Proportional Reasoning in Grade 8 Mathematics Learning
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Grade
8
Module
4
2
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M
o
d
u
l
e
4
Proportional
Reasoning
3
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W
h
i
c
h
I
s
a
B
e
t
t
e
r
B
u
y
?
12
tickets
for
$15.00
or
20
tickets
for
$23.00?
4
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W
h
i
c
h
i
s
a
B
e
t
t
e
r
B
u
y
?
What
is
your
answer?
How
did
you
obtain
your
answer?
What
are
some
strategies
that
your
students
might
use?
5
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W
h
i
c
h
I
s
a
B
e
t
t
e
r
B
u
y
?
If
a
student
values
each
ticket
as
worth
$1.00,
what
might
the
student
say
about
each
deal
using…
Additive
reasoning
Proportional
reasoning
6
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P
r
o
p
o
r
t
i
o
n
a
l
T
h
i
n
k
i
n
g
As
different
ways
to
think
about
proportions
are
considered
and
discussed,
teachers
should
help
students
recognize
when
and
how
various
ways
of
reasoning
about
proportions
might
be
appropriate
to
solve
problems
PSSM,
2000
7
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C
a
p
s
t
o
n
e
o
f
t
h
e
C
u
r
r
i
c
u
l
u
m
!
“Proportional
reasoning
has
been
referred
to
as
the
capstone
for
the
elementary
curriculum
and
the
cornerstone
of
algebra
and
beyond.”
Van
de
Walle,
J.
A
(2004).
Elementary
and
Middle
School
Mathematics:
Teaching
Developmentally
.
Pearson
Learning
Inc.
8
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g
A
m
a
z
i
n
g
–
I
s
n
’
t
I
t
?
“It
is
estimated
that
more
than
half
of
the
adult
population
cannot
be
viewed
as
proportional
thinkers.
That
means
that
we
do
not
acquire
the
habits
and
skills
of
proportional
reasoning
simply
by
getting
older.”
Van
de
Walle,
J.
A
(2004).
Elementary
and
Middle
School
Mathematics:
Teaching
Developmentally
.
Pearson
Learning
Inc.
9
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R
e
s
e
a
r
c
h
S
a
y
s
…
“Research
indicates
that
instruction
can
have
an
effect,
especially
if
rules
and
algorithms
for
fraction
computation,
for
comparing
ratios,
and
for
solving
proportions
are
delayed.
Premature
use
of
rules
encourages
students
to
apply
rules
without
thinking
and,
thus,
the
ability
to
reason
proportionally
often
does
not
develop.”
Van
de
Walle,
J.
A
(2004).
Elementary
and
Middle
School
Mathematics:
Teaching
Developmentally
.
Pearson
Learning
Inc.
10
T
e
a
c
h
i
n
g
E
f
f
e
c
t
i
v
e
l
y
Instruction
in
solving
proportions
should
include
methods
that
have
a
strong
intuitive
basis
PSSM,
2000
In
a
group
of
students
who
can
successfully
apply
an
algorithm,
how
can
you
distinguish
between
those
who
can
reason
proportionally
and
those
who
cannot?
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11
“
O
n
e
I
n
c
h
T
a
l
l
”
b
y
S
h
e
l
S
i
l
v
e
r
s
t
e
i
n
If
you
were
only
one
inch
tall,
you’d
ride
a
worm
to
school
The
teardrop
of
a
crying
ant
would
be
your
swimming
pool.
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12
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W
o
u
l
d
T
h
i
s
B
e
T
r
u
e
?
If
you
were
only
one
inch
tall,
you
could
wear
a
thimble
on
your
head
13
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H
o
w
A
b
o
u
t
T
h
i
s
?
If
you
were
only
one
inch
tall,
it
would
take
about
a
month
to
get
down
to
the
store
14
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L
e
t
’
s
I
n
v
e
s
t
i
g
a
t
e
!
If
you
were
only
one
inch
tall
…
Could
you
ride
a
worm
to
school?
