connected mathematics project 3 preparing for the

Connected Mathematics Project 3 Preparing for the Transition:

Supporting Teachers to Successfully Implement CMP3

Professional DevelopmentParticiPant Workbook

Connect Mathematics Project 3: Preparing for the Transition: Supporting Teachers to Successfully Implement CMP3, Participant Workbook

© 2013 Pearson, Inc.2

For Professional Development resources and programs, visit www.pearsonpd.com.

Pearson School achievement Servicesconnected Mathematics Project 3Preparing for the transition: Supporting teachers to Successfully implement cMP3Participant Workbook

Pearson provides these materials for the expressed purpose of training district and school personnel on the effective implementation of Pearson products within classrooms, and other professional development topics. these materials may not be used for any other purpose, and may not be reproduced, distributed, or stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without Pearson’s express written permission.

Excel® is a trademark of the Microsoft group of companies.

Published by Pearson School achievement Services, a division of Pearson, inc.1900 E. Lake ave., Glenview, iL 60025

© 2013 Pearson, inc.all rights reserved.Printed in the United States of america.



Table of Contents

agenda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Section 1: Focus, coherence, and rigor: How cMP3 Meets the content and Pedagogical Shifts of the common core. . . . . . . . . . . . . . . . . . 8

Section 2: the inquiry-based Learning Structure of cMP3 . . . . . . . . . . . . . . . . 21

Section 3: Supporting teachers to build Mathematical communities capable of cMP3’s inquiry-based Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Section 4: inquiry-based classrooms: a Shift in Student Experience . . . . . . . . 33

Section 5: Developing Your Plan for Supporting teachers. . . . . . . . . . . . . . . . . 39

Reflection and Closing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48



Agenda

Section

introduction

1. Focus, coherence, and rigor: How cMP3 Meets the content and Pedagogical Shifts of the common core

2. the inquiry-based Learning Structure of cMP3

3. Supporting teachers to build Mathematical communities capable of cMP3’s inquiry-based Learning

4. inquiry-based classrooms: a Shift in Student Experience

5. Developing Your Plan for Supporting teachers

Reflection and Closing



Outcomes

at the conclusion of this workshop, you will be able to

• understand the CMP3 inquiry-based lesson structure;

• investigate how CMP3 aligns with the focus, coherence, and rigor of the content and pedagogical shifts of the Common Core;

• apply best practices to support teachers to implement CMP3 with fidelity;

• support students to successfully adopt inquiry-based instruction; and

• develop a plan for supporting teachers to implement CMP3 in their classrooms throughout the school year.



Introduction

Year One Professional Development Goals • Support teachers in learning how to teach with focus, coherence, and rigor of the Common

core Learning Standards in Mathematics (ccLSM) through the cMP3 curriculum.

• Learn strategies and best practices to support student learning through inquiry-based instruction.

• Support teachers in making instructional decisions within the CMP3 curriculum through an analysis of student and teacher work.

• Deepen teacher content knowledge through the effective implementation of the CMP3 curriculum.

Year One Professional Development Components • Intensive Sessions (A–E)

• Intensive Webinars (synchronous and recorded)

• Teacher Sessions (1–4)

• Teacher Webinars (synchronous and recorded)

• Teacher Tutorials (static, online Web resources)



Introduction

Mathematical Task a. Mirari conjectures that, for any three consecutive numbers, one number would be divisible

by 3. Do you think Mirari is correct? Explain.

b. Gia claims that the sum of any three consecutive whole numbers is divisible by 6. is this true? Explain.

c. kim claims that the product of any three consecutive whole numbers is divisible by 6. is this true? Explain.

d. Does the product of any four consecutive whole numbers have any interesting properties? Explain.

(Pearson Education, inc. 2014c, 85)



Section 1: Focus, Coherence, and Rigor: How CMP3 Meets the Content and Pedagogical Shifts of the Common Core

Quick Write: The ShiftFocus:

coherence:

rigor:




Focus: Two Out of Three Ain’t Badthe following chart is an excerpt from Student achievement Partners (n.d.). cross out the one area of major focus that does not represent an area of major focus for the indicated grade.

Grade Which 2 of the following represent areas of major focus for the indicated grade?

Kcompare numbers Use tally marks Understand meaning of

addition and subtraction

1add and subtract within 20 Measure lengths indirectly

and by iterating length unitscreate and extend patterns and sequences

2Work with equal groups of objects to gain foundations for multiplication

Understand place value identify line of symmetry in two dimensional

3Multiply and divide within 100

identify the measures of central tendency and distribution

Develop understanding of fractions as numbers

4Examine transformations on the coordinate plane

Generalize place value understanding for multi-digit whole numbers

Extend understanding of fraction equivalence and ordering

5

Understand and calculate probability of single events

Understand the place value system

apply and extend previous understandings of multiplication and division to multiply and divide fractions

6

Understand ratio concepts and use ratio reasoning to solve problems

identify and utilize rules of divisibility

apply and extend previous understandings of arithmetic to algebraic expressions

7

apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers

Use properties of operations to generate equivalent expressions

Generate the prime factorization of numbers to solve

8Standard form of a linear equation

Define, evaluate, and compare functions

Understand and apply the Pythagorean theorem

Alg.1Quadratic inequalities Linear and quadratic

functionscreating equations to model situations

Alg.2

Exponential and logarithmic functions

Polar coordinates Using functions to model situations




Stoplight Highlighting: Where is the focus?

(PAARC 2011, 29–30)




CM

P3 F

ocus

es W

here

the

Stan

dard

s Fo

cus

Gra

de 6

Maj

or C

lust

erC

onte

nt o

f Maj

or C

lust

erLi

st a

nd T

ally

CM

P3 In

vest

igat

ions

on

Con

tent

of M

ajor

Clu

ster

6.R

P.1–

3: U

nder

stan

d ra

tio

conc

epts

and

use

ratio

reas

onin

g to

sol

ve p

robl

ems.

• U

nder

stan

d an

d us

e la

ngua

ge o

f rat

ios

to d

escr

ibe

the

rela

tions

hip

betw

een

two

quan

titie

s.•

Und

erst

and

unit

rate

s an

d us

e it

in th

e co

ntex

t of a

ratio

rela

tions

hip.

• S

olve

real

-wor

ld p

robl

ems

with

ratio

tabl

es, t

ape

diag

ram

s, d

oubl

e nu

mbe

r lin

e di

agra

ms,

or e

quat

ions

, inc

ludi

ng p

robl

ems

with

uni

t pric

ing,

per

cent

, sp

eed,

and

uni

ts o

f mea

sure

men

t.

num

ber o

f inv

estig

atio

ns: _

____

____

6.n

S.1

: app

ly a

nd e

xten

d pr

evio

us u

nder

stan

ding

s of

m

ultip

licat

ion

and

divi

sion

to

divi

de fr

actio

ns b

y fra

ctio

ns.

• In

terp

ret a

nd c

ompu

te q

uotie

nts

of fr

actio

ns.

