Fair Sharing and Equipartitioning in Mathematics Learning
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G
r
a
d
e
T
w
o
M
o
d
u
l
e
3
2
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E
q
u
i
p
a
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i
t
i
o
n
i
n
g
Equipartitioning
means
“Sharing
Fairly”
What
experiences
have
students
had
with
sharing
fairly?
3
Partners
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W
h
a
t
H
a
p
p
e
n
s
i
n
K
a
n
d
1
s
t
G
r
a
d
e
s
?
Share
collections
of
objects
fairly
and
reassemble
Kindergarten:
share
between
2
people
1
st
grade:
share
between
2-6
people
Share
rectangles
and
circles
among
two
or
four
people
and
reassemble
1
st
grade
4
W
h
a
t
H
a
p
p
e
n
s
i
n
2
n
d
G
r
a
d
e
?
Look
at
the
Essential
Standards
Fairly
share
collections
and
reassemble
them
Fairly
share
a
rectangle
or
circle
among
two,
four,
eight;
three
and
six
people
and
reassemble
it
N
a
m
e
t
h
e
s
h
a
r
e
a
s
“
1
/
n
t
h
o
f
t
h
e
w
h
o
l
e
”
Predict
that
the
size
of
a
fair
share
decreases
as
the
number
of
shares
increases,
and
vice
versa
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5
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E
q
u
i
p
a
r
t
i
t
i
o
n
i
n
g
Parts
of
a
whole
Continuous
or
area
models
Equal
shares
by
sub-dividing
a
whole
P
o
l
a
n
d
F
r
a
n
c
e
6
E
q
u
i
p
a
r
t
i
t
i
o
n
i
n
g
Parts
of
a
group
Discrete
or
set
models
Dividing
into
subsets
of
equal
size
(equal
numbers
of
elements)
7
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E
q
u
i
p
a
r
t
i
t
i
o
n
i
n
g
,
D
i
v
i
s
i
o
n
,
F
r
a
c
t
i
o
n
s
All
focus
on
equal
portions
Understanding
the
“whole”
is
important
Often
referred
to
as
“fair
shares”
Models
may
look
different
depending
on
the
context
8
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E
q
u
i
p
a
r
t
i
t
i
o
n
i
n
g
,
D
i
v
i
s
i
o
n
,
F
r
a
c
t
i
o
n
s
How
could
2
people
fairly
share
the
cake?
The
gumballs?
How
could
you
divide
the
cake
and
the
gumballs
into
2
equal
parts?
Show
1/2
of
the
cake
and
1/2
of
the
gumballs
9
Partners
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D
i
v
i
s
i
o
n
a
s
E
q
u
i
p
a
r
t
i
t
i
o
n
i
n
g
Get
at
least
15
two-color
counters
Use
them
to
solve
these
problems:
Larry
and
3
friends
are
sharing
12
cupcakes.
How
many
cupcakes
will
each
child
get
if
each
gets
a
fair
share?
Larry,
Mary,
Devin
and
Martez
each
have
3
brownies.
How
many
brownies
did
Larry’s
mom
bake?
10
Partners
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S
t
u
d
e
n
t
T
h
i
n
k
i
n
g
Tom,
Renee,
and
Sara
collected
some
apples.
Each
person
got
five
apples.
What
was
the
total
number
of
apples
collected?
What
conceptual
understanding
related
to
equipartitioning
do
students
have?
How
do
we
start
to
build
understanding
of
the
relationships
between
the
whole
and
the
equal
parts?
11
Partners
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R
e
l
a
t
i
o
n
s
h
i
p
s
:
W
h
o
l
e
s
a
n
d
P
a
r
t
s
Needs
to
be
made
explicit
Whole
was
15
apples
5
apples
represented
1/3
of
the
whole
5
represented
a
fair
share
for
3
people
Bringing
together
each
of
the
three
thirds
makes
the
whole
If
5
people
shared
the
same
whole,
the
fair
share
would
be
smaller
12
C
l
a
s
s
r
o
o
m
P
r
o
b
l
e
m
s
Write
several
problems
for
2
nd
graders
that
address
division
as
equipartitioning
Ideas
to
include
Equal
sized
groups
Determine
initial
size
of
a
group
given
the
equal
parts
Describe
what
happens
when
collections
are
divided
fairly
and
what
happens
when
reassembled
Partners
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Partners
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C
o
n
s
i
d
e
r
i
n
g
S
e
t
M
o
d
e
l
s
The
whole
is
the
total
set
of
objects
and
subsets
of
the
whole
are
fractional
parts
Number
of
objects
not
size
is
important
Examples:
counters,
people,
M
&
Ms,
any
discrete
objects
14
F
a
i
r
S
h
a
r
e
s
:
R
e
c
t
a
n
g
l
e
s
Fold
a
square
in
half
Cut
your
square
How
do
you
know
each
half
is
equal?
