Geometric Concepts in Grade Five Mathematics
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1
PARTNERS
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Grade
Five
Module
4
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B
i
g
I
d
e
a
s
i
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G
e
o
m
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t
r
y
Shapes
or
groups
of
shapes
can
be
classified
by
their
properties
Two-dimensional
shapes
are
combined
to
make
three-dimensional
shapes
Area,
perimeter,
and
volumes
are
examples
of
measurable
attributes
in
geometry
2
B
i
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I
d
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a
s
i
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G
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Shapes
can
be
described
in
terms
of
their
location
and
viewed
from
different
perspectives;
geometric
figures
can
be
moved
in
a
plane
without
changing
their
size
or
shape
Coordinate
systems
can
be
used
to
describe
locations
precisely
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3
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4
Exploring
Attributes
of
Shapes
Developing
Vocabulary
Classifying
Shapes
S
h
a
p
e
s
&
P
r
o
p
e
r
t
i
e
s
Refining
Concepts
Partners
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Level
0
Level
1
Level
2
Level
3
Visualization
(Recognition)
Description/Analysis
Abstract/Relational
Formal
Deduction
(high
school
geometry)
Level
4
Rigor
(college
level
geometry)
5
V
a
n
H
i
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L
e
v
e
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s
o
f
G
e
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m
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r
i
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T
h
o
u
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h
t
Partners
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S
h
a
p
e
s
a
n
d
P
r
o
p
e
r
t
i
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s
Look
at
this
shape
What
are
some
of
its
properties?
6
S
h
a
p
e
s
a
n
d
P
r
o
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e
r
t
i
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s
Find
these
properties
in
your
environment:
Parallel
lines
Right
angles
Shapes
with
“dents”
(concave)
Solids
like
a
cylinder
Solids
like
a
pyramid
Shapes
with
rotational
symmetry
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8
W
h
a
t
i
s
a
P
o
l
y
g
o
n
?
All
of
these
are
Polygons
None
of
these
are
Polygons
W
h
a
t
i
s
a
P
o
l
y
g
o
n
?
Which
of
these
is
a
Polygon?
What
attributes
does
a
polygon
have
that
makes
it
a
polygon?
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C
o
n
s
t
r
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i
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P
o
l
y
g
o
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s
Create
a
polygon
to
fit
each
description
4
sides
and
4
right
angles
3
sides
and
1
right
angle
5
sides
12
sides
4
sides
with
exactly
2
sides
parallel
4
sides
with
2
pairs
of
sides
perpendicular
3
sides
with
2
sides
perpendicular
10
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R
e
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I
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u
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a
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P
o
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y
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o
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s
These
are
regular
polygons
These
are
irregular
polygons
11
T
r
i
a
n
g
l
e
s
Sort
the
triangles
into
three
groups
so
that
no
two
triangles
belong
in
more
than
one
group
Write
a
description
of
each
group
Now
sort
the
triangles
again
into
three
different
groups
so
that
no
triangle
belongs
in
two
groups
Write
a
description
of
each
of
these
groups
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T
r
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Fill
in
the
chart
below
with
a
sketch
of
a
triangle
that
fits
both
labels
Are
any
impossible
ones?
13
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Q
u
a
d
r
i
l
a
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a
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s
Find
all
the
quadrilaterals
Sort
them
into
groups
Are
there
overlaps?
Draw
a
Venn
diagram
to
sort
the
shapes
Can
you
find
a
different
way
to
sort
them?
14
\
D
i
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Q
u
a
d
r
i
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a
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s
Draw
the
diagonals
in
each
quadrilateral
What
properties
can
you
identify
in
the
diagonals?
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D
i
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o
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Q
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16
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D
i
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o
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a
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Q
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i
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a
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a
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s
17
Partners
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D
i
a
g
o
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a
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s
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Q
u
a
d
r
i
l
a
t
e
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a
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s
How
many
diagonals
in
these
quadrilaterals?
18
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D
i
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Q
u
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i
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Concave
quadrilaterals
do
have
2
diagonals.
What
is
different?
19
Q
u
a
d
r
i
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a
t
e
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a
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s
Look
at
this
set
of
parallelograms.
