precal math 2009 cover - home | san antonio isd€¦ · · 2009-08-18identify and sketch graphs...
TRANSCRIPT
Mathematics Precalculus
Curriculum and Instruction Professional Learning Conference
August 17 and 18, 2009 © San Antonio Independent School District
The San Antonio Independent School District does not discriminate on the basis of race, color, gender, religion, national origin, age, disability, or any other basis prohibited by law.
Literacy with an Attitude: Tools for the Thinking Classroom Teaching and Learning Professional Learning Conference San Antonio Independent School District, August 2009
AGENDA
August 17 and 18, 2009 Department: Mathematics Presenter: Mark Bell & Ed Gordon Grade Level: High School Contact Information: Mark Bell & Ed Gordon
Content NOTES
Norms
Participate actively o Ask questions o Take care of your neighbor o One person talks at a time
Take care of yourself. Electronic devices off or silent. It’s OK to have FUN!
Expectations
Preparation for Math Curriculum for 2009-2010 Academic Year Increase knowledge and encourage the use of best practices
Teaching and learning aligned with curriculum and assessments
Objectives & Overview
Speaker PM Aug 17: Frank Wang, “Beauty and Mathematics – Mathematician’s Search for Pattern and Order”
Curriculum Guide Update (Aug 17)
Writing in Math Class (Aug 17)
Vocabulary Tools (Aug 17 and Aug 18)
Cooperative Learning Structures (Aug 18)
Planning the First Week (Aug 18)
Q&A/Evaluation
Mat
hem
atic
s –
Pre
calc
ulu
s
Un
it o
f S
tud
y: F
un
ctio
ns
and
th
eir
Gra
ph
s
Fir
st G
rad
ing
Per
iod
: W
eeks
1-
4 (1
5 d
ays
)
CURR
ICULU
M O
VERV
IEW
En
du
rin
g U
nd
erst
and
ing
s (B
ig I
dea
s)
Un
it R
atio
nal
e
Mod
els
allo
w u
s to
test
or p
redi
ct re
al w
orld
phe
nom
ena.
Und
erst
andi
ng w
hen
and
how
to u
se fu
nctio
ns a
s m
odel
s of
real
wor
ld p
heno
men
a ar
e ke
ys to
the
valid
ity o
f pro
blem
sol
ving
and
pre
dict
ing
in to
day’
s w
orld
, esp
ecia
lly in
the
field
s of
sci
ence
, eng
inee
ring
and
econ
omic
s. A
chie
ving
this
end
requ
ires
a th
orou
gh
know
ledg
e of
the
char
acte
ristic
s of
func
tions
, whe
ther
they
are
sta
ted
verb
ally
, in
num
eric
al o
r tab
ular
form
, gra
phic
ally
or b
y eq
uatio
n or
form
ula.
Ess
enti
al Q
ues
tio
ns
Gu
idin
g Q
ues
tio
ns
W
hat d
egre
e of
pre
cisi
on a
nd a
ccur
acy
mus
t exi
st b
etw
een
a re
al w
orld
phe
nom
enon
an
d its
mod
el fo
r the
mod
el to
be
cons
ider
ed v
alid
?
Is a
gra
phin
g ca
lcul
ator
itse
lf a
mod
el?
W
hat a
ncho
rs a
ny c
oord
inat
e sy
stem
?
Wha
t mat
hem
atic
al re
quire
men
t of a
pro
blem
situ
atio
n m
ust e
xist
for a
line
ar
func
tion
to s
erve
as
a m
odel
?
Wha
t are
four
way
s to
exp
ress
a fu
nctio
nal r
elat
ions
hip
betw
een
two
varia
bles
?
Wha
t are
pos
sibl
e un
ique
poi
nts
for a
line
ar fu
nctio
n gr
aphe
d on
a C
arte
sian
co
ordi
nate
sys
tem
?
Wha
t are
the
poss
ible
diff
eren
ces
betw
een
a pa
rent
func
tion
and
a tra
nsfo
rmed
fu
nctio
n?
H
ow a
re th
e te
rms
slop
e an
d ra
te o
f cha
nge
rela
ted?
Why
do
we
stud
y co
mpo
sitio
ns o
f fun
ctio
ns?
W
hy d
o w
e st
udy
inve
rse
of fu
nctio
ns?
H
ow w
ere
mos
t of t
he fo
rmul
as in
sci
ence
dev
elop
ed?
W
hat i
s re
gres
sion
?
How
doe
s a
scat
terp
lot d
iffer
from
a g
raph
?
TE
KS
(S
tan
dar
ds)
T
EK
S S
pec
ific
ity
- In
ten
ded
Ou
tco
me
Concepts
P.1
The
stu
dent
def
ines
func
tions
, des
crib
es c
hara
cter
istic
s of
func
tions
, and
tra
nsla
tes
amon
g ve
rbal
, num
eric
al, g
raph
ical
, and
sym
bolic
repr
esen
tatio
ns o
f fu
nctio
ns, i
nclu
ding
pol
ynom
ial,
ratio
nal,
pow
er (i
nclu
ding
radi
cal),
exp
onen
tial,
loga
rithm
ic, t
rigon
omet
ric, a
nd p
iece
wis
e-de
fined
func
tions
. The
stu
dent
is e
xpec
ted
to: P
.1A
des
crib
e pa
rent
func
tions
sym
bolic
ally
and
gra
phic
ally
, inc
ludi
ng f(
x) =
xn ,
f(x) =
ln x
, f(x
) = lo
g a x
, f(x
) = 1
/x, f
(x) =
ex , f
(x) =
|x|,
f(x) =
ax ,
f(x) =
sin
x, f
(x) =
arc
sin
x, e
tc.;
P.1
B d
eter
min
e th
e do
mai
n an
d ra
nge
of fu
nctio
ns u
sing
gra
phs,
tabl
es, a
nd
sym
bols
;
P.1
C d
escr
ibe
sym
met
ry o
f gra
phs
of e
ven
and
odd
func
tions
; and
” I
CA
N”
stat
emen
ts h
igh
ligh
ted
in
yel
low
sh
ou
ld b
e d
isp
laye
d f
or
stu
den
ts.
I can
:
desc
ribe
pare
nt fu
nctio
ns b
y st
atin
g th
e eq
uatio
n an
d sk
etch
ing
the
grap
h (P
.1A)
dete
rmin
e th
e al
low
able
val
ues
of th
e in
depe
nden
t and
dep
ende
nt v
aria
bles
whe
ther
th
e fu
nctio
n is
pre
sent
ed g
raph
ical
ly, n
umer
ical
ly o
r sym
bolic
ally
(P.1
B)
re
cogn
ize
and
stat
e a
func
tion’
s pr
oper
ty o
f sym
met
ry, i
f any
(P.1
C)
st
ate
grap
hica
lly (c
oord
inat
es) o
r sym
bolic
alnc
tion
– ly
the
sign
ifica
nt v
alue
s of
a fu
zero
s, m
axim
a or
min
ima,
infle
ctio
n, e
tc. (
P.1
D)
tra
nsla
te, r
efle
ct o
r dila
te a
par
ent f
unct
ion
(P.2
A)
de
term
ine
com
posi
tions
and
inve
rses
of f
unct
ions
and
des
crib
e th
e re
sults
ver
bally
, sy
mbo
lical
ly, n
umer
ical
ly a
nd g
raph
ical
ly (P
.2A
& P.
2B)
SA
ISD
© 2
009-
10 –
Firs
t Gra
ding
Per
iod
M
athe
mat
ics
Pre
calc
ulus
Pag
e 1
of 1
2
Pow
er S
tand
ards
repr
esen
t the
ess
entia
l kno
wle
dge
and
skill
s st
uden
ts n
eed
for s
ucce
ss in
hig
h sc
hool
and
bey
ond.
Pow
er S
tand
ards
mus
t be
mas
tere
d to
suc
cess
fully
pas
s th
e re
quire
d as
sess
men
ts a
t eac
h gr
ade
leve
l. A
ll TA
KS
elig
ible
kno
wle
dge
and
skill
s ar
e id
entif
ied
as P
ower
Sta
ndar
ds.
SA
ISD
© 2
009-
10 –
Firs
t Gra
ding
Per
iod
M
athe
mat
ics
Pre
calc
ulus
Pag
e 2
of 1
2
Pow
er S
tand
ards
repr
esen
t the
ess
entia
l kno
wle
dge
and
skill
s st
uden
ts n
eed
for s
ucce
ss in
hig
h sc
hool
and
bey
ond.
Pow
er S
tand
ards
mus
t be
mas
tere
d to
suc
cess
fully
pas
s th
e re
quire
d as
sess
men
ts a
t eac
h gr
ade
leve
l. A
ll TA
KS
elig
ible
kno
wle
dge
and
skill
s ar
e id
entif
ied
as P
ower
Sta
ndar
ds.