Could
you
wear
a
thimble
on
your
head?
Would
it
take
a
month
to
get
to
the
store?
15
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J
u
s
t
T
h
e
F
a
c
t
s
,
M
a
’
a
m
!
What
information
will
you
need
to
solve
these
problems?
16
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T
i
m
e
T
o
I
n
v
e
s
t
i
g
a
t
e
17
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g
L
e
t
’
s
T
a
l
k
What
are
our
conclusions?
Could
you
ride
a
worm
to
school?
Could
you
wear
a
thimble
on
your
head?
Would
it
take
a
month
to
get
to
the
store?
18
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I
n
c
r
e
a
s
e
d
S
t
u
d
e
n
t
U
n
d
e
r
s
t
a
n
d
i
n
g
Problems
that
involve
constructing
or
interpreting
scale
drawings
offer
students
opportunities
to
use
and
increase
their
knowledge
of
similarity,
ratio,
and
proportionality
PSSM
19
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g
T
a
n
g
r
a
m
T
i
m
e
!
20
Partners
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o
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a
t
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c
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L
e
a
r
n
i
n
g
T
a
n
g
r
a
m
T
i
m
e
!
21
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g
I
n
v
e
s
t
i
g
a
t
i
o
n
a
n
d
E
x
p
l
o
r
a
t
i
o
n
“Students
who
learned
through
investigation
and
exploration
were
not
only
more
successful
at
giving
correct
responses
to
proportional
reasoning
tasks
but
also
better
able
to
justify
those
answers.”
Fey,
J.T.,
Miller,
J.L.
(2000).
Proportional
Reasoning.
Mathematics
Teaching
in
the
Middle
School
.
5
(5),
312
22
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C
o
n
t
i
n
u
i
n
g
O
u
r
I
n
v
e
s
t
i
g
a
t
i
o
n
s
…
23
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T
h
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S
i
e
r
p
i
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s
k
i
T
r
i
a
n
g
l
e
A
c
t
i
v
i
t
y
24
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S
i
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k
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STAGE
0
Using
dot
paper,
construct
an
equilateral
triangle
of
side
length
16
What
is
the
area
of
this
triangle?
25
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S
i
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k
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i
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l
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STAGE
1
Mark
the
midpoint
of
each
side
of
the
triangle
Join
the
midpoints
to
form
4
smaller
triangles
26
T
h
e
S
i
e
r
p
i
n
s
k
i
T
r
i
a
n
g
l
e
STAGE
1
Determine
some
relationships
between
the
new
triangles
and
the
original
triangle
Similarity?
Congruence?
What
is
the
area
of
each
new
triangle?
What
fraction
of
the
original
triangle
does
each
new
triangle
represent?
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27
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S
i
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p
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s
k
i
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r
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a
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g
l
e
STAGE
1
Remove
(shade)
the
center
triangle
What
fraction
of
the
original
area
is
not
shaded?
Update
the
chart
28
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S
i
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k
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l
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STAGE
2
Bisect
each
side
of
the
“new”
(unshaded)
triangles
Join
the
midpoints
in
each
to
form
a
total
of
12
smaller
triangles
(16
if
you
divided
the
larger
shaded
triangle)
29
T
h
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S
i
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r
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k
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i
a
n
g
l
e
STAGE
2
Determine
some
relationships
between
the
new
triangles
and
the
original
triangle
Remove
(shade)
the
center
triangles
What
is
the
area
of
the
new
triangle?
What
fraction
of
the
original
area
is
not
shaded?
Partners
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30
Partners
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S
i
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k
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a
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l
e
STAGE
3
Bisect
each
side
of
the
“new”
(unshaded)
triangles
Join
the
midpoints
in
each
to
form
a
total
of
_?_
smaller
triangles
31
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S
i
e
r
p
i
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s
k
i
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r
i
a
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g
l
e
STAGE
3
Remove
(shade)
the
center
triangles
What
fraction
of
the
original
area
is
not
shaded?
