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About Teaching Mathematics,
Third Edition
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NEW!
Fourth Edition
Available
April 2015
Special Features
PART
What’s in About Teaching Mathematics,
Fourth Edition?
1
Prob
le
Inves m-Solving
tigat
ions
This
first
part
of
the
book
reflects
the
major
s
t
overhaul of About Teaching Mathematics.
Poin
Here Marilyn addresses twenty-three issues
Part 2
and presents her current thinking. The Starting Points are organized into six categories:
Problem-Solving
• General Pedagogical Issues
• Number and Operations:
Investigations
The Mathematical Cornerstone
Sample Whole-Class
Lessons
2
PART
Part 1
Starting Points
AT A G
LANC
E
Overv
iew
X
X
Plann
in
XX
Proble g Whole-C
lass Le
m-So
lving
Proble
Inves ssons with
X
X
tigatio
m
-Solvin
nt,
XX
ns
Conte
g Inve
XX
Proble
sues Connecting
s
t
m
igatio
-Solvin
XX
d:
n
s
:
g
Starte struction
P
M
r
In
oblem
easur
d
vestig
XX
d In
erstan ers
emen
-Solvin
lum
XX
ations
es, an
’t Und
rn
urricu
t
: Data
g Inve
Proble
the C at You Don e Math Lea
X
g
n
X
s
ri
tigatio
m-So
e
v
h
ti
X
c
W
e
e
X
n
h
ff
lv
and A
ns: G
ac
eE
ersto
XX
lgebr ing Investig
eome
an’t Te nts Becom
l Corn
a
a
e
t
ic
r
d
X
ic
a
y
t
t
X
tu
T
a
io
hinkin
Proble
ns: Paan introduction about how to plan
ng S
em
etic
g After
m
tterns
Math on Arithm•
XX
From Arithmetic
to Problem Solving
Numb Solving In
XX
: The
ve
e
problem-solving
lessons,
specific ideas are
v
ations : A Perspecti Common
r
estiga
X
a
X
n
d
Oper
e
t
io
n
s
soning dents Mak
•
Assessing
Understanding
and
Skills
ations
:
organized into five areas
Stu
ns
XX of the curriculum:
y Mea tanding
hy Do Errors?
XX
It Reall
tic
•ers Suggestions
for Classroom Instruction
ithme Sign: What ping Und
•
Measurement
XX
ual
velo
he Eq blems: De ns
Sample Lesson
• Connecting to English Language Arts
XX
tio
rns
Pro
ra
• Data
e
rd
p
o
merical patte
O
W
XX
mes
metic
looking for nu
ing:
ro
in
d
th
lv
ts
g
in
ri
Instruction
o
en
al
A
S
P
in
ud
f
st
s
o
ular way.
XX
ation engage • beGeometry
m Solv s to Problem
rs in a partic
This investig
roble
m
adding num
XX
om
fr
ic to P ord Proble
s
ge
e
er
u
em
s
Is
that
Detailed plans are provided for
using problem-solving investigations
in whole-class lessons. These instructional suggestions are offered for all
the content areas in Part 2 and for
all arithmetic topics in Part 3.
Over the years, teachers have written to Marilyn asking for help with
some of the investigations in About
Teaching Mathematics. In these
sections, Marilyn addresses the
underlying mathematics for those
problem-solving investigation.
NEW
Part 4
Questions
Teachers Ask
• Addition and Subtraction
• Introducing Multiplication
• Introducing Division
Race fo
lem.
smaller prob
r 20 • Extending Multiplication and Division
similar but
a
they
Play w
s
or
A
m
e.
le
om
ob
ith a pa
of the pr
rt Here
Marilyn
responds
todr
pedagogical
into a palin
rtner.•P Fractions
2. Pose a pa
The fir
k. Then show
to change 48
or
y
w
tr
r
ts
ei
en
th
k
ud
layers t
chec
st
e PalinHave the st questions
they canreceived
from
teachers
48 on th
ake tur
second player sta•
the board so she’s fo
68, and
rts Decimals
ns coun
sults r 13,
work, do it on
and co
player
re
the Math
e
th
rd
co
ting on
to re the years. She chose more than two About e pointed out that
co
unts eit
over
or two
class how
e
gu
th
llea
l
e
uesPercents
co
al
A
e
t.
h
o
at
ee
more n ntin•
ig
e
r two n
ding Sh
of the
p is to invest
umbers with the next r “One” or
umbers
drome Recordozen questions
or grou
the digits in each es in my
reflect
common
share themes
ch pairthat
to
Ea
w
Extens
“
n.