Evi
den
ce o
f L
earn
ing
(S
um
mat
ive
Ass
essm
ent)
A
t lea
st 8
0% o
f the
tim
e, th
e st
uden
ts d
emon
stra
te o
rally
, in
writ
ing
or u
se m
odel
s to
sho
w th
ey c
an:
de
scrib
e pa
rent
func
tions
by
stat
ing
the
equa
tion
and
sket
chin
g th
e gr
aph
de
term
ine
the
allo
wab
le v
alue
s of
the
inde
pend
ent a
nd d
epen
dent
var
iabl
es w
heth
er th
e fu
nctio
n is
pre
sent
ed g
raph
ical
ly, n
umer
ical
ly o
r sym
bolic
ally
reco
gniz
e an
d st
ate
a fu
nctio
n’s
prop
erty
of s
ymm
etry
, if a
ny
st
ate
grap
hica
lly (c
oord
inat
es) o
r sym
bolic
ally
the
sign
ifica
nt v
alue
s of
a fu
nctio
n –
zero
s, m
axim
a or
min
ima,
infle
ctio
n, e
tc.
tra
nsla
te, r
efle
ct o
r dila
te a
par
ent f
unct
ion
det
erm
ine
com
posi
tions
and
inve
rses
of f
unct
ions
and
des
crib
e th
e re
sults
ver
bally
, sym
bolic
ally
, num
eric
ally
and
gra
phic
ally
P.1
D re
cogn
ize
and
use
conn
ectio
ns a
mon
g si
gnifi
cant
val
ues
of a
func
tion
(zer
os, m
axim
um v
alue
s, m
inim
um v
alue
s, e
tc.),
poi
nts
on th
e gr
aph
of a
func
tion,
an
d th
e sy
mbo
lic re
pres
enta
tion
of a
func
tion.
P
.2 T
he s
tude
nt in
terp
rets
the
mea
ning
of t
he s
ymbo
lic re
pres
enta
tion
of fu
nctio
ns
and
oper
atio
ns o
n fu
nctio
ns to
sol
ve m
eani
ngfu
l pro
blem
s. T
he s
tude
nt is
exp
ecte
d to
: P.2
A a
pply
bas
ic tr
ansf
orm
atio
ns, i
nclu
ding
a •
f(x),
f(x
) + d
, f(x
– c
), f(b
• x)
, and
com
posi
tions
with
abs
olut
e va
lue
func
tions
, inc
ludi
ng
|f(x)
|and
f(|x
|), to
the
pare
nt fu
nctio
ns; a
nd
P.2
B p
erfo
rm o
pera
tions
incl
udin
g co
mpo
sitio
ns o
n fu
nctio
ns, f
ind
inve
rses
, and
de
scrib
e th
ese
proc
edur
es a
nd re
sults
ver
bally
, num
eric
ally
, sym
bolic
ally
, and
gr
aphi
cally
.
CURR
ICULU
M G
UID
E E
ssen
tial
Pre
-req
uis
ite
Ski
lls
Alge
bra
2
iden
tify
the
mat
hem
atic
al d
omai
ns a
nd ra
nges
of f
unct
ions
and
det
erm
ine
reas
onab
le d
omai
n an
d ra
nge
valu
es fo
r con
tinuo
us a
nd d
iscr
ete
situ
atio
ns (2
A.1
A)
id
entif
y an
d sk
etch
gra
phs
of p
aren
t fun
ctio
ns, i
nclu
ding
line
ar (f
(x) =
x),
quad
ratic
(f(x
) = x
2 ),
expo
nent
ial (
f(x) =
ax
), an
d lo
garit
hmic
(f(x
) = lo
ga x
) fun
ctio
ns, a
bsol
ute
valu
e of
x
(f(x)
= I
x I )
, squ
are
root
of x
(f(x
) = √
x),
and
reci
proc
al o
f x (f
(x) =
1/x
) (2
A.4
A)
ex
tend
par
ent f
unct
ions
with
par
amet
ers
such
as
a in
f(x)
= a
/x a
nd d
escr
ibe
the
effe
cts
of p
aram
eter
cha
nges
on
the
grap
h of
par
ent f
unct
ions
(2A
.4B
)
desc
ribe
and
anal
yze
the
rela
tions
hip
betw
een
a fu
nctio
n an
d its
inve
rse
(2A
.4C
)
use
the
pare
nt fu
nctio
n to
inve
stig
ate,
des
crib
e, a
nd p
redi
ct th
e ef
fect
s of
cha
nges
in a
, h, a
nd k
on
the
grap
hs o
f y =
a(x
– h
)2 +
k fr
om o
f a fu
nctio
n in
app
lied
and
pure
ly
mat
hem
atic
al s
ituat
ions
(2A
.7B
)
use
func
tions
to m
odel
and
mak
e pr
edic
tions
in p
robl
em s
ituat
ions
invo
lvin
g di
rect
and
inve
rse
varia
tion
(2A
.10G
) G
eom
etry
se
lect
an
appr
opria
te re
pres
enta
tion
(con
cret
e, p
icto
rial,
grap
hica
l, ve
rbal
, or s
ymbo
lic) i
n or
der t
o so
lve
prob
lem
s (G
.4)
Th
e T
each
ing
Pla
n
Inst
ruct
ion
al M
od
el &
Tea
cher
Dir
ecti
on
s T
he
teac
her
will
…
So
stu
den
ts c
an d
emo
nst
rate
co
mp
eten
cy,
th
e st
ud
ents
will
…
9 W
eeks
Pro
ject
: T
his
proj
ect i
s to
be
assi
gned
on
Day
1 w
ith th
e st
uden
t cho
ices
sub
mitt
ed to
the
teac
her b
y th
e en
d of
the
4th w
eek
and
the
final
pro
duct
due
at t
he e
nd o
f the
8th
wee
k. S
ee th
e lin
k be
low
for t
able
of s
tude
nt c
hoic
es a
nd g
radi
ng ru
bric
. (G
rad
able
Act
ivit
y)
RA
FT
: S
tude
nts
are
prov
ided
cho
ices
for t
he w
ays
in w
hich
they
sho
w th
at th
ey k
now
and
can
do
thei
r ass
essm
ent t
ask.
The
tabl
e is
des
igne
d as
follo
ws.
R
ole
A
ud
ien
ce
Fo
rmat
T
op
ic
Mat
h St
uden
t Pr
inci
pal
Sale
s Br
ochu
re
Com
plex
Num
bers
Mat
h Te
ache
r (e
xper
ienc
ed)
Col
lege
Pro
fess
or
Out
line
of a
less
on p
lan
Com
plet
ing
the
Squa
re
Mat
h Te
xtbo
ok A
utho
r G
raph
ing
Cal
cula
tor
Sale
sper
son
Car
toon
– a
t lea
st th
ree
pane
ls.
Dire
ct V
aria
tion
Par
ent
Sub
stitu
te T
each
er
Lette
r Tr
ansl
atio
n ac
ross
th
e y-
axis
on
the
grap
hing
cal
cula
tor
Stu
dent
s ca
n ra
ndom
ly c
hoos
e th
e R
ole,
Aud
ienc
e, F
orm
at a
nd T
opic
. Th
ey a
re to
com
plet
e a
writ
ing
task
bas
ed o
n th
e ch
oice
s th
ey h
ave
mad
e. S
ince
thes
e ar
e th
e to
pics
stu
died
th
roug
hout
the
9 w
eeks
, thi
s pr
ojec
t is
due
near
the
end
of th
e ni
ne w
eeks
. U
se th
e fo
llow
ing
to li
nks
to a
ssig
n an
d sc
ore:
R
ubric
. A
stu
dent
han
dout
for t
his
assi
gnm
ent i
s th
e la
st
page
of t
his
curri
culu
m g
uide
. D
ay
1 –
Exp
ecta
tio
ns
and
Pro
ced
ure
s E
ng
age/
Exp
lore
Hav
e th
e st
uden
ts d
raw
a la
rge
cloc
k on
a s
heet
of b
lank
pap
er
H
ave
them
dra
w o
n th
is c
lock
the
num
bers
for t
he 1
2 ho
urs.
By
mix
ing
with
the
rest
of t
he c
lass
, hav
e th
em s
et u
p “a
ppoi
ntm
ents
” with
six
oth
er
clas
smat
es fo
r the
eve
n-nu
mbe
red
hour
s. E
ach
pairi
ng s
houl
d w
rite
thei
r nam
es o
n th
e sa
me
hour
of t
he p
artn
er’s
clo
ck.
A
llow
eac
h pa
iring
onl
y tw
o m
inut
es to
reco
rd e
ach
othe
r’s n
ames
at t
he s
ame
hour
and
br
iefly
dis
cuss
pre
viou
s m
ath
cour
ses
take
n in
hig
h sc
hool
.
Whe
n fin
ishe
d, e
ach
stud
ent s
houl
d ha
ve s
ix d
iffer
ent s
tude
nt n
ames
for
th
e ev
en-n
umbe
red
hour
s.
Da
y 1
– E
xpec
tati
on
s an
d P
roce
du
res
En
gag
e/E
xplo
re
D
raw
a la
rge
cloc
k on
a s
heet
of b
lank
pap
er
D
raw
on
this
clo
ck th
e nu
mbe
rs fo
r the
12
hour
s.
B
y m
ixin
g w
ith th
e re
st o
f the
cla
ss, s
et u
p “a
ppoi
ntm
ents
” with
six
oth
er
clas
smat
es fo
r the
eve
n-nu
mbe
red
hour
s. E
ach
pairi
ng s
houl
d w
rite
thei
r nam
es o
n th
e sa
me
hour
of t
he p
artn
er’s
clo
ck.
A
llow
eac
h pa
iring
a m
axim
um o
f tw
o m
inut
es to
reco
rd e
ach
othe
r’s n
ames
at t
he
sam
e ho
ur p
ositi
on a
nd b
riefly
dis
cuss
pre
viou
s m
ath
cour
ses
take
n in
hig
h sc
hool
.
Whe
n fin
ishe
d, e
ach
stud
ent s
houl
d ha
ve s
ix d
iffer
ent s
tude
nt n
ames
for t
he
even
-num
bere
d ho
urs.