Update
the
chart
32
Partners
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T
h
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i
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k
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r
i
a
n
g
l
e
Is
an
example
of
a
fractal
(a
self-similar
object)
In
general,
a
fractal
is
a
geometric
object
whose
parts
are
reduced
-
sized
copies
of
the
whole
Give
some
real-life
examples
of
fractals
33
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l
e
Determine
a
mathematical
relationship
for
The
number
of
triangles
at
each
stage
The
area
of
each
new
triangle
The
fraction
of
the
original
area
that
each
new
triangle
represents
The
fraction
of
the
original
area
that
is
not
shaded
34
T
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l
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What
patterns
emerge?
What
if
the
iterations
continued…
What
would
be
the
area
of
one
of
the
smallest
triangles
in
the
4th
iteration?
What
fraction
of
the
original
triangle
would
not
be
shaded
at
this
stage?
What
about
the
100th
iteration?
What
is
happening
to
the
area
of
the
un-
shaded
region
as
the
number
of
iterations
grows?
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35
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l
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What
will
your
students
think
of
this
activity?
What
mathematical
concepts
are
covered
in
this
activity?
Will
you
give
this
activity
a
try?
36
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T
h
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K
o
c
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S
n
o
w
f
l
a
k
e
37
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S
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o
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f
l
a
k
e
STAGE
0
Using
dot
paper,
construct
an
equilateral
triangle
of
side
length
9
What
is
the
perimeter
of
this
triangle?
38
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T
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K
o
c
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S
n
o
w
f
l
a
k
e
STAGE
1
Trisect
each
side
of
the
triangle
Remove
(erase)
the
middle
segment
of
each
side
39
T
h
e
K
o
c
h
S
n
o
w
f
l
a
k
e
STAGE
1
Create
a
new
equilateral
triangle
on
each
side
of
the
original
triangle
by
adding
two
segments
of
the
same
length
as
the
erased
segment
onto
the
side
The
erased
segment
will
be
the
base
of
the
new
equilateral
triangle
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n
g
40
Partners
f
o
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M
a
t
h
e
m
a
t
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c
s
L
e
a
r
n
i
n
g
T
h
e
K
o
c
h
S
n
o
w
f
l
a
k
e
The
new
shape
will
be
a
six-pointed
star
What
is
the
perimeter
of
the
star?
41
Partners
f
o
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L
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a
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n
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n
g
T
h
e
K
o
c
h
S
n
o
w
f
l
a
k
e
STAGE
2
Reiterate
the
process
described
in
Stage
1
First
trisect
of
each
side
of
the
triangle
Remove
(erase)
the
middle
segment
of
each
side
42
Partners
f
o
r
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t
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m
a
t
i
c
s
L
e
a
r
n
i
n
g
T
h
e
K
o
c
h
S
n
o
w
f
l
a
k
e
STAGE
2
Create
a
new
equilateral
triangle
on
each
side
of
the
6-pointed
star
by
adding
two
segments
of
the
same
length
as
the
erased
segment
onto
the
side
The
erased
segment
will
be
the
base
of
the
new
equilateral
triangle
43
Partners
f
o
r
M
a
t
h
e
m
a
t
i
c
s
L
e
a
r
n
i
n
g
T
h
e
K
o
c
h
S
n
o
w
f
l
a
k
e
STAGE
2
The
new
figure
should
look
like
a
snowflake
What
is
the
perimeter
of
the
new
figure?
44
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
T
h
e
K
o
c
h
S
n
o
w
f
l
a
k
e
Look
at
the
value
of
the
perimeter
at
each
stage
Is
there
a
pattern
here?
The
perimeter
of
each
figure
is
__?__
times
the
perimeter
of
the
previous
figure
45
Partners
f
o
r
M
a
t
h
e
m
a
t
i
c
s
L
e
a
r
n
i
n
g
T
h
e
K
o
c
h
S
n
o
w
f
l
a
k
e
How
many
iterations
would
it
take
to
obtain
a
perimeter
of
100
units?