ho
.
n
io
O
.
de
T
at
u
ci
ig
de
n
m
palindrom
h
st
ld
e
ve
ing
e
b
ou
in
ult
,
e
sh
w
res
e
ions
t
th
ey
r
t
w
as
in
th
t
Organized similarly
to the
an instructional
o
esen
ner is t
same—44
99. First
ro to archives.
s on the char
1. Cha
he play d counts on e .” The 3. Pr
from zeher
examp les were the not
bersfrom
lvidual record
m
co
di
nu
in
t,
e
nge the
ar
t
th
ep
ch
tha
ke
e
it
99
t’s
to
r who
a 0–
in Part 3, assessments
also
er on
and 1111. Bu
eryone is
Winsuggestions
says, “are h
r,
to record on
for 21
. The number
ning N
the work. Ev
ith one colo
ch student is
Twenty e
always the case a twoor Rac included
umbfor
lindromes w
ted. Also, ea
ra
pa
.
st
y
on
is
on
ad
each
topic
to
inform
instruction.
.”
e
le,
m
so
re
al
d
de
r
mp
e
e
. Inste
for 3
67, for exa
her color, an
s
.
bers that ar
-p
te in 19
75! The layer game in
nu
g
Introducing
1. Present or
review conc
epts.
the same fo
rward and ba A palindrome is a numbe
r that reads
ckward, such
number that
as 44, 252,
is not a palin
or 8,008. A
drome, such
palindrome
as 13, can be
by us
changed into
and add. Dem ing a particular procedur
a
e: You revers
onstrate on
the board ho
e the digits
one addition,
w
to
13 is a one-st
do
th
is.
Si
nc
ep palindrom
e it takes
e.
Some numbe
rs take mor
with 67, whi
e than one
ch is a two-st
addition. Dem
ep palindrom
onstrate
e.
Part 3
Teaching Arithmetic
ot
m
ct them
oring the nu
omes with an
to use. Dire
e-step palindr
on the colors
tackle
e
coloring in on
to
re
t
ag
no
ld
em
ou
th
they sh
ts. Also, tell
tion.
ar
di
ch
As a group,
ad
th
ith
bo
w
ut
patterns on
r a serious bo
fo
ome
y
dr
to look for
ad
lin
re
pa
e
ng
less they ar
with a resulti
98 and 89 un
ty-four steps
en
tw
s
ke
ta
r
(Each numbe
023,188!)
of 8,813,200,
Palindrome
s
This investig
ation engage
s students in
that emerge
looking for nu
from adding
merical patte
numbers in a
rns
particular w
ay.
Materials
large 0–99 ch
art, 1 per stud
ent
Palindrome
See Reproduc
Recording Sh
ible R.13
eet, 1 per stud
crayons, colo
ent
red pencils, or
See Reproduc
markers
ible R.14
NEW
• Patterns and Algebraic Thinking
nd
mW
XX
9: Fro ader View Strategies a
ible R.13
g
in
ls
A Bro
a
lv
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ri
o
te
X
e
a
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s
M
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•entNumber
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er Sen th Skills
t, 1 per stud
ar
t
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ud
99
st
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r
e
10: Pro ing Numb
pe
a
rg
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la
t,
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ee
l
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ta
ng
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en
uil
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includes an overview of the
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t 11: B veloping M
XX
Palindrome
ncils, or m
De
Skills sessment
s, colored pe
d
on
X
ay
n
X
cr
nt 12:
a
s
g
A
mathematical
focus, sample rewhole-class
e
in
v
d
ti
ads
ns
XX
orma
rstan
number that
ing
Unde Integrating F m Teaching om Discussio on
uc
e is a investigations
od
om
tr
dr
In
X
lessons,
additional
lin
X
ti
pa
o
:
A
c
ro
.