Talk
with
a
neighbor
Is
your
half
equivalent
to
your
neighbor’s?
What
key
ideas
do
you
want
2
nd
graders
to
understand
in
this
activity?
Partners
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15
K
e
y
I
d
e
a
s
Each
part
is
one-half
of
the
whole
square
The
two
parts
make
the
square
Why
are
these
not
halves
(non-examples)?
Why
may
some
students
think
they
are?
If
we
started
with
different
size
squares,
would
the
one-halves
be
equivalent?
Partners
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16
Partners
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g
O
n
e
H
a
l
f
?
H
o
w
M
a
n
y
W
a
y
s
?
17
Partners
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O
n
e
H
a
l
f
?
H
o
w
M
a
n
y
W
a
y
s
?
18
Partners
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R
e
c
t
a
n
g
l
e
s
a
n
d
C
i
r
c
l
e
s
2nd
graders
share
rectangles
and
circles
among
2,
4,
8,
3,
and
6
people
What
experiences
do
we
need
to
provide
students?
Context
Models
19
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P
e
o
p
l
e
F
r
a
c
t
i
o
n
s
20
Partners
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G
o
o
f
y
F
r
a
c
t
i
o
n
s
22
Partners
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L
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g
C
o
n
t
e
x
t
Maria’s
mom
baked
a
pan
of
brownies
How
can
Maria
share
the
brownies
with
3
friends
and
herself?
Partners
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G
e
t
t
o
3
0
•
Play the game
•
What mathematics is addressed
by the game?
23
Partners
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C
o
n
t
e
x
t
u
a
l
P
r
o
b
l
e
m
s
With
a
partner
brainstorm
problems
that
match
the
Essential
Standard:
“Explain
the
division
of
rectangles
and
circles
to
accommodate
different
numbers
of
people”
Share
your
ideas
24
Partners
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L
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n
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g
M
a
k
i
n
g
S
e
n
s
e
o
f
F
r
a
c
t
i
o
n
s
Students
need:
Multiple
models
Experience
with
multiple
contexts
Time
Language
25
F
r
a
c
t
i
o
n
s
:
R
e
a
s
o
n
s
f
o
r
D
i
f
f
i
c
u
l
t
i
e
s
Content
is
new
(proportional
vs.
additive
reasoning)
and
notation
is
different
Conceptual-development
experiences
are
limited;
symbols
are
often
emphasized
Instruction
is
Too
abstract
Too
procedural
Without
connections
With
limited
models
Without
meaningful
contexts
26
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F
r
a
c
t
i
o
n
R
e
s
e
a
r
c
h
Students’
difficulties
in
understanding
fractions
stem
from
learning
fractions
by
using
rote
memorization
of
procedures
without
a
connection
with
informal
ways
of
solving
problems
involving
fractions.
Steffe
and
Olive,
2002
27
I
n
2
n
d
G
r
a
d
e
:
N
o
S
y
m
b
o
l
s
f
o
r
F
r
a
c
t
i
o
n
s
1
2
Partners
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28
Partners
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M
a
t
h
&
L
i
t
e
r
a
t
u
r
e
What
happens
to
each
child’s
share
as
the
doorbell
rings?
What
happens
to
each
child’s
share
when
Grandma
arrives?
29
Partners
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g
C
o
m
p
e
n
s
a
t
o
r
y
P
r
i
n
c
i
p
l
e
The
principle
states
that
the
bigger
the
unit,
the
smaller
the
number
of
that
unit
needed
1/8
is
fewer
than
1/4;
1/4
is
few
cookies
than
1/2
of
the
cookies
What
experiences
might
lead
students
in
developing
an
understanding
of
this
idea?
30
W
h
e
n
C
o
n
c
e
p
t
s
A
r
e
N
o
t
W
e
l
l
-
D
e
v
e
l
o
p
e
d
…
NAEP
Question:
Estimate
the
sum
of
12/13
and
7/8
a.
1
b.
2
c.
19
d.
21
How
did
you
decide
your
estimate?
Which
answer
did
most
13
year
old
students
choose?
Partners
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31
W
h
e
n
C
o
n
c
e
p
t
s
A
r
e
N
o
t
W
e
l
l
-
D
e
v
e
l
o
p
e
d
…
N
A
E
P
Q
u
e
s
t
i
o
n
Estimate
the
sum
of
12/13
and
7/8
a.
