly a
nd d
ivid
e al
gebr
aic
frac
tions
and
com
plex
fra
ctio
ns.
(I-1
0)
Sim
plif
y, a
dd, s
ubtr
act a
nd m
ultip
ly r
adic
als,
and
rat
iona
lize
deno
min
ator
s.(I
- 1
1)
Solv
e di
rect
and
inve
rse
vari
atio
n pr
oble
ms.
(1-
12)
Solv
e el
emen
tary
wor
d pr
oble
ms.
(1-
13)
Use
for
mul
as. (
1-14
)
Solv
e an
d gr
aph
abso
lute
val
ue p
robl
ems.
(I-
15)
Solv
e qu
adra
tic e
quat
ions
. (1-
16)
Und
erst
and
the
conc
ept o
f fu
nctio
ns a
nd th
eir
grap
hs. (
1-17
)
Eva
luat
e al
gebr
aic
expr
essi
ons.
(11
-1 /
172
83
Exa
mpl
es o
f In
dica
tors 17
3
Stan
dard
5: A
lgeb
ra (
cont
inue
d)
Stud
ent O
utco
mes
Use
law
s of
exp
onen
ts to
cha
nge
the
form
s of
exp
ress
ions
invo
lvin
gin
tegr
al
expo
nent
s an
d ra
tiona
l exp
onen
ts.
(11-
2)
Sim
plif
y ra
dica
ls. (
II-3
)
Cha
nge
from
fra
ctio
nal e
xpon
ents
tora
dica
l for
m o
r ra
dica
l for
m to
frac
tiona
l
expo
nent
s w
hene
ver
nece
ssar
y.(1
1-4)
Solv
e fo
rmul
as f
or s
peci
fied
sym
bols
. (11
-5)
Solv
e eq
uatio
ns a
nd in
equa
litie
sin
volv
ing
abso
lute
val
ue. (
11-6
)
Perf
orm
ope
ratio
ns w
ith p
olyn
omia
lsin
clud
ing
synt
hetic
div
isio
n. (
11-7
)
Fact
or v
ario
us ty
pes
ofpo
lyno
mia
ls. (
11-8
)
Solv
e qu
adra
tic e
quat
ions
by
fact
orin
g an
d us
ing
the
quad
ratic
form
ula.
(11
-9)
Solv
e qu
adra
tic in
equa
litie
s.(I
I- 1
0)
Perf
orm
ope
ratio
ns w
ith r
atio
nal
expr
essi
ons.
(11
-1 1
)
Solv
e eq
uatio
ns a
nd in
equa
litie
sin
volv
ing
ratio
nal e
xpre
ssio
ns.
(11-
12)
Perf
orm
ope
ratio
ns w
ithco
mpl
ex n
umbe
rs. (
11-1
3)
Wor
k w
ith f
orm
s of
the
equa
tion
of a
line
and
the
conc
ept o
f th
esl
ope
of a
line
.
(11-
14)
17 4
Solv
e lin
ear
syst
ems
ofeq
uatio
ns in
thre
e va
riab
les.
(11-
15)
84
Exa
mpl
es o
f In
dica
tors
Stan
dard
5: A
lgeb
ra (
cont
inue
d)
Stud
ent O
utco
mes
Wor
k w
ith th
e fu
nctio
n co
ncep
t and
fun
ctio
n no
tatio
n. (
11-1
6)
Use
the
met
hod
of c
ompl
etin
g th
e sq
uare
to f
ind
the
vert
ex o
f a
para
bola
and
then
gra
ph. (
11-1
7 )
Solv
e ex
pone
ntia
l and
loga
rith
mic
equ
atio
ns. (
11-1
8)
Wor
k w
ith a
rith
met
ic a
nd g
eom
etri
c pr
ogre
ssio
ns. (
II-1
9)
Use
a c
alcu
lato
r to
obt
ain
two-
deci
mal
-pla
ce a
ppro
xim
atio
ns to
the
answ
er f
orra
dica
l, ex
pone
ntia
l or
loga
rith
mic
for
ms.
(I1
-20)
Wor
k w
ith c
onic
sec
tions
, and
use
alg
ebra
to s
olve
pro
blem
s in
volv
ing
coor
dina
te g
eom
etry
. (11
-21
)
Solv
e eq
uatio
ns a
nd p
robl
ems
whi
ch r
ely
on p
rope
rtie
s of
loga
rith
mic
and
expo
nent
ial f
unct
ions
. (11
-22)
Des
crib
e an
d gr
aph
one-
to-o
ne f
unct
ions
and
thei
r in
vers
es. (
11-2
8)
Wor
k w
ith in
vers
e tr
igon
omet
ric
func
tions
. (11
-29)
Ver
ify
trig
onom
etri
c id
entit
ies
and
solv
e tr
igon
omet
ric
equa
tions
. (11
-30)
176
85
Exa
mpl
es o
f In
dica
tors
177
9-12
Stan
dard
6: F
unct
ions
Func
tions
are
spe
cial
cor
resp
onde
nces
bet
wee
n th
e el
emen
ts o
f tw
o se
ts in
a g
iven
fie
ld o
f m
athe
mat
ics.
The
y ar
e re
clus
ive
with
in a
giv
en f
ield
but
may
occu
r as
oper
atio
ns in
oth
er m
athe
mat
ical
are
as. T
he f
unct
ion
conc
ept i
s im
port
ant b
ecau
se it
is a
mat
hem
atic
al r
epre
sent
atio
n of
man
y in
put-
outp
ut s
ituat
ions
fou
nd in
the
real
wor
ld, i
nclu
ding
thos
e th
at r
ecen
tly h
ave
aris
en a
s a
resu
lt of
tech
nolo
gica
l adv
ance
s.
Stud
ent O
utco
mes
Exp
lain
var
iabi
lity
of s
tatis
tical
mea
sure
s in
term
s of
sam
plin
g pr
oces
s or
popu
latio
n di
ffer
ence
s. (
1V-3
)
Iden
tify,
inte
rpre
t and
con
stru
ct g
raph
s of
non
linea
r re
latio
ns s
uch
as p
arab
o-la
s an
d ci
rcle
s. (
V1-
5)
Dis
tingu
ish
betw
een
indu
ctiv
e an
d de
duct
ive
reas
onin
g an
d ex
plai
nho
w e
ach
is m
ost a
ppro
pria
tely
use
d. (
VII
-1)
Add
ition
al C
ours
es a
nd S
kills
Req
uire
d fo
r U
nive
rsity
Adm
issi
on
Upo
n co
mpl
etin
g co
urse
s in
Alg
ebra
I. A
lgeb
ra I
I an
d G
eom
etry
, a h
igh
scho
ol s
tude
nt s
houl
d ha
ve m
aste
red
the
follo
win
g ou
tcom
es:
Find
the
slop
e of
a li
ne a
nd w
rite
equ
atio
ns o
f lin
es in
sta
ndar
d an
d sl
ope
inte
rcep
t for
m. (
1-8)
Sim
plif
y, a
dd. s
ubtr
act i
tnd
mul
tiply
rad
ical
s, a
nd r
atio
naliz
e de
nom
inat
ors.
(I-1
)
Und
erst
and
the
conc
ept o
f fu
nctio
ns a
nd th
eir
grap
hs. (
1-17
)
Perf
orm
ope
ratio
ns w
ith c
ompl
ex n
umbe
rs. (
11-1
3)
Wor
k w
ith f
orm
s of
the
equa
tion
of a
line
and
the
conc
ept o
f th
e sl
ope
of a
line.
(11
- 14
)
Wor
k w
ith th
e fu
nctio
n co
ncep
t and
fun
ctio
n no
tatio
n. (
11-
1 6)
Use
con
cept
s of
con
grue
nce
and
sim
ilari
ty. (
111-
6)
Exa
mpl
es o
f In
dica
tors
Mod
el r
eal-
wor
ld p
robl
ems
with
a v
arie
ty o
f fu
nctio
ns.
Rep
rese
nt a
nd a
naly
ze r
elat
ions
hips
usi
ng ta
bles
, ver
bal r
ules
, equ
atio
nsan
d gr
aphs
.
Tra
nsla
te a
mon
g ta
bula
r, s
ymbo
lic a
nd g
raph
ical
rep
rese
ntat
ions
of
func
-tio
ns.
Dem
onst
rate
that
a v
arie
ty o
f pr
oble
m s
ituat
ions
can
be
mod
eled
by
the
sam
e ty
pe o
f fu
nctio
n.
Det
erm
ine
the
effe
cts
of p
aram
eter
cha
nges
on
the
grap
hs o
f fu
nctio
ns.
And
so
that
, in
addi
tion,
col
lege
-int
endi
ng s
tude
nts
can
Perf
orm
ope
ratio
ns o
n, a
nd th
e ge
nera
l pro
pert
ies
and
beha
vior
of,
cla
sses
of f
unct
ions
.
179
Stan
dard
7: G
eom
etry
fro
m a
Syn
thet
ic P
ersp
ectiv
e
Geo
met
ry p
rovi
des
expe
rien
ces
that
dee
pen
stud
ents
' und
erst
andi
ng o
f sh
apes
and
thei
r pr
oper
ties,
with
an
emph
asis
on
thei
r w
ide
appl
icab
ility
in h
uman
act
ivity
.Ph
ysic
al m
odel
s, th
ree-
dim
ensi
onal
fig
ures
and
oth
er r
eal-
wor
ld o
bjec
ts s
houl
d be
use
d to
pro
vide
a s
tron
g ba
se f
or th
e de
velo
pmen
t of
geom
etri
c in
tuiti
on th
at w
illhe
use
d in
wor
king
with
abs
trac
t ide
as.
Stud
ent O
utco
mes
Use
app
ropr
iate
sta
ndar
d un
its o
f m
easu
rem
ent t
o m
ake
reas
onab
le e
stim
ates
of li
near
. are
a, v
olum
e an
d w
eigh
t mea
sure
s of
obj
ects
com
mon
ly e
ncou
nter
edin
dai
ly li
fe. (
11-3
)
Iden
tify
and
dist
ingu
ish
betw
een
geom
etri
c el
emen
ts, s
uch
as li
nes,
ang
les,
plan
es, p
olyg
ons.
cir
cles
and
reg
ular
sol
ids.
(II
I- 1
)
Und
erst
and
and
appl
y ba
sic
geom
etri
c re
latio
nshi
ps s
uch
as p
aral
lel,
inte
rsec
t-in
g. p
erpe
ndic
ular
, sim
ilari
ty, p
ropo
rtio
nalit
y an
d co
ngru
ence
. (11
1-2)
Cal
cula
te a
reas
of
regu
lar
poly
gons
and
cir
cles
and
vol
umes
of
rect
angu
lar
solid
s. c
ylin
ders
and
sph
eres
. (I1
1-3
)
Use
com
pass
. pro
trac
tor
and
rule
r to
per
form
sta
ndar
d ge
omet
ric
cons
tnic
-tio
ns. (
1114
)
App
ly th
e Py
thag
orea
n th
eore
m in
sol
ving
pro
blem
s. (
11I-
5)
Iden
tify
and
dist
ingu
ish
betw
een
arith
met
ic a
nd g
eom
etri
c pr
ogre
ssio
ns a
ndst
ate
a ru
le o
r fo
rmul
a fo
r th
e ge
nera
l ter
m o
f su
ch a
pro
gres
sion
. (V
1-1
)
Dis
tingu
ish
betw
een
indu
ctiv
e an
d de
duct
iv-
reas
onin
g an
d ex
plai
n ho
w e
ach
is m
ost a
ppro
pria
tely
use
d. (
VII
- 1
)
Rec
ogni
ie w
hen
cond
ition
s of
def
initi
ons
are
satis
fied
. ( V
11-2
)
'se
dedu
ctiv
e re
ason
ing
to g
ener
ate
conc
lusi
ons.
