Practices Make Proficient Students Academic Success Institute Sheila Walters, [email protected] Session Website: http://bit.ly/1DjtVnl
Practices Make Proficient Students
Academic Success Institute
Sheila Walters, [email protected]
Session Website: http://bit.ly/1DjtVnl
Updated 10/18/10 1
Mathematics | Standards for Mathematical Practice
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be
Updated 10/18/10 2
gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 x 8 equals the well-remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.
The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.
In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.
Impl
emen
ting
Stan
dard
s for
Mat
hem
atic
al P
ract
ices
In
stitu
te fo
r A
dvan
ced
Stud
y/Pa
rk C
ity M
athe
mat
ics
Inst
itute
/ Cr
eate
d by
Lea
rnin
g Se
rvic
es, M
odifi
ed b
y M
elis
a H
anco
ck, 2
013
#1 M
ake
sens
e of
pro
blem
s an
d pe
rsev
ere
in s
olvi
ng t
hem
. Su
mm
ary
of S
tand
ards
for
Mat
hem
atic
al P
ract
ice
Que
stio
ns t
o D
evel
op M
athe
mat
ical
Thi
nkin
g
1. M
ake
sens
e of
pro
blem
s an
d pe
rsev
ere
in s
olvi
ng th
em.
• In
terp
ret a
nd m
ake
mea
ning
of t
he p
robl
em lo
okin
g fo
r st
artin
g po
ints
. A
naly
ze
wha
t is
give
n to
exp
lain
to th
emse
lves
the
mea
ning
of t
he p
robl
em.
• Pl
an a
sol
utio
n pa
thw
ay in
stea
d of
jum
ping
to a
sol
utio
n.
• M
onito
r th
e pr
ogre
ss a
nd c
hang
e th
e ap
proa
ch if
nec
essa
ry.
• Se
e re
latio
nshi
ps b
etw
een
vari
ous
repr
esen
tatio
ns.
• Re
late
cur
rent
situ
atio
ns to
con
cept
s or
ski
lls p
revi
ousl
y le
arne
d an
d co
nnec
t m
athe
mat
ical
idea
s to
one
ano
ther
.
• St
uden
ts a
sk th
emse
lves
, “D
oes
this
mak
e se
nse?
” an
d un
ders
tand
var
ious
ap
proa
ches
to s
olut
ions
.
H
ow w
ould
you
des
crib
e th
e pr
oble
m in
you
r ow
n w
ords
?
How
wou
ld y
ou d
escr
ibe
wha
t you
are
tryi
ng to
find
?
W
hat d
o yo
u no
tice
abou
t...?
Wha
t inf
orm
atio
n is
giv
en in
the
prob
lem
?
D
escr
ibe
the
rela
tions
hip
betw
een
the
quan
titie
s.
D
escr
ibe
wha
t you
hav
e al
read
y tr
ied.
Wha
t mig
ht y
ou c
hang
e?
Ta
lk m
e th
roug
h th
e st
eps
you’
ve u
sed
to th
is p
oint
.
Wha
t ste
ps in
the
proc
ess
are
you
mos
t con
fiden
t abo
ut?
W
hat a
re s
ome
othe
r st
rate
gies
you
mig
ht tr
y?
W
hat a
re s
ome
othe
r pr
oble
ms
that
are
sim
ilar
to th
is o
ne?
H
ow m
ight
you
use
one
of y
our
prev
ious
pro
blem
s to
hel
p y
ou b
egin
?
How
els
e m
ight
you
org
aniz
e...r
epre
sent
...sh
ow...
?
Impl
emen
tati
on C
hara
cter
isti
cs: W
hat
does
it lo
ok li
ke in
pla
nnin
g an
d de
liver
y?
Task
: ele
men
ts to
kee
p in
min
d w
hen
dete
rmin
ing
lear
ning
exp
erie
nces
Teac
her:
act
ions
that
furt
her
the
deve
lopm
ent o
f mat
h pr
actic
es w
ithin
thei
r st
uden
ts
Task
:
� Re
quir
es s
tude
nts
to e
ngag
e w
ith c
once
ptua
l ide
as th
at u
nder
lie th
e pr
oced
ures
to c
ompl
ete
the
task
and
dev
elop
und
erst
andi
ng.
