Objective
1
To review concepts and units of volume.
materials
Teaching the Lesson
Key Activities
Students read and discuss an essay about volume in the Student Reference Book. Students
use base-10 blocks and metersticks to visualize the sizes of various metric cubic units. They
also make open boxes and fill them with centimeter cubes to determine their volume.
Key Concepts and Skills
• Use multiplication to solve volume problems. [Operations and Computation Goal 3]
• Find the area of the base of a rectangular prism. [Measurement and Reference Frames Goal 2]
• Count unit cubes to find the volume of a rectangular prism.
[Measurement and Reference Frames Goal 2]
• Calculate the surface area of a rectangular prism.
Student Reference Book, p. 137
Study Link 11 3
Teaching Aid Master (Math Masters,
p. 403)
base-10 blocks
centimeter cubes
tape
metersticks
slate
scissors
traffic cones and string (optional)
[Measurement and Reference Frames Goal 2]
• Write number models with parentheses.
[Patterns, Functions, and Algebra Goal 3]
Key Vocabulary cubic units • volume • dimensions • surface area
Ongoing Assessment: Recognizing Student Achievement Use Mental Math and Reflexes.
computer with Internet access
(optional)
See Advance Preparation
[Operations and Computation Goal 3]
2
materials
Ongoing Learning & Practice
Students play the Credits/Debits Game to practice addition of integers.
Students practice and maintain skills through Math Boxes and Study Link activities.
3
Students use 24 cubes to
build as many rectangular
prisms as possible, each
with a different base.
ENRICHMENT
Students create penticubes
and compare their surface
areas.
ELL SUPPORT
Students add volume and
cubic units to their Math
Word Banks.
Additional Information
Advance Preparation For the cubic meter demonstration in Part 1, you need 3 metersticks;
for the alternative demonstration, you need 4 metersticks, 4 traffic cones, and string or
2 metersticks, 2 traffic cones, string, and tape. See the illustrations on pages 868 and 869.
866
Unit 11 3-D Shapes, Weight, Volume, and Capacity
Study Link Master (Math Masters,
p. 330)
Game Master (Math Masters, p. 468)
materials
Differentiation Options
READINESS
Math Journal 2, p. 297
Student Reference Book, p. 238
Teaching Aid Master (Math Masters,
p. 389)
Differentiation Handbook
centimeter cubes
Technology
Assessment Management System
Mental Math and Reflexes
See the iTLG.
See the Web site on page 870.
Getting Started
Mental Math and Reflexes
Math Message
Pose multiplication and division facts. Suggestions:
248
5 7 35
10 6 60
20 4 5
36 6 6
45 5 9
Read page 137 of the
Student Reference Book. Be prepared
to explain why there is a picture of a
sandbox on the page.
9 9 81
8 7 56
6 9 54
48 6 8
63 7 9
72 9 8
3 9 27
4 8 32
6 4 24
42 7 6
64 8 8
28 4 7
Study Link 11 3
Follow-Up
Working in small groups,
have students compare answers
and pose the riddle they wrote.
Ongoing Assessment:
Recognizing Student Achievement
Mental Math
and
Reflexes
Use Mental Math and Reflexes to assess students’ ability to solve
multiplication and division facts. Students are making adequate progress
if they demonstrate automaticity with the multiplication facts and proficiency
with the division facts in the
,
, and
problems. Some students
may demonstrate automaticity with the division facts.
[Operations and Computation Goal 3]
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Student Reference Book, p. 137)
Student Page
Measurement
Review the information on Student Reference Book, page 137.
Once students explain the significance of the picture of the
sandbox, ask them to give other examples in which it is useful to
know the volume of an object. For example:
Buying a cooler—to decide whether it is big enough to hold the
food for a camping trip
Renting a car—to decide if the trunk is large enough to hold
the family’s luggage
Tell students that in this lesson they will review units of volume and
explore how to use cubes to find the volume of a rectangular prism.
Volume and Capacity
Volume
The volume of a solid object such as a brick or a ball is a
measure of how much space the object takes up. The volume
of a container such as a freezer is a measure of how much the
container will hold.
1 cubic centimeter
(actual size)
Volume is measured in cubic units. A base-10 cube has sides
that are 1 centimeter long; it is called a cubic centimeter.
A cube with 1-inch sides is called a cubic inch.
Other cubic units are used to measure large volumes. A cubic
foot has 1-foot sides. A cubic yard has 1-yard sides and can
hold 27 cubic feet. A cubic meter has 1-meter sides and can
hold more than 35 cubic feet.