What
are
its
properties
of
sides?
angles?
symmetry?
diagonals?
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Partners
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Q
u
a
d
r
i
l
a
t
e
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a
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s
Classify
the
quadrilaterals
by
labeling
the
parts
of
this
Venn
diagram:
21
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Q
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a
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22
T
r
u
e
o
r
F
a
l
s
e
?
If
it
is
a
square,
it
is
also
a
rhombus
Some
parallelograms
are
rectangles
All
rectangles
are
squares
If
it
has
exactly
two
lines
of
symmetry,
then
it
must
be
a
quadrilateral
No
triangles
have
diagonals
All
triangles
have
3
congruent
sides
All
trapezoids
have
exactly
2
parallel
sides
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T
r
u
e
o
r
F
a
l
s
e
?
If
it
is
a
square,
it
is
also
a
rhombus.
T
Some
parallelograms
are
rectangles.
T
All
rectangles
are
squares.
F
If
it
has
exactly
two
lines
of
symmetry,
then
it
must
be
a
quadrilateral.
F
All
triangles
have
no
diagonals.
T
All
triangles
have
3
congruent
sides.
F
All
trapezoids
have
exactly
2
parallel
sides.
T
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24
A
l
g
e
b
r
a
C
o
n
n
e
c
t
i
o
n
:
D
i
a
g
o
n
a
l
s
How
many
diagonals
in
a
triangle?
a
quadrilateral?
A
pentagon?
Any
polygon?
H
o
w
m
a
n
y
d
i
a
g
o
n
a
l
s
i
n
a
p
o
l
y
g
o
n
w
i
t
h
n
sides?
What’s
the
rule?
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25
A
n
g
l
e
s
D
E
B
A
F
What
are
angles?
What
makes
conceptualizing
the
size
of
an
angle
challenging
for
students?
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A
n
g
l
e
s
“The
protractor
is
one
of
the
most
poorly
understood
measuring
instruments
in
school.”
John
Van
de
Walle
Why
is
measuring
the
size
of
an
angle
so
difficult
for
students?
27
Partners
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C
h
a
l
l
e
n
g
e
s
o
f
a
P
r
o
t
r
a
c
t
o
r
Units
(degrees)
are
very
small
No
angles
are
shown
on
protractor;
only
little
marks
around
the
edge
Numbers
go
both
clockwise
and
counterclockwise
on
a
typical
protractor
28
Partners
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A
n
g
l
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S
i
z
e
Students
need
to
practice
telling
the
difference
between
a
small
and
a
large
angle
prior
to
measuring
angles
What
activities
might
provide
this?
29
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R
e
a
d
i
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g
a
P
r
o
t
r
a
c
t
o
r
How
do
you
help
students
understand
how
to
use
a
typical
protractor?
What
experiences
need
to
come
before
students
try
to
use
a
protractor?
30
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A
n
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U
n
i
t
s
Use
a
straightedge
to
draw
a
narrow
angle
on
your
card
Cut
it
out
Use
the
wedge
as
a
unit
to
measure
angles,
counting
the
number
of
wedges
that
fit
into
a
particular
angle
31
M
a
k
i
n
g
a
P
r
o
t
r
a
c
t
o
r
Fold
the
piece
of
waxed
paper
in
half,
creasing
the
fold
tightly
Fold
in
half
again
so
that
the
folded
edges
match
Fold
along
the
diagonal
from
the
folded
corner
Fold
again
from
the
folded
corner
to
bring
together
the
two
sides
that
form
the
folded
corner
Cut
or
tear
off
the
edge
about
4-5
inches
from
the
vertex
and
unfold
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32
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M
a
k
i
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g
a
T
o
o
l
f
o
r
M
e
a
s
u
r
i
n
g
A
n
g
l
e
s
Compare
your
waxed
paper
tool
to
a
traditional
protractor
33
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M
e
a
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A
n
g
l
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s
Make
your
answer
card
Decide
on
the
angle
you
want
to
use
Draw
an
angle
on
the
index
card,
measuring
very
carefully
Plan
good
“incorrect”
answers
Place
the
answer
choices
in
appropriate
locations
34
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35
M
e
a
s
u
r
e
m
e
n
t
a
n
d
G
e
o
m
e
t
r
y
How
does
geometry
overlap
with
measuring
angles?