• S

olve

wor

d pr

oble

ms

usin

g di

visi

on o

f fra

ctio

ns b

y fra

ctio

ns.

• U

se v

isua

l fra

ctio

n m

odel

s an

d eq

uatio

ns to

repr

esen

t pro

blem

s an

d ju

stify

con

cept

ual u

nder

stan

ding

of t

he s

ituat

ion.

num

ber o

f inv

estig

atio

ns: _

____

____

6.N

S.5

–8: A

pply

and

ext

end

prev

ious

und

erst

andi

ngs

of

num

bers

to th

e sy

stem

of r

atio

nal

num

bers

.

• U

nder

stan

d po

sitiv

e an

d ne

gativ

e nu

mbe

rs in

real

-wor

ld c

onte

xt.

• U

nder

stan

d a

ratio

nal n

umbe

r as

a po

int o

n th

e nu

mbe

r lin

e.•

Ext

end

both

num

ber l

ine

diag

ram

s an

d co

ordi

nate

axe

s to

incl

ude

ratio

nal

num

bers

.•

Und

erst

and

orde

ring

and

abso

lute

val

ue o

f rat

iona

l num

bers

.•

Sol

ve re

al-w

orld

and

mat

hem

atic

al p

robl

ems

by g

raph

ing.

num

ber o

f inv

estig

atio

ns: _

____

____

6.E

E.1

–4: A

pply

and

ext

end

prev

ious

und

erst

andi

ngs

of a

rithm

etic

to a

lgeb

raic

ex

pres

sion

s.

• W

rite

and

eval

uate

num

eric

al e

xpre

ssio

ns w

ith w

hole

-num

ber e

xpon

ents

.•

Writ

e, re

ad, a

nd e

valu

ate

expr

essi

ons

in w

hich

lette

rs s

tand

for n

umbe

rs.

• A

pply

the

dist

ribut

ive

prop

erty

and

com

bine

like

term

s to

gen

erat

e eq

uiva

lent

exp

ress

ions

.•

Iden

tify

two

equi

vale

nt e

xpre

ssio

ns

num

ber o

f inv

estig

atio

ns: _

____

____

6.E

E.5

–8: R

easo

n ab

out a

nd

solv

e on

e-va

riabl

e eq

uatio

ns a

nd

ineq

ualit

ies.

• U

nder

stan

d so

lvin

g an

equ

atio

n or

ineq

ualit

y as

det

erm

inin

g w

hich

val

ues

mak

e it

true.

• W

ritin

g an

d so

lvin

g eq

uatio

ns a

nd in

equa

litie

s w

ith o

ne v

aria

bles

.•

Sol

ving

real

-wor

ld a

nd m

athe

mat

ical

pro

blem

s by

writ

ing

and

solv

ing

equa

tions

with

non

nega

tive

ratio

nal n

umbe

rs.

• R

epre

sent

sol

utio

ns o

f ine

qual

ities

on

num

ber l

ine

diag

ram

s.

num

ber o

f inv

estig

atio

ns: _

____

____

6.E

E.9

: rep

rese

nt a

nd a

naly

ze

quan

titat

ive

rela

tions

hips

bet

wee

n de

pend

ent a

nd in

depe

nden

t va

riabl

es.

• R

epre

sent

two

real

-wor

ld q

uant

ities

that

cha

nge

in re

latio

nshi

p to

eac

h ot

her w

ith d

epen

dent

and

inde

pend

ent v

aria

bles

.•

Ana

lyze

the

rela

tions

hip

betw

een

depe

nden

t and

inde

pend

ent v

aria

bles

us

ing

grap

hs, t

able

s, a

nd e

quat

ions

.

num

ber o

f inv

estig

atio

ns: _

____

____




Gra

de 7

Maj

or C

lust

erC

onte

nt o

f Maj

or C

lust

erLi

st a

nd T

ally

CM

P3 In

vest

igat

ions

on

Con

tent

of M

ajor

Clu

ster

7.R

P.1–

3: A

naly

ze p

ropo

rtion

al

rela

tions

hips

and

use

them

to

solv

e re

al-w

orld

and

mat

hem

atic

al

prob

lem

s.

• C

ompu

te u

nit r

ates

ass

ocia

ted

with

ratio

s of

frac

tions

, inc

ludi

ng le

ngth

s,

area

s an

d ot

her q

uant

ities

mea

sure

d in

like

or d

iffer

ent u

nits

.•

Rec

ogni

ze a

nd re

pres

ent p

ropo

rtion

al re

latio

nshi

ps.

• Te

st fo

r equ

ival

ent r

atio

s in

a ta

ble

or g

raph

ing

on th

e co

ordi

nate

pla

ne to

de

term

ine

if th

e gr

aph

pass

es th

roug

h th

e or

igin

.•

Iden

tify

the

cons

tant

of p

ropo

rtion

ality

in ta

bles

, gra

phs,

equ

atio

ns,

diag

ram

s, a

nd v

erba

l des

crip

tions

of p

ropo

rtion

al re

latio

nshi

ps.

• R

epre

sent

pro

porti

onal

rela

tions

hips

by

equa

tions

.•

Sol

ve m

ulti-

step

ratio

and

per

cent

pro

blem

s.

num

ber o

f inv

estig

atio

ns: _

____

___

7.N

S.1

–3: A

pply

and

ext

end

prev

ious

und

erst

andi

ngs

of

oper

atio

ns w

ith fr

actio

ns to

add

, su

btra

ct, m

ultip

ly a

nd d

ivid

e ra

tiona

l num

bers

.

• A

dd a

nd s

ubtra

ct ra

tiona

l num

bers

.•

Rep

rese

nt a

dditi

on a

nd s

ubtra

ctio

n of

ratio

nal n

umbe

rs o

n ho

rizon

tal a

nd

verti

cal n

umbe

r lin

e di

agra

ms.

• U

nder

stan

d su

btra

ctio

n as

add

ing

the

inve

rse.

• M

ultip

ly a

nd d

ivid

e ra

tiona

l num

bers

.•

Con

vert

a ra

tiona

l num

ber t

o a

deci

mal

usi

ng lo

ng d

ivis

ion

and

know

why

de

cim

als

can

term

inat

e or

repe

at.

• S

olve

real

-wor

ld a

nd m

athe

mat

ical

pro

blem

s us

ing

the

four

ope

ratio

ns o

n ra

tiona

l num

bers

.

num

ber o

f inv

estig

atio

ns: _

____

___

7.E

E.1

–2: U

se p

rope

rties

of

oper

atio

ns to

gen

erat

e eq

uiva

lent

ex

pres

sion

s.

• A

dd, s

ubtra

ct, f

acto

r, an

d ex

pand

line

ar e

xpre

ssio

ns w

ith ra

tiona

l co

effic

ient

s.•

Und

erst

and

that

rew

ritin

g ex

pres

sion

s in

diff

eren

t for

ms

can

help

to b

ette

r un

ders

tand

how

qua

ntiti

es a

re re

late

d.

num

ber o

f inv

estig

atio

ns: _

____

___

7.E

E.3

–4: S

olve

real

-life

an

d m

athe

mat

ical

pro

blem

s us

ing

num

eric

al a

nd a

lgeb

raic

ex

pres

sion

s an

d eq

uatio

ns.

