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Session #1, October 2014
Grade 1
Packet Contents (Selected pages relevant to session work)
Content Standards
Standards for Mathematical Practice
California Mathematical Framework
Kansas CTM Flipbook
Learning Outcomes
Sample Assessment Items
CCSS-M Teacher Professional Learning
Grade 1
Operations and Algebraic Thinking 1.OA Represent and solve problems involving addition and subtraction.
1. Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.2
2. Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
Understand and apply properties of operations and the relationship between addition and subtraction.
3. Apply properties of operations as strategies to add and subtract.3 Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
4. Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.
Add and subtract within 20.
5. Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). 6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use
strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
Work with addition and subtraction equations.
7. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
8. Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = � – 3, 6 + 6 = �.
Number and Operations in Base Ten 1.NBT Extend the counting sequence.
1. Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.
Understand place value.
2. Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
a. 10 can be thought of as a bundle of ten ones — called a “ten.”
2 See Glossary, Table 1. 3 Students need not use formal terms for these properties.
1
Prepublication Version, April 2013 California Department of Education 12 |
Com
mon
Cor
e St
ate
Stan
dard
s - M
athe
mat
ics
Stan
dard
s for
Mat
hem
atic
al P
ract
ices
– 1
st G
rade
Stan
dard
for M
athe
mat
ical
Pra
ctic
e 1st
Gra
de
1: M
ake
sens
e of
pro
blem
s and
per
seve
re in
solv
ing
them
. M
athe
mat
ical
ly p
rofic
ient
stud
ents
star
t by
expl
aini
ng to
them
selv
es th
e m
eani
ng o
f a p
robl
em
and
look
ing
for e
ntry
poi
nts t
o its
solu
tion.
The
y an
alyz
e gi
vens
, con
stra
ints
, rel
atio
nshi
ps, a
nd
goal
s. T
hey
mak
e co
njec
ture
s abo
ut th
e fo
rm a
nd m
eani
ng o
f the
solu
tion
and
plan
a so
lutio
n pa
thw
ay ra
ther
than
sim
ply
jum
ping
into
a so
lutio
n at
tem
pt. T
hey
cons
ider
ana
logo
us p
robl
ems,
and
try
spec
ial c
ases
and
sim
pler
form
s of t
he o
rigin
al p
robl
em in
ord
er to
gai
n in
sight
into
its
solu
tion.
The
y m
onito
r and
eva
luat
e th
eir p
rogr
ess a
nd c
hang
e co
urse
if n
eces
sary
. Old
er
stud
ents
mig
ht, d
epen
ding
on
the
cont
ext o
f the
pro
blem
, tra
nsfo
rm a
lgeb
raic
exp
ress
ions
or
chan
ge th
e vi
ewin
g w
indo
w o
n th
eir g
raph
ing
calc
ulat
or to
get
the
info
rmat
ion
they
nee
d.
Mat
hem
atic
ally
pro
ficie
nt st
uden
ts c
an e
xpla
in c
orre
spon
denc
es b
etw
een
equa
tions
, ver
bal
desc
riptio
ns, t
able
s, a
nd g
raph
s or d
raw
dia
gram
s of i
mpo
rtan
t fea
ture
s and
rela
tions
hips
, gra
ph
data
, and
sear
ch fo
r reg
ular
ity o
r tre
nds.
You
nger
stud
ents
mig
ht re
ly o
n us
ing
conc
rete
obj
ects
or
pic
ture
s to
help
con
cept
ualiz
e an
d so
lve
a pr
oble
m. M
athe
mat
ical
ly p
rofic
ient
stud
ents
che
ck
thei
r ans
wer
s to
prob
lem
s usin
g a
diffe
rent
met
hod,
and
they
con
tinua
lly a
sk th
emse
lves
, “Do
es
this
mak
e se
nse?
” Th
ey c
an u
nder
stan
d th
e ap
proa
ches
of o
ther
s to
solv
ing
com
plex
pro
blem
s an
d id
entif
y co
rres
pond
ence
s bet
wee
n di
ffere
nt a
ppro
ache
s.
In fi
rst g
rade
, stu
dent
s rea
lize
that
doi
ng m
athe
mat
ics i
nvol
ves
solv
ing
prob
lem
s and
disc
ussin
g ho
w th
ey so
lved
them
. Stu
dent
s ex
plai
n to
them
selv
es th
e m
eani
ng o
f a p
robl
em a
nd lo
ok
for w
ays t
o so
lve
it. Y
oung
er
stud
ents
may
use
con
cret
e ob
ject
s or p
ictu
res t
o he
lp th
em
conc
eptu
alize
and
solv
e pr
oble
ms.
The
y m
ay c
heck
thei
r th
inki
ng b
y as
king
them
selv
es, -
Does
this
mak
e se
nse?
The
y ar
e w
illin
g to
try
othe
r app
roac
hes.
2: R
easo
n ab
stra
ctly
and
qua
ntita
tivel
y.