1
b.
2
c.
19
d.
21
Students
look
at
fractions
as
though
they
represent
two
separate
whole
numbers
This
leads
to
misinterpretation
and
an
inability
to
access
the
reasonableness
of
results
Partners
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32
Partners
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i
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g
C
h
i
n
e
s
e
P
r
o
v
e
r
b
“
T
e
l
l
m
e
a
n
d
I
'
l
l
f
o
r
g
e
t
;
s
h
o
w
m
e
a
n
d
I
m
a
y
r
e
m
e
m
b
e
r
;
i
n
v
o
l
v
e
m
e
a
n
d
I
'
l
l
u
n
d
e
r
s
t
a
n
d
.
”
33
Partners
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W
h
a
t
i
s
M
e
a
s
u
r
e
m
e
n
t
?
A
count
of
how
many
units
are
needed
to
fill,
cover,
or
match
the
attribute
of
the
object
being
measured
A
fundamental
mathematical
process
interwoven
throughout
all
strands
of
mathematics
34
W
h
y
M
e
a
s
u
r
e
m
e
n
t
?
An
essential
link
between
mathematics
and
other
disciplines
such
as
science,
art,
music,
and
social
studies
It
makes
mathematics
real
and
tangible
for
students
giving
them
a
handle
on
their
world
What
are
the
implications
for
teaching
measurement?
35
M
e
a
s
u
r
e
m
e
n
t
:
B
i
g
I
d
e
a
s
An
object
can
be
described
and
categorized
in
multiple
ways
(attributes)
The
measurement
of
a
specific
numerical
attribute
tells
the
number
of
units
The
process
of
measurement
is
similar
for
all
attributes,
but
the
measurement
system
and
tool
vary
according
to
the
attribute
Measurements
are
accurate
to
the
extent
that
the
appropriate
unit/tool
is
used
properly
36
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M
e
a
s
u
r
e
m
e
n
t
S
t
a
n
d
a
r
d
s
Look
at
the
Essential
Standards
for
Measurement
Look
at
the
Clarifying
Objectives
37
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A
t
t
r
i
b
u
t
e
s
What
are
the
attributes
of
the
present?
Which
of
these
are
measurable?
38
Partners
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o
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t
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m
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t
i
c
s
L
e
a
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n
i
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g
P
r
o
c
e
s
s
o
f
M
e
a
s
u
r
e
m
e
n
t
Determine
the
attribute
to
be
measured
Choose
an
appropriate
unit
that
has
the
same
attribute
Choose
the
tool
Determine
how
many
of
that
unit
is
needed
by
filling,
covering,
or
matching
the
object
39
Partners
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o
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t
h
e
m
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t
i
c
s
L
e
a
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n
i
n
g
A
C
l
o
a
k
f
o
r
a
D
r
e
a
m
e
r
Covering
space
with
nonstandard
units
leaving
no
gaps
or
overlaps
40
Partners
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L
e
a
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n
i
n
g
M
e
a
s
u
r
i
n
g
a
S
q
u
a
r
e
Use
the
pattern
blocks
to
cover
the
square
Only
use
one
type
of
pattern
block
41
C
o
m
p
u
t
e
r
A
c
t
i
v
i
t
i
e
s
www.arcytech.org/java/patterns/patterns_j.shtml
Partners
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g
42
Partners
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t
i
c
s
L
e
a
r
n
i
n
g
M
e
a
s
u
r
e
m
e
n
t
C
o
m
p
o
n
e
n
t
s
Conservation
Transitivity
Partitioning
Unit
Iteration
Compensatory
Principle
43
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C
o
m
p
o
n
e
n
t
s
o
f
M
e
a
s
u
r
e
m
e
n
t
C
o
n
s
e
r
v
a
t
i
o
n
:
a
n
o
b
j
e
c
t
m
a
i
n
t
a
i
n
s
t
h
e
same
size
if
it
is
rearranged,
transformed,
or
divided
in
various
ways
44
C
o
n
s
e
r
v
a
t
i
o
n
E
x
a
m
p
l
e
s
Students
explore
conservation
Take
a
clay
ball,
cut
it
in
half.
Then
roll
one
half
into
a
ball
and
other
into
a
snake.
Do
they
have
the
same
mass?
Pour
the
same
amount
of
liquid
from
one
container
to
another.
How
tall
is
the
liquid
in
the
container?
Does
the
amount
stay
the
same?
