Practices Make Proficient Students

Academic Success Institute

Sheila Walters, Sheila_Walters@sccoe.org

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

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path

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wel

l as,

mul

tiple

rep

rese

ntat

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, suc

h as

vis

ual d

iagr

ams,

man

ipul

ativ

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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

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to a

cces

s re

leva

nt k

now

ledg

e an

d ex

peri

ence

s an

d m

ake

appr

opri

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use

of th

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wor

king

thro

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the

task

.

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quir

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tude

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d ju

stify

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lutio

ns.

Teac

her:

� A

llow

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uden

ts ti

me

to in

itiat

e a

plan

; use

s qu

estio

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ompt

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nee

ded

to a

ssis

t stu

dent

s in

dev

elop

ing

a pa

thw

ay.

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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

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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.

sgonzales

Typewritten Text

10

sgonzales

Typewritten Text

12

sgonzales

Typewritten Text

13

sgonzales

Typewritten Text

14

sgonzales

Typewritten Text

15

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