• S

olve

mul

ti-st

ep re

al-li

fe a

nd m

athe

mat

ical

pro

blem

s w

ith p

ositi

ve a

nd

nega

tive

ratio

nal n

umbe

rs re

pres

ente

d as

who

le n

umbe

rs, f

ract

ions

, and

/or

dec

imal

s.•

Con

vert

betw

een

who

le n

umbe

rs, f

ract

ions

, and

dec

imal

s an

d kn

ow h

ow

to a

sses

s th

e re

ason

able

ness

of s

olut

ions

.•

Use

var

iabl

es to

repr

esen

t qua

ntiti

es in

real

-wor

ld a

nd m

athe

mat

ical

pr

oble

ms.

• C

onst

ruct

sim

ple

equa

tions

and

ineq

ualit

ies

to s

olve

pro

blem

s by

re

ason

ing

abou

t qua

ntiti

es.

• S

olve

wor

d pr

oble

ms

with

equ

atio

ns o

f the

form

px

+ q

= r a

nd p

(x +

q) =

r,

whe

re p

, q, a

nd r

are

spec

ified

ratio

nal n

umbe

rs.

• S

olve

wor

d pr

oble

ms

with

ineq

ualit

ies

of th

e fo

rm p

x +

q >

r or p

x +

q <

r, w

here

p, q

, and

r ar

e sp

ecifi

ed ra

tiona

l num

bers

.•

Gra

ph s

olut

ion

sets

of i

nequ

aliti

es.

num

ber o

f inv

estig

atio

ns: _

____

___




Gra

de 8

Maj

or C

lust

erC

onte

nt o

f Maj

or C

lust

erLi

st a

nd T

ally

CM

P3 In

vest

igat

ions

on

Con

tent

of M

ajor

Clu

ster

8.E

E.1

–4: W

ork

with

radi

cals

and

in

tege

r exp

onen

ts.

• A

pply

pro

perti

es o

f int

eger

exp

onen

ts.

• U

se s

quar

e ro

ots

and

cube

root

s to

sol

ve e

quat

ions

of t

he fo

rm x

2 = p

and

x3 =

p,

whe

re p

is a

pos

itive

ratio

nal n

umbe

r.•

Find

squ

are

root

s of

sm

all p

erfe

ct s

quar

es a

nd c

ube

root

s of

sm

all p

erfe

ct c

ubes

.•

Exp

ress

a n

umbe

r as

a w

hole

num

ber t

imes

a w

hole

-num

ber p

ower

of t

en.

• P

erfo

rm o

pera

tions

with

num

bers

exp

ress

ed in

sci

entifi

c no

tatio

n, in

clud

ing

prob

lem

s w

here

bot

h de

cim

al a

nd s

cien

tific

nota

tion

are

used

.n

umbe

r of i

nves

tigat

ions

: ___

__

8.E

E.5

–6: U

nder

stan

d th

e co

nnec

tions

bet

wee

n pr

opor

tiona

l re

latio

nshi

ps, l

ines

and

line

ar

equa

tions

.

• G

raph

pro

porti

onal

rela

tions

hips

and

und

erst

and

the

unit

rate

as

the

slop

e.

• U

se m

ultip

le re

pres

enta

tions

to c

ompa

re tw

o di

ffere

nt p

ropo

rtion

al re

latio

nshi

ps.

• U

se s

imila

r tria

ngle

s to

exp

lain

con

stan

t slo

pe o

f a n

on-v

ertic

al li

ne o

n th

e co

ordi

nate

pla

ne.

• D

eriv

e y

= m

x as

a li

ne th

roug

h th

e or

igin

and

y =

mx

+ b

as a

line

inte

rcep

ting

the

verti

cal a

xis

at b

.n

umbe

r of i

nves

tigat

ions

: ___

__

8.E

E.7

–8: a

naly

ze a

nd s

olve

line

ar

equa

tions

and

pai

rs o

f sim

ulta

neou

s lin

ear e

quat

ions

.

• S

olve

line

ar e

quat

ions

in o

ne v

aria

ble.

• D

eter

min

e if

linea

r equ

atio

ns in

one

var

iabl

e ha

ve o

ne, i

nfini

tely

man

y, o

r no

solu

tions

.•

Sol

ve li

near

equ

atio

ns w

ith ra

tiona

l coe

ffici

ents

.•

Ana

lyze

and

sol

ve p

airs

of s

imul

tane

ous

linea

r equ

atio

ns a

nd u

nder

stan

d th

e so

lutio

ns. c

orre

spon

d to

poi

nts

of in

ters

ectio

n of

thei

r gra

phs.

• S

olve

sys

tem

s al

gebr

aica

lly a

nd e

stim

ate

by g

raph

ing.

•Sol

ve re

al-w

orld

and

m

athe

mat

ical

pro

blem

s.

num

ber o

f inv

estig

atio

ns: _

____

8.F.

1 –3:

Defi

ne, e

valu

ate

and

com

pare

func

tions

.•

Und

erst

and

a fu

nctio

n as

a ru

le th

at a

ssig

ns e

ach

inpu

t exa

ctly

one

out

put.

• C

ompa

re p

rope

rties

of t

wo

func

tions

alg

ebra

ical

ly, g

raph

ical

ly, n

umer

ical

ly, in

ta

bles

, or b

y ve

rbal

des

crip

tion.

• Id

entif

y y

= m

x +

b as

a li

near

func

tion

with

the

grap

h of

a s

traig

ht li

ne.

num

ber o

f inv

estig

atio

ns:

____

_

8.G

.1–5

: Und

erst

and

cong

ruen

ce

and

sim

ilarit

y us

ing

phys

ical

mod

els,

tra

nspa

renc

ies

or g

eom

etry

sof

twar

e.

• Ve

rify

expe

rimen

tally

the

prop

ertie

s of

rota

tions

, refl

ectio

ns, a

nd tr

ansl

atio

ns.

• U

nder

stan

d th

at tw

o-di

men

sion

al fi

gure

s ar

e co

ngru

ent i

f one

can

be

obta

ined

from

th

e ot

her t

hrou

gh ro

tatio

ns, r

eflec

tions

, and

tran

slat

ions

.•

Use

coo

rdin

ates

to d

escr

ibe

the

effe

ct o

f dila

tions

, tra

nsla

tions

, rot

atio

ns, a

nd

refle

ctio

ns.

• U

nder

stan

d th

at tw

o-di

men

sion

al fi

gure

s ar

e si

mila

r if o

ne c

an b

e ob

tain

ed fr

om th

e ot

her t

hrou

gh d

ilatio

ns, r

otat

ions

, refl

ectio

ns, a

nd tr

ansl

atio

ns.