The volume of an object can be
very useful to know. Suppose
you wanted to buy sand to fill
an empty sandbox. To estimate
how much sand to buy, you
would measure the length, width, and height of the sandbox.
The length, width, and height are called the dimensions of
the box. You would then use these dimensions to calculate
how many cubic feet (or cubic yards) of sand to order. You
could do similar calculations to determine how much concrete
would be needed to build a patio, or how much gravel to buy
for a path in the backyard.
Capacity
We often measure things that are poured into or out of
containers such as liquids, grains, salt, and so on. The
volume of a container that is filled with a liquid or a solid
that can be poured is often called its capacity.
Capacity is usually measured in units such as gallons,
quarts, pints, cups, fluid ounces, liters, and milliliters.
These are standard units, but they are not cubic units.
The tables at the right compare different units of capacity.
1 cubic inch
(actual size)
Jupiter is the largest planet
in the solar system. The
volume of Jupiter is 1,300
times the volume of Earth.
U.S. Customary Units
1 gallon (gal) 4 quarts (qt)
1 gallon 2 half-gallons
1 half-gallon 2 quarts
1 quart 2 pints (pt)
1 pint 2 cups (c)
1 cup 8 fluid ounces (fl oz)
1 pint 16 fluid ounces
1 quart 32 fluid ounces
1 half-gallon 64 fluid
ounces
1 gallon 128 fluid ounces
Metric Units
1 liter (L) 1,000 milliliters (mL)
1
1 milliliter 1,000 liter
Student Reference Book, p. 137
Lesson 11 4
867
Visualizing Metric Cubic Units
WHOLE-CLASS
ACTIVITY
Discuss the following:
Linear measurements are usually given in standard units (such
as feet or meters), and area measurements are often given in
squares of those units (such as square feet or square meters).
Many volume measurements are given in cubes of standard
units, or cubic units.
The area of a closed 2-dimensional figure is the number of unit
squares and fractions of unit squares needed to cover the
interior of the figure. The volume of a 3-dimensional object is
the number of unit cubes and fractions of unit cubes needed to
fill the space taken up by the object. To support English
language learners, discuss the everyday and mathematical
meanings of volume.
Use base -10 blocks and metersticks to help students visualize the
sizes of various metric cubic units.
Hold up a cm cube. Point out that each edge is 1 centimeter
long, so the volume of a cm cube is 1 cubic centimeter.
Hold up a “big cube.” Explain that each edge is 10 centimeters,
or 1 decimeter, long, so the volume of a big cube is 1 cubic
decimeter.
Ask: How many cubic centimeters are in 1 cubic decimeter? Have
students use base -10 blocks to “prove” their answers. There are
10 cubic centimeters in 1 long; there are 10 longs, or 100 cubic
centimeters, in 1 flat. You can fill a cubic decimeter container with
10 flats, or 1,000 cubic centimeters. Therefore, 1 cubic decimeter
equals 1,000 cubic centimeters.
You will not have enough base -10 blocks to build a 1-meter cube,
but you can help students visualize such a cube using one of these
methods:
Method 1: Place two metersticks on a flat surface at right angles
to each other. Hold up a third meterstick perpendicular to the
other metersticks so that all three sticks meet in one corner.
1 meter
1m
ete
1 meter
r
868
With the help of this partial frame, students can imagine a
cube whose edges are the length of a meterstick. The volume
of this cube is 1 cubic meter.
Method 2: Place four hollow traffic cones on the floor at the
corners of a square with 1-meter sides. Put a meterstick through
the top of each cone so that each stick stands straight up. Connect
the tops of the metersticks with string to form a square. The
string should be as close to the top of the metersticks as possible.
(See margin.)
Unit 11 3-D Shapes, Weight, Volume, and Capacity
Method 3: A variation of the above method uses two cones and
two metersticks. Place the cones 1 meter from a wall and 1 meter
apart. Connect the tops of the metersticks with string. Run a
string from the top of each meterstick to the wall at a height of
1 meter, and tape the string to the wall.
1m
e te
r
1 meter
1 meter
Adjusting the Activity
Ask: How many cubic decimeters are in 1 cubic meter? 1,000 How
many cubic centimeters are in 1 cubic meter? 100 100 100 1,000,000
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
V I S U A L
Remind students of alternative ways of writing square units:
square m, sq m, or m2; square cm, sq cm, or cm2; and square in.,
sq in., or in2. Similarly, cubic units may be written as m3, cm3, or
in3. These are read as “cubic meter,” “cubic centimeter,” and
“cubic inch.”
