Mathematics – Grade Level Assessments and Content Expectations
Session 4 ~ Understanding Volume
Participant Packet
Module developed by Macomb County Teachers
under the leadership of Marie Copeland
Developers
M-GLAnCE Project Directors
Debbie Ferry
Macomb ISD
Mathematics Consultant
Carol Nowakowski
Retired
Mathematics Consultant
K-4 Project Coordinator
Marie Copeland
Warren Consolidated
Macomb MSTC
5-8 Project Coordinator
2004 Project Contributors
David Andrews
Chippewa Valley Schools
William Ashton
Fraser Public Schools
Lynn Bieszki
Chippewa Valley Schools
Sharon Chriss
Romeo Schools
Kimberly DeShon
Anchor Bay School District
Barbara Diliegghio
Retired, Math Consultant
Kimberly Dolan
Anchor Bay School District
Jodi Giraud
L’Anse Creuse Schools
Julie Hessell
Romeo Schools
Amy Holloway
Clintondale Schools
Barbara Lipinski
Anchor Bay School District
Linda Mayle
Romeo Schools
Therese Miekstyn
Chippewa Valley Schools
James Navetta
Chippewa Valley Schools
Gene Ogden
Anchor Bay School District
Rebecca Phillion
Richmond Comm. Schools
Charlene Pitrucelle
Anchor Bay School District
Shirley Starman
Van Dyke Public Schools
Ronald Studley
Anchor Bay School District
2005 and 2006 Session/Module Developers
Carol Nowakowski
Retired, Math Consultant
Deb Barnett
Luann Murray
Lake Shore Public Schools Genesee ISD
Kathy Albrecht
Retired, Math Consultant
Jo-Anne Schimmelpfenneg
Terri Faitel
Trenton Public Schools
Debbie Ferry
Macomb ISD
Retired, Math Consultant
Marie Copeland
Warren Consolidated
Grade 5: Understand the concept of volume.
M.TE.05.08
M.TE.05.09
M.PS.05.10
Build solids with
unit cubes and
state their
volumes.
Use filling (unit
cubes or liquid),
and counting or
measuring to find
the volume of a
cube and
rectangular
prism.
Grade 5: Know, and convert among, measurement units within a given system.
M.UN.05.01
M.UN.05.02
M.UN.05.03
M.UN.05.04
Solve applied
problems about
the volumes of
rectangular
prisms using
multiplication
and division and
using the
appropriate units.
Recognize the
equivalence of 1
liter, 1,000 ml
and 1,000 cm3
and include
conversions
among liters,
milliliters, and
cubic
centimeters.
Know the units of
measure of
volume: cubic
centimeter, cubic
meter, cubic
inches, cubic feet,
cubic yards, and
use their
abbreviation s
(cm3, m3, in3, ft3,
yd3).
Compare the
relative sizes of
one cubic inch to
one cubic foot,
and one cubic
centimeter to one
cubic meter.
Convert
measurements of
length, weight,
area, volume, and
time within a
given system
using easily
manipulated
numbers.
INSTRUCTIONAL SEQUENCE:
Make a cubic net of a cubic decimeter.
Discover the volume of a cubic decimeter by filling and counting.
Develop the relationship of a cubic centimeter
to a cubic decimeter to a cubic meter.
Build rectangular prisms of a given volume.
Identify the length, width, and height of rectangular prisms.
Identify the top, front, and side layers of rectangular prisms.
Find the volume of rectangular prisms by filling layers,
counting, and/or decomposing.
M-GLAnCE 5th Grade – Session 4 – Understanding Volume – Participant Packet
Page 3 Revised 10.15.07
Fifth Grade Supporting Volume Measurement GLCEs
Grade 5: Multiply and divide whole numbers.
N.FL.05.04
Multiply a multiply a multi-digit
number by a two digit number;
recognize and be able to explain
computational errors such as not
accounting for place value.
Grade 5: Understand the meaning of decimal fractions and
percentages.
N.FL.05.05
N.ME.05.08
Solve applied problems
Understand the relative magnitude of ones, tenths, and hundredths and
involving multiplication
the relationship of each place value to the place to its right, e.g., 1 is
and division of whole
10 tenths, one tenth is 10 hundredths.
numbers.
Grade 5: Multiply and divide by powers of ten.
N.MR.05.15
N.FL.05.16
Multiply a whole number by powers of 10:
Divide numbers by 10s, 100s, 1000s, using
mental strategies.
0.01, 0.1, 1,10,100,1,000; and identify
patterns.