• U

se in

form

al a

rgum

ents

to e

stab

lish

fact

s ab

out a

ngle

sum

and

ext

erio

r ang

le

of tr

iang

les,

ang

les

crea

ted

whe

n pa

ralle

l lin

es a

re c

ut b

y a

trans

vers

al, a

nd a

a tri

angl

e si

mila

rity.

num

ber o

f inv

estig

atio

ns:

____

_

8.G

.6–8

: Und

erst

and

and

appl

y th

e P

ytha

gore

an t

heor

em.

• E

xpla

in a

pro

of o

f Pyt

hago

rean

The

orem

and

of i

ts c

onve

rse.

• A

pply

Pyt

hago

rean

The

orem

.•

Use

Pyt

hago

rean

The

orem

to fi

nd d

ista

nce

betw

een

two

poin

ts in

the

coor

dina

te

syst

em.

num

ber o

f inv

estig

atio

ns:

____

_

(New

Yor

k S

tate

Mat

hem

atic

s C

omm

on C

ore

Wor

kgro

up, n

.d.,

36–3

8, 4

1–42

, 46–

48)




Notes on the Content of Major Clusters • Grade 6

• Grade 7

• Grade 8

Discussion on CMP3’s FocusWhere do you see the focus of the ccLSM in cMP3?




Discussion on CMP3’s CoherenceWhere do you see the coherence of the ccLSM in cMP3?

CMP3’s Rigor of the Common Core

Quick WriteGiven the descriptions of the instructional shifts in the “crosswalk of common core instructional Shifts: Mathematics” document, what will you see when there is rigor in the classroom?




Achieving Rigor through the Math PracticesUse the following UrL: http://www.p12.nysed.gov/ciai/common_core_standards/pdfdocs/nysp12cclsmath.pdf

Component of RigorIdentify One Math Practice for Each

Component of Rigor, and Explain How Students Will Use the Math Practice to Achieve the Rigor of the Common Core

Fluency: Students are expected to have speed and accuracy with simple calculations; teachers structure class time and/or homework time for students to memorize, through repetition, core functions (found in the attached list of fluencies) such as multiplication tables so that they are more able to understand and manipulate more complex concepts.

Deep Understanding: teachers teach more than “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of core math concepts by applying them to new situations as well as writing and speaking about their understanding.

Application: Students are expected to use math and choose the appropriate concept for application even when they are not prompted to do so. teachers provide opportunities at all grade levels for students to apply math concepts in “real world” situations. teachers in content areas outside of math, particularly science, ensure that students are using math—at all grade levels—to make meaning of and access content.

Dual Intensity: Students are practicing and understanding. there is more than a balance between these two things in the classroom—both are occurring with intensity. teachers create opportunities for students to participate in “drills” and make use of those skills through extended application of math concepts. the amount of time and energy spent practicing and understanding learning environments is driven by the specific mathematical concept and therefore, varies throughout the given school year.

(nYSED, n.d.)

Rig

or: r

equi

re fl

uenc

y, a

pplic

atio

n, a

nd d

eep

unde

rsta

ndin

g




Growing, Growing, Growing: Problem 4.1 a. the paper chen starts with has an area of 64 square inches. copy and complete the table

to show the area of a ballot after each of the fi rst 10 cuts.

b. How does the area of a ballot change with each cut?

c. Write an equation for the area a of a ballot after any cut n.

Number of Cuts Area (in.2)

0 64

1 32

2 16

3

4

5

6

7

8

9

10




D. Make a graph of the data.

(Pearson Education, inc. 2014c)




In the following dialogue, from a CMP class, students compare their graphs for Problem 4.1 to their graphs of an inverse variation relationship.

Collin: i looked at one of the inverse graphs as well and i realized something, that this one has a Y intercept and inverse graph never had a Y intercept.

Teacher: Let’s investigate that idea. Does anybody remember one of the situations that was an inverse proportion relationship? that could help us talk about it.

James: it’s like, uh, factor pairs of each other that makes the graph. Like if there were 24, you know, if you were going to buy 24, um, 240,000 square feet of land, or something, um, then you could. . .you’d find all the factor pairs, like, you’d. . . Like, whatever the X is, like if the X was 2, then, then it’d be like 120,000 because, yeah, because the one would be 240,000, so then, yeah, it’s, it’s like factor pairs of each other, 2 times 120,000, that’s 240,000. . .and then you just like multiply the X by the Y and get 240,000.

Teacher: So you’re thinking of that problem back in the last unit where there were the pieces of land - it could be a 2 by 120,000. and then you listed several of those and that’s the factor pairs you were talking about. is that pattern also true on the graph and table for today’s problem?

James: Um, no. these are not the factor pairs idea that i was talking about.

Teacher: So it is a different relationship between variables. Why isn’t there a Y intercept on the inverse proportion graph, like collin said?

Sarah: Um, well, um, zero times nothing would ever give you 240,000, so there can’t be a y-intercept.

Teacher: Does a zero exist for this situation? at zero cuts, did he have an area?

Sara: Yup. 64.

Teacher: So there would be a Y intercept on this one, and the relationship between the X and the Y for an inverse proportion relationship is not exactly the same relationship between the X and the Y that we’re finding here. But the shapes of the graphs look alike.


Rigor and Math Practices in CMP3annotate your work and the cMP3 classroom dialogue to indicate where and how you use rigor and the math practices in your problem solving and the cMP3 dialogue. Space is also provided here for notes.




Revisit the Section 1 Big Questions • In what ways does CMP3 align with the focus, coherence, and rigor of the CCLSM?

• How do students experience rigor of the CCLSM in the investigation of CMP3 problems?



Section 2: The Inquiry-Based Learning Structure of CMP3

What is inquiry-based learning?

Types of Problem SolvingThe following excerpt appears in Van de Walle, Karp, and Bay-Williams’ Elementary and Middle School Mathematics: Teaching Developmentally (2013, 32):

in a classic publication on the types of teaching related to problem solving, Schroeder and Lester (1989) identified three types of approaches to problem solving:

1. Teaching for problem solving. this approach can be summarized as teaching a skill so that a student can later problem solve. teaching for problem solving often starts with learning the abstract concept and then moving to solving problems as a way to apply the learned skills. For example, students learn the algorithm for adding fractions and, once that is mastered, solve story problems that involve adding fractions. (this approach is used in many textbooks and is likely familiar to you.)

2. Teaching about problem solving. this second approach involves teaching students how to problem solve, which can include teaching the process (understand, design a strategy, implement, look back) or strategies for solving a problem. an example of a strategy is “draw a picture,” in which students use a picture or diagram to help solve a problem. See “teaching about Problem Solving” in this chapter.

3. Teaching through problem solving. this approach generally means that students learn mathematics through real contexts, problems, situations, and models. the contexts and models allow students to build meaning for the concepts so that they can move to abstract concepts. teaching through problem solving might be described as upside down from teaching for problem solving—with the problem(s) presented at the beginning of a lesson and skills emerging from working with the problem(s). For example, in exploring the situation of combining 1/2 and 1/3 feet of ribbon to figure out how long the ribbon is, students would be led to discover the procedure for adding fractions.