Mat
hem
atic
ally
pro
ficie
nt st
uden
ts m
ake
sens
e of
qua
ntiti
es a
nd th
eir r
elat
ions
hips
in p
robl
em
situa
tions
. The
y br
ing
two
com
plem
enta
ry a
bilit
ies t
o be
ar o
n pr
oble
ms i
nvol
ving
qua
ntita
tive
rela
tions
hips
: the
abi
lity
to d
econ
text
ualiz
e-to
abs
trac
t a g
iven
situ
atio
n an
d re
pres
ent i
t sy
mbo
lical
ly a
nd m
anip
ulat
e th
e re
pres
entin
g sy
mbo
ls as
if th
ey h
ave
a lif
e of
thei
r ow
n, w
ithou
t ne
cess
arily
att
endi
ng to
thei
r ref
eren
ts-a
nd th
e ab
ility
to c
onte
xtua
lize,
to p
ause
as n
eede
d du
ring
the
man
ipul
atio
n pr
oces
s in
orde
r to
prob
e in
to th
e re
fere
nts f
or th
e sy
mbo
ls in
volv
ed.
Qua
ntita
tive
reas
onin
g en
tails
hab
its o
f cre
atin
g a
cohe
rent
repr
esen
tatio
n of
the
prob
lem
at
hand
; con
sider
ing
the
units
invo
lved
; att
endi
ng to
the
mea
ning
of q
uant
ities
, not
just
how
to
com
pute
them
; and
kno
win
g an
d fle
xibl
y us
ing
diffe
rent
pro
pert
ies o
f ope
ratio
ns a
nd o
bjec
ts.
Youn
ger s
tude
nts r
ecog
nize
that
a
num
ber r
epre
sent
s a sp
ecifi
c qu
antit
y. T
hey
conn
ect t
he
quan
tity
to w
ritte
n sy
mbo
ls.
Qua
ntita
tive
reas
onin
g en
tails
cr
eatin
g a
repr
esen
tatio
n of
a
prob
lem
whi
le a
tten
ding
to th
e m
eani
ngs o
f the
qua
ntiti
es.
3: C
onst
ruct
via
ble
argu
men
ts a
nd c
ritiq
ue th
e re
ason
ing
of o
ther
s.
Mat
hem
atic
ally
pro
ficie
nt st
uden
ts u
nder
stan
d an
d us
e st
ated
ass
umpt
ions
, def
initi
ons,
and
prev
ious
ly e
stab
lishe
d re
sults
in c
onst
ruct
ing
argu
men
ts. T
hey
mak
e co
njec
ture
s and
bui
ld a
lo
gica
l pro
gres
sion
of st
atem
ents
to e
xplo
re th
e tr
uth
of th
eir c
onje
ctur
es. T
hey
are
able
to
anal
yze
situa
tions
by
brea
king
them
into
cas
es, a
nd c
an re
cogn
ize a
nd u
se c
ount
erex
ampl
es.
They
just
ify th
eir c
oncl
usio
ns, c
omm
unic
ate
them
to o
ther
s, a
nd re
spon
d to
the
argu
men
ts o
f ot
hers
. The
y re
ason
indu
ctiv
ely
abou
t dat
a, m
akin
g pl
ausib
le a
rgum
ents
that
take
into
acc
ount
th
e co
ntex
t fro
m w
hich
the
data
aro
se. M
athe
mat
ical
ly p
rofic
ient
stud
ents
are
also
abl
e to
co
mpa
re th
e ef
fect
iven
ess o
f tw
o pl
ausib
le a
rgum
ents
, dist
ingu
ish c
orre
ct lo
gic
or re
ason
ing
from
that
whi
ch is
flaw
ed, a
nd-if
ther
e is
a fla
w in
an
argu
men
t-ex
plai
n w
hat i
t is.
Ele
men
tary
st
uden
ts c
an c
onst
ruct
arg
umen
ts u
sing
conc
rete
refe
rent
s suc
h as
obj
ects
, dra
win
gs, d
iagr
ams,
an
d ac
tions
. Suc
h ar
gum
ents
can
mak
e se
nse
and
be c
orre
ct, e
ven
thou
gh th
ey a
re n
ot
gene
raliz
ed o
r mad
e fo
rmal
unt
il la
ter g
rade
s. L
ater
, stu
dent
s lea
rn to
det
erm
ine
dom
ains
to
whi
ch a
n ar
gum
ent a
pplie
s. S
tude
nts a
t all
grad
es c
an li
sten
or r
ead
the
argu
men
ts o
f oth
ers,
de
cide
whe
ther
they
mak
e se
nse,
and
ask
use
ful q
uest
ions
to c
larif
y or
impr
ove
the
argu
men
ts.