M-GLAnCE 5th Grade – Session 4 – Understanding Volume – Participant Packet
Page 4 Revised 10.15.07
N.MR.05.17
Multiply one-digit whole numbers by
decimals up to two decimal places.
Grade 5: Understand the concept of volume.
M.TE.05.08
M.TE.05.09
M.PS.05.10
Build solids with
unit cubes and
state their
volumes.
Use filling (unit
cubes or liquid),
and counting or
measuring to find
the volume of a
cube and
rectangular
prism.
Solve applied
problems about
the volumes of
rectangular
prisms using
multiplication
and division and
using the
appropriate units.
Grade 5: Know, and convert among, measurement units within a given system.
M.UN.05.01
M.UN.05.02
M.UN.05.03
M.UN.05.04
Recognize the
equivalence of 1
liter, 1,000 ml
and 1,000 cm3
and include
conversions
among liters,
milliliters, and
cubic
centimeters.
Know the units of
measure of
volume: cubic
centimeter, cubic
meter, cubic
inches, cubic feet,
cubic yards, and
use their
abbreviation s
(cm3, m3, in3, ft3,
yd3).
Compare the
relative sizes of
one cubic inch to
one cubic foot,
and one cubic
centimeter to one
cubic meter.
Convert
measurements of
length, weight,
area, volume, and
time within a
given system
using easily
manipulated
numbers.
Grade 5: Multiply and divide whole numbers.
Grade 5: Understand the meaning of decimal fractions and percentages.
N.FL.05.04
N.FL.05.05
N.ME.05.08
Solve applied problems
Understand the relative magnitude of ones, tenths, and hundredths and
Multiply a multiply a multi-digit
involving multiplication
the relationship of each place value to the place to its right, e.g., 1 is
number by a two digit number;
and division of whole
10 tenths, one tenth is 10 hundredths.
recognize and be able to explain
computational errors such as not
numbers.
accounting for place value.
Grade 5: Multiply and divide by powers of ten.
N.MR.05.15
N.FL.05.16
Multiply a whole number by powers of 10:
Divide numbers by 10s, 100s, 1000s, using
0.01, 0.1, 1,10,100,1,000; and identify
mental strategies.
patterns.
M-GLAnCE 5th Grade – Session 4 – Understanding Volume – Participant Packet
Page 5 Revised 10.15.07
N.MR.05.17
Multiply one-digit whole numbers by
decimals up to two decimal places.
Fifth Grade Participation Packet Session 4
Focus on: Volume
Name of Activity
I.
II.
Review of Metric Square
Units
Introduction to Volume
Description of Activity
!
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th
Review square meter, square decimeter, and square
centimeter. Draw lines dividing poster board into 100
sq. decimeters. (Need 2 or 3 sheets.) Shade in and
label one square decimeter. Shade in and label one
square centimeter.
To begin volume start with a 10 cm x 10 cm x 10 cm
cube net. Have students make 2 cubes (without tops)
from tag board. Cut them out carefully. Before
putting them together, cut 10 square decimeters from
cm paper and either glue or tape them onto the sides
of the boxes.
Have students place one row of centimeter cubes
along an inside edge of one of their boxes so they can
visualize the space filled. Ask questions such as:
How many centimeter cubes are needed to fill in just
one row? How many rows do you think will be
needed to cover just the bottom layer? How many
centimeter cubes will be needed to fill this layer?
Since we don’t have enough cubic centimeters to fill
in the bottom layer of everyone’s cubes, which size
base-ten block could you use next?
Materials/Handouts
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Key Tips for Presenter
2-3 sheets of poster
board
Base-ten blocks
Square centimeter
!
Refer back to this 2-dimensional poster, as
necessary, for comparison when building the 3dimensional model.
Tagboard with
decimeter net or net
template that can be
traced on to tag
board.
Scissors
Tape
Glue
Base-ten thousand
cubes, flats, rods and
centimeter cubes
2 - Square centimeter
grid paper (at least 10
square decimeters per
student)
!
Students may have never worked with nets before
so it would be a good time to show how to
develop a net by using a thousand cube from the
base-ten blocks as a model.
Tape 6 square decimeters together to form a net
that will fold into a cubic decimeter around the
thousand cube.
Show at least three ways that the net can be
flattened out and how it is helpful to have tabs to
glue together when forming the cube.
Fill the rest of the bottom layer with rods. How many
cubic centimeters would this be? How many rods
would this be equivalent to? Which size base-ten
block would the bottom layer be equivalent to?
Discuss: How many flats would I need to fill the
whole cubic decimeter? How many rods? Etc.
Fill the rest with flats.
