pacc cc pa math crosswalk hs 2-11-13a...
TRANSCRIPT
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
A
lge
bra
1
CC
.2.1
.HS
.F.1
Ap
ply
an
d e
xte
nd
th
e p
rop
ert
ies
of
exp
on
en
ts t
o
solv
e p
rob
lem
s w
ith
ra
tio
na
l e
xpo
ne
nts
.
N.R
N.
1.
Exp
lain
ho
w t
he
de
fin
itio
n o
f th
e m
ea
nin
g o
f ra
tio
na
l e
xpo
ne
nts
fo
llo
ws
fro
m
ext
en
din
g t
he
pro
pe
rtie
s o
f in
teg
er
exp
on
en
ts t
o t
ho
se v
alu
es,
all
ow
ing
fo
r a
no
tati
on
fo
r
rad
ica
ls i
n t
erm
s o
f ra
tio
na
l e
xpo
ne
nts
. F
or
exa
mp
le,
we
de
fin
e 5
1/3
to
be
th
e c
ub
e r
oo
t o
f
5 b
eca
use
we
wa
nt
(51
/3)3
= 5
(1/3
)3 t
o h
old
, so
(5
1/3
)3 m
ust
eq
ua
l 5
.
2.
Re
wri
te e
xpre
ssio
ns
invo
lvin
g r
ad
ica
ls a
nd
ra
tio
na
l e
xpo
ne
nts
usi
ng
th
e p
rop
ert
ies
of
exp
on
en
ts.
2.1
.A1
.A
Mo
de
l a
nd
co
mp
are
va
lue
s o
f ir
rati
on
al
nu
mb
ers
.
2.2
.A1
.C
Eva
lua
te n
um
eri
cal
exp
ress
ion
s th
at
incl
ud
e t
he
fo
ur
ba
sic
op
era
tio
ns
an
d o
pe
rati
on
s o
f p
ow
ers
an
d r
oo
ts,
reci
pro
cals
,
op
po
site
s, a
nd
ab
solu
te v
alu
es.
2.8
.A1
.B
Eva
lua
te a
nd
sim
pli
fy c
om
ple
x a
lge
bra
ic e
xpre
ssio
ns,
fo
r
exa
mp
le:
su
ms
of
po
lyn
om
ials
, p
rod
uct
s/q
uo
tie
nts
of
exp
on
en
tia
l te
rms
an
d p
rod
uct
of
a b
ino
mia
l ti
me
s a
tri
no
mia
l;
solv
e a
nd
gra
ph
lin
ea
r e
qu
ati
on
s a
nd
in
eq
ua
liti
es.
CC
.2.1
.HS
.F.2
Ap
ply
pro
pe
rtie
s o
f ra
tio
na
l a
nd
irr
ati
on
al
nu
mb
ers
to s
olv
e r
ea
l w
orl
d o
r m
ath
em
ati
cal
pro
ble
ms.
N.R
N.
3.
Exp
lain
wh
y t
he
su
m o
r p
rod
uct
of
two
ra
tio
na
l n
um
be
rs i
s ra
tio
na
l; t
ha
t th
e s
um
of
a
rati
on
al
nu
mb
er
an
d a
n i
rra
tio
na
l n
um
be
r is
irr
ati
on
al;
an
d t
ha
t th
e p
rod
uct
of
a n
on
zero
rati
on
al
nu
mb
er
an
d a
n i
rra
tio
na
l n
um
be
r is
irr
ati
on
al.
2.1
.A1
.A
Mo
de
l a
nd
co
mp
are
va
lue
s o
f ir
rati
on
al
nu
mb
ers
.
2.1
.A1
.B
Use
fa
cto
rin
g t
o c
rea
te e
qu
iva
len
t fo
rms
of
po
lyn
om
ials
.
2.1
.A1
.E
Ap
ply
th
e c
on
cep
ts o
f p
rim
e a
nd
co
mp
osi
te m
on
om
ials
to
de
term
ine
GC
Fs
(Gre
ate
st C
om
mo
n F
act
or)
an
d L
CM
s (
Lea
st
Co
mm
on
Mu
ltip
le)
of
mo
no
mia
ls.
2.1
.A1
.F
Ext
en
d t
he
co
nce
pt
an
d u
se o
f in
ve
rse
op
era
tio
ns
to d
ete
rmin
e
un
kn
ow
n q
ua
nti
tie
s in
lin
ea
r a
nd
po
lyn
om
ial
eq
ua
tio
ns.
2.2
.A1
.C
Eva
lua
te n
um
eri
cal
exp
ress
ion
s th
at
incl
ud
e t
he
fo
ur
ba
sic
op
era
tio
ns
an
d o
pe
rati
on
s o
f p
ow
ers
an
d r
oo
ts,
reci
pro
cals
,
op
po
site
s, a
nd
ab
solu
te v
alu
es.
2.8
.A1
.B
Eva
lua
te a
nd
sim
pli
fy c
om
ple
x a
lge
bra
ic e
xpre
ssio
ns,
fo
r
exa
mp
le:
su
ms
of
po
lyn
om
ials
, p
rod
uct
s/q
uo
tie
nts
of
exp
on
en
tia
l te
rms
an
d p
rod
uct
of
bin
om
ial
tim
es
a t
rin
om
ial;
solv
e a
nd
gra
ph
lin
ea
r e
qu
ati
on
s a
nd
in
eq
ua
liti
es.
CC
.2.1
.HS
.F.3
Ap
ply
qu
an
tita
tive
re
aso
nin
g t
o c
ho
ose
an
d I
nte
rpre
t
un
its
an
d s
cale
s in
fo
rmu
las,
gra
ph
s a
nd
da
ta
dis
pla
ys.
N.Q
.
1.
Use
un
its
as
a w
ay
to
un
de
rsta
nd
pro
ble
ms
an
d t
o g
uid
e t
he
so
luti
on
of
mu
lti-
ste
p
pro
ble
ms;
ch
oo
se a
nd
in
terp
ret
un
its
con
sist
en
tly
in
fo
rmu
las;
ch
oo
se a
nd
in
terp
ret
the
sca
le a
nd
th
e o
rig
in i
n g
rap
hs
an
d d
ata
dis
pla
ys.
2.
De
fin
e a
pp
rop
ria
te q
ua
nti
tie
s fo
r th
e p
urp
ose
of
de
scri
pti
ve
mo
de
lin
g.
3.
Ch
oo
se a
le
ve
l o
f a
ccu
racy
ap
pro
pri
ate
to
lim
ita
tio
ns
on
me
asu
rem
en
t w
he
n r
ep
ort
ing
qu
an
titi
es.
2.1
.A1
.F
Ext
en
d t
he
co
nce
pt
an
d u
se o
f in
ve
rse
op
era
tio
ns
to d
ete
rmin
e
un
kn
ow
n q
ua
nti
tie
s in
lin
ea
r a
nd
po
lyn
om
ial
eq
ua
tio
ns.
2.8
.A1
.E
Use
co
mb
ina
tio
ns
of
sym
bo
ls a
nd
nu
mb
ers
to
cre
ate
exp
ress
ion
s, e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
in t
wo
or
mo
re
va
ria
ble
s, s
yst
em
s o
f e
qu
ati
on
s, a
nd
in
eq
ua
liti
es,
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.1
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.1
.HS
.F.4
Use
un
its
as
a w
ay
to
un
de
rsta
nd
pro
ble
ms
an
d t
o
gu
ide
th
e s
olu
tio
n o
f m
ult
i-st
ep
pro
ble
ms.
N.Q
.
1.
Use
un
its
as
a w
ay
to
un
de
rsta
nd
pro
ble
ms
an
d t
o g
uid
e t
he
so
luti
on
of
mu
lti-
ste
p
pro
ble
ms;
ch
oo
se a
nd
in
terp
ret
un
its
con
sist
en
tly
in
fo
rmu
las;
ch
oo
se a
nd
in
terp
ret
the
sca
le a
nd
th
e o
rig
in i
n g
rap
hs
an
d d
ata
dis
pla
ys.
2.
De
fin
e a
pp
rop
ria
te q
ua
nti
tie
s fo
r th
e p
urp
ose
of
de
scri
pti
ve
mo
de
lin
g.
2.1
.A1
.F
Ext
en
d t
he
co
nce
pt
an
d u
se o
f in
ve
rse
op
era
tio
ns
to d
ete
rmin
e
un
kn
ow
n q
ua
nti
tie
s in
lin
ea
r a
nd
po
lyn
om
ial
eq
ua
tio
ns.
2.8
.A1
.E
Use
co
mb
ina
tio
ns
of
sym
bo
ls a
nd
nu
mb
ers
to
cre
ate
exp
ress
ion
s, e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
in t
wo
or
mo
re
va
ria
ble
s, s
yst
em
s o
f e
qu
ati
on
s, a
nd
in
eq
ua
liti
es,
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
2.8
.A1
.F
Inte
rpre
t th
e r
esu
lts
of
solv
ing
eq
ua
tio
ns,
in
eq
ua
liti
es,
sy
ste
ms
of
eq
ua
tio
ns,
an
d i
ne
qu
ali
tie
s in
th
e c
on
text
of
the
sit
ua
tio
ns
tha
t m
oti
va
ted
th
e m
od
el.
CC
.2.1
.HS
.F.5
Ch
oo
se a
le
ve
l o
f a
ccu
racy
ap
pro
pri
ate
to
lim
ita
tio
ns
on
me
asu
rem
en
t w
he
n r
ep
ort
ing
qu
an
titi
es.
N.Q
.
3.
Ch
oo
se a
le
ve
l o
f a
ccu
racy
ap
pro
pri
ate
to
lim
ita
tio
ns
on
me
asu
rem
en
t w
he
n r
ep
ort
ing
qu
an
titi
es.
2.1
.A1
.F
Ext
en
d t
he
co
nce
pt
an
d u
se o
f in
ve
rse
op
era
tio
ns
to d
ete
rmin
e
un
kn
ow
n q
ua
nti
tie
s in
lin
ea
r a
nd
po
lyn
om
ial
eq
ua
tio
ns.
2.8
.A1
.E
Use
co
mb
ina
tio
ns
of
sym
bo
ls a
nd
nu
mb
ers
to
cre
ate
exp
ress
ion
s, e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
in t
wo
or
mo
re
va
ria
ble
s, s
yst
em
s o
f e
qu
ati
on
s, a
nd
in
eq
ua
liti
es,
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
2.8
.A1
.F
Inte
rpre
t th
e r
esu
lts
of
solv
ing
eq
ua
tio
ns,
in
eq
ua
liti
es,
sy
ste
ms
of
eq
ua
tio
ns,
an
d i
ne
qu
ali
tie
s in
th
e c
on
text
of
the
sit
ua
tio
n
tha
t m
oti
va
ted
th
e m
od
el.
CC
.2.1
.HS
.F.6
Ext
en
d t
he
kn
ow
led
ge
of
ari
thm
eti
c o
pe
rati
on
s a
nd
ap
ply
to
co
mp
lex
nu
mb
ers
.
Inte
nti
on
all
y B
lan
k2
.8.A
1.B
Eva
lua
te a
nd
sim
pli
fy c
om
ple
x a
lge
bra
ic e
xpre
ssio
ns,
fo
r
exa
mp
le:
su
ms
of
po
lyn
om
ials
, p
rod
uct
s/q
uo
tie
nts
of
exp
on
en
tia
l te
rms
an
d p
rod
uct
of
a b
ino
mia
l ti
me
s a
tri
no
mia
l;
solv
e a
nd
gra
ph
lin
ea
r e
qu
ati
on
s a
nd
in
eq
ua
liti
es.
CC
.2.1
.HS
.F.7
Ap
ply
co
nce
pts
of
com
ple
x n
um
be
rs i
n p
oly
no
mia
l
ide
nti
tie
s a
nd
qu
ad
rati
c e
qu
ati
on
s to
so
lve
pro
ble
ms.
Inte
nti
on
all
y B
lan
k2
.8.A
1.B
Eva
lua
te a
nd
sim
pli
fy c
om
ple
x a
lge
bra
ic e
xpre
ssio
ns,
fo
r
exa
mp
le:
su
ms
of
po
lyn
om
ials
, p
rod
uct
s/q
uo
tie
nts
of
exp
on
en
tia
l te
rms
an
d p
rod
uct
of
a b
ino
mia
l ti
me
s a
tri
no
mia
l;
solv
e a
nd
gra
ph
lin
ea
r e
qu
ati
on
s a
nd
in
eq
ua
liti
es.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.2
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.D.1
Inte
rpre
t th
e s
tru
ctu
re o
f e
xpre
ssio
ns
to r
ep
rese
nt
a
qu
an
tity
in
te
rms
of
its
con
text
.
A.S
SE
.
1.
Inte
rpre
t e
xpre
ssio
ns
tha
t re
pre
sen
t a
qu
an
tity
in
te
rms
of
its
con
text
.
a.
Inte
rpre
t p
art
s o
f a
n e
xpre
ssio
n,
such
as
term
s, f
act
ors
, a
nd
coe
ffic
ien
ts.
b.
Inte
rpre
t co
mp
lica
ted
exp
ress
ion
s b
y v
iew
ing
on
e o
r m
ore
of
the
ir
pa
rts
as
a s
ing
le e
nti
ty.
Fo
r e
xam
ple
, in
terp
ret
P(1
+r)
n a
s th
e p
rod
uct
of
P a
nd
a f
act
or
no
t d
ep
en
din
g o
n P
.
2.
Use
th
e s
tru
ctu
re o
f a
n e
xpre
ssio
n t
o i
de
nti
fy w
ay
s to
re
wri
te i
t. F
or
exa
mp
le,
see
x4 –
y4 a
s (x
2)2
– (
y2)2
, th
us
reco
gn
izin
g i
t a
s a
dif
fere
nce
of
squ
are
s th
at
can
be
fa
cto
red
as
(x2
– y
2)(
x2 +
y2).
2.1
.A1
.A
Mo
de
l a
nd
co
mp
are
va
lue
s o
f ir
rati
on
al
nu
mb
ers
.
2.1
.A1
.B
Use
fa
cto
rin
g t
o c
rea
te e
qu
iva
len
t fo
rms
of
po
lyn
om
ials
.
2.1
.A1
.E
Ap
ply
th
e c
on
cep
ts o
f p
rim
e a
nd
co
mp
osi
te m
on
om
ials
to
de
term
ine
GC
Fs
(Gre
ate
st C
om
mo
n F
act
or)
an
d L
CM
s (
Lea
st
Co
mm
on
Mu
ltip
le)
of
mo
no
mia
ls.
2.2
.A1
.C
Eva
lua
te n
um
eri
cal
exp
ress
ion
s th
at
incl
ud
e t
he
fo
ur
ba
sic
op
era
tio
ns
an
d o
pe
rati
on
s o
f p
ow
ers
an
d r
oo
ts,
reci
pro
cals
,
op
po
site
s, a
nd
ab
solu
te v
alu
es.
2.8
.A1
.B
Eva
lua
te a
nd
sim
pli
fy c
om
ple
x a
lge
bra
ic e
xpre
ssio
ns,
fo
r
exa
mp
le:
su
ms
of
po
lyn
om
ials
, p
rod
uct
s/q
uo
tie
nts
of
exp
on
en
tia
l te
rms
an
d p
rod
uct
of
a b
ino
mia
l ti
me
s a
tri
no
mia
l;
solv
e a
nd
gra
ph
lin
ea
r e
qu
ati
on
s a
nd
in
eq
ua
liti
es.
CC
.2.2
.HS
.D.2
Wri
te e
xpre
ssio
ns
in e
qu
iva
len
t fo
rms
to s
olv
e
pro
ble
ms.
A.S
SE
.
3.
Ch
oo
se a
nd
pro
du
ce a
n e
qu
iva
len
t fo
rm o
f a
n e
xpre
ssio
n t
o r
eve
al
an
d e
xpla
in
pro
pe
rtie
s o
f th
e q
ua
nti
ty r
ep
rese
nte
d b
y t
he
exp
ress
ion
.
a.
Fa
cto
r a
qu
ad
rati
c e
xpre
ssio
n t
o r
eve
al
the
ze
ros
of
the
fu
nct
ion
it
de
fin
es.
b.
Co
mp
lete
th
e s
qu
are
in
a q
ua
dra
tic
exp
ress
ion
to
re
ve
al
the
ma
xim
um
or
min
imu
m v
alu
e o
f th
e f
un
ctio
n i
t d
efi
ne
s.
c. U
se t
he
pro
pe
rtie
s o
f e
xpo
ne
nts
to
tra
nsf
orm
exp
ress
ion
s fo
r
exp
on
en
tia
l fu
nct
ion
s. F
or
exa
mp
le t
he
exp
ress
ion
1.1
5t ca
n b
e r
ew
ritt
en
as
(1.1
51/1
2)1
2t ≈
1.0
12
12
t to
re
ve
al
the
ap
pro
xim
ate
eq
uiv
ale
nt
mo
nth
ly i
nte
rest
ra
te i
f th
e a
nn
ua
l ra
te i
s
15
%.
2.1
.A1
.A
Mo
de
l a
nd
co
mp
are
va
lue
s o
f ir
rati
on
al
nu
mb
ers
.
2.1
.A1
.B
Use
fa
cto
rin
g t
o c
rea
te e
qu
iva
len
t fo
rms
of
po
lyn
om
ials
.
2.1
.A1
.E
Ap
ply
th
e c
on
cep
ts o
f p
rim
e a
nd
co
mp
osi
te m
on
om
ials
to
de
term
ine
GC
Fs
(Gre
ate
st C
om
mo
n F
act
or)
an
d L
CM
s (
Lea
st
Co
mm
on
Mu
ltip
le)
of
mo
no
mia
ls.
2.2
.A1
.C
Eva
lua
te n
um
eri
cal
exp
ress
ion
s th
at
incl
ud
e t
he
fo
ur
ba
sic
op
era
tio
ns
an
d o
pe
rati
on
s o
f p
ow
ers
an
d r
oo
ts,
reci
pro
cals
,
op
po
site
s, a
nd
ab
solu
te v
alu
es.
2.8
.A1
.B
Eva
lua
te a
nd
sim
pli
fy a
lge
bra
ic e
xpre
ssio
ns,
fo
r e
xam
ple
: s
um
s
of
po
lyn
om
ials
, p
rod
uct
s/q
uo
tie
nts
of
exp
on
en
tia
l te
rms
an
d
pro
du
ct o
f a
bin
om
ial
tim
es
a t
rin
om
ial;
so
lve
an
d g
rap
h l
ine
ar
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.3
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.D.3
Ext
en
d t
he
kn
ow
led
ge
of
ari
thm
eti
c o
pe
rati
on
s a
nd
ap
ply
to
po
lyn
om
ials
.
A.A
PR
.
1.
Un
de
rsta
nd
th
at
po
lyn
om
ials
fo
rm a
sy
ste
m a
na
log
ou
s to
th
e i
nte
ge
rs,
na
me
ly,
the
y
are
clo
sed
un
de
r th
e o
pe
rati
on
s o
f a
dd
itio
n,
sub
tra
ctio
n,
an
d m
ult
ipli
cati
on
; a
dd
,
sub
tra
ct,
an
d m
ult
iply
po
lyn
om
ials
.
2.1
.A1
.A
Mo
de
l a
nd
co
mp
are
va
lue
s o
f ir
rati
on
al
nu
mb
ers
.
2.1
.A1
.B
Use
fa
cto
rin
g t
o c
rea
te e
qu
iva
len
t fo
rms
of
po
lyn
om
ials
.
2.1
.A1
.E
Ap
ply
th
e c
on
cep
ts o
f p
rim
e a
nd
co
mp
osi
te m
on
om
ials
to
de
term
ine
GC
Fs
(Gre
ate
st C
om
mo
n F
act
or)
an
d L
CM
s (
Lea
st
Co
mm
on
Mu
ltip
le)
of
mo
no
mia
ls.
2.2
.A1
.C
Eva
lua
te n
um
eri
cal
exp
ress
ion
s th
at
incl
ud
e t
he
fo
ur
ba
sic
op
era
tio
ns
an
d o
pe
rati
on
s o
f p
ow
ers
an
d r
oo
ts,
reci
pro
cals
,
op
po
site
s, a
nd
ab
solu
te v
alu
es.
2.8
.A1
.B
Eva
lua
te a
nd
sim
pli
fy c
om
ple
x a
lge
bra
ic e
xpre
ssio
ns,
fo
r
exa
mp
le:
su
ms
of
po
lyn
om
ials
, p
rod
uct
s/q
uo
tie
nts
of
exp
on
en
tia
l te
rms
an
d p
rod
uct
of
a b
ino
mia
l ti
me
s a
tri
no
mia
l;
solv
e a
nd
gra
ph
lin
ea
r e
qu
ati
on
s a
nd
in
eq
ua
liti
es.
CC
.2.2
.HS
.D.4
Un
de
rsta
nd
th
e r
ela
tio
nsh
ip b
etw
ee
n z
ero
s a
nd
fact
ors
of
po
lyn
om
ials
to
ma
ke
ge
ne
rali
zati
on
s a
bo
ut
fun
ctio
ns
an
d t
he
ir g
rap
hs.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.D.5
Use
po
lyn
om
ial
ide
nti
tie
s to
so
lve
pro
ble
ms.
Inte
nti
on
all
y B
lan
k2
.1.A
1.A
Mo
de
l a
nd
co
mp
are
va
lue
s o
f ir
rati
on
al
nu
mb
ers
.
2.1
.A1
.B
Use
fa
cto
rin
g t
o c
rea
te e
qu
iva
len
t fo
rms
of
po
lyn
om
ials
.
2.1
.A1
.E
Ap
ply
th
e c
on
cep
ts o
f p
rim
e a
nd
co
mp
osi
te m
on
om
ials
to
de
term
ine
GC
Fs
(Gre
ate
st C
om
mo
n F
act
or)
an
d L
CM
s (
Lea
st
Co
mm
on
Mu
ltip
le)
of
mo
no
mia
ls.
2.2
.A1
.C
Eva
lua
te n
um
eri
cal
exp
ress
ion
th
at
incl
ud
e t
he
fo
ur
ba
sic
op
era
tio
ns
an
d o
pe
rati
on
s o
f p
ow
ers
an
d r
oo
ts,
reci
pro
cals
,
op
po
site
s, a
nd
ab
solu
te v
alu
es.
2.8
.A1
.B
Eva
lua
te a
nd
sim
pli
fy a
lge
bra
ic e
xpre
ssio
ns,
fo
r e
xam
ple
: s
um
s
of
po
lyn
om
ials
, p
rod
uct
s/q
uo
tie
nts
of
exp
on
en
tia
l te
rms
an
d
pro
du
ct o
f a
bin
om
ial
tim
es
trin
om
ial;
so
lve
an
d g
rap
h l
ine
ar
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.4
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.D.6
Ext
en
d t
he
kn
ow
led
ge
of
rati
on
al
fun
ctio
ns
to
rew
rite
in
eq
uiv
ale
nt
form
s.
Inte
nti
on
all
y B
lan
k2
.1.A
1.A
Mo
de
l a
nd
co
mp
are
va
lue
s o
f ir
rati
on
al
nu
mb
ers
.
2.1
.A1
.B
Use
fa
cto
rin
g t
o c
rea
te e
qu
iva
len
t fo
rms
of
po
lyn
om
ials
.
2.1
.A1
.E
Ap
ply
th
e c
on
cep
ts o
f p
rim
e a
nd
co
mp
osi
te m
on
om
ials
to
de
term
ine
GC
Fs
(Gre
ate
st C
om
mo
n F
act
or)
an
d L
CM
s (
Lea
st
Co
mm
on
Mu
ltip
le)
of
mo
no
mia
ls.
2.2
.A1
.C
Eva
lua
te n
um
eri
cal
exp
ress
ion
s th
at
incl
ud
e t
he
fo
ur
ba
sic
op
era
tio
ns
an
d o
pe
rati
on
s o
f p
ow
ers
an
d r
oo
ts,
reci
pro
cals
,
op
po
site
s, a
nd
ab
solu
te v
alu
es.
2.8
.A1
.B
Eva
lua
te a
nd
sim
pli
fy c
om
ple
x a
lge
bra
ic e
xpre
ssio
ns,
fo
r
exa
mp
le:
su
ms
of
po
lyn
om
ials
, p
rod
uct
/qu
oti
en
ts o
f
exp
on
en
tia
l te
rms
an
d p
rod
uct
of
a b
ino
mia
l ti
me
s a
tri
no
mia
l;
solv
e a
nd
gra
ph
lin
ea
r e
qu
ati
on
s a
nd
in
eq
ua
liti
es.
CC
.2.2
.HS
.D.7
Cre
ate
an
d g
rap
h e
qu
ati
on
s o
r in
eq
ua
liti
es
to
de
scri
be
nu
mb
ers
or
rela
tio
nsh
ips.
A.C
ED
.
1.
Cre
ate
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s in
on
e v
ari
ab
le a
nd
use
th
em
to
so
lve
pro
ble
ms.
Incl
ud
e e
qu
ati
on
s a
risi
ng
fro
m l
ine
ar
an
d q
ua
dra
tic
fun
ctio
ns,
an
d s
imp
le r
ati
on
al
an
d
exp
on
en
tia
l fu
nct
ion
s.
2.
Cre
ate
eq
ua
tio
ns
in t
wo
or
mo
re v
ari
ab
les
to r
ep
rese
nt
rela
tio
nsh
ips
be
twe
en
qu
an
titi
es;
gra
ph
eq
ua
tio
ns
on
co
ord
ina
te a
xes
wit
h l
ab
els
an
d s
cale
s.