M
e
a
s
u
r
e
m
e
n
t
“Perhaps
the
biggest
error
in
measurement
instruction
is
the
failure
to
recognize
and
separate
two
types
of
objectives:
first,
understanding
the
meaning
and
technique
of
measuring
a
particular
attribute
unit
and,
second,
learning
about
the
standard
units
commonly
used
to
measure
that
attribute.”
John
Van
de
Walle,
Teaching
Student-Centered
Mathematics
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T
r
i
a
n
g
l
e
s
Use
a
straight
edge
to
make
a
large
triangle
Place
a
visible
dot
on
each
vertex
Rip
off
each
of
the
angles
Carefully
join
the
angles
at
the
dots
and
tape
or
glue
them
down
What
do
you
notice?
37
A
n
g
l
e
S
u
m
s
Measure
each
angle
of
each
polygon
Record
your
findings
in
the
appropriate
chart
Find
the
sum
of
the
angles
for
each
figure
Compare
your
results
to
those
of
your
classmates
What
conclusion(s)
about
the
sum
of
the
angles
do
you
draw
from
the
results?
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38
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A
n
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S
u
m
s
The
sum
of
the
angle
measures
of
a
triangle
is
180
°
39
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A
n
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S
u
m
s
The
sum
of
the
angle
measures
of
any
quadrilateral
is
360
°
Any
quadrilateral
can
be
divided
into
two
triangles,
each
with
an
angle
sum
or
180
°
40
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n
i
n
g
A
n
g
l
e
S
u
m
s
Divide
a
pentagon
into
triangles
Use
only
diagonal
lines
and
do
not
cross
any
lines
How
many
triangles
did
you
make?
How
does
that
impact
the
sum
of
the
angles
of
the
pentagon?
Do
the
same
with
a
hexagon
What
is
the
pattern?
41
D
e
s
c
r
i
b
e
t
h
e
P
o
l
y
g
o
n
Create
a
“Wanted”
poster
for
one
of
the
“culprits”
below:
A
right
scalene
triangle
42
A
rhombus
A
trapezoid
An
obtuse
isosceles
triangle
A
regular
pentagon
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N
e
t
s
f
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r
3
-
D
S
h
a
p
e
s
A
pentomino
is
made
from
5
congruent
squares,
with
squares
touching
only
by
a
whole
side
(“edge
to
edge
construction”)
Make
as
many
different
pentomino
shapes
as
you
can.
How
many
can
you
make?
OK
Not
OK
43
Partners
f
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s
L
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r
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a
t
i
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g
N
e
t
s
f
o
r
3
-
D
S
h
a
p
e
s
Are
these
pentominoes
different
or
congruent?
44
Partners
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C
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a
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N
e
t
s
f
o
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3
-
D
S
h
a
p
e
s
Which
of
the
pentomino
pieces
can
be
folded
into
a
topless
box?
What
rectangles
can
be
created
from
the
pentomino
pieces
that
can
be
folded
into
a
topless
box?
45
Partners
f
o
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m
a
t
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c
s
L
e
a
r
n
i
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g
A
r
e
a
a
n
d
P
e
r
i
m
e
t
e
r
What
is
the
relationship
between
the
area
of
the
T-shaped
net
and
the
surface
area
of
the
topless
box
it
creates?
Are
the
perimeter
of
the
net
and
the
sum
of
the
lengths
of
the
edges
of
the
3-D
topless
box
also
equivalent?
46
V
o
l
u
m
e
:
R
e
c
t
a
n
g
u
l
a
r
P
r
i
s
m
s
Use
centimeter
cubes
to
create
rectangular
prisms
How
many
different
prisms
can
you
make
with
these
cubes:
24
32
36
Record
the
dimensions
of
each
prism
that
you
make
and
the
total
number
of
cubes
used
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47
C
o
n
n
e
c
t
i
o
n
s
Which
of
the
process
standards
did
we
use?
Problem
Solving
Reasoning
and
Proof
Communication
Connections
Representation
In
Grade
5
what
are
strong
connections
between
geometry
and
measurement?
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48
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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.