� Re
quir
es c
ogni
tive
effo
rt -
whi
le p
roce
dure
s m
ay b
e fo
llow
ed, t
he a
ppro
ach
or p
athw
ay is
not
exp
licitl
y su
gges
ted
by th
e ta
sk, o
r ta
sk in
stru
ctio
ns a
nd m
ultip
le e
ntry
poi
nts
are
avai
labl
e. T
he p
robl
em fo
cuse
s st
uden
ts’ a
tten
tion
on a
mat
hem
atic
al id
ea a
nd p
rovi
des
an o
ppor
tuni
ty to
dev
elop
and
/or
use
mat
hem
atic
al h
abits
of m
ind.
� A
llow
s fo
r m
ultip
le e
ntry
poi
nts
and
solu
tion
path
s as
wel
l as,
mul
tiple
rep
rese
ntat
ions
, suc
h as
vis
ual d
iagr
ams,
man
ipul
ativ
es, s
ymbo
ls, a
nd p
robl
em s
ituat
ions
. M
akin
g co
nnec
tions
am
ong
mul
tiple
rep
rese
ntat
ions
to d
evel
op m
eani
ng.
� Re
quir
es s
tude
nts
to a
cces
s re
leva
nt k
now
ledg
e an
d ex
peri
ence
s an
d m
ake
appr
opri
ate
use
of th
em in
wor
king
thro
ugh
the
task
.
� Re
quir
es s
tude
nts
to d
efen
d an
d ju
stify
thei
r so
lutio
ns.
Teac
her:
� A
llow
s st
uden
ts ti
me
to in
itiat
e a
plan
; use
s qu
estio
n pr
ompt
s as
nee
ded
to a
ssis
t stu
dent
s in
dev
elop
ing
a pa
thw
ay.
� Co
ntin
ually
ask
s st
uden
ts if
thei
r pl
ans
and
solu
tions
mak
e se
nse.
� Q
uest
ions
stu
dent
s to
see
con
nect
ions
to p
revi
ous
solu
tion
atte
mpt
s an
d/or
task
s to
mak
e se
nse
of c
urre
nt p
robl
em.
� Co
nsis
tent
ly a
sks
to d
efen
d an
d ju
stify
thei
r so
lutio
n by
com
pari
ng s
olut
ion
path
s.
� D
iffer
entia
tes
to k
eep
adva
nced
stu
dent
s ch
alle
nged
dur
ing
wor
k tim
e
Impl
emen
ting
Stan
dard
s for
Mat
hem
atic
al P
ract
ices
In
stitu
te fo
r A
dvan
ced
Stud
y/Pa
rk C
ity M
athe
mat
ics
Inst
itute
/ Cr
eate
d by
Lea
rnin
g Se
rvic
es, M
odifi
ed b
y M
elis
a H
anco
ck, 2
013
#3 C
onst
ruct
via
ble
argu
men
ts a
nd c
riti
que
the
reas
onin
g of
oth
ers.
Sum
mar
y of
Sta
ndar
ds fo
r M
athe
mat
ical
Pra
ctic
e Q
uest
ions
to
Dev
elop
Mat
hem
atic
al T
hink
ing
3. C
onst
ruct
via
ble
argu
men
ts a
nd c
riti
que
the
reas
onin
g of
oth
ers.
•
Ana
lyze
pro
blem
s an
d us
e st
ated
mat
hem
atic
al a
ssum
ptio
ns, d
efin
ition
s, a
nd
esta
blis
hed
resu
lts in
con
stru
ctin
g ar
gum
ents
.
• Ju
stify
con
clus
ions
with
mat
hem
atic
al id
eas.
• Li
sten
to th
e ar
gum
ents
of o
ther
s an
d as
k us
eful
que
stio
ns t
o de
term
ine
if an
ar
gum
ent m
akes
sen
se.