8,008. A for whole3
u
ts
s
ro
or
1
s
s
ep
tr
t
2,
s
s
nc
la
25
co
in
la
,
In
C
o
as 44
review
a
Math iews
into C ortance of
into investigations.
ward, suchand independent
1. Present or
XX
p
lessons,
and back
rv
n be changed
ng into
d class
The Im rating Writi n-One Inte
gits
such as 13, ca
same forwar
di
:
e,
e
e
th
4
th
om
e
1
dr
X
rs
t
lin
ve
X
re
a pa
Poin
is notAbout
corpo ts of One-o
ocedure: You
Mathematics
number that
ular pr
fi
15: In
nce it takes for some
Sisections
XX
particthe
Point : The Bene
w to do this.
ction
e by using a
ho
u
om
d
r
dr
ar
t
lin
bo
6
s
pa
e
1
X
th
t
s
In
X
on
s
in
te
la
of
the
investigations
give
insights into the
m
ra
o
st
P
C
o
on
g
ls
ssro
Math
and add. Dem is a one-step palindrome.
ateria n
for Cla g Games inFocusing
tive M tron
othe cornerstone
13 underlying math.
s
of
elementary
ti
n,
la
n
c
tio
u
g
di
u
io
in
ad
ip
in
t
e
g
s
on
an
Ins
Us
XX
Mana
le of M tervention
int 17:
g and curriculum, the ideas in this
ng Po 18: The Ro s for mathematics
In
anizin
X
t
rg
X
ie
in
O
g
o
r
te
ing P
: Stra
tures fo oare
XX
models
for Minstruction
that builds
oint 19 Four Struc part
1
cti n
ath
ting P
uction
Instru
0:
2
m
t
o
XX
in
ro
s Instr Reading to
o
t
s
P
r
s
g
la
A
C
in
e
m
rt
students’
understanding,
confidence,
and
g
o
a
fr
X
u
Demonstrate
X
e
g
one addition.
h Lan aching Bridg
an
s
th
li
e
or
g
m
n
E
Te
lary in arithmetic:
rs take
competence
e.
ocabu
ding a
ing to
Some numbe
step palindrom
nnect Point 21: Buil hing Math V Literature
hich is a twow
,
c
67
a
’s
g
ith
e
w
n
re Beginning Number Concepts
Startin Point 22: T
hild•
sing C
g
Startin Point 23: U
9
•
Place Value
g
Startin
2 or an
a
Chang
e How
y other d of Race for
Many N
numbe
20, pla
rule so
r.
y Race
u
it’s pos
sible to mbers a Play
try it c
er Can
ountin
c
o
u
n
t
C
o
g on on
e, two, n one, two, o ount. Change
r three
three, o
the
nu
r four n
I includ
umbers mbers. Then
ed this
.
two
wro
Sample Less
ssoonn
that result
step palindrome
in 484.
Number and
Operations
271
Marilyn’s Notes
Marilyn’s personal (and dog eared)
copy of the Third Edition is filled
with notes, some scribbled into the
margins and some on sticky notes. In
this new Fourth Edition, Marilyn has
reproduced many of these notes, added
new ones as well, and organized them
into categories to make them more
useful. Look for notes titled Teaching
Tip, Management Suggestion, When
Observing, Vocabulary Suggestion,
Literature Connection, and more.
2. Pose a pa
rt of the pr
oblem or a
Have the st
similar but
uden
smaller prob
work, do it on ts try to change 48 into
lem.
a palindrom
the board so
e. As they
they can chec
the class how
k their work.
to record the
Then show
results for 13
drome Recor
, 68, and 48
ding Sheet.
on the Palin3. Present th
e investigat
ion. Each pa
the numbers
ir or group is
from zero to
to investigat
99. First they
the work. Ev
e all
About the Math
should decide
eryone is to
how to share
keep individu
demonstrate
A colleague point
al
d. Also, each
re
ed
co
Teaching
Tip
rd
s
on
the chart as
student is to
the digits in each out that
oring the nu
record on a 0–
mbe
of the
99
for
Square
Up,
point
res
ch
ult
coloring in on rs that are alreAs
ar
ing
t,
ad
pa
collindromes in my
y palindrom
e-step palindr
es with one
exa
mp
omes
les
As a group,
were the
outwith
to students
they color,
another cothat
they should
and 1111. But’s th same—44
lor, and so on
agree on the
to look for
at not
.
colors
can
form
squares
on
the
patterns on
to
alw
us
ay
e. Direct them
s the case. The
both charts.
98 and 89 un
A
lso
,
less they are
te
67
ll
,
for examp le, is number
them si
without
dtesto tackle
no
ready diagonal,
(Each numbe
for a serious
step palindrome a twor takes twen
When
Observing
bo
ut
w
ith
ty
that results
-four st
parallel
to athe sides of ad
the
of 8,813,200,
dition.
eps with
in 484.