(or
as
close
to
100
as
you
can
get)
As
you
perform
more
and
more
iterations,
what
happens
to
the
value
of
the
perimeter
and
the
area?
Partners
f
o
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L
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a
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n
i
n
g
46
T
h
e
K
o
c
h
S
n
o
w
f
l
a
k
e
An
infinite
perimeter
encloses
a
finite
area
…
now
that’s
amazing!
What
will
your
students
think
of
this
activity?
Will
you
give
this
activity
a
try?
47
Partners
f
o
r
M
a
t
h
e
m
a
t
i
c
s
L
e
a
r
n
i
n
g
S
u
m
m
a
r
i
z
i
n
g
t
h
e
W
o
r
k
What
mathematical
concepts
and
skills
are
addressed
in
these
activities?
Understanding
of
and
computation
with
real
numbers
Understanding
of
and
use
of
measurement
concepts
Understand
of
and
use
properties
and
relationships
in
geometry
48
Partners
f
o
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a
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c
s
L
e
a
r
n
i
n
g
P
r
o
p
o
r
t
i
o
n
a
l
R
e
a
s
o
n
i
n
g
A
c
t
i
v
i
t
i
e
s
One
Inch
Tall
Tangrams
Sierpinski
Triangle
Koch
Snowflake
What
are
your
favorite
proportional
reasoning
activities?
49
W
h
a
t
B
i
g
I
d
e
a
s
A
r
e
A
d
d
r
e
s
s
e
d
?
Fluency
with
different
types
of
reasoning
(quantitative,
additive,
multiplicative,
proportional)
is
necessary
for
mathematical
development
Fluency
(accuracy,
efficiency,
flexibility)
using
operations
with
rational
numbers
becomes
solidified
in
the
middle
grades
Partners
f
o
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M
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m
a
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c
s
L
e
a
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n
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50
Partners
f
o
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M
a
t
h
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m
a
t
i
c
s
L
e
a
r
n
i
n
g
…
B
I
G
I
d
e
a
s
Two
dimensional
figures
are
viewed
in
the
rectangular
coordinate
plane
and
transformations
of
two
dimensional
figures
within
the
plane
may
produce
figures
that
are
similar
and/or
congruent
to
the
original
figure
51
E
s
s
e
n
t
i
a
l
S
t
a
n
d
a
r
d
s
How
do
these
activities
address
NC’s
Essential
Standards?
In
what
ways
do
these
activities
offer
opportunities
for
students
with
strong
visualization
skills
to
be
successful
in
mathematics
classes?
What
final
comments
do
you
have
about
proportional
reasoning?
Partners
f
o
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M
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t
h
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m
a
t
i
c
s
L
e
a
r
n
i
n
g
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.
Yolanda
Sawyer
Penny
Shockley
Pat
Sickles
Nancy
Teague
Michelle
Tucker
Kaneka
Turner
Bob
Vorbroker
Jan
Wessell
Daniel
Wicks
Carol
Williams
Stacy
Wozny
Tery
Gunter
Barbara
Hardy
Kathy
Harris
Julie
Kolb
Renee
Matney
Tina
McSwain
Marilyn
Michue
Amanda
Northrup
Kayonna
Pitchford
Ron
Powell
Susan
Riddle
Judith
Rucker
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
Cara
Gordon
Shana
Runge
Julia
Cazin
Ruafika
Cobb
Anna
Corbett
Gail
Cotton
Jeanette
Cox
Leanne
Daughtry
Lisa
Davis
Ryan
Dougherty
Shakila
Faqih
Patricia
Essick
Donna
Godley
2
0
0
9
W
r
i
t
e
r
s
Kathy
Harris
Rendy
King
Tery
Gunter
Judy
Rucker
Penny
Shockley
Nancy
Teague
Jan
Wessell
Stacy
Wozny
Amanda
Baucom
Julie
Kolb
P
a
r
t
n
e
r
s
S
t
a
f
f
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
f
o
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M
a
t
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e
m
a
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c
s
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e
a
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n
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PARTNERS
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o
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m
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L
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a
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n
g
Grade
8
Module
4
Dive into the world of proportional reasoning in Grade 8 mathematics with a focus on identifying better buys, strategies for decision-making, and the importance of developing proportional thinking skills. Discover the significance of proportional reasoning as a key element in the curriculum, its implications on algebraic concepts, and the need for effective instructional practices to foster students' ability to reason proportionally.