( V
II-3
)
1:-)
o87
Exa
mpl
es o
f In
dica
tors
Inte
rpre
t and
dra
w th
ree-
dim
ensi
onal
obj
ects
.
Rep
rese
nt p
robl
em s
ituat
ions
with
geo
met
ric
mod
els
and
appl
y pr
oper
ties
of f
igur
es.
Cla
ssif
y fi
gure
s in
term
s of
con
grue
nce
and
sim
ilari
ty a
nd a
pply
thes
ere
latio
nshi
ps.
Ded
uce
prop
ertie
s of
. and
rel
atio
nshi
ps b
etw
een,
fig
ures
fro
m g
iven
as-
sum
ptio
ns.
181
Stan
dard
7: G
eom
etry
fro
m a
Syn
thet
ic P
ersp
ectiv
e (c
ontin
ued)
Stud
ent O
utco
mes
Iden
tify
valid
and
inva
lid a
rgum
ents
. (V
II-5
)
Add
ition
al C
ours
es a
nd S
kills
Req
uire
d fo
r U
nive
rsity
Adm
issi
on
Upo
n co
mpl
etin
g co
urse
s in
Alg
ebra
I, A
lgeb
ra I
I an
d G
eom
etry
, a h
igh
scho
olst
uden
t sho
uld
have
mas
tere
d th
e fo
llow
ing
outc
omes
:
Und
erst
and
degr
ee a
nd r
adia
n m
easu
re o
f an
gles
. (11
-23)
Wor
k w
ith tr
igon
omet
ric
func
tions
as
ratio
s of
sid
es o
f ri
ght t
rian
gles
and
as
circ
ular
fun
ctio
ns. (
11-2
4)
I. 's
e ca
lcul
ator
s an
d ta
bles
to a
ppro
xim
ate
valu
es o
f tr
igon
omet
ric
func
tions
.(
11-2
6)
Der
ive
exac
t val
ues
of tr
igon
omet
ric
func
tions
ass
ocia
ted
with
ang
les
of 3
0,45
, 60
and
90 d
egre
es. (
11-2
7)
Solv
e ri
ght t
rian
gles
and
use
the
law
of
sine
s an
d th
e la
w o
f co
sine
s. (
11-3
1)
l'se
defi
nitio
ns, p
ostu
late
s an
d th
eore
ms
to d
evel
op p
roof
s. (
111-
1)
Solv
e w
ord
prob
lem
s. (
I11-
2)
Re
form
al. l
ogic
al r
eam
min
g to
sol
ve a
bstr
act.
theo
retic
al a
nd p
ract
ical
prob
lem
s. (
111-
4)
Und
erst
and
angl
e an
d lin
e re
latio
nshi
ps. e
.g..
vert
ical
ang
les,
par
alle
l or
perp
endi
cula
r lin
es, a
nd s
uppl
emen
tary
and
com
plem
enta
ry a
ngle
s. (
111-
5)
1288
9-12
Exa
mpl
es o
f In
dica
tors
And
so
that
, in
addi
tion,
col
lege
-inte
ndin
g st
uden
ts c
an-
Dev
elop
an
unde
rsta
ndin
g of
an
axio
mat
ic s
yste
m th
roug
h in
vest
igat
ing
and
com
pari
ng v
ario
us g
eom
etri
es.
183
Stan
dard
7: G
eom
etry
fro
m a
Syn
thet
ic P
ersp
ectiv
e (c
ontin
ued)
Stud
ent O
utco
mes
Use
con
cept
s of
con
grue
nce
and
sim
ilari
ty. (
111-
6)
Exp
lore
pol
ygon
al r
elat
ions
hips
to in
clud
e an
d id
entif
y co
mm
on g
eom
etri
csh
apes
and
des
crib
e th
eir
angl
e m
easu
res.
(11
1-7)
Exp
lore
cir
cles
and
arc
cha
ract
eris
tics
to d
eter
min
e th
e m
easu
re o
f ar
cs a
ndth
eir
rela
ted
angl
es a
nd c
hord
s, ta
ngen
ts, a
nd s
ecan
ts. (
I11-
8)
Kno
w a
nd u
se th
e Py
thag
orea
n th
eore
m. (
111-
9)
Find
the
peri
met
er a
nd a
rea
of a
pla
ne f
igur
e. (
111-
10)
Find
vol
ume
and
surf
ace
area
of
a so
lid. (
111-
1I
)
Mak
e an
d ap
ply
geom
etri
c co
nstr
uctio
n. (
111-
12)
Lke
rig
ht tr
iang
le tr
igon
omet
ry to
sol
ve p
robl
ems.
(II
I- 1
4)
S 4
89
Exa
mpl
es o
f In
dica
tors
185
9-12
Stan
dard
8: G
eom
etry
fro
m a
n A
lgeb
raic
Per
spec
tive
One
of
the
mos
t im
port
ant c
onne
ctio
ns in
all
of m
athe
mat
ics
is th
at b
etw
een
geom
etry
and
alg
ebra
. The
inte
rpla
y be
twee
n ge
omet
ry a
nd a
lgeb
ra s
trep
gthe
nsst
uden
ts' a
bilit
y to
for
mul
ate
and
anal
yze
prob
lem
s fr
om a
var
iety
of
pers
pect
ives
.
Stud
ent O
utco
mes
Cal
cula
te a
reas
of
regu
lar
poly
gons
and
cir
cles
and
vol
umes
of
rect
angu
lar
solid
s, c
ylin
ders
and
sph
eres
. (II
I-3)
Add
ition
al C
ours
es a
nd S
kills
Req
uire
d fo
r U
nive
rsity
Adm
issi
on
Upo
n co
mpl
etin
g co
urse
s in
Alg
ebra
I. A
lgeb
ra I
I an
d G
eom
etry
, a h
igh
scho
olst
uden
t sho
uld
have
mas
tere
d th
e fo
llow
ing
outc
omes
:
Wor
k w
ith c
onic
sec
tions
, and
use
alg
ebra
to s
olve
pro
blem
s in
volv
ing
coor
dina
te g
eom
etry
. (11
-21
)
Solv
e w
ord
prob
lem
s.(I
II-2)
Perf
orm
alg
ebra
ic m
anip
ulat
ions
. (II
I-3)
Hnd
the
peri
met
er a
nd a
rea
of a
pla
ne f
igur
e.(I
ll-I0
)
Hnd
vol
ume
and
surf
ace
area
of a
sol
id. (
Ill-1
I
Sol
ve p
robl
ems
appl
ying
the
met
hods
of c
oord
inat
e ge
omet
ry,
e.g.
. the
dk-
tanc
leal
)rm
ula.
the
mid
poin
t for
mul
a an
d fi
ndin
g th
e eq
uatio
n of
a li
ne.
(111
- I 3
).
90
Exa
mpl
es o
f In
dica
tors
Tra
nsla
te b
etw
een
synt
hetic
and
coo
rdin
ate
repr
esen
tatio
ns.
Det
erm
ine
prop
ertie
s of
fig
ures
usi
ng tr
ansf
orm
atio
ns a
nd u
sing
coo
rdi-
nate
s.
Iden
tify
cong
ruen
t and
sim
ilar
fiQ
ures
usi
ng tr
ansf
orm
atio
ns.
Ana
lyze
pro
pert
ies
of E
uclid
ean
tran
sfor
mat
ions
and
rel
ate
tran
slat
ions
tove
ctor
s.
And
so
that
, in
addi
tion,
col
lege
-inte
ndin
g st
uden
ts c
an
Det
erm
ine
prop
ertie
s of
figu
res
usin
g ve
ctor
s.
Use
tran
sfor
mat
ions
, coo
rdin
ates
and
vec
tors
in p
robl
em s
olvi
ng.
187
Stan
dard
9: T
rigo
nom
etry
Tri
gono
met
ry is
a s
tudy
of
tria
ngul
ar r
elat
ions
hips
and
cir
cula
r fu
nctio
ns.
engi
neer
ing,
thro
ugh
whi
ch it
har
mon
izes
geo
met
ry a
nd a
lgeb
ra.
Stud
ent O
utco
mes
Iden
tify,
inte
rpre
t and
con
stru
ct g
raph
s of
non
linea
r re
latio
ns s
uch
as p
arab
olas
and
circ
les.
(V
1-5)
Add
ition
al C
ours
es a
nd S
kills
Req
uire
d fo
r U
nive
rsity
Adm
issi
on
Upo
n co
mpl
etin
g (m
urse
s in
Alg
ebra
I. A
lgeb
ra 1
1 an
d G
eom
etry
, a h
igh
scho
olst
uden
t sho
uld
have
mas
tere
d th
e fo
llow
ing
outc
omes
:
Und
erst
and
degr
ee a
nd r
adia
n m
easu
re o
f an
gles
. (II
-23)
Wor
k w
ith tr
igon
omet
ric
func
tions
as
ratio
s of
sid
es o
f ri
ght t
rian
gles
and
.As
circ
ular
fun
ctio
ns. (
Gra
ph tr
igon
omet
ric
func
tions
. (11
-25)
Use
cal
cula
tors
and
tabl
es to
app
roxi
mat
e va
lues
of
trig
onom
etri
c fu
nctio
ns.
(l1-
26)
Der
ive
e \ a
ct v
alue
s of
trig
onom
etri
c fu
nctio
ns a
ssoc
iate
d w
ith
angl
es o
f 30
.45,
60 a
nd 9
0 de
gree
s. (
11-2
7)
Wor
k w
ith im
erse
trig
onom
etri
c fu
nctio
ns. (
11-2
9
Ver
ify
trig
onom
etri
c id
entit
ies
and
solv
e tr
ig(
omet
ric
equa
tions
. (11
-30)
Sok
e ri
ght t
rian
gles
and
use
the
law
of
sine
s an
d th
e la
w o
f co
sine
s. (
II-3
1)
;,,c
righ
t tri
angl
e tr
igon
omet
ry to
sol
ve p
robl
ems.
(1
-1)
186
9-12
Its
man
y re
al-w
orld
app
licat
ions
incl
ude
navi
gatio
n, s
urve
ying
, arc
hite
ctur
e an
d
91
Exa
mpl
es o
f In
dica
tors
App
ly tr
igon
omet
ry to
sol
ve p
robl
ems
invo
lvin
g tr
iang
les.
Rep
rese
nt p
erio
dic
real
-wor
ld s
ituat
ions
usi
ng th
e si
ne a
nd c
osin
e fu
nc-
tions
.
And
so
that
, in
addi
tion,
col
lege
-int
endi
ng s
tude
nts
can
Show
the
conn
ectio
n be
twee
n tr
iang
ular
and
cir
cula
r tr
igon
omet
ric
func
-tio
ns.