3.
Re
pre
sen
t co
nst
rain
ts b
y e
qu
ati
on
s o
r in
eq
ua
liti
es,
an
d b
y s
yst
em
s o
f e
qu
ati
on
s a
nd
/or
ine
qu
ali
tie
s, a
nd
in
terp
ret
solu
tio
ns
as
via
ble
or
no
n-v
iab
le o
pti
on
s in
a m
od
eli
ng
co
nte
xt.
Fo
r e
xam
ple
, re
pre
sen
t in
eq
ua
liti
es
de
scri
bin
g n
utr
itio
na
l a
nd
co
st c
on
stra
ints
on
com
bin
ati
on
s o
f d
iffe
ren
t fo
od
s.
4.
Re
arr
an
ge
fo
rmu
las
to h
igh
lig
ht
a q
ua
nti
ty o
f in
tere
st,
usi
ng
th
e s
am
e r
ea
son
ing
as
in
solv
ing
eq
ua
tio
ns.
Fo
r e
xam
ple
, re
arr
an
ge
Oh
m’s
la
w V
= I
R t
o h
igh
lig
ht
resi
sta
nce
R.
Lin
ea
r, q
ua
dra
tic,
an
d e
xpo
ne
nti
al
(in
teg
er
inp
uts
on
ly);
fo
r A
.CE
D.3
lin
ea
r o
nly
.
2.1
.A1
.E
Ap
ply
th
e c
on
cep
ts o
f p
rim
e a
nd
co
mp
osi
te m
on
om
ials
to
de
term
ine
GC
Fs
(Gre
ate
st C
om
mo
n F
act
or)
an
d L
CM
s (L
ea
st
Co
mm
on
Mu
ltip
le)
of
mo
no
mia
ls.
2.1
.A1
.F
Ext
en
d t
he
co
nce
pt
an
d u
se o
f in
ve
rse
op
era
tio
ns
to d
ete
rmin
e
un
kn
ow
n q
ua
nti
tie
s in
lin
ea
r a
nd
po
lyn
om
ial
eq
ua
tio
ns.
2.8
.A1
.B
Eva
lua
te a
nd
sim
pli
fy c
om
ple
x a
lge
bra
ic e
xpre
ssio
ns,
fo
r
exa
mp
le:
su
ms
of
po
lyn
om
ials
, p
rod
uct
s/q
uo
tie
nts
of
exp
on
en
tia
l te
rms
an
d p
rod
uct
of
a b
ino
mia
l ti
me
s a
tri
no
mia
l;
solv
e a
nd
gra
ph
lin
ea
r e
qu
ati
on
s a
nd
in
eq
ua
liti
es.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.5
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.D.8
Ap
ply
in
ve
rse
op
era
tio
ns
to s
olv
e e
qu
ati
on
s o
r
form
ula
s fo
r a
giv
en
va
ria
ble
.
A.R
EI.
1.
Exp
lain
ea
ch s
tep
in
so
lvin
g a
sim
ple
eq
ua
tio
n a
s fo
llo
win
g f
rom
th
e e
qu
ali
ty o
f
nu
mb
ers
ass
ert
ed
at
the
pre
vio
us
ste
p,
sta
rtin
g f
rom
th
e a
ssu
mp
tio
n t
ha
t th
e o
rig
ina
l
eq
ua
tio
n h
as
a s
olu
tio
n.
Co
nst
ruct
a v
iab
le a
rgu
me
nt
to j
ust
ify
a s
olu
tio
n m
eth
od
.
2.
So
lve
sim
ple
ra
tio
na
l a
nd
ra
dic
al
eq
ua
tio
ns
in o
ne
va
ria
ble
, a
nd
giv
e e
xam
ple
s sh
ow
ing
ho
w e
xtra
ne
ou
s so
luti
on
s m
ay
ari
se.
2.1
.A1
.E
Ap
ply
th
e c
on
cep
ts o
f p
rim
e a
nd
co
mp
osi
te m
on
om
ials
to
de
term
ine
GC
Fs
(Gre
ate
st C
om
mo
n F
act
or)
an
d L
CM
s (L
ea
st
Co
mm
on
Mu
ltip
le)
of
mo
no
mia
ls.
2.8
.A1
.E
Use
co
mb
ina
tio
n o
f sy
mb
ols
an
d n
um
be
rs t
o c
rea
te
exp
ress
ion
s, e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
in t
wo
or
mo
re
va
ria
ble
s, s
yst
em
s o
f e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
2.8
.A1
.F
Inte
rpre
t th
e r
esu
lts
of
solv
ing
eq
ua
tio
ns,
in
eq
ua
liti
es,
sy
ste
ms
of
eq
ua
tio
ns,
an
d i
ne
qu
ali
tie
s in
th
e c
on
text
of
the
sit
ua
tio
n
tha
t m
oti
va
ted
th
e m
od
el.
CC
.2.2
.HS
.D.9
Use
re
aso
nin
g t
o s
olv
e e
qu
ati
on
s a
nd
ju
stif
y t
he
solu
tio
n m
eth
od
.
A.R
EI.
1.
Exp
lain
ea
ch s
tep
in
so
lvin
g a
sim
ple
eq
ua
tio
n a
s fo
llo
win
g f
rom
th
e e
qu
ali
ty o
f
nu
mb
ers
ass
ert
ed
at
the
pre
vio
us
ste
p,
sta
rtin
g f
rom
th
e a
ssu
mp
tio
n t
ha
t th
e o
rig
ina
l
eq
ua
tio
n h
as
a s
olu
tio
n.
Co
nst
ruct
a v
iab
le a
rgu
me
nt
to j
ust
ify
a s
olu
tio
n m
eth
od
.
2.
So
lve
sim
ple
ra
tio
na
l a
nd
ra
dic
al
eq
ua
tio
ns
in o
ne
va
ria
ble
, a
nd
giv
e e
xam
ple
s sh
ow
ing
ho
w e
xtra
ne
ou
s so
luti
on
s m
ay
ari
se.
2.1
.A1
.A
Mo
de
l a
nd
co
mp
are
va
lue
s o
f ir
rati
on
al
nu
mb
ers
.
2.1
.A1
.B
Use
fa
cto
rin
g t
o c
rea
te e
qu
iva
len
t fo
rms
of
po
lyn
om
ials
.
2.1
.A1
.E
Ap
ply
th
e c
on
cep
ts o
f p
rim
e a
nd
co
mp
osi
te m
on
om
ials
to
de
term
ine
GC
Fs
(Gre
ate
st C
om
mo
n F
act
or)
an
d L
CM
s (
Lea
st
Co
mm
on
Mu
ltip
le)
of
mo
no
mia
ls.
2.1
.A1
.F
Ext
en
d t
he
co
nce
pt
an
d u
se o
f in
ve
rse
op
era
tio
ns
to d
ete
rmin
e
un
kn
ow
n q
ua
nti
tie
s in
lin
ea
r a
nd
po
lyn
om
ial
eq
ua
tio
ns.
2.2
.A1
.C
Eva
lua
te n
um
eri
cal
exp
ress
ion
s th
at
incl
ud
e t
he
fo
ur
ba
sic
op
era
tio
ns
an
d o
pe
rati
on
s o
f p
ow
ers
an
d r
oo
ts,
reci
pro
cals
,
op
po
site
s, a
nd
ab
solu
te v
alu
es.
2.8
.A1
.E
Use
co
mb
ina
tio
ns
of
sym
bo
ls a
nd
nu
mb
ers
to
cre
ate
exp
ress
ion
s, e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
in t
wo
or
mo
re
va
ria
ble
s, s
yst
em
s o
f e
qu
ati
on
s, a
nd
in
eq
ua
liti
es,
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
2.8
.A1
.F
Inte
rpre
t th
e r
esu
lts
of
solv
ing
eq
ua
tio
ns,
in
eq
ua
liti
es,
sy
ste
ms
of
eq
ua
tio
ns,
an
d i
ne
qu
ali
tie
s in
th
e c
on
text
of
the
sit
ua
tio
n
tha
t m
oti
va
ted
th
e m
od
el.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.6
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.D.1
0
Re
pre
sen
t, s
olv
e a
nd
in
terp
ret
eq
ua
tio
ns/
ine
qu
ali
tie
s
an
d s
yst
em
s o
f e
qu
ati
on
s/in
eq
ua
liti
es
alg
eb
raic
all
y
an
d g
rap
hic
all
y.
A.R
EI.
3.
So
lve
lin
ea
r e
qu
ati
on
s a
nd
in
eq
ua
liti
es
in o
ne
va
ria
ble
, in
clu
din
g e
qu
ati
on
s w
ith
coe
ffic
ien
ts r
ep
rese
nte
d b
y l
ett
ers
.
4.
So
lve
qu
ad
rati
c e
qu
ati
on
s in
on
e v
ari
ab
le.
a.
Use
th
e m
eth
od
of
com
ple
tin
g t
he
sq
ua
re t
o t
ran
sfo
rm a
ny
qu
ad
rati
c e
qu
ati
on
in
x
into
an
eq
ua
tio
n o
f th
e f
orm
(x
– p
)2 =
q t
ha
t h
as
the
sa
me
so
luti
on
s. D
eri
ve
th
e q
ua
dra
tic
form
ula
fro
m t
his
fo
rm.
b.
So
lve
qu
ad
rati
c e
qu
ati
on
s b
y i
nsp
ect
ion
(e
.g.,
fo
r x2
= 4
9),
ta
kin
g s
qu
are
ro
ots
,
com
ple
tin
g t
he
sq
ua
re,
the
qu
ad
rati
c fo
rmu
la a
nd
fa
cto
rin
g,
as
ap
pro
pri
ate
to
th
e i
nit
ial
form
of
the
eq
ua
tio
n.
Re
cog
niz
e w
he
n t
he
qu
ad
rati
c fo
rmu
la g
ive
s co
mp
lex
solu
tio
ns
an
d
wri
te t
he
m a
s a
± b
i fo
r re
al
nu
mb
ers
a a
nd
b.
5.
Pro
ve
th
at,
giv
en
a s
yst
em
of
two
eq
ua
tio
ns
in t
wo
va
ria
ble
s, r
ep
laci
ng
on
e e
qu
ati
on
by
the
su
m o
f th
at
eq
ua
tio
n a
nd
a m
ult
iple
of
the
oth
er
pro
du
ces
a s
yst
em
wit
h t
he
sa
me
solu
tio
ns.
6.
So
lve
sy
ste
ms
of
lin
ea
r e
qu
ati
on
s e
xact
ly a
nd
ap
pro
xim
ate
ly (
e.g
., w
ith
gra
ph
s),
focu
sin
g o
n p
air
s o
f li
ne
ar
eq
ua
tio
ns
in t
wo
va
ria
ble
s.
7.
So
lve
a s
imp
le s
yst
em
co
nsi
stin
g o
f a
lin
ea
r e
qu
ati
on
an
d a
qu
ad
rati
c e
qu
ati
on
in
tw
o
va
ria
ble
s a
lge
bra
ica
lly
an
d g
rap
hic
all
y.
Fo
r e
xam
ple
, fi
nd
th
e p
oin
ts o
f in
ters
ect
ion
be
twe
en
th
e l
ine
y =
–3
x a
nd
th
e c
ircl
e x
2 +
y2 =
3.
10
. U
nd
ers
tan
d t
ha
t th
e g
rap
h o
f a
n e
qu
ati
on
in
tw
o v
ari
ab
les
is t
he
se
t o
f a
ll i
ts s
olu
tio
ns
plo
tte
d i
n t
he
co
ord
ina
te p
lan
e,
oft
en
fo
rmin
g a
cu
rve
(w
hic
h c
ou
ld b
e a
lin
e).
11
. E
xpla
in w
hy
th
e x
-co
ord
ina
tes
of
the
po
ints
wh
ere
th
e g
rap
hs
of
the
eq
ua
tio
ns
y =
f(x
)
an
d y
= g
(x)
inte
rse
ct a
re t
he
so
luti
on
s o
f th
e e
qu
ati
on
f(x
) =
g(x
); f
ind
th
e s
olu
tio
ns
ap
pro
xim
ate
ly,
e.g
., u
sin
g t
ech
no
log
y t
o g
rap
h t
he
fu
nct
ion
s, m
ak
e t
ab
les
of
va
lue
s, o
r
fin
d s
ucc
ess
ive
ap
pro
xim
ati
on
s. I
ncl
ud
e c
ase
s w
he
re f
(x)
an
d/o
r g
(x)
are
lin
ea
r,
po
lyn
om
ial,
ra
tio
na
l, a
bso
lute
va
lue
, e
xpo
ne
nti
al,
an
d l
og
ari
thm
ic f
un
ctio
ns.
12
. G
rap
h t
he
so
luti
on
s to
a l
ine
ar
ine
qu
ali
ty i
n t
wo
va
ria
ble
s a
s a
ha
lf-p
lan
e (
exc
lud
ing
the
bo
un
da
ry i
n t
he
ca
se o
f a
str
ict
ine
qu
ali
ty),
an
d g
rap
h t
he
so
luti
on
se
t to
a s
yst
em
of
lin
ea
r in
eq
ua
liti
es
in t
wo
va
ria
ble
s a
s th
e i
nte
rse
ctio
n o
f th
e c
orr
esp
on
din
g h
alf
-pla
ne
s.
2.1
.A1
.F
Ext
en
d t
he
co
nce
pt
an
d u
se o
f in
ve
rse
op
era
tio
ns
to d
ete
rmin
e
un
kn
ow
n q
ua
nti
tie
s in
lin
ea
r a
nd
po
lyn
om
ial
eq
ua
tio
ns.
2.8
.A1
.B
Eva
lua
te a
nd
sim
pli
fy a
lge
bra
ic e
xpre
ssio
ns,
fo
r e
xam
ple
s:
sum
s o
f p
oly
no
mia
ls,
pro
du
cts/
qu
oti
en
ts o
f e
xpo
ne
nti
al
term
s
an
d p
rod
uct
of
a b
ino
mia
l ti
me
s a
tri
no
mia
l; s
olv
e a
nd
gra
ph
lin
ea
r e
qu
ati
on
s a
nd
in
eq
ua
liti
es.
2.8
.A1
.E
Use
co
mb
ina
tio
ns
of
sym
bo
ls a
nd
nu
mb
ers
to
cre
ate
exp
ress
ion
s, e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
in t
wo
or
mo
re
va
ria
ble
s, s
yst
em
s o
f e
qu
ati
on
s, a
nd
in
eq
ua
liti
es,
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
2.8
.A1
.F
Inte
rpre
t th
e r
esu
lts
of
solv
ing
eq
ua
tio
ns,
in
eq
ua
liti
es,
sy
ste
ms
of
eq
ua
tio
ns,
an
d i
ne
qu
ali
tie
s in
th
e c
on
text
of
the
sit
ua
tio
n
tha
t m
oti
va
ted
th
e m
od
el.
CC
.2.2
.HS
.C.1
Use
th
e c
on
cep
t a
nd
no
tati
on
of
fun
ctio
ns
to
inte
rpre
t a
nd
ap
ply
th
em
in
te
rms
of
the
ir c
on
text
.
F.I
F.
1.
Un
de
rsta
nd
th
at
a f
un
ctio
n f
rom
on
e s
et
(ca
lle
d t
he
do
ma
in)
to a
no
the
r se
t (c
all
ed
th
e
ran
ge
) a
ssig
ns
to e
ach
ele
me
nt
of
the
do
ma
in e
xact
ly o
ne
ele
me
nt
of
the
ra
ng
e.
If f
is
a
fun
ctio
n a
nd
x i
s a
n e
lem
en
t o
f it
s d
om
ain
, th
en
f(x
) d
en
ote
s th
e o
utp
ut
of
f
corr
esp
on
din
g t
o t
he
in
pu
t x.
Th
e g
rap
h o
f f
is t
he
gra
ph
of
the
eq
ua
tio
n y
= f
(x).
2.
Use
fu
nct
ion
no
tati
on
, e
va
lua
te f
un
ctio
ns
for
inp
uts
in
th
eir
do
ma
ins,
an
d i
nte
rpre
t
sta
tem
en
ts t
ha
t u
se f
un
ctio
n n
ota
tio
n i
n t
erm
s o
f a
co
nte
xt.
3.
Re
cog
niz
e t
ha
t se
qu
en
ces
are
fu
nct
ion
s, s
om
eti
me
s d
efi
ne
d r
ecu
rsiv
ely
, w
ho
se d
om
ain
is a
su
bse
t o
f th
e i
nte
ge
rs.
Fo
r e
xam
ple
, th
e F
ibo
na
cci
seq
ue
nce
is
de
fin
ed
re
curs
ive
ly b
y
f(0
) =
f(1
) =
1,
f(n
+1
) =
f(n
) +
f(n
-1)
for
n ≥
1.
Lea
rn a
s g
en
era
l p
rin
cip
le;
focu
s o
n l
ine
ar
an
d e
xpo
ne
nti
al
an
d o
n a
rith
me
tic
an
d
ge
om
etr
ic s
eq
ue
nce
s.
2.6
.A1
.C
Se
lect
or
calc
ula
te t
he
ap
pro
pri
ate
me
asu
re o
f ce
ntr
al
ten
de
ncy
, ca
lcu
late
an
d a
pp
ly t
he
in
terq
ua
rtil
e r
an
ge
fo
r o
ne
-
va
ria
ble
da
ta,
an
d c
on
stru
ct a
lin
e o
f b
est
fit
an
d c
alc
ula
te i
ts
eq
ua
tio
n f
or
two
va
ria
ble
da
ta.
2.8
.A1
.D
De
mo
nst
rate
an
un
de
rsta
nd
ing
an
d a
pp
ly p
rop
ert
ies
of
fun
ctio
ns
(do
ma
in,
ran
ge
) a
nd
ch
ara
cte
rist
ics
of
lin
ea
r
fun
ctio
ns.
2.9
.A1
.C
Use
te
chn
iqu
es
fro
m c
oo
rdin
ate
ge
om
etr
y t
o e
sta
bli
sh
pro
pe
rtie
s o
f li
ne
s a
nd
2-d
ime
nsi
on
al
sha
pe
s a
nd
so
lid
s.
2.1
1.A
1.B
De
scri
be
ra
tes
of
cha
ng
e a
s m
od
ele
d b
y l
ine
ar
eq
ua
tio
ns.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.7
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.C.2
Gra
ph
an
d a
na
lyze
fu
nct
ion
s a
nd
use
th
eir
pro
pe
rtie
s
to m
ak
e c
on
ne
ctio
ns
be
twe
en
th
e d
iffe
ren
t
rep
rese
nta
tio
ns.
F.I
F.
4.
Fo
r a
fu
nct
ion
th
at
mo
de
ls a
re
lati
on
ship
be
twe
en
tw
o q
ua
nti
tie
s, i
nte
rpre
t k
ey
fea
ture
s o
f g
rap
hs
an
d t
ab
les
in t
erm
s o
f th
e q
ua
nti
tie
s, a
nd
sk
etc
h g
rap
hs
sho
win
g k
ey
fea
ture
s g
ive
n a
ve
rba
l d
esc
rip
tio
n o
f th
e r
ela
tio
nsh
ip.
Ke
y f
ea
ture
s in
clu
de
: in
terc
ep
ts;
inte
rva
ls w
he
re t
he
fun
ctio
n i
s in
cre
asi
ng
, d
ecr
ea
sin
g,
po
siti
ve
, o
r n
eg
ati
ve
; re
lati
ve
ma
xim
um
s a
nd
min
imu
ms;
sy
mm
etr
ies;
en
d b
eh
avio
r; a
nd
pe
rio
dic
ity
.
5.
Re
late
th
e d
om
ain
of
a f
un
ctio
n t
o i
ts g
rap
h a
nd
, w
he
re a
pp
lica
ble
, to
th
e q
ua
nti
tati
ve
rela
tio
nsh
ip i
t d
esc
rib
es.
Fo
r e
xam
ple
, if
th
e f
un
ctio
n h
(n)
giv
es
the
nu
mb
er
of
pe
rso
n-
ho
urs
it
tak
es
to a
sse
mb
le n
en
gin
es
in a
fa
cto
ry,
the
n t
he
po
siti
ve
in
teg
ers
wo
uld
be
an
ap
pro
pri
ate
do
ma
in f
or
the
fu
nct
ion
.
6.
Ca
lcu
late
an
d i
nte
rpre
t th
e a
ve
rag
e r
ate
of
cha
ng
e o
f a
fu
nct
ion
(p
rese
nte
d
sym
bo
lica
lly
or
as
a t
ab
le)
ove
r a
sp
eci
fie
d i
nte
rva
l. E
stim
ate
th
e r
ate
of
cha
ng
e f
rom
a
gra
ph
.
7.
Gra
ph
fu
nct
ion
s e
xpre
sse
d s
ym
bo
lica
lly
an
d s
ho
w k
ey
fe
atu
res
of
the
gra
ph
, b
y h
an
d i
n
sim
ple
ca
ses
an
d u
sin
g t
ech
no
log
y f
or
mo
re c
om
pli
cate
d c
ase
s.
a.
Gra
ph
lin
ea
r a
nd
qu
ad
rati
c fu
nct
ion
s a
nd
sh
ow
in
terc
ep
ts,
ma
xim
a,
an
d m
inim
a.
b.
Gra
ph
sq
ua
re r
oo
t, c
ub
e r
oo
t, a
nd
pie
cew
ise
-de
fin
ed
fu
nct
ion
s,
incl
ud
ing
ste
p f
un
ctio
ns
an
d a
bso
lute
va
lue
fu
nct
ion
s.
e.
Gra
ph
exp
on
en
tia
l a
nd
lo
ga
rith
mic
fu
nct
ion
s, s
ho
win
g i
nte
rce
pts
an
d e
nd
be
ha
vio
r, a
nd
tri
go
no
me
tric
fu
nct
ion
s, s
ho
win
g p
eri
od
, m
idli
ne
, a
nd
am
pli
tud
e.
8.
Wri
te a
fu
nct
ion
de
fin
ed
by
an
exp
ress
ion
in
dif
fere
nt
bu
t e
qu
iva
len
t fo
rms
to r
eve
al
an
d e
xpla
in d
iffe
ren
t p
rop
ert
ies
of
the
fu
nct
ion
.
a.
Use
th
e p
roce
ss o
f fa
cto
rin
g a
nd
co
mp
leti
ng
th
e s
qu
are
in
a
qu
ad
rati
c fu
nct
ion
to
sh
ow
ze
ros,
ext
rem
e v
alu
es,
an
d s
ym
me
try
of
the
gra
ph
, a
nd
inte
rpre
t th
ese
in
te
rms
of
a c
on
text
.
b.
Use
th
e p
rop
ert
ies
of
exp
on
en
ts t
o i
nte
rpre
t e
xpre
ssio
ns
for
exp
on
en
tia
l fu
nct
ion
s. F
or
exa
mp
le,
ide
nti
fy p
erc
en
t ra
te o
f ch
an
ge
in
fu
nct
ion
s su
ch a
s y
= (
1.0
2)t
, y
= (
0.9
7)t
, y
= (
1.0
1)1
2t,
y =
(1
.2)t
/10
, a
nd
cla
ssif
y t
he
m a
s re
pre
sen
tin
g
exp
on
en
tia
l g
row
th o
r d
eca
y
9.
Co
mp
are
pro
pe
rtie
s o
f tw
o f
un
ctio
ns
ea
ch r
ep
rese
nte
d i
n a
dif
fere
nt
wa
y
2.8
.A1
.D
De
mo
nst
rate
an
un
de
rsta
nd
ing
an
d a
pp
ly p
rop
ert
ies
of
fun
ctio
ns
(do
ma
in,
ran
ge
) a
nd
ch
ara
cte
rist
ics
of
lin
ea
r
fun
ctio
ns.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.8
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.C.3
Wri
te f
un
ctio
ns
or
seq
ue
nce
s th
at
mo
de
l
rela
tio
nsh
ips
be
twe
en
tw
o q
ua
nti
tie
s.
F.B
F.
1.
Wri
te a
fu
nct
ion
th
at
de
scri
be
s a
re
lati
on
ship
be
twe
en
tw
o q
ua
nti
tie
s.
a.
De
term
ine
an
exp
lici
t e
xpre
ssio
n,
a r
ecu
rsiv
e p
roce
ss,
or
ste
ps
for
calc
ula
tio
n f
rom
a c
on
text
.
b.
Co
mb
ine
sta
nd
ard
fu
nct
ion
ty
pe
s u
sin
g a
rith
me
tic
op
era
tio
ns.
Fo
r
exa
mp
le,
bu
ild
a f
un
ctio
n t
ha
t m
od
els
th
e t
em
pe
ratu
re o
f a
co
oli
ng
bo
dy
by
ad
din
g a
con
sta
nt
fun
ctio
n t
o a
de
cay
ing
exp
on
en
tia
l, a
nd
re
late
th
ese
fu
nct
ion
s to
th
e m
od
el.
2.
Wri
te a
rith
me
tic
an
d g
eo
me
tric
se
qu
en
ces
bo
th r
ecu
rsiv
ely
an
d w
ith
an
exp
lici
t
form
ula
, u
se t
he
m t
o m
od
el
situ
ati
on
s, a
nd
tra
nsl
ate
be
twe
en
th
e t
wo
fo
rms.