Ana

lyzi

ng C

MP3

’s L

esso

n St

ruct

ure:

Who

’s d

oing

wha

t whe

n?

Laun

chEx

plor

eSu

mm

ariz

e

Stud

ents

Wha

t are

stu

dent

s do

ing?

Wha

t are

stu

dent

s sa

ying

?

Use

one

wor

d to

des

crib

e a

stud

ent’s

role

.

Teac

hers

Wha

t is

the

teac

her d

oing

?

Wha

t is

the

teac

her s

ayin

g?

Use

one

wor

d to

des

crib

e th

e te

ache

r’s ro

le.




Scrambled Sentencesthe role of a student during the Launch phase:

the role of a student during the Explore phase:

the role of a student during the Summarize phase:

the role of the teacher during the Launch phase:

the role of the teacher during the Explore phase:

the role of the teacher during the Summarize phase:




Revisit the Section 2 Big Questions • What is CMP3’s lesson structure?

• What are the roles of the teacher and the students in each phase of a CMP3 inquiry-based lesson?

• What is CMP3’s instructional philosophy?



Section 3: Supporting Teachers to Build Mathematical Communities Capable of CMP3’s Inquiry-Based Learning

Quick Write: What is the difference?

Traditional Classroom CMP3 Classroom

curriculum is presented part to whole, with emphasis on basic skills.

curriculum is presented whole to part, with the emphasis on the coherence of concepts per the ccLSM.

Strict adherence to fixed curriculum is highly valued.

Problems have multiple entry points, and students can use multiple pathways to find the solution. the most important aspect of the classroom is ongoing assessment of students’ thinking.

Students are viewed as ‘blank slates’ onto which information is etched by the teacher.

Multiple entry points to problems and the Launch-Explore-Summarize (LES) lesson structure provide opportunities for students to activate, construct, and critique their own knowledge while they investigate new concepts.

teachers generally behave in a didactic manner, disseminating information to students.

teachers generally launch the lesson to make sure students understand the Focus Question, and then they act as a guide or a facilitator as students construct and critique their own ideas as they discover new concepts.

teachers seek the correct answer to validate student learning.

teachers seek students’ thinking to determine misconceptions and students’ solution methods that employ multiple representations to discuss and make connections during the Summarize phase.

assessment of student learning is viewed as separate from teaching and occurs almost entirely through testing.

assessment of students’ learning occurs constantly through observation of student work, presentation, and discourse. Student notebooks are a key source of ongoing formative assessment.

Students primarily work alone. Students work in groups as a community of mathematicians engaged in mathematical discourse.

(brooks and brooks quoted in brahier 2009, 60)




How will teachers’ cMP3 classrooms be different from their past classrooms?

What strategies will teachers need to employ to transform their classrooms to function as cMP3 classrooms?




Supp

ortin

g Te

ache

rs to

Bui

ld M

athe

mat

ical

Com

mun

ities

Gui

delin

es fo

r Bui

ldin

g a

Mat

hem

atic

al C

omm

unity

“Tee

n-Fr

iend

ly”

Cla

ssro

om

Nor

ms

Bes

t Pra

ctic

es fo

r Pro

mot

ing

Nor

ms

CM

P3 C

ompo

nent

s fo

r Pr

omot

ing

Nor

ms

View

our

cla

ss a

s a

com

mun

ity

in w

hich

eac

h pe

rson

wan

ts a

ll of

th

e ot

hers

to b

e su

cces

sful

in th

eir

lear

ning

exp

erie

nces

. try

not

to

see

the

clas

sroo

m a

s a

com

petit

ive

envi

ronm

ent i

n w

hich

you

r rol

e is

to

outd

o ot

hers

.

crit

iciz

e id

eas,

not

peo

ple

(e.g

., sa

y,

“i di

sagr

ee w

ith th

e w

ay y

ou s

olve

d th

at p

robl

em b

ecau

se .

. .,”

rath

er

than

, “Yo

u’re

stu

pid;

I ca

n’t b

elie

ve

you

got t

hat a

nsw

er!”)

.

Mak

e fre

quen

t con

tribu

tions

to

clas

sroo

m d

iscu

ssio

ns b

y as

king

qu

estio

ns, a

nsw

erin

g qu

estio

ns,

and

reaf

firm

ing

or d

isag

reei

ng w

ith

com

men

ts m

ade

by o

ther

s.

take

resp

onsi

bilit

y fo

r the

lear

ning

of

othe

r stu

dent

s. if

you

und

erst

and

a co

ncep

t, ta

ke it

upo

n yo

urse

lf to

hel

p ot

hers

(at y

our t

able

or i

n th

e w

hole

cl

ass)

und

erst

and

it as

wel

l.

ask

que

stio

ns a

nd le

t the

teac

her

and

team

mat

es k

now

whe

n yo

u do

n’t

unde

rsta

nd s

omet

hing

. rem

embe

r, th

at n

o on

e ca

n re

ad y

our m

ind—

you

will

nee

d to

com

mun

icat

e yo

ur la

ck

of u

nder

stan

ding

to g

et s

omeo

ne to

he

lp y

ou.

(con

tinue

d on

nex

t pag

e)




Gui

delin

es fo

r Bui

ldin

g a

Mat

hem

atic

al C

omm

unity

“Tee

n-Fr

iend

ly”

Cla

ssro

om

Nor

ms

Bes

t Pra

ctic

es fo

r Pro

mot

ing

Nor

ms

CM

P3 C

ompo

nent

s fo

r Pr

omot

ing

Nor

ms

Enc

oura

ge c

lass

mat

es to

par

ticip

ate.

D

on’t

let i

ndiv

idua

ls s

it, d

ay a

fter d

ay,

with

out c

ontri

butin

g th

eir t

houg

hts

(e.g

., en

cour

age

the

pers

on s

ittin

g ne

xt to

you

to ra

ise

the

ques

tion

with

the

clas

s th

at th

e in

divi

dual

has

ex

pres

sed

to y

ou).

rec

ogni

ze th

at th

ere

is n

o su

ch th

ing

as a

wro

ng a

nsw

er in

a m

athe

mat

ics

clas

sroo

m. i

t has

bee

n sa

id th

at

stud

ents

nev

er g

ive

a w

rong

ans

wer

; th

ey s

impl

y an

swer

a q

uest

ion

diffe

rent

from

the

one

the

teac

her

inte

nded

. See

min

gly

wro

ng a

nsw

ers

are

actu

ally

opp

ortu

nitie

s fo

r the

cl

ass

to e

xplo

re n

ew id

eas.

rea

lize

that

it is

nat

ural

to fe

ar fa

ilure

in

the

clas

sroo

m, b

ut re

cogn

ize

that

yo

ur c

lass

mat

es h

ave

this

sam

e fe

ar

and

that

risk

-taki

ng is

impo

rtant

for

succ

ess.