Use
cir
cula
r fu
nctio
ns to
mod
el p
erio
dic
real
-wor
ld s
ituat
ions
.
Gra
ph tr
igon
omet
ric
func
tions
.
Solv
e tr
igon
omet
ric
equa
tions
and
ver
ify
trig
onom
etri
c id
entit
ies.
Dem
onst
rate
the
conn
ectio
ns b
etw
een
trig
onom
etri
c fu
nctio
ns a
nd p
olar
coor
dina
tes,
com
plex
num
bers
and
ser
ies. 189
9-12
Stan
dard
10:
Sta
tistic
s
Col
lect
ing,
rep
rese
ntin
g an
d pr
oces
sing
dat
a ar
e ac
tiviti
es o
f m
ajor
impo
rtan
ce to
con
tem
pora
ry s
ocie
ty. S
tude
nts
shou
ld r
ecog
nize
that
sta
tistic
s pl
ay a
n im
port
ant
inte
rmed
iate
rol
e be
twee
n th
e ex
actn
ess
of m
athe
mat
ical
stu
dies
and
the
equi
voca
l nat
ure
of a
wor
ld d
epen
dent
larg
ely
on in
divi
dual
opi
nion
.
Stud
ent O
utco
mes
Sele
ct a
nd u
se a
ppro
pria
te p
rinc
iple
s of
cou
ntin
g co
llect
ions
and
arr
ange
men
tsof
obj
ects
or
outc
omes
of
sequ
entia
l pro
cedu
res.
(IV
- I)
Sele
ct a
nd u
se a
ppro
pria
te s
tatis
tical
mea
sure
s to
des
crib
e se
ts o
f da
ta. (
IV-2
)
Exp
lain
var
iabi
lity
of s
tatis
tical
mea
sure
s in
term
s of
sam
plin
g pr
oces
s or
popu
latio
n di
ffer
ence
s. (
IV
-3 )
Iden
tify
and
expl
ain
mis
uses
of
stat
istic
s. (
IV-5
)
Iden
tify
and
expl
ain
grap
hic
mis
repr
esen
tatio
n or
dis
tort
ions
of
sets
of
data
.(
VI-
4)
Use
sta
tistic
al r
easo
ning
to te
st h
ypot
hese
s in
form
ally
. (V
II-4
)
Iden
tify
valid
and
inva
lid a
rgum
ents
. ( V
II-5
)
92
Exa
mpl
es o
f In
dica
tors
Con
stru
ct a
nd d
raw
infe
renc
es f
rom
cha
rts,
tabl
es a
nd g
raph
s th
at s
umm
a-ri
ze d
ata
from
rea
l-w
orld
situ
atio
ns.
Use
cur
ve f
ittin
g to
pre
dict
fro
m d
ata.
App
ly m
easu
res
of c
entr
al te
nden
cy, v
aria
bilit
y an
d co
rrel
atio
n.
Use
sam
plin
g to
exp
lain
its
role
in s
tatis
tical
cla
ims.
Des
ign
a st
atis
tical
exp
erim
ent t
o st
udy
a pr
oble
m, c
ondu
ct th
e ex
peri
men
tan
d in
terp
ret a
nd c
omm
unic
ate
the
outc
omes
.
Exp
lain
the
effe
cts
of d
ata
tran
sfor
mat
ions
on
mea
sure
s of
cen
tral
tend
ency
and
vari
abili
ty.
A n
d so
that
, in
addi
tion,
col
lege
-int
endi
ng s
tude
nts
can
Tra
nsfo
rm d
ata
to a
id in
dat
a in
terp
reta
tion
and
pred
ictio
n.
Tes
t hyp
othe
ses
usin
g ap
prop
riat
e st
atis
tics.
1 91
Stan
dard
11:
Pro
babi
lity
Prob
abili
ty p
rovi
des
conc
epts
and
met
hods
for
dea
ling
with
unc
erta
inty
and
for
inte
rpre
ting
pred
ictio
nsba
sed
on u
ncer
tain
ty. I
ts s
tudy
pro
vide
s a
stud
ent w
ith a
bas
is
of u
nder
stan
ding
fro
m w
hich
to m
ake
info
rmed
obs
erva
tions
abo
ut th
e lik
elih
ood
of e
vent
s an
d to
inte
rpre
t the
val
idity
of
stat
istic
al c
laim
s.
Stud
ent O
utco
mes
Sele
ct a
nd u
se a
ppro
pria
te p
rinc
iple
s of
cou
ntin
g co
llect
ions
and
arr
ange
men
tsof
obj
ects
or
outc
omes
of
sequ
entia
l pro
cedu
res.
(IV
-I)
Use
the
conc
ept o
f m
athe
mat
ical
exp
ecta
tion
to e
stim
ate
outc
omes
of
rand
ompr
oces
ses.
(IV
-4)
Use
exp
erim
enta
l obs
erva
tions
to e
stim
ate
empi
rica
l pro
babi
litie
s an
d po
pula
-tio
n pa
ram
eter
s. (
IV-6
)
Dis
tingu
ish
betw
een
inde
pend
ent a
nd d
epen
dent
eve
nts
and
use
cond
ition
alpr
obab
ilitie
s. (
IV-7
)
1_ 9
2
93
Exa
mpl
es o
f In
dica
tors
Use
exp
erim
enta
l or
theo
retic
al p
roba
bilit
y, a
s ap
prop
riat
e, to
rep
rese
nt a
nd
solv
e pr
oble
ms
invo
lvin
g un
cert
aint
y.
Use
sim
ulat
ions
to e
stim
ate
prob
abili
ties.
Exp
lain
the
conc
ept o
f a
rand
om v
aria
ble.
Cre
ate
and
inte
rpre
t dis
cret
e pr
obab
ility
dis
trib
utio
ns.
Des
crib
e, in
gen
eral
term
s, th
e no
rmal
cur
ve a
nd u
se it
s pr
oper
ties
to a
nsw
erqu
estio
ns a
bout
set
s of
dat
a th
at a
re a
ssum
ed to
be
norm
ally
dis
trib
uted
.
And
so
that
, in
addi
tion,
col
lege
-int
endi
ng s
tude
nts
can-
Use
the
conc
ept o
f a
rand
om v
aria
ble
to g
ener
ate
and
inte
rpre
t pro
babi
lity
dist
ribu
tions
incl
udin
g bi
nom
ial,
unif
orm
, nor
mal
and
chi
squ
are.
193
9-12
Stan
dard
12:
Dis
cret
e M
athe
mat
ics
The
non
mat
eria
l wor
ld o
f in
form
atio
n pr
oces
sing
req
uire
s th
e us
e of
dis
cret
e (d
isco
ntin
uous
) m
athe
mat
ics.
Com
pute
rs a
re e
ssen
tially
fin
ite, d
iscr
ete
mac
hine
s,th
e to
pics
fro
m d
iscr
ete
mat
hem
atic
s ar
e es
sent
ial t
o so
lvin
g pr
oble
ms
usin
g co
mpu
ter
met
hods
. Dis
cret
e m
athe
mat
ics
is th
e fi
eld
of m
athe
mat
ical
pro
pert
ies
of s
ets
and
syst
ems
that
hav
e on
ly a
fin
ite n
umbe
r of
ele
men
ts.
Stud
ent O
utco
mes
Stan
dard
I 2
: Dis
cret
e M
athe
mat
ics
is a
n N
CT
M s
tand
ard
that
is n
ot c
urre
ntly
addr
esse
d in
the
Ari
zona
Ess
entia
l Ski
lls. W
hen
the
Ess
entia
l Ski
lls a
re r
evis
edan
d th
e im
port
ance
of
this
are
a in
crea
ses,
it w
ill h
e ad
dres
sed.
19,4
94
Exa
mpl
es o
f In
dica
tors
Rep
rese
nt p
robl
em s
ituat
ions
usi
ng d
iscr
ete
stru
ctur
es s
uch
as f
inite
gra
phs,
mat
rice
s, s
eque
nces
and
rec
urre
nce
rela
tions
.
Rep
rese
nt a
nd a
naly
ze f
inite
gra
phs
usin
g m
atri
ces.
Dev
elop
and
ana
lyze
alg
orith
ms.
Solv
e en
umer
atio
n an
d fi
nite
pro
babi
lity
prob
lem
s.
And
so
that
, in
addi
tion,
col
lege
-int
endi
ng s
tude
nts
can
Rep
rese
nt a
nd s
olve
pro
blem
s us
ing
linea
r pr
ogra
mm
ing
and
diff
eren
ceeq
uatio
ns.
Inve
stig
ate
prob
lem
situ
atio
ns th
at a
rise
in c
onne
ctio
n w
ith c
ompu
ter
valid
atio
n an
d th
e ap
plic
atio
n of
alg
orith
ms.
1 9
5
9-12
Stan
dard
13:
Con
cept
ual U
nder
pinn
ing
of C
alcu
lus
Thi
s st
anda
rd c
alls
for
the
oppo
rtun
ity f
or s
tude
nts
to s
yste
mat
ical
ly, b
ut in
form
ally
, inv
estig
ate
the
cent
ral i
deas
of
calc
ulus
: lim
it, th
e ar
ea u
nder
a c
urve
, the
rate
of c
hang
e an
d th
e sl
ope
of a
tang
ent l
ine.
Thi
s st
anda
rd w
ill e
xten
d st
uden
ts' k
now
ledg
e of
fun
ctio
n ch
arac
teri
stic
s an
d in
trod
uce
them
to th
e m
ode
ofin
fini
te p
roce
sses
.Su
ch a
stu
dy w
ill d
eepe
n a
stud
ent's
und
erst
andi
ng o
f fu
nctio
n an
d its
util
ity in
rep
rese
ntin
g an
d an
swer
ing
ques
tions
abo
ut r
eal-
wor
ld s
ituat
ions
.
Stud
ent O
utco
mes
Stan
dard
13:
Con
cept
ual U
nder
pinn
ing
of C
alcu
lus
is a
n N
CT
M s
tand
ard
that
is n
ot c
urre
ntly
add
ress
ed in
the
Ari
zona
Ess
entia
l Ski
lls. W
hen
the
Ess
entia
lSk
ills
are
revi
sed
and
the
impo
rtan
ce o
f th
is a
rea
incr
ease
s, th
e co
ncep
tual
unde
rpin
ning
s of
cal
culu
s w
ill b
e ad
dres
sed.
196
95
Exa
mpl
es o
f In
dica
tors
Det
erm
ine
max
imum
and
min
imum
poi
nts
of a
gra
ph a
nd in
terp
ret t
here
sults
in p
robl
em s
ituat
ions
.
Inve
stig
ate
limiti
ng p
roce
sses
by
exam
inin
g in
fini
te s
eque
nces
, ser
ies
and
area
s un
der
curv
es.
And
so
that
, in
addi
tion,
col
lege
-int
endi
ng s
tude
nts
can
Und
erst
and
the
conc
eptu
al f
ound
atio
ns o
f lim
it, th
e ar
ea u
nder
a c
urve
, the
rate
of
chan
ge. t
he s
lope
of
a ta
ngen
t lin
e an
d th
e ap
plic
atio
ns o
f ea
ch in
othe
r di
scip
lines
.
Ana
lyze
the
grap
hs o
f po
lyno
mia
l, ra
tiona
l, ra
dica
l and
tran
scen
dent
alfu
nctio
ns.