2.6
.A1
.C
Se
lect
or
calc
ula
te t
he
ap
pro
pri
ate
me
asu
re o
f ce
ntr
al
ten
de
ncy
, ca
lcu
late
an
d a
pp
ly t
he
in
terq
ua
rtil
e r
an
ge
fo
r o
ne
-
va
ria
ble
da
ta,
an
d c
on
stru
ct a
lin
e o
f b
est
fit
an
d c
alc
ula
te i
ts
eq
ua
tio
n f
or
two
va
ria
ble
da
ta.
2.8
.A1
.C
Ide
nti
fy a
nd
re
pre
sen
t p
att
ern
s a
lge
bra
ica
lly
an
d/o
r g
rap
hic
all
y.
2.8
.A1
.D
De
mo
nst
rate
an
un
de
rsta
nd
ing
an
d a
pp
ly p
rop
ert
ies
of
fun
ctio
ns
(do
ma
in,
ran
ge
) a
nd
ch
ara
cte
rist
ics
of
lin
ea
r
fun
ctio
ns.
2.8
.A1
.F
Inte
rpre
t th
e r
esu
lts
of
solv
ing
eq
ua
tio
ns,
in
eq
ua
liti
es,
sy
ste
ms
of
eq
ua
tio
ns,
an
d i
ne
qu
ali
tie
s in
th
e c
on
text
of
the
sit
ua
tio
n
tha
t m
oti
va
ted
th
e m
od
el.
2.9
.A1
.C
Use
te
chn
iqu
es
fro
m c
oo
rdin
ate
ge
om
etr
y t
o e
sta
bli
sh
pro
pe
rtie
s o
f li
ne
s a
nd
2-d
ime
nsi
on
al
sha
pe
s a
nd
so
lid
s.
2.1
1.A
1.B
De
scri
be
ra
tes
of
cha
ng
e a
s m
od
ele
d b
y l
ine
ar
eq
ua
tio
ns.
CC
.2.2
.HS
.C.4
Inte
rpre
t th
e e
ffe
cts
tra
nsf
orm
ati
on
s h
ave
on
fun
ctio
ns
an
d f
ind
th
e i
nve
rse
s o
f fu
nct
ion
s.
F.B
F.
3.
Ide
nti
fy t
he
eff
ect
on
th
e g
rap
h o
f re
pla
cin
g f
(x)
by
f(x
) +
k,
k f
(x),
f(k
x),
an
d f
(x +
k)
for
spe
cifi
c va
lue
s o
f k
(b
oth
po
siti
ve
an
d n
eg
ati
ve
); f
ind
th
e v
alu
e o
f k
giv
en
th
e g
rap
hs.
Exp
eri
me
nt
wit
h c
ase
s a
nd
ill
ust
rate
an
exp
lan
ati
on
of
the
eff
ect
s o
n t
he
gra
ph
usi
ng
tech
no
log
y.
Incl
ud
e r
eco
gn
izin
g e
ve
n a
nd
od
d f
un
ctio
ns
fro
m t
he
ir g
rap
hs
an
d a
lge
bra
ic
exp
ress
ion
s fo
r th
em
.
4.
Fin
d i
nve
rse
fu
nct
ion
s.
a.
So
lve
an
eq
ua
tio
n o
f th
e f
orm
f(x
) =
c f
or
a s
imp
le f
un
ctio
n f
tha
t h
as
an
in
ve
rse
an
d w
rite
an
exp
ress
ion
fo
r th
e i
nve
rse
. F
or
exa
mp
le,
f(x)
=2
x3
or
f(x)
= (
x+1
)/(x
–1
) fo
r x
≠ 1
.
2.8
.A1
.D
De
mo
nst
rate
an
un
de
rsta
nd
ing
an
d a
pp
ly p
rop
ert
ies
of
fun
ctio
ns
(do
ma
in,
ran
ge
) a
nd
ch
ara
cte
rist
ics
of
lin
ea
r
fun
ctio
ns.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.9
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.C.5
Co
nst
ruct
an
d c
om
pa
re l
ine
ar,
qu
ad
rati
c a
nd
exp
on
en
tia
l m
od
els
to
so
lve
pro
ble
ms.
F.L
E.
1.
Dis
tin
gu
ish
be
twe
en
sit
ua
tio
ns
tha
t ca
n b
e m
od
ele
d w
ith
lin
ea
r fu
nct
ion
s a
nd
wit
h
exp
on
en
tia
l fu
nct
ion
s.
a.
Pro
ve
th
at
lin
ea
r fu
nct
ion
s g
row
by
eq
ua
l d
iffe
ren
ces
ove
r e
qu
al
inte
rva
ls,
an
d t
ha
t e
xpo
ne
nti
al
fun
ctio
ns
gro
w b
y e
qu
al
fact
ors
ove
r e
qu
al
inte
rva
ls.
b.
Re
cog
niz
e s
itu
ati
on
s in
wh
ich
on
e q
ua
nti
ty c
ha
ng
es
at
a c
on
sta
nt
rate
pe
r u
nit
in
terv
al
rela
tive
to
an
oth
er.
c. R
eco
gn
ize
sit
ua
tio
ns
in w
hic
h a
qu
an
tity
gro
ws
or
de
cay
s b
y a
con
sta
nt
pe
rce
nt
rate
pe
r u
nit
in
terv
al
rela
tive
to
an
oth
er
2.
Co
nst
ruct
lin
ea
r a
nd
exp
on
en
tia
l fu
nct
ion
s, i
ncl
ud
ing
ari
thm
eti
c a
nd
ge
om
etr
ic
seq
ue
nce
s, g
ive
n a
gra
ph
, a
de
scri
pti
on
of
a r
ela
tio
nsh
ip,
or
two
in
pu
t-o
utp
ut
pa
irs
(in
clu
de
re
ad
ing
th
ese
fro
m a
ta
ble
).
3.
Ob
serv
e u
sin
g g
rap
hs
an
d t
ab
les
tha
t a
qu
an
tity
in
cre
asi
ng
exp
on
en
tia
lly
eve
ntu
all
y
exc
ee
ds
a q
ua
nti
ty i
ncr
ea
sin
g l
ine
arl
y,
qu
ad
rati
c a
lly
, o
r (m
ore
ge
ne
rall
y)
as
a p
oly
no
mia
l
fun
ctio
n.
2.8
.A1
.D
De
mo
nst
rate
an
un
de
rsta
nd
ing
an
d a
pp
ly p
rop
ert
ies
of
fun
ctio
ns
(do
ma
in,
ran
ge
) a
nd
ch
ara
cte
rist
ics
of
lin
ea
r
fun
ctio
ns.
2.9
.A1
.C
Use
te
chn
iqu
es
fro
m c
oo
rdin
ate
ge
om
etr
y t
o e
sta
bli
sh
pro
pe
rtie
s o
f li
ne
s a
nd
2-d
ime
nsi
on
al
sha
pe
s a
nd
so
lid
s.
2.1
1.A
1.B
De
scri
be
ra
tes
of
cha
ng
e a
s m
od
ele
d b
y l
ine
ar
eq
ua
tio
ns.
CC
.2.2
.HS
.C.6
Inte
rpre
t fu
nct
ion
s in
te
rms
of
the
sit
ua
tio
n t
he
y
mo
de
l.
F.L
E.
5.
Inte
rpre
t th
e p
ara
me
ters
in
a l
ine
ar
or
exp
on
en
tia
l fu
nct
ion
in
te
rms
of
a c
on
text
.
2.8
.A1
.D
De
mo
nst
rate
an
un
de
rsta
nd
ing
an
d a
pp
ly p
rop
ert
ies
of
fun
ctio
ns
(do
ma
in,
ran
ge
) a
nd
ch
ara
cte
rist
ics
of
lin
ea
r
fun
ctio
ns.
2.9
.A1
.C
Use
te
chn
iqu
es
fro
m c
oo
rdin
ate
ge
om
etr
y t
o e
sta
bli
sh
pro
pe
rtie
s o
f li
ne
s a
nd
2-d
ime
nsi
on
al
sha
pe
s a
nd
so
lid
s.
2.1
1.A
1
De
scri
be
ra
tes
of
cha
ng
e a
s m
od
ele
d b
y l
ine
ar
eq
ua
tio
ns.
CC
.2.2
.HS
.C.7
Ap
ply
ra
dia
n m
ea
sure
of
an
an
gle
an
d t
he
un
it c
ircl
e
to a
na
lyze
th
e t
rig
on
om
etr
ic f
un
ctio
ns.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.C.8
Ch
oo
se t
rig
on
om
etr
ic f
un
ctio
ns
to m
od
el
pe
rio
dic
ph
en
om
en
a a
nd
de
scri
be
th
e p
rop
ert
ies
of
the
gra
ph
s.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.C.9
Pro
ve
th
e P
yth
ag
ore
an
id
en
tity
an
d u
se i
t to
calc
ula
te t
rig
on
om
etr
ic r
ati
os.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.1
Use
ge
om
etr
ic f
igu
res
an
d t
he
ir p
rop
ert
ies
to
rep
rese
nt
tra
nsf
orm
ati
on
s in
th
e p
lan
e.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.2
Ap
ply
rig
id t
ran
sfo
rma
tio
ns
to d
ete
rmin
e a
nd
exp
lain
con
gru
en
ce.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.10
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.3
.HS
.A.3
Ve
rify
an
d a
pp
ly g
eo
me
tric
th
eo
rem
s a
s th
ey
re
late
to g
eo
me
tric
fig
ure
s.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.4
Ap
ply
th
e c
on
cep
t o
f co
ng
rue
nce
to
cre
ate
ge
om
etr
ic
con
stru
ctio
ns.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.5
Cre
ate
ju
stif
ica
tio
ns
ba
sed
on
tra
nsf
orm
ati
on
s to
est
ab
lish
sim
ila
rity
of
pla
ne
fig
ure
s.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.6
Ve
rify
an
d a
pp
ly t
he
ore
ms
invo
lvin
g s
imil
ari
ty a
s th
ey
rela
te t
o p
lan
e f
igu
res.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.7
Ap
ply
tri
go
no
me
tric
ra
tio
s to
so
lve
pro
ble
ms
invo
lvin
g r
igh
t tr
ian
gle
s.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.8
Ap
ply
ge
om
etr
ic t
he
ore
ms
to v
eri
fy p
rop
ert
ies
of
circ
les.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.9
Ext
en
d t
he
co
nce
pt
of
sim
ila
rity
to
de
term
ine
arc
len
gth
s a
nd
are
as
of
sect
ors
of
circ
les.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.1
0
Tra
nsl
ate
be
twe
en
th
e g
eo
me
tric
de
scri
pti
on
an
d t
he
eq
ua
tio
n f
or
a c
on
ic s
ect
ion
.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.1
1
Ap
ply
co
ord
ina
te g
eo
me
try
to
pro
ve
sim
ple
ge
om
etr
ic t
he
ore
ms
alg
eb
raic
all
y.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.1
2
Exp
lain
vo
lum
e f
orm
ula
s a
nd
use
th
em
to
so
lve
pro
ble
ms.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.1
3
An
aly
ze r
ela
tio
nsh
ips
be
twe
en
tw
o-d
ime
nsi
on
al
an
d
thre
e-d
ime
nsi
on
al
ob
ject
s.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.1
4
Ap
ply
ge
om
etr
ic c
on
cep
ts t
o m
od
el
an
d s
olv
e r
ea
l
wo
rld
pro
ble
ms.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.11
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.4
.HS
.B.1
Su
mm
ari
ze,
rep
rese
nt,
an
d i
nte
rpre
t d
ata
on
a s
ing
le
cou
nt
or
me
asu
rem
en
t va
ria
ble
.
S.I
D.
1.
Re
pre
sen
t d
ata
wit
h p
lots
on
th
e r
ea
l n
um
be
r li
ne
(d
ot
plo
ts,
his
tog
ram
s, a
nd
bo
x
plo
ts).
2.
Use
sta
tist
ics
ap
pro
pri
ate
to
th
e s
ha
pe
of
the
da
ta d
istr
ibu
tio
n t
o c
om
pa
re c
en
ter
(me
dia
n,
me
an
) a
nd
sp
rea
d (
inte
rqu
art
ile
ra
ng
e,
sta
nd
ard
de
via
tio
n)
of
two
or
mo
re
dif
fere
nt
da
ta s
ets
.
3.
Inte
rpre
t d
iffe
ren
ces
in s
ha
pe
, ce
nte
r, a
nd
sp
rea
d i
n t
he
co
nte
xt o
f th
e d
ata
se
ts,
acc
ou
nti
ng
fo
r p
oss
ible
eff
ect
s o
f e
xtre
me
da
ta p
oin
ts (
ou
tlie
rs).
2.6
.A1
.E
Ma
ke
pre
dic
tio
ns
ba
sed
on
lin
es
of
be
st f
it o
r d
raw
co
ncl
usi
on
s
on
th
e v
alu
e o
f a
va
ria
ble
in
a p
op
ula
tio
n b
ase
d o
n t
he
re
sult
s o
f
a s
am
ple
.
2.7
.A1
.A
Ca
lcu
late
pro
ba
bil
itie
s fo
r in
de
pe
nd
en
t, d
ep
en
de
nt,
or
com
po
un
d e
ve
nts
.
CC
.2.4
.HS
.B.2
Su
mm
ari
ze,
rep
rese
nt,
an
d i
nte
rpre
t d
ata
on
tw
o
cate
go
rica
l a
nd
qu
an
tita
tive
va
ria
ble
s.
S.I
D.
5.
Su
mm
ari
ze c
ate
go
rica
l d
ata
fo
r tw
o c
ate
go
rie
s in
tw
o-w
ay
fre
qu
en
cy t
ab
les.
In
terp
ret
rela
tive
fre
qu
en
cie
s in
th
e c
on
text
of
the
da
ta (
incl
ud
ing
jo
int,
ma
rgin
al,
an
d c
on
dit
ion
al
rela
tive
fre
qu
en
cie
s).
Re
cog
niz
e p
oss
ible
ass
oci
ati
on
s a
nd
tre
nd
s in
th
e d
ata
.
6.
Re
pre
sen
t d
ata
on
tw
o q
ua
nti
tati
ve
va
ria
ble
s o
n a
sca
tte
r p
lot,
an
d d
esc
rib
e h
ow
th
e
va
ria
ble
s a
re r
ela
ted
.
a.
Fit
a f
un
ctio
n t
o t
he
da
ta;
use
fu
nct
ion
s fi
tte
d t
o d
ata
to
so
lve
pro
ble
ms
in t
he
co
nte
xt o
f th
e d
ata
. U
se g
ive
n f
un
ctio
ns
or
cho
ose
a f
un
ctio
n s
ug
ge
ste
d
by
th
e c
on
text
. E
mp
ha
size
lin
ea
r, q
ua
dra
tic,
an
d e
xpo
ne
nti
al
mo
de
ls.
b.
Info
rma
lly
ass
ess
th
e f
it o
f a
fu
nct
ion
by
plo
ttin
g a
nd
an
aly
zin
g
resi
du
als
.
c. F
it a
lin
ea
r fu
nct
ion
fo
r a
sca
tte
r p
lot
tha
t su
gg
est
s a
lin
ea
r
ass
oci
ati
on
.
2.8
.A1
.D
De
mo
nst
rate
an
un
de
rsta
nd
ing
an
d a
pp
ly p
rop
ert
ies
of
fun
ctio
ns
(do
ma
in,
ran
ge
) a
nd
ch
ara
cte
rist
ics
of
lin
ea
r
fun
ctio
ns.
2.9
.A1
.C
Use
te
chn
iqu
es
fro
m c
oo
rdin
ate
ge
om
etr
y t
o e
sta
bli
sh
pro
pe
rtie
s o
f li
ne
s a
nd
2-d
ime
nsi
on
al
sha
pe
s a
nd
so
lid
s.
2.1
1.A
1
De
scri
be
ra
tes
of
cha
ng
e a
s m
od
ele
d b
y l
ine
ar
eq
ua
tio
ns.
CC
.2.4
.HS
.B.3
An
aly
ze l
ine
ar
mo
de
ls t
o m
ak
e i
nte
rpre
tati
on
s b
ase
d
on
th
e d
ata
.
S.I
D.
7.
Inte
rpre
t th
e s
lop
e (
rate
of
cha
ng
e)
an
d t
he
in
terc
ep
t (c
on
sta
nt
term
) o
f a
lin
ea
r m
od
el
in t
he
co
nte
xt o
f th
e d
ata
.
8.
Co
mp
ute
(u
sin
g t
ech
no
log
y)
an
d i
nte
rpre
t th
e c
orr
ela
tio
n c
oe
ffic
ien
t o
f a
lin
ea
r fi
t.
9.
Dis
tin
gu
ish
be
twe
en
co
rre
lati
on
an
d c
au
sati
on
.
2.6
.A1
.C
Se
lect
or
calc
ula
te t
he
ap
pro
pri
ate
me
asu
re o
f ce
ntr
al
ten
de
ncy
, ca
lcu
late
an
d a
pp
ly t
he
in
terq
ua
rtil
e r
an
ge
fo
r o
ne
-
va
ria
ble
da
ta,
an
d c
on
stru
ct a
lin
e o
f b
est
fit
an
d c
alc
ula
te i
ts
eq
ua
tio
n f
or
two
va
ria
ble
da
ta.
2.7
.A1
.A
Ca
lcu
late
pro
ba
bil
itie
s fo
r in
de
pe
nd
en
t, d
ep
en
de
nt,
or
com
po
un
d e
ve
nts
.
2.9
.A1
.C
Use
te
chn
iqu
es
fro
m c
oo
rdin
ate
ge
om
etr
y t
o e
sta
bli
sh
pro
pe
rtie
s o
f li
ne
s a
nd
2-d
ime
nsi
on
al
sha
pe
s a
nd
so
lid
s.
2.1
1.A
1.B
De
scri
be
ra
tes
of
cha
ng
e a
s m
od
ele
d b
y l
ine
ar
eq
ua
tio
ns.
CC
.2.4
.HS
.B.4
Re
cog
niz
e a
nd
eva
lua
te r
an
do
m p
roce
sse
s
un
de
rly
ing
sta
tist
ica
l e
xpe
rim
en
ts.
Inte
nti
on
all
y B
lan
k2
.7.A
1.A
Ca
lcu
late
pro
ba
bil
itie
s fo
r in
de
pe
nd
en
t, d
ep
en
de
nt,
or
com
po
un
d e
ve
nts
.
CC
.2.4
.HS
.B.5
Ma
ke
in
fere
nce
s a
nd
ju
stif
y c
on
clu
sio
ns
ba
sed
on
sam
ple
su
rve
ys,
exp
eri
me
nts
, a
nd
ob
serv
ati
on
al
stu
die
s.
Inte
nti
on
all
y B
lan
k2
.7.A
1.A
Ca
lcu
late
pro
ba
bil
itie
s fo
r in
de
pe
nd
en
t, d
ep
en
de
nt,
or
com
po
un
d e
ve
nts
.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.12
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.4
.HS
.B.6
Use
th
e c
on
cep
ts o
f in
de
pe
nd
en
ce a
nd
co
nd
itio
na
l
pro
ba
bil
ity
to
in
terp
ret
da
ta.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.4
.HS
.B.7
Ap
ply
th
e r
ule
s o
f p
rob
ab
ilit
y t
o c
om
pu
te
pro
ba
bil
itie
s o
f co
mp
ou
nd
eve
nts
in
a u
nif
orm
pro
ba
bil
ity
mo
de
l.
S.C
P.
6.
Fin
d t
he
co
nd
itio
na
l p
rob
ab
ilit
y o
f A
giv
en
B a
s th
e f
ract
ion
of
B’s
ou
tco
me
s th
at
als
o
be
lon
g t
o A
, a
nd
in
terp
ret
the
an
swe
r in
te
rms
of
the
mo
de
l.
7.
Ap
ply
th
e A
dd
itio
n R
ule
, P
(A o
r B
) =
P(A
) +
P(B
) –
P(A
an
d B
), a
nd
in
terp
ret
the
an
swe
r
in t
erm
s o
f th
e m
od
el.
2.7
.A1
.A
Ca
lcu
late
pro
ba
bil
itie
s fo
r in
de
pe
nd
en
t, d
ep
en
de
nt,
or
com
po
un
d e
ve
nts
.
Alg
eb
ra 2
CC
.2.1
.HS
.F.1
Ap
ply
an
d e
xte
nd
th
e p
rop
ert
ies
of
exp
on
en
ts t
o
solv
e p
rob
lem
s w
ith
ra
tio
na
l e
xpo
ne
nts
.
Inte
nti
on
all
y B
lan
k2
.1.A
2.A
Mo
de
l a
nd
co
mp
are
va
lue
s o
f co
mp
lex
nu
mb
ers
.
2.1
.A2
.D
Use
exp
on
en
tia
l n
ota
tio
n t
o r
ep
rese
nt
an
y r
ati
on
al
nu
mb
er.
2.1
.A2
.F
Un
de
rsta
nd
th
e c
on
cep
ts o
f e
xpo
ne
nti
al
an
d l
og
ari
thm
ic f
orm
s
an
d u
se t
he
in
ve
rse
re
lati
on
ship
s b
etw
ee
n e
xpo
ne
nti
al
an
d
log
ari
thm
ic e
xpre
ssio
n t
o d
ete
rmin
e u
nk
no
wn
qu
an
titi
es
in
eq
ua
tio
ns.
2.2
.A2
.C
Eva
lua
te n
um
eri
cal
exp
ress
ion
s o
f co
mp
lex
nu
mb
ers
th
at
incl
ud
e t
he
fo
ur
ba
sic
op
era
tio
ns
an
d o
pe
rati
on
s o
f p
ow
ers
,
op
po
site
s, c
on
jug
ate
s, a
nd
ab
solu
te v
alu
es.
CC
.2.1
.HS
.F.2
Ap
ply
pro
pe
rtie
s o
f ra
tio
na
l a
nd
irr
ati
on
al
nu
mb
ers
to s
olv
e r
ea
l w
orl
d o
r m
ath
em
ati
cal
pro
ble
ms.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.13
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.1
.HS
.F.3
Ap
ply
qu
an
tita
tive
re
aso
nin
g t
o c
ho
ose
an
d I
nte
rpre
t
un
its
an
d s
cale
s in
fo
rmu
las,
gra
ph
s a
nd
da
ta
dis
pla
ys.
Inte
nti
on
all
y B
lan
k2
.3.A
2.E
De
scri
be
ho
w a
ch
an
ge
in
th
e v
alu
e o
f o
ne
va
ria
ble
in
fo
rmu
las
aff
ect
s th
e v
alu
e o
f th
e m
ea
sure
me
nt.
2.6
A2
.C
Co
nst
ruct
a l
ine
of
be
st f
it a
nd
ca
lcu
late
its
eq
ua
tio
n f
or
lin
ea
r
an
d n
on
lin
ea
r tw
o-v
ari
ab
le d
ata
.
2.6
.A2
.E
Ma
ke
pre
dic
tio
ns
ba
sed
on
lin
es
be
st f
it o
r d
raw
co
ncl
usi
on
s o
n
the
va
lue
of
a v
ari
ab
le i
n a
po
pu
lati
on
ba
sed
on
th
e r
esu
lts
of
a
sam
ple
.
2.7
.A2
.A
Use
pro
ba
bil
ity
to
pre
dic
t th
e l
ike
lih
oo
d o
f a
n o
utc
om
e i
n a
n
exp
eri
me
nt.
2.7
.A2
.C
Co
mp
are
od
ds
an
d p
rob
ab
ilit
y.
2.7
.A2
.E
Use
pro
ba
bil
ity
to
ma
ke
ju
dg
me
nts
ab
ou
t th
e l
ike
lih
oo
d o
f
va
rio
us
ou
tco
me
s.
2.8
.A2
.B
Eva
lua
te a
nd
sim
pli
fy a
lge
bra
ic e
xpre
ssio
ns:
pro
du
cts/
qu
oti
en
ts o
f p
oly
no
mia
ls,
log
ari
thm
ic e
xpre
ssio
ns
an
d
com
ple
x fr
act
ion
s; a
nd
so
lve
an
d g
rap
h,
lin
ea
r, q
ua
dra
tic,
exp
on
en
tia
l a
nd
lo
ga
rith
mic
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s, a
nd
solv
e a
nd
gra
ph
sy
ste
ms
of
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s.
2.8
.A2
.D
De
mo
nst
rate
an
d u
nd
ers
tan
din
g a
nd
ap
ply
pro
pe
rtie
s o
f
fun
ctio
ns
(do
ma
in,
ran
ge
, in
ve
rse
s) a
nd
ch
ara
cte
rist
ics
of
fam
ilie
s o
f fu
nct
ion
s (l
ine
ar,
po
lyn
om
ial,
ra
tio
na
l, e
xpo
ne
nti
al,
log
ari
thm
ic).
2.8
.A2
.E
Us
com
bin
ati
on
s o
f sy
mb
ols
an
d n
um
be
rs t
o c
rea
te
exp
ress
ion
s, e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
in t
wo
or
mo
re
va
ria
ble
s, s
yst
em
s o
f e
qu
ati
on
s a
nd
in
eq
ua
liti
es,
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
2.1
1A
2.A
De
term
ine
an
d i
nte
rpre
t m
axi
mu
m a
nd
min
imu
m v
alu
es
of
a
fun
ctio
n o
ve
r a
sp
eci
fie
d i
nte
rva
l.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.14
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.1
.HS
.F.4
Use
un
its
as
a w
ay
to
un
de
rsta
nd
pro
ble
ms
an
d t
o
gu
ide
th
e s
olu
tio
n o
f m
ult
i-st
ep
pro
ble
ms.