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PARTNERS PARTNERS forMathematicsLearning Grade8 Module4 Partners Partners forMathematicsLearning
2 Module4 ProportionalReasoning Partners Partners forMathematicsLearning
3 WhichIsaBetterBuy? 12ticketsfor$15.00or20ticketsfor $23.00? Partners Partners forMathematicsLearning
4 WhichisaBetterBuy? Whatisyouranswer? Howdidyouobtainyouranswer? Whataresomestrategiesthatyour studentsmightuse? Partners Partners forMathematicsLearning
5 WhichIsaBetterBuy? Ifastudentvalueseachticketasworth $1.00,whatmightthestudentsayabout eachdealusing Additivereasoning Proportionalreasoning Partners Partners forMathematicsLearning
6 ProportionalThinking Asdifferentwaystothinkaboutproportions areconsideredanddiscussed,teachers shouldhelpstudentsrecognizewhenand howvariouswaysofreasoningabout proportionsmightbeappropriatetosolve problems PSSM,2000 Partners Partners forMathematicsLearning
7 CapstoneoftheCurriculum! Proportionalreasoninghasbeenreferred toasthecapstonefortheelementary curriculumandthecornerstoneofalgebra andbeyond. VandeWalle,J.A(2004).ElementaryandMiddleSchool Mathematics:TeachingDevelopmentally.PearsonLearningInc. Partners Partners forMathematicsLearning
8 Amazing Isn tIt? Itisestimatedthatmorethanhalfofthe adultpopulationcannotbeviewedas proportionalthinkers.Thatmeansthatwe donotacquirethehabitsandskillsof proportionalreasoningsimplybygetting older. VandeWalle,J.A(2004).ElementaryandMiddleSchool Mathematics:TeachingDevelopmentally.PearsonLearningInc. Partners Partners forMathematicsLearning
9 ResearchSays Researchindicatesthatinstructioncanhavean effect,especiallyifrulesandalgorithmsfor fractioncomputation,forcomparingratios,and forsolvingproportionsaredelayed.Premature useofrulesencouragesstudentstoapplyrules withoutthinkingand,thus,theabilitytoreason proportionallyoftendoesnotdevelop. VandeWalle,J.A(2004).ElementaryandMiddleSchool Mathematics:TeachingDevelopmentally.PearsonLearningInc. Partners Partners forMathematicsLearning
10 TeachingEffectively Instructioninsolvingproportionsshouldinclude methodsthathaveastrongintuitivebasis PSSM,2000 Inagroupofstudentswhocansuccessfully applyanalgorithm,howcanyoudistinguish betweenthosewhocanreasonproportionally andthosewhocannot? Partners Partners forMathematicsLearning
11 OneInchTall byShelSilverstein Ifyouwereonlyoneinchtall,you dridea wormtoschool Theteardropofacryingant wouldbeyourswimmingpool. Partners Partners forMathematicsLearning
12 WouldThisBeTrue? Ifyouwereonlyoneinchtall,youcouldwear athimbleonyourhead Partners Partners forMathematicsLearning
13 HowAboutThis? Ifyouwereonlyoneinchtall,itwouldtake aboutamonthtogetdowntothestore Partners Partners forMathematicsLearning
14 Let sInvestigate! Ifyouwereonlyoneinchtall Couldyourideawormtoschool? Couldyouwearathimbleonyourhead? Wouldittakeamonthtogettothestore? Partners Partners forMathematicsLearning
15 JustTheFacts,Ma am! Whatinformationwillyouneedtosolve theseproblems? Partners Partners forMathematicsLearning
16 TimeToInvestigate Partners Partners forMathematicsLearning
17 Let sTalk Whatareourconclusions? Couldyourideawormtoschool? Couldyouwearathimbleonyourhead? Wouldittakeamonthtogettothestore? Partners Partners forMathematicsLearning