197
9-12
Stan
dard
14:
Mat
hem
atic
al S
truc
ture
The
str
uctu
re o
f m
athe
mat
ics
is th
e fr
amew
ork
upon
whi
ch it
s va
riou
s fi
elds
are
bui
lt. D
esca
rte
calle
d m
athe
mat
ics
the
-sci
ence
of
orde
r."
Stud
ents
sho
uld
beco
me
awar
e of
this
str
uctu
re, h
ow it
pro
vide
s a
stro
ng f
ound
atio
n on
whi
ch a
var
iety
of
cont
ent s
tran
ds a
rebu
ilt a
nd h
ow it
sim
ulta
neou
sly
hold
s th
ese
diff
eren
t str
ands
toge
ther
. Eac
h fi
eld
offe
rs it
s ow
n pa
ttern
of
orde
r. A
s a
scie
nce
of p
atte
rns,
mat
hem
atic
s is
a m
ode
of in
quir
y th
at r
evea
lsfu
ndam
enta
l tru
th a
bout
the
orde
r of
our
wor
ld. A
n aw
aren
ess
of th
ese
broa
d st
ruct
urin
g pr
inci
ples
fre
es s
tude
nts
to ta
ke a
mor
e co
nstr
uctiv
e ap
proa
ch to
new
mat
hem
atic
al to
pics
. The
degr
ee o
f fo
rmal
ism
mus
t be
cons
iste
nt w
ith th
e st
uden
t's le
vel o
f m
athe
mat
ical
mat
urity
.
Stud
ent O
utco
mes
Use
the
four
ari
thm
etic
ope
ratio
ns o
n ra
tiona
l num
bers
(ex
pres
sed
as c
omm
onfr
actio
ns, d
ecim
al f
ract
ions
or
perc
ents
) w
ith a
ccur
acy
and
reas
onab
le s
peed
,us
ing
penc
il an
d pa
per
or c
alcu
lato
r. (
1-1)
Perf
orm
men
tal c
alcu
latio
ns o
f th
e fo
ur o
pera
tions
with
app
ropr
iate
leve
ls o
fpr
ecis
ion.
(1-
2)
Und
erst
and
the
conc
ept o
f or
deri
ng th
e re
al n
umbe
rs b
y lo
catin
g th
eir
corr
e-sp
ondi
ng p
oint
s on
the
num
ber
line.
(1-
3)
Use
sta
ndar
d ru
les
for
orde
r of
ope
ratio
ns to
geth
er w
ith s
ymbo
ls o
f in
clus
ion
to c
orre
ctly
eva
luat
e nu
mer
ical
exp
ress
ions
and
for
mul
as in
a v
arie
ty o
fpr
oble
m s
ettin
gs. (
1-5)
Und
erst
and
and
use
prop
ertie
s of
equ
ality
and
ineq
ualit
y. (
1-6)
Und
erst
and
the
appr
oxim
ate
natu
re o
f m
easu
rem
ent a
nd a
pply
the
prec
isio
n of
mea
sure
men
ts to
the
resu
lting
acc
urac
y of
cal
cula
tions
. (II
- 1
)
Mak
e un
it co
nver
sion
s w
ith a
ppro
pria
te le
vels
of
accu
racy
bet
wee
n an
d w
ithin
stan
dard
sys
tem
s of
mea
sure
s. (
11-2
)
198
96
Exa
mpl
es o
f In
dica
tors
Com
pare
and
con
tras
t the
rea
l num
ber
syst
em a
nd it
s va
riou
s su
bsys
tem
sw
ith r
egar
d to
thei
r st
ruct
ural
cha
ract
eris
tics.
App
ly th
e lo
gic
of a
lgeb
raic
pro
cedu
res.
Com
pare
see
min
gly
diff
eren
t mat
hem
atic
al s
yste
ms
to d
eter
min
e if
they
are
esse
ntia
lly th
e sa
me.
And
so
that
, in
addi
tion,
col
lege
-int
endi
ng s
tude
nts
can
Dev
elop
the
com
plex
num
ber
syst
em a
nd d
emon
stra
te f
acili
ty w
ith it
sop
erat
ions
.
Prov
e el
emen
tary
theo
rem
s w
ithin
var
ious
mat
hem
atic
al s
truc
ture
s, s
uch
asgr
oups
and
fie
lds.
Exp
lain
the
natu
re a
nd p
urpo
se o
f ax
iom
atic
sys
tem
s.
199
CO
AL
3. I
nteg
rate
Stu
dent
Ass
essm
ent w
ith th
e L
earn
ing
Proc
ess
The
mos
t acc
urat
e re
flect
ion
of s
tude
nts'
pro
gres
s in
mat
hem
atic
s is
evi
denc
edby
the
stud
ents
' per
form
ance
in th
e cl
assr
oom
. In
orde
r fo
r as
sess
men
t to
beva
lidit
mus
t be
inte
grat
ed in
to th
e le
arni
ng p
roce
ss. T
he in
tegr
atio
n of
asse
ssm
ent a
nd le
arni
ng in
Ariz
ona
scho
ols
is b
eing
str
ongl
y im
pact
ed b
y th
eA
rizon
a S
tude
nt A
sses
smen
t Pro
gram
and
the
Dis
tric
t Ass
essm
ent P
lan.
Ari
zona
Stu
dent
Ass
essm
ent P
rogr
am (
ASA
P)
Wor
king
as
part
ners
. the
Joi
nt L
egis
lativ
e C
omm
ittee
on
Goa
ls fo
r E
duca
tiona
lE
xcel
lenc
e, th
e st
ate
Boa
rd o
f Edu
catio
n an
d th
e A
rizon
a D
epar
tmen
t of
Edu
catio
n es
tabl
ishe
d go
als
for
impr
ovin
g K
- I 2
stu
dent
ach
ieve
men
t. T
heco
mm
ittee
ado
pted
the
stat
e B
oard
-app
rove
d m
athe
mat
ics
Ess
entia
l Ski
lls a
sth
e hi
gh s
tand
ard
for
achi
evem
ent i
n A
rizon
a. T
he A
rizon
a S
tude
nt A
sses
s-m
ent P
rogr
am w
as th
en d
evel
oped
to a
sses
s A
rizon
a's
stud
ent p
rogr
ess
inm
aste
ring
the
Ess
entia
l Ski
lls. T
he m
athe
mat
ics
asse
ssnl
ents
gav
e de
pth
to th
eE
ssen
tial S
kills
and
a n
ew w
ay o
f loo
king
at a
nd th
inki
ng a
bout
cur
ricul
um.
inst
ruct
ion
and
asse
ssm
ent.
But
the
Ariz
ona
mat
hem
atic
s E
ssen
tial S
kills
wer
ein
terp
rete
d by
man
y di
stric
ts a
s a
list o
f iso
late
d sk
ills
and
wer
e as
sess
ed
98
acco
rdin
gly.
The
refo
re, t
he 1
987
docu
men
t was
ref
orm
atte
d to
illu
stra
te th
atth
e m
athe
mat
ics
Ess
entia
l Ski
lls a
re m
ultif
acet
ed a
nd a
ligne
d w
ith th
e A
SA
Pm
athe
mat
ics
asse
ssm
ents
in b
oth
cont
ent a
nd p
roce
ss.
The
Ariz
ona
Stu
dent
Ass
essm
ent P
rogr
am, a
com
preh
ensi
ve s
tate
wid
e pr
o-gr
am, w
as in
itiat
ed to
rai
se th
e st
anda
rds
for
curr
icul
um, i
nstr
uctio
nan
d
asse
ssm
ent.
It se
ts w
orld
-cla
ss s
tand
ards
in m
athe
mat
ics.
It is
evi
dent
that
the
trad
ition
al s
tand
ardi
zed
test
s al
one,
whi
ch m
easu
re s
tude
nt a
gain
st s
tude
nt o
nis
olat
ed s
kills
onc
e a
year
, will
not
suf
fice
as th
e in
dica
tor
of h
ow p
repa
red
stud
ents
rea
lly a
re fo
r th
e w
orld
that
aw
aits
them
. It i
s im
pera
tive
to e
xam
ine
our
stud
ents
aga
inst
yet
ano
ther
sta
ndar
d, a
sta
ndar
d th
at v
alue
sbo
thco
nten
tan
d pr
oces
s,a
stan
dard
that
can
ref
lect
the
dept
h of
the
stud
ents
' mat
hem
atic
alun
ders
tand
ing,
a s
tand
ard
that
is ta
ught
in th
e cl
assr
oom
eve
ry d
ay a
nd is
docu
men
ted
as s
tude
nt p
erfo
rman
ce.
The
sta
te le
vel o
f AS
AP
con
sist
s of
mat
hem
atic
s as
sess
men
ts w
hich
eva
luat
eth
e pr
ogre
ss o
f Ariz
ona
scho
ols
in p
repa
ring
stud
ents
to r
easo
n, a
pply
mat
h-em
atic
al c
once
pts
and
com
mun
icat
e th
eir
thin
king
on
a sa
mpl
e of
Ess
entia
lS
kills
.
2 )i
The
fol
low
ing
are
exam
ples
of
the
type
s of
ass
essm
ent i
tem
s in
clud
ed in
the
ASA
P m
athe
mat
ics
asse
ssm
ents
at t
he s
tate
leve
l. T
hese
que
stio
ns a
re f
rom
the
eigh
th g
rade
ASA
P sa
mpl
e as
sess
men
t fro
m R
iver
side
Pub
lishi
ng C
ompa
ny.
Thi
s as
sess
men
t tak
es p
lace
in a
n in
dust
rial
art
s cl
assr
oom
. The
pro
blem
sill
ustr
ate
how
fra
ctio
ns a
re m
ultif
acet
ed a
nd a
sses
s th
e de
pth
of th
e st
uden
ts'
unde
rsta
ndin
g. T
he s
tude
nts
mus
t be
able
to d
emon
stra
te a
bas
ic u
nder
stan
ding
of th
e re
latio
nshi
ps b
etw
een
frac
tiona
l siz
es, d
efin
e th
e pr
oble
m a
nd c
ompu
te to
dete
rmin
e th
e re
sults
, illu
stra
te a
vis
ual u
nder
stan
ding
of
the
prob
lem
, and
expl
ain
thei
r th
inki
ng.
In s
our
indu
stri
al tu
ts c
lass
you
are
mak
ing
proj
ects
with
woo
d. Y
ou a
nd th
ree
of y
our
frie
nds
Jan,
Kei
sha
and
lleci
de to
mah
e bi
rdho
uses
for
you
r fi
rst p
rote
ct.
Ese
icis
e A
You
r te
as h
er, M
r R
amir
ei,g
is e
s th
e fo
ur o
t you
one
pie
ce o
f w
ood
b4 in
ches
long
and
ins
esea
t hO
U a
slip
ot p
aper
that
rea
ds.
"Mea
sure
the
hoar
d an
d cu
t oft
1/4
ol
99
Firs
t Jan
mea
sure
s th
e ho
ard
and
cuts
off
1/4
of
it. S
he g
ives
the
rest
of
the
hoar
d to
Kei
sha.
Nes
t. K
eish
a m
easu
res
the
hoar
d an
d C
MS
ofr
114
of it
. She
giv
es th
e re
st o
r th
e ho
ard
to E
arl.