Inte
nti
on
all
y B
lan
k2
.3.A
2.E
De
scri
be
ho
w a
ch
an
ge
in
th
e v
alu
e o
f o
ne
va
ria
ble
in
fo
rmu
las
aff
ect
s th
e v
alu
e o
f th
e m
ea
sure
me
nt.
2.8
.A2
.B
Eva
lua
te a
nd
sim
pli
fy a
lge
bra
ic e
xpre
ssio
ns,
fo
r e
xam
ple
:
pro
du
cts/
qu
oti
en
ts o
f p
oly
no
mia
ls,
log
ari
thm
ic e
xpre
ssio
ns
an
d
com
ple
x fr
act
ion
s; a
nd
so
lve
an
d g
rap
h l
ine
ar,
qu
ad
rati
c,
exp
on
en
tia
l a
nd
lo
ga
rith
mic
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s, a
nd
solv
e a
nd
gra
ph
sy
ste
ms
of
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s.
2.8
.A2
.D
De
mo
nst
rate
an
un
de
rsta
nd
ing
an
d a
pp
ly p
rop
ert
ies
of
fun
ctio
na
l (d
om
ain
, ra
ng
e,
inve
rse
s) a
nd
ch
ara
cte
rist
ics
of
fam
ilie
s o
f fu
nct
ion
s (l
ine
ar,
po
lyn
om
ial,
ra
tio
na
l, e
xpo
ne
nti
al,
log
ari
thm
ic).
2.8
.A2
.E
Use
co
mb
ina
tio
ns
of
sym
bo
ls a
nd
nu
mb
ers
to
cre
ate
exp
ress
ion
s, e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
in t
wo
or
mo
re
va
ria
ble
s, s
yst
em
s o
f e
qu
ati
on
s a
nd
in
eq
ua
liti
es,
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
2.1
1.A
2.A
De
term
ine
an
d i
nte
rpre
t m
axi
mu
m a
nd
min
imu
m v
alu
es
of
a
fun
ctio
n o
ve
r a
sp
eci
fie
d i
nte
rva
l.
CC
.2.1
.HS
.F.5
Ch
oo
se a
le
ve
l o
f a
ccu
racy
ap
pro
pri
ate
to
lim
ita
tio
ns
on
me
asu
rem
en
t w
he
n r
ep
ort
ing
qu
an
titi
es.
Inte
nti
on
all
y B
lan
k2
.6.A
2.C
Co
nst
ruct
a l
ine
of
be
st f
it a
nd
ca
lcu
late
its
eq
ua
tio
n f
or
lin
ea
r
an
d n
on
lin
ea
r tw
o-v
ari
ab
le d
ata
.
2.6
.A2
.E
Ma
ke
pre
dic
tio
ns
ba
sed
on
lin
es
of
be
st f
it o
r d
raw
co
ncl
usi
on
s
on
th
e v
alu
e o
f a
va
ria
ble
in
a p
op
ula
tio
n b
ase
d o
n t
he
re
sult
s o
f
a s
am
ple
.
2.7
.A2
.A
Use
pro
ba
bil
ity
to
pre
dic
t th
e l
ike
lih
oo
d o
f a
n o
utc
om
e i
n a
n
exp
eri
me
nt.
2.7
.A2
.C
Co
mp
are
od
ds
an
d p
rob
ab
ilit
y.
2.7
.A2
.E
Use
pro
ba
bil
ity
to
ma
ke
ju
dg
me
nts
ab
ou
t th
e l
ike
lih
oo
d o
f
va
rio
us
ou
tco
me
s.
CC
.2.1
.HS
.F.6
Ext
en
d t
he
kn
ow
led
ge
of
ari
thm
eti
c o
pe
rati
on
s a
nd
ap
ply
to
co
mp
lex
nu
mb
ers
.
N.C
N.
1.
Kn
ow
th
ere
is
a c
om
ple
x n
um
be
r i
such
th
at
i2 =
–1
, a
nd
eve
ry c
om
ple
x n
um
be
r h
as
the
fo
rm a
+ b
i w
ith
a a
nd
b r
ea
l.
2.
Use
th
e r
ela
tio
n i
2 =
–1
an
d t
he
co
mm
uta
tive
, a
sso
cia
tive
, a
nd
dis
trib
uti
ve
pro
pe
rtie
s
to a
dd
, su
btr
act
, a
nd
mu
ltip
ly c
om
ple
x n
um
be
rs.
2.1
.A2
.A
Mo
de
l a
nd
co
mp
are
va
lue
s o
f co
mp
lex
nu
mb
ers
.
2.2
.A2
.C
Eva
lua
te n
um
eri
cal
exp
ress
ion
s o
f co
mp
lex
nu
mb
ers
th
at
incl
ud
e t
he
fo
ur
ba
sic
op
era
tio
ns
an
d o
pe
rati
on
s o
f p
ow
ers
,
op
po
site
s, c
on
jug
ate
s, a
nd
ab
solu
te v
alu
es.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.15
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.1
.HS
.F.7
Ap
ply
co
nce
pts
of
com
ple
x n
um
be
rs i
n p
oly
no
mia
l
ide
nti
tie
s a
nd
qu
ad
rati
c e
qu
ati
on
s to
so
lve
pro
ble
ms.
N.C
N.
7.
So
lve
qu
ad
rati
c e
qu
ati
on
s w
ith
re
al
coe
ffic
ien
ts t
ha
t h
ave
co
mp
lex
solu
tio
ns.
8.
(+)
Ext
en
d p
oly
no
mia
l id
en
titi
es
to t
he
co
mp
lex
nu
mb
ers
. F
or
exa
mp
le,
rew
rite
x2 +
4
as
(x +
2i)
(x –
2i)
.
9.
(+)
Kn
ow
th
e F
un
da
me
nta
l T
he
ore
m o
f A
lge
bra
; sh
ow
th
at
it i
s tr
ue
fo
r q
ua
dra
tic
po
lyn
om
ials
.
Inte
nti
on
all
y B
lan
k
CC
.2.2
.HS
.D.1
Inte
rpre
t th
e s
tru
ctu
re o
f e
xpre
ssio
ns
to r
ep
rese
nt
a
qu
an
tity
in
te
rms
of
its
con
text
.
A.S
SE
.
1.
Inte
rpre
t e
xpre
ssio
ns
tha
t re
pre
sen
t a
qu
an
tity
in
te
rms
of
its
con
text
.
a.
Inte
rpre
t p
art
s o
f a
n e
xpre
ssio
n,
such
as
term
s, f
act
ors
, a
nd
coe
ffic
ien
ts.
b.
Inte
rpre
t co
mp
lica
ted
exp
ress
ion
s b
y v
iew
ing
on
e o
r m
ore
of
the
ir
pa
rts
as
a s
ing
le e
nti
ty.
Fo
r e
xam
ple
, in
terp
ret
P(1
+r)
n a
s th
e p
rod
uct
of
P a
nd
a f
act
or
no
t d
ep
en
din
g o
n P
.
2.
Use
th
e s
tru
ctu
re o
f a
n e
xpre
ssio
n t
o i
de
nti
fy w
ay
s to
re
wri
te i
t. F
or
exa
mp
le,
see
x4
–
y4
as
(x2
)2 –
(y
2)2
, th
us
reco
gn
izin
g i
t a
s a
dif
fere
nce
of
squ
are
s th
at
can
be
fa
cto
red
as
(x2
– y
2)(
x2 +
y2
).
2.1
.A2
.B
Use
fa
cto
rin
g t
o c
rea
te e
qu
iva
len
t fo
rms
of
po
lyn
om
ials
.
2.1
.A2
.D
Use
exp
on
en
tia
l n
ota
tio
n t
o r
ep
rese
nt
an
y r
ati
on
al
nu
mb
er.
2.1
.A2
.F
Un
de
rsta
nd
th
e c
on
cep
ts o
f e
xpo
ne
nti
al
an
d l
og
ari
thm
ic f
orm
s
an
d u
se t
he
in
ve
rse
re
lati
on
ship
s b
etw
ee
n e
xpo
ne
nti
al
an
d
log
ari
thm
ic e
xpre
ssio
n t
o d
ete
rmin
e u
nk
no
wn
qu
an
titi
es
in
eq
ua
tio
ns.
2.2
.A2
.C
Eva
lua
te n
um
eri
cal
exp
ress
ion
s o
f co
mp
lex
nu
mb
ers
th
at
incl
ud
e t
he
fo
ur
ba
sic
op
era
tio
ns
an
d o
pe
rati
on
s o
f p
ow
ers
,
op
po
site
s, c
on
jug
ate
s, a
nd
ab
solu
te v
alu
es.
CC
.2.2
.HS
.D.2
Wri
te e
xpre
ssio
ns
in e
qu
iva
len
t fo
rms
to s
olv
e
pro
ble
ms.
A.S
SE
.
4.
De
rive
th
e f
orm
ula
fo
r th
e s
um
of
a f
init
e g
eo
me
tric
se
rie
s (w
he
n t
he
com
mo
n r
ati
o i
s n
ot
1),
an
d u
se t
he
fo
rmu
la t
o s
olv
e p
rob
lem
s. F
or
exa
mp
le,
calc
ula
te
mo
rtg
ag
e p
ay
me
nts
.
2.1
.A2
.B
Use
fa
cto
rin
g t
o c
rea
te e
qu
iva
len
t fo
rms
of
po
lyn
om
ials
.
2.1
.A2
.D
Use
exp
on
en
tia
l n
ota
tio
n t
o r
ep
rese
nt
an
y r
ati
on
al
nu
mb
er.
2.1
.A2
.F
Un
de
rsta
nd
th
e c
on
cep
ts o
f e
xpo
ne
nti
al
an
d l
og
ari
thm
ic f
orm
s
an
d u
se t
he
in
ve
rse
re
lati
on
ship
s b
etw
ee
n e
xpo
ne
nti
al,
an
d
log
ari
thm
ic e
xpre
ssio
n t
o d
ete
rmin
e u
nk
no
wn
qu
an
titi
es
in
eq
ua
tio
ns.
2.2
.A2
.C
Eva
lua
te n
um
eri
cal
exp
ress
ion
s o
f co
mp
lex
nu
mb
ers
th
at
incl
ud
e f
ou
r b
asi
c o
pe
rati
on
s a
nd
op
era
tio
ns
of
po
we
rs,
op
po
site
s, c
on
jug
ate
s, a
nd
ab
solu
te v
alu
es.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.16
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.D.3
Ext
en
d t
he
kn
ow
led
ge
of
ari
thm
eti
c o
pe
rati
on
s a
nd
ap
ply
to
po
lyn
om
ials
.
A.A
PR
.
1.
Un
de
rsta
nd
th
at
po
lyn
om
ials
fo
rm a
sy
ste
m a
na
log
ou
s to
th
e i
nte
ge
rs,
na
me
ly,
the
y
are
clo
sed
un
de
r th
e o
pe
rati
on
s o
f a
dd
itio
n,
sub
tra
ctio
n,
an
d m
ult
ipli
cati
on
; a
dd
,
sub
tra
ct,
an
d m
ult
iply
po
lyn
om
ials
.
2.1
.A2
.B
Use
fa
cto
rin
g t
o c
rea
te e
qu
iva
len
t fo
rms
of
po
lyn
om
ials
.
2.1
.A2
.D
Use
exp
on
en
tia
l n
ota
tio
n t
o r
ep
rese
nt
an
y r
ati
on
al
nu
mb
er.
2.1
.A2
.F
Un
de
rsta
nd
th
e c
on
cep
ts o
f e
xpo
ne
nti
al
an
d l
og
ari
thm
ic f
orm
s
an
d u
se t
he
in
ve
rse
re
lati
on
ship
s b
etw
ee
n e
xpo
ne
nti
al
an
d
log
ari
thm
ic e
xpre
ssio
n t
o d
ete
rmin
e u
nk
no
wn
qu
an
titi
es
in
eq
ua
tio
ns.
2.2
.A2
.C
Eva
lua
te n
um
eri
cal
exp
ress
ion
s o
f co
mp
lex
nu
mb
ers
th
at
incl
ud
e t
he
fo
ur
ba
sic
op
era
tio
ns
an
d o
pe
rati
on
s o
f p
ow
ers
,
op
po
site
s, c
on
jug
ate
s, a
nd
ab
solu
te v
alu
es.
CC
.2.2
.HS
.D.4
Un
de
rsta
nd
th
e r
ela
tio
nsh
ip b
etw
ee
n z
ero
s a
nd
fact
ors
of
po
lyn
om
ials
to
ma
ke
ge
ne
rali
zati
on
s a
bo
ut
fun
ctio
ns
an
d t
he
ir g
rap
hs.
A.A
PR
.
2.
Kn
ow
an
d a
pp
ly t
he
Re
ma
ind
er
Th
eo
rem
: F
or
a p
oly
no
mia
l p
(x)
an
d a
nu
mb
er
a,
the
rem
ain
de
r o
n d
ivis
ion
by
x –
a i
s p
(a),
so
p(a
) =
0 i
f a
nd
on
ly i
f (x
– a
) is
a f
act
or
of
p(x
).
3.
Ide
nti
fy z
ero
s o
f p
oly
no
mia
ls w
he
n s
uit
ab
le f
act
ori
zati
on
s a
re a
va
ila
ble
, a
nd
use
th
e
zero
s to
co
nst
ruct
a r
ou
gh
gra
ph
of
the
fu
nct
ion
de
fin
ed
by
th
e p
oly
no
mia
l.
2.1
.A2
.B
Use
fa
cto
rin
g t
o c
rea
te e
qu
iva
len
t fo
rms
of
po
lyn
om
ials
.
2.1
.A2
.D
Use
exp
on
en
tia
l n
ota
tio
n t
o r
ep
rese
nt
an
y r
ati
on
al
nu
mb
er.
2.1
.A2
.F
Un
de
rsta
nd
th
e c
on
cep
ts o
f e
xpo
ne
nti
al
an
d l
og
ari
thm
ic f
orm
s
an
d u
se t
he
in
ve
rse
re
lati
on
ship
s b
etw
ee
n e
xpo
ne
nti
al
an
d
log
ari
thm
ic e
xpre
ssio
n t
o d
ete
rmin
e u
nk
no
wn
qu
an
titi
es
in
eq
ua
tio
ns.
2.2
.A2
.C
Eva
lua
te n
um
eri
cal
exp
ress
ion
s o
f co
mp
lex
nu
mb
ers
th
at
incl
ud
e t
he
fo
ur
ba
sic
op
era
tio
ns
an
d o
pe
rati
on
s o
f p
ow
ers
,
op
po
site
s, c
on
jug
ate
s, a
nd
ab
solu
te v
alu
es.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.17
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.D.5
Use
po
lyn
om
ial
ide
nti
tie
s to
so
lve
pro
ble
ms.
A.A
PR
.
4.
Pro
ve
po
lyn
om
ial
ide
nti
tie
s a
nd
use
th
em
to
de
scri
be
nu
me
rica
l re
lati
on
ship
s. F
or
exa
mp
le,
the
po
lyn
om
ial
ide
nti
ty (
x2 +
y2
)2 =
(x2
– y
2)2
+ (
2xy
)2 c
an
be
use
d t
o g
en
era
te
Py
tha
go
rea
n t
rip
les.
5.
(+)
Kn
ow
an
d a
pp
ly t
he
Bin
om
ial
Th
eo
rem
fo
r th
e e
xpa
nsi
on
of
(x +
y)n
in p
ow
ers
of
x
an
d y
fo
r a
po
siti
ve
in
teg
er
n,
wh
ere
x a
nd
y a
re a
ny
nu
mb
ers
, w
ith
co
eff
icie
nts
de
term
ine
d f
or
exa
mp
le b
y P
asc
al’
s T
ria
ng
le.1
2.1
.A2
.B
Use
fa
cto
rin
g t
o c
rea
te e
qu
iva
len
t fo
rms
of
po
lyn
om
ials
.
2.1
.A2
.D
Use
exp
on
en
tia
l n
ota
tio
n t
o r
ep
rese
nt
an
y r
ati
on
al
nu
mb
er.
2.1
.A2
.F
Un
de
rsta
nd
th
e c
on
cep
ts o
f e
xpo
ne
nti
al
an
d l
og
ari
thm
ic f
orm
s
an
d u
se t
he
in
ve
rse
re
lati
on
ship
s b
etw
ee
n e
xpo
ne
nti
al
an
d
log
ari
thm
ic e
xpre
ssio
n t
o d
ete
rmin
e u
nk
no
wn
qu
an
titi
es
in
eq
ua
tio
ns.
2.2
.A2
.C
Eva
lua
te n
um
eri
cal
exp
ress
ion
s o
f co
mp
lex
nu
mb
ers
th
at
incl
ud
e t
he
fo
ur
ba
sic
op
era
tio
ns
an
d o
pe
rati
on
s o
f p
ow
ers
,
op
po
site
s, c
on
jug
ate
s, a
nd
ab
solu
te v
alu
es.
2.3
.A2
.C
So
lve
a f
orm
ula
fo
r a
giv
en
va
ria
ble
usi
ng
alg
eb
raic
pro
cess
es.
2.8
.A2
.B
Eva
lua
te a
nd
sim
pli
fy a
lge
bra
ic e
xpre
ssio
ns,
fo
r e
xam
ple
:
pro
du
cts/
qu
oti
en
ts o
f p
oly
no
mia
ls,
log
ari
thm
ic e
xpre
ssio
ns
an
d
com
ple
x fr
act
ion
s; a
nd
so
lve
an
d g
rap
h l
ine
ar,
qu
ad
rati
c,
exp
on
en
tia
l a
nd
lo
ga
rith
mic
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s, a
nd
solv
e a
nd
gra
ph
sy
ste
ms
of
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s.
2.8
.A2
.D
De
mo
nst
rate
an
un
de
rsta
nd
ing
an
d a
pp
ly p
rop
ert
ies
of
fun
ctio
ns
(do
ma
in,
ran
ge
, in
ve
rse
s) a
nd
ch
ara
cte
rist
ics
of
fam
ilie
s o
f fu
nct
ion
s (l
ine
ar,
po
lyn
om
ial,
ra
tio
na
l, e
xpo
ne
nti
al,
log
ari
thm
ic).
2.8
.A2
.E
Use
co
mb
ina
tio
ns
of
sym
bo
ls a
nd
nu
mb
ers
to
cre
ate
exp
ress
ion
s, e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
in t
wo
or
mo
re
va
ria
ble
s, s
yst
em
s o
f e
qu
ati
on
s a
nd
in
eq
ua
liti
es,
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
2.8
.A2
.F
Inte
rpre
t th
e r
esu
lts
of
solv
ing
eq
ua
tio
ns,
in
eq
ua
liti
es,
sy
ste
ms
of
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s in
th
e c
on
text
of
the
sit
ua
tio
n t
ha
t
mo
tiva
ted
th
e m
od
el.
2.1
1.A
2.B
An
aly
ze a
nd
in
terp
ret
rate
s g
row
th/d
eca
y
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.18
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.D.6
Ext
en
d t
he
kn
ow
led
ge
of
rati
on
al
fun
ctio
ns
to
rew
rite
in
eq
uiv
ale
nt
form
s.
A.A
PR
.
6.
Re
wri
te s
imp
le r
ati
on
al
exp
ress
ion
s in
dif
fere
nt
form
s; w
rite
a(x
)/b
(x)
in t
he
fo
rm q
(x)
+
r(x)
/b(x
), w
he
re a
(x),
b(x
), q
(x),
an
d r
(x)
are
po
lyn
om
ials
wit
h t
he
de
gre
e o
f r(
x) l
ess
th
an
the
de
gre
e o
f b
(x),
usi
ng
in
spe
ctio
n,
lon
g d
ivis
ion
, o
r, f
or
the
mo
re c
om
pli
cate
d e
xam
ple
s,
a c
om
pu
ter
alg
eb
ra s
yst
em
.
7.
(+)
Un
de
rsta
nd
th
at
rati
on
al
exp
ress
ion
s fo
rm a
sy
ste
m a
na
log
ou
s to
th
e r
ati
on
al
nu
mb
ers
, cl
ose
d u
nd
er
ad
dit
ion
, su
btr
act
ion
, m
ult
ipli
cati
on
, a
nd
div
isio
n b
y a
no
nze
ro
rati
on
al
exp
ress
ion
; a
dd
, su
btr
act
, m
ult
iply
, a
nd
div
ide
ra
tio
na
l e
xpre
ssio
ns.
2.1
.A2
.F
Un
de
rsta
nd
th
e c
on
cep
ts o
f e
xpo
ne
nti
al
an
d l
og
ari
thm
ic f
orm
s
an
d u
se t
he
in
ve
rse
re
lati
on
ship
s b
etw
ee
n e
xpo
ne
nti
al
an
d
log
ari
thm
ic e
xpre
ssio
n t
o d
ete
rmin
e u
nk
no
wn
qu
an
titi
es
in
eq
ua
tio
ns.
2.3
.A2
.C
So
lve
a f
orm
ula
fo
r a
giv
en
va
ria
ble
usi
ng
alg
eb
raic
pro
cess
es.
2.3
.A2
.E
De
scri
be
ho
w a
ch
an
ge
in
th
e v
alu
e o
f o
ne
va
ria
ble
in
fo
rmu
las
aff
ect
s th
e v
alu
e o
f th
e m
ea
sure
me
nt.
2.8
.A2
.B
Eva
lua
te a
nd
sim
pli
fy a
lge
bra
ic e
xpre
ssio
ns,
fo
r e
xam
ple
:
pro
du
cts/
qu
oti
en
ts o
f p
oly
no
mia
ls,
log
ari
thm
ic e
xpre
ssio
ns
an
d
com
ple
x fr
act
ion
s; a
nd
so
lve
an
d g
rap
h l
ine
ar,
qu
ad
rati
c,
exp
on
en
tia
l a
nd
lo
ga
rith
mic
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s, a
nd
solv
e a
nd
gra
ph
sy
ste
ms
of
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s.
2.8
.A2
.E
Use
co
mb
ina
tio
ns
of
sym
bo
ls a
nd
nu
mb
ers
to
cre
ate
exp
ress
ion
s, e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
in t
wo
or
mo
re
va
ria
ble
s, s
yst
em
s o
f e
qu
ati
on
s a
nd
in
eq
ua
liti
es,
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
2.8
.A2
.F
Inte
rpre
t th
e r
esu
lts
of
solv
ing
eq
ua
tio
ns,
in
eq
ua
liti
es,
sy
ste
ms
of
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s in
th
e c
on
text
of
the
sit
ua
tio
n t
ha
t
mo
tiva
ted
th
e m
od
el.
2.1
1.A
2.B
An
aly
ze a
nd
in
terp
ret
rate
s g
row
th/d
eca
y.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.19
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.D.7
Cre
ate
an
d g
rap
h e
qu
ati
on
s o
r in
eq
ua
liti
es
to
de
scri
be
nu
mb
ers
or
rela
tio
nsh
ips.
A.C
ED
.
1.
Cre
ate
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s in
on
e v
ari
ab
le a
nd
use
th
em
to
so
lve
pro
ble
ms.
Incl
ud
e e
qu
ati
on
s a
risi
ng
fro
m l
ine
ar
an
d q
ua
dra
tic
fun
ctio
ns,
an
d s
imp
le r
ati
on
al
an
d
exp
on
en
tia
l fu
nct
ion
s.
2.
Cre
ate
eq
ua
tio
ns
in t
wo
or
mo
re v
ari
ab
les
to r
ep
rese
nt
rela
tio
nsh
ips
be
twe
en
qu
an
titi
es;
gra
ph
eq
ua
tio
ns
on
co
ord
ina
te a
xes
wit
h l
ab
els
an
d s
cale
s.
3.
Re
pre
sen
t co
nst
rain
ts b
y e
qu
ati
on
s o
r in
eq
ua
liti
es,
an
d b
y s
yst
em
s o
f e
qu
ati
on
s a
nd
/or
ine
qu
ali
tie
s, a
nd
in
terp
ret
solu
tio
ns
as
via
ble
or
no
n-v
iab
le o
pti
on
s in
a m
od
eli
ng
co
nte
xt.
Fo
r e
xam
ple
, re
pre
sen
t in
eq
ua
liti
es
de
scri
bin
g n
utr
itio
na
l a
nd
co
st c
on
stra
ints
on
com
bin
ati
on
s o
f d
iffe
ren
t fo
od
s.
4.
Re
arr
an
ge
fo
rmu
las
to h
igh
lig
ht
a q
ua
nti
ty o
f in
tere
st,
usi
ng
th
e s
am
e r
ea
son
ing
as
in
solv
ing
eq
ua
tio
ns.
Fo
r e
xam
ple
, re
arr
an
ge
Oh
m’s
la
w V
= I
R t
o h
igh
lig
ht
resi
sta
nce
R.
2.1
.A2
.F
Un
de
rsta
nd
th
e c
on
cep
ts o
f e
xpo
ne
nti
al
an
d l
og
ari
thm
ic f
orm
s
an
d u
se t
he
in
ve
rse
re
lati
on
ship
s b
etw
ee
n e
xpo
ne
nti
al
an
d
log
ari
thm
ic e
xpre
ssio
n t
o d
ete
rmin
e u
nk
no
wn
qu
an
titi
es
in
eq
ua
tio
ns.
2.3
.A2
.C
So
lve
a f
orm
ula
fo
r a
giv
en
va
ria
ble
usi
ng
alg
eb
raic
pro
cess
es.
2.3
.A2
.E
De
scri
be
ho
w a
ch
an
ge
in
th
e v
alu
e o
f o
ne
va
ria
ble
in
fo
rmu
las
aff
ect
s th
e v
alu
e o
f th
e m
ea
sure
me
nt.