SA
ISD
© 2
009-
10 –
Firs
t Gra
ding
Per
iod
M
athe
mat
ics
Pre
calc
ulus
Pag
e 3
of 1
2
Pow
er S
tand
ards
repr
esen
t the
ess
entia
l kno
wle
dge
and
skill
s st
uden
ts n
eed
for s
ucce
ss in
hig
h sc
hool
and
bey
ond.
Pow
er S
tand
ards
mus
t be
mas
tere
d to
suc
cess
fully
pas
s th
e re
quire
d as
sess
men
ts a
t eac
h gr
ade
leve
l. A
ll TA
KS
elig
ible
kno
wle
dge
and
skill
s ar
e id
entif
ied
as P
ower
Sta
ndar
ds.
H
ave
the
stud
ents
sto
re th
is c
lock
in a
saf
e pl
ace
sinc
e it
will
be
used
repe
ated
ly
thro
ugho
ut th
e 9-
wee
k gr
adin
g pe
riod
for p
air d
esig
natio
ns.
Exp
lain
Brie
f the
stu
dent
on
your
exp
ecta
tions
and
dai
ly p
roce
dure
s to
incl
ude
at a
min
imum
: 1.
9-
wee
k pr
ojec
t 2.
TA
KS re
adin
ess
3.
Jour
nals
4.
M
ajor
new
topi
cs to
be
seen
for t
he fi
rst t
ime
H
ave
stud
ents
jour
nal t
heir
cour
se e
xpec
tatio
ns
S
tore
this
clo
ck in
a s
afe
plac
e si
nce
it w
ill b
e us
ed re
peat
edly
thro
ugho
ut th
e 9-
wee
k gr
adin
g pe
riod
for p
air d
esig
natio
ns.
Exp
lain
Take
not
es o
n:
5.
9-w
eek
proj
ect
6.
TAKS
read
ines
s 7.
Jo
urna
ls
8.
Maj
or n
ew to
pics
to b
e se
en fo
r the
firs
t tim
e
Jour
nal y
our c
ours
e ex
pect
atio
ns
Day
2 -
Sec
tio
n 1
.1 R
ecta
ng
ula
r C
oo
rdin
ates
E
ng
age/
Exp
lore
Usi
ng a
glo
be o
r a w
orld
map
, ask
stu
dent
s ho
w th
ey w
ould
tell
som
eone
thei
r spe
cific
sh
ip’s
loca
tion
if th
ey w
ere
in th
e m
iddl
e of
the
Pac
ific
Oce
an.
You
are
tryi
ng to
elic
it th
e ne
ed fo
r a re
fere
nce
or c
oord
inat
e sy
stem
, of w
hich
latit
ude/
long
itude
is o
ne e
xam
ple.
H
ow a
bout
in o
uter
spa
ce?
C
onsi
der r
elat
ing
how
Des
carte
s fo
rmul
ated
the
conc
ept o
f a c
oord
inat
e sy
stem
.
E
xpla
in
Pr
ogre
ssin
g fro
m e
xam
ple
1 th
roug
h 9
in s
ectio
n 1.
1(at
a m
inim
um, e
xam
ples
2, 4
, 6, 8
an
d 9)
stre
ss th
e im
porta
nce
of d
efin
ition
s an
d fo
rmul
as in
side
the
blue
box
es.
D
o E
xam
ple
2 on
pag
e 3
with
a g
raph
ing
calc
ulat
or (s
catte
rplo
t)
Ens
ure
that
the
stud
ents
real
ize
how
the
Pyt
hago
rean
The
orem
and
the
Dis
tanc
e Fo
rmul
a ar
e re
late
d by
der
ivin
g th
e la
tter f
rom
the
form
er.
Ela
bo
rate
H
ave
stud
ents
com
plet
e th
e “W
ritin
g ab
out M
athe
mat
ics”
on
page
8
(tran
sfor
mat
ions
).
E
valu
ate
O
bser
ve e
ach
stud
ent’s
com
plet
ion
of th
e sc
atte
rplo
t pro
cess
in E
xam
ple
2 on
pag
e 3.
Diff
eren
tiate
:
Stru
gglin
g le
arne
r – d
ecid
e be
twee
n m
anua
l plo
tting
or t
he u
se o
f a
grap
hing
cal
cula
tor
for p
lotti
ng a
nd s
tudy
onl
y th
e “c
ritic
al” e
xam
ples
.
On
leve
l lea
rner
– d
urin
g th
e ta
sk a
ssig
ned,
ask
que
stio
ns o
f par
tner
or t
each
er w
hen
a ne
ed fo
r cla
rific
atio
n ar
ises
Adv
ance
d le
arne
r – c
ompl
ete
thos
e ex
erci
ses
in th
e te
xt w
ith th
e te
chno
logy
icon
for a
ch
alle
nge
Day
2 -
Sec
tio
n 1
.1 R
ecta
ng
ula
r C
oo
rdin
ates
E
ng
age
an
swer
how
you
wou
ld te
ll so
meo
ne y
our l
ocat
ion
if yo
u w
ere
in th
e m
iddl
e of
the
ocea
n or
in o
uter
spa
ce
Exp
lore
on a
pie
ce o
f gra
ph p
aper
with
axe
s dr
awn,
labe
l all
the
parts
of a
gra
ph
ex
plai
n w
hy y
ou th
ink
that
the
text
book
aut
hors
use
d di
ffere
nt c
olor
s on
pag
es 2
an
d 4
Exp
lain
take
not
es a
s th
e in
stru
ctor
exp
lain
s so
me
of th
e ex
ampl
es
1.
Ex
1 –
plot
ting
poin
ts
2.
Ex
2 –
cons
truct
ing
a sc
atte
r plo
t (us
e gr
aphi
ng c
alcu
lato
r) 3.
E
x 3
– us
ing
the
dist
ance
form
ula
(P.1
D)
4.
Ex 4
– d
ista
nce
form
ula
appl
icat
ion
– is
it a
righ
t tria
ngle
? (P
.1D
) 5.
Ex
5 –
find
ing
the
mid
poin
t of a
line
seg
men
t (P
.1D
) (P
.1D
) 6.
Ex
6 –
dis
tanc
e fo
rmul
a ap
plic
atio
n –
how
long
was
the
pass
? (P
.1D
) 7.
E
x 7
– m
idpo
int a
pplic
atio
n –
estim
ate
reve
nue
(P.1
D)
8.
Ex
8 –
trans
latio
n of
poi
nts
on a
gra
ph (P
.2A
)
Der
ive
the
dist
ance
form
ula
from
the
Pyt
hago
rean
The
orem
E
lab
ora
te
C
ompl
ete
the
writ
ing
assi
gnm
ent t
o ex
plai
n th
e th
ree
give
n tra
nsla
tions
(P.2
A)
Day
3 -
Sec
tio
n 1
.2 G
rap
hs
of
Eq
uat
ion
s E
ng
age
A
sk th
e st
uden
ts h
ow a
sca
tter p
lot d
iffer
s fro
m a
gra
ph.
Day
3 -
Sec
tio
n 1
.2 G
rap
hs
of
Eq
uat
ion
s E
ng
age
an
swer
how
a s
catte
rplo
t diff
ers
from
a g
raph
SA
ISD
© 2
009-
10 –
Firs
t Gra
ding
Per
iod
M
athe
mat
ics
Pre
calc
ulus
Pag
e 4
of 1
2
Pow
er S
tand
ards
repr
esen
t the
ess
entia
l kno
wle
dge
and
skill
s st
uden
ts n
eed
for s
ucce
ss in
hig
h sc
hool
and
bey
ond.
Pow
er S
tand
ards
mus
t be
mas
tere
d to
suc
cess
fully
pas
s th
e re
quire
d as
sess
men
ts a
t eac
h gr
ade
leve
l. A
ll TA
KS
elig
ible
kno
wle
dge
and
skill
s ar
e id
entif
ied
as P
ower
Sta
ndar
ds.
Exp
lore
Con
side
r hav
ing
pairs
of s
tude
nts
(2 o
’clo
ck a
ppoi
ntm
ent)
perfo
rm E
xam
ple
2 or
Exa
mpl
e 3
(ske
tchi
ng a
gra
ph g
iven
the
equa
tion)
on
page
s 15
and
16,
one
on
grap
h pa
per a
nd th
e ot
her o
n th
e gr
aphi
ng c
alcu
lato
r. H
ave
them
dis
cuss
the
proc
edur
es th
at th
ey fo
llow
ed
and
conn
ect w
hich
man
ual s
teps
are
equ
ival
ent t
o th
e gr
aphi
ng c
alcu
lato
r ste
ps.
Exp
lain
Pro
ceed
thro
ugh
som
e of
the
exam
ples
on
the
boar
d or
ove
rhea
d
H
ave
the
stud
ents
jour
nal t
he s
teps
to g
raph
on
page
15
for e
mph
asis
S
tress
the
impo
rtanc
e of
kno
win
g th
e gr
aphi
ng c
alcu
lato
r pro
cedu
res
cold
whi
le
disc
ussi
ng th
e S
tudy
Tip
on
page
21
(alw
ays
know
at l
east
two
solu
tions
met
hods
for e
ach
type
of p
robl
em)
Ela
bo
rate
Hav
e th
e st
uden
ts re
ad th
e st
udy
tips
alou
d an
d di
scus
s w
ith th
eir 2
o’c
lock
app
oint
men
t pa
rtner
. E
valu
ate
Fo
r stu
dent
s w
orki
ng p
robl
ems
on th
e bo
ard,
ove
rhea
d or
cha
rt pa
per,
mix
it u
p by
so
met
imes
ask
ing
for v
olun
teer
s an
d ot
her t
imes
by
assi
gnin
g st
uden
ts b
y na
me.