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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
2
0
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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
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S
t
a
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f
Freda
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Webmaster
Anita
Bowman,
Outside
Evaluator
Ana
Floyd,
Reviewer
Meghan
Griffith,
Administrative
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Ed
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Reviewer
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Stein,
Higher
Education
Please
give
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Mathematics
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project
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Jeane
Joyner
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PARTNERS
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Grade
Five
Module
4
Geometry
Dive into the world of geometry with Grade Five Mathematics! This module covers topics such as shapes classification, properties of two-dimensional and three-dimensional shapes, coordinate systems, polygon construction, and Van Hiele levels of geometric thought. Engage with hands-on activities, visualizations, and vocabulary development to enhance your understanding of geometry concepts.
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Presentation Transcript
1 PARTNERS forMathematicsLearning GradeFive Module4 Partners forMathematicsLearning
2 BigIdeasinGeometry Shapesorgroupsofshapescanbe classifiedbytheirproperties Two-dimensionalshapesarecombined tomakethree-dimensionalshapes Area,perimeter,andvolumesare examplesofmeasurableattributesin geometry Partners forMathematicsLearning
3 BigIdeasinGeometry Shapescanbedescribedintermsoftheir locationandviewedfromdifferent perspectives;geometricfigurescanbe movedinaplanewithoutchangingtheir sizeorshape Coordinatesystemscanbeusedto describelocationsprecisely Partners forMathematicsLearning
4 Shapes&Properties Exploring Attributes ofShapes Refining Concepts Classifying Shapes Developing Vocabulary Partners forMathematicsLearning
5 VanHieleLevelsofGeometricThought Visualization(Recognition) Description/Analysis Abstract/Relational FormalDeduction(highschool geometry) Level4Rigor(collegelevelgeometry) Level0 Level1 Level2 Level3 Partners forMathematicsLearning
6 ShapesandProperties Lookatthisshape Whataresomeofitsproperties? Partners forMathematicsLearning
7 ShapesandProperties Findthesepropertiesinyourenvironment: Parallellines Rightangles Shapeswith dents (concave) Solidslikeacylinder Solidslikeapyramid Shapeswithrotationalsymmetry Partners forMathematicsLearning
8 WhatisaPolygon? AllofthesearePolygons NoneofthesearePolygons Partners forMathematicsLearning
9 WhatisaPolygon? WhichoftheseisaPolygon? Whatattributesdoesapolygonhavethat makesitapolygon? Partners forMathematicsLearning
10 ConstructingPolygons Createapolygontofiteachdescription 4sidesand4rightangles 3sidesand1rightangle 5sides 12sides 4sideswithexactly2sidesparallel 4sideswith2pairsofsidesperpendicular 3sideswith2sidesperpendicular
11 RegularandIrregularPolygons Theseareregularpolygons Theseareirregularpolygons Partners forMathematicsLearning
12 Triangles Sortthetrianglesintothreegroupssothat notwotrianglesbelonginmorethanone group Writeadescriptionofeachgroup Nowsortthetrianglesagainintothree differentgroupssothatnotrianglebelongs intwogroups Writeadescriptionofeachofthesegroups Partners forMathematicsLearning
13 Triangles Fillinthechartbelowwithasketchofa trianglethatfitsbothlabels Areanyimpossibleones? Partners forMathematicsLearning