Firs
t gra
ders
con
stru
ct a
rgum
ents
us
ing
conc
rete
refe
rent
s, su
ch a
s ob
ject
s, pi
ctur
es, d
raw
ings
, and
ac
tions
. The
y al
so p
ract
ice
thei
r m
athe
mat
ical
com
mun
icat
ion
skill
s as t
hey
part
icip
ate
in
mat
hem
atic
al d
iscus
sions
in
volv
ing
ques
tions
like
-How
did
yo
u ge
t tha
t? -E
xpla
in y
our
thin
king
, and
-Why
is th
at tr
ue?‖
Th
ey n
ot o
nly
expl
ain
thei
r ow
n th
inki
ng, b
ut li
sten
to o
ther
s’
expl
anat
ions
. The
y de
cide
if th
e ex
plan
atio
ns m
ake
sens
e an
d as
k qu
estio
ns.
4: M
odel
with
mat
hem
atic
s.
Mat
hem
atic
ally
pro
ficie
nt st
uden
ts c
an a
pply
the
mat
hem
atic
s the
y kn
ow to
solv
e pr
oble
ms
arisi
ng in
eve
ryda
y lif
e, so
ciet
y, a
nd th
e w
orkp
lace
. In
early
gra
des,
this
mig
ht b
e as
sim
ple
as
writ
ing
an a
dditi
on e
quat
ion
to d
escr
ibe
a sit
uatio
n. In
mid
dle
grad
es, a
stud
ent m
ight
app
ly
prop
ortio
nal r
easo
ning
to p
lan
a sc
hool
eve
nt o
r ana
lyze
a p
robl
em in
the
com
mun
ity. B
y hi
gh
scho
ol, a
stud
ent m
ight
use
geo
met
ry to
solv
e a
desig
n pr
oble
m o
r use
a fu
nctio
n to
des
crib
e ho
w o
ne q
uant
ity o
f int
eres
t dep
ends
on
anot
her.
Mat
hem
atic
ally
pro
ficie
nt st
uden
ts w
ho c
an
appl
y w
hat t
hey
know
are
com
fort
able
mak
ing
assu
mpt
ions
and
app
roxi
mat
ions
to si
mpl
ify a
co
mpl
icat
ed si
tuat
ion,
real
izing
that
thes
e m
ay n
eed
revi
sion
late
r. Th
ey a
re a
ble
to id
entif
y im
port
ant q
uant
ities
in a
pra
ctic
al si
tuat
ion
and
map
thei
r rel
atio
nshi
ps u
sing
such
tool
s as
diag
ram
s, tw
o-w
ay ta
bles
, gra
phs,
flow
char
ts a
nd fo
rmul
as. T
hey
can
anal
yze
thos
e re
latio
nshi
ps m
athe
mat
ical
ly to
dra
w c
oncl
usio
ns. T
hey
rout
inel
y in
terp
ret t
heir
mat
hem
atic
al
resu
lts in
the
cont
ext o
f the
situ
atio
n an
d re
flect
on
whe
ther
the
resu
lts m
ake
sens
e, p
ossib
ly
impr
ovin
g th
e m
odel
if it
has
not
serv
ed it
s pur
pose
.
In e
arly
gra
des,
stud
ents
ex
perim
ent w
ith re
pres
entin
g pr
oble
m si
tuat
ions
in m
ultip
le
way
s inc
ludi
ng n
umbe
rs, w
ords
(m
athe
mat
ical
lang
uage
), dr
awin
g pi
ctur
es, u
sing
obje
cts,
ac
ting
out,
mak
ing
a ch
art o
r list
, cr
eatin
g eq
uatio
ns, e
tc. S
tude
nts
need
opp
ortu
nitie
s to
conn
ect
the
diffe
rent
repr
esen
tatio
ns a
nd
expl
ain
the
conn
ectio
ns. T
hey
shou
ld b
e ab
le to
use
all
of th
ese
repr
esen
tatio
ns a
s nee
ded.
5: U
se a
ppro
pria
te to
ols s
trat
egic
ally
. M
athe
mat
ical
ly p
rofic
ient
stud
ents
con
sider
the
avai
labl
e to
ols w
hen
solv
ing
a m
athe
mat
ical
pr
oble
m. T
hese
tool
s mig
ht in
clud
e pe
ncil
and
pape
r, co
ncre
te m
odel
s, a
rule
r, a
prot
ract
or, a
ca
lcul
ator
, a sp
read
shee
t, a
com
pute
r alg
ebra
syst
em, a
stat
istic
al p
acka
ge, o
r dyn
amic
ge
omet
ry so
ftw
are.