45
C
o
m
p
o
n
e
n
t
s
o
f
M
e
a
s
u
r
e
m
e
n
t
T
r
a
n
s
i
t
i
v
i
t
y
:
t
w
o
o
b
j
e
c
t
s
c
a
n
b
e
c
o
m
p
a
r
e
d
in
terms
of
a
measurable
attribute
using
a
third
object
If
A
=
B
and
B
=
C,
then
A
=
C
If
A
<
B
and
B
<
C,
then
A
<
C
If
A
>
B
and
B
>
C,
then
A
>
C
46
Partners
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s
L
e
a
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n
i
n
g
P
a
r
t
i
t
i
o
n
i
n
g
Larger
units
can
be
subdivided
into
equivalent
units…mentally
subdividing
Choose
a
place
in
the
room
(table,
wall)
Mentally
divide
it
in
half
Choose
a
unit
of
measure
(marker,
handspan)
Determine
the
measurement
and
double
it
47
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s
L
e
a
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n
i
n
g
C
o
m
p
o
n
e
n
t
s
o
f
M
e
a
s
u
r
e
m
e
n
t
U
n
i
t
s
:
t
h
e
a
t
t
r
i
b
u
t
e
b
e
i
n
g
m
e
a
s
u
r
e
d
dictates
the
type
of
unit
used;
the
unit
must
have
the
same
attribute
as
the
attribute
to
be
measured
48
Partners
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i
c
s
L
e
a
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n
i
n
g
C
o
m
p
o
n
e
n
t
s
o
f
M
e
a
s
u
r
e
m
e
n
t
U
n
i
t
i
t
e
r
a
t
i
o
n
:
u
n
i
t
s
m
u
s
t
b
e
r
e
p
e
a
t
e
d
i
n
order
to
determine
the
measure
49
I
t
e
r
a
t
i
o
n
E
x
p
e
r
i
e
n
c
e
s
What
questions
might
you
ask
to
encourage
these
students
to
focus
on
the
placement
of
units?
How
can
you
provide
opportunities
for
students
to
use
iteration?
Partners
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s
L
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i
n
g
50
C
o
m
p
e
n
s
a
t
o
r
y
P
r
i
n
c
i
p
l
e
T
h
e
c
o
m
p
e
n
s
a
t
o
r
y
p
r
i
n
c
i
p
l
e
s
t
a
t
e
s
t
h
a
t
the
bigger
the
unit,
the
smaller
the
number
of
that
unit
is
needed
Turn
to
your
partner
and
give
an
example
of
what
this
means
What
happens
when
the
unit
used
to
measure
is
smaller?
Do
you
need
more
or
fewer
units?
Partners
f
o
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a
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L
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a
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51
Partners
f
o
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M
a
t
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e
m
a
t
i
c
s
L
e
a
r
n
i
n
g
C
o
m
p
e
n
s
a
t
o
r
y
P
r
i
n
c
i
p
l
e
i
n
A
c
t
i
o
n
Students
measure
the
top
of
the
cabinet
and
it
is
60
cubes
long
Then
they
measure
it
in
craft
sticks
and
find
out
that
it
is
5
sticks
long
It
takes
more
of
the
smaller
units
(cubes)
to
measure
the
cabinet
than
the
larger
unit
(sticks)
52
M
e
a
s
u
r
e
m
e
n
t
E
s
t
i
m
a
t
e
Choose
two
spots
on
a
surface
Mark
each
spot
Estimate
the
halfway
point
between
the
two
spots
and
mark
it
Cut
a
piece
of
string
to
represent
the
distance
between
the
two
spots
Fold
it
in
half
and
use
it
to
measure
the
halfway
point
How
close
was
your
estimate?
Partners
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53
Partners
f
o
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m
a
t
i
c
s
L
e
a
r
n
i
n
g
C
o
n
n
e
c
t
i
n
g
D
a
t
a
&
M
e
a
s
u
r
e
m
e
n
t
What
took
place
in
the
Giant
Step
activity?
How
could
iteration
have
been
highlighted
during
this
activity?
How
could
the
compensatory
principle
have
been
more
explicit
during
this
activity?