2.8
.A2
.B
Eva
lua
te a
nd
sim
pli
fy a
lge
bra
ic e
xpre
ssio
ns,
fo
r e
xam
ple
:
pro
du
cts/
qu
oti
en
ts o
f p
oly
no
mia
ls,
log
ari
thm
ic e
xpre
ssio
ns
an
d
com
ple
x fr
act
ion
s; a
nd
so
lve
an
d g
rap
h l
ine
ar,
qu
ad
rati
c,
exp
on
en
tia
l a
nd
lo
ga
rith
mic
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s, s
olv
e
an
d g
rap
h s
yst
em
s o
f e
qu
ati
on
s a
nd
in
eq
ua
liti
es.
2.8
.A2
.D
De
mo
nst
rate
an
un
de
rsta
nd
ing
an
d a
pp
ly p
rop
ert
ies
of
fun
ctio
ns
(do
ma
in,
ran
ge
, in
ve
rse
s) a
nd
ch
ara
cte
rist
ics
of
fam
ilie
s o
f fu
nct
ion
s (l
ine
ar,
po
lyn
om
ial,
ra
tio
na
l, e
xpo
ne
nti
al,
log
ari
thm
ic).
2.8
.A2
.E
Use
co
mb
ina
tio
ns
of
sym
bo
ls a
nd
nu
mb
ers
to
cre
ate
exp
ress
ion
s, e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
in t
wo
or
mo
re
va
ria
ble
s, s
yst
em
s o
f e
qu
ati
on
s a
nd
in
eq
ua
liti
es,
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
2.8
.A2
.F
Inte
rpre
t th
e e
qu
ati
on
s, i
ne
qu
ali
tie
s, s
yst
em
s o
f e
qu
ati
on
s a
nd
ine
qu
ali
tie
s in
th
e c
on
text
of
the
th
at
situ
ati
on
th
at
mo
tiva
ted
the
mo
de
l.
2.1
1.A
2.A
De
term
ine
an
d i
nte
rpre
t m
axi
mu
m a
nd
min
imu
m v
alu
es
of
a
fun
ctio
n o
ve
r a
sp
eci
fie
d i
nte
rva
l.
2.1
1.A
2.B
An
aly
ze a
nd
in
terp
ret
rate
s g
row
th/d
eca
y.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.20
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.D.8
Ap
ply
in
ve
rse
op
era
tio
ns
to s
olv
e e
qu
ati
on
s o
r
form
ula
s fo
r a
giv
en
va
ria
ble
.
A.R
EI.
2.
So
lve
sim
ple
ra
tio
na
l a
nd
ra
dic
al
eq
ua
tio
ns
in o
ne
va
ria
ble
, a
nd
giv
e e
xam
ple
s sh
ow
ing
ho
w e
xtra
ne
ou
s so
luti
on
s m
ay
ari
se.
2.1
.A2
.F
Un
de
rsta
nd
th
e c
on
cep
ts o
f e
xpo
ne
nti
al
an
d l
og
ari
thm
ic f
orm
s
an
d u
se t
he
in
ve
rse
re
lati
on
ship
s b
etw
ee
n e
xpo
ne
nti
al
an
d
log
ari
thm
ic e
xpre
ssio
n t
o d
ete
rmin
e u
nk
no
wn
qu
an
titi
es
in
eq
ua
tio
ns.
2.3
.A2
.C
So
lve
a f
orm
ula
fo
r a
giv
en
va
ria
ble
usi
ng
alg
eb
raic
pro
cess
es.
2.3
.A2
.E
De
scri
be
ho
w a
ch
an
ge
in
th
e v
alu
e o
f o
ne
va
ria
ble
in
fo
rmu
las
aff
ect
s th
e v
alu
e o
f th
e m
ea
sure
me
nt.
2.8
.A2
.B
Eva
lua
te a
nd
sim
pli
fy a
lge
bra
ic e
xpre
ssio
ns,
fo
r e
xam
ple
:
pro
du
cts/
qu
oti
en
ts o
f p
oly
no
mia
ls,
log
ari
thm
ic e
xpre
ssio
ns
an
d
com
ple
x fr
act
ion
s; a
nd
so
lve
an
d g
rap
h l
ine
ar,
qu
ad
rati
c,
exp
on
en
tia
l a
nd
lo
ga
rith
mic
eq
ua
tio
ns,
an
d i
ne
qu
ali
tie
s, a
nd
solv
e a
nd
gra
ph
sy
ste
ms
of
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s.
2.8
.A2
.E
Use
co
mb
ina
tio
ns
of
sym
bo
ls a
nd
nu
mb
ers
to
cre
ate
exp
ress
ion
s, e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
in t
wo
or
mo
re
va
ria
ble
s, s
yst
em
s o
f e
qu
ati
on
s a
nd
in
eq
ua
liti
es,
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
2.8
.A2
.F
Inte
rpre
t th
e r
esu
lts
of
solv
ing
eq
ua
tio
ns,
in
eq
ua
liti
es,
sy
ste
ms
of
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s in
th
e c
on
text
of
the
sit
ua
tio
n t
ha
t
mo
tiva
ted
th
e m
od
el.
2.1
1.A
2.B
An
aly
ze a
nd
in
terp
ret
rate
s g
row
th/d
eca
y.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.21
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.D.9
Use
re
aso
nin
g t
o s
olv
e e
qu
ati
on
s a
nd
ju
stif
y t
he
solu
tio
n m
eth
od
.
A.R
EI.
2.
So
lve
sim
ple
ra
tio
na
l a
nd
ra
dic
al
eq
ua
tio
ns
in o
ne
va
ria
ble
, a
nd
giv
e e
xam
ple
s sh
ow
ing
ho
w e
xtra
ne
ou
s so
luti
on
s m
ay
ari
se.
2.1
.A2
.F
Un
de
rsta
nd
th
e c
on
cep
ts o
f e
xpo
ne
nti
al
an
d l
og
ari
thm
ic f
orm
s
an
d u
se t
he
in
ve
rse
re
lati
on
ship
s b
etw
ee
n e
xpo
ne
nti
al
an
d
log
ari
thm
ic e
xpre
ssio
n t
o d
ete
rmin
e u
nk
no
wn
qu
an
titi
es
in
eq
ua
tio
ns.
2.3
.A2
.C
So
lve
a f
orm
ula
fo
r a
giv
en
va
ria
ble
usi
ng
alg
eb
raic
pro
cess
es.
2.3
.A2
.E
De
scri
be
ho
w a
ch
an
ge
in
th
e v
alu
e o
f o
ne
va
ria
ble
in
fo
rmu
las
aff
ect
s th
e v
alu
e o
f th
e m
ea
sure
me
nt.
2.8
.A2
.B
Eva
lua
te a
nd
sim
pli
fy a
lge
bra
ic e
xpre
ssio
ns,
fo
r e
xam
ple
:
pro
du
cts/
qu
oti
en
ts o
f p
oly
no
mia
ls,
log
ari
thm
ic e
xpre
ssio
ns
an
d
com
ple
x fr
act
ion
s; a
nd
so
lve
lin
ea
r, q
ua
dra
tic,
exp
on
en
tia
l a
nd
log
ari
thm
ic e
qu
ati
on
s, a
nd
in
eq
ua
liti
es,
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
2.8
.A2
.F
Inte
rpre
t th
e r
esu
lts
of
solv
ing
eq
ua
tio
ns,
in
eq
ua
liti
es,
sy
ste
ms
of
eq
ua
tio
ns,
an
d i
ne
qu
ali
tie
s in
th
e c
on
text
of
the
sit
ua
tio
n
tha
t m
oti
va
ted
th
e m
od
el.
2.1
1.A
2.B
An
aly
ze a
nd
in
terp
ret
rate
s o
f g
row
th/d
eca
y.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.22
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.D.1
0
Re
pre
sen
t, s
olv
e a
nd
in
terp
ret
eq
ua
tio
ns/
ine
qu
ali
tie
s
an
d s
yst
em
s o
f e
qu
ati
on
s/in
eq
ua
liti
es
alg
eb
raic
all
y
an
d g
rap
hic
all
y.
A.R
EI.
11
. E
xpla
in w
hy
th
e x
-co
ord
ina
tes
of
the
po
ints
wh
ere
th
e g
rap
hs
of
the
eq
ua
tio
ns
y =
f(x
)
an
d y
= g
(x)
inte
rse
ct a
re t
he
so
luti
on
s o
f th
e e
qu
ati
on
f(x
) =
g(x
); f
ind
th
e s
olu
tio
ns
ap
pro
xim
ate
ly,
e.g
., u
sin
g t
ech
no
log
y t
o g
rap
h t
he
fu
nct
ion
s, m
ak
e t
ab
les
of
va
lue
s, o
r
fin
d s
ucc
ess
ive
ap
pro
xim
ati
on
s. I
ncl
ud
e c
ase
s w
he
re f
(x)
an
d/o
r g
(x)
are
lin
ea
r,
po
lyn
om
ial,
ra
tio
na
l, a
bso
lute
va
lue
, e
xpo
ne
nti
al,
an
d l
og
ari
thm
ic f
un
ctio
ns.
2.1
.A2
.F
Un
de
rsta
nd
th
e c
on
cep
ts o
f e
xpo
ne
nti
al
an
d l
og
ari
thm
ic f
orm
s
an
d u
se t
he
in
ve
rse
re
lati
on
ship
s b
etw
ee
n e
xpo
ne
nti
al
an
d
log
ari
thm
ic e
xpre
ssio
n t
o d
ete
rmin
e u
nk
no
wn
qu
an
titi
es
in
eq
ua
tio
ns.
2.3
.A2
.C
So
lve
a f
orm
ula
fo
r a
giv
en
va
ria
ble
usi
ng
alg
eb
raic
pro
cess
es.
2.3
.A2
.E
De
scri
be
ho
w a
ch
an
ge
in
th
e v
alu
e o
f o
ne
va
ria
ble
in
fo
rmu
las
aff
ect
s th
e v
alu
e o
f th
e m
ea
sure
me
nt.
2.8
.A2
.B
Eva
lua
te a
nd
sim
pli
fy a
lge
bra
ic e
xpre
ssio
ns,
fo
r e
xam
ple
:
pro
du
cts/
qu
oti
en
ts o
f p
oly
no
mia
ls,
log
ari
thm
ic e
xpre
ssio
ns
an
d
com
ple
x fr
act
ion
s; a
nd
so
lve
an
d g
rap
h l
ine
ar,
qu
ad
rati
c,
exp
on
en
tia
l a
nd
lo
ga
rith
mic
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s, a
nd
solv
e a
nd
gra
ph
sy
ste
ms
of
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s.
2.8
.A2
.E
Use
co
mb
ina
tio
ns
of
sym
bo
ls a
nd
nu
mb
ers
to
cre
ate
exp
ress
ion
s, e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
in t
wo
or
mo
re
va
ria
ble
s, s
yst
em
s o
f e
qu
ati
on
s a
nd
in
eq
ua
liti
es,
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
2.8
.A2
.F
Inte
rpre
t th
e r
esu
lts
of
solv
ing
eq
ua
tio
ns,
in
eq
ua
liti
es,
sy
ste
ms
of
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s in
th
e c
on
text
of
the
sit
ua
tio
n t
ha
t
mo
tiva
ted
th
e m
od
el.
2.1
1.A
2.B
An
aly
ze a
nd
in
terp
ret
rate
s g
row
th/d
eca
y.
CC
.2.2
.HS
.C.1
Use
th
e c
on
cep
t a
nd
no
tati
on
of
fun
ctio
ns
to
inte
rpre
t a
nd
ap
ply
th
em
in
te
rms
of
the
ir c
on
text
.
Inte
nti
on
all
y B
lan
k2
.8.A
2.C
Re
cog
niz
e,
de
scri
be
an
d g
en
era
lize
pa
tte
rns
usi
ng
se
qu
en
ces
an
d s
eri
es
to p
red
ict
lon
g-t
erm
ou
tco
me
s.
2.8
.A2
.D
De
mo
nst
rate
an
un
de
rsta
nd
ing
an
d a
pp
ly p
rop
ert
ies
of
fun
ctio
ns
(do
ma
in,
ran
ge
, in
ve
rse
s) a
nd
ch
ara
cte
rist
ics
of
fam
ilie
s o
f fu
nct
ion
s (l
ine
ar,
po
lyn
om
ial,
ra
tio
na
l, e
xpo
ne
nti
al,
log
ari
thm
ic).
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.23
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.C.2
Gra
ph
an
d a
na
lyze
fu
nct
ion
s a
nd
use
th
eir
pro
pe
rtie
s
to m
ak
e c
on
ne
ctio
ns
be
twe
en
th
e d
iffe
ren
t
rep
rese
nta
tio
ns.
F.I
F.
4.
Fo
r a
fu
nct
ion
th
at
mo
de
ls a
re
lati
on
ship
be
twe
en
tw
o q
ua
nti
tie
s, i
nte
rpre
t k
ey
fea
ture
s o
f g
rap
hs
an
d t
ab
les
in t
erm
s o
f th
e q
ua
nti
tie
s, a
nd
sk
etc
h g
rap
hs
sho
win
g k
ey
fea
ture
s g
ive
n a
ve
rba
l d
esc
rip
tio
n o
f th
e r
ela
tio
nsh
ip.
Ke
y f
ea
ture
s in
clu
de
: in
terc
ep
ts;
inte
rva
ls w
he
re t
he
fun
ctio
n i
s in
cre
asi
ng
, d
ecr
ea
sin
g,
po
siti
ve
, o
r n
eg
ati
ve
; re
lati
ve
ma
xim
um
s a
nd
min
imu
ms;
sy
mm
etr
ies;
en
d b
eh
avio
r; a
nd
pe
rio
dic
ity
.
5.
Re
late
th
e d
om
ain
of
a f
un
ctio
n t
o i
ts g
rap
h a
nd
, w
he
re a
pp
lica
ble
, to
th
e q
ua
nti
tati
ve
rela
tio
nsh
ip i
t d
esc
rib
es.
Fo
r e
xam
ple
, if
th
e f
un
ctio
n h
(n)
giv
es
the
nu
mb
er
of
pe
rso
n-
ho
urs
it
tak
es
to a
sse
mb
le n
en
gin
es
in a
fa
cto
ry,
the
n t
he
po
siti
ve
in
teg
ers
wo
uld
be
an
ap
pro
pri
ate
do
ma
in f
or
the
fu
nct
ion
.
6.
Ca
lcu
late
an
d i
nte
rpre
t th
e a
ve
rag
e r
ate
of
cha
ng
e o
f a
fu
nct
ion
(p
rese
nte
d
sym
bo
lica
lly
or
as
a t
ab
le)
ove
r a
sp
eci
fie
d i
nte
rva
l. E
stim
ate
th
e r
ate
of
cha
ng
e f
rom
a
gra
ph
.
7.
Gra
ph
fu
nct
ion
s e
xpre
sse
d s
ym
bo
lica
lly
an
d s
ho
w k
ey
fe
atu
res
of
the
gra
ph
, b
y h
an
d i
n
sim
ple
ca
ses
an
d u
sin
g t
ech
no
log
y f
or
mo
re c
om
pli
cate
d c
ase
s.
a.
Gra
ph
lin
ea
r a
nd
qu
ad
rati
c fu
nct
ion
s a
nd
sh
ow
in
terc
ep
ts,
ma
xim
a,
an
d m
inim
a.
b.
Gra
ph
sq
ua
re r
oo
t, c
ub
e r
oo
t, a
nd
pie
cew
ise
-de
fin
ed
fu
nct
ion
s,
incl
ud
ing
ste
p f
un
ctio
ns
an
d a
bso
lute
va
lue
fu
nct
ion
s.
c. G
rap
h p
oly
no
mia
l fu
nct
ion
s, i
de
nti
fyin
g z
ero
s w
he
n s
uit
ab
le
fact
ori
zati
on
s a
re a
va
ila
ble
, a
nd
sh
ow
ing
en
d b
eh
avio
r.
d.
(+)
Gra
ph
ra
tio
na
l fu
nct
ion
s, i
de
nti
fyin
g z
ero
s a
nd
asy
mp
tote
s
wh
en
su
ita
ble
fa
cto
riza
tio
ns
are
ava
ila
ble
, a
nd
sh
ow
ing
en
d b
eh
avio
r.
e.
Gra
ph
exp
on
en
tia
l a
nd
lo
ga
rith
mic
fu
nct
ion
s, s
ho
win
g i
nte
rce
pts
an
d e
nd
be
ha
vio
r, a
nd
tri
go
no
me
tric
fu
nct
ion
s, s
ho
win
g p
eri
od
, m
idli
ne
, a
nd
am
pli
tud
e.
8.
Wri
te a
fu
nct
ion
de
fin
ed
by
an
exp
ress
ion
in
dif
fere
nt
bu
t e
qu
iva
len
t fo
rms
to r
eve
al
an
d e
xpla
in d
iffe
ren
t p
rop
ert
ies
of
the
fu
nct
ion
.
a.
Use
th
e p
roce
ss o
f fa
cto
rin
g a
nd
co
mp
leti
ng
th
e s
qu
are
in
a
qu
ad
rati
c fu
nct
ion
to
sh
ow
ze
ros,
ext
rem
e v
alu
es,
an
d s
ym
me
try
of
the
gra
ph
, a
nd
inte
rpre
t th
ese
in
te
rms
of
a c
on
text
.
b.
Use
th
e p
rop
ert
ies
of
exp
on
en
ts t
o i
nte
rpre
t e
xpre
ssio
ns
for
exp
on
en
tia
l fu
nct
ion
s. F
or
exa
mp
le,
ide
nti
fy p
erc
en
t ra
te o
f ch
an
ge
in
fu
nct
ion
s su
ch a
s y
= (
1.0
2)t
, y
= (
0.9
7)t
, y
= (
1.0
1)1
2t,
y =
(1
.2)t
/10
, a
nd
cla
ssif
y t
he
m a
s re
pre
sen
tin
g
exp
on
en
tia
l g
row
th o
r d
eca
y
9.
Co
mp
are
pro
pe
rtie
s o
f tw
o f
un
ctio
ns
ea
ch r
ep
rese
nte
d i
n a
dif
fere
nt
wa
y
2.8
.A2
.B
Eva
lua
te a
nd
sim
pli
fy a
lge
bra
ic e
xpre
ssio
ns,
fo
r e
xam
ple
:
pro
du
cts/
qu
oti
en
ts o
f p
oly
no
mia
ls,
log
ari
thm
ic e
xpre
ssio
ns
an
d
com
ple
x fr
act
ion
s; a
nd
so
lve
an
d g
rap
h l
ine
ar,
qu
ad
rati
c,
exp
on
en
tia
l a
nd
lo
ga
rith
mic
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s, a
nd
solv
e a
nd
gra
ph
sy
ste
ms
of
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s.
2.8
.A2
.C
Re
cog
niz
e,
de
scri
be
an
d g
en
era
lize
pa
tte
rns
usi
ng
se
qu
en
ces
an
d s
eri
es
to p
red
ict
lin
g-t
erm
ou
tco
me
s.
2.8
.A2
.D
De
mo
nst
rate
an
un
de
rsta
nd
ing
an
d a
pp
ly p
rop
ert
ies
of
fun
ctio
ns
(do
ma
in,
ran
ge
, in
ve
rse
s) a
nd
ch
ara
cte
rist
ics
of
fam
ilie
s o
f fu
nct
ion
s (l
ine
ar,
po
lyn
om
ial,
ra
tio
na
l, e
xpo
ne
nti
al,
log
ari
thm
ic).
2.8
.A2
.E
Use
co
mb
ina
tio
ns
of
sym
bo
ls a
nd
nu
mb
ers
to
cre
ate
exp
ress
ion
s, e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
in t
wo
or
mo
re
va
ria
ble
s, s
yst
em
s o
f e
qu
ati
on
s a
nd
in
eq
ua
liti
es,
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
2.2
.A2
.F
Inte
rpre
t th
e r
esu
lts
solv
ing
eq
ua
tio
ns,
in
eq
ua
liti
es,
sy
ste
ms
of
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s in
th
e c
on
text
of
the
sit
ua
tio
n t
ha
t
mo
tiva
ted
th
e m
od
el.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.24
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.C.3
Wri
te f
un
ctio
ns
or
seq
ue
nce
s th
at
mo
de
l
rela
tio
nsh
ips
be
twe
en
tw
o q
ua
nti
tie
s.
F.B
F.
1.
Wri
te a
fu
nct
ion
th
at
de
scri
be
s a
re
lati
on
ship
be
twe
en
tw
o q
ua
nti
tie
s.
a.
De
term
ine
an
exp
lici
t e
xpre
ssio
n,
a r
ecu
rsiv
e p
roce
ss,
or
ste
ps
for
calc
ula
tio
n f
rom
a c
on
text
.
b.
Co
mb
ine
sta
nd
ard
fu
nct
ion
ty
pe
s u
sin
g a
rith
me
tic
op
era
tio
ns.
Fo
r
exa
mp
le,
bu
ild
a f
un
ctio
n t
ha
t m
od
els
th
e t
em
pe
ratu
re o
f a
co
oli
ng
bo
dy
by
ad
din
g a
con
sta
nt
fun
ctio
n t
o a
de
cay
ing
exp
on
en
tia
l, a
nd
re
late
th
ese
fu
nct
ion
s to
th
e m
od
el.
2.1
.A2
.F
Un
de
rsta
nd
th
e c
on
cep
ts o
f e
xpo
ne
nti
al
an
d l
og
ari
thm
ic f
orm
s
an
d u
se t
he
in
ve
rse
re
lati
on
ship
s b
etw
ee
n e
xpo
ne
nti
al
an
d
log
ari
thm
ic e
xpre
ssio
n t
o d
ete
rmin
e u
nk
no
wn
qu
an
titi
es
in
eq
ua
tio
ns.
2.8
.A2
.B
Eva
lua
te a
nd
sim
pli
fy a
lge
bra
ic e
xpre
ssio
ns,
fo
r e
xam
ple
:
pro
du
cts/
qu
oti
en
ts o
f p
oly
no
mia
ls,
log
ari
thm
ic e
xpre
ssio
ns
an
d
com
ple
x fr
act
ion
s; a
nd
so
lve
an
d g
rap
h l
ine
ar,
qu
ad
rati
c,
exp
on
en
tia
l a
nd
lo
ga
rith
mic
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s, a
nd
solv
e a
nd
gra
ph
sy
ste
ms
of
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s.
2.8
.A2
.C
Re
cog
niz
e,
de
scri
be
an
d g
en
era
lize
pa
tte
rns
usi
ng
se
qu
en
ces
an
d s
eri
es
to p
red
ict
lon
g-t
erm
ou
tco
me
s.
2.8
.A2
.E
Use
co
mb
ina
tio
ns
of
sym
bo
ls a
nd
nu
mb
ers
to
cre
ate
exp
ress
ion
s, e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
in t
wo
or
mo
re
va
ria
ble
s, s
yst
em
s o
f e
qu
ati
on
s a
nd
in
eq
ua
liti
es,
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
2.8
.A2
.F
Inte
rpre
t th
e r
esu
lts
of
solv
ing
eq
ua
tio
ns,
in
eq
ua
liti
es,
sy
ste
ms
of
eq
ua
tio
ns,
an
d i
ne
qu
ali
tie
s in
th
e c
on
text
of
the
sit
ua
tio
n
tha
t m
oti
va
ted
th
e m
od
el.
2.1
1.A
2.A
De
term
ine
an
d i
nte
rpre
t m
axi
mu
m a
nd
min
imu
m v
alu
es
of
a
fun
ctio
n o
ve
r a
sp
eci
fie
d i
nte
rva
l.
2.1
1.A
2.B
An
aly
ze a
nd
in
terp
ret
rate
s g
row
th/d
eca
y.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.25
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.C.4
Inte
rpre
t th
e e
ffe
cts
tra
nsf
orm
ati
on
s h
ave
on
fun
ctio
ns
an
d f
ind
th
e i
nve
rse
s o
f fu
nct
ion
s.
F.B
F.
3.
Ide
nti
fy t
he
eff
ect
on
th
e g
rap
h o
f re
pla
cin
g f
(x)
by
f(x
) +
k,
k f
(x),
f(k
x),
an
d f
(x +
k)
for
spe
cifi
c va
lue
s o
f k
(b
oth
po
siti
ve
an
d n
eg
ati
ve
); f
ind
th
e v
alu
e o
f k
giv
en
th
e g
rap
hs.
Exp
eri
me
nt
wit
h c
ase
s a
nd
ill
ust
rate
an
exp
lan
ati
on
of
the
eff
ect
s o
n t
he
gra
ph
usi
ng
tech
no
log
y.
Incl
ud
e r
eco
gn
izin
g e
ve
n a
nd
od
d f
un
ctio
ns
fro
m t
he
ir g
rap
hs
an
d a
lge
bra
ic
exp
ress
ion
s fo
r th
em
.
4.
Fin
d i
nve
rse
fu
nct
ion
s.
a.
So
lve
an
eq
ua
tio
n o
f th
e f
orm
f(x
) =
c f
or
a s
imp
le f
un
ctio
n f
tha
t h
as
an
in
ve
rse
an
d w
rite
an
exp
ress
ion
fo
r th
e i
nve
rse
. F
or
exa
mp
le,
f(x)
=2
x3
or
f(x)
= (
x+1
)/(x
–1
) fo
r x
≠ 1
.