18 IncreasedStudentUnderstanding Problemsthatinvolveconstructingor interpretingscaledrawingsofferstudents opportunitiestouseandincreasetheir knowledgeofsimilarity,ratio,and proportionality PSSM Partners Partners forMathematicsLearning
19 TangramTime! Partners Partners forMathematicsLearning
20 TangramTime! Partners Partners forMathematicsLearning
21 InvestigationandExploration Studentswholearnedthrough investigationandexplorationwerenot onlymoresuccessfulatgivingcorrect responsestoproportionalreasoning tasksbutalsobetterabletojustify thoseanswers. Fey,J.T.,Miller,J.L.(2000).ProportionalReasoning.Mathematics TeachingintheMiddleSchool.5(5),312 Partners Partners forMathematicsLearning
22 ContinuingOurInvestigations Partners Partners forMathematicsLearning
23 TheSierpinskiTriangleActivity Partners Partners forMathematicsLearning
24 TheSierpinskiTriangle STAGE0 Usingdotpaper,constructanequilateral triangleofsidelength16 Whatistheareaofthistriangle? Partners Partners forMathematicsLearning
25 TheSierpinskiTriangle STAGE1 Markthemidpointofeachsideofthe triangle Jointhemidpointsto form4smallertriangles Partners Partners forMathematicsLearning
26 TheSierpinskiTriangle STAGE1 Determinesomerelationshipsbetweenthenew trianglesandtheoriginaltriangle Similarity? Congruence? Whatistheareaofeachnewtriangle? Whatfractionoftheoriginaltriangledoeseach newtrianglerepresent? Partners Partners forMathematicsLearning
27 TheSierpinskiTriangle STAGE1 Remove(shade)thecentertriangle Whatfractionoftheoriginal areaisnotshaded? Updatethechart Partners Partners forMathematicsLearning
28 TheSierpinskiTriangle STAGE2 Bisecteachsideofthe new (unshaded) triangles Jointhemidpointsineachtoformatotal of12smallertriangles(16ifyoudivided thelargershadedtriangle) Partners Partners forMathematicsLearning
29 TheSierpinskiTriangle STAGE2 Determinesomerelationships betweenthenewtriangles andtheoriginaltriangle Remove(shade)thecentertriangles Whatistheareaofthenewtriangle? Whatfractionoftheoriginalareaisnot shaded? Partners Partners forMathematicsLearning
30 TheSierpinskiTriangle STAGE3 Bisecteachsideofthe new (unshaded) triangles Jointhemidpointsineachtoformatotal of_?_smallertriangles Partners Partners forMathematicsLearning
31 TheSierpinskiTriangle STAGE3 Remove(shade)thecentertriangles Whatfractionoftheoriginalareaisnot shaded? Updatethechart Partners Partners forMathematicsLearning
32 TheSierpinskiTriangle Isanexampleofafractal(aself-similar object) Ingeneral,afractalisageometricobject whosepartsarereduced-sizedcopiesof thewhole Givesomereal-lifeexamplesoffractals Partners Partners forMathematicsLearning
33 TheSierpinskiTriangle Determineamathematicalrelationshipfor Thenumberoftrianglesateachstage Theareaofeachnewtriangle Thefractionoftheoriginalareathateachnew trianglerepresents Thefractionoftheoriginalareathatisnot shaded Partners Partners forMathematicsLearning
34 TheSierpinskiTriangle Whatpatternsemerge? Whatiftheiterationscontinued Whatwouldbetheareaofoneofthesmallest trianglesinthe4thiteration? Whatfractionoftheoriginaltrianglewouldnot beshadedatthisstage? Whataboutthe100thiteration? Whatishappeningtotheareaoftheun- shadedregionasthenumberofiterations grows? Partners Partners forMathematicsLearning