Exp
lore
with
a p
artn
er, o
ne p
lots
the
grap
h fro
m E
x 2
or E
x 3
(ske
tchi
ng a
gra
ph g
iven
the
equa
tion)
man
ually
on
grap
h pa
per a
nd th
e ot
her g
raph
s on
the
grap
hing
ca
lcul
ator
(gc)
will
dis
cuss
as
a pa
ir ho
w e
ach
of th
eir p
roce
ss s
teps
cor
resp
onds
– m
anua
l vs
gc
Exp
lain
mak
e a
jour
nal e
ntry
– S
ketc
hing
Gra
phs
on p
age
14
ta
ke n
otes
on
teac
her’s
pre
sent
atio
n 1.
E
x 4
- fin
d th
e x-
and
y-in
terc
epts
of g
raph
s of
equ
atio
ns (T
EK
S
P1D
) 2.
E
x 5/
6 - u
se s
ymm
etry
to s
ketc
h gr
aphs
of e
quat
ions
(TE
KS
P
1C)
3.
Ex
7 –
grap
hing
abs
olut
e va
lue
func
tion
(P.2
A)
4.
Ex
8 –
equa
tion
of a
circ
le (P
.1D
) E
lab
ora
te
R
ead
stud
y tip
s al
oud
and
disc
uss
with
par
tner
Da
y 4
– S
ect
ion
1.3
Lin
ear
Eq
uat
ion
s in
Tw
o V
aria
ble
s E
ng
age/
Exp
lore
Dra
w th
ree
lines
with
two
bein
g pa
ralle
l and
the
third
not
par
alle
l. A
sk s
tude
nt p
airs
(4
o’cl
ock
appo
intm
ent)
to c
ompa
re a
nd c
ontra
st th
e 3
lines
. Y
ou a
re e
liciti
ng a
t a m
inim
um
stud
ent r
ecog
nitio
n of
line
seg
men
ts, p
aral
lel l
ines
, lin
es n
ot p
aral
lel a
nd th
e co
ncep
t of
slop
e. W
heth
er y
ou d
raw
the
lines
in fr
ee s
pace
or o
n a
coor
dina
te g
rid w
ill h
ave
a si
gnifi
cant
impa
ct o
n th
e ric
hnes
s of
the
disc
ussi
ons.
E
xpla
in
P
roce
edin
g th
roug
h S
ectio
n 1.
3, e
nsur
e th
at th
e st
uden
ts u
nder
stan
d w
hy a
ver
tical
line
’s
slop
e is
des
igna
ted
as u
ndef
ined
, th
e di
ffere
nce
betw
een
the
slop
e-in
terc
ept f
orm
and
the
poin
t-slo
pe
form
of t
he e
quat
ion
of a
line
, and
w
hy th
e St
udy
Tip
on p
age
29 is
true
.
Ela
bo
rate
Ask
the
stud
ents
the
impa
ct o
n E
xam
ple
6 if
we
wer
e as
ked
to fi
nd th
e “p
rofit
” the
co
mpa
ny m
akes
rath
er th
an th
e “c
ost.”
Wha
t els
e w
ould
we
need
to k
now
?
E
valu
ate
V
erba
l qui
zzin
g of
gro
ups
on s
lope
det
erm
inat
ion,
equ
atio
n bu
ildin
g an
d ra
te o
f cha
nge
(ver
sus
slop
e) d
urin
g cl
ass.
Da
y 4
– S
ect
ion
1.3
Lin
ear
Eq
uat
ion
s in
Tw
o V
aria
ble
s E
ng
age/
Exp
lore
with
par
tner
ana
lyze
3 li
ne g
roup
ing
give
n by
teac
her (
P.1D
) E
xpla
in
ta
ke n
otes
or f
ollo
w in
text
1.
sl
ope-
inte
rcep
t for
m –
Ex
1 (P
.1D
) 2.
sl
ope
give
n 2
poin
ts –
Ex
2 (P
.1D
) 3.
po
int-s
lope
form
– E
x 3
(P.1
D)
4.
para
llel/p
erpe
ndic
ular
– E
x 4
(P.1
A)
5.
slop
e as
a ra
tio –
Ex
5 (P
.1D
) 6.
sl
ope
as ra
te o
f cha
nge
– E
x 6
& 7
& 8
(P.1
D)
Ela
bo
rate
addr
ess
the
chan
ge to
Ex
6 m
ade
by th
e te
ache
r (P
.1D
)
Eva
luat
e
resp
ond
to v
erba
l que
stio
ns fr
om in
stru
ctor
nea
r end
of c
lass
SA
ISD
© 2
009-
10 –
Firs
t Gra
ding
Per
iod
M
athe
mat
ics
Pre
calc
ulus
Pag
e 5
of 1
2
Pow
er S
tand
ards
repr
esen
t the
ess
entia
l kno
wle
dge
and
skill
s st
uden
ts n
eed
for s
ucce
ss in
hig
h sc
hool
and
bey
ond.
Pow
er S
tand
ards
mus
t be
mas
tere
d to
suc
cess
fully
pas
s th
e re
quire
d as
sess
men
ts a
t eac
h gr
ade
leve
l. A
ll TA
KS
elig
ible
kno
wle
dge
and
skill
s ar
e id
entif
ied
as P
ower
Sta
ndar
ds.
Da
y 5
– Q
uiz
ove
r S
ecti
on
s 1
.1 –
1.3
E
xpla
in
R
evie
w a
ny h
omew
ork
or g
ener
al q
uest
ions
from
stu
dent
s E
valu
ate
A
dmin
iste
r qui
z en
surin
g al
ignm
ent w
ith g
uidi
ng T
EK
S
Da
y 5
– Q
uiz
ove
r S
ecti
on
s 1
.1 –
1.3
E
xpla
in
G
et a
ny li
nger
ing
conc
erns
or q
uest
ions
ans
wer
ed
Eva
luat
e
Take
qui
z D
ay
6 -
Sec
tio
n 1
.4 F
un
ctio
ns
En
gag
e/E
xplo
re
C
onsi
der a
n ai
rcra
ft ca
rrier
. S
ituat
ion
1: o
ne g
ood
jet i
s on
cat
apul
t #1,
the
cata
pult
fires
an
d on
e go
od je
t goe
s fly
ing
– go
od fu
nctio
ning
! S
ituat
ion
2: o
ne g
ood
jet i
s on
cat
apul
t #2
, the
cat
apul
t fire
s an
d on
e go
od je
t doe
s no
t go
flyin
g –
mal
func
tioni
ng.
Her
e w
e ha
ve
the
sam
e in
put w
ith d
iffer
ent o
utpu
ts fr
om th
e sa
me
proc
ess.
Hav
e st
uden
ts d
iscu
ss in
a
mat
h co
ntex
t. E
xpla
in –
Pro
ceed
ing
thro
ugh
sect
ion
1.4,
Exa
mpl
es 1
-9 (a
t a m
inim
um, 2
b, 3
c, 4
, 5d,
7):
The
Four
Way
s to
repr
esen
t a F
unct
ion
box
on p
age
41 c
anno
t be
emph
asiz
ed
enou
gh!!!
H
ave
stud
ents
reco
rd in
jour
nal.
A
thor
ough
und
erst
andi
ng o
f fun
ctio
n no
tatio
n (p
ages
42/
43) p
recl
udes
a lo
t of s
tude
nt
pain
in fu
ture
mat
h co
urse
s, a
s w
ell a
s th
is c
ours
e!
N
otic
e th
at E
xam
ples
3 a
nd 6
are
pre
view
s of
com
posi
tions
of f
unct
ions
.
Not
ice
that
Exa
mpl
e 9
fore
shad
ows
deriv
ativ
es.
R
ead
with
stu
dent
s th
e Fu
nctio
n Te
rmin
olog
y bo
x on
pag
e 47
.
Ela
bo
rate
Ass
ign
the
stud
ents
the
ques
tion
in th
e Te
chno
logy
box
on
page
44
rela
ted
to d
omai
n an
d ra
nge
of a
radi
cal.
E
valu
ate
A
ssig
n ra
ndom
Exa
mpl
es o
r Che
ckpo
ints
for s
tude
nts
to w
ork
on th
e bo
ard
or o
verh
ead
chec
k as
sign
ed h
omew
ork
Da
y 6
- S
ecti
on
1.4
Fu
nct
ion
s E
ng
age/
Exp
lore
answ
er fu
nctio
n re
late
d qu
estio
n in
volv
ing
cata
pult
shot
s of
f an
airc
raft
carri
er (P
,1)
Exp
lain
take
not
es a
s te
ache
r pro
ceed
s th
roug
h ex
ampl
es
1.
Ex
1 –
func
tion
test
– v
erba
l, nu
mer
ical
, gra
phic
al (P
.1A
) 2.
E
x 2
– fu
nctio
n te
st –
alg
ebra
ic (P
.1A
) 3.
E
x 3
– fu
nctio
n ev
alua
tion
(P.1
D, P
.2B
) 4.
Ex
4 –
pie
cew
ise
func
tion
eval
uatio
n (P
.1D
) 5.
E
x 5
– do
mai
n de
term
inat
ion
from
alg
ebra
ic re
pres
enta
tion
(P.1
B)
6.