M-GLAnCE 5 Grade – Session 4 – Understanding Volume – Participant Packet
Page 6 Revised 10.15.07
!
!
!
You might want to borrow base-ten blocks from
other teachers in order for each student to fill their
own boxes.
!
Rather than make your own decimeter cubes, you
could purchase the following 4x4x4 boxes in
either white or brown for $29 per 100 from:
Action Bag Company
501 N. Edgewood Avenue
Wood Dale, IL 60191-1410
(800) 824-2247
www.actionbag.com
Name of Activity
Description of Activity
!
!
!
III. Make a Cubic Meter
Class Activity
!
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!
th
Pass out a thousand-cube block to each group for the
students to compare with their filled cubes.
Discuss that the model they created shows the filling
of a thousand cube, while the base-ten model shows
only the surface area.
Rename the thousand cube as a cubic decimeter since
it is 1 dm in length, 1 dm in width, and 1 dm in
height.
CLASS DISCUSSION: How many cubic decimeters
will fit along edge of a meter stick? How many
would it take to stack as tall as a meter?
In an open corner of the classroom, if possible, use
the cubic decimeters they made and stack 10 high.
Relate this to the meter stick.
Lay 9 cubes on the floor from the corner going along
the wall (perpendicular). Compare this length to the
meter stick. Ask: Why am I only adding 9 cubes?
Stack 9 cubes on top of the last cube. (Continue to
compare to the meter stick.)
Lay 9 more cubes on the floor coming out at a 90degree angle (perpendicular) from the wall. (Use
paper clips to hold cubes together.) Use as many
cubes as you can to show the edges of the new cubic
meter.
Materials/Handouts
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On the wall behind the cubic meter you just made,
hang the poster board you made earlier with the sq.
decimeters marked. Use this for the comparison of
the length and height of the cubic units.
!
!
Substitute a meter stick for the 10 cubic decimeters
stacked in the corner and against the walls.
!
!
When done showing length, width, and height with
the cubic decimeters, hang from the ceiling with
poster board at the top (marked in square decimeters
and with one square decimeter marked in square
centimeters) as the lid.
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!
Decimeter cubes
Meter sticks
Paper clips to hold
the cubic decimeters
together.
Poster board with 100
sq. dm marked and
one of the square
decimeters marked in
square centimeters
Meter sticks or meter
stick representations
made from sentence
strips.
String or fishing line
Magnifying glasses
Salt
Centimeter rulers
with millimeters
M-GLAnCE 5 Grade – Session 4 – Understanding Volume – Participant Packet
Page 7 Revised 10.15.07
Key Tips for Presenter
!
The goal is to make a very visual representation
of a cubic meter using the individual parts that it
is made of. If it is possible, it will be beneficial to
leave the cubic meter up for awhile in order for
students to refer to.
!
The relationship between the cubic inch, cubic
foot, and cubic yard could be shown in a similar
manner.
!
There are other topics and strands that can be
discussed and interwoven in this unit with rich
vocabulary: angles, perpendicular, parallel, etc.
!
Make labels with both words and abbreviations
Name of Activity
Description of Activity
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!
IV.
Building Rectangular
Prisms to Find Volume
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Materials/Handouts
Hang a cubic millimeter representation (here you
could use a piece of salt), a cubic cm, and a cubic dm
all labeled within the cubic meter, as a mobile.
Hang a second cubic dm with each side labeled with
its different equivalences: 1 liter, 1000 cubic cm,
1000 ml & 1000 g. (The students may find it
interesting to see that filling a one-liter bottle is
equivalent to one cubic decimeter.)
Give each pair of students at least 48 centimeter
cubes and allow them 5 minutes to discover two
different rectangular prisms using 24 of the cubes.
Have students describe their findings.
The teacher needs to model the 24 cm x 1 cm x 1 cm
prism (flat) on the overhead and then have each
student make the same model with their cubes. (Ask
the students to build their prism flat on the desk and
not as a tower.)
Explain that the prism has a length, a width, and a
height. Using different orientations of the prism, ask
what the different lengths/widths/ heights would be.
Choose the view that will be easiest for you to draw
on the overhead. Have the students record this view’s
length, width/depth, and height on Recording Sheet
#1. Have the students fill in the number of cubes
used, the volume, and then model how to make a
sketch. Have the students make this sketch on the
recording sheet.
Ask students to build a rectangular prism that has a
length, width, or height of one. Quickly review their
findings and choose one for all students to build in
the same orientation. Have all of the students record
this view’s length, width/depth, height, number of
cubes, and volume. Model with the students how to
make this view’s sketch on Recording Sheet #1.