Fina
lly, E
arl m
easu
res
the
hoar
d an
d cu
ts o
ff 1
14 o
f it.
He
give
s th
e re
st o
f it
to y
ou.
You
dis
cos
er th
at y
ou h
ave
muc
h m
ore
woo
d th
an y
ou n
eed.
Kei
sha
and
Ear
l com
plai
n th
at th
ey d
on't
have
enou
gh.
Dra
w a
pic
ture
ol t
he 6
4.in
ch h
oard
on
the
grid
bel
ow a
nd s
how
whe
re th
e st
uden
ts c
ut it
. Sho
w y
our
calc
ulat
ions
.
2.E
xpla
in w
hat w
as c
onfu
sing
abo
ut M
r. k
annt
re,'s
dir
ectio
ns a
nd te
ll ho
w h
e co
uld
has
e m
ade
them
cle
arer
.
2 3
The
fol
low
ing
are
exam
ples
of
stud
ent p
rodu
cts
from
the
eigh
th g
rade
sam
ple
asse
ssm
ent.
Exa
min
e th
em to
see
how
eac
h st
uden
t atte
mpt
ed to
sol
ve th
e pr
oble
ms.
oble
ms.
Wha
t con
clus
ions
can
be
mad
e ab
out t
he s
tude
nts
unde
rsta
ndin
g of
the
prob
lem
s an
d th
e m
athe
mat
ical
con
cept
s ne
eded
to s
olve
them
?
RN
A w
rrm
r..1
757,
11'
1,, .
1 11
1411
.1 1
fik p
h It
heal
porlp
col
d he
hm a
nd Ip
m 7
,ht.
e th
e lo
plen
r. 1
111
It N
hor7
1II,
pl t
rim,
Ked
...th
a.E
arC
.nt
e,
, 2.3
v r,
7ir
,z.3
dtI t
ola
1.3
a.a.
J. l
a 7,
tr J
...71
111
x=
9.75
4-7
-an.
..
20 .13,
Y.5
eare
.
2.F
xpl
ain
ii lii
i liii
t P
oIns
pi,7
alm
ni \
h. R
amire
z's
du e
r.Ip
mr.
and
tell
hom
Pul
a ha
ir m
ade
them
kat
er.
I ct
. the
,17,
7 I l
ira I
p.t M
o. to
hel
p 11
1.1}
, pm
.r a
ns..
rom
plet
e.
LJhL
Lw
as5
corif
iL5i
ngct
hoth
..tM
r. R
einu
'rez%
)..i
d4tr
e.c6
:511
,5w
a.s
464:
dta
inea
sare
andc
rth
the
bcan
l znt
o1.
/Jha
t6i
add
/wvd
/.4/
a.5
.c r
zi. t
he b
oon
infr
fau
rt.h
,5. ,
Or
ihe.
.boa
ndT
rivr
ths
cod
6her
l cuf 2
4
Ihr.
1 .1
pp
Pp
C p
l llie
64.
1117
17 h
oard
MI i
he g
rill h
elm
, and
....
het 7
. the
Inde
rtp.
rtN
1141
.11
1011
1 1.
.111
111
lulls
.
112
43.
.5
2.I x
plai
n 77
hat
was
con
losi
ng a
hm. '
Or.
Ham
ner'x
m h
ons
and
tell
Ihe
001d
lune
111
.11I
V
Mem
cle
arer
.cr
. the
Nm
iew
Che
t It
I 10
1111
10.1
ip h
elp
mak
e sm
. ans
uer
tem
plet
,
*fia
niiii
tAtu
dttr
Ad,
kla9
41.J
f1i2
11h'
il .t
hiud
ztJ1
4Ax4
az44
.A
7)jS
141&
11 e
tiu.A
fici
azed
iattr
apag
i ,pa
chi
Aid
A,_
,2tA
hsh*
.eat
at.&
10.1
4.1t
al_L
ithul
d_.A
gilb
t
f
100
wIL
.C.e
kL/F
L-a
firt
aid.
frA
gm.
AP
1.D
raw
a p
ictu
re o
f the
64.
inch
hoa
rd o
n he
grid
Otto
1 a
nti s
how
st h
ere
the
stud
ent,
cut I
t. S
how
your
cal
cula
tions
.
!
.1-t
s ,c
1
Com
puta
tion
Box
Me,
7;
'171
' ill'
I I
11r
11
4..6
6.:1
:I7
4 3
e1-
2.F
%pl
ain
wha
t was
onf
usin
g ab
out M
r. R
amire
z's
dire
r (s
ous
arid
tell
110,
, he
coul
d ho
se m
ade
them
cle
arer
. l ti
he R
twie
w C
heck
Lis
t bel
ow to
hel
p m
ake
sour
ans
ner.
onot
plet
e.
1..R
amoi
.c1,
AO
sa.
far.
hc.L
Air
Yi4
'Li'
.t4
Y.4
19u.
a1al
tLtin
c._.
..-4r
benn
a.6
Atf
ss.V
1.41
t%%
.S,
4sL
i. 4
_Fic
evip
.4.4
. -4a
2hir
_u_g
iula
riel
_cili
ent-
-1A
hbo_
____
hbo_
Lot
dc:k
hitu
g, I
NIc
iAlt"
s,i,
JIL
i r
eip.
)64.
aei
p.m
orSi
.
Id, c
ut lt
. Sho
w
ould
hos
e m
adt
plel
e.
21)5
The
stu
dent
s' r
espo
nses
are
sco
red
usin
g a
gene
ric
4-po
int r
ubri
c, a
sta
ndar
d se
tc
f cr
iteri
a de
scri
bing
eac
h sc
ore
poin
t. A
hea
dnot
e is
incl
uded
for
eac
h ite
m in
the
asse
ssm
ent,
desc
ribi
ng th
e es
sent
ial c
ompo
nent
s an
d/or
an
exam
ple
of o
nepo
ssib
le w
ay to
sol
ve th
e pr
oble
m. B
oth
of th
ese
item
s ar
e sc
ored
usi
ng th
ege
neri
c ru
bric
.
Obs
erva
tion
1
Acc
epta
ble
resp
onse
s:T
he d
iagr
am b
elow
sho
ws
an a
ccep
tabl
e re
spon
se. T
he w
idth
of
the
boar
d m
ayva
ry. T
he s
cale
of
the
stud
ents
' dia
gram
s m
ay b
e di
ffer
ent.
..,
.
:4.
...4.
_.4.
.;
.t__4
...
;
-;
1--
The
com
puta
tion
box
shou
ld c
onta
in th
e fo
llow
ing
com
puta
tions
(or
equ
iva-
lent
s) o
r co
mpu
tatio
ns w
hich
cor
rect
ly m
atch
the
scal
e th
e st
uden
ts u
sed
in th
edi
agra
m:
1/4
(64)
=16
1/4
(64-
16)=
1/4
(48)
=12
1/4
(48-
12)=
114
(36)
=9
21)
6
Obs
erva
tion
2
Acc
epta
ble
resp
onse
s: A
n ac
cept
able
res
pons
e w
ould
add
ress
the
follo
win
gtw
o pa
rts
of th
e pr
oble
m:
Wha
t was
con
fusi
ng a
bout
the
dire
ctio
ns (
i.e.,
The
stu
dent
s di
d no
tre
aliz
e th
at M
r. R
amir
ez m
eant
1/4
of
the
who
le b
oard
.)?
How
cou
ld M
r. R
amir
ez h
ave
mad
e th
e di
rect
ions
cle
arer
(i.e
., H
eco
uld
have
sug
gest
ed th
at th
e st
uden
ts m
easu
re th
e bo
ard
and
divi
de it
into
fou
rths
bef
ore
doin
g an
y cu
tting
.)? 2
7
Gen
eric
4-p
oint
rub
ric
A 4
res
pons
e re
pres
ents
an
effe
ctiv
e so
lutio
n. I
t sho
ws
com
plet
e un
ders
tand
ing
of th
e pr
oble
m, t
horo
ughl
y ad
dres
ses
all p
oint
s re
leva
nt to
the
solu
tion,
sho
ws
logi
cal r
easo
ning
and
val
id c
oncl
usio
ns, c
omm
unic
ates
eff
ectiv
ely
and
clea
r] y
thro
ugh
wri
ting
and/
or d
iagr
ams,
and
incl
udes
ade
quat
e an
d co
rrec
t com
puta
-tio
ns a
nd/o
r se
t up.
A 3
res
pons
e co
ntai
ns m
inor
fla
ws.
Alth
ough
it s
how
s an
und
erst
andi
ng o
f th
epr
oble
m, c
omm
unic
ates
ade
quat
ely
thro
ugh
wri
ting
and/
or d
iagr
ams,
and
gene
rally
rea
ches
rea
sona
ble
conc
lusi
ons,
it s
how
s m
inor
fla
ws
in r
easo
ning
and/
or c
ompu
tatio
n or
neg
lect
s to
add
ress
som
e as
pect
of
the
prob
lem
.
A 2
res
pons
e sh
ows
gaps
in u
nder
stan
ding
and
/or
exec
utio
n. I
t sho
ws
one
orso
me
com
bina
tion
of th
e fo
llow
ing
flaw
s: a
n in
com
plet
e un
ders
tand
ing
of th
epr
oble
m, f
ailu
re to
add
ress
all
aspe
cts
of th
e pr
oble
m. f
aulty
rea
soni
ng, w
eak
conc
lusi
ons,
unc
lear
com
mun
icat
ion
in w
ritin
g an
d/or
dia
gram
s, o
r a
poor
unde
rsta
ndin
g of
rel
evan
t mat
hem
atic
al p
roce
dure
s or
con
cept
s.
A 1
res
pons
e sh
ows
som
e ef
fort
bey
ond
rest
atin
g th
e pr
oble
m o
r co
pyin
g gi
ven
data
. It s
how
s so
me
com
bina
tion
of th
e fo
llow
ing
flaw
s: li
ttle
unde
rsta
ndin
gof
the
prob
lem
, fai
lure
to a
ddre
ss m
ost a
spec
ts o
f th
e pr
oble
m, m
ajor
fla
ws
inre
ason
ing
that
lead
to in
valid
con
clus
ions
, or
a la
ck o
f un
ders
tand
ing
ofre
leva
nt m
athe
mat
ical
pro
cedu
res
or c
once
pts.
20b
102
Ass
ign
a 0
if th
e re
spon
se s
how
s no
und
erst
andi
ng o
f th
e pr
oble
m o
r if
the
stud
ent f
ails
to r
espo
nd to
the
item
.
Ass
ign
an N
S (N
ot S
cora
ble)
if th
e re
spon
se is
ille
gibl
e or
wri
tten
in a
lang
uage
othe
r th
an E
nglis
h.
Whe
n sc
orin
g st
uden
ts' p
aper
s, y
ou w
ill f
ind
that
som
e ar
e cl
earl
y a
4 an
d so
me
are
clea
rly
a 3,
2 o
r a
1. T
hese
pap
ers
are
easy
to s
core
; but
mor
etim
es th
an n
ot,
ther
e is
not
a c
lear
cut
del
inea
tion.