Ex
6 –
volu
me
of c
ylin
der a
s f(r
) the
n f(h
) (P
.2B
) 7.
E
x 7
– ba
llist
ic a
pplic
atio
n (q
uadr
atic
) (P
.1D
) 8.
Ex
8 –
mak
ing
pred
ictio
n fro
m p
iece
wis
e fu
nctio
n (P
.1D
) 9.
E
x 9
– ev
alua
ting
a di
ffere
nce
quot
ient
(P.2
A)
10.
Func
tion
Term
inol
ogy
box
on p
age
49.
Ela
bo
rate
answ
er q
uest
ions
in T
echn
olog
y bo
x on
pag
e 45
E
valu
ate
pr
oble
ms
as a
ssig
ned
in c
lass
or f
or h
omew
ork
Day
7
– S
ecti
on
1.5
An
alyz
ing
Gra
ph
s o
f F
un
ctio
ns
En
gag
e/E
xplo
re
H
ave
stud
ent p
airs
(6 o
’clo
ck a
ppoi
ntm
ent)
read
the
intro
box
on
page
54
(Wha
t & W
hy to
Le
arn)
and
dis
cuss
whe
ther
any
topi
c se
ems
like
it co
uld
be c
halle
ngin
g E
xpla
in
P
roce
ed th
roug
h se
lect
ed E
xam
ples
put
ting
parti
cula
r em
phas
is o
n th
e fo
llow
ing:
1.
Ex
1 –
inte
rval
not
atio
n 2.
E
x 2
– pa
rt c,
a d
isco
ntin
uous
func
tion
3.
Ex
4 - i
ncre
asin
g an
d de
crea
sing
sig
nifie
s w
hat t
he fu
nctio
n (o
r y) i
s do
ing
as x
incr
ease
s –
this
is s
omet
imes
con
fusi
ng to
stu
dent
s 4.
Ex
6 o
r 7 –
get
ting
stud
ents
read
y fo
r cal
culu
s, a
vera
ge ra
te o
f cha
nge
Ela
bo
rate
A
ssig
n st
uden
ts in
pai
rs (6
o’c
lock
app
oint
men
t) th
e E
xplo
ratio
n bo
x (e
ven,
odd
or
neith
er fu
nctio
ns) o
n pa
ge 6
0 E
valu
ate
A
ssig
n ho
mew
ork
P
erio
dic
quer
ying
of s
tude
nt p
airs
or o
bser
vatio
n
Day
7
– S
ecti
on
1.5
An
alyz
ing
Gra
ph
s o
f F
un
ctio
ns
En
gag
e/E
xplo
re
re
ad a
nd d
iscu
ss w
ith p
artn
er to
pics
to b
e le
arne
d in
toda
y’s
less
on a
nd d
eter
min
e w
heth
er a
ny m
ight
app
ear t
o be
cha
lleng
ing
Exp
lain
take
not
es o
n th
e te
ache
r pre
sent
atio
n of
exa
mpl
es fr
om g
raph
ical
repr
esen
tatio
ns
of fu
nctio
ns 1.
Ex 1
– d
omai
n an
d ra
nge
(P.1
B)
2.
Ex
2 –
verti
cal l
ine
test
for f
unct
ion
(P.1
A)
3.
Ex
3 –
findi
ng z
eros
(P.1
D)
4.
Ex 4
– in
crea
sing
and
dec
reas
ing
func
tions
(P.1
D)
5.
Ex
5 –
rela
tive
min
ima
(P.1
D)
6.
Ex
6 –
aver
age
rate
of c
hang
e of
a fu
nctio
n (P
.1A
) 7.
E
x 7
– av
erag
e ra
te o
f cha
nge
of s
peed
(P.1
A)
8.
Ex
8 –
even
and
odd
func
tions
(P.1
C, P
.1D
) E
lab
ora
te/E
valu
ate
an
swer
que
stio
ns in
Exp
lora
tion
box
on p
age
64 (P
.1D
)
com
plet
e ho
mew
ork
assi
gnm
ent
SA
ISD
© 2
009-
10 –
Firs
t Gra
ding
Per
iod
M
athe
mat
ics
Pre
calc
ulus
Pag
e 6
of 1
2
Pow
er S
tand
ards
repr
esen
t the
ess
entia
l kno
wle
dge
and
skill
s st
uden
ts n
eed
for s
ucce
ss in
hig
h sc
hool
and
bey
ond.
Pow
er S
tand
ards
mus
t be
mas
tere
d to
suc
cess
fully
pas
s th
e re
quire
d as
sess
men
ts a
t eac
h gr
ade
leve
l. A
ll TA
KS
elig
ible
kno
wle
dge
and
skill
s ar
e id
entif
ied
as P
ower
Sta
ndar
ds.
Diff
eren
tiate
:
Stru
gglin
g le
arne
r – u
se p
artn
er a
nd te
ache
r as
reso
urce
s w
hen
chal
leng
es o
ccur
; giv
e al
l of
ass
igne
d pr
oble
ms
a try
and
ask
for a
ssis
tanc
e w
hen
clar
ifica
tion
is n
eede
d
On
leve
l lea
rner
– p
roce
ed w
ith a
ssig
ned
wor
k an
d us
e pa
rtner
and
teac
her f
or a
ssis
tanc
e
Ad
vanc
ed le
arne
r – s
ort t
hrou
gh th
e pr
oble
ms
with
the
tech
nolo
gy ic
on o
n pa
ges
62-6
5 fo
r a
chal
leng
e: a
ltern
ativ
ely,
try
the
othe
r Syn
thes
is p
robl
ems
on p
age
65.
Day
8
– S
ecti
on
1.6
A L
ibra
ry o
f P
aren
t F
un
ctio
ns
En
gag
e/E
xplo
re
A
sk th
e cl
ass
wha
t a p
aren
t fun
ctio
n is
and
whi
ch o
nes
do th
ey a
lread
y kn
ow –
list
on
char
t pap
er a
s th
ey a
nsw
er.
Exp
lain
For e
ach
pare
nt fu
nctio
n, d
iscu
ss th
e gr
aph,
dom
ain
and
rang
e, a
nd “s
peci
al” p
oint
s
1.
linea
r par
ent f
unct
ion
and
Ex
1 2.
qu
adra
tic p
aren
t fun
ctio
n 3.
cu
bic
pare
nt fu
nctio
n 4.
sq
uare
root
par
ent f
unct
ion
5.
reci
proc
al p
aren
t fun
ctio
n 6.
st
ep a
nd p
iece
wis
e fu
nctio
ns a
nd E
x 2
Ela
bo
rate
Hav
e st
uden
ts p
ut p
aren
t fun
ctio
n eq
uatio
ns a
nd g
raph
s fro
m p
age
70 in
jour
nal
Eva
luat
e
Obs
erve
stu
dent
s du
ring
guid
ed p
ract
ice
A
ssig
n ho
mew
ork
Day
8
– S
ecti
on
1.6
A L
ibra
ry o
f P
aren
t F
un
ctio
ns
En
gag
e/E
xplo
re
lis
t cur
rent
kno
wle
dge
on p
aren
t fun
ctio
ns (P
.1A
) E
xpla
in
ta
ke n
otes
on
teac
her p
rese
ntat
ion
pa
rent
func
tions
– li
near
, qua
drat
ic, c
ubic
, abs
olut
e va
lue,
squ
are
root
, rec
ipro
cal
and
step
func
tions
(P.1
A, P
.1B
) E
lab
ora
te
en
ter i
nto
jour
nal t
he g
raph
s an
d do
mai
n an
d ra
nge
of p
aren
t fun
ctio
ns o
n pa
ge 7
0 (P
.1A
, P.1
B)
Eva
luat
e
an
swer
teac
her q
uest
ions
in c
lass
or o
n ho
mew
ork
assi
gnm
ents
Da
y 9
– S
ect
ion
1.7
Tra
nsf
orm
atio
ns
of
Fu
nct
ion
s
En
gag
e/E
xplo
re
ha
ve s
tude
nt in
pai
rs (1
2 o’
cloc
k ap
poin
tmen
t) pl
ot o
n th
eir g
raph
ing
calc
ulat
ors
the
follo
win
g e
quat
ions
: 1.
y
= x2
2.
y =
(x –
3)2
3.
y =
x2 –3
H
ave
them
dis
cuss
how
they
are
the
sam
e an
d ho
w th
ey d
iffer
Exp
lain
Mod
el o
n th
e bo
ard
or o
verh
ead
the
Verti
cal a
nd H
oriz
onta
l (rig
id) S
hifts
con
tain
ed in
the
box
on p
age
74 a
nd E
x 1
– ha
ve s
tude
nts
put f
unct
ions
on
thei
r gra
phin
g ca
lcul
ator
s as
yo
u pr
esen
t the
m
M
odel
on
the
boar
d or
ove
rhea
d th
e R
efle
ctio
ns in
the
Coo
rdin
ate
Axe
s co
ntai
ned
in th
e bo
x on
pag
e 76
and
Ex
2 &
3
Mod
el o
n th
e bo
ard
or o
verh
ead
the
Non
rigid
Tra
nsfo
rmat
ions
con
tain
ed o
n pa
ge 7
8
Da
y 9
– S
ect
ion
1.7
Tra
nsf
orm
atio
ns
of
Fu
nct
ion
s E
ng
age/
Exp
lore
plot
the
thre
e as
sign
ed fu
nctio
ns o
n yo
ur g
raph
ing
calc
ulat
or (P
.2A
)
disc
uss
with
par
tner
the
sim
ilarit
ies
and
diffe
renc
es o
f the
se g
raph
s (P
.2A
) E
xpla
in
ta
ke n
otes
on
the
teac
her’s
pre
sent
atio
n an
d pu
t the
var
ious
func
tion
on y
our
grap
hing
cal
cula
tor
1.
trans
latio
ns (P
.2A
) 2.
re
flect
ions
(P.2
A)
3.
stre
tch
and
shrin
k (P
.2A
) E
valu
ate
co
mpl
ete
hom
ewor
k as
sign
men
t and
hav
e qu
estio
ns re
ady
for t
each
er if
you
hav
e di
fficu
lty
SA
ISD
© 2
009-
10 –
Firs
t Gra
ding
Per
iod
M
athe
mat
ics
Pre
calc
ulus
Pag
e 7
of 1
2
Pow
er S
tand
ards
repr
esen
t the
ess
entia
l kno
wle
dge
and
skill
s st
uden
ts n
eed
for s
ucce
ss in
hig
h sc
hool
and
bey
ond.