Pro
ficie
nt st
uden
ts a
re su
ffici
ently
fam
iliar
with
tool
s app
ropr
iate
for t
heir
grad
e or
cou
rse
to m
ake
soun
d de
cisio
ns a
bout
whe
n ea
ch o
f the
se to
ols m
ight
be
help
ful,
reco
gnizi
ng b
oth
the
insig
ht to
be
gain
ed a
nd th
eir l
imita
tions
. For
exa
mpl
e, m
athe
mat
ical
ly
prof
icie
nt h
igh
scho
ol st
uden
ts a
naly
ze g
raph
s of f
unct
ions
and
solu
tions
gen
erat
ed u
sing
a gr
aphi
ng c
alcu
lato
r. Th
ey d
etec
t pos
sible
err
ors b
y st
rate
gica
lly u
sing
estim
atio
n an
d ot
her
mat
hem
atic
al k
now
ledg
e. W
hen
mak
ing
mat
hem
atic
al m
odel
s, th
ey k
now
that
tech
nolo
gy c
an
enab
le th
em to
visu
alize
the
resu
lts o
f var
ying
ass
umpt
ions
, exp
lore
con
sequ
ence
s, a
nd
com
pare
pre
dict
ions
with
dat
a. M
athe
mat
ical
ly p
rofic
ient
stud
ents
at v
ario
us g
rade
leve
ls ar
e ab
le to
iden
tify
rele
vant
ext
erna
l mat
hem
atic
al re
sour
ces,
such
as d
igita
l con
tent
loca
ted
on a
w
ebsit
e, a
nd u
se th
em to
pos
e
In fi
rst g
rade
, stu
dent
s beg
in to
co
nsid
er th
e av
aila
ble
tool
s (in
clud
ing
estim
atio
n) w
hen
solv
ing
a m
athe
mat
ical
pro
blem
an
d de
cide
whe
n ce
rtai
n to
ols
mig
ht b
e he
lpfu
l. Fo
r ins
tanc
e,
first
gra
ders
dec
ide
it m
ight
be
best
to u
se c
olor
ed c
hips
to
mod
el a
n ad
ditio
n pr
oble
m.
6: A
tten
d to
pre
cisi
on.
Mat
hem
atic
ally
pro
ficie
nt st
uden
ts tr
y to
com
mun
icat
e pr
ecise
ly to
oth
ers.
The
y tr
y to
use
cle
ar
defin
ition
s in
disc
ussio
n w
ith o
ther
s and
in th
eir o
wn
reas
onin
g. T
hey
stat
e th
e m
eani
ng o
f the
sy
mbo
ls th
ey c
hoos
e, in
clud
ing
usin
g th
e eq
ual s
ign
cons
isten
tly a
nd a
ppro
pria
tely
. The
y ar
e ca
refu
l abo
ut sp
ecify
ing
units
of m
easu
re, a
nd la
belin
g ax
es to
cla
rify
the
corr
espo
nden
ce w
ith
quan
titie
s in
a pr
oble
m. T
hey
calc
ulat
e ac
cura
tely
and
effi
cien
tly, e
xpre
ss n
umer
ical
ans
wer
s w
ith a
deg
ree
of p
reci
sion
appr
opria
te fo
r the
pro
blem
con
text
. In
the
elem
enta
ry g
rade
s,
stud
ents
giv
e ca
refu
lly fo
rmul
ated
exp
lana
tions
to e
ach
othe
r. By
the
time
they
reac
h hi
gh
scho
ol th
ey h
ave
lear
ned
to e
xam
ine
clai
ms a
nd m
ake
expl
icit
use
of d
efin
ition
s.
As y
oung
chi
ldre
n be
gin
to
deve
lop
thei
r mat
hem
atic
al
com
mun
icat
ion
skill
s, th
ey tr
y to
us
e cl
ear a
nd p
reci
se la
ngua
ge in
th
eir d
iscus
sions
with
oth
ers a
nd
whe
n th
ey e
xpla
in th
eir o
wn
reas
onin
g.
7: L
ook
for a
nd m
ake
use
of st
ruct
ure.
M
athe
mat
ical
ly p
rofic
ient
stud
ents
look
clo
sely
to d
iscer
n a
patt
ern
or st
ruct
ure.
You
ng
stud
ents
, for
exa
mpl
e, m
ight
not
ice
that
thre
e an
d se
ven
mor
e is
the
sam
e am
ount
as s
even
and
th
ree
mor
e, o
r the
y m
ay so
rt a
col
lect
ion
of sh
apes
acc
ordi
ng to
how
man
y sid
es th
e sh
apes
ha
ve. L
ater
, stu
dent
s will
see
7 ×
8 eq
uals
the
wel
l rem
embe
red
7 ×
5 +
7 ×
3, in
pre
para
tion
for
lear
ning
abo
ut th
e di
strib
utiv
e pr
oper
ty. I
n th
e ex
pres
sion
x2 +
9x
+ 14
, old
er st
uden
ts c
an se
e th
e 14
as 2
× 7
and
the
9 as
2 +
7. T
hey
reco
gnize
the
signi
fican
ce o
f an
exist
ing
line
in a
ge
omet
ric fi
gure
and
can
use
the
stra
tegy
of d
raw
ing
an a
uxili
ary
line
for s
olvi
ng p
robl
ems.