35 TheSierpinskiTriangle Whatwillyourstudentsthinkofthis activity? Whatmathematicalconceptsarecovered inthisactivity? Willyougivethisactivityatry? Partners Partners forMathematicsLearning
36 TheKochSnowflake Partners Partners forMathematicsLearning
37 TheKochSnowflake STAGE0 Usingdotpaper,constructanequilateral triangleofsidelength9 Whatistheperimeterofthistriangle? Partners Partners forMathematicsLearning
38 TheKochSnowflake STAGE1 Trisecteachsideofthetriangle Remove(erase)themiddlesegmentof eachside Partners Partners forMathematicsLearning
39 TheKochSnowflake STAGE1 Createanewequilateraltriangleoneach sideoftheoriginaltrianglebyaddingtwo segmentsofthesamelengthasthe erasedsegmentontotheside Theerasedsegmentwillbethebaseof thenewequilateraltriangle Partners Partners forMathematicsLearning
40 TheKochSnowflake Thenewshapewillbea six-pointedstar Whatistheperimeter ofthestar? Partners Partners forMathematicsLearning
41 TheKochSnowflake STAGE2 ReiteratetheprocessdescribedinStage1 Firsttrisectofeachsideofthetriangle Remove(erase)themiddlesegmentof eachside Partners Partners forMathematicsLearning
42 TheKochSnowflake STAGE2 Createanewequilateraltriangleoneach sideofthe6-pointedstarbyaddingtwo segmentsofthesamelengthasthe erasedsegmentontotheside Theerasedsegmentwillbethebaseof thenewequilateraltriangle Partners Partners forMathematicsLearning
43 TheKochSnowflake STAGE2 Thenewfigureshouldlooklikea snowflake Whatistheperimeterofthenewfigure? Partners Partners forMathematicsLearning
44 TheKochSnowflake Lookatthevalueoftheperimeter ateachstage Isthereapatternhere? Theperimeterofeachfigureis__?__ timestheperimeterofthepreviousfigure Partners Partners forMathematicsLearning
45 TheKochSnowflake Howmanyiterationswouldittaketoobtain aperimeterof100units?(orascloseto 100asyoucanget) Asyouperformmoreandmoreiterations, whathappenstothevalueoftheperimeter andthearea? Partners Partners forMathematicsLearning
46 TheKochSnowflake Aninfiniteperimeterenclosesafinitearea nowthat samazing! Whatwillyourstudentsthinkofthisactivity? Willyougivethisactivityatry? Partners Partners forMathematicsLearning
47 SummarizingtheWork Whatmathematicalconceptsandskillsare addressedintheseactivities? Understandingofandcomputationwithreal numbers Understandingofanduseofmeasurement concepts Understandofandusepropertiesand relationshipsingeometry Partners Partners forMathematicsLearning
48 ProportionalReasoningActivities OneInchTall Tangrams SierpinskiTriangle KochSnowflake Whatareyourfavoriteproportional reasoningactivities? Partners Partners forMathematicsLearning
49 WhatBigIdeasAreAddressed? Fluencywithdifferenttypesofreasoning (quantitative,additive,multiplicative, proportional)isnecessaryformathematical development Fluency(accuracy,efficiency,flexibility) usingoperationswithrationalnumbers becomessolidifiedinthemiddlegrades Partners Partners forMathematicsLearning
50 BIGIdeas Twodimensionalfiguresareviewedinthe rectangularcoordinateplaneand transformationsoftwodimensionalfigures withintheplanemayproducefiguresthat aresimilarand/orcongruenttotheoriginal figure Partners Partners forMathematicsLearning