Pow
er S
tand
ards
mus
t be
mas
tere
d to
suc
cess
fully
pas
s th
e re
quire
d as
sess
men
ts a
t eac
h gr
ade
leve
l. A
ll TA
KS
elig
ible
kno
wle
dge
and
skill
s ar
e id
entif
ied
as P
ower
Sta
ndar
ds.
Eva
luat
e
Ass
ign
hom
ewor
k as
des
ired
D
ay
10
– Q
uiz
ove
r S
ecti
on
s 1
.4 –
1.7
E
xpla
in
R
evie
w a
ny h
omew
ork
or g
ener
al q
uest
ions
from
stu
dent
s E
valu
ate
A
dmin
iste
r qui
z en
surin
g al
ignm
ent w
ith g
uidi
ng T
EK
S
Da
y 1
0 –
Qu
iz o
ver
Sec
tio
ns
1.4
– 1
.7
Exp
lain
Get
any
ling
erin
g co
ncer
ns o
r que
stio
ns a
nsw
ered
E
valu
ate
Ta
ke q
uiz
Da
y 1
1 –
Sec
tio
n 1
.8 C
om
bin
atio
ns
of
Fu
nct
ion
s:
Co
mp
osi
te F
un
ctio
ns
En
gag
e/E
xplo
re
H
ave
stud
ents
in p
airs
(10
o’cl
ock
appo
intm
ent)
revi
ew “W
hat y
ou s
houl
d le
arn”
on
page
84
In a
who
le c
lass
dis
cuss
ion
form
at, a
sk fo
r ant
icip
ated
cha
lleng
es
Exp
lain
Pre
sent
the
very
stra
ight
forw
ard
Sum
, Diff
eren
ce, P
rodu
ct a
nd Q
uotie
nt F
unct
ions
in th
e bo
x on
pag
e 84
Ass
ign
2 st
uden
ts a
t ran
dom
to p
rese
nt e
xam
ples
1 &
2 to
the
clas
s
Not
e th
e ra
nge
and
dom
ain
cons
ider
atio
ns d
iscu
ssed
on
page
85
just
prio
r to
pres
entin
g Ex
ampl
e 3.
Pre
sent
Ex
4 an
d 6
afte
r dis
cuss
ing
the
cont
ents
of t
he b
lue
box
on p
age
86 –
Def
initi
on o
f C
ompo
sitio
n of
Tw
o Fu
nctio
ns.
B
e aw
are
that
the
func
tion
nota
tion
asso
ciat
ed w
ith c
ompo
sitio
ns is
usu
ally
ver
y co
nfus
ing
to s
tude
nts.
E
lab
ora
te
A
ssig
n to
gro
ups
prob
lem
31
on p
age
89 fo
r ind
epen
dent
pra
ctic
e. A
sk fo
r a v
olun
teer
to
pres
ent t
o th
e cl
ass.
D
iffe
ren
tiat
e:
St
rugg
ling
lear
ner –
use
par
tner
and
teac
her a
s re
sour
ces
whe
n ch
alle
nges
occ
ur; g
ive
all
of a
ssig
ned
prob
lem
s a
try a
nd a
sk fo
r ass
ista
nce
whe
n cl
arifi
catio
n is
nee
ded
O
n le
vel l
earn
er –
pro
ceed
with
ass
igne
d w
ork
and
use
partn
er a
nd te
ache
r for
ass
ista
nce
Adva
nced
lear
ner –
sor
t thr
ough
the
prob
lem
s w
ith th
e te
chno
logy
icon
on
page
s 89
-92
for
a ch
alle
nge:
alte
rnat
ivel
y, tr
y th
e S
ynth
esis
pro
blem
s on
pag
e 92
.
Da
y 1
1 –
Sec
tio
n 1
.8 C
om
bin
atio
ns
of
Fu
nct
ion
s:
Co
mp
osi
te F
un
ctio
ns
En
gag
e/E
xplo
re
R
ead
“Wha
t you
sho
uld
lear
n” o
n pa
ge 8
4 an
d co
nsid
er a
ny a
ntic
ipat
ed c
halle
nges
E
xpla
in
Ta
ke n
otes
as
teac
her p
rese
nts
com
bina
tions
and
com
posi
tions
of f
unct
ions
E
x 1
– S
um o
f 2 fu
nctio
ns –
stu
dent
pre
sent
ed (P
.2B
) E
x 2
– D
iffer
ence
of 2
func
tions
– s
tude
nt p
rese
nted
(P.2
B)
Ex
3 –
Dom
ains
of q
uotie
nt o
f fun
ctio
ns (P
.2B
) E
x 4
– C
ompo
sitio
n of
func
tions
(P.2
B)
Ex
5 –
Dom
ain
of c
ompo
site
func
tion
(P.2
B)
Ex
6 –
Dec
ompo
sing
a c
ompo
site
func
tion
(P.2
B)
Ex
7 –
App
licat
ion:
bac
teria
cou
nt (P
.2B
) E
lab
ora
te
W
ork
prob
lem
# 3
1 on
pag
e 89
and
con
side
r pre
sent
ing
it to
the
clas
s (P
.2B
)
Da
y 1
2 –
Sec
tio
n 1
.9 In
vers
e F
un
ctio
ns
En
gag
e/E
xplo
re
A
sk th
e st
uden
ts to
writ
e do
wn
the
spec
ific
step
s (o
pera
tions
) to
solv
e
y =
2x –
3
for x
. W
hen
they
hav
e co
mpl
eted
this
, com
pare
thei
r des
crip
tions
to f-1
(x) =
(x +
3)/2
. E
xpla
in
P
rese
nt in
vers
e fu
nctio
n m
ater
ial o
n pa
ge 9
3 fo
llow
ed b
y E
x 1
and
mat
eria
l in
box
on
page
94
follo
wed
by
Ex
2.
In
trodu
ce g
raph
ical
sol
utio
ns (s
wap
coo
rdin
ates
) fol
low
ed b
y E
x 3
& 4
.
Pre
sent
one
-to-o
ne fu
nctio
ns o
n pa
ge 9
6 fo
llow
ed b
y E
x 5.
Sho
w s
tude
nts
an a
lgeb
raic
app
roac
h to
find
ing
the
inve
rse
follo
wed
by
Ex
7.
Da
y 1
2 –
Sec
tio
n 1
.9 In
vers
e F
un
ctio
ns
En
gag
e/E
xplo
re
lis
t spe
cific
ste
ps to
sol
ve p
robl
em g
iven
, the
n co
mpa
re to
func
tion
give
n (P
.2B
) E
xpla
in
ta
ke n
otes
on
teac
her p
rese
ntat
ion
1.
swap
ping
coo
rdin
ates
of o
rder
ed p
airs
to g
et in
vers
e: E
x 1
(P.2
B)
2.
perfo
rm c
ompo
site
of f
unct
ion
and
inve
rse
to e
nd u
p w
ith x
: Ex
2 (P
.2B
) 3.
gr
aphi
cally
det
erm
ine
and
swap
coo
rdin
ates
– s
ymm
etric
abo
ut y
=
x: E
x 3
& 4
(P.2
B)
4. O
ne-to
One
func
tions
– h
oriz
onta
l lin
e te
st: E
x 5
(P.2
B)
5. F
indi
ng in
vers
e al
gebr
aica
lly –
see
box
on
page
97:
Ex
6 &
7 (P
.2B)
SA
ISD
© 2
009-
10 –
Firs
t Gra
ding
Per
iod
M
athe
mat
ics
Pre
calc
ulus
Pag
e 8
of 1
2
Pow
er S
tand
ards
repr
esen
t the
ess
entia
l kno
wle
dge
and
skill
s st
uden
ts n
eed
for s
ucce
ss in
hig
h sc
hool
and
bey
ond.
Pow
er S
tand
ards
mus
t be
mas
tere
d to
suc
cess
fully
pas
s th
e re
quire
d as
sess
men
ts a
t eac
h gr
ade
leve
l. A
ll TA
KS
elig
ible
kno
wle
dge
and
skill
s ar
e id
entif
ied
as P
ower
Sta
ndar
ds.