For

this

reas

on, n

ever

laug

h at

the

resp

onse

of a

cla

ssm

ate:

la

ught

er e

rode

s co

nfide

nce

and

feed

s th

at fe

ar. a

lso,

you

may

be

in

the

sam

e po

sitio

n on

the

next

day

.U

se fi

rst n

ames

. It c

reat

es a

muc

h m

ore

supp

ortiv

e en

viro

nmen

t whe

n yo

u sa

y, “i

thin

k Fr

ance

s is

cor

rect

, bu

t i d

isag

ree

with

Jos

eph’

s an

swer

, an

d he

re’s

why

. . .

” tha

n to

refe

r to

clas

smat

es s

impl

y as

“he”

and

“she

.”S

uppo

rt on

e an

othe

r. W

hen

you

resp

ond

in a

cla

ss d

iscu

ssio

n, m

ake

use

of p

revi

ous

poin

ts m

ade

by

sayi

ng, “

i agr

ee w

ith M

ark.

and

i al

so

thin

k th

at .

. .” i

f som

eone

com

es u

p w

ith a

uni

que

appr

oach

or s

olut

ion

to a

pro

blem

, it i

s ap

prop

riate

to

appl

aud

and

affir

m th

at p

erso

n.

(bra

hier

200

9, 1

90)




Best Practices for Promoting Mathematical Communities 1. Equity Sticks: Write the names of students on popsicle sticks, and place them in a cup or

bag. When calling on students to present or answer a question, randomly select a popsicle stick.

2. Excel® Random Generator: Enter the names of students into an Excel spreadsheet, and assign a student to use the random generator to select a student to present or answer a question. if possible, project the Excel random generator so that students see the fairness.

3. Student Fishbowl: Select students to act out group work, mathematical discourse, teamwork on a partner quiz, and so on while the other students observe and take notes on a graphic organizer. conclude the Fishbowl activity with a whole-group discussion of how students in the fishbowl interacted and functioned.

4. Assigning Roles in Group Work: By formally assigning roles, students have a specific set of responsibilities. Some commonly used roles are timekeeper, Gatekeeper/taskmaster, Facilitator, Skeptic, recorder, Summarizer, and Presenter.

5. Random Group Grading: randomly choose one member of a group to grade and assign that grade to the entire group. assign zeros for copying.

6. Graded Individual Exit Slips: at the conclusion of the Summarize phase, have students independently complete graded exit slips to assess the day’s learning.

7. Three Before Me: Have a standing rule that a student must pose his or her question to three other students before he or she can ask the teacher.

8. Questioning to Promote Teamwork: redirect all inquiries back to other group members in a form of a question (for example, say, “What can you say about tony’s question?”, or “Look at what Sara did. How can this help answer tony’s question?”)

9. Display Student Work: Display students’ work, and refer to it to help students make connections between multiple representations and mathematical concepts.

10. Word Wall: Display a word wall with the definition, multiple representations, and examples of student work on the concept in context.

11. Sentence Starters Wall: Display a wall with sentence starters for seeking clarification, asking for help, acknowledging someone else’s idea, affirming someone else’s thoughts or ideas, reporting a partner’s idea, presenting a group’s idea, paraphrasing what someone else says, predicting, making a suggestion, giving an opinion, and respectfully disagreeing.

12. Think–Pair–Shares: Students have the opportunity to sort out their own ideas and discuss them with a peer before the whole class engages in a discussion on a mathematical question. this strategy reduces the anxiety around contributing to classroom discourse and promotes student-student interaction.

13. KWL Charts: Students activate prior knowledge and construct questions, and teachers formatively assess students’ thinking and prerequisite concepts.

14. Peer Assessment: Students manage the learning of their classroom, and the process promotes student-student mathematical discourse.




15. Student Behavior Self-Assessment: Use a rubric for students to grade how well their behavior was in accordance to the classroom norms.

16. Multiple Representations: as opposed to traditional direct instruction, encourage students to use their own math skills to solve problems and represent their solutions with any representation they can.

17. Talk Moves: When orchestrating classroom discussion, use revoicing, rephrasing, reasoning, elaboration, and wait time (Van de Walle, Karp, and Bay-Williams 2013, 43).

18. Star Student of the Week: recognize students when they help other students and/or contribute to building a community of mathematicians in the classroom. Post students’ pictures with examples of why they have been awarded the honor.

19. Got It!–Almost–Not Yet: Have these three sections somewhere on a wall near the door where students exit, and have a small name tag (magnetic or pin-up) of each student’s name. ask students to place their name tags under the Got it!, almost, or not Yet to describe their level of understanding of the day’s learning or a specified question or concept. Use this as a formative assessment for planning.

20. Display a Daily Participation Grade: Develop a participation grading rubric that aligns with class norms. Make it simple and transparent, post participation grades daily for students to see, and make the weight enough for students to see that good participation grades will raise their average.

21. What time is it?: Make a sign with Launch, Explore, Summarize, and any other phase(s) of a normal day (warm-up, exit slip, independent assessment, and so on). Label each phase with 0, 1, or 2, where 0 means no one talks, 1 means one person at a time talks, and 2 means everyone can use accountable talk. Use some way to indicate the phase of the lesson throughout each class.

22. Student Work Analysis: include presentation, discussion, and analysis of incorrect student work. Use this as both an opportunity for students to understand that incorrect solutions are just a way to rule out what does not work, as well as to make connections to the difference with correct solutions.

23. Equity Monitors: Have students designated as Equity coaches monitor other students’ participation. these students can motivate, assist, or even assess participation depending on the makeup of the class.




Brainstorm of Additional Best Practices for Promoting Communities of Mathematicians

CMP3 Components That Promote Mathematical Communities a. Partner quizzes

b. Problems with multiple entry points

c. Student-centered, collaborative Explore phase

D. Summarize phase in which students discuss and critique the work of multiple students

E. application problems from applications-connections-Extensions (acE) used as exit slips

F. Sample student work in teacher Place for students to analyze to make connections with their own work

G. Value of student thinking over correctness in LES lesson structure

H. cooperative learning groups in every lesson

i. Students ask questions of either the teacher or their group members during all three phases

J. Summarize phase in which the teacher sequences the discussion of multiple representations of solutions

K. Mathematical Reflections and Common Core Mathematical Reflections for students to reflect on how their groups investigated problems together

L. Problems with real-world context that engage students in discussion of the problem situation

M. teachers monitoring dynamics throughout the Explore phase

n. teachers orchestrating classroom discourse in the Summarize phase at the end of every lesson

o. teacher Place functionalities for keeping notes on students and communicating directly with families




Revisit the Section 3 Big Questions • What are the hallmarks of a CMP3 classroom?

• What best practices can teachers use to build a mathematical community capable of the inquiry-based learning of cMP3?



Section 4: Inquiry-Based Classrooms: A Shift in Student Experience

What Students Need to Be Successful with the Shifts • Engaging, real-world problems

• Motivation to engage

• Communication skills

• Problem-solving skills




Motivation to Engage ProjectRead the following excerpt from Slavin (1996, 53–55), and then design a tool or rubric that teachers can use to motivate every student in every group every day.