14 Quadrilaterals Findallthequadrilaterals Sortthemintogroups Arethereoverlaps? DrawaVenndiagramtosorttheshapes Canyoufindadifferentwaytosortthem? Partners forMathematicsLearning
15 DiagonalsinQuadrilaterals Drawthediagonalsineachquadrilateral \ Whatpropertiescanyouidentifyinthe diagonals? Partners forMathematicsLearning
16 DiagonalsinQuadrilaterals Parallelogram Rectangle Rhombus Square Trapezoid Kite Properties of Diagonals form congruent triangles bisecteach other are congruent Are perpendicular Bisect opposite angles Partners forMathematicsLearning
17 DiagonalsinQuadrilaterals Parallelogram Rectangle Rhombus Square Trapezoid Kite Properties of Diagonals form congruent triangles bisecteach other are congruent Are perpendicular Bisect opposite angles 1pair yes yes 2pairs (adjacen t) 2opposite pairs 2opposite pairs yes yes yes yes yes yes yes yes yes yes yes yes yes Partners forMathematicsLearning
18 DiagonalsinQuadrilaterals Howmanydiagonalsinthese quadrilaterals? Partners forMathematicsLearning
19 DiagonalsinQuadrilaterals Concavequadrilateralsdohave2 diagonals. Whatisdifferent? Partners forMathematicsLearning
20 Quadrilaterals Lookatthissetofparallelograms. Whatareitspropertiesofsides?angles? symmetry?diagonals? Partners forMathematicsLearning
21 Quadrilaterals Classifythequadrilateralsbylabelingthe partsofthisVenndiagram: Partners forMathematicsLearning
22 Quadrilaterals Partners forMathematicsLearning
23 TrueorFalse? Ifitisasquare,itisalsoarhombus Someparallelogramsarerectangles Allrectanglesaresquares Ifithasexactlytwolinesofsymmetry,then itmustbeaquadrilateral Notriangleshavediagonals Alltriangleshave3congruentsides Alltrapezoidshaveexactly2parallelsides Partners forMathematicsLearning
24 TrueorFalse? Ifitisasquare,itisalsoarhombus.T Someparallelogramsarerectangles.T Allrectanglesaresquares.F Ifithasexactlytwolinesofsymmetry,thenit mustbeaquadrilateral.F Alltriangleshavenodiagonals.T Alltriangleshave3congruentsides.F Alltrapezoidshaveexactly2parallelsides.T Partners forMathematicsLearning
25 AlgebraConnection:Diagonals Howmanydiagonalsinatriangle? aquadrilateral? Apentagon? Anypolygon? Howmanydiagonalsinapolygonwithn sides? What stherule? Partners forMathematicsLearning
26 Angles D B E A F Whatareangles? Whatmakesconceptualizingthesizeofan anglechallengingforstudents? Partners forMathematicsLearning
27 Angles Theprotractorisoneofthemostpoorly understoodmeasuringinstrumentsin school. JohnVandeWalle Whyismeasuringthesizeofanangleso difficultforstudents? Partners forMathematicsLearning
28 ChallengesofaProtractor Units(degrees)areverysmall Noanglesareshownonprotractor;only littlemarksaroundtheedge Numbersgobothclockwiseand counterclockwiseonatypicalprotractor Partners forMathematicsLearning
29 AngleSize Studentsneedtopracticetellingthe differencebetweenasmallandalarge anglepriortomeasuringangles Whatactivitiesmightprovidethis? Partners forMathematicsLearning
30 ReadingaProtractor Howdoyouhelpstudentsunderstandhow touseatypicalprotractor? Whatexperiencesneedtocomebefore studentstrytouseaprotractor? Partners forMathematicsLearning
31 AngleUnits Useastraightedgetodrawanarrowangle onyourcard Cutitout Usethewedgeasaunittomeasure angles,countingthenumberofwedges thatfitintoaparticularangle Partners forMathematicsLearning
32 MakingaProtractor Foldthepieceofwaxedpaperinhalf,creasing thefoldtightly Foldinhalfagainsothatthefoldededges match Foldalongthediagonalfromthefoldedcorner Foldagainfromthefoldedcornertobring togetherthetwosidesthatformthefolded corner Cutortearofftheedgeabout4-5inchesfrom thevertexandunfold Partners forMathematicsLearning
33 MakingaToolforMeasuringAngles Compareyourwaxedpapertooltoa traditionalprotractor Partners forMathematicsLearning
34 MeasuringAngles Makeyouranswercard Decideontheangleyouwanttouse Drawanangleontheindexcard, measuringverycarefully Plangood incorrect answers Placetheanswerchoicesinappropriate locations Partners forMathematicsLearning