Take students step-by-step through the process of
drawing this prism on the isometric paper. (It is
Key Tips for Presenter
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for all of the different parts of the mobile ahead of
time.
Students can use magnifying glasses to look at the
table salt and choose a grain that comes closest to
a cubic millimeter.
Each student should have a net of a cubic
decimeter labeled with 1 liter, 1000 cubic
centimeters, 1000 millimeters and 1 gram. This
concept can be experimented with or given as a
fact. Have students realize that this is where
volume, capacity, and weight are compared as a
scientific standard.
Enough cubic
centimeters for each
pair of students to
have at least 48
1 cm isometric dot
paper (2 sheets per
student)
1 ruler per student
marked in
centimeters
Overhead
Recording Sheet #1
M-GLAnCE 5th Grade – Session 4 – Understanding Volume – Participant Packet
Page 8 Revised 10.15.07
!
Note: One-centimeter isometric paper has equal
spacing of 1-cm between all diagonal and vertical
Name of Activity
Description of Activity
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!
V.
The Layering
Perspective of Volume
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ASSESSMENT: Give students 12 cubes and ask
them to build, sketch, label, and record all possible
prisms.
!
Working with a partner, have students build the 4 x 3
x 2 prism like they had previously drawn on their
isometric paper.
In the second column of Recording Sheet #2 write
“top”.
Beginning with the top layer of the 4 x 3 x 2 prism
that they built, have students count how many cubes
there are and record it on the recording sheet.
With the teacher modeling on the overhead, have the
students color in the top layer of the 4 x 3 x 2 prism
on the teacher prepared isometric paper.
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Materials/Handouts
easiest to draw the height first, and then draw the
length or width on the diagonal. It is also better to
complete the visible edges of the prism before
drawing in the individual cubes.)
As students are filling in the individual cubes on the
isometric paper, walk around and give assistance to
any who are having difficulties.
Have students label on these drawings l = , w = , and
h=.
Discuss each step as you go.
Continue following the same procedure of building,
recording, sketching, and drawing on isometric paper
with the rest of the prisms.
When done, have students complete the bottom of the
recording sheet and discuss it.
Key Tips for Presenter
dots. These are the dots used to make the
drawings. (Do not use horizontal lines in your
drawings.)
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Teacher prepared
isometric paper with
3 identical drawings
of the 4 x 3 x 2
rectangular prism
from previous
activity
Centimeter cubes (24
per pair)
Colored pencils or
markers
Rulers
Recording Sheet #2
These questions are to get students thinking, so take
all answers, as some students may not be able to
M-GLAnCE 5th Grade – Session 4 – Understanding Volume – Participant Packet
Page 9 Revised 10.15.07
!
There are a total of 6 different rectangular prisms
with a volume of 24. It is best to begin with
doing the 4 prisms that have one as either the
length, width, or the height before going on to the
other two.
!
CHALLENGE: The students could be asked to
draw the prisms on isometric paper.
Name of Activity
Description of Activity
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Materials/Handouts
visualize the layers at this point. Ask questions like:
How many layers are underneath the top layer? How
many layers total? How many cubes would be in
each layer? Does the number of cubes correspond to
any of the measures of the prism? Does the number of
layers correspond to any of the measures of the
prism?
Next have students remove the top layer from their
model keeping the 12 cubes intact. Look at the next
layer. Count how many are in that layer. Remove
that layer intact. Count the number of cubes in the
bottom layer. Discuss.
With students, add up the number of cubes from each
layer as 12 + 12 = 24 and record on recording sheet.
Then show this repeated addition as a multiplication
problem (4 x 3) x 2 = 12 x 2 = 24cm3.
Discuss that the volume of the prism is 24 cubic
centimeters, which corresponds to the answer of the
addition and multiplication problem.
Continue by having students build the same 4 x 3 x 2
prism. Only this time in the second column of
Recording Sheet #3 write “front”.
Focusing on the front layer of the 4 x 3 x 2 prism that
they built, have students count how many cubes there
are and record it on the recording sheet.
With the teacher modeling on the overhead, have the
students color in the front layer. Discuss similar
questions as before with same purpose.
Remove the front layer keeping the cubes intact.
Count how many are in that layer and record that
number, the number of layers, the addition,
multiplication, and the labeled volume.
Continue by having students build the same 4 x 3 x 2
prism. Only this time in the second column of
Recording Sheet #2 write “side”.
Focusing on the side layer of the 4 x 3 x 2 prism that
they built, have students count how many cubes there
are and record it on the recording sheet.