2.8
.A2
.D
De
mo
nst
rate
an
un
de
rsta
nd
ing
an
d a
pp
ly p
rop
ert
ies
of
fun
ctio
ns
(do
ma
in,
ran
ge
, in
ve
rse
s) a
nd
ch
ara
cte
rist
ics
of
fam
ilie
s o
f fu
nct
ion
s (l
ine
ar,
po
lyn
om
ial,
ra
tio
na
l, e
xpo
ne
nti
al,
log
ari
thm
ic).
2.8
.A2
.E
Use
co
mb
ina
tio
ns
of
sym
bo
ls a
nd
nu
mb
ers
to
cre
ate
exp
ress
ion
s, e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
in t
wo
or
mo
re
va
ria
ble
s, s
yst
em
s o
f e
qu
ati
on
s a
nd
in
eq
ua
liti
es,
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
2.8
.A2
.F
Inte
rpre
t th
e r
esu
lts
of
solv
ing
eq
ua
tio
ns,
in
eq
ua
liti
es,
sy
ste
ms
of
eq
ua
tio
ns,
an
d i
ne
qu
ali
tie
s in
th
e c
on
text
of
the
sit
ua
tio
n
tha
t m
oti
va
ted
th
e m
od
el.
2.1
1.A
2.A
De
term
ine
an
d i
nte
rpre
t m
axi
mu
m a
nd
min
imu
m v
alu
es
of
a
fun
ctio
n o
ve
r a
sp
eci
fie
d i
nte
rva
l.
2.1
1.A
2.B
An
aly
ze a
nd
in
terp
ret
rate
s g
row
th/d
eca
y.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.26
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.C.5
Co
nst
ruct
an
d c
om
pa
re l
ine
ar,
qu
ad
rati
c a
nd
exp
on
en
tia
l m
od
els
to
so
lve
pro
ble
ms.
F.L
E.
2.
Co
nst
ruct
lin
ea
r a
nd
exp
on
en
tia
l fu
nct
ion
s, i
ncl
ud
ing
ari
thm
eti
c a
nd
ge
om
etr
ic
seq
ue
nce
s, g
ive
n a
gra
ph
, a
de
scri
pti
on
of
a r
ela
tio
nsh
ip,
or
two
in
pu
t-o
utp
ut
pa
irs
(in
clu
de
re
ad
ing
th
ese
fro
m a
ta
ble
).
3.
Ob
serv
e u
sin
g g
rap
hs
an
d t
ab
les
tha
t a
qu
an
tity
in
cre
asi
ng
exp
on
en
tia
lly
eve
ntu
all
y
exc
ee
ds
a q
ua
nti
ty i
ncr
ea
sin
g l
ine
arl
y,
qu
ad
rati
c a
lly
, o
r (m
ore
ge
ne
rall
y)
as
a p
oly
no
mia
l
fun
ctio
n.
4.
Fo
r e
xpo
ne
nti
al
mo
de
ls,
exp
ress
as
a l
og
ari
thm
th
e s
olu
tio
n t
o a
bct
= d
wh
ere
a,
c, a
nd
d a
re n
um
be
rs a
nd
th
e b
ase
b i
s 2
, 1
0,
or
e;
eva
lua
te t
he
lo
ga
rith
m u
sin
g t
ech
no
log
y.
2.1
.A2
.F
Un
de
rsta
nd
th
e c
on
cep
ts o
f e
xpo
ne
nti
al
an
d l
og
ari
thm
ic f
orm
s
an
d u
se t
he
in
ve
rse
re
lati
on
ship
s b
etw
ee
n e
xpo
ne
nti
al
an
d
log
ari
thm
ic e
xpre
ssio
n t
o d
ete
rmin
e u
nk
no
wn
qu
an
titi
es
in
eq
ua
tio
ns.
2.8
.A2
.B
Eva
lua
te a
nd
sim
pli
fy a
lge
bra
ic e
xpre
ssio
ns,
fo
r e
xam
ple
:
pro
du
cts/
qu
oti
en
ts o
f p
oly
no
mia
ls,
log
ari
thm
ic e
xpre
ssio
ns
an
d
com
pe
ls f
ract
ion
s; a
nd
so
lve
an
d g
rap
h l
ine
ar,
qu
ad
rati
c,
exp
on
en
tia
l a
nd
lo
ga
rith
mic
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s, a
nd
solv
e a
nd
gra
ph
sy
ste
ms
of
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s.
2.8
.A2
.C
Re
cog
niz
e,
de
scri
be
an
d g
en
era
lize
pa
tte
rns
usi
ng
se
qu
en
ces
an
d s
eri
es
to p
red
ict
lon
g-t
erm
ou
tco
me
s.
2.8
.A2
.D
De
mo
nst
rate
an
un
de
rsta
nd
ing
an
d a
pp
ly p
rop
ert
ies
of
fun
ctio
ns
(do
ma
in,
ran
ge
, in
ve
rse
s) a
nd
ch
ara
cte
rist
ics
of
fam
ilie
s o
f fu
nct
ion
s (l
ine
ar,
po
lyn
om
ial,
ra
tio
na
l, e
xpo
ne
nti
al,
log
ari
thm
ic.
2.8
.A2
.E
Use
co
mb
ina
tio
ns
of
sym
bo
ls a
nd
nu
mb
ers
to
cre
ate
exp
ress
ion
s, e
qu
ati
on
s, a
nd
in
eq
ua
liti
es
in t
wo
or
mo
re
va
ria
ble
s, s
yst
em
s o
f e
qu
ati
on
s a
nd
in
eq
ua
liti
es,
an
d f
un
ctio
na
l
rela
tio
nsh
ips
tha
t m
od
el
pro
ble
m s
itu
ati
on
s.
2.8
.A2
.F
Inte
rpre
t th
e r
esu
lts
of
solv
ing
eq
ua
tio
ns,
in
eq
ua
liti
es,
sy
ste
ms
of
eq
ua
tio
ns
an
d i
ne
qu
ali
tie
s in
th
e c
on
text
of
the
sit
ua
tio
n t
ha
t
mo
tiva
ted
th
e m
od
el.
2.8
.A2
.D
De
mo
nst
rate
an
un
de
rsta
nd
ing
an
d a
pp
ly p
rop
ert
ies
of
fun
ctio
ns
(do
ma
in,
ran
ge
, in
ve
rse
s) a
nd
ch
ara
cte
rist
ics
of
fam
ilie
s o
f fu
nct
ion
s (l
ine
ar,
po
lyn
om
ial,
ra
tio
na
l, e
xpo
ne
nti
al,
log
ari
thm
ic).
2.1
1.A
2.B
An
aly
ze a
nd
in
terp
ret
rate
s g
row
th/d
eca
y.
CC
.2.2
.HS
.C.6
Inte
rpre
t fu
nct
ion
s in
te
rms
of
the
sit
ua
tio
n t
he
y
mo
de
l.
Inte
nti
on
all
y B
lan
k2
.8.A
2.C
Re
cog
niz
e,
de
scri
be
an
d g
en
era
lize
pa
tte
rns
usi
ng
se
qu
en
ces
an
d s
eri
es
to p
red
ict
lon
g-t
erm
ou
tco
me
s.
2.8
.A2
.D
De
mo
nst
rate
an
un
de
rsta
nd
ing
an
d a
pp
ly p
rop
ert
ies
of
fun
ctio
ns
(do
ma
in,
ran
ge
, in
ve
rse
s) a
nd
ch
ara
cte
rist
ics
of
fam
ilie
s o
f fu
nct
ion
s (l
ine
ar,
po
lyn
om
ial,
ra
tio
na
l, e
xpo
ne
nti
al,
log
ari
thm
ic).
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.27
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.C.7
Ap
ply
ra
dia
n m
ea
sure
of
an
an
gle
an
d t
he
un
it c
ircl
e
to a
na
lyze
th
e t
rig
on
om
etr
ic f
un
ctio
ns.
F.T
F.
1.
Un
de
rsta
nd
ra
dia
n m
ea
sure
of
an
an
gle
as
the
le
ng
th o
f th
e a
rc o
n t
he
un
it c
ircl
e
sub
ten
de
d b
y t
he
an
gle
.
2.
Exp
lain
ho
w t
he
un
it c
ircl
e i
n t
he
co
ord
ina
te p
lan
e e
na
ble
s th
e e
xte
nsi
on
of
trig
on
om
etr
ic f
un
ctio
ns
to a
ll r
ea
l n
um
be
rs,
inte
rpre
ted
as
rad
ian
me
asu
res
of
an
gle
s
tra
ve
rse
d c
ou
nte
rclo
ckw
ise
aro
un
d t
he
un
it c
ircl
e.
Inte
nti
on
all
y B
lan
k
CC
.2.2
.HS
.C.8
Ch
oo
se t
rig
on
om
etr
ic f
un
ctio
ns
to m
od
el
pe
rio
dic
ph
en
om
en
a a
nd
de
scri
be
th
e p
rop
ert
ies
of
the
gra
ph
s
F.T
F.
5.
Ch
oo
se t
rig
on
om
etr
ic f
un
ctio
ns
to m
od
el
pe
rio
dic
ph
en
om
en
a w
ith
sp
eci
fie
d
am
pli
tud
e,
fre
qu
en
cy,
an
d m
idli
ne
.
Inte
nti
on
all
y B
lan
k
CC
.2.2
.HS
.C.9
Pro
ve
th
e P
yth
ag
ore
an
id
en
tity
an
d u
se i
t to
calc
ula
te t
rig
on
om
etr
ic r
ati
os.
F.T
F.
8.
Pro
ve
th
e P
yth
ag
ore
an
id
en
tity
sin
2(θ
) +
co
s2(θ
) =
1 a
nd
use
it
to f
ind
sin
(θ),
co
s(θ
), o
r
tan
(θ)
giv
en
sin
(θ),
co
s(θ
), o
r ta
n(θ
) a
nd
th
e q
ua
dra
nt
of
the
an
gle
.
Inte
nti
on
all
y B
lan
k
CC
.2.3
.HS
.A.1
Use
ge
om
etr
ic f
igu
res
an
d t
he
ir p
rop
ert
ies
to
rep
rese
nt
tra
nsf
orm
ati
on
s in
th
e p
lan
e.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.2
Ap
ply
rig
id t
ran
sfo
rma
tio
ns
to d
ete
rmin
e a
nd
exp
lain
con
gru
en
ce.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.3
Ve
rify
an
d a
pp
ly g
eo
me
tric
th
eo
rem
s a
s th
ey
re
late
to g
eo
me
tric
fig
ure
s.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.4
Ap
ply
th
e c
on
cep
t o
f co
ng
rue
nce
to
cre
ate
ge
om
etr
ic
con
stru
ctio
ns.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.5
Cre
ate
ju
stif
ica
tio
ns
ba
sed
on
tra
nsf
orm
ati
on
s to
est
ab
lish
sim
ila
rity
of
pla
ne
fig
ure
s.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.6
Ve
rify
an
d a
pp
ly t
he
ore
ms
invo
lvin
g s
imil
ari
ty a
s th
ey
rela
te t
o p
lan
e f
igu
res.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.7
Ap
ply
tri
go
no
me
tric
ra
tio
s to
so
lve
pro
ble
ms
invo
lvin
g r
igh
t tr
ian
gle
s.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.8
Ap
ply
ge
om
etr
ic t
he
ore
ms
to v
eri
fy p
rop
ert
ies
of
circ
les.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.9
Ext
en
d t
he
co
nce
pt
of
sim
ila
rity
to
de
term
ine
arc
len
gth
s a
nd
are
as
of
sect
ors
of
circ
les.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.28
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.3
.HS
.A.1
0
Tra
nsl
ate
be
twe
en
th
e g
eo
me
tric
de
scri
pti
on
an
d t
he
eq
ua
tio
n f
or
a c
on
ic s
ect
ion
.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.1
1
Ap
ply
co
ord
ina
te g
eo
me
try
to
pro
ve
sim
ple
ge
om
etr
ic t
he
ore
ms
alg
eb
raic
all
y.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.1
2
Exp
lain
vo
lum
e f
orm
ula
s a
nd
use
th
em
to
so
lve
pro
ble
ms.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.1
3
An
aly
ze r
ela
tio
nsh
ips
be
twe
en
tw
o-d
ime
nsi
on
al
an
d
thre
e-d
ime
nsi
on
al
ob
ject
s.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.3
.HS
.A.1
4
Ap
ply
ge
om
etr
ic c
on
cep
ts t
o m
od
el
an
d s
olv
e r
ea
l
wo
rld
pro
ble
ms.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.4
.HS
.B.1
Su
mm
ari
ze,
rep
rese
nt,
an
d i
nte
rpre
t d
ata
on
a s
ing
le
cou
nt
or
me
asu
rem
en
t va
ria
ble
.
S.I
D.4
4.
Use
th
e m
ea
n a
nd
sta
nd
ard
de
via
tio
n o
f a
da
ta s
et
to f
it i
t to
a n
orm
al
dis
trib
uti
on
an
d
to e
stim
ate
po
pu
lati
on
pe
rce
nta
ge
s. R
eco
gn
ize
th
at
the
re a
re d
ata
se
ts f
or
wh
ich
su
ch a
pro
ced
ure
is
no
t a
pp
rop
ria
te.
Use
ca
lcu
lato
rs,
spre
ad
she
ets
, a
nd
ta
ble
s to
est
ima
te a
rea
s
un
de
r th
e n
orm
al
curv
e.
Inte
nti
on
all
y B
lan
k
CC
.2.4
.HS
.B.2
Su
mm
ari
ze,
rep
rese
nt,
an
d i
nte
rpre
t d
ata
on
tw
o
cate
go
rica
l a
nd
qu
an
tita
tive
va
ria
ble
s.
S.I
D.
5.
Su
mm
ari
ze c
ate
go
rica
l d
ata
fo
r tw
o c
ate
go
rie
s in
tw
o-w
ay
fre
qu
en
cy t
ab
les.
In
terp
ret
rela
tive
fre
qu
en
cie
s in
th
e c
on
text
of
the
da
ta (
incl
ud
ing
jo
int,
ma
rgin
al,
an
d c
on
dit
ion
al
rela
tive
fre
qu
en
cie
s).
Re
cog
niz
e p
oss
ible
ass
oci
ati
on
s a
nd
tre
nd
s in
th
e d
ata
.
6.
Re
pre
sen
t d
ata
on
tw
o q
ua
nti
tati
ve
va
ria
ble
s o
n a
sca
tte
r p
lot,
an
d d
esc
rib
e h
ow
th
e
va
ria
ble
s a
re r
ela
ted
.
a.
Fit
a f
un
ctio
n t
o t
he
da
ta;
use
fu
nct
ion
s fi
tte
d t
o d
ata
to
so
lve
pro
ble
ms
in t
he
co
nte
xt o
f th
e d
ata
. U
se g
ive
n f
un
ctio
ns
or
cho
ose
a f
un
ctio
n s
ug
ge
ste
d
by
th
e c
on
text
. E
mp
ha
size
lin
ea
r, q
ua
dra
tic,
an
d e
xpo
ne
nti
al
mo
de
ls.
b.
Info
rma
lly
ass
ess
th
e f
it o
f a
fu
nct
ion
by
plo
ttin
g a
nd
an
aly
zin
g
resi
du
als
.
c. F
it a
lin
ea
r fu
nct
ion
fo
r a
sca
tte
r p
lot
tha
t su
gg
est
s a
lin
ea
r
ass
oci
ati
on
.
2.6
.A2
.C
Co
nst
ruct
a l
ine
of
be
st f
it a
nd
ca
lcu
late
its
eq
ua
tio
n f
or
lin
ea
r
an
d n
on
lin
ea
r tw
o-v
ari
ab
le d
ata
.
2.6
.A2
.E
Ma
ke
pre
dic
tio
ns
ba
sed
on
lin
es
of
be
st f
it o
r d
raw
co
ncl
usi
on
s
on
th
e v
alu
e o
f a
va
ria
ble
in
a p
op
ula
tio
n b
ase
d o
n t
he
re
sult
s o
f
a s
am
ple
.
2.7
.A2
.A
Use
pro
ba
bil
ity
to
pre
dic
t th
e l
ike
lih
oo
d o
f a
n o
utc
om
e i
n a
n
exp
eri
me
nt.
2.7
.A2
.E
Use
pro
ba
bil
ity
to
ma
ke
ju
dg
me
nts
ab
ou
t th
e l
ike
lih
oo
d o
f
va
rio
us
ou
tco
me
s.
2.8
.A2
.C
Re
cog
niz
e,
de
scri
be
an
d g
en
era
lize
pa
tte
rns
usi
ng
se
qu
en
ces
an
d s
eri
es
to p
red
ict
lon
g-t
erm
ou
tco
me
s.
2.8
.A2
.D
De
mo
nst
rate
an
un
de
rsta
nd
ing
an
d a
pp
ly p
rop
ert
ies
of
fun
ctio
na
l (d
om
ain
, ra
ng
e,
inve
rse
s) a
nd
ch
ara
cte
rist
ics
of
fam
ilie
s o
f fu
nct
ion
s (l
ine
ar,
po
lyn
om
ial,
ra
tio
na
l, e
xpo
ne
nti
al,
log
ari
thm
ic).
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.29
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.4
.HS
.B.3
An
aly
ze l
ine
ar
mo
de
ls t
o m
ak
e i
nte
rpre
tati
on
s b
ase
d
on
th
e d
ata
.
Inte
nti
on
all
y B
lan
k2
.6.A
2.C
Co
nst
ruct
a l
ine
of
be
st f
it a
nd
ca
lcu
late
its
eq
ua
tio
n f
or
lin
ea
r
an
d n
on
lin
ea
r tw
o-v
ari
ab
le d
ata
.
2.6
.A2
.E
Ma
ke
pre
dic
tio
ns
ba
sed
on
lin
es
of
be
st f
it o
r d
raw
co
ncl
usi
on
s
on
th
e v
alu
e o
f a
va
ria
ble
in
a p
op
ula
tio
n b
ase
d o
n t
he
re
sult
s o
f
a s
am
ple
.
2.7
.A2
.A
Use
pro
ba
bil
ity
to
pre
dic
t th
e l
ike
lih
oo
d o
f a
n o
utc
om
e i
n a
n
exp
eri
me
nt.
2.7
.A2
.C
Co
mp
are
od
ds
an
d p
rob
ab
ilit
y.
2.7
.A2
.E
Use
pro
ba
bil
ity
to
ma
ke
ju
dg
me
nts
ab
ou
t th
e l
ike
lih
oo
d o
f
va
rio
us
ou
tco
me
s.
CC
.2.4
.HS
.B.4
Re
cog
niz
e a
nd
eva
lua
te r
an
do
m p
roce
sse
s
un
de
rly
ing
sta
tist
ica
l e
xpe
rim
en
ts.
S.I
C.
1.
Un
de
rsta
nd
sta
tist
ics
as
a p
roce
ss f
or
ma
kin
g i
nfe
ren
ces
ab
ou
t p
op
ula
tio
n p
ara
me
ters
ba
sed
on
a r
an
do
m s
am
ple
fro
m t
ha
t p
op
ula
tio
n.
2.
De
cid
e i
f a
sp
eci
fie
d m
od
el
is c
on
sist
en
t w
ith
re
sult
s fr
om
a g
ive
n d
ata
-ge
ne
rati
ng
pro
cess
, e
.g.,
usi
ng
sim
ula
tio
n.
Fo
r e
xam
ple
, a
mo
de
l sa
ys
a s
pin
nin
g c
oin
fa
lls
he
ad
s u
p
wit
h p
rob
ab
ilit
y 0
.5.
Wo
uld
a r
esu
lt o
f 5
ta
ils
in a
ro
w c
au
se y
ou
to
qu
est
ion
th
e m
od
el?
2.6
.A2
.C
Co
nst
ruct
a l
ine
of
be
st f
it a
nd
ca
lcu
late
its
eq
ua
tio
n f
or
lin
ea
r
an
d n
on
lin
ea
r tw
o-v
ari
ab
le d
ata
.
2.6
.A2
.E
Ma
ke
pre
dic
tio
ns
ba
sed
on
lin
es
of
be
st f
it o
r d
raw
co
ncl
usi
on
s
on
th
e v
alu
e o
f a
va
ria
ble
in
a p
op
ula
tio
n b
ase
d o
n t
he
re
sult
s o
f
a s
am
ple
.
2.7
.A2
.A
Use
pro
ba
bil
ity
to
pre
dic
t li
ke
lih
oo
d o
f a
n o
utc
om
e i
n a
n
exp
eri
me
nt.
2.7
.A2
.C
Co
mp
are
od
ds
an
d p
rob
ab
ilit
y.
2.7
.A2
.E
Use
pro
ba
bil
ity
to
ma
ke
ju
dg
me
nts
ab
ou
t th
e l
ike
lih
oo
d o
f
va
rio
us
ou
tco
me
s.C
C.2
.4.H
S.B
.5
Ma
ke
in
fere
nce
s a
nd
ju
stif
y c
on
clu
sio
ns
ba
sed
on
sam
ple
su
rve
ys,
exp
eri
me
nts
, a
nd
ob
serv
ati
on
al
stu
die
s.
S.I
C.
3.
Re
cog
niz
e t
he
pu
rpo
ses
of
an
d d
iffe
ren
ces
am
on
g s
am
ple
su
rve
ys,
exp
eri
me
nts
, a
nd
ob
serv
ati
on
al
stu
die
s; e
xpla
in h
ow
ra
nd
om
iza
tio
n r
ela
tes
to e
ach
4.
Use
da
ta f
rom
a s
am
ple
su
rve
y t
o e
stim
ate
a p
op
ula
tio
n m
ea
n o
r p
rop
ort
ion
; d
eve
lop
a
ma
rgin
of
err
or
thro
ug
h t
he
use
of
sim
ula
tio
n m
od
els
fo
r ra
nd
om
sa
mp
lin
g.
5.
Use
da
ta f
rom
a r
an
do
miz
ed
exp
eri
me
nt
to c
om
pa
re t
wo
tre
atm
en
ts;
use
sim
ula
tio
ns
to d
eci
de
if
dif
fere
nce
s b
etw
ee
n p
ara
me
ters
are
sig
nif
ica
nt.
6.
Eva
lua
te r
ep
ort
s b
ase
d o
n d
ata
.
2.6
.A2
.C
Co
nst
ruct
a l
ine
of
be
st f
it a
nd
ca
lcu
late
its
eq
ua
tio
n f
or
lin
ea
r
an
d n
on
lin
ea
r tw
o-v
ari
ab
le d
ata
.
2.6
.A2
.E
Ma
ke
pre
dic
tio
ns
ba
sed
on
lin
es
of
be
st f
it o
r d
raw
co
ncl
usi
on
s
on
th
e v
alu
e o
f a
va
ria
ble
in
a p
op
ula
tio
n b
ase
d o
n t
he
re
sult
s o
f
a s
am
ple
.
2.7
.A2
.A
Use
pro
ba
bil
ity
to
pre
dic
t th
e l
ike
lih
oo
d o
f a
n o
utc
om
e i
n a
n
exp
eri
me
nt.
2.7
.A2
.C
Co
mp
are
od
ds
an
d p
rob
ab
ilit
y.
2.7
.A2
.E
Use
pro
ba
bil
ity
to
ma
ke
ju
dg
me
nts
ab
ou
t th
e l
ike
lih
oo
d o
f
va
rio
us
ou
tco
me
s.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.30
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.4
.HS
.B.6
Use
th
e c
on
cep
ts o
f in
de
pe
nd
en
ce a
nd
co
nd
itio
na
l
pro
ba
bil
ity
to
in
terp
ret
da
ta.
Inte
nti
on
all
y B
lan
k2
.6.A
2.C
Co
nst
ruct
a l
ine
of
be
st f
it a
nd
ca
lcu
late
its
eq
ua
tio
n f
or
lin
ea
r
an
d n
on
lin
ea
r tw
o-v
ari
ab
le d
ata
.
2.6
.A2
.E
Ma
ke
pre
dic
tio
ns
ba
sed
on
lin
es
of
be
st f
it o
r d
raw
co
ncl
usi
on
s
on
th
e v
alu
e o
f a
va
ria
ble
in
a p
op
ula
tio
n b
ase
d o
n t
he
re
sult
s o
f
a s
am
ple
.
2.7
.A2
.A
Use
pro
ba
bil
ity
to
pre
dic
t th
e l
ike
lih
oo
d o
f a
n o
utc
om
e i
n a
n
exp
eri
me
nt.
2.7
.A2
.C
Co
mp
are
od
ds
an
d p
rob
ab
ilit
y.
2.7
.A2
.E
Use
pro
ba
bil
ity
to
ma
ke
ju
dg
me
nts
ab
ou
t th
e l
ike
lih
oo
d o
f
va
rio
us
ou
tco
me
s.
CC
.2.4
.HS
.B.7
Ap
ply
th
e r
ule
s o
f p
rob
ab
ilit
y t
o c
om
pu
te
pro
ba
bil
itie
s o
f co
mp
ou
nd
eve
nts
in
a u
nif
orm
pro
ba
bil
ity
mo
de
l.
S.M
D.
6.
(+)
Use
pro
ba
bil
itie
s to
ma
ke
fa
ir d
eci
sio
ns
(e.g
., d
raw
ing
by
lo
ts,
usi
ng
a r
an
do
m
nu
mb
er
ge
ne
rato
r).
7.