The
y al
so c
an st
ep b
ack
for a
n ov
ervi
ew a
nd sh
ift p
ersp
ectiv
e. T
hey
can
see
com
plic
ated
thin
gs, s
uch
as so
me
alge
brai
c ex
pres
sions
, as s
ingl
e ob
ject
s or a
s bei
ng c
ompo
sed
of se
vera
l obj
ects
. For
ex
ampl
e, th
ey c
an se
e 5
– 3(
x –
y)2
as 5
min
us a
pos
itive
num
ber t
imes
a sq
uare
and
use
that
to
real
ize th
at it
s val
ue c
anno
t be
mor
e th
an 5
for a
ny re
al n
umbe
rs x
and
y.
Firs
t gra
ders
beg
in to
disc
ern
a nu
mbe
r pat
tern
or s
truc
ture
. For
in
stan
ce, i
f stu
dent
s rec
ogni
ze 1
2 +
3 =
15, t
hen
they
also
kno
w 3
+
12 =
15.
(Com
mut
ativ
e pr
oper
ty
of a
dditi
on.)
To a
dd 4
+ 6
+ 4
, the
fir
st tw
o nu
mbe
rs c
an b
e ad
ded
to m
ake
a te
n, so
4 +
6 +
4 =
10
+ 4
= 14
.
8: L
ook
for a
nd e
xpre
ss re
gula
rity
in re
peat
ed re
ason
ing.
M
athe
mat
ical
ly p
rofic
ient
stud
ents
not
ice
if ca
lcul
atio
ns a
re re
peat
ed, a
nd lo
ok b
oth
for g
ener
al
met
hods
and
for s
hort
cuts
. Upp
er e
lem
enta
ry st
uden
ts m
ight
not
ice
whe
n di
vidi
ng 2
5 by
11
that
th
ey a
re re
peat
ing
the
sam
e ca
lcul
atio
ns o
ver a
nd o
ver a
gain
, and
con
clud
e th
ey h
ave
a re
peat
ing
deci
mal
. By
payi
ng a
tten
tion
to th
e ca
lcul
atio
n of
slop
e as
they
repe
ated
ly c
heck
w
heth
er p
oint
s are
on
the
line
thro
ugh
(1, 2
) with
slop
e 3,
mid
dle
scho
ol st
uden
ts m
ight
ab
stra
ct th
e eq
uatio
n (y
– 2
)/(x
– 1
) = 3
. Not
icin
g th
e re
gula
rity
in th
e w
ay te
rms c
ance
l whe
n ex
pand
ing
(x –
1)(x
+ 1
), (x
– 1
)(x2
+ x
+ 1)
, and
(x –
1)(x
3 +
x2 +
x +
1) m
ight
lead
them
to th
e ge
nera
l for
mul
a fo
r the
sum
of a
geo
met
ric se
ries.
As th
ey w
ork
to so
lve
a pr
oble
m,
mat
hem
atic
ally
pro
ficie
nt st
uden
ts m
aint
ain
over
sight
of t
he p
roce
ss, w
hile
att
endi
ng to
the
deta
ils. T
hey
cont
inua
lly e
valu
ate
the
reas
onab
lene
ss o
f the
ir in
term
edia
te re
sults
.
In th
e ea
rly g
rade
s, st
uden
ts
notic
e re
petit
ive
actio
ns in
co
untin
g an
d co
mpu
tatio
n, e
tc.
Whe
n ch
ildre
n ha
ve m
ultip
le
oppo
rtun
ities
to a
dd a
nd su
btra
ct
-ten
and
mul
tiple
s of -
ten
they
no
tice
the
patt
ern
and
gain
a
bett
er u
nder
stan
ding
of p
lace
va
lue.
Stu
dent
s con
tinua
lly c
heck
th
eir w
ork
by a
skin
g th
emse
lves
, -D
oes t
his m
ake
sens
e?
State Board of Education-Adopted Grade One Page 14 of 41
Common Misconceptions.
• Some students misunderstand the meaning of the equal sign. The equal sign means “is the same as,”
but many primary students think the equal sign means “the answer is coming up” to the right of the
equal sign. When students see only examples of number sentences with the operation to the left of
the equal sign and the answer to the right, they overgeneralize the meaning of the equal sign, which
creates this misconception. First graders should see equations written multiple ways, for example 5 +
7 = 12 and 12 = 5 + 7. The Put Together/Take Apart Both (with addends unknown) problems—such
as “Robbie puts 12 balls in a basket. 4 are white and the rest are black. How many are black?”—are
particularly helpful for eliciting equations such as 12 = 5 + 7. These equations can begin in
kindergarten with small numbers (5 = 4 + 1) and they should be used throughout grade one for such
problems.
• Many students assume key words or phrases in a problem suggest the same operation every time.
For example, students might assume the word “left” always means subtract to find a solution. To help
students avoid this misconception include problems in which key words represent different
operations. For example, Joe took 8 stickers he no longer wanted and gave them to Anna. Now Joe
has 11 stickers “left”. How many stickers did Joe have to begin with? Facilitate students’
understanding of scenarios represented in word problems. Students should analyze word problems
(MP.1, MP.2) and not rely on key words.