M-GLAnCE 5th Grade – Session 4 – Understanding Volume – Participant Packet
Page 10 Revised 10.15.07
Key Tips for Presenter
!
By taking off the layers we are decomposing the
prism to make it more visual.
Name of Activity
Description of Activity
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!
!
VI.
Assessment
!
!
Materials/Handouts
With the teacher modeling on the overhead, have the
students color in the side layer.
Ask the same questions as above with the same
purpose in mind.
Remove the side layer keeping the cubes intact.
Count how many are in that layer and record that
number. Continue decomposing the prism of its side
layers. Then record the number of layers, the
addition, multiplication, and the labeled volume, as
before.
Using the following questions as a guide, have
students respond by journal writing or in some other
format, in order for you to check the depth of their
understanding.
o How many layers are underneath/next to/behind
the top/side/front layer? How many layers total?
How many cubes would be in each layer? Does
the number of cubes correspond to any of the
measures of the prism? Does the number of
layers correspond to any of the measures of the
prism?
Using Recording Sheet #2 as a guide provide students
with a prepared worksheet showing several drawings
of prisms on isometric paper. These drawings should
have only the visible edges showing. The students
should be able to accurately fill in the recording sheet
by building (if necessary), decomposing into layers or
using any other method they know.
o Ask for a response to questions such as: Did you
need to draw in the cubes in order to see a layer?
Does it help to count in layers to get the volume?
Have you discovered an easier way to get the
volume? Describe the method you used to find
the volume using pictures and words.
!
!
Teacher prepared
prism drawings on
isometric paper
showing only visible
edges
Rulers
M-GLAnCE 5th Grade – Session 4 – Understanding Volume – Participant Packet
Page 11 Revised 10.15.07
Key Tips for Presenter
!
Students are made aware of the volume formula in
fifth grade, but they do not formally use it.
!
Be sure the prisms have a variety of dimensions.
!
Estimated volumes could be found using catalogs
for real world examples of 3-d objects.
Fifth Grade Volume
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TIPS
When students disagree with each other, encourage them
to justify their reasoning to each other or the class. Often
they are correct in their thinking but they are not
answering the question given.
Sometimes it may be necessary for the class to come to
some consensus.
Make every opportunity so ALL children have a turn at
participating: changing roles in the activities from
recorders to measurers, explaining their rationale,
illustrating their answer for their group/ and or class, etc.
It is too easy to let the more vocal students respond so we
can move on.
Solving problems without giving students time to think
about them often leads to “fragile knowledge”.
Be sure students know and use the abbreviations for the
measures of volume.
The purpose of the questions asked on the worksheets is
not for a grade but for discovery, reasoning, justifying,
and communication practice.
The purpose of the questions asked on the worksheets is
not always for a grade but for discovery, reasoning, and
communication practice.
When working with volume, it is important to rename and
refer to the base-ten blocks cubes.
It is important for students to visualize prisms in a variety
of orientations (views) and to understand how to name the
length, width, and height, even though when they use the
formula, it will not make a difference.
COMMON MISCONCEPTIONS
! Students often confuse the area
measure of one square inch or one
square centimeter for the volume
measure of one cubic inch or one
cubic centimeter.
! Be aware that many students are
unable to visualize and/or draw 3dimensional objects in a 2dimensional format.
M-GLAnCE 5th Grade – Session 4 – Understanding Volume – Participant Packet
Page 12 Revised 10.15.07
Fifth Grade Presentation
Focus on: Volume
Name of Activity
II.
III.
Review of Metric Square
Units
Introduction to Volume
Description of Activity
!
!
Materials/Handouts
Review square meter, square decimeter, square
centimeter. Draw lines dividing poster board into 100
sq. decimeters. (Need 2 or 3 sheets.) Shade in and
label one square decimeter. Shade in and label one
square centimeter.
!
To begin volume start with a 10 cm x 10 cm x 10 cm
cube net. Have students make 2 cubes (without tops)
from tag board. Cut carefully and tape or glue
together.
!
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!
!
!
!
!
!
!
!
!
Have students place one row of centimeter cubes
along an inside edge of one of their boxes so they can
visualize the space filled. Ask questions such as:
How many centimeter cubes are needed to fill in just
one row? How many rows do you think will be
needed to cover just the bottom layer? How many
centimeter cubes will be needed to fill this layer?
Since we don’t have enough cubic centimeters to fill
in the bottom layer of everyone’s cubes, which size
base-ten block could you use next?
!
Key Tips for Presenter
2-3 sheets of poster
board
Base-ten blocks
Square centimeter
!
Refer back to this 2-dimensional poster, as
necessary, for comparison when building the 3dimensional model.