54
C
a
p
a
c
i
t
y
a
n
d
V
o
l
u
m
e
Capacity
refers
to
the
amount
a
container
will
hold
Pitchers
and
bottles
Scoops,
cups,
quarts
measure
liquids
Volume
is
often
though
of
as
the
amount
of
space
an
object
takes
up
Non-Standard
units
such
wooden
cubes,
marbles,
tennis
balls
teddy
bear
counters
C
a
p
a
c
i
t
y
i
s
t
h
e
f
o
c
u
s
f
o
r
p
r
i
m
a
r
y
c
h
i
l
d
r
e
n
Partners
f
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s
L
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a
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n
i
n
g
55
N
o
n
s
t
a
n
d
a
r
d
U
n
i
t
s
o
f
C
a
p
a
c
i
t
y
Containers
or
scoops:
small
containers
that
are
filled
and
poured
repeatedly
into
the
container
being
measured;
the
number
of
scoops
provides
the
measure
Solid
units:
small
congruent
objects
that
fill
the
container
being
measured;
the
number
of
small
objects
provides
the
measure
Partners
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56
C
a
p
a
c
i
t
y
:
C
o
m
p
e
n
s
a
t
o
r
y
P
r
i
n
c
i
p
l
e
How
can
the
compensatory
principle
be
developed
through
the
concept
of
capacity
in
second
grade?
Partners
f
o
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i
c
s
L
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a
r
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i
n
g
57
M
a
s
s
Mass
is
the
amount
of
matter
in
an
object
Weight
is
a
measure
of
the
pull
or
force
of
gravity
on
an
object
On
the
moon
there
is
less
gravity
so
an
object
weighs
less
The
mass
of
an
object
is
identical
on
the
moon
and
on
Earth
Partners
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58
Partners
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o
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m
a
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i
c
s
L
e
a
r
n
i
n
g
E
x
p
l
o
r
i
n
g
M
a
s
s
Children
can
use
their
hands
to
estimate
which
of
two
objects
is
heavier
Children
place
marbles
on
one
side
of
a
balance
scale
to
find
the
mass
of
a
book
using
marbles
as
a
non-standard
measure
59
Partners
f
o
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t
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m
a
t
i
c
s
L
e
a
r
n
i
n
g
C
l
a
s
s
r
o
o
m
E
x
p
e
r
i
e
n
c
e
s
Look
at
the
measurement
handouts
Each
group
will
work
through
the
Scavenger
Hunt
and
the
Measuring
Rectangles
As
you
work
discuss
how
the
activity
addresses
the
Essential
Standards
60
M
e
a
s
u
r
i
n
g
R
e
c
t
a
n
g
l
e
s
How
would
students
order
the
rectangles?
What
attributes
do
they
use?
Do
they
realize
that
when
they
consider
different
attributes,
the
rectangles
may
be
ordered
differently?
Can
they
cover
it
with
no
gaps
or
overlaps?
Other
ideas?
Partners
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s
L
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61
Partners
f
o
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t
h
e
m
a
t
i
c
s
L
e
a
r
n
i
n
g
S
c
a
v
e
n
g
e
r
H
u
n
t
How
could
you
extend
the
activity?
62
Partners
f
o
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a
t
h
e
m
a
t
i
c
s
L
e
a
r
n
i
n
g
M
e
a
s
u
r
e
m
e
n
t
J
i
g
s
a
w
Roll
This
Weigh
Side
by
Side
Guess
and
Count
Paper
Clip
Measurement
Which
Takes
Longer?
63
Partners
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o
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m
a
t
i
c
s
L
e
a
r
n
i
n
g
Marker
board
P
a
t
h
s
chair
bookcase
2
nd
Grade
Essential
Understanding
Describe
relationships
of
objects
using
proximity,
position,
direction,
and
turns
Look
at
the
handout,
“Paths
Lesson”
64
N
C
T
M
A
p
p
l
e
t
s
http://illuminations.nctm.org/
Hiding
Ladybug
Adventures
Ladybug
Mazes
Turtle
Pond
Partners
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L
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g
65
Partners
f
o
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M
a
t
h
e
m
a
t
i
c
s
L
e
a
r
n
i
n
g
M
e
a
s
u
r
e
m
e
n
t
Is
one
of
the
most
widely
used
applications
of
mathematics
Connects
to
geometry,
data,
number,
and
problem
solving
Experience
provide
opportunities
to
learn
and
understand
66
P
e
r
s
o
n
a
l
R
e
f
l
e
c
t
i
o
n
Opportunities
to
learn
and
understand
in
an
environment
that
is
interesting,
challenging,
and
enjoyable
provide
students
with
strong
foundations
upon
which
to
build
What
strengths
does
your
instructional
program
currently
have?
What
three
challenges
will
you
take
from
this
module
for
your
classroom?
Partners
f
o
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t
h
e
m
a
t
i
c
s
L
e
a
r
n
i
n
g
67
D
P
I
M
a
t
h
e
m
a
t
i
c
s
S
t
a
f
f
Everly
Broadway,
Chief
Consultant
Renee
Cunningham
Kitty
Rutherford
Robin
Barbour
Mary
H.