195
Students can collaborate in small groups to develop problem solving strategies. Grade 196
one students use a variety of strategies and models, such as drawings, words, and 197
equations with symbols for the unknown numbers, to find the solutions. Students 198
explain, write, and reflect on their problem solving strategies. (MP.1, MP.2, MP.3, MP.4, 199
MP.6) For example, each student could write or draw a problem in which three whole 200
things are to be combined. Students might exchange their problems with other students, 201
solve them individually, and then discuss their models and solution strategies. The 202
students work together to solve each problem using a different strategy. The level of 203
difficulty for these problems can also be differentiated by using smaller numbers (up to 204
10) or larger numbers (up to 20). 205
206
Operations and Algebraic Thinking 1.OA Understand and apply properties of operations and the relationship between addition and
The Mathematics Framework was adopted by the California State Board of Education on November 6, 2013. The Mathematics Framework has not been edited for publication.
State Board of Education-Adopted Grade One Page 15 of 41
subtraction. 3. Apply properties of operations as strategies to add and subtract.3 Examples: If 8 + 3 = 11 is known,
then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
4. Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.
207
First grade students build their understanding of the relationship between addition and 208
subtraction. Instruction should include opportunities for students to investigate, identify 209
and then apply a pattern or structure in mathematics. For example, pose a string of 210
addition and subtraction problems involving the same three numbers chosen from the 211
numbers 0 to 20 (e.g., 4 + 6 = 10 and 6 + 4 = 10, 10 – 6 = 4 and 10 – 4 = 6 ). These are 212
related facts—a set of three numbers that can be expressed with an addition or 213
subtraction equation. Related facts help develop an understanding of the relationship 214
between addition and subtraction and the commutative and associative properties. 215
216
Students apply properties of operations as strategies to add and subtract (1.OA.3▲). 217
Although it is not necessary for grade one students to learn the names of the properties, 218
students need to understand the important ideas of the following properties: 219
• Identity property of addition (e.g., 6 = 6 + 0). “Adding 0 to a number results in the 220
same number.” 221
• Identity property of subtraction (e.g., 9 – 0 = 9). “Subtracting 0 from a number 222
results in the same number.” 223
• Commutative property of addition (e.g., 4 + 5 = 5 + 4). “The order in which you 224
add numbers doesn’t matter.” 225
• Associative property of addition (e.g., 3 + (9 + 1) = (3 + 9) +1 = 12 + 1 =13). 226
“When adding more than two numbers, it doesn’t matter which numbers you add 227
together first.” 228
229
Example.
Students build a tower of 8 green cubes and 3 yellow cubes, and another tower of 3 yellow and 8 green
3 Students need not use formal terms for these properties. The Mathematics Framework was adopted by the California State Board of Education on November 6, 2013. The Mathematics Framework has not been edited for publication.
State Board of Education-Adopted Grade One Page 16 of 41
cubes to show that order does not change the result in the operation of addition. Students can also use
cubes of 3 different colors to demonstrate that (2 + 6) + 4 is equivalent to 2 + (6 + 4) and then to prove 2
+ (6 + 4) = 2 + 10.
230
[Note; Sidebar] 231
Focus, Coherence, and Rigor Students apply the commutative and associative properties as strategies to solve addition problems
(1.OA.3▲) (these properties do not apply to subtraction). They use mathematical tools, such as cubes
and counters, and visual models (e.g., drawings and a 100 chart) to model and explain their thinking.
Students can share, discuss, and compare their strategies as a class. (MP.2, MP.7, and MP.8)
232
Students understand subtraction as an unknown-addend problem. (1.OA.4▲). Word 233
problems such as Put Together/Take Apart (with addend unknown) afford students a 234
context to see subtraction as the opposite of addition by finding an unknown addend. 235
Understanding subtraction as an unknown-addend addition problem is one of the 236
essential understandings students will need in middle school to extend arithmetic to 237
negative rational numbers (Adapted from Arizona 2010 and Progressions, K-5 CC and 238
OA 2011). 239
240
Common Misconceptions.
Students may assume that the commutative property applies to subtraction. After students have
discovered and applied the commutative property of addition, ask them to investigate whether this
property works for subtraction. Have students share and discuss their reasoning and guide them to
conclude that the commutative property does not apply to subtraction (Adapted from KATM 1st FlipBook
2012).
This can be challenging because students might think they can switch the addends in subtraction
equations because of their work with related fact equations using the commutative property for addition,
but students need to understand they cannot switch the total and an addend.
241
Operations and Algebraic Thinking 1.OA Add and subtract within 20.
5. Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). 6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use
strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between
The Mathematics Framework was adopted by the California State Board of Education on November 6, 2013. The Mathematics Framework has not been edited for publication.
8
Domain: Operations and Algebraic Thinking (OA)
Cluster: Understand and apply properties of operations and the relationship between addition
and subtraction.
Standard: 1.OA.3 Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is
known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative
property of addition.) (Students need not use formal terms for these properties.)