Tagboard with
decimeter net or net
template that can be
traced on to tag
board.
Scissors
Tape/glue
Base-ten thousand
cubes, flats, rods and
centimeter cubes
Square centimeter
grid paper (at least 10
square decimeters per
student)
!
Students may have never worked with nets before
so it would be a good time to show how to
develop a net by using a thousand cube from the
base-ten blocks as a model.
Tape 6 square decimeters together to form a net
that will fold into a cubic decimeter around the
thousand cube.
Show at least three ways that the net can be
flattened out and how it is helpful to have tabs to
glue together when forming the cube.
Fill the rest of the bottom layer with rods. How many
cubic centimeters would this be? How many rods
would this be equivalent to? Which size base-ten
block would the bottom layer be equivalent to?
Discuss: How many flats would I need to fill the
whole cubic decimeter? How many rods? Etc.
Fill the rest with flats.
Pass out a thousand cube block to each group for the
students to compare with their filled cubes.
M-GLAnCE 5th Grade – Session 4 – Understanding Volume – Participant Packet
Page 13 Revised 10.15.07
!
!
!
You might want to borrow base-ten blocks from
other teachers in order for each student to fill their
own boxes.
!
4x4x4 boxes come in either white or brown and
can be bought for $29 per 100 from:
Action Bag Company
501 N. Edgewood Avenue
Wood Dale, IL 60191-1410
(800) 824-2247
www.actionbag.com
Name of Activity
Description of Activity
!
!
III. Make a Cubic Meter
Class Activity
!
!
!
!
!
Materials/Handouts
!
!
!
On the wall behind the cubic meter you just made,
hang the poster board you made earlier with the sq.
decimeters marked. Use this for the comparison of
the length and height of the cubic units.
!
!
Substitute a meter stick for the 10 cubic decimeters
stacked in the corner and against the walls.
When done showing length, width, and height with
the cubic decimeters, hang from the ceiling with
poster board at the top (marked in square decimeters
and with one square decimeter marked in square
centimeters) as the lid.
!
!
Decimeter cubes
Meter sticks
Paper clips to hold
the cubic decimeters
together.
Key Tips for Presenter
!
!
!
!
th
Discuss that the model they created shows the filling
of a thousand cube, while the base-ten model shows
only the surface area.
Rename the thousand cube as a cubic decimeter since
it is 1 dm in length, 1 dm in width, and 1 dm in
height.
CLASS DISCUSSION: How many cubic decimeters
will fit along edge of a meter stick? How many
would it take to stack as tall as a meter?
In an open corner of the classroom, if possible, use
the cubic decimeters they made and stack 10 high.
Relate this to the meter stick.
Lay 9 cubes on the floor from the corner going along
the wall (perpendicular). Compare this length to the
meter stick. Ask: Why am I only adding 9 cubes?
Stack 9 cubes on top of the last cube. (Continue to
compare to the meter stick.)
Lay 9 more cubes on the floor coming out at a 90
degree angle (perpendicular) from the wall. (Use
paper clips to hold cubes together.) Use as many
cubes as you can to show the edges of the new cubic
meter.
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Poster board with 100
sq. dm marked and
one of the square
decimeters marked in
square centimeters
Meter sticks or meter
stick representations
made from sentence
strips.
String or fishing line
Magnifying glasses
Salt
Centimeter rulers
with millimeters
Hang a cubic millimeter representation (here you
could use a piece of salt), a cubic cm, and a cubic dm
all labeled within the cubic meter, as a mobile.
M-GLAnCE 5 Grade – Session 4 – Understanding Volume – Participant Packet
Page 14 Revised 10.15.07
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The goal is to make a very visual representation
of a cubic meter using the individual parts that it
is made of. If it is possible, it will be beneficial to
leave the cubic meter up for awhile in order for
students to refer to.
There are other topics and strands that can be
discussed and interwoven in this unit with rich
vocabulary: angles, perpendicular, parallel, etc.
Make labels with both words and abbreviations
for all of the different parts of the mobile ahead of
time.
Students can use magnifying glasses to look at the
Name of Activity
Description of Activity
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IV.
Building Rectangular
Prisms to Find Volume
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Materials/Handouts
Hang a second cubic dm with each side labeled with
its different equivalences: 1 liter, 1000 cubic cm,
1000 ml & 1000 g.
Give each pair of students at least 48 centimeter
cubes and allow them 5 minutes to discover two
different rectangular prisms using 24 of the cubes.
Have students describe their findings.
The teacher needs to model the 24cm x 1cm x 1cm
prism (flat) on the overhead and then have each
student make the same model with their cubes. (Ask
the students to build their prism flat on the desk and
not as a tower.)