WHAT FACTORS CONTRIBUTE TO ACHIEVEMENT EFFECTS OFCOOPERATIVE LEARNING?1

Research on cooperative learning has moved beyond the question of whether cooperative learning is effective in accelerating student achievement to focus on the conditions under which it is optimally effective. The foregoing discussion describes alternative overarching theories to explain cooperative learning effects, and an integration of these theories. Beyond this, it is important to understand in more detail the factors that contribute to or detract from the effectiveness of cooperative learning. There are two primary ways to learn about factors that contribute to the effectiveness of cooperative learning. One is to compare the outcomes of studies of alternative methods. For example, if programs that incorporated group rewards produced stronger or more consistent positive effects (in comparison to control groups) than programs that did not, then this would provide one kind of evidence that group rewards enhance the outcomes of cooperative learning. The problem with such comparisons is that the studies being compared usually differ in measures, durations, subjects, and many other factors that could explain differing outcomes. Better evidence is provided by studies that compared alternative forms of cooperative learning. In such studies, most factors, other than the ones being studied, can be held constant. The following sections discuss both types of studies to further explore factors that contribute to the effectiveness of cooperative learning for increasing achievement.

Group Goals and Individual Accountability

As noted earlier, reviewers of the cooperative learning literature have long concluded that cooperative learning has its greatest effects on student learning when groups are recognized or rewarded based on individual learning of their members (Slavin, 1983a, 1983b, 1989, 1992, 1995; Ellis & Fouts, 1993; Newmann & Thompson, 1987; Manning & Lucking, 1991; Davidson, 1985; Mergendoller & Packer, 1989). For example, methods of this type may give groups certificates based on the average of individual quiz scores of group members, where group members could not help each other on the quizzes. Alternatively, group members might be chosen at random to represent the group, and the whole group might be rewarded based on the selected member’s performance. In contrast, methods lacking group goals give students only individual grades or other individual feedback, and there is no group consequence for doing well as a group. Methods lacking individual accountability might reward groups for doing well, but the basis for this reward would be a single project, worksheet, quiz, or other product that could theoretically have been done by only one group member. The importance of group goals and individual accountability is in providing students with an incentive to help each other and to encourage each other to put forth maximum effort (Slavin, 1995). If students value doing well as a group, and the group can succeed only by ensuring that all group members have learned the material, then group members will be motivated to teach each other. Studies of behaviors within groups that relate most to achievement gains consistently show that students who give each other explanations (and less consistently, those who receive such explanations) are the students who learn the most in cooperative learning. Giving or receiving answers without explanation generally reduces achievement (Webb, 1989, 1992). At least in theory, group goals and individual accountability should motivate students to engage in the behaviors that increase achievement and avoid those that reduce it. If a group member wants her group to be successful, she must teach her groupmates (and learn the material herself). If she simply tells her groupmates the answers, they will fail the quiz that they must take individually. If she ignores a groupmate who is not understanding the material, the groupmate will fail and the group will fail as well. In groups lacking individual accountability, one or two students may do the group’s work, while others engage in “social loafing” (Latane, Williams, & Harkins, 1979). For example, in a group asked to complete a single project or solve a single problem, some students may be discouraged from participating. A group trying to complete a common problem may not want to stop and explain what is going on to a groupmate who doesn’t understand, or may feel it is useless or counterproductive to try to involve certain groupmates.

1 These sections are adapted from Slavin (1995).




Communication Skills ProjectWrite at least three sentence starters for each of the bullet points below, and display these sentence starters in every cMP3 classroom to provide models for students.

• Seeking clarification

• Asking for help

• Acknowledging someone else’s idea

• Affirming someone else’s thoughts or ideas

• Reporting on a partner’s idea

• Presenting on a group’s idea

• Paraphrasing what someone else says

• Making a prediction

• Making a suggestion

• Giving an opinion

• Respectfully disagreeing

For example, a sentence starter for paraphrasing might be, “So, you are saying that. . .”, and a sentence starter for asking for help might be, “i do not understand what they did, can you explain that to me?”

Finally, plan an exercise to make students aware of the display of sentence starters and to teach them how to use the sentence starters in the LES lesson structure of their cMP3 inquiry-based classrooms.




Problem-Solving Skills ProjectRead the following excerpt from Van de Walle, Karp, and Bay-Williams (2013, 33–34). Choose a cMP3 problem in teacher Place, and use Polya’s four-step method to solve the problem and employ one or more of the problem-solving strategies outlined in the reading. include how you can support students to learn how to use the four-step problem-solving method and the problem-solving strategies in your presentation. if time permits, conclude your presentation by outlining additional problem-solving strategies to employ in a cMP3 classroom.




Brainstorm School-Wide Efforts for Supporting Students through the Shift

notes:




Revisit the Section 4 Big Questions • What skills do students need to be successful in CMP3 inquiry-based classrooms?

• How can you support teachers to implement CMP3 with fidelity?



Section 5: Developing Your Plan for Supporting Teachers

Collaborating with ColleaguesMany teachers have found it valuable to plan with a colleague before, during, and after teaching the unit. Very often, student work is a focus for their discussions, as it provides a platform for discussing the mathematics in the Unit, investigation, or Problem. Discussion can also cover effective teaching strategies and other issues related to teaching. the following sets of summary questions can be useful for working either alone or with colleagues.


How, when, and where will you collaborate with colleagues?review the tool for collaborating before a Lesson and the tool for collaborating after a Lesson. Discuss at your tables how, when, and where you can use these at your school during the school year. complete the following table based on your table discussion.

Tool How? When? Where?

Before a Lesson

After a Lesson




Tool for Collaborating Before a LessonMathematical Goals

What is the focus question for this lesson? (Primary learning goal)

What are some secondary mathematical goals that may arise?

Materials Vocabulary, Processes-notes

Launch How will i get kids to buy into the problem?

is there any prior knowledge that kids will need to do the problem?

Do i need to introduce any mathematics? any contextual information?

is there any way i can connect to the previous problems?

How can i keep from “giving away” how to do the problem?

What is the most effective arrangement (group them) for this problem? (individual, pair, group, whole class, combination)

How will i have them report out/share their learning from the Explore portion of the lesson?

Explore What do i expect to see students doing?

What struggles do I anticipate? (Areas of difficulty or misconceptions)

What questions might i ask to help kids sort out the ideas? (How will you scaffold?)

What might i ask to redirect a student’s thinking if they are off track?

What questions can i ask to check for understanding? extend learning?

Summarize How will i have students share what they learned from the problem?

What thinking went on in the individuals/pairs/groups that the whole class should hear?

What order should they hear it?

Does the order matter?

What are key mathematical questions that need to be answered to pull-out the mathematical opportunities in the problem? (to get at big ideas, strategies, skill practice)

What questions do i want to ask to check for understanding?

What questions do i want to ask to extend their learning?

How can i get students to: Listen to each others thinking,ask questions of me and each other, challenge ideas that are not clear/incomplete/incorrect,take notes on the essential ideas for future reference

How will i know if my students are understanding the mathematics?

What should i do tomorrow? next week? next unit?

HomeworkWhat homework is appropriate to assign and for what students?