Standards for Mathematical Practice (MP):
MP.2. Reason abstractly and quantitatively. MP.7. Look for and make use of structure.
MP.8. Look for and express regularity in repeated reasoning.
Connections: This cluster is connected to the First Grade Critical Area of Focus #1, Developing understanding of
addition, subtraction, and strategies for addition and subtraction within 20.
This cluster is connected to Understand addition as putting together and adding to, and understand
subtraction as taking apart and taking from in Kindergarten, to Add and subtract within 20 and Use place
value understanding and properties of operations to add and subtract in Grade 1 and to Use place value
understanding and properties of operations to add and subtract in Grade 2.
Explanations and Examples: 1.OA.3 calls for students to apply properties of operations as strategies to add and subtract.
Students do not need to use formal terms for these properties. Students should use mathematical tools, such as cubes and counters, and representations such as the number line
and a 100 chart to model these ideas.
Example:
Student can build a tower of 8 green cubes and 3 yellow cubes and another tower of 3 yellow and 8 green
cubes to show that order does not change the result in the operation of addition. Students can also use
cubes of 3 different colors to prove that (2 + 6) + 4 is equivalent to
2 + (6 + 4) and then to prove 2 + 6 + 4 = 2 + 10. Students should understand the important ideas of the
following properties:
Identity property of addition (e.g., 6 = 6 + 0)
Identity property of subtraction (e.g., 9 – 0 = 9)
Commutative property of addition--Order does not matter when you add numbers.
e.g. 4 + 5 = 5 + 4)
Associative property of addition--When adding a string of numbers you can add any two numbers
first. (e.g., 3 + 9 + 1 = 3 + 10 = 13)
Student 1
Using a number balance to investigate the commutative property. If I put a weight on 8 first and then 2,
I think that it will balance if I put a weight on 2 first this time then on 8.
Students need several experiences investigating whether the commutative property works with
subtraction. The intent is not for students to experiment with negative numbers but only to recognize that
taking 5 from 8 is not the same as taking 8 from 5. Students should recognize that they will be working
with numbers later on that will allow them to subtract larger numbers from smaller numbers. However, in
first grade we do not work with negative numbers.
9
Instructional Strategies (1.AO. 3-4)
Instruction needs to focus on lessons that help students to discover and apply the commutative and associative properties as strategies for solving addition problems. It is not necessary for students to learn the names for these properties. It is important for students to share, discuss
and compare their strategies as a class. The second focus is using the relationship between addition and subtraction as a strategy to solve unknown-addend problems. Students naturally
connect counting on to solving subtraction problems. For the problem ―15 – 7 = ?‖ they think about the number they have to add to 7 to get to 15. First graders should be working with sums and differences less than or equal to 20 using the numbers 0 to 20.
Provide investigations that require students to identify and then apply a pattern or structure in mathematics. For example, pose a string of addition and subtraction problems involving the
same three numbers chosen from the numbers 0 to 20, like 4 + 13 = 17 and 13 + 4 = 17. Students analyze number patterns and create conjectures or guesses. Have students choose other combinations of three numbers and explore to see if the patterns work for all numbers 0
to 20. Students then share and discuss their reasoning. Be sure to highlight students’ uses of the commutative and associative properties and the relationship between addition and
subtraction. Expand the student work to three or more addends to provide the opportunities to change the order and/or groupings to make tens. This will allow the connections between place-value
models and the properties of operations for addition to be seen. Understanding the commutative and associative properties builds flexibility for computation and estimation, a key element of
number sense. Provide multiple opportunities for students to study the relationship between addition and subtraction in a variety of ways, including games, modeling and real-world situations. Students
need to understand that addition and subtraction are related, and that subtraction can be used to solve problems where the addend is unknown.
Common Misconceptions: A common misconception is that the commutative property applies to subtraction. After students have discovered and applied the commutative property for addition, ask them to investigate whether this property works for subtraction. Have students share and discuss their
reasoning and guide them to conclude that the commutative property does not apply to subtraction.
First graders might have informally encountered negative numbers in their lives, so they think they can take away more than the number of items in a given set, resulting in a negative number below zero. Provide many problems situations where students take away all objects
from a set, e.g. 19 - 19 = 0 and focus on the meaning of 0 objects and 0 as a number. Ask students to discuss whether they can take away more objects than what they have.
1.OA.3
10
Domain: Operations and Algebraic Thinking (OA)
Cluster: Understand and apply properties of operations and the relationship between addition
and subtraction.
Standard: 1.OA.4 Understand subtraction as an unknown-addend problem.
For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. Add and subtract
within 20. Standards for Mathematical Practice (MP): MP.2. Reason abstractly and quantitatively. MP.7. Look for and make use of structure. MP.8. Look for and express regularity in repeated reasoning.