Explain that the prism has a length, a width, and a
height. Using different orientations of the prism, ask
what the different lengths/widths/ heights would be.
Choose the view that will be easiest for you to draw
on the overhead. Have the students record this view’s
length, width/depth, and height on Recording Sheet
#1. Have the students fill in the number of cubes
used, the volume, and then model how to make a
sketch. Have the students make this sketch on the
recording sheet.
Ask students to build a rectangular prism that has a
length, width, or height of one. Quickly review their
findings and choose one for all students to build in
the same orientation. Have all of the students record
this view’s length, width/depth, height, number of
cubes, and volume. Model with the students how to
make this view’s sketch on Recording Sheet #1.
Take students step-by-step through the process of
drawing this prism on the isometric paper. (It is
easiest to draw the height first, then draw the length
or width on the diagonal. It is better to complete the
visible edges of the prism before drawing in the
Key Tips for Presenter
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Enough cubic
centimeters for each
pair of students to
have at least 48
1 cm isometric dot
paper (2 sheets per
student)
1 ruler per student
marked in
centimeters
Overhead
Recording Sheet #1
M-GLAnCE 5th Grade – Session 4 – Understanding Volume – Participant Packet
Page 15 Revised 10.15.07
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table salt and choose a grain that comes closest to
a cubic millimeter.
Each student should have a net of a cubic
decimeter labeled with 1 liter, 1000 cubic
centimeters, 1000 millimeters and 1 gram. This
concept can be experimented with or given as a
fact. Have students realize that this is where
volume, capacity and weight are compared as a
scientific standard.
Note: One-centimeter isometric paper has equal
spacing of 1-cm between all diagonal and vertical
dots. These are the dots used to make the
drawings. (Do not use horizontal lines in your
drawings.)
Name of Activity
Description of Activity
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V.
The Layering
Perspective of Volume
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Materials/Handouts
individual cubes.)
As students are filling in the individual cubes, walk
around and give assistance to any who are having
difficulties.
Have students label on these drawings l = , w = , and
h=.
Have students label on these drawings l = , w = , and
h=.
Continue following the same procedure of building,
recording, sketching, and drawing on isometric paper
with the rest of the prisms.
Key Tips for Presenter
!
There are a total of 6 different rectangular prisms
with a volume of 24. It is best to begin with
doing the 4 prisms that have one as either the
length, width, or the height before going on to the
other two.
!
CHALLENGE: The students could be asked to
draw the prisms on isometric paper.
ASSESSMENT: Give students 12 cubes and ask
them to build, sketch, label, and record all possible
prisms.
Working with a partner, have students build the 4 x 2
x 3 prism like they had previously drawn on their
isometric paper.
In the second column of Recording Sheet #2 write
“top”.
Beginning with the top layer of the 4 x 2 x 3 prism
that they built, have students count how many cubes
there are and record it on the recording sheet.
With the teacher modeling on the overhead, have the
students color in the top layer of the 4 x 2 x 3 prism
on the teacher prepared isometric paper.
These questions are to get students thinking, so take
all answers, as some students may not be able to
visualize the layers at this point. Ask questions like:
How many layers are underneath the top layer? How
many layers total? How many cubes would be in
each layer? Does the number of cubes correspond to
any of the measures of the prism? Does the number of
layers correspond to any of the measures of the
!
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Teacher prepared
isometric paper with
3 identical drawings
of the 4 x 2 x 3
rectangular prism
from previous
activity
Centimeter cubes (24
per pair)
Colored pencils or
markers
Rulers
Recording Sheet #2
M-GLAnCE 5th Grade – Session 4 – Understanding Volume – Participant Packet
Page 16 Revised 10.15.07
Name of Activity
Description of Activity
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Materials/Handouts
prism?
Next have students remove the top layer from their
model keeping the 8 cubes intact. Look at the next
layer. Count how many are in that layer. Remove
that layer intact. Count the number of cubes in the
bottom layer. Discuss.
With students add up the number of cubes from each
layer as 8 + 8 + 8 = 24 and record on recording sheet.
Then show this repeated addition as a multiplication
problem 8 x 3 = 24.
Discuss that the volume of the prism is 24 cubic
centimeters, which corresponds to the answer of the
addition and multiplication problem.
Continue by having students build the same 4 x 2 x 3
prism. Only this time in the second column of
Recording Sheet #3 write “front”.
Focusing on the front layer of the 4 x 2 x 3 prism that
they built, have students count how many cubes there
are and record it on the recording sheet.