Russell
Carmella
Fair
Johannah
Maynor
Amy
Smith
Partners
for
Mathematics
Learning
is
a
Mathematics-Science
Partnership
Project
funded
by
the
NC
Department
of
Public
Instruction.
Permission
is
granted
for
the
use
of
these
materials
in
professional
development
in
North
Carolina
Partners
school
districts.
Partners
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s
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i
n
g
68
Partners
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o
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t
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e
m
a
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i
c
s
L
e
a
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n
i
n
g
P
M
L
D
i
s
s
e
m
i
n
a
t
i
o
n
C
o
n
s
u
l
t
a
n
t
s
Susan
Allman
Julia
Cazin
Ruafika
Cobb
Anna
Corbett
Gail
Cotton
Jeanette
Cox
Leanne
Daughtry
Lisa
Davis
Ryan
Dougherty
Shakila
Faqih
Patricia
Essick
Donna
Godley
Cara
Gordon
Tery
Gunter
Barbara
Hardy
Kathy
Harris
Julie
Kolb
Renee
Matney
Tina
McSwain
Marilyn
Michue
Amanda
Northrup
Kayonna
Pitchford
Ron
Powell
Susan
Riddle
Judith
Rucker
Shana
Runge
Yolanda
Sawyer
Penny
Shockley
Pat
Sickles
Nancy
Teague
Michelle
Tucker
Kaneka
Turner
Bob
Vorbroker
Jan
Wessell
Daniel
Wicks
Carol
Williams
Stacy
Wozny
69
2
0
0
9
W
r
i
t
e
r
s
P
a
r
t
n
e
r
s
S
t
a
f
f
Kathy
Harris
Rendy
King
Tery
Gunter
Judy
Rucker
Penny
Shockley
Nancy
Teague
Jan
Wessell
Stacy
Wozny
Amanda
Baucom
Julie
Kolb
Freda
Ballard,
Webmaster
Anita
Bowman,
Outside
Evaluator
Ana
Floyd,
Reviewer
Meghan
Griffith,
Administrative
Assistant
Tim
Hendrix,
Co-PI
and
Higher
Ed
Ben
Klein
,
Higher
Education
Katie
Mawhinney,
Co-PI
and
Higher
Ed
Wendy
Rich,
Reviewer
Catherine
Stein,
Higher
Education
Please
give
appropriate
credit
to
the
Partners
for
Mathematics
Learning
project
when
using
the
materials.
Jeane
Joyner
,
Co-PI
a
nd
Project
Director
Partners
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L
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Partners
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a
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Delve into the concept of equipartitioning and fair sharing in mathematics, focusing on experiences of students in kindergarten through 2nd grade. Understand how to divide objects fairly among different numbers of people, predict size changes in fair shares, and explore division as equipartitioning through engaging activities. Discover the importance of understanding wholes and equal portions in fractions and division, all while fostering critical thinking and problem-solving skills in young learners.
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Presentation Transcript
PARTNERS forMathematicsLearning GradeTwo Module3 Partners forMathematicsLearning
2 Equipartitioning Equipartitioningmeans SharingFairly Whatexperiences havestudentshad withsharingfairly? Partners forMathematicsLearning
3 WhatHappensinKand1stGrades? Sharecollectionsofobjectsfairlyandreassemble Kindergarten:sharebetween2people 1stgrade:sharebetween2-6people Sharerectanglesandcirclesamongtwoorfour peopleandreassemble 1stgrade Partners forMathematicsLearning
4 WhatHappensin2ndGrade? LookattheEssentialStandards Fairlysharecollectionsandreassemblethem Fairlysharearectangleorcircleamongtwo, four,eight;threeandsixpeopleand reassembleit Nametheshareas 1/nthofthewhole Predictthatthesizeofafairsharedecreases asthenumberofsharesincreases,andvice versa Partners forMathematicsLearning
5 Equipartitioning Partsofawhole Poland France Continuousorareamodels Equalsharesbysub-dividingawhole Partners forMathematicsLearning
6 Equipartitioning Partsofagroup Discreteorsetmodels Dividingintosubsetsofequalsize(equal numbersofelements)
7 Equipartitioning,Division,Fractions Allfocusonequalportions Understandingthe whole isimportant Oftenreferredtoas fairshares Modelsmaylookdifferentdependingon thecontext Partners forMathematicsLearning
8 Equipartitioning,Division,Fractions Howcould2peoplefairlysharethecake? Thegumballs? Howcouldyoudividethecakeandthe gumballsinto2equalparts? Show1/2ofthecakeand1/2ofthe gumballs Partners forMathematicsLearning
9 DivisionasEquipartitioning Getatleast15two-colorcounters Usethemtosolvetheseproblems: Larryand3friendsaresharing12cupcakes. Howmanycupcakeswilleachchildgetifeach getsafairshare? Larry,Mary,DevinandMartezeachhave3 brownies.HowmanybrowniesdidLarry s mombake? Partners forMathematicsLearning