023,188!)
resulting
I’ve
found
that
groups
pa
lin
dr
om
geoboard.
e
Management Suggestion
To compile students’ work, I
ask students to choose one
of their shapes, copy it onto
large geoboard dot paper, cut
it out, and post it. I ask them
to post only shapes that
haven’t already been posted.
sometimes start by dividing
up the numbers and working
individually, but then shift to
d Operations
collaborating Ninumabedirfan
ferent
way as they begin to notice
patterns, such as about
which numbers are one-step
palindromes.
Includes more than 250 math investigations!
271
Special Features
PART
What’s in About Teaching Mathematics,
Fourth Edition?
1
Prob
le
Inves m-Solving
tigat
ions
This
first
part
of
the
book
reflects
the
major
s
t
overhaul of About Teaching Mathematics.
Poin
Here Marilyn addresses twenty-three issues
Part 2
and presents her current thinking. The Starting Points are organized into six categories:
Problem-Solving
• General Pedagogical Issues
• Number and Operations:
Investigations
The Mathematical Cornerstone
Sample Whole-Class
Lessons
2
PART
Part 1
Starting Points
AT A G
LANC
E
Overv
iew
X
X
Plann
in
XX
Proble g Whole-C
lass Le
m-So
lving
Proble
Inves ssons with
X
X
tigatio
m
-Solvin
nt,
XX
ns
Conte
g Inve
XX
Proble
sues Connecting
s
t
m
igatio
-Solvin
XX
d:
n
s
:
g
Starte struction
P
M
r
In
oblem
easur
d
vestig
XX
d In
erstan ers
emen
-Solvin
lum
XX
ations
es, an
’t Und
rn
urricu
t
: Data
g Inve
Proble
the C at You Don e Math Lea
X
g
n
X
s
ri
tigatio
m-So
e
v
h
ti
X
c
W
e
e
X
n
h
ff
lv
and A
ns: G
ac
eE
ersto
XX
lgebr ing Investig
eome
an’t Te nts Becom
l Corn
a
a
e
t
ic
r
d
X
ic
a
y
t
t
X
tu
T
a
io
hinkin
Proble
ns: Paan introduction about how to plan
ng S
em
etic
g After
m
tterns
Math on Arithm•
XX
From Arithmetic
to Problem Solving
Numb Solving In
XX
: The
ve
e
problem-solving
lessons,
specific ideas are
v
ations : A Perspecti Common
r
estiga
X
a
X
n
d
Oper
e
t
io
n
s
soning dents Mak
•
Assessing
Understanding
and
Skills
ations
:
organized into five areas
Stu
ns
XX of the curriculum:
y Mea tanding
hy Do Errors?
XX
It Reall
tic
•ers Suggestions
for Classroom Instruction
ithme Sign: What ping Und
•
Measurement
XX
ual
velo
he Eq blems: De ns
Sample Lesson
• Connecting to English Language Arts
XX
tio
rns
Pro
ra
• Data
e
rd
p
o
merical patte
O
W
XX
mes
metic
looking for nu
ing:
ro
in
d
th
lv
ts
g
in
ri
Instruction
o
en
al
A
S
P
in
ud
f
st
s
o
ular way.
XX
ation engage • beGeometry
m Solv s to Problem
rs in a partic
This investig
roble
m
adding num
XX
om
fr
ic to P ord Proble
s
ge
e
er
u
em
s
Is
that
Detailed plans are provided for
using problem-solving investigations
in whole-class lessons. These instructional suggestions are offered for all
the content areas in Part 2 and for
all arithmetic topics in Part 3.
Over the years, teachers have written to Marilyn asking for help with
some of the investigations in About
Teaching Mathematics. In these
sections, Marilyn addresses the
underlying mathematics for those
problem-solving investigation.
NEW
Part 4
Questions
Teachers Ask
• Addition and Subtraction
• Introducing Multiplication
• Introducing Division
Race fo
lem.
smaller prob
r 20 • Extending Multiplication and Division
similar but
a
they
Play w
s
or
A
m
e.
le
om
ob
ith a pa
of the pr
rt Here
Marilyn
responds
todr
pedagogical
into a palin
rtner.•P Fractions
2. Pose a pa
The fir
k. Then show
to change 48
or
y
w
tr
r
ts
ei
en
th
k
ud
layers t
chec
st
e PalinHave the st questions
they canreceived
from
teachers
48 on th
ake tur
second player sta•
the board so she’s fo
68, and
rts Decimals
ns coun
sults r 13,
work, do it on
and co
player
re
the Math
e
th
rd
co
ting on
to re the years. She chose more than two About e pointed out that
co
unts eit
over
or two
class how
e
gu
th
llea
l
e
uesPercents
co
al
A
e
t.