Ela
bo
rate
/Eva
luat
e
A
ssig
n W
ritin
g ab
out M
athe
mat
ics
on p
age
98 fo
r hom
ewor
k (w
hy th
e gi
ven
func
tions
do
or d
o no
t hav
e in
vers
es)
Ela
bo
rate
/Eva
luat
e
com
plet
e W
ritin
g in
Mat
hem
atic
s as
sign
men
t on
inve
rses
(P.2
B)
Da
y 1
3 –
Sec
tio
n 1
.10
Mat
hem
atic
al M
od
elin
g a
nd
Var
iati
on
E
ng
age/
Exp
lore
Ass
ign
stud
ents
in p
airs
(8 o
’clo
ck a
ppoi
ntm
ent)
to m
ake
two
tabl
es re
late
d to
d =
rt,
dist
ance
equ
als
rate
tim
es ti
me
1.
d ve
rsus
t fo
r a c
onst
ant r
ate
of 1
0 fe
et p
er s
econ
d: d
= 1
0t
2.
r ver
sus
t for
a c
onst
ant d
ista
nce
of 1
00 fe
et: r
= 1
00/t
di
rect
the
pair
to d
iscu
ss h
ow th
e va
riabl
e va
ry in
eac
h si
tuat
ion
in te
rms
of th
e te
rms
“incr
easi
ng” a
nd “d
ecre
asin
g”
Exp
lain
assi
gn E
x 1
to b
e pe
rform
ed b
y st
uden
ts o
n th
e gr
aphi
ng c
alcu
lato
r as
a re
gres
sion
pr
oble
m
pr
esen
t dire
ct v
aria
tion
as y
= m
x +
b w
here
b =
0 re
sulti
ng in
a s
tate
men
t of p
ropo
rtion
, us
ually
rew
ritte
n as
y =
kx
pr
esen
t dire
ct v
aria
tion
as a
n nt
h po
wer
with
Ex
4
pres
ent i
nver
se a
nd jo
int v
aria
tion
with
Ex
5 &
6.
Eva
luat
e
Assi
gn th
e V
ocab
ular
y C
heck
on
page
109
for h
omew
ork.
Da
y 1
3 –
Sec
tio
n 1
.10
Mat
hem
atic
al M
od
elin
g a
nd
Var
iati
on
E
ng
age/
Exp
lore
com
plet
e co
mpa
rison
and
dis
cuss
ion
with
a p
artn
er o
f tw
o si
tuat
ions
usi
ng th
e fo
rmul
a d
= rt
(P.1
A)
Exp
lain
take
not
es a
s te
ache
r mod
els
varia
tions
– d
irect
, inv
erse
and
join
t 1.
E
x 1
& 2
– m
odel
s w
ith g
raph
ing
calc
ulat
ors
(regr
essi
ons)
(P.1
A)
2.
Ex
3 &
4 -
dire
ct v
aria
tion
(P.1
A)
3.
Ex
5 –
dire
ct &
inve
rse
varia
tion
(P.1
A)
4.
Ex
6 –
join
t var
iatio
n (P
.1A
) E
valu
ate
co
mpl
ete
voca
bula
ry c
heck
on
page
109
: 1-9
(P.1
A)
Da
y 1
4 –
Un
it R
evi
ew
En
gag
e
Sea
rchi
ng fo
r stu
dent
pro
blem
are
as, h
ave
the
stud
ents
sca
n pa
ges
115
and
116
to
rem
ind
them
of w
hat h
as b
een
cove
red.
In a
who
le c
lass
con
text
, ask
the
stud
ents
to li
st th
e pr
oble
m a
reas
.
Add
ress
the
prob
lem
are
as fi
rst c
once
ptua
lly, t
hen
by w
orki
ng a
ppro
pria
te p
robl
ems
from
pa
ges
117-
122.
Con
side
r gro
upin
g st
uden
ts w
ith c
omm
on c
halle
nges
with
pee
r tut
ors,
E
valu
ate
Ti
me
allo
win
g, a
ssig
n pr
oble
ms
to in
divi
dual
s to
be
post
ed d
urin
g cl
ass
perio
d –
one
from
ea
ch s
ectio
n th
at h
as n
ot y
et b
een
disc
usse
d.
W
rap
up w
ith o
ne la
st c
all f
or q
uest
ions
.
Da
y 1
4 –
Un
it R
evi
ew
En
gag
e
revi
ew p
ages
115
and
116
to s
ee w
hat h
as b
een
cove
red
in th
is u
nit a
nd to
de
term
ine
whe
re q
uest
ions
mig
ht s
till e
xist
cont
ribut
e ch
alle
nges
to th
e cl
ass
list
Eva
luat
e
ensu
re th
at h
is/h
er p
robl
em a
rea(
s) is
(are
) add
ress
ed
pa
rtici
pate
in p
eer t
utor
gro
upin
gs
Da
y 1
5 –
Un
it A
sse
ssm
ent
Ad
min
iste
r ass
essm
ent e
nsur
ing
alig
nmen
t with
gui
ding
TE
KS
En
sure
that
nee
ded
mat
eria
ls a
re o
n ha
nd
1.
grap
hing
cal
cula
tors
2.
pa
per
3.
penc
ils
Da
y 1
5 –
Un
it A
sse
ssm
ent
ta
ke a
sses
smen
t
SA
ISD
© 2
009-
10 –
Firs
t Gra
ding
Per
iod
M
athe
mat
ics
Pre
calc
ulus
Pag
e 9
of 1
2
Pow
er S
tand
ards
repr
esen
t the
ess
entia
l kno
wle
dge
and
skill
s st
uden
ts n
eed
for s
ucce
ss in
hig
h sc
hool
and
bey
ond.
Pow
er S
tand
ards
mus
t be
mas
tere
d to
suc
cess
fully
pas
s th
e re
quire
d as
sess
men
ts a
t eac
h gr
ade
leve
l. A
ll TA
KS
elig
ible
kno
wle
dge
and
skill
s ar
e id
entif
ied
as P
ower
Sta
ndar
ds.
Da
y 1
6 –
Un
it A
sse
ssm
ent
Re
view
, Ret
each
, R
efle
ct
En
gag
e
retu
rn a
sses
smen
ts to
stu
dent
s
give
sta
tistic
s as
soci
ated
with
ass
essm
ent s
core
s E
xpla
in
as
sign
stu
dent
s ra
ndom
ly in
gro
ups
of 3
or 4
inst
ruct
them
to to
geth
er re
view
the
prob
lem
s m
isse
d an
d se
e if
they
can
now
sol
ve th
em
fo
r any
pro
blem
s le
ft ov
er, s
how
sol
utio
ns to
stu
dent
s E
lab
ora
te
as
sign
stu
dent
s to
eva
luat
e in
thei
r jou
rnal
s w
hy th
ey re
ceiv
ed th
e sc
ore
that
they
go
t E
valu
ate
as
sign
as
hom
ewor
k to
turn
in a
fully
cor
rect
ed e
xam
Da
y 1
6 –
Un
it A
sse
ssm
ent
Re
view
, Ret
each
, R
efle
ct
En
gag
e
rece
ive
indi
vidu
al a
sses
smen
t and
cla
ss a
vera
ge
Exp
lain
in g
roup
s, w
ork
on s
olut
ions
to p
robl
ems
mis
sed
Ela
bo
rate
jour
nal h
ow h
e/sh
e ea
rned
the
grad
e re
ceiv
ed
Eva
luat
e
corre
ct a
ll pr
oble
ms
mis
sed
Vo
cab
ula
ry:
quad
rant
s
solu
tion
in
terc
epts
sym
met
ry
st
anda
rd fo
rm o
f an
equa
tion
tra
nsla
tion
fu
nctio
n
do
mai
n
rang
e
inde
pend
ent
de
pend
ent
ve
rtica
l lin
e te
st
co
mbi
natio
ns o
f fun
ctio
ns
co
mpo
sitio
n
inve
rse
func
tion
re
gres
sion
leas
t squ
ares
corre
latio
n
slop
e
poin
t-slo
pe fo
rm
pa
ralle
l
perp
endi
cula
r
rate
of c
hang
e
ratio
mod
el
ze
ros
of a
func
tion
re
lativ
e m
inim
um
re
lativ
e m
axim
um
av
erag
e ra
te o
f cha
nge
ev
en a
nd o
dd fu
nctio
ns
tra
nsfo
rmat
ion
TA
KS
Vo
cab
ula
ry (
verb
s):
defin
e
trans
late
s
desc
ribe
de
term
ine
re
cogn
ize
in
terp
rets
solv
e
appl
y
perfo
rm
Res
ou
rces
: P
rec
alcu
lus
wit
h L
imit
s S
ectio
ns
1.1
Rec
tang
ular
Coo
rdin
ates
1.
2 G
raph
s of
Equ
atio
ns
1.3
Line
ar E
quat
ions
in T
wo
Var
iabl
es
1.4
Func
tions
1.
5 A
naly
zing
Gra
phs
of F
unct
ions
1.
6 A
Lib
rary
of P
aren
t Fun
ctio
ns
1.7
Tran
sfor
mat
ion
of F
unct
ions
1.
8 C
ombi
natio
ns o
f Fun
ctio
ns: C
ompo
site
Fun
ctio
ns
1.9
Inve
rse
Func
tions
1.
10 M
athe
mat
ical
Mod
elin
g an
d Va
riatio
n
SA
ISD
© 2
009-
10 –
Firs
t Gra
ding
Per
iod
M
athe
mat
ics
Pre
calc
ulus
Pag
e 10
of 1
2
Pow
er S
tand
ards
repr
esen
t the
ess
entia
l kno
wle
dge
and
skill
s st
uden
ts n
eed
for s
ucce
ss in
hig
h sc
hool
and
bey
ond.