(+)
An
aly
ze d
eci
sio
ns
an
d s
tra
teg
ies
usi
ng
pro
ba
bil
ity
co
nce
pts
(e
.g.,
pro
du
ct t
est
ing
,
me
dic
al
test
ing
, p
ull
ing
a h
ock
ey
go
ali
e a
t th
e e
nd
of
a g
am
e).
2.7
.A2
.A
Use
pro
ba
bil
ity
to
pre
dic
t th
e l
ike
lih
oo
d o
f a
n o
utc
om
e i
n a
n
exp
eri
me
nt.
2.7
.A2
.C
Co
mp
are
od
ds
an
d p
rob
ab
ilit
y.
2.7
.A2
.E
Use
pro
ba
bil
ity
to
ma
ke
ju
dg
me
nts
ab
ou
t th
e l
ike
lih
oo
d o
f
va
rio
us
ou
tco
me
s.
Ge
om
etr
y
CC
.2.1
.HS
.F.1
Ap
ply
an
d e
xte
nd
th
e p
rop
ert
ies
of
exp
on
en
ts t
o
solv
e p
rob
lem
s w
ith
ra
tio
na
l e
xpo
ne
nts
.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.1
.HS
.F.2
Ap
ply
pro
pe
rtie
s o
f ra
tio
na
l a
nd
irr
ati
on
al
nu
mb
ers
to s
olv
e r
ea
l w
orl
d o
r m
ath
em
ati
cal
pro
ble
ms.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.1
.HS
.F.3
Ap
ply
qu
an
tita
tive
re
aso
nin
g t
o c
ho
ose
an
d I
nte
rpre
t
un
its
an
d s
cale
s in
fo
rmu
las,
gra
ph
s a
nd
da
ta
dis
pla
ys.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.1
.HS
.F.4
Use
un
its
as
a w
ay
to
un
de
rsta
nd
pro
ble
ms
an
d t
o
gu
ide
th
e s
olu
tio
n o
f m
ult
i-st
ep
pro
ble
ms.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.1
.HS
.F.5
Ch
oo
se a
le
ve
l o
f a
ccu
racy
ap
pro
pri
ate
to
lim
ita
tio
ns
on
me
asu
rem
en
t w
he
n r
ep
ort
ing
qu
an
titi
es.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.31
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.1
.HS
.F.6
Ext
en
d t
he
kn
ow
led
ge
of
ari
thm
eti
c o
pe
rati
on
s a
nd
ap
ply
to
co
mp
lex
nu
mb
ers
.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.1
.HS
.F.7
Ap
ply
co
nce
pts
of
com
ple
x n
um
be
rs i
n p
oly
no
mia
l
ide
nti
tie
s a
nd
qu
ad
rati
c e
qu
ati
on
s to
so
lve
pro
ble
ms.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.D.1
Inte
rpre
t th
e s
tru
ctu
re o
f e
xpre
ssio
ns
to r
ep
rese
nt
a
qu
an
tity
in
te
rms
of
its
con
text
.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.D.2
Wri
te e
xpre
ssio
ns
in e
qu
iva
len
t fo
rms
to s
olv
e
pro
ble
ms.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.D.3
Ext
en
d t
he
kn
ow
led
ge
of
ari
thm
eti
c o
pe
rati
on
s a
nd
ap
ply
to
po
lyn
om
ials
.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.D.4
Un
de
rsta
nd
th
e r
ela
tio
nsh
ip b
etw
ee
n z
ero
s a
nd
fact
ors
of
po
lyn
om
ials
to
ma
ke
ge
ne
rali
zati
on
s a
bo
ut
fun
ctio
ns
an
d t
he
ir g
rap
hs.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.D.5
Use
po
lyn
om
ial
ide
nti
tie
s to
so
lve
pro
ble
ms.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.D.6
Ext
en
d t
he
kn
ow
led
ge
of
rati
on
al
fun
ctio
ns
to
rew
rite
in
eq
uiv
ale
nt
form
s.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.D.7
Cre
ate
an
d g
rap
h e
qu
ati
on
s o
r in
eq
ua
liti
es
to
de
scri
be
nu
mb
ers
or
rela
tio
nsh
ips.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.D.8
Ap
ply
in
ve
rse
op
era
tio
ns
to s
olv
e e
qu
ati
on
s o
r
form
ula
s fo
r a
giv
en
va
ria
ble
.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.D.9
Use
re
aso
nin
g t
o s
olv
e e
qu
ati
on
s a
nd
ju
stif
y t
he
solu
tio
n m
eth
od
.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.D.1
0
Re
pre
sen
t, s
olv
e a
nd
in
terp
ret
eq
ua
tio
ns/
ine
qu
ali
tie
s
an
d s
yst
em
s o
f e
qu
ati
on
s/in
eq
ua
liti
es
alg
eb
raic
all
y
an
d g
rap
hic
all
y.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.32
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.C.1
Use
th
e c
on
cep
t a
nd
no
tati
on
of
fun
ctio
ns
to
inte
rpre
t a
nd
ap
ply
th
em
in
te
rms
of
the
ir c
on
text
.
Inte
nti
on
all
y B
lan
k2
.3.G
.C
Use
pro
pe
rtie
s o
f g
eo
me
tric
fig
ure
s a
nd
me
asu
rem
en
t fo
rmu
las
to s
olv
e f
or
a m
issi
ng
qu
an
tity
(e
.g.,
th
e m
ea
sure
of
a s
pe
cifi
c
an
gle
cre
ate
d b
y p
ara
lle
l li
ne
s a
nd
is
a t
ran
sve
rsa
l).
2.3
.G.E
De
scri
be
ho
w a
ch
an
ge
in
th
e v
alu
e o
f o
ne
va
ria
ble
in
are
a a
nd
vo
lum
e f
orm
ula
s a
ffe
ct t
he
va
lue
of
the
me
asu
rem
en
t.
2.7
.G.A
Use
ge
om
etr
ic f
igu
res
an
d t
he
co
nce
pt
of
are
a t
o c
alc
ula
te
pro
ba
bil
ity
.
2.1
1.G
.A
Fin
d t
he
me
asu
res
of
the
sid
es
of
a p
oly
go
n w
ith
a g
ive
n
pe
rim
ete
r th
at
wil
l m
axi
miz
e t
he
are
a o
f th
e p
oly
go
n.
2.1
1.G
.C
Use
su
ms
of
are
as
of
sta
nd
ard
sh
ap
es
to e
stim
ate
th
e a
rea
s o
f
com
ple
x sh
ap
es.
CC
.2.2
.HS
.C.2
Gra
ph
an
d a
na
lyze
fu
nct
ion
s a
nd
use
th
eir
pro
pe
rtie
s
to m
ak
e c
on
ne
ctio
ns
be
twe
en
th
e d
iffe
ren
t
rep
rese
nta
tio
ns.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.C.3
Wri
te f
un
ctio
ns
or
seq
ue
nce
s th
at
mo
de
l
rela
tio
nsh
ips
be
twe
en
tw
o q
ua
nti
tie
s.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.C.4
Inte
rpre
t th
e e
ffe
cts
tra
nsf
orm
ati
on
s h
ave
on
fun
ctio
ns
an
d f
ind
th
e i
nve
rse
s o
f fu
nct
ion
s.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.C.5
Co
nst
ruct
an
d c
om
pa
re l
ine
ar,
qu
ad
rati
c a
nd
exp
on
en
tia
l m
od
els
to
so
lve
pro
ble
ms.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.C.6
Inte
rpre
t fu
nct
ion
s in
te
rms
of
the
sit
ua
tio
n t
he
y
mo
de
l.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.C.7
Ap
ply
ra
dia
n m
ea
sure
of
an
an
gle
an
d t
he
un
it c
ircl
e
to a
na
lyze
th
e t
rig
on
om
etr
ic f
un
ctio
ns.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.2
.HS
.C.8
Ch
oo
se t
rig
on
om
etr
ic f
un
ctio
ns
to m
od
el
pe
rio
dic
ph
en
om
en
a a
nd
de
scri
be
th
e p
rop
ert
ies
of
the
gra
ph
s
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.33
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.2
.HS
.C.9
Pro
ve
th
e P
yth
ag
ore
an
id
en
tity
an
d u
se i
t to
calc
ula
te t
rig
on
om
etr
ic r
ati
os.
Inte
nti
on
all
y B
lan
k2
.10
.G.A
Ide
nti
fy,
cre
ate
, a
nd
so
lve
pra
ctic
al
pro
ble
ms
invo
lvin
g r
igh
t
tria
ng
les
usi
ng
th
e t
rig
on
om
etr
ic f
un
ctio
ns
an
d t
he
Py
tha
go
rea
n
Th
eo
rem
.
CC
.2.3
.HS
.A.1
Use
ge
om
etr
ic f
igu
res
an
d t
he
ir p
rop
ert
ies
to
rep
rese
nt
tra
nsf
orm
ati
on
s in
th
e p
lan
e.
G.C
O.
1.
Kn
ow
pre
cise
de
fin
itio
ns
of
an
gle
, ci
rcle
, p
erp
en
dic
ula
r li
ne
, p
ara
lle
l li
ne
, a
nd
lin
e
seg
me
nt,
ba
sed
on
th
e u
nd
efi
ne
d n
oti
on
s o
f p
oin
t, l
ine
, d
ista
nce
alo
ng
a l
ine
, a
nd
dis
tan
ce a
rou
nd
a c
ircu
lar
arc
.
2.
Re
pre
sen
t tr
an
sfo
rma
tio
ns
in t
he
pla
ne
usi
ng
, e
.g.,
tra
nsp
are
nci
es
an
d g
eo
me
try
soft
wa
re;
de
scri
be
tra
nsf
orm
ati
on
s a
s fu
nct
ion
s th
at
tak
e p
oin
ts i
n t
he
pla
ne
as
inp
uts
an
d g
ive
oth
er
po
ints
as
ou
tpu
ts.
Co
mp
are
tra
nsf
orm
ati
on
s th
at
pre
serv
e d
ista
nce
an
d
an
gle
to
th
ose
th
at
do
no
t (e
.g.,
tra
nsl
ati
on
ve
rsu
s h
ori
zon
tal
stre
tch
).
3.
Giv
en
a r
ect
an
gle
, p
ara
lle
log
ram
, tr
ap
ezo
id,
or
reg
ula
r p
oly
go
n,
de
scri
be
th
e r
ota
tio
ns
an
d r
efl
ect
ion
s th
at
carr
y i
t o
nto
its
elf
.
4.
De
ve
lop
de
fin
itio
ns
of
rota
tio
ns,
re
fle
ctio
ns,
an
d t
ran
sla
tio
ns
in t
erm
s o
f a
ng
les,
cir
cle
s,
pe
rpe
nd
icu
lar
lin
es,
pa
rall
el
lin
es,
an
d l
ine
se
gm
en
ts.
5.
Giv
en
a g
eo
me
tric
fig
ure
an
d a
ro
tati
on
, re
fle
ctio
n,
or
tra
nsl
ati
on
, d
raw
th
e
tra
nsf
orm
ed
fig
ure
usi
ng
, e
.g.,
gra
ph
pa
pe
r, t
raci
ng
pa
pe
r, o
r g
eo
me
try
so
ftw
are
. S
pe
cify
a s
eq
ue
nce
of
tra
nsf
orm
ati
on
s th
at
wil
l ca
rry
a g
ive
n f
igu
re o
nto
an
oth
er.
Inte
nti
on
all
y B
lan
k
CC
.2.3
.HS
.A.2
Ap
ply
rig
id t
ran
sfo
rma
tio
ns
to d
ete
rmin
e a
nd
exp
lain
con
gru
en
ce.
G.C
O.
6.
Use
ge
om
etr
ic d
esc
rip
tio
ns
of
rig
id m
oti
on
s to
tra
nsf
orm
fig
ure
s a
nd
to
pre
dic
t th
e
eff
ect
of
a g
ive
n r
igid
mo
tio
n o
n a
giv
en
fig
ure
; g
ive
n t
wo
fig
ure
s, u
se t
he
de
fin
itio
n o
f
con
gru
en
ce i
n t
erm
s o
f ri
gid
mo
tio
ns
to d
eci
de
if
the
y a
re c
on
gru
en
t.
7.
Use
th
e d
efi
nit
ion
of
con
gru
en
ce i
n t
erm
s o
f ri
gid
mo
tio
ns
to s
ho
w t
ha
t tw
o t
ria
ng
les
are
co
ng
rue
nt
if a
nd
on
ly i
f co
rre
spo
nd
ing
pa
irs
of
sid
es
an
d c
orr
esp
on
din
g p
air
s o
f a
ng
les
are
co
ng
rue
nt.
8.
Exp
lain
ho
w t
he
cri
teri
a f
or
tria
ng
le c
on
gru
en
ce (
AS
A,
SA
S,
an
d S
SS
) fo
llo
w f
rom
th
e
de
fin
itio
n o
f co
ng
rue
nce
in
te
rms
of
rig
id m
oti
on
s.
2.1
.G.C
Use
ra
tio
an
d p
rop
ort
ion
to
mo
de
l re
lati
on
ship
s b
etw
ee
n
qu
an
titi
es.
2.4
.G.A
Wri
te f
orm
al
pro
ofs
(d
ire
ct p
roo
fs,
ind
ire
ct p
roo
f/p
roo
fs b
y
con
tra
dic
tio
n,
use
of
cou
nte
r-e
xam
ple
s, t
ruth
ta
ble
s, e
tc.)
to
va
lid
ate
co
nje
ctu
res
or
arg
um
en
ts.
2.7
.G.A
Use
ge
om
etr
ic f
igu
res
an
d t
he
co
nce
pt
of
are
a t
o c
alc
ula
te
pro
ba
bil
ity
.
2.9
.G.B
Use
arg
um
en
ts b
ase
d o
n t
ran
sfo
rma
tio
ns
to e
sta
bli
sh
con
gru
en
ce o
r si
mil
ari
ty o
f 2
dim
en
sio
na
l sh
ap
es.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.34
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.3
.HS
.A.3
Ve
rify
an
d a
pp
ly g
eo
me
tric
th
eo
rem
s a
s th
ey
re
late
to g
eo
me
tric
fig
ure
s.
G.C
O.
9.
Pro
ve
th
eo
rem
s a
bo
ut
lin
es
an
d a
ng
les.
Th
eo
rem
s in
clu
de
: ve
rtic
al
an
gle
s a
re
con
gru
en
t; w
he
n a
tra
nsv
ers
al
cro
sse
s p
ara
lle
l li
ne
s, a
lte
rna
te i
nte
rio
r a
ng
les
are
con
gru
en
t a
nd
co
rre
spo
nd
ing
an
gle
s a
re c
on
gru
en
t; p
oin
ts o
n a
pe
rpe
nd
icu
lar
bis
ect
or
of
a l
ine
se
gm
en
t a
re e
xact
ly t
ho
se e
qu
idis
tan
t fr
om
th
e s
eg
me
nt’
s e
nd
po
ints
.
10
. P
rove
th
eo
rem
s a
bo
ut
tria
ng
les.
Th
eo
rem
s in
clu
de
: m
ea
sure
s o
f in
teri
or
an
gle
s o
f a
tria
ng
le s
um
to
18
0°;
ba
se a
ng
les
of
iso
sce
les
tria
ng
les
are
co
ng
rue
nt;
th
e s
eg
me
nt
join
ing
mid
po
ints
of
two
sid
es
of
a t
ria
ng
le i
s p
ara
lle
l to
th
e t
hir
d s
ide
an
d h
alf
th
e l
en
gth
;
the
me
dia
ns
of
a t
ria
ng
le m
ee
t a
t a
po
int.
11
. P
rove
th
eo
rem
s a
bo
ut
pa
rall
elo
gra
ms.
Th
eo
rem
s in
clu
de
: o
pp
osi
te s
ide
s a
re
con
gru
en
t, o
pp
osi
te a
ng
les
are
co
ng
rue
nt,
th
e d
iag
on
als
of
a p
ara
lle
log
ram
bis
ect
ea
ch o
the
r, a
nd
co
nve
rse
ly,
rect
an
gle
s a
re p
ara
lle
log
ram
s w
ith
con
gru
en
t d
iag
on
als
.
2.1
.G.C
Use
ra
tio
an
d p
rop
ort
ion
to
mo
de
l re
lati
on
ship
s b
etw
ee
n
qu
an
titi
es.
2.3
.G.C
Use
pro
pe
rtie
s o
f g
eo
me
tric
fig
ure
s a
nd
me
asu
rem
en
t fo
rmu
las
to s
olv
e f
or
a m
issi
ng
qu
an
tity
(e
.g.,
th
e m
ea
sure
of
a s
pe
cifi
c
an
gle
cre
ate
d b
y p
ara
lle
l li
ne
s a
nd
is
tra
nsv
ers
al.
2.3
.G.E
De
scri
be
ho
w a
ch
an
ge
in
th
e v
alu
e o
ne
va
ria
ble
in
are
a a
nd
vo
lum
e f
orm
ula
s a
ffe
ct t
he
va
lue
of
the
me
asu
rem
en
t.
2.4
.G.A
Wri
te f
orm
al
pro
ofs
(d
ire
ct p
roo
fs,
ind
ire
ct p
roo
fs/p
roo
fs b
y
con
tra
dic
tio
n,
use
of
cou
nte
r-e
xam
ple
s, t
ruth
ta
ble
s, e
tc.)
to
va
lid
ate
co
nje
ctu
res
or
arg
um
en
ts.
2.4
.G.B
Use
sta
tem
en
ts,
con
ve
rse
s, i
nve
rse
s, a
nd
co
ntr
ap
osi
tive
to
con
stru
ct v
ali
d a
rgu
me
nts
or
to v
ali
da
te a
rgu
me
nts
re
lati
ng
to
ge
om
etr
ic t
he
ore
ms.
2.7
.G.A
Use
ge
om
etr
ic f
igu
res
an
d t
he
co
nce
pt
of
are
a t
o c
alc
ula
te
pro
ba
bil
ity
.
2.9
.G.B
Use
arg
um
en
ts b
ase
d o
n t
ran
sfo
rma
tio
n t
o e
sta
bli
sh
con
gru
en
ce o
r si
mil
ari
ty o
f 2
dim
en
sio
na
l sh
ap
es.
2.1
1.G
.A
Fin
d t
he
me
asu
res
of
the
sid
es
of
a p
oly
go
n w
ith
a g
ive
n
pe
rim
ete
r th
at
wil
l m
axi
miz
e t
he
are
a o
f th
e p
oly
go
n.
2.1
1.G
.C
Use
su
ms
of
are
as
of
Sta
nd
ard
Sh
ap
es
to e
stim
ate
th
e a
rea
s o
f
com
ple
x sh
ap
es.
CC
.2.3
.HS
.A.4
Ap
ply
th
e c
on
cep
t o
f co
ng
rue
nce
to
cre
ate
ge
om
etr
ic
con
stru
ctio
ns.
G.C
O.
12
. M
ak
e f
orm
al
ge
om
etr
ic c
on
stru
ctio
ns
wit
h a
va
rie
ty o
f to
ols
an
d m
eth
od
s (c
om
pa
ss
an
d s
tra
igh
ted
ge
, st
rin
g,
refl
ect
ive
de
vic
es,
pa
pe
r fo
ldin
g,
dy
na
mic
ge
om
etr
ic s
oft
wa
re,
etc
.).
Co
py
ing
a s
eg
me
nt;
co
py
ing
an
an
gle
;
bis
ect
ing
a s
eg
me
nt;
bis
ect
ing
an
an
gle
; co
nst
ruct
ing
pe
rpe
nd
icu
lar
lin
es,
in
clu
din
g t
he
pe
rpe
nd
icu
lar
bis
ect
or
of
a l
ine
se
gm
en
t; a
nd
co
nst
ruct
ing
a l
ine
pa
rall
el
to a
giv
en
lin
e
thro
ug
h a
po
int
no
t o
n t
he
lin
e.
13
. C
on
stru
ct a
n e
qu
ila
tera
l tr
ian
gle
, a
sq
ua
re,
an
d a
re
gu
lar
he
xag
on
in
scri
be
d i
n a
cir
cle
.
Inte
nti
on
all
y B
lan
k
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.35
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.3
.HS
.A.5
Cre
ate
ju
stif
ica
tio
ns
ba
sed
on
tra
nsf
orm
ati
on
s to
est
ab
lish
sim
ila
rity
of
pla
ne
fig
ure
s.
G.S
RT
.
1.
Ve
rify
exp
eri
me
nta
lly
th
e p
rop
ert
ies
of
dil
ati
on
s g
ive
n b
y a
ce
nte
r a
nd
a s
cale
fa
cto
r:
a.
A d
ila
tio
n t
ak
es
a l
ine
no
t p
ass
ing
th
rou
gh
th
e c
en
ter
of
the
dil
ati
on
to
a p
ara
lle
l li
ne
, a
nd
le
ave
s a
lin
e p
ass
ing
th
rou
gh
th
e c
en
ter
un
cha
ng
ed
.
b.
Th
e d
ila
tio
n o
f a
lin
e s
eg
me
nt
is l
on
ge
r o
r sh
ort
er
in t
he
ra
tio
giv
en
by
th
e s
cale
fa
cto
r.
2.
Giv
en
tw
o f
igu
res,
use
th
e d
efi
nit
ion
of
sim
ila
rity
in
te
rms
of
sim
ila
rity
tra
nsf
orm
ati
on
s
to d
eci
de
if
the
y a
re s
imil
ar;
exp
lain
usi
ng
sim
ila
rity
tra
nsf
orm
ati
on
s th
e m
ea
nin
g o
f
sim
ila
rity
fo
r tr
ian
gle
s a
s th
e e
qu
ali
ty o
f a
ll c
orr
esp
on
din
g p
air
s o
f a
ng
les
an
d t
he
pro
po
rtio
na
lity
of
all
co
rre
spo
nd
ing
pa
irs
of
sid
es.
3.
Use
th
e p
rop
ert
ies
of
sim
ila
rity
tra
nsf
orm
ati
on
s to
est
ab
lish
th
e A
A c
rite
rio
n f
or
two
tria
ng
les
to b
e s
imil
ar.
2.1
.G.C
Use
ra
tio
an
d p
rop
ort
ion
to
mo
de
l re
lati
on
ship
s b
etw
ee
n
qu
an
titi
es.
2.4
.G.A
Wri
te f
orm
al
pro
ofs
(d
ire
ct p
roo
fs,
ind
ire
ct p
roo
f/p
roo
fs b
y
con
tra
dic
tio
n,
use
of
cou
nte
r-e
xam
ple
s, t
ruth
ta
ble
s, e
tc.)
to
va
lid
ate
co
nje
ctu
res
or
arg
um
en
ts.
2.7
.G.A
Use
arg
um
en
ts b
ase
d o
n t
ran
sfo
rma
tio
ns
to e
sta
bli
sh
con
gru
en
ce o
r si
mil
ari
ty o
f 2
dim
en
sio
na
l sh
ap
es.
CC
.2.3
.HS
.A.6
Ve
rify
an
d a
pp
ly t
he
ore
ms
invo
lvin
g s
imil
ari
ty a
s th
ey
rela
te t
o p
lan
e f
igu
res.
G.S
RT
.
4.
Pro
ve
th
eo
rem
s a
bo
ut
tria
ng
les.
Th
eo
rem
s in
clu
de
: a
lin
e p
ara
lle
l to
on
e s
ide
of
a
tria
ng
le d
ivid
es
the
oth
er
two
pro
po
rtio
na
lly
, a
nd
co
nve
rse
ly;
the
Py
tha
go
rea
n T
he
ore
m
pro
ve
d u
sin
g t
ria
ng
le s
imil
ari
ty.
5.
Use
co
ng
rue
nce
an
d s
imil
ari
ty c
rite
ria
fo
r tr
ian
gle
s to
so
lve
pro
ble
ms
an
d t
o p
rove
rela
tio
nsh
ips
in g
eo
me
tric
fig
ure
s.
2.1
.G.C
Use
ra
tio
an
d p
rop
ort
ion
to
mo
de
l re
lati
on
ship
s b
etw
ee
n
qu
an
titi
es.
2.4
.G.A
Wri
te f
orm
al
pro
ofs
(d
ire
ct p
roo
fs,
ind
ire
ct p
roo
f/p
roo
fs b
y
con
tra
dic
tio
n,
use
of
cou
nte
r-e
xam
ple
s, t
ruth
ta
ble
s, e
tc.)
to
va
lid
ate
co
nje
ctu
res
or
arg
um
en
ts.
2.7
.G.A
Use
sta
tem
en
ts,
con
ve
rse
, in
ve
rse
s, a
nd
co
ntr
ap
osi
tive
s to
con
stru
ct v
ali
d a
rgu
me
nts
or
to v
ali
da
te a
rgu
me
nts
re
lati
ng
to
ge
om
etr
ic t
he
ore
ms.
2.9
.G.B
Use
arg
um
en
ts b
ase
d o
n t
ran
sfo
rma
tio
n t
o e
sta
bli
sh
con
gru
en
ce o
r si
mil
ari
ty o
f 2
dim
en
sio
na
l sh
ap
es.
CC
.2.3
.HS
.A.7
Ap
ply
tri
go
no
me
tric
ra
tio
s to
so
lve
pro
ble
ms
invo
lvin
g r
igh
t tr
ian
gle
s.
GIS
T.
6.
Un
de
rsta
nd
th
at
by
sim
ila
rity
, si
de
ra
tio
s in
rig
ht
tria
ng
les
are
pro
pe
rtie
s o
f th
e a
ng
les
in t
he
tri
an
gle
, le
ad
ing
to
de
fin
itio
ns
of
trig
on
om
etr
ic r
ati
os
for
acu
te a
ng
les.