h
o
at
ee
more n ntin•
ig
e
r two n
ding Sh
of the
p is to invest
umbers with the next r “One” or
umbers
drome Recordozen questions
or grou
the digits in each es in my
reflect
common
share themes
ch pairthat
to
Ea
w
Extens
“
n.
ho
.
n
io
O
.
de
T
at
u
ci
ig
de
n
m
palindrom
h
st
ld
e
ve
ing
e
b
ou
in
ult
,
e
sh
w
res
e
ions
t
th
ey
r
t
w
as
in
th
t
Organized similarly
to the
an instructional
o
esen
ner is t
same—44
99. First
ro to archives.
s on the char
1. Cha
he play d counts on e .” The 3. Pr
from zeher
examp les were the not
bersfrom
lvidual record
m
co
di
nu
in
t,
e
nge the
ar
t
th
ep
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the same fo
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r that reads
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and add. Dem ing a particular procedur
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onstrate on
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13 is a one-st
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Some numbe
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e than one
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onstrate
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Part 3
Teaching Arithmetic
ot
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tion.
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As a group,
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98 and 89 un
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023,188!)
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This investig
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that emerge
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rns
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ay.
Materials
large 0–99 ch
art, 1 per stud
ent
Palindrome
See Reproduc
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crayons, colo
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See Reproduc
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rs take
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nnect Point 21: Buil hing Math V Literature
hich is a twow
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re Beginning Number Concepts
Startin Point 22: T
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•
Place Value
g
Startin
2 or an
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Many N
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sible to mbers a Play
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Sample Less
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that result
step palindrome
in 484.
Number and
Operations
271
Marilyn’s Notes
Marilyn’s personal (and dog eared)
copy of the Third Edition is filled
with notes, some scribbled into the
margins and some on sticky notes. In
this new Fourth Edition, Marilyn has
reproduced many of these notes, added
new ones as well, and organized them
into categories to make them more
useful. Look for notes titled Teaching
Tip, Management Suggestion, When
Observing, Vocabulary Suggestion,
Literature Connection, and more.
2. Pose a pa
rt of the pr
oblem or a
Have the st
similar but
uden
smaller prob
work, do it on ts try to change 48 into
lem.
a palindrom
the board so
e. As they
they can chec
the class how
k their work.
to record the
Then show
results for 13
drome Recor
, 68, and 48
ding Sheet.
on the Palin3. Present th
e investigat
ion. Each pa
the numbers
ir or group is
from zero to
to investigat
99. First they
the work. Ev
e all
About the Math
should decide
eryone is to
how to share
keep individu
demonstrate
A colleague point
al
d. Also, each
re
ed
co
Teaching
Tip
rd
s
on
the chart as
student is to
the digits in each out that
oring the nu
record on a 0–
mbe
of the
99
for
Square
Up,
point
res
ch
ult
coloring in on rs that are alreAs
ar
ing
t,
ad
pa
collindromes in my
y palindrom
e-step palindr
es with one
exa
mp
omes
les
As a group,
were the
outwith
to students
they color,
another cothat
they should
and 1111. But’s th same—44
lor, and so on
agree on the
to look for
at not
.
colors
can
form
squares
on
the
patterns on
to
alw
us
ay
e. Direct them
s the case. The
both charts.
98 and 89 un
A
lso
,
less they are
te
67
ll
,
for examp le, is number
them si
without
dtesto tackle
no
ready diagonal,
(Each numbe
for a serious
step palindrome a twor takes twen
When
Observing
bo
ut
w
ith
ty
that results
-four st
parallel
to athe sides of ad
the
of 8,813,200,
dition.
eps with
in 484.
023,188!)
resulting
I’ve
found
that
groups
pa
lin
dr
om
geoboard.
e
Management Suggestion
To compile students’ work, I
ask students to choose one
of their shapes, copy it onto
large geoboard dot paper, cut
it out, and post it. I ask them
to post only shapes that
haven’t already been posted.
sometimes start by dividing
up the numbers and working
individually, but then shift to
d Operations
collaborating Ninumabedirfan
ferent
way as they begin to notice
patterns, such as about
which numbers are one-step
palindromes.
Includes more than 250 math investigations!
271
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