• A
sk c
lari
fyin
g qu
estio
ns o
r su
gges
t ide
as to
impr
ove/
revi
se th
e ar
gum
ent.
• Co
mpa
re t
wo
argu
men
ts a
nd d
eter
min
e co
rrec
t or
flaw
ed lo
gic.
W
hat m
athe
mat
ical
evi
denc
e su
ppor
ts y
our
solu
tion?
How
can
you
be
sure
that
...?
/ H
ow c
ould
you
pro
ve th
at...
? W
ill it
stil
l wor
k if.
..?
W
hat w
ere
you
cons
ider
ing
whe
n...?
How
did
you
dec
ide
to tr
y th
at s
trat
egy?
How
did
you
test
whe
ther
you
r ap
proa
ch w
orke
d?
H
ow d
id y
ou d
ecid
e w
hat t
he p
robl
em w
as a
skin
g yo
u to
find
? (W
hat w
as u
nkno
wn?
)
Did
you
try
a m
etho
d th
at d
id n
ot w
ork?
Why
did
n’t i
t wor
k? W
ould
it e
ver
wor
k?
W
hy o
r w
hy n
ot?
W
hat i
s th
e sa
me
and
wha
t is
diff
eren
t abo
ut...
?
How
cou
ld y
ou d
emon
stra
te a
cou
nter
-exa
mpl
e?
Impl
emen
tati
on C
hara
cter
isti
cs: W
hat
does
it lo
ok li
ke in
pla
nnin
g an
d de
liver
y?
Task
: ele
men
ts to
kee
p in
min
d w
hen
dete
rmin
ing
lear
ning
exp
erie
nces
Teac
her:
act
ions
that
furt
her
the
deve
lopm
ent o
f mat
h pr
actic
es w
ithin
thei
r st
uden
ts
Task
:
� Is
str
uctu
red
to b
ring
out
mul
tiple
rep
rese
ntat
ions
, app
roac
hes,
or
erro
r an
alys
is.
� Em
beds
dis
cuss
ion
and
com
mun
icat
ion
of r
easo
ning
and
just
ifica
tion
with
oth
ers.
� Re
quir
es s
tude
nts
to p
rovi
de e
vide
nce
to e
xpla
in th
eir
thin
king
bey
ond
mer
ely
usin
g co
mpu
tatio
nal s
kills
to fi
nd a
sol
utio
n.
� Ex
pect
s st
uden
ts to
giv
e fe
edba
ck a
nd a
sk q
uest
ions
of o
ther
s’ s
olut
ions
.
Teac
her:
� En
cour
ages
stu
dent
s to
use
pro
ven
mat
hem
atic
al u
nder
stan
ding
s, (d
efin
ition
s, p
rope
rtie
s, c
onve
ntio
ns, t
heor
ems,
etc
.), to
sup
port
thei
r re
ason
ing.
� Q
uest
ions
stu
dent
s so
they
can
tell
the
diff
eren
ce b
etw
een
assu
mpt
ions
and
logi
cal c
onje
ctur
es.
� A
sks
ques
tions
that
req
uire
stu
dent
s to
just
ify th
eir
solu
tion
and
thei
r so
luti
on p
athw
ay.
� Pr
ompt
s st
uden
ts to
res
pect
fully
eva
luat
e pe
er a
rgum
ents
whe
n so
lutio
ns a
re s
hare
d.
� A
sks
stud
ents
to c
ompa
re a
nd c
ontr
ast v
ario
us s
olut
ion
met
hods
.
� Cr
eate
s va
riou
s in
stru
ctio
nal o
ppor
tuni
ties
for
stud
ents
to e
ngag
e in
mat
hem
atic
al d
iscu
ssio
ns (w
hole
gro
up, s
mal
l gro
up, p
artn
ers,
etc
.).
Impl
emen
ting
Stan
dard
s for
Mat
hem
atic
al P
ract
ices
In
stitu
te fo
r A
dvan
ced
Stud
y/Pa
rk C
ity M
athe
mat
ics
Inst
itute
/ Cr
eate
d by
Lea
rnin
g Se
rvic
es, M
odifi
ed b
y M
elis
a H
anco
ck, 2
013
#6 A
tten
d to
pre
cisi
on.