For
exa
mpl
e, a
pap
er w
ill o
ften
hav
ech
arac
teri
stic
s of
bot
h a
3 pa
per
and
a 2
pape
r. I
n th
is c
ase,
the
deci
sion
for
scor
ing
beco
mes
mor
e co
mpl
icat
ed. T
his
pape
r ne
eds
to b
e ex
amin
ed m
ore
care
fully
. Doe
s th
e pa
per
have
mor
e ch
arac
teri
stic
s of
a 3
pap
er o
r do
es it
hav
em
ore
char
acte
rist
ics
of a
2 p
aper
? Fe
w p
aper
s w
ill h
ave
all
the
char
acte
rist
ics
of a
spe
cifi
c sc
ore
poin
t; bu
t if
the
pape
r ha
s m
ore
char
acte
rist
ics
of o
ne p
oint
than
ano
ther
, tha
t is
the
scor
e it
shou
ld b
e as
sign
ed. E
xam
ine
the
stud
ents
'pr
oduc
ts O
n th
e pr
evio
us p
age.
How
wou
ld y
ou e
valu
ate
and
scor
e ea
ch o
fth
em?
Usi
ng p
erfo
rman
ce-b
ased
ass
essm
ents
, we
can
now
beg
in to
und
erst
and
the
dept
h of
the
stud
ents
' mat
hem
atic
al a
bilit
y an
d th
eir
abili
ty to
app
ly th
ese
conc
epts
in a
pro
blem
-sol
ving
con
text
.
2 0;
)
Dis
tric
t Ass
essm
ent P
lan
(DA
P)
Dis
tric
ts a
re r
equi
red
by la
w to
file
a D
istr
ict A
sses
smen
t Pla
n w
ith th
e A
rizo
naD
epar
tmen
t of
Edu
catio
n. T
his
plan
est
ablis
hes
the
dist
rict
's e
xpec
tatio
ns a
ndth
e st
anda
rds
for
stud
ent a
chie
vem
ent.
The
DA
P w
ill h
ave
a si
gnif
ican
tim
pact
on
mat
hem
atic
s in
stru
ctio
n. D
istr
icts
are
exp
ecte
d to
bui
ld a
sses
smen
tpr
ogra
ms
that
are
alig
ned
to th
e st
anda
rds
set b
y th
e A
rizo
na E
ssen
tial S
kills
for
Mat
hem
atic
s in
bot
h co
nten
t and
pro
cess
to th
e de
pth
dem
onst
rate
d by
the
ASA
P m
athe
mat
ics
asse
ssm
ents
. Dis
tric
ts m
ust d
ecid
e w
hat s
trat
egie
s th
eyw
ill u
se to
det
erm
ine
stud
ents
' pro
gres
s in
mee
ting
the
stan
dard
s.
The
DA
P im
pact
s m
ore
than
just
dis
tric
t ass
essm
ent p
ract
ices
. Rev
isin
g or
crea
ting
asse
ssm
ent i
nstr
umen
ts a
lso
dire
ctly
aff
ects
oth
er c
ompo
nent
s: d
is-
tric
t obj
ectiv
es, i
nstr
uctio
nal m
ater
ials
and
inst
ruct
iona
l str
ateg
ies.
Dis
tric
tO
bjec
tives
Inte
rpre
t Mat
hE
ssen
tial S
kills
as
Mul
tifac
eted
All
four
of
thes
e co
mpo
nent
s ar
e in
tegr
al, a
nd a
n ef
fect
ive
plan
can
not b
ede
velo
ped
with
out a
ddre
ssin
g al
l fou
r si
mul
tane
ousl
y an
d ke
epin
g th
em in
alig
nmen
t. If
stu
dent
s ar
e to
mee
t the
sta
ndar
ds s
et b
y di
stri
cts,
wha
t is
asse
ssed
shou
ld b
e al
igne
d w
ith w
hat i
s ta
ught
; and
inst
ruct
iona
l str
ateg
ies
utili
zed
in th
ecl
assr
oot.
..hou
ld r
einf
orce
wha
t is
valu
ed.
Cur
ricu
lum
Alig
nmen
t Con
tinuu
m
Inst
ruct
iona
lM
ater
ials
Inve
stig
atio
ns.
Proj
ects
, Tas
ks
Inst
ruct
iona
lSt
rate
gies
Act
ive
Part
icip
atio
n,C
oope
rativ
e G
roup
s,E
x !o
ratio
n
Dis
tric
tA
sses
smen
t
Perf
orm
ance
-Bas
edA
sses
smen
t
Mat
hem
atic
sE
ssen
tial S
kills
( M
ES)
Sta
ndar
dSe
t by
the
Stat
e
ASA
PPe
rfor
man
ce-
Bas
edA
sses
smen
t
Mat
h E
ssen
tial S
kills
Inte
rpre
ted
as a
Lis
t
210
Tex
tboo
k
103
Tea
cher
mpa
rtin
gIn
form
atio
n (P
assi
veL
earn
ing)
CR
T'
211
Man
y di
stri
cts
have
alr
eady
alig
ned
all f
our
com
pone
nts.
How
ever
, dis
tric
tsT
his
dist
rict
may
hav
e al
l fou
r co
mpo
nent
salig
ned
but s
its O
n th
e bo
ttom
of
the
mus
t exa
min
e w
here
thei
r pr
esen
t sys
tem
lies
on th
e co
ntin
uum
and
det
erm
ine
cont
inuu
m. T
his
plan
doe
s no
t ass
ess
stud
ents
' pro
gres
sin
mee
ting
the
stan
dard
s
if it
mee
ts th
e st
anda
rds
set b
y th
e m
athe
mat
ics
Ess
entia
l Ski
lls a
nd th
e A
SAP.
set b
y th
e st
ate.
It h
as in
terp
rete
d th
em
athe
mat
ics
Ess
entia
l Ski
lls a
s a
list,
taug
ht
it ac
cord
ingl
y an
d as
sess
ed s
tude
nts'
pro
gres
sin
isol
atio
n.
Dis
tric
tO
bjec
tives
Whe
re d
oes
your
dis
tric
t fit
on th
e co
ntin
uum
? W
illth
e pl
acem
ent e
nabl
e yo
ur
dist
rict
to r
each
the
stan
dard
s se
t by
the
stat
e?
Cur
ricu
l.im
Alig
nmen
t Con
tinuu
m
Inst
ruct
iona
lM
ater
ials
Inst
ruct
iona
lS
trat
egie
sD
istr
ict
Ass
essm
ent
ME
SS
tund
ard
Set
by
Sta
te
AS
AP
Per
form
ance
-B
ased
Ass
essm
ent
1"--
Dis
tric
tO
bjec
tives
104
2j3
As
dist
rict
s m
ake
deci
sion
s ab
out h
ow th
ey w
ill a
sses
sth
eir
stud
ents
, the
y m
ust
be c
ogni
zant
of
how
thei
r as
sess
men
t pra
ctic
es in
terr
elat
ew
ith th
e ot
her
thre
e
com
pone
nts.
If
a di
stri
ct e
lect
s to
use
aco
mbi
natio
n of
CR
Ts
and
port
folio
s, it
rais
es d
istr
ict a
sses
smen
t up
the
cont
inuu
m. T
here
is n
o lo
nger
an
alig
nmen
t
amon
g al
l fou
r co
mpo
nent
s.
Dis
tric
tO
bjec
tives
ME
SS
tand
ard
Set
by
Sta
te
EObD
istr
ict
ject
ives
214
The
use
of
port
folio
s ca
lls f
or th
e co
llect
ion
of s
tude
nt w
ork
invo
lvin
g sh
ort
task
s, p
roje
cts,
exp
lana
tions
and
/or
inve
stig
atio
ns.U
nles
s ch
ange
s ar
e m
ade
to
the
rem
aini
ng th
ree
com
pone
nts,
inac
cura
teco
nclu
sion
s w
ill b
e dr
awn
by th
e
dist
rict
s. W
hat i
s be
ing
asse
ssed
is n
ot b
eing
taug
ht.
Cur
ricu
lum
Alig
nmen
t Con
tinuu
m
Inst
ruct
iona
lM
ater
ials
Inst
ruct
iona
lS
trat
egie
sD
istr
ict
Ass
essm
ent
105
AS
AP
Per
form
ance
-B
ased
Ass
essm
ent
215
In o
rder
for
stud
ents
to m
eet t
he n
ew s
tand
ards
set
by
the
dist
rict,
all t
he fo
urco
mpo
nent
s m
ust b
e re
alig
ned.
Dis
tric
t obj
ectiv
es m
ust b
e re
inte
rpre
ted
for
dept
h, a
dditi
onal
cla
ssro
om m
ater
ials
mus
t he
adde
d to
pro
vide
exp
erie
nce
with
proj
ects
and
inve
stig
atio
ns, a
nd te
ache
rs m
ust r
ecei
ve a
dditi
onal
trai
ning
tofa
cilit
ate
new
type
s of
lear
ning
exp
erie
nces
tbr
the
stud
ents
in th
e cl
assr
oom
.
21i;
Dis
tric
tO
bjec
tives
The
Dis
tric
t Ass
essm
ent P
lan
can
he a
pow
erfu
l ins
trum
ent f
or c
hang
ing
mat
hem
atic
s fr
om a
sta
tic d
isci
plin
e to
a d
ynam
ic p
roce
ss a
nd e
mpo
wer
ing
stud
ents
mat
hem
atic
ally
.
Cur
ricul
um A
lignm
ent C
ontin
uum
inst
ruct
iona
lM
ater
ials
Inst
ruct
iona
lS
trat
egie
sD
istr
ict
Ass
essm
ent
into
mm
omm
o.vi
sman
tos
M E
SS
tand
ard
Set
by
Sta
te
AS
AP
Per
form
ance
-B
ased
Ass
essm
ent
Dis
tric
tO
bjec
tives
Inte
rpre
ted
with
Som
e 1)
epth
Tex
tboo
ks a
ndP
robk
m-S
olvi
ngM
ater
ials
Coo
pera
tive
C r
oups
,D
iscu
ssio
ns a
ndLe
ctur
e
Por
tIblio
s
106
217
Perf
orm
ance
in th
e C
lass
room
Mos
t mat
hem
atic
s ed
ucat
ors
have
a d
ream
of
wha
t stu
dent
sid
eally
sho
uld
beex
peri
enci
ng in
the
clas
sroo
m, e
.g.,
wor
king
in s
mal
l gro
ups
orin
depe
nden
tly;
desi
gnin
g an
d do
ing
inve
stig
atio
ns a
nd p
roje
cts;
usi
ng to
ols
such
as
man
ipul
a-
tive
mat
eria
ls, c
alcu
lato
rs, c
ompu
ters
, ass
orte
d te
xtbo
oks
and
othe
r re
fere
nce
mat
eria
ls; c
onsu
lting
with
eac
h ot
her
and
with
the
teac
her;
and
keep
ing
note
book
s an
d ot
her
wri
tten
repo
rts
of th
eir
prog
ress
, pro
blem
san
d po
tent
ial
solu
tions
. The
ent
ire
clas
s m
ay d
iscu
ss a
par
ticul
ar p
oint
ofv
iew
or
prob
lem
.T
he c
urri
culu
m is
ric
h in
rea
l pro
blem
sol
ving
and
incl
udes
afu
ll ra
nge
ofm
athe
mat
ical
idea
s. M
athe
mat
ical
pow
er is
at w
ork
in e
very
cor
ner
of th
ecl
assr
oom
. Our
vis
ion
of in
tegr
atin
g th
e in
stru
ctio
nal p
roce
ssan
d as
sess
men
t
supp
orts
this
dre
am.