Tool for Collaborating After a LessonGeneral

a. How do you think the lesson went compared to what you expected?

b. What went well? What happened to make you feel that way?

c. Were there times in the lesson when students were struggling with the mathematical ideas?

d. What would you change or modify next time?

Overviewa. How comfortable are you with the level of sophistication your students achieved with ________________________

__________________________________________________________________________________________?

b. What ideas will you continue to emphasize?

c. What do students understand about _____________________________________________________________?

d. What were you thinking when you asked _________________________________________________________?

How did __________’s response compare to what you expected?

e. What do you think ____________ was thinking when he/she asked ________________________________________________________________________________________________________________________________?

f. Why do you think _____________ said ______________________________________________________________________________________________________________________________________________________?

g. What sense do you think ____________ is making from the ideas in this unit? How will you assess his/her level of understanding?

h. Where will you go with this idea tomorrow? in the future?

i. How would you modify the lesson if you had the chance to repeat the class?

Launch a. What did you have to think about when you planned the launch?

b. How did the launch engage the students in the problem?

c. What information/background did the students have to help them engage the problem?

d. What did you observe from the launch? Was there too much information? too little? Just right?




Explore a. Why did you decide to have students work individually, with a partner, in groups, or as a whole class?

b. What did you do to provide the individual differences in the class? (scaffold, extend)

c. What did students struggle with? What did they make sense of?

d. What did you observe during explore time that helped you shape the summary?

Summarizea. What did the students learn today? What is your evidence?

b. What misconceptions became apparent during class?

c. How did the summary compare to what you had anticipated?

d. Did all major mathematical ideas of the lesson surface during the summary?

no? When will you revisit them?

Yes? is there a way to extend students thinking?

e. What questions did the summary raise for students? For you?





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Revisit the Section 5 Big Question • What processes do you need to put into place to support teachers in their classrooms

throughout the school year?



Reflection and Closing

Homework to Complete Prior to the Next Intensive Session • Use the provided planning template to plan and collaborate with teachers to support

implementation.

Suggested Ideas • Following Teacher Session 1, select a lesson from CMP3. Write a one-page narrative of

how you envision the lesson will play out in the classroom. once others complete the same assignment, collaborate to compare and contrast the narratives.



Appendix

(nYSED, n.d.)

Crosswalk of Common Core Instructional Shifts: Mathematics

6 Shifts: EngagenYwww.engageny.org

3 Shifts: Student achievement Partners www.achievethecore.org

1: Focus: Teachers use the power of the eraser and signifi cantly narrow and deepen the scope of how time and energy is spent in the math classroom. they do so in order to focus deeply on only the concepts that are prioritized in the standards so that students reach strong foundational knowledge and deep conceptual understanding and are able to transfer mathematical skills and understanding across concepts and grades.

2: Coherence: Principals and teachers carefully connect the learning within and across grades so that, for example, fractions or multiplication spiral across grade levels and students can build new understanding onto foundations built in previous years. teachers can begin to count on deep conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning.

3: Fluency: Students are expected to have speed and accuracy with simple calculations; teachers structure class time and/or homework time for students to memorize, through repetition, core functions (found in the attached list of fl uencies) such as multiplication tables so that they are more able to understand and manipulate more complex concepts.

4: Deep Understanding: teachers teach more than “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than a set of mnemonics of discrete procedures. Students demonstrate deep conceptual understanding of core math concepts by applying them to new situations as well as writing and speaking about their understanding.

5: Application: Students are expected to use math and choose the appropriate concept for application even when they are not prompted to do so. teachers provide opportunities at all grade levels for students to apply math concepts in “real world” situations. teachers in content areas outside of math, particularly science, ensure that students are using math — at all grade levels — to make meaning of and access content.

6: Dual Intensity: Students are practicing and understanding. there is more than a balance between these two things in the classroom — both are occurring with intensty. teachers create opportunities for students to participate in “drills” and make use of those skills through extended application of math concepts. the amount of time and energy spent practicing and understanding learning environments is driven by the specifi c mathematical concept and therefore, varies throughout the given school year.

1: Focus strongly where the Standards focus

2: Coherence: Think across grades, and link to major topics within grades

3: Rigor: require fl uency, application and deep understanding

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References

brahier, Daniel J. 2009. Teaching Secondary and Middle School Mathematics, 3rd ed. Upper Saddle river, nJ: Pearson Education, inc.

Lappan, Glenda, and Dennis Raskin. 2013. “CMP3 Lesson Structure.” Pearson Education, Inc.; 9 min., 22 sec. MP4.

national Governors association center for best Practices (nGa center), council of chief State School Officers (CCSSO). 2010. “Common Core State Standards for Mathematics.” Washington, Dc: national Governors association center for best Practices, council of Chief State School Officers. Accessed June 28, 2012. http://www.corestandards.org/assets/ccSSi_Math%20Standards.pdf.

new York State Education Department (nYSED). n.d. “crosswalk of common core instructional Shifts: Mathematics.” accessed May 28, 2013. http://schools.nyc.gov/nr/rdonlyres/9375E046-3913-4aF5-9FE3-D21baE8FEE8D/0/commoncoreinstructionalShifts_Mathematics.pdf.

new York State Mathematics common core Workgroup. n.d. “P-12 common core Learning Standards for Mathematics.” new York, nY: new York State Mathematics common core Workgroup. accessed May 28, 2013. http://www.p12.nysed.gov/ciai/common_core_standards/pdfdocs/nysp12cclsmath.pdf.

Partnership for assessment of readiness for college and careers (Parcc). 2011. “Parcc Model Framework Mathematics 3-11.” accessed May 28, 2013. http://ok.gov/sde/sites/ok.gov.sde/files/C3PARCC%20MCF%20for%20Mathematics_Fall%202011%20release.pdf.

Pearson Education, inc. n.d. CMP3: Core Middle Grades Mathematics Program. Upper Saddle river, nJ: Pearson Education, inc.

———. 2014a. CMP3: Growing, Growing, Growing. Upper Saddle river, nJ: Pearson Education, inc.

———. 2014b. CMP3: Prime Time. Upper Saddle river, nJ: Pearson Education, inc.

———. 2014c. CMP3 Program and Implementation Guide. Upper Saddle river, nJ: Pearson Education, inc.

Slavin, robert E. 1996. “research for the Future: research on cooperative Learning and achievement: What We know, What We need to know.” Contemporary Educational Psychology 21, no. 0004: 43–69. Accessed May 28, 2013. http://www.konferenslund.se/pp/taPPS_Slavin.pdf.

Student achievement Partners. n.d. “Practicing with the Shifts: common core State Standards for Mathematics.” accessed May 28, 2013. http://www.achievethecore.org/math-common-core/professional-development/introduction-math-shifts.

Van de Walle, John A., Karen S. Karp, and Jennifer M. Bay-Williams. 2013. Elementary and Middle School Mathematics: Teaching Developmentally, 8th ed. Upper Saddle river, nJ: Pearson Education, inc.

connected mathematics project 3 preparing for the

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