Sum
mar
y of
Sta
ndar
ds fo
r M
athe
mat
ical
Pra
ctic
e Q
uest
ions
to
Dev
elop
Mat
hem
atic
al T
hink
ing
6. A
tten
d to
pre
cisi
on.
• Co
mm
unic
ate
prec
isel
y w
ith o
ther
s an
d tr
y to
use
cle
ar m
athe
mat
ical
lang
uage
whe
n di
scus
sing
thei
r re
ason
ing.
• U
nder
stan
d m
eani
ngs
of s
ymbo
ls u
sed
in m
athe
mat
ics
and
can
labe
l qua
ntiti
es
appr
opri
atel
y.
• Ex
pres
s nu
mer
ical
ans
wer
s w
ith
a de
gree
of p
reci
sion
app
ropr
iate
for
the
prob
lem
co
ntex
t.
• Ca
lcul
ate
effic
ient
ly a
nd a
ccur
atel
y.
W
hat m
athe
mat
ical
term
s ap
ply
in th
is s
ituat
ion?
How
did
you
kno
w y
our
solu
tion
was
rea
sona
ble?
Exp
lain
how
you
mig
ht s
how
that
you
r so
lutio
n an
swer
s th
e pr
oble
m.
Is
ther
e a
mor
e ef
ficie
nt s
trat
egy?
How
are
you
sho
win
g th
e m
eani
ng o
f the
qua
ntiti
es?
W
hat s
ymbo
ls o
r m
athe
mat
ical
not
atio
ns a
re im
port
ant i
n th
is p
robl
em?
W
hat m
athe
mat
ical
lang
uage
..., d
efin
ition
s...,
pro
pert
ies
can
you
use
to e
xpla
in...
?
H
ow c
ould
you
test
you
r so
lutio
n to
see
if it
ans
wer
s th
e pr
oble
m?
Impl
emen
tati
on C
hara
cter
isti
cs: W
hat
does
it lo
ok li
ke in
pla
nnin
g an
d de
liver
y?
Task
: ele
men
ts to
kee
p in
min
d w
hen
dete
rmin
ing
lear
ning
exp
erie
nces
Teac
her:
act
ions
that
furt
her
the
deve
lopm
ent o
f mat
h pr
actic
es w
ithin
thei
r st
uden
ts
Task
:
� Re
quir
es s
tude
nts
to u
se p
reci
se v
ocab
ular
y (in
wri
tten
and
ver
bal r
espo
nses
) whe
n co
mm
unic
atin
g m
athe
mat
ical
idea
s.
� Ex
pect
s st
uden
ts to
use
sym
bols
app
ropr
iate
ly.
� Em
beds
exp
ecta
tions
of h
ow p
reci
se th
e so
lutio
n ne
eds
to b
e (s
ome
may
mor
e ap
prop
riat
ely
be e
stim
ates
). Te
ache
r:
� Co
nsis
tent
ly d
eman
ds a
nd m
odel
s pr
ecis
ion
in c
omm
unic
atio
n an
d in
mat
hem
atic
al s
olut
ions
. (u
ses
and
mod
els
corr
ect c
onte
nt te
rmin
olog
y).
� Ex
pect
s st
uden
ts to
use
pre
cise
mat
hem
atic
al v
ocab
ular
y du
ring
mat
hem
atic
al c
onve
rsat
ions
. (id
entif
ies
inco
mpl
ete
resp
onse
s an
d as
ks s
tude
nts
to re
vise
thei
r res
pons
e).