10 StudentThinking Tom,Renee,andSaracollectedsome apples.Eachpersongotfiveapples.What wasthetotalnumberofapplescollected? Whatconceptualunderstandingrelatedto equipartitioningdostudentshave? Howdowestarttobuildunderstandingofthe relationshipsbetweenthewholeandtheequal parts? Partners forMathematicsLearning
11 Relationships:WholesandParts Needstobemadeexplicit Wholewas15apples 5applesrepresented1/3ofthewhole 5representedafairsharefor3people Bringingtogethereachofthethreethirds makesthewhole If5peoplesharedthesamewhole, thefairsharewouldbesmaller Partners forMathematicsLearning
12 ClassroomProblems Writeseveralproblemsfor2ndgradersthat addressdivision asequipartitioning Ideastoinclude Equalsizedgroups Determineinitialsizeofagroupgiven theequalparts Describewhathappenswhencollections aredividedfairlyandwhathappens whenreassembled Partners forMathematicsLearning
13 ConsideringSetModels Thewholeisthetotalsetofobjectsand subsetsofthewholearefractionalparts Numberofobjectsnotsizeisimportant Examples:counters,people,M&Ms, anydiscreteobjects Partners forMathematicsLearning
14 FairShares:Rectangles Foldasquareinhalf Cutyoursquare Howdoyouknoweachhalfisequal?Talk withaneighbor Isyourhalfequivalenttoyourneighbor s? Whatkeyideasdoyouwant2ndgradersto understandinthisactivity? Partners forMathematicsLearning
15 KeyIdeas Eachpartisone-halfofthewholesquare Thetwopartsmakethesquare Whyarethesenothalves(non-examples)? Whymaysomestudentsthinktheyare? Ifwestartedwithdifferentsizesquares, wouldtheone-halvesbeequivalent? Partners forMathematicsLearning
16 OneHalf?HowManyWays? Partners forMathematicsLearning
17 OneHalf?HowManyWays? Partners forMathematicsLearning
18 RectanglesandCircles 2ndgradersshare rectanglesandcircles among2,4,8,3,and6people Whatexperiencesdoweneedtoprovide students? Context Models Partners forMathematicsLearning
19 PeopleFractions Partners forMathematicsLearning
20 Goofy Fractions Partners forMathematicsLearning
22 Context Maria smombaked apanofbrownies HowcanMariasharethebrownieswith3 friendsandherself? Partners forMathematicsLearning
Get to 30 Play the game What mathematics is addressed by the game? Partners forMathematicsLearning
23 ContextualProblems Withapartnerbrainstormproblemsthat matchtheEssentialStandard: Explainthedivisionofrectanglesand circlestoaccommodatedifferentnumbers ofpeople Shareyourideas Partners forMathematicsLearning
24 MakingSenseofFractions Studentsneed: Multiplemodels Experiencewith multiplecontexts Time Language Partners forMathematicsLearning
25 Fractions:ReasonsforDifficulties Contentisnew(proportionalvs.additive reasoning)andnotationisdifferent Conceptual-developmentexperiencesarelimited; symbolsareoftenemphasized Instructionis Tooabstract Tooprocedural Withoutconnections Withlimitedmodels Withoutmeaningfulcontexts
26 FractionResearch Students difficultiesinunderstanding fractionsstemfromlearningfractionsby usingrotememorizationofprocedures withoutaconnectionwithinformalways ofsolvingproblemsinvolvingfractions. SteffeandOlive,2002 Partners forMathematicsLearning
27 In2ndGrade: NoSymbolsforFractions 1 2 Partners forMathematicsLearning
28 Math&Literature Whathappenstoeachchild sshare asthedoorbellrings? Whathappenstoeach child ssharewhen Grandmaarrives? Partners forMathematicsLearning
29 CompensatoryPrinciple Theprinciplestatesthatthebiggerthe unit,thesmallerthenumberofthatunit needed 1/8isfewerthan1/4;1/4isfewcookies than1/2ofthecookies Whatexperiencesmightleadstudentsin developinganunderstandingofthisidea? Partners forMathematicsLearning
30 WhenConceptsAreNot Well-Developed NAEPQuestion: Estimatethesumof12/13and7/8 a.1 b.2 c.19 d.21 Howdidyoudecideyourestimate? Whichanswerdidmost13yearold studentschoose? Partners forMathematicsLearning
31 WhenConceptsAreNot Well-Developed NAEPQuestion Estimatethesumof12/13and7/8 a.1b.2 c.19d.21 Studentslookatfractionsasthoughthey representtwoseparatewholenumbers Thisleadstomisinterpretationandan inabilitytoaccessthereasonablenessof results Partners forMathematicsLearning
32 ChineseProverb TellmeandI'llforget; showmeandImayremember; involvemeandI'll understand. Partners forMathematicsLearning
33 WhatisMeasurement? Acountofhowmanyunitsareneededto fill,cover,ormatchtheattributeofthe objectbeingmeasured Afundamentalmathematicalprocess interwoventhroughoutallstrandsof mathematics Partners forMathematicsLearning
34 WhyMeasurement? Anessentiallinkbetween mathematicsandotherdisciplinessuchas science,art,music,andsocialstudies Itmakesmathematicsrealandtangiblefor studentsgivingthemahandleontheir world Whataretheimplicationsforteaching measurement?