Pow
er S
tand
ards
mus
t be
mas
tere
d to
suc
cess
fully
pas
s th
e re
quire
d as
sess
men
ts a
t eac
h gr
ade
leve
l. A
ll TA
KS
elig
ible
kno
wle
dge
and
skill
s ar
e id
entif
ied
as P
ower
Sta
ndar
ds.
st
retc
h an
d sh
rink
ho
rizon
tal l
ine
test
one-
to-o
ne
ho
rizon
tal l
ine
test
dire
ct v
aria
tion
in
vers
e va
riatio
n
jo
int v
aria
tion
Evi
den
ce o
f L
earn
ing
F
orm
ativ
e M
ini A
sses
smen
ts
TA
KS
C
olle
ge-
Rea
din
ess
i.e.,
An
tici
pat
ed S
kill
s fo
r S
AT
/AC
T/A
P/C
aree
r/L
ife
TA
KS
2004
Exi
t Lev
el It
em
SAT
A
, B, C
, and
D a
re p
oint
s on
a lin
e, w
ith D
the
mid
poin
t of
segm
ent B
C.
The
leng
ths o
f seg
men
ts A
B, A
C, a
nd B
C a
re
10, 2
, and
12,
resp
ectiv
ely.
Wha
t is t
he le
ngth
of s
egm
ent
AD
?
(A)
2
(B
) 4
(C)
6
(D
) 10
(E)
12
SAT
x
f(x)
0
a 1
24
2 b
The
tabl
e ab
ove
show
s som
e va
lues
for t
he fu
nctio
n f(
x).
If
f(x)
is a
line
ar fu
nctio
n, w
hat i
s the
val
ue o
f a +
b ?
(
A)
24
(B
) 36
(
C)
48
(D
) 72
(
E)
It ca
nnot
be
dete
rmin
ed fr
om th
e in
form
atio
n gi
ven.
SA
T
Let t
he o
pera
tions
Δ a
nd Џ
be
defin
ed fo
r all
num
bers
a
and
b a
s fol
low
s:
a
Δ b
= a
+ 3
b
a Џ
b =
a +
4b
If 4
Δ 5
y =
(5y
) Џ
4,
wha
t is t
he v
alue
of y
?
SA
ISD
© 2
009-
10 –
Firs
t Gra
ding
Per
iod
M
athe
mat
ics
Pre
calc
ulus
Pag
e 11
of 1
2
Pow
er S
tand
ards
repr
esen
t the
ess
entia
l kno
wle
dge
and
skill
s st
uden
ts n
eed
for s
ucce
ss in
hig
h sc
hool
and
bey
ond.
Pow
er S
tand
ards
mus
t be
mas
tere
d to
suc
cess
fully
pas
s th
e re
quire
d as
sess
men
ts a
t eac
h gr
ade
leve
l. A
ll TA
KS
elig
ible
kno
wle
dge
and
skill
s ar
e id
entif
ied
as P
ower
Sta
ndar
ds.
SA
ISD
© 2
009-
10 –
Firs
t Gra
ding
Per
iod
M
athe
mat
ics
Pre
calc
ulus
Pag
e 12
of 1
2
Pow
er S
tand
ards
repr
esen
t the
ess
entia
l kno
wle
dge
and
skill
s st
uden
ts n
eed
for s
ucce
ss in
hig
h sc
hool
and
bey
ond.
Pow
er S
tand
ards
mus
t be
mas
tere
d to
suc
cess
fully
pas
s th
e re
quire
d as
sess
men
ts a
t eac
h gr
ade
leve
l. A
ll TA
KS
elig
ible
kno
wle
dge
and
skill
s ar
e id
entif
ied
as P
ower
Sta
ndar
ds.
1st
9-
Wee
ks W
riti
ng
Pro
ject
Th
is p
roje
ct is
ass
igne
d on
Day
1 o
f the
cou
rse
with
the
stud
ent R
AFT
cho
ices
to b
e su
bmitt
ed to
the
teac
her b
y S
ep 4
th a
nd th
e fin
al
prod
uct d
ue o
n O
ct 1
6th.
S
tude
nts
are
prov
ided
the
follo
win
g ch
oice
s (R
AFT
) for
the
deta
ils o
f how
they
indi
cate
a s
olid
kno
wle
dge
of th
e to
pic.
Ro
le
Au
die
nce
F
orm
at
To
pic
Mat
h S
tude
nt
Prin
cipa
l S
ales
Bro
chur
e C
ompl
ex N
umbe
rs
Mat
h Te
ache
r (e
xper
ienc
ed)
Col
lege
Pro
fess
or
Out
line
of a
less
on
plan
Com
plet
ing
the
Squ
are
Mat
h Te
xtbo
ok
Aut
hor
Gra
phin
g C
alcu
lato
r S
ales
pers
on
Car
toon
– a
t lea
st
thre
e pa
nels
.
Dire
ct V
aria
tion
Par
ent
Sub
stitu
te T
each
er
Lette
r
Tran
slat
ion
acro
ss th
e y-
axis
on
the
grap
hing
cal
cula
tor
S
tude
nts
may
rand
omly
cho
ose
the
Rol
e, A
udie
nce,
For
mat
and
Top
ic.
The
writ
ing
task
is to
be
com
plet
ed b
ased
on
the
stud
ent’s
su
bmitt
ed c
hoic
es.
Sin
ce th
ese
are
the
topi
cs s
tudi
ed th
roug
hout
the
9 w
eeks
, thi
s pr
ojec
t is
due
late
r in
the
nine
wee
ks.
Kagan’s Cooperative Learning Structures © 1998
1. Circle-the-Sage: Students who know, stand to become sages; teammates each gather around a different sage to learn. Students return to teams to compare notes.
2. Find My Rule: The teacher places items in a frame (two boxes, Venn, on a line); Students induce the rule.
a. Two Box Introduction b. What’s My Line c. Crack My Venn
3. Find Someone Who: Students circulate, finding others who can contribute to their worksheet.
a. People Hunt: Students circulate, finding others who match their own characteristics.
b. Fact Bingo: Find Someone Who played on a bingo worksheet. 4. Find the Fib: Teammates try to determine which of three statements is a fib.
a. Fact or Fiction: Teammates try to determine if a statement is true or false. 5. Inside/Outside Circle: Students in concentric circles rotate to face a partner to
answer the teacher’s questions or those of the partner. 6. Jigsaw Problem Solving: Each teammate has part of the answer or a clue
card; teammates must put their info together to solve the team problem. 7. Match Mine: Receivers arrange objects to match those of Senders whose
objects are hidden by a barrier. a. Draw-What-I-Say: Receiver draws what sender describes. b. Build-What-I-Write: Receiver constructs what Sender has described in
writing. 8. Mix-Freeze-Group: Students rush to form groups of a specific size, hoping not
to land in “Lost and Found” (with the teacher). 9. Mix-Pair-Discuss: Students pair with classmate, to discuss question posed by
the teacher. 10. Numbered Heads Together: Students huddle to make sure all can respond, a
number is called, the student with the number responds. a. Paired Heads Together: Students in pairs huddle to make sure they both
can respond, an “A” or “B” is called, the student with that letter responds. b. Traveling Heads Together: Students in Numbered heads travel to new
teams to share response. 11. One Stray: The teacher calls a number; students with that number “stray” to
join another team, often to share. a. Two Stray: Two students stray to another team, often to share and to
listen. 12. Pairs Check: Students work first in pairs each doing a problem and receiving
coaching from their partner; then pairs check and celebrate after every two problems.
13. Pairs Compare: Pairs generate ideas or answers, compare their answers with another pair, and then see if working together the can come up with additional responses neither pair alone had.
14. Partners: Pairs work to prepare a presentation, then present to the other pair in their team.
15. Rally Table: Students in pairs take turns, writing, drawing, pasting. (2 erasers, 2 pencils per team)
16. Rotating Review: Teams discuss topic, chart their thoughts, rotate to the next chart to discuss and chart their thoughts.
17. Sages Share: Students take turns interviewing the “sages”-those who can explain an idea.
18. Send-(Trade) a-Problem: Teammates make problems whicj are sent around the class for other teams to solve.
19. Showdown: Teammates each write an answer, then there is a “showdown” as they show their answers to each other. Teammates verify answers.
20. Teammates Consult: For each of a series of questions, students place pens in a cup, share and discuss their answers, and then pick up pens to write answer in own words.
21. Team-Pair-Solo: Students solve problems first as a team, then as a pair, finally alone.
22. Team Stand-N-Share: All teams stand. Teams share ideas and record ideas from other teams. Teams sit when all ideas are share and continue to record until all teams sit.
23. Team Statements: Students think, discuss in pairs, write an individual statement, RoundRobin individual statements, and then work together to arrive at team statement they all endorse more strongly than their individual statements.
24. Team Mind Map: Students draw and label the central image, brainstorm, draw and label main ideas radiating out of the central image, and finally add details using colors, images, branches and key words.
25. Think-Pair-Share: Students think about their response to a question, discuss answer in pairs, and share their own or partner’s answer with the class.
26. Three-Pair-Share: Students share on a topic three times, once with each teammate.
27. Three-Step Interview: Students share with a partner, the partner shares with them, and then they RoundRobin share their partner’s response with the other teammates.
28. Timed Pair Share: Students share with a partner for a predetermined amount of time and then the partner shares with them for the same amount of time.
29. What Am I? Students attempt to determine their secret identity (taped on their back) by circulating asking “yes-no” questions of classmates. They are allowed three questions per classmate (or unlimited questions until they receive a no response). They then find a new classmate to question. When the student guesses his/her identify, he/she becomes a consultant to give clues to those who have not yet found their identity.