With the teacher modeling on the overhead, have the
students color in the front layer. Discuss similar
questions as before with same purpose.
Remove the front layer keeping the cubes intact.
Count how many are in that layer and record that
number, the number of layers, the addition,
multiplication, and the labeled volume.
Continue by having students build the same 4 x 2 x 3
prism. Only this time in the second column of
Recording Sheet #2 write “side”.
Focusing on the side layer of the 4 x 2 x 3 prism that
they built, have students count how many cubes there
are and record it on the recording sheet.
With the teacher modeling on the overhead, have the
students color in the side layer.
Ask the same questions as above with the same
purpose in mind.
Remove the side layer keeping the cubes intact.
Count how many are in that layer and record that
M-GLAnCE 5th Grade – Session 4 – Understanding Volume – Participant Packet
Page 17 Revised 10.15.07
Key Tips for Presenter
!
By taking off the layers we are decomposing the
prism to make it more visual.
Name of Activity
Description of Activity
Materials/Handouts
Key Tips for Presenter
number. Continue decomposing the prism of its side
layers. Then record the number of layers, the
addition, multiplication, and the labeled volume, as
before.
VI.
Assessment
!
Using the following questions as a guide, have
students respond by journal writing or in some other
format, in order for you to check the depth of their
understanding.
o How many layers are underneath/next to/behind
the top/side/front layer? How many layers total?
How many cubes would be in each layer? Does
the number of cubes correspond to any of the
measures of the prism? Does the number of
layers correspond to any of the measures of the
prism?
!
Using Recording Sheet #2 as a guide provide students
with a prepared worksheet showing several drawings
of prisms on isometric paper. These drawings should
have only the visible edges showing. The students
should be able to accurately fill in the recording sheet
by building (if necessary), decomposing into layers or
using any other method they know.
o Ask for a response to questions such as: Did you
need to draw in the cubes in order to see a layer?
Does it help to count in layers to get the volume?
Have you discovered an easier way to get the
volume? Describe the method you used to find
the volume using pictures and words.
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Teacher prepared
prism drawings on
isometric paper
showing only visible
edges
Rulers
M-GLAnCE 5th Grade – Session 4 – Understanding Volume – Participant Packet
Page 18 Revised 10.15.07
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Be sure the prisms have a variety of dimensions.
Volume
SHAPE
LENGTH
WIDTH/
DEPTH
HEIGHT
# of
Cubes
Needed
VOLUM
E
Prism A
24 x 1 x 1
Prism B
__ x __ x __
1. What is the relationship between the number of cubes and the volume?
2. What do you notice about the 3 dimensions (length, width, height) and the volume?
3. If you see a relationship, how could you test that theory?
M-GLAnCE 5th Grade – Session 4 – Understanding Volume – Participant Packet
Page 19 Revised 10.15.07
SKETCH
Volume
FIGURE
VIEW:
Top, Front, or
Side Layer
# OF CUBES
IN ONE
LAYER
TOTAL # OF
LAYERS IN
FIGURE
VOLUME BY
ADDITION
M-GLAnCE 5th Grade – Session 4 – Understanding Volume – Participant Packet
Page 20 Revised 10.15.07
VOLUME BY
MULTIPLICATION
VOLUME
Literature and Resources
ESTIMATION
! Betcha! by Stuart Murphy
! Dumpling Soup by Jama Kim Rattigan
MEASUREMENT & LENGTH
! Jim & The Beanstalk by Raymond Briggs
! Counting On Frank by Rod Clement
! Length by Henry Pluckrose
! How Tall? How Long? How Big? (Scale drawings/facts/records, etc.) by Nicholas Harris
ALL MEASUREMENT
o Polly’s Pen Pal by Stuart Murphy
o Measuring Penny by Loreen Leedy
AREA & PERIMETER
! A Cloak For A Dreamer by Aileen Friedman
! Chickens on the Move by Pam Pollack and Meg Belviso
! Spaghetti & Meatballs For All by Marilyn Burns
! Bigger, Better, Best by Stuart Murphy
RESOURCES:
o Math For Fun: “Measuring Sizes” by Andrew King
o Let’s Investigate Area & Volume by Marion Smoothey
o Super-sized Science Projects with Volume by Robert Gardner
o Measuring Up by Sandra Markle
www.Illuminations.nctm.org/index.aspx
www.nctm.org
www.hbschool.com/menus/math_advantage.html (Go to e-lab and Grade level for student activities)
www.aplusmath.com
M-GLAnCE 5th Grade – Session 4 – Understanding Volume – Participant Packet
Page 21 Revised 10.15.07