The
dre
am r
equi
res
inte
grat
ing
the
inst
ruct
iona
l pro
cess
and
asse
ssm
ent.
The
ASA
P an
d D
AP
rein
forc
e th
is in
tegr
atio
n an
d ar
e de
sign
ed to
enc
oura
gecl
assr
oom
teac
hers
to a
ddre
ss m
athe
mat
ics
from
apr
oble
m-s
olvi
ng p
ersp
ec-
tive.
Pro
blem
sol
ving
is c
ompl
ex. I
t inv
olve
s th
ere
call
of f
acts
, the
use
of
ava
riet
y of
ski
lls a
nd p
roce
dure
s, th
e ab
ility
to e
valu
ate
one'
s ow
nth
inki
ng a
nd
prog
ress
whi
le s
olvi
ng th
e pr
oble
m,
and
the
abili
ty to
com
mun
icat
e on
e's
thin
king
. Fur
ther
mor
e, s
ucce
ss in
pro
blem
sol
ving
als
ode
pend
s on
the
stu-
dent
s in
tere
st, m
otiv
atio
n, a
nd s
elf-
conf
iden
ce.
The
NC
TM
Cur
ricu
lum
and
Eva
luat
ion
Stan
dard
sha
s an
ent
ire
stan
dard
devo
ted
to m
athe
mat
ical
dis
posi
tion.
Mat
hem
atic
al d
ispo
sitio
n is
the
stud
ent's
ent
ire
appr
oach
to s
olvi
ng a
giv
en ta
sk o
rpr
oble
m. T
he N
CT
M
Stan
dard
s de
fine
s m
athe
mat
ical
dis
posi
tion
as
-con
fide
nce
in u
sing
mat
hem
atic
s-f
lexi
bilit
y in
c.
flor
atio
n-w
illin
gnes
s to
per
seve
re-i
nter
est,
curi
osity
, and
inve
ntiv
enes
s-i
nclin
atio
n to
mon
itor
and
refl
ect O
n on
e's
own
perf
orm
ance
-val
uing
the
appl
icat
ion
of m
athe
mat
ics
-app
reci
atio
n of
the
role
of
mat
hem
atic
s in
cul
ture
218
107
Stud
ents
' mat
hem
atic
al d
ispo
sitio
n th
en r
elat
es to
how
they
pos
ition
them
-se
lves
whe
n fa
ced
with
a p
robl
em in
volv
ing
mat
hem
atic
s.
Inst
ruct
ion
mus
t now
incl
ude
the
deve
lopm
ent a
ndre
fine
men
t of
thin
king
skill
s su
ch a
s un
ders
tand
ing
and
form
ulat
ing
ques
tions
,con
ditio
ns, v
aria
bles
,co
njec
ture
s an
d m
etho
ds o
f an
alyz
ing
data
. Stu
dent
pro
blem
solv
ers
need
to b
e
able
to a
sses
s th
e re
ason
able
ness
of
thei
r an
swer
s.T
his
requ
ires
dev
elop
ing
good
est
imat
ion
tech
niqu
es. T
he in
clus
ion
ofpr
oble
m-s
olvi
ng s
trat
egie
s in
inst
ruct
ion
will
str
engt
hen
stud
ents
' con
fide
nce
and
help
them
deve
lop
posi
-
tive
attit
udes
and
bel
iefs
con
cern
ing
thei
r ab
ilitie
s.St
uden
ts n
eed
to b
e ta
ught
how
to u
se s
peci
fic
mat
hem
atic
al k
now
ledg
e as
wel
l as
whe
n to
use
that
know
ledg
e. I
t is
impo
rtan
t for
stu
dent
s to
dev
elop
the
skill
s th
at e
nabl
e th
em
to s
olve
com
plex
pro
blem
s su
cces
sful
ly.
Ass
essm
ent m
ust p
aral
lel a
nd b
ein
terw
oven
with
that
inst
ruct
ion.
Evi
denc
e or
indi
cato
rsof
stu
dent
per
for-
man
ce in
eac
h ph
ase
of p
robl
emso
lvin
g be
com
e th
e as
sess
men
t. T
echn
ique
sva
ry f
rom
sim
ple
teac
her
obse
rvat
ions
tofo
rmal
eva
luat
ions
of
stud
ent w
ork
usin
g a
scor
ing
rubr
ic.
219
1 th
em-
hink
ing
Lri
able
s,ed
to b
eel
opin
geg
ies
inap
pos
i-e
taug
htus
e th
at)l
e th
eman
d be
perf
or-
:hni
ques
!nt w
ork
The
re a
re s
ever
al te
chni
ques
for
ass
essi
ng th
e cr
itica
l rel
atio
nshi
p be
twee
npe
rfor
man
ce, a
ttitu
des
and
belie
fs r
egar
ding
pro
blem
sol
ving
:
obse
rvin
g an
d qu
estio
ning
stu
dent
s-
eval
uatin
g st
uden
t pro
duct
s-
usin
g st
uden
t sel
f-as
sess
men
ts
Eac
h of
thes
e te
chni
ques
acc
esse
s in
form
atio
n ab
out t
he s
tude
nt th
roug
h a
form
of c
omm
unic
atio
n. T
he k
ey f
acto
r in
any
ass
essm
ent i
s w
hat t
he s
tude
ntco
mm
unic
ates
. Jus
t as
we
reco
gniz
e m
any
diff
eren
t lea
rnin
g st
yles
, the
re a
real
so m
any
diff
eren
t way
s th
at s
tude
nts
com
mun
icat
e w
hat t
hey
do a
nd d
o no
tun
ders
tand
. Thu
s, w
e ha
ve a
sses
smen
t tec
hniq
ues
in w
hich
an
impa
rtia
lob
serv
er (
the
teac
her)
rec
ords
obs
erve
d ac
tions
and
com
men
ts, a
nd w
e us
edi
rect
que
stio
ning
to d
eter
min
e m
easu
red
resp
onse
s. T
he s
tude
nt h
as th
eop
port
unity
to th
ink
thro
ugh
the
proc
esse
s of
pro
blem
sol
ving
in w
ritin
gst
uden
t rep
orts
and
in a
nsw
erin
g m
any
of th
e qu
estio
ns in
per
form
ance
-bas
edas
sess
men
ts. S
ince
eac
h of
thes
e te
chni
ques
add
ress
es d
iffe
rent
are
as, i
t is
logi
cal t
o us
e a
vari
ety
of te
chni
ques
with
eac
h st
uden
t. In
pre
pari
ng s
tude
nts
for
perf
orm
ance
-bas
ed a
sses
smen
t, te
ache
rs n
eed
to s
truc
ture
exp
erie
nces
that
teac
h st
uden
ts to
ana
lyze
thei
r ow
n th
inki
ng p
roce
sses
. Stu
dent
s m
ust b
egin
toun
ders
tand
why
thos
e sk
ills
are
need
ed, r
esha
pe a
ttitu
des
and
deve
lop
habi
tsth
at c
ontin
ue th
e us
e of
thos
e cr
itica
l thi
nkin
g sk
ills.
2Z.t)
og
In th
e w
orld
of
wor
k, p
eopl
e ar
e va
lued
for
the
task
s or
pro
ject
s th
ey d
o w
ell,
thei
r ab
ility
to w
ork
with
oth
ers
and
thei
r re
spon
se to
pro
blem
situ
atio
ns. T
opr
epar
e st
uden
ts f
or f
utur
e su
cces
s, b
oth
curr
icul
um a
nd a
sses
smen
t mus
tpr
omot
e th
is k
ind
of p
erfo
rman
ce.
As
we
mov
e in
to p
erfo
rman
ce-b
ased
ass
essm
ent,
we
mov
e ou
t of
the
trad
i-tio
nal "
grad
ing-
sys
tem
. Stu
dent
wor
k lo
oks
diff
eren
t fro
m s
kill-
orie
nted
assi
gnm
ents
. The
inte
nt is
now
to d
raw
out
stu
dent
thin
king
pro
cess
es. F
orex
ampl
e, th
e fo
llow
ing
pape
rs a
re th
e re
sult
of a
n as
sign
men
t giv
en to
sev
eral
diff
eren
t cla
sses
whe
re s
tude
nts
wer
e as
ked
to c
ount
the
num
ber
of p
eopl
e in
thei
r cl
ass
and
then
det
erm
ine
how
man
y ha
ndsh
akes
ther
e w
ould
be
if e
ach
stud
ent w
ere
to s
hake
han
ds o
nce
with
eve
ry o
ther
stu
dent
.
The
pro
blem
was
pos
ed v
erba
lly b
y th
e te
ache
r: a
nd th
e st
uden
ts w
ere
aske
dto
wri
te th
e qu
estio
n in
thei
r ow
n w
ords
, wor
k ou
t a s
olut
ion,
sho
w th
eir
wor
kan
d ex
plai
n th
e st
rate
gies
they
use
d. E
xam
ine
the
stud
ent p
rodu
cts
on th
efo
llow
ing
page
. Wha
t do
the
resp
onse
s sa
y ab
out t
he s
tude
nts'
mat
hem
atic
aldi
spos
ition
and
thei
r ab
ility
to c
omm
unic
ate
thei
r un
ders
tand
ing? 22
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223
Usi
ng a
n as
sess
men
t por
tfol
io w
ill e
nabl
e te
ache
rs to
ass
ess
stud
ent p
rogr
ess
mor
e ea
sily
. Por
tfol
ios
give
the
teac
hers
the
oppo
rtun
ity to
exa
min
e ho
wst
uden
ts a
re p
erfo
rmin
g in
divi
dual
ly a
nd/o
r in
coo
pera
tive
grou
ps, u
sing
bot
hfo
rmal
and
info
rmal
mea
ns o
f as
sess
men
t. Po
rtfo
lios
allo
w te
ache
rs to
ana
lyze
the
stud
ents
' lev
el o
f un
ders
tand
ing
and
abili
ty to
app
ly m
athe
mat
ical
con
cept
sin
con
text
. Man
y te
ache
rs a
re a
ccus
tom
ed to
mai
ntai
ning
stu
dent
fol
ders
,ho
usin
g th
eir
wor
k fo
r a
spec
ific
per
iod
of ti
me.
The
por
tfol
io d
iffe
rs f
rom
the
fold
er in
that
it b
ecom
es a
per
man
ent r
ecor
d of
exa
mpl
es o
f st
uden
t wor
k th
atsp
ecif
ical
ly d
emon
stra
te s
kill
and
perf
orm
ance
cap
abili
ties.
A s
tude
nt p
ortf
o-lio
will
als
o co
ntai
n w
ritte
n te
ache
r ob
serv
atio
ns a
nd s
tude
nt s
elf-
anal
ysis
. The
port
folio
is a
his
tori
cal r
ecor
d w
hich
doc
umen
ts th
e st
uden
ts' g
row
th, s
kill
deve
lopm
ent a
nd m
athe
mat
ical
dis
posi
tion
as le
arne
rs.