35 Measurement:BigIdeas Anobjectcanbedescribedand categorizedinmultipleways(attributes) Themeasurementofaspecificnumerical attributetellsthenumberofunits Theprocessofmeasurementissimilarfor allattributes,butthemeasurementsystem andtoolvaryaccordingtotheattribute Measurementsareaccuratetotheextent thattheappropriateunit/toolisused properly
36 MeasurementStandards LookattheEssential Standardsfor Measurement LookattheClarifying Objectives Partners forMathematicsLearning
37 Attributes Whataretheattributesofthepresent? Whichofthesearemeasurable? Partners forMathematicsLearning
38 ProcessofMeasurement Determinetheattribute tobemeasured Chooseanappropriateunit thathasthesameattribute Choosethetool Determinehowmanyofthatunitisneeded byfilling,covering,ormatchingtheobject Partners forMathematicsLearning
39 ACloakforaDreamer Coveringspacewith nonstandardunits leavingnogaps oroverlaps Partners forMathematicsLearning
40 MeasuringaSquare Usethepatternblockstocoverthesquare Onlyuseonetypeofpatternblock Partners forMathematicsLearning
41 ComputerActivities www.arcytech.org/java/patterns/patterns_j.shtml Partners forMathematicsLearning
42 MeasurementComponents Conservation Transitivity Partitioning UnitIteration CompensatoryPrinciple Partners forMathematicsLearning
43 ComponentsofMeasurement Conservation:anobjectmaintainsthe samesizeifitisrearranged,transformed, ordividedinvariousways Partners forMathematicsLearning
44 ConservationExamples Studentsexploreconservation Takeaclayball,cutitinhalf.Thenrollone halfintoaballandotherintoasnake.Dothey havethesamemass? Pourthesameamountofliquidfromone containertoanother.Howtallistheliquidin thecontainer?Doesthe amountstaythesame?
45 ComponentsofMeasurement Transitivity:twoobjectscanbecompared intermsofameasurableattributeusinga thirdobject IfA=BandB=C,thenA=C IfA<BandB<C,thenA<C IfA>BandB>C,thenA>C
46 Partitioning Largerunitscanbesubdividedinto equivalentunits mentallysubdividing Chooseaplaceintheroom(table,wall) Mentallydivideitinhalf Chooseaunitofmeasure(marker,handspan) Determinethemeasurementanddoubleit Partners forMathematicsLearning
47 ComponentsofMeasurement Units:theattributebeingmeasured dictatesthetypeofunitused;theunitmust havethesameattributeastheattributeto bemeasured Partners forMathematicsLearning
48 ComponentsofMeasurement Unititeration:unitsmustberepeatedin ordertodeterminethemeasure Partners forMathematicsLearning
49 IterationExperiences Whatquestionsmight youasktoencourage thesestudentstofocus ontheplacementof units? Howcanyouprovideopportunitiesfor studentstouseiteration? Partners forMathematicsLearning
50 CompensatoryPrinciple Thecompensatoryprinciplestatesthat thebiggertheunit,thesmallerthenumber ofthatunitisneeded Turntoyourpartnerandgiveanexample ofwhatthismeans Whathappenswhentheunitusedto measureissmaller?Doyouneedmoreor fewerunits? Partners forMathematicsLearning