Connections: See 1.0A.3
Explanations and Examples: 1.OA.4 asks for students to use subtraction in the context of unknown addend problems. When determining the answer to a subtraction problem, 12 - 5, students think, ―If I have 5, how many
more do I need to make 12?‖ Encouraging students to record this symbolically, 5 + ? = 12, will develop
their understanding of the relationship between addition and subtraction. Some strategies they may use
are counting objects, creating drawings, counting up, using number lines or 10 frames to determine an
answer. Refer to Table 1 to consider the level of difficulty of this standard.
Example:
12 – 5 = __ could be expressed as 5 + __ = 12. Students should use cubes and counters, and
representations such as the number line and the100 chart, to model and solve problems involving the
inverse relationship between addition and subtraction.
Student 1
I used a ten frame. I started with 5 counters. I now that I had to have 12, which is one full ten fram
and two left overs. I needed 7 counters, so 12 – 5 = 7
Student 2
I used a part-part-whole diagram. I put 5 counters on one side. I wrote 12 above the diagram. I put
counters into the other side until there were 12 in all. I know I put 7 counters into the other side,
so 12 - 5 = 7.
12
5 7
Student 3 Draw a number line I started at 5 and counted up until I reached 12. I counted 7 numbers, so I knew that 12 – 5 = 7.
Instructional Strategies:
See 1.OA.3 Common Misconceptions:
See 1.OA.3
1.OA.4
SCUSD 1st Grade Curriculum Map
Unit 1: Adding & Subtracting Within 20 Domain: Operations and Algebraic Thinking
Cluster: Understand and apply properties of operations and the relationship between addition and subtraction
Sequence of Learning Outcomes 1.OA.3 – 1.OA.4
3. Identify properties of addition and subtraction such as adding or subtracting zero to or from a number resulting in the same number (e.g., 6 = 6 + 0; 6 – 0 = 6).
4. Apply and understand commutative (e.g., 4 + 5 = 5 + 4) and associate (e.g., 3 + (9 + 1) = (3 + 9) + 1; (3 + 9) + 1= 12 + 1 = 13) properties of addition.
5. Investigate, identify, and apply a pattern or structure in addition and subtraction (e.g., the relationship between numbers 4, 6, and 10).
6. Understand and solve subtraction problems as unknown-addend (e.g., 10 minus 8 can be solved by asking 8 plus what equals 10).
SCUSD 1st Grade Textbook
Sequence of Learning Outcomes 1.OA.3-4
1-7 Children will learn to add in any order.
2-1 Children will solve problems by finding the missing part.
2-2 Children will find a missing part of 8 when one part is known.
2-3 Children will use subtraction to find the missing part of 9 when one part is known.
3-4 Children will use counters and a part-part-whole mat to find missing parts of 10.
4-1 Children will count on to add, starting with the greater number.
4-7 Children will learn to use doubles addition facts to master related subtraction facts.
4-8 Children will understand how addition facts to 8 relate to subtraction facts to 8.
4-9 Children will write related addition and subtraction facts to 12.
5-8 Children will use the associative and commutative properties to add three numbers.
6-3 Find subtraction facts to 18 and learn the relationship between addition and subtraction. 6-4 Use a part-part-whole model to find the subtraction facts and addition facts in a fact family.
Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. 1
Topic 1Test
NameMark the best answer.
1. Which number does the picture show?
𝖠 5
𝖡 9
𝖢 10
𝖣 11
2. 2 crayons are inside the box. 5 crayons are outside the box. How many crayons are there in all?
𝖠 2
𝖡 3
𝖢 5
𝖣 7
3. Do the choices below show parts of 8? Mark Yes or No.
𝖠 Yes 𝖡 No
𝖠 Yes 𝖡 No
𝖠 Yes 𝖡 No
𝖠 Yes 𝖡 No
4. How many rabbits are there in all?
𝖠 10
𝖡 9
𝖢 6
𝖣 3
1 of 2Topic 1
Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. 1
Name
Use the picture to solve. Write the parts. Then write an addition sentence.
5. Ted drew 3 trees.Then he drew 3 more. How many trees did he draw in all? Write an addition sentence.
+
+ =
6. 6 birds are in a tree. Some more birds join them. Now there are 8 birds in all. How many birds came to join? Write an addition sentence.
+ =
7. Use counters to solve.Write the missing addend to complete each addition sentence.
+ 1 = 7
1 + = 7
Topic 1 2 of 2
Name
Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. 1
Quick Check
1-7
1. Which number sentencestell about the counters?
𝖠 3 + 3 = 6 4 + 2 = 6
𝖡 3 + 4 = 7 4 + 3 = 7
𝖢 4 + 4 = 8 5 + 3 = 8
𝖣 7 + 2 = 9 6 + 3 = 9
2. Kevin has 4 pennies in his left pocket.He has the same number of penniesin his right pocket.How many pennies are in Kevin’s pocketsin all?
𝖠 10
𝖡 8
𝖢 6
𝖣 4
3. Complete the picture to solve.Write the number sentences.
Marni has some hair ribbons.She has 7 pink ribbons. The rest are purple ribbons. She has 10 hair ribbons in all.
______ + ______ = ______
______ + ______ = ______Q 1•7