The
foc
us o
f an
ass
essm
ent p
ortf
olio
mus
t be
dete
rmin
ed b
y th
e st
uden
t and
the
teac
her
usin
g di
stri
ct g
uide
lines
. It s
houl
d m
easu
re th
e pr
ogre
ss th
e st
uden
ts a
rem
akin
g in
rea
chin
g th
e st
anda
rds
set b
y th
e di
stri
ct. A
gen
eric
rub
ric
coul
d be
used
to a
sses
s th
e st
uden
ts' p
rogr
ess
agai
nst t
he s
tand
ard.
The
fol
low
ing
exam
ple
lists
item
s w
hich
mig
ht b
e in
clud
ed in
an
asse
ssm
ent p
ortf
olio
. Apo
rtfo
lio n
eeds
to h
ave
a cl
eor
focu
s an
d di
rect
ion
in o
rder
to d
eter
min
e cl
earl
yN
tude
ntm
athe
mat
ical
dis
posi
tion
and
empo
wer
men
t.
224
1 10
(gA
rrlif
r,.f
'N'N
c'O
rcIn
c
r,4
%)`
"lk
(Xt
(1=
CLe
c.-S
rttc
tt`.
..14T
hf
rYC
tVir,
,'c'
MA
PPIN
G A
PO
RT
FOL
IO
.-t,
r;sr
,o,r
1^.
1 \rN
101r
ncr
ri
The
foc
us in
stu
dent
port
folio
s is
on:
skoc
cnk
'-Irr
(\c\
-..7
,4s,
Ckf
rk (
c,,c
. e (
-)ce
__
'.\C
`
_..'1
,.. O
CE
INt
CvI
(LC
_AV
")
rt.'f
r(
c,-I
lect
i On
C't
("\a
ke"c
',, W
OO
V...
)
CA
CC
)'<tlk
%--
Ica.
st..,
.,rY
teit
earn
ee-
crec
klk
st-
Ark a, Vw
2lec
l,
nocx
t.pto
r,
1'4
3c_
cr,r
c ck
t or1
5r\
6kca
kec'
,
V-
cyC
ctix
r cc
cric
r1t
-rS.
.o(
itc'
;
nAA
rvcr
vat
\Are
eS s
-ko&
nt3b
1 k
141)
kc-
.rc
cco\
rNtr
Crr
cc2t
errc
rr, 22
5
A b
lank
por
tfol
io m
ap is
pro
vide
d fo
r yo
ur u
se.
L L
MA
PPIN
G A
PO
RT
FOL
IO*
\ \ ..---
----
----
-
/T
he f
ocus
in s
tude
nt p
ortf
olio
s is
on:
/ /*F
rom
Far
Wes
t Lab
orat
ory
226
-]
\ \22
7
Scor
ing
The
mat
hem
atic
s as
sess
men
ts f
or A
SAP
are
now
usi
ng a
gen
eric
rub
ric
for
scor
ing.
Thi
s ru
bric
set
s th
e st
anda
rds
and
expe
ctat
ions
for
Ari
zona
stu
dent
s.It
is u
sed
to s
core
all
of th
e A
SAP
mat
hem
atic
s as
sess
men
ts. T
he g
ener
ic r
ubri
cw
as a
dopt
ed to
giv
e ed
ucat
ors
mor
e op
port
uniti
es f
or g
ivin
g st
uden
ts c
redi
t for
unus
ual a
nd u
niqu
e re
spon
ses
that
are
mat
hem
atic
ally
sou
nd.
Sinc
e th
is r
ubri
c is
gen
eric
, its
ver
satil
ity g
oes
beyo
nd th
e A
SAP
asse
ssm
ents
.It
set
s a
stan
dard
and
an
expe
ctat
ion
for
stud
ents
' wor
k. I
t can
be
used
to a
sses
sm
athe
mat
ical
task
s, p
roje
cts,
inve
stig
atio
ns, p
ortf
olio
s an
d/or
dis
tric
t obj
ec-
tives
.
Gen
eric
4-P
oint
Rub
ric
A 4
res
pons
e re
pres
ents
an
effe
ctiv
e so
lutio
n. I
t sho
ws
com
plet
e un
ders
tand
ing
of th
e pr
oble
m, t
horo
ughl
y ad
dres
ses
all p
oint
s re
leva
nt to
the
solu
tion,
sho
ws
logi
cal r
easo
ning
and
val
id c
oncl
usio
ns, c
omm
unic
ates
eff
ectiv
ely
and
clea
rly
thro
ugh
wri
ting
and/
or d
iagr
ams,
and
incl
udes
ade
quat
e an
d co
rrec
t com
puta
-tio
ns a
nd/o
r se
t up.
2 2
S
112
A 3
res
pons
e co
ntai
ns m
inor
fla
ws.
Alth
ough
it s
how
s an
und
erst
andi
ng o
f th
epr
oble
m. c
omm
unic
ates
ade
quat
ely
thro
ugh
wri
ting
and/
or d
iagr
ams,
and
gene
rally
rea
ches
rea
sona
ble
conc
lusi
ons,
it s
how
s m
inor
fla
ws
in r
easo
ning
and/
or c
ompu
tatio
n or
neg
lect
s to
add
ress
som
e as
pect
of
the
prob
lem
.
A 2
res
pons
e sh
ows
gaps
in u
nder
stan
ding
and
/or
exec
utio
n. I
t sho
ws
one
orso
me
com
bina
tion
of th
e fo
llow
ing
flaw
s: a
n in
com
plet
e un
ders
tand
ing
of th
epr
oble
m, f
ailu
re to
add
ress
all
aspe
cts
of th
e pr
oble
m, f
aulty
rea
soni
ng, w
eak
conc
lusi
ons,
unc
lear
com
mun
icat
ion
in w
ritin
g an
d/or
dia
gram
s, o
r a
poor
unde
rsta
ndin
g of
rel
evan
t mat
hem
atic
al p
roce
dure
s or
con
cept
s.
A I
res
pons
e sh
ows
som
e ef
fort
bey
ond
rest
atin
g th
e pr
oble
m o
r co
pyin
g gi
ven
data
. It s
how
s so
me
com
bina
tion
of th
e fo
llow
ing
flaw
s: li
ttle
unde
rsta
ndin
gof
the
prob
lem
, fai
lure
to a
ddre
ss m
ost a
spec
ts o
f th
e pr
oble
m, m
ajor
fla
ws
inre
ason
ing
that
lead
to in
valid
con
clus
ions
. or
a la
ck o
f un
ders
tand
ing
ofre
leva
nt m
athe
mat
ical
pro
cedu
res
or c
once
pts.
Ass
ign
a 0
if th
e re
spon
se s
how
s no
und
erst
andi
ng o
f th
e pr
oble
m o
r if
the
stud
ent f
ails
to r
espo
nd to
the
item
.
2 4.
2::J
In r
evie
win
g th
is r
ubric
, it b
ecom
es o
bvio
us th
at c
urric
ulum
and
inst
ruct
ion
will
need
to lo
ok d
iffer
ent.
We
mus
t ana
lyze
the
rubr
ic c
aref
ully
and
see
how
itco
mpa
res
to th
e co
mpo
nent
s de
fined
und
er m
athe
mat
ical
pow
er, e
.g.,
mat
h-em
atic
al th
inki
ng, m
athe
mat
ical
und
erst
andi
ng. t
ools
and
tech
niqu
es, a
ndco
mm
unic
atio
n sk
ills.
We
mus
t dev
elop
gui
delin
es to
hel
p ou
r st
uden
ts m
eet t
hene
w s
tand
ards
and
sho
w th
em h
ow th
is tr
ansl
ates
into
stu
dent
per
form
ance
.
Mat
hem
atic
al T
hink
ing:
Stu
dent
s m
ust f
irst u
se th
eir
know
ledg
e an
dun
ders
tand
ing
to a
naly
ze th
e pr
oble
m. T
hey
shou
ld u
se in
duct
ive
and
dedu
ctiv
ere
ason
ing
to e
valu
ate
and
inte
rpre
t the
maj
or is
sues
. mak
e co
njec
ture
s an
dde
velo
p a
stra
tegy
that
will
yie
ld a
n ef
fect
ive
solu
tion
to th
e pr
oble
m.
Mat
hem
atic
al U
nder
stan
ding
:S
tude
nts
mus
t app
ly th
eir
unde
rsta
nd-
ing
of m
athe
mat
ical
con
cept
s an
d pr
oced
ures
in s
olvi
ng th
e pr
oble
m. b
e ab
le to
expl
ain
the
proc
esse
s us
ed a
nd d
eter
min
e th
e si
gnifi
canc
e of
the
data
they
hav
e
obta
ined
. Too
ls a
nd T
echn
ique
s:S
tuck
nts
ofte
n ne
ed to
be
able
to d
emon
stra
teth
eir
mat
hem
atic
al u
nder
stan
ding
in d
iffer
ent w
ays.
ran
ging
from
mod
elin
gco
ncep
tual
und
erst
andi
ng th
roug
h th
e us
e of
han
ds-o
n m
ater
ials
to a
hig
h le
vel
of a
pplic
atio
n us
ing
com
pute
rs a
nd/o
r ca
lcul
ator
s in
bot
h in
stru
ctio
nal a
ndas
sess
men
t set
tings
.
230
Com
mun
icat
ion
Skill
s:C
omm
unic
atin
g m
athe
mat
ical
ly is
a c
ruci
alpa
rt o
f mat
hem
atic
al e
mpo
wer
men
t. A
n ex
plan
atio
n of
wha
t the
res
ults
are
, wha
tth
ey m
ean
and
how
the
stud
ents
kno
w, i
s as
impo
rtan
t as
the
proc
ess
itsel
f.E
ffect
ivel
y co
mm
unic
atin
g th
e so
lutio
n to
a p
robl
em e
nabl
es o
ther
s to
und
er-
stan
d ex
actly
wha
t the
stu
dent
s ha
ve d
one,
the
dept
h of
thei
r un
ders
tand
ing
and
thei
r ab
ility
to m
ake
mat
hem
atic
al c
onne
ctio
ns. R
espo
nses
can
ran
ge fr
omve
rbal
or
writ
ten
expl
anat
ions
; dia
gram
s, ta
bles
or
grap
hs; n
umer
ical
rep
rese
n-ta
tions
of t
he p
robl
em; a
nd/o
r a
com
bina
tion
of a
ny o
f the
abo
ve. T
he a
bilit
y to
com
mun
icat
e m
athe
mat
ical
ly is
a n
eces
sary
and
pow
erfu
l too
l.
Usi
ng th
ese
guid
elin
es, w
e ca
n be
gin
to m
ove
our
stud
ents
clo
ser
to m
athe
mat
i-ca
l em
pow
erm
ent b
y he
lpin
g th
em b
ecom
e th
inki
ng in
divi
dual
s w
ho c
anan
alyz
e, in
terp
ret,
conj
ectu
re. e
xpla
in a
nd d
raw
logi
cal c
oncl
usio
ns.
231
Res
ourc
es
Cal
ifor
nia
Dep
artm
ent o
f E
duca
tion.
(19
91).
A S
ampl
er o
f M
athe
mat
ics
Ass
essm
ent.
Sacr
amen
to: O
ffic
e of
Sta
te P
rint
ing.
Car
negi
e C
omm
issi
on o
n Sc
ienc
e, T
echn
olog
y, a
nd G
over
nmen
t. (S
epte
m-
ber
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2 3
4
116