35 MeasurementandGeometry Howdoesgeometryoverlapwith measuringangles? Partners forMathematicsLearning
36 Measurement Perhapsthebiggesterrorinmeasurement instructionisthefailuretorecognizeand separatetwotypesofobjectives:first, understandingthemeaningandtechnique ofmeasuringaparticularattributeunitand, second,learningaboutthestandardunits commonlyusedtomeasurethatattribute. JohnVandeWalle,TeachingStudent-CenteredMathematics Partners forMathematicsLearning
37 Triangles Useastraightedgetomakealarge triangle Placeavisibledotoneachvertex Ripoffeachoftheangles Carefullyjointheanglesatthedotsand tapeorgluethemdown Whatdoyounotice? Partners forMathematicsLearning
38 AngleSums Measureeachangleofeachpolygon Recordyourfindingsintheappropriate chart Findthesumoftheanglesforeachfigure Compareyourresultstothoseofyour classmates Whatconclusion(s)aboutthesumofthe anglesdoyoudrawfromtheresults? Partners forMathematicsLearning
39 AngleSums Thesumoftheanglemeasuresofa triangleis180 Partners forMathematicsLearning
40 AngleSums Thesumoftheanglemeasuresofany quadrilateralis360 Anyquadrilateralcanbedividedintotwo triangles,eachwithananglesumor180 Partners forMathematicsLearning
41 AngleSums Divideapentagonintotriangles Useonlydiagonallinesanddonotcrossany lines Howmanytrianglesdidyoumake? Howdoesthatimpactthesumoftheanglesof thepentagon? Dothesamewithahexagon Whatisthepattern? Partners forMathematicsLearning
42 DescribethePolygon Createa Wanted posterforoneofthe culprits below: Arightscalenetriangle Arhombus Atrapezoid Anobtuseisoscelestriangle Aregularpentagon Partners forMathematicsLearning
43 CreatingNetsfor3-DShapes Apentominoismadefrom5congruent squares,withsquarestouchingonlybya wholeside( edgetoedgeconstruction ) Makeasmanydifferentpentominoshapes asyoucan.Howmanycanyoumake? OKNotOK Partners forMathematicsLearning
44 CreatingNetsfor3-DShapes Arethesepentominoesdifferentor congruent? Partners forMathematicsLearning
45 CreatingNetsfor3-DShapes Whichofthepentominopiecescanbe foldedintoatoplessbox? Whatrectanglescanbecreatedfromthe pentominopiecesthatcanbefoldedintoa toplessbox? Partners forMathematicsLearning
46 AreaandPerimeter Whatistherelationship betweentheareaofthe T-shapednetandthesurfaceareaofthe toplessboxitcreates? Aretheperimeterofthenetandthesumof thelengthsoftheedgesofthe3-Dtopless boxalsoequivalent? Partners forMathematicsLearning
47 Volume:RectangularPrisms Usecentimetercubestocreaterectangular prisms Howmanydifferentprismscanyoumake withthesecubes: 24 32 Recordthedimensionsofeachprismthat youmakeandthetotalnumberofcubes used 36 Partners forMathematicsLearning
48 Connections Whichoftheprocessstandardsdidwe use? ProblemSolving ReasoningandProof Communication Connections Representation InGrade5whatarestrongconnections betweengeometryandmeasurement? Partners forMathematicsLearning
DPIMathematicsStaff EverlyBroadway,ChiefConsultant ReneeCunninghamKittyRutherford RobinBarbourMaryH.Russell CarmellaFairJohannahMaynor AmySmith PartnersforMathematicsLearningisaMathematics-Science PartnershipProjectfundedbytheNCDepartmentofPublicInstruction. Permissionisgrantedfortheuseofthesematerialsinprofessional developmentinNorthCarolinaPartnersschooldistricts. Partners forMathematicsLearning
PMLDisseminationConsultants SusanAllman JuliaCazin RuafikaCobb AnnaCorbett GailCotton JeanetteCox LeanneDaughtry LisaDavis RyanDougherty ShakilaFaqih PatriciaEssick DonnaGodley ShanaRunge YolandaSawyer PennyShockley PatSickles NancyTeague MichelleTucker KanekaTurner BobVorbroker JanWessell DanielWicks CarolWilliams StacyWozny CaraGordon TeryGunter BarbaraHardy KathyHarris JulieKolb ReneeMatney TinaMcSwain MarilynMichue AmandaNorthrup KayonnaPitchford RonPowell SusanRiddle JudithRucker Partners forMathematicsLearning