� Q
uest
ions
stu
dent
s to
iden
tify
sym
bols
, qua
ntiti
es, a
nd u
nits
in a
cle
ar m
anne
r
Stud
ent
Vita
l Act
ions
Prin
cipl
es
All
stud
ents
par
tici
pate
(e.g
., bo
ys a
nd g
irls
, ELL
and
sp
ecia
l nee
ds s
tude
nts)
, not
just
the
hand
-rai
sers
. Eq
uity
requ
ires
part
icip
atio
n.A
➤
Stud
ents
say
a s
econ
d se
nten
ce (s
pont
aneo
usly
or
prom
pted
by
the
teac
her
or a
noth
er s
tude
nt) t
o ex
tend
and
exp
lain
thei
r th
inki
ng.
CCSS
-M p
ract
ices
1 |
2 |
3 |
6
Logi
c co
nnec
ts
sent
ence
s.B ➤
Stud
ents
tal
k ab
out
each
oth
er’s
thi
nkin
g (n
ot ju
st th
eir
own)
.
CCSS
-M p
ract
ices
1 |
2 |
3 |
6 |
7 |
8
Und
erst
andi
ng e
ach
othe
r’s re
ason
ing
deve
lops
reas
onin
g pr
ofici
ency
. C ➤
Stud
ents
rev
ise
thei
r th
inki
ng, a
nd th
eir
wri
tten
wor
k
incl
udes
revi
sed
expl
anat
ions
and
just
ifica
tions
. CC
SS-M
pra
ctic
es 1
| 2
| 3
| 4
Revi
sing
exp
lana
tions
sol
idifi
es
unde
rsta
ndin
g.D
➤
Stud
ents
look
for
mor
e pr
ecis
e w
ays
of e
xpre
ssin
g th
eir
thin
king
, en
cour
agin
g ea
ch o
ther
to lo
ok fo
r an
d us
e ac
adem
ic la
ngua
ge.
CCS
S-M
pra
ctic
es 3
| 6
Acad
emic
lang
uage
pro
mot
es
prec
ise
thin
king
.E ➤
Engl
ish
lear
ners
pro
duce
lang
uage
that
com
mun
icat
es
idea
s an
d re
ason
ing,
eve
n w
hen
that
lang
uage
is im
perf
ect.
CCSS
-M p
ract
ices
1 |
2 |
3 |
6
ELLs
dev
elop
lang
uage
thro
ugh
expl
anat
ion.
F ➤
Stud
ents
eng
age
and
pers
ever
e
at p
oint
s of
diffi
culty
, cha
lleng
e, o
r er
ror.
C
CSS-
M p
ract
ice
1
Prod
uctiv
e st
rugg
le
prod
uces
gro
wth
.G
➤
1.OA.2: 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.
Winter Fun
There were 17 children playing in the snow at recess, 8 of them were girls and the rest were boys. How many were boys?
Answer ________________
Explain thinking in numbers, words, or pictures.
CCSS-4.NF.3a,d: Understand a fraction a/b with a > 1 as a sum of fractions 1/b. a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
Who Walked Farther?
Tray walked ¾ of a mile to Jose’s house and then walked back home. His sister Maria walked ½ of a mile to the candy store, ½ of a mile to visit her grandma and then walked ½ of a mile home. Tray said he walked farther, but Maria said they walked the same total distance. Who is correct?
Answer ______________________________
Explain your answer using words, diagrams and/or numbers.
8.EE.8: Analyze and solve pairs of simultaneous linear equations
Name __________________________________________________________ Date ________________
Furnace Repairs Antonio’s furnace has quit working during the coldest part of the year. He decides to call some furnace specialists to see what it might cost to have the furnace fixed. Since he is unsure of the parts he needs, he decides to compare the cost based only on the service fee and labor cost. Below are the price estimates he has gathered. Each company has also given him an estimate of the time it will take to fix the furnace. Company A charges $35 per hour Company B charges a $20 service fee for coming out to the house and then $25
per each additional hour. For which time intervals should Antonio choose Company A, Company B? Answer: ___________________________________________________________
Support your decision with sound reasoning and representations. Consider including equations, tables, and/or graphs.
Squares Upon Squares
Task provided by Ruth Parker and the Mathematics Education Collaborative, MEC, http://www.mec-‐math.org
• How do you see the shapes growing? Illustrate how you see it growing
Case 1
Case 3
Case 2