7.
Exp
lain
an
d u
se t
he
re
lati
on
ship
be
twe
en
th
e s
ine
an
d c
osi
ne
of
com
ple
me
nta
ry
an
gle
s.
8.
Use
tri
go
no
me
tric
ra
tio
s a
nd
th
e P
yth
ag
ore
an
Th
eo
rem
to
so
lve
rig
ht
tria
ng
les
in
ap
pli
ed
pro
ble
ms.
2.8
.G.B
Use
alg
eb
raic
re
pre
sen
tati
on
s to
so
lve
pro
ble
ms
usi
ng
coo
rdin
ate
ge
om
etr
y.
2.9
.G.C
Use
te
chn
iqu
es
fro
m c
oo
rdin
ate
ge
om
etr
y t
o e
sta
bli
sh
pro
pe
rtie
s o
f li
ne
s a
nd
2-d
ime
nsi
on
al
sha
pe
s a
nd
so
lid
s.
2.1
0.G
.A
Ide
nti
fy,
cre
ate
, a
nd
so
lve
pra
ctic
al
pro
ble
ms
invo
lvin
g r
igh
t
tria
ng
les
usi
ng
th
e t
rig
on
om
etr
ic f
un
ctio
ns
an
d t
he
Py
tha
go
rea
n
Th
eo
rem
.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.36
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.3
.HS
.A.8
Ap
ply
ge
om
etr
ic t
he
ore
ms
to v
eri
fy p
rop
ert
ies
of
circ
les.
GM
..
1.
Pro
ve
th
at
all
cir
cle
s a
re s
imil
ar.
2.
Ide
nti
fy a
nd
de
scri
be
re
lati
on
ship
s a
mo
ng
in
scri
be
d a
ng
les,
ra
dii
, a
nd
ch
ord
s. I
ncl
ud
e
the
re
lati
on
ship
be
twe
en
ce
ntr
al,
in
scri
be
d,
an
d c
ircu
msc
rib
ed
an
gle
s; i
nsc
rib
ed
an
gle
s o
n
a d
iam
ete
r a
re r
igh
t a
ng
les;
th
e r
ad
ius
of
a c
ircl
e i
s p
erp
en
dic
ula
r to
th
e t
an
ge
nt
wh
ere
the
ra
diu
s in
ters
ect
s th
e c
ircl
e.
3.
Co
nst
ruct
th
e i
nsc
rib
ed
an
d c
ircu
msc
rib
ed
cir
cle
s o
f a
tri
an
gle
, a
nd
pro
ve
pro
pe
rtie
s o
f
an
gle
s fo
r a
qu
ad
rila
tera
l in
scri
be
d i
n a
cir
cle
.
4.
(+)
Co
nst
ruct
a t
an
ge
nt
lin
e f
rom
a p
oin
t o
uts
ide
a g
ive
n c
ircl
e t
o t
he
cir
cle
.
2.4
.G.A
Wri
te f
orm
al
pro
ofs
(d
ire
ct p
roo
fs,
ind
ire
ct p
roo
fs/p
roo
fs b
y
con
tra
dic
tio
n,
use
of
cou
nte
r-e
xam
ple
s, t
ruth
ta
ble
s, e
tc.)
to
va
lid
ate
co
nje
ctu
res
or
arg
um
en
ts.
2.4
.G.B
Use
sta
tem
en
ts,
con
ve
rse
s, i
nve
rse
s, a
nd
co
ntr
ap
osi
tive
s to
con
stru
ct v
ali
d a
rgu
me
nts
or
to v
ali
da
te a
rgu
me
nts
re
lati
ng
to
ge
om
etr
ic t
he
ore
ms.
2.9
.G.B
Use
arg
um
en
ts b
ase
d o
n t
ran
sfo
rma
tio
ns
to e
sta
bli
sh
con
gru
en
ce o
r si
mil
ari
ty o
f 2
dim
en
sio
na
l sh
ap
es.
CC
.2.3
.HS
.A.9
Ext
en
d t
he
co
nce
pt
of
sim
ila
rity
to
de
term
ine
arc
len
gth
s a
nd
are
as
of
sect
ors
of
circ
les.
GM
..
5.
De
rive
usi
ng
sim
ila
rity
th
e f
act
th
at
the
le
ng
th o
f th
e a
rc i
nte
rce
pte
d b
y a
n a
ng
le i
s
pro
po
rtio
na
l to
th
e r
ad
ius,
an
d d
efi
ne
th
e r
ad
ian
me
asu
re o
f th
e a
ng
le a
s th
e c
on
sta
nt
of
pro
po
rtio
na
lity
; d
eri
ve
th
e f
orm
ula
fo
r th
e a
rea
of
a s
ect
or.
2.3
.G.C
Use
pro
pe
rtie
s o
f g
eo
me
tric
fig
ure
s a
nd
me
asu
rem
en
t fo
rmu
las
to s
olv
e f
or
a m
issi
ng
qu
an
tity
(e
.g.,
th
e m
ea
sure
of
a s
pe
cifi
c
an
gle
cre
ate
d b
y p
ara
lle
l li
ne
s a
nd
is
a t
ran
sve
rsa
l).
CC
.2.3
.HS
.A.9
Ext
en
d t
he
co
nce
pt
of
sim
ila
rity
to
de
term
ine
arc
len
gth
s a
nd
are
as
of
sect
ors
of
circ
les.
GM
..
5.
De
rive
usi
ng
sim
ila
rity
th
e f
act
th
at
the
le
ng
th o
f th
e a
rc i
nte
rce
pte
d b
y a
n a
ng
le i
s
pro
po
rtio
na
l to
th
e r
ad
ius,
an
d d
efi
ne
th
e r
ad
ian
me
asu
re o
f th
e a
ng
le a
s th
e c
on
sta
nt
of
pro
po
rtio
na
lity
; d
eri
ve
th
e f
orm
ula
fo
r th
e a
rea
of
a s
ect
or.
2.4
.G.A
Wri
te f
orm
al
pro
ofs
(d
ire
ct p
roo
fs,
ind
ire
ct p
roo
fs/p
roo
fs b
y
con
tra
dic
tio
n,
use
of
cou
nte
r-e
xam
ple
s, t
ruth
ta
ble
s, e
tc.)
to
va
lid
ate
co
nje
ctu
res
or
arg
um
en
ts.
CC
.2.3
.HS
.A.9
Ext
en
d t
he
co
nce
pt
of
sim
ila
rity
to
de
term
ine
arc
len
gth
s a
nd
are
as
of
sect
ors
of
circ
les.
GM
..
5.
De
rive
usi
ng
sim
ila
rity
th
e f
act
th
at
the
le
ng
th o
f th
e a
rc i
nte
rce
pte
d b
y a
n a
ng
le i
s
pro
po
rtio
na
l to
th
e r
ad
ius,
an
d d
efi
ne
th
e r
ad
ian
me
asu
re o
f th
e a
ng
le a
s th
e c
on
sta
nt
of
pro
po
rtio
na
lity
; d
eri
ve
th
e f
orm
ula
fo
r th
e a
rea
of
a s
ect
or.
2.9
.G.A
Ide
nti
fy a
nd
use
pro
pe
rtie
s a
nd
re
lati
on
s o
f g
eo
me
tric
fig
ure
s;
cre
ate
ju
stif
ica
tio
ns
for
arg
um
en
ts r
ela
ted
to
ge
om
etr
ic
rela
tio
ns.
CC
.2.3
.HS
.A.9
Ext
en
d t
he
co
nce
pt
of
sim
ila
rity
to
de
term
ine
arc
len
gth
s a
nd
are
as
of
sect
ors
of
circ
les.
GM
..
5.
De
rive
usi
ng
sim
ila
rity
th
e f
act
th
at
the
le
ng
th o
f th
e a
rc i
nte
rce
pte
d b
y a
n a
ng
le i
s
pro
po
rtio
na
l to
th
e r
ad
ius,
an
d d
efi
ne
th
e r
ad
ian
me
asu
re o
f th
e a
ng
le a
s th
e c
on
sta
nt
of
pro
po
rtio
na
lity
; d
eri
ve
th
e f
orm
ula
fo
r th
e a
rea
of
a s
ect
or.
2.1
1.G
.A
Fin
d t
he
me
asu
res
of
the
sid
es
of
a p
oly
go
n w
ith
a g
ive
n
pe
rim
ete
r th
at
wil
l m
axi
miz
e t
he
are
a o
f th
e p
oly
go
n.
CC
.2.3
.HS
.A.1
0
Tra
nsl
ate
be
twe
en
th
e g
eo
me
tric
de
scri
pti
on
an
d t
he
eq
ua
tio
n f
or
a c
on
ic s
ect
ion
.
GA
GE
.
1.
De
rive
th
e e
qu
ati
on
of
a c
ircl
e o
f g
ive
n c
en
ter
an
d r
ad
ius
usi
ng
th
e P
yth
ag
ore
an
Th
eo
rem
; co
mp
lete
th
e s
qu
are
to
fin
d t
he
ce
nte
r a
nd
ra
diu
s o
f a
cir
cle
giv
en
by
an
eq
ua
tio
n.
2.
De
rive
th
e e
qu
ati
on
of
a p
ara
bo
la g
ive
n a
fo
cus
an
d d
ire
ctri
x.
2.9
.G.C
Use
te
chn
iqu
es
fro
m c
oo
rdin
ate
ge
om
etr
y t
o e
sta
bli
sh
pro
pe
rtie
s o
f li
ne
s a
nd
2-d
ime
nsi
on
al
sha
pe
s a
nd
so
lid
s.
2.1
1.G
.C
Use
su
ms
of
are
as
of
sta
nd
ard
sh
ap
es
to e
stim
ate
th
e a
rea
s o
f
com
ple
x sh
ap
es.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.37
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.3
.HS
.A.1
1
Ap
ply
co
ord
ina
te g
eo
me
try
to
pro
ve
sim
ple
ge
om
etr
ic t
he
ore
ms
alg
eb
raic
all
y.
G.G
PE
.
4.
Use
co
ord
ina
tes
to p
rove
sim
ple
ge
om
etr
ic t
he
ore
ms
alg
eb
raic
all
y.
Fo
r e
xam
ple
, p
rove
or
dis
pro
ve
th
at
a f
igu
re d
efi
ne
d b
y f
ou
r g
ive
n p
oin
ts i
n t
he
co
ord
ina
te p
lan
e i
s a
rect
an
gle
; p
rove
or
dis
pro
ve
th
at
the
po
int
(1, √
3)
lies
on
th
e c
ircl
e c
en
tere
d a
t th
e o
rig
in
an
d c
on
tain
ing
th
e p
oin
t (0
, 2
).
5.
Pro
ve
th
e s
lop
e c
rite
ria
fo
r p
ara
lle
l a
nd
pe
rpe
nd
icu
lar
lin
es
an
d u
se t
he
m t
o s
olv
e
ge
om
etr
ic p
rob
lem
s (e
.g.,
fin
d t
he
eq
ua
tio
n o
f a
lin
e p
ara
lle
l o
r p
erp
en
dic
ula
r to
a g
ive
n
lin
e t
ha
t p
ass
es
thro
ug
h a
giv
en
po
int)
.
6.
Fin
d t
he
po
int
on
a d
ire
cte
d l
ine
se
gm
en
t b
etw
ee
n t
wo
giv
en
po
ints
th
at
pa
rtit
ion
s th
e
seg
me
nt
in a
giv
en
ra
tio
.
7.
Use
co
ord
ina
tes
to c
om
pu
te p
eri
me
ters
of
po
lyg
on
s a
nd
are
as
of
tria
ng
les
an
d
rect
an
gle
s, e
.g.,
usi
ng
th
e d
ista
nce
fo
rmu
la.
2.4
.G.B
Use
sta
tem
en
ts,
con
ve
rse
s, i
nve
rse
s, a
nd
co
ntr
ap
osi
tive
s to
con
stru
ct v
ali
d a
rgu
me
nts
or
to v
ali
da
te a
rgu
me
nts
re
lati
ng
to
ge
om
etr
ic t
he
ore
ms.
2.8
.G.B
Use
alg
eb
raic
re
pre
sen
tati
on
s to
so
lve
pro
ble
ms
usi
ng
coo
rdin
ate
ge
om
etr
y.
2.1
0.G
.A
Ide
nti
fy,
cre
ate
, a
nd
so
lve
pra
ctic
al
pro
ble
ms
invo
lvin
g r
igh
t
tria
ng
les
usi
ng
th
e t
rig
on
om
etr
ic f
un
ctio
ns
an
d t
he
Py
tha
go
rea
n
Th
eo
rem
.
CC
.2.3
.HS
.A.1
2
Exp
lain
vo
lum
e f
orm
ula
s a
nd
use
th
em
to
so
lve
pro
ble
ms.
G.G
MD
.
1.
Giv
e a
n i
nfo
rma
l a
rgu
me
nt
for
the
fo
rmu
las
for
the
cir
cum
fere
nce
of
a c
ircl
e,
are
a o
f a
circ
le,
vo
lum
e o
f a
cy
lin
de
r, p
yra
mid
, a
nd
co
ne
. U
se d
isse
ctio
n a
rgu
me
nts
, C
ava
lie
ri’s
pri
nci
ple
, a
nd
in
form
al
lim
it a
rgu
me
nts
.
3.
Use
vo
lum
e f
orm
ula
s fo
r cy
lin
de
rs,
py
ram
ids,
co
ne
s, a
nd
sp
he
res
to s
olv
e p
rob
lem
s.
2.3
.G.C
Use
pro
pe
rtie
s o
f g
eo
me
tric
fig
ure
s a
nd
me
asu
rem
en
t fo
rmu
las
to s
olv
e f
or
a m
issi
ng
qu
an
tity
(e
.g.,
th
e m
ea
sure
of
a s
pe
cifi
c
an
gle
cre
ate
d b
y p
ara
lle
l li
ne
s a
nd
is
a t
ran
sve
rsa
l).
2.3
.G.E
De
scri
be
ho
w a
ch
an
ge
in
th
e v
alu
e o
f o
ne
va
ria
ble
in
are
a a
nd
vo
lum
e f
orm
ula
s a
ffe
ct t
he
va
lue
of
the
me
asu
rem
en
t.
CC
.2.3
.HS
.A.1
3
An
aly
ze r
ela
tio
nsh
ips
be
twe
en
tw
o-d
ime
nsi
on
al
an
d
thre
e-d
ime
nsi
on
al
ob
ject
s.
G.G
MD
.
4.
Ide
nti
fy t
he
sh
ap
es
of
two
-dim
en
sio
na
l cr
oss
-se
ctio
ns
of
thre
e-d
ime
nsi
on
al
ob
ject
s,
an
d i
de
nti
fy t
hre
e-d
ime
nsi
on
al
ob
ject
s g
en
era
ted
by
ro
tati
on
s o
f tw
o-d
ime
nsi
on
al
ob
ject
s.
2.3
.G.C
Use
pro
pe
rtie
s o
f g
eo
me
tric
fig
ure
s a
nd
me
asu
rem
en
t fo
rmu
las
to s
olv
e f
or
a m
issi
ng
qu
an
tity
(e
.g.,
th
e m
ea
sure
of
a s
pe
cifi
c
an
gle
cre
ate
d b
y p
ara
lle
l li
ne
s a
nd
is
a t
ran
sve
rsa
l).
2.9
.G.A
Ide
nti
fy a
nd
use
pro
pe
rtie
s a
nd
re
lati
on
s o
f g
eo
me
tric
fig
ure
s;
cre
ate
ju
stif
ica
tio
ns
for
arg
um
en
ts r
ela
ted
to
ge
om
etr
ic
rela
tio
ns.
CC
.2.3
.HS
.A.1
4
Ap
ply
ge
om
etr
ic c
on
cep
ts t
o m
od
el
an
d s
olv
e r
ea
l
wo
rld
pro
ble
ms.
G.M
G.
1.
Use
ge
om
etr
ic s
ha
pe
s, t
he
ir m
ea
sure
s, a
nd
th
eir
pro
pe
rtie
s to
de
scri
be
ob
ject
s (e
.g.,
mo
de
lin
g a
tre
e t
run
k o
r a
hu
ma
n t
ors
o a
s a
cy
lin
de
r).
2.
Ap
ply
co
nce
pts
of
de
nsi
ty b
ase
d o
n a
rea
an
d v
olu
me
in
mo
de
lin
g s
itu
ati
on
s (e
.g.,
pe
rso
ns
pe
r sq
ua
re m
ile
, B
TU
s p
er
cub
ic f
oo
t).
3.
Ap
ply
ge
om
etr
ic m
eth
od
s to
so
lve
de
sig
n p
rob
lem
s (e
.g.,
de
sig
nin
g a
n o
bje
ct o
r
stru
ctu
re t
o s
ati
sfy
ph
ysi
cal
con
stra
ints
or
min
imiz
e c
ost
; w
ork
ing
wit
h t
yp
og
rap
hic
gri
d
syst
em
s b
ase
d o
n r
ati
os)
.
2.3
.G.E
De
scri
be
ho
w a
ch
an
ge
in
th
e v
alu
e o
f o
ne
va
ria
ble
in
are
a a
nd
vo
lum
e f
orm
ula
s a
ffe
ct t
he
va
lue
of
the
me
asu
rem
en
t.
2.7
.G.A
Use
ge
om
etr
ic f
igu
res
an
d t
he
co
nce
pt
of
are
a t
o c
alc
ula
te
pro
ba
bil
ity
.
2.1
1.G
.A
Fin
d t
he
me
asu
res
of
the
sid
es
of
a p
oly
go
n w
ith
a g
ive
n
pe
rim
ete
r th
at
wil
l m
axi
miz
e t
he
are
a o
f th
e p
oly
go
n.
2.1
1.G
.C
Use
su
ms
of
are
as
of
sta
nd
ard
sh
ap
es
to e
stim
ate
th
e a
rea
s o
f
com
ple
x sh
ap
es.
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.38
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.4
.HS
.B.1
Su
mm
ari
ze,
rep
rese
nt,
an
d i
nte
rpre
t d
ata
on
a s
ing
le
cou
nt
or
me
asu
rem
en
t va
ria
ble
.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.4
.HS
.B.2
Su
mm
ari
ze,
rep
rese
nt,
an
d i
nte
rpre
t d
ata
on
tw
o
cate
go
rica
l a
nd
qu
an
tita
tive
va
ria
ble
s.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.4
.HS
.B.3
An
aly
ze l
ine
ar
mo
de
ls t
o m
ak
e i
nte
rpre
tati
on
s b
ase
d
on
th
e d
ata
.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.4
.HS
.B.4
Re
cog
niz
e a
nd
eva
lua
te r
an
do
m p
roce
sse
s
un
de
rly
ing
sta
tist
ica
l e
xpe
rim
en
ts.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.4
.HS
.B.5
Ma
ke
in
fere
nce
s a
nd
ju
stif
y c
on
clu
sio
ns
ba
sed
on
sam
ple
su
rve
ys,
exp
eri
me
nts
, a
nd
ob
serv
ati
on
al
stu
die
s.
Inte
nti
on
all
y B
lan
kIn
ten
tio
na
lly
Bla
nk
CC
.2.4
.HS
.B.6
Use
th
e c
on
cep
ts o
f in
de
pe
nd
en
ce a
nd
co
nd
itio
na
l
pro
ba
bil
ity
to
in
terp
ret
da
ta.
S.C
P.
1.
De
scri
be
eve
nts
as
sub
sets
of
a s
am
ple
sp
ace
(th
e s
et
of
ou
tco
me
s) u
sin
g
cha
ract
eri
stic
s (o
r ca
teg
ori
es)
of
the
ou
tco
me
s, o
r a
s u
nio
ns,
in
ters
ect
ion
s, o
r
com
ple
me
nts
of
oth
er
eve
nts
(“o
r,”
“an
d,”
“n
ot”
).
2.
Un
de
rsta
nd
th
at
two
eve
nts
A a
nd
B a
re i
nd
ep
en
de
nt
if t
he
pro
ba
bil
ity
of
A a
nd
B
occ
urr
ing
to
ge
the
r is
th
e p
rod
uct
of
the
ir p
rob
ab
ilit
ies,
an
d u
se t
his
ch
ara
cte
riza
tio
n t
o
de
term
ine
if
the
y a
re i
nd
ep
en
de
nt.
3.
Un
de
rsta
nd
th
e c
on
dit
ion
al
pro
ba
bil
ity
of
A g
ive
n B
as
P(A
an
d B
)/P
(B),
an
d i
nte
rpre
t
ind
ep
en
de
nce
of
A a
nd
B a
s sa
yin
g t
ha
t th
e c
on
dit
ion
al
pro
ba
bil
ity
of
A g
ive
n B
is
the
sam
e a
s th
e p
rob
ab
ilit
y o
f A
, a
nd
th
e c
on
dit
ion
al
pro
ba
bil
ity
of
B g
ive
n A
is
the
sa
me
as
the
pro
ba
bil
ity
of
B.
4.
Co
nst
ruct
an
d i
nte
rpre
t tw
o-w
ay
fre
qu
en
cy t
ab
les
of
da
ta w
he
n t
wo
ca
teg
ori
es
are
ass
oci
ate
d w
ith
ea
ch o
bje
ct b
ein
g c
lass
ifie
d.
Use
th
e t
wo
-wa
y t
ab
le a
s a
sa
mp
le s
pa
ce t
o
de
cid
e i
f e
ve
nts
are
in
de
pe
nd
en
t a
nd
to
ap
pro
xim
ate
co
nd
itio
na
l p
rob
ab
ilit
ies.
Fo
r
exa
mp
le,
coll
ect
da
ta f
rom
a r
an
do
m s
am
ple
of
stu
de
nts
in
yo
ur
sch
oo
l o
n t
he
ir f
avo
rite
sub
ject
am
on
g m
ath
, sc
ien
ce,
an
d E
ng
lish
. E
stim
ate
th
e p
rob
ab
ilit
y t
ha
t a
ra
nd
om
ly
sele
cte
d s
tud
en
t fr
om
yo
ur
sch
oo
l w
ill
favo
r sc
ien
ce g
ive
n t
ha
t th
e s
tud
en
t is
in
te
nth
gra
de
. D
o t
he
sa
me
fo
r o
the
r su
bje
cts
an
d c
om
pa
re t
he
re
sult
s.
5.
Re
cog
niz
e a
nd
exp
lain
th
e c
on
cep
ts o
f co
nd
itio
na
l p
rob
ab
ilit
y a
nd
in
de
pe
nd
en
ce i
n
eve
ryd
ay
la
ng
ua
ge
an
d e
ve
ryd
ay
sit
ua
tio
ns.
Fo
r e
xam
ple
, co
mp
are
th
e c
ha
nce
of
ha
vin
g
lun
g c
an
cer
if y
ou
are
a s
mo
ke
r w
ith
th
e c
ha
nce
of
be
ing
a s
mo
ke
r if
yo
u h
ave
lu
ng
can
cer.
Inte
nti
on
all
y B
lan
k
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.39
DR
AF
T P
A C
om
mo
n C
ore
- C
om
mo
n C
ore
- P
A A
cad
em
ic S
tan
da
rds
Cro
ssw
alk
fo
r H
igh
Sch
oo
l
DR
AF
T
PA
Co
mm
on
Co
re S
tan
da
rds
Co
mm
on
Co
re S
tate
Sta
nd
ard
sP
A A
cad
em
ic S
tan
da
rds
CC
.2.4
.HS
.B.7
Ap
ply
th
e r
ule
s o
f p
rob
ab
ilit
y t
o c
om
pu
te
pro
ba
bil
itie
s o
f co
mp
ou
nd
eve
nts
in
a u
nif
orm
pro
ba
bil
ity
mo
de
l.
S.C
P.
6.
Fin
d t
he
co
nd
itio
na
l p
rob
ab
ilit
y o
f A
giv
en
B a
s th
e f
ract
ion
of
B’s
ou
tco
me
s th
at
als
o
be
lon
g t
o A
, a
nd
in
terp
ret
the
an
swe
r in
te
rms
of
the
mo
de
l.
7.
Ap
ply
th
e A
dd
itio
n R
ule
, P
(A o
r B
) =
P(A
) +
P(B
) –
P(A
an
d B
), a
nd
in
terp
ret
the
an
swe
r
in t
erm
s o
f th
e m
od
el.
S.M
D.
6.
(+)
Use
pro
ba
bil
itie
s to
ma
ke
fa
ir d
eci
sio
ns
(e.g
., d
raw
ing
by
lo
ts,
usi
ng
a r
an
do
m
nu
mb
er
ge
ne
rato
r).
7.
(+)
An
aly
ze d
eci
sio
ns
an
d s
tra
teg
ies
usi
ng
pro
ba
bil
ity
co
nce
pts
(e
.g.,
pro
du
ct t
est
ing
,
me
dic
al
test
ing
, p
ull
ing
a h
ock
ey
go
ali
e a
t th
e e
nd
of
a g
am
e).
Inte
nti
on
all
y B
lan
k
5/2
0/2
01
3T
his
cro
ssw
alk
is
de
sig
ne
d t
o a
ssis
t e
du
cato
rs a
s th
ey
ali
gn
cu
rric
ulu
m.
Th
e a
lig
nm
en
t o
f P
AC
CS
an
d P
A A
cad
em
ic S
tan
da
rds
are
ba
sed
up
on
Ke
yst
on
e E
lig
ible
Co
nte
nt
an
d t
he
CC
SS
are
ba
sed
up
on
th
e M
ath
em
ati
cs A
pp
en
dix
A.40