Using Cubes to Find the Volume
PARTNER
ACTIVITY
of a Rectangular Prism
(Math Masters, p. 403)
Model the following activity for students before they work in
partnerships to make their own boxes:
1. On a sheet of centimeter grid paper (Math Masters, page 403),
draw a pattern for an open box. For example, the bottom of the
box might be a rectangle 4 centimeters long and 3 centimeters
wide, and the box might be 2 centimeters high.
2. Cut out the pattern. Fold up the sides, and tape them together.
3. Fill the box with centimeter cubes. The number of cubes
needed to fill the box is the volume of the box.
2 cm
3 cm
4 cm
P attern for open box
Partners may make boxes with any dimensions they choose,
but the height must be at least 2 centimeters.
Lesson 11 4
869
Student Page
Date
Time
LESSON
Technology Link Alternatively, have students visit the
Web site at http://www.illuminations.nctm.org/tools/tool_
detail.aspx?id6 to create boxes of varying dimensions and
manipulate and count unit cubes, rows of unit cubes, or layers
of unit cubes.
Math Boxes
11 4
1. The object below has the shape of a
2. Draw the figure after it is rotated
1
counterclockwise 4-turn.
geometric solid. Name the solid.
cylinder
101
4. Insert , , or to make a true number
the answer. Then correctly place the
decimal point.
sentence.
a.
.
a. 7.56 4 3 0 2 4
Number model:
8 4 32
b.
.
c.
b. 563.2 ÷ 4 1 4 0 8
Number model:
600 4 150
d.
5. Round each number to the nearest tenth.
1
4
c. 4.35
d. 9.60
e. 22.89
2
5
A U D I T O R Y
1
3
6
K I N E S T H E T I C
T A C T I L E
V I S U A L
60
6. Jake can ride his bike 5 miles in
40 minutes. At this rate, how long
does it take him to ride 1 mile? Circle
the best answer.
3.5
0.7
4.4
9.6
22.9
b. 0.71
Have students imagine that each of the boxes has a lid. Have them
calculate the surface area of the closed boxes by determining the sum of the
areas of the faces. The surface area of the sample box on page 869 is
52 square centimeters: (2 ∗ (2 ∗ 3)) (2 ∗ (2 ∗ 4)) (2 ∗ (3 ∗ 4)) 52.
6
123 241
8.9 5.7
14
3
e. 9
a. 3.46
Adjusting the Activity
106 107
3. Write a number model to estimate
A. 200 minutes
B. 40 minutes
2 Ongoing Learning & Practice
C. 20 minutes
D. 8 minutes
182 183
297
Math Journal 2, p. 297
Playing the Credits/Debits Game
PARTNER
ACTIVITY
(Student Reference Book, p. 238; Math Masters, p. 468)
Students play the Credits/Debits Game to practice adding positive
and negative numbers. See Lesson 10-6 for additional information.
Math Boxes 11 4
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 297)
Mixed Practice Math Boxes in this lesson are linked
with Math Boxes in Lessons 11-2 and 11-6. The skills
in Problems 5 and 6 preview Unit 12 content.
Study Link Master
Name
Date
STUDY LINK
Time
Volume
11 4
Cut out the pattern below and tape it together to form an open box.
137 138
1.
Find and record two items in your home that have volumes equal to
1
about 2 of the volume of the open box.
2.
Find and record two items in your home that have about the same volume
as the open box.
3.
Find and record two items in your home that have volumes equal to
about 2 times the volume of the open box.
Answers vary.
Answers vary.
Writing/Reasoning Have students write a response to the
following: How did you round each number to the nearest
tenth in Problem 5? Sample answer: I found the digit in
the tenths place. Then I looked at the digit to the right. If it was
less than 5, I kept the digit in the tenths place the same. If the
number was 5 or greater, I rounded up the number in the
tenths place.
Answers vary.
Practice
232
6. 8
24
29
5.
86 / 5 17
7.
43
5
8
Study Link 11 4
1
R1, or 17 5
2
1
89 R2, or 89 4, or 89 2
TAB
TAB
TAB
96 ÷ 4 TAB
4.
(Math Masters, p. 330)
Math Masters, p. 330
870
INDEPENDENT
ACTIVITY
Unit 11 3-D Shapes, Weight, Volume, and Capacity
Home Connection Students cut out and assemble an
open box. They search for items at home that have
1
volumes equal to about 2 of, the same as, and 2 times
the volume of the open box.
3 Differentiation Options
READINESS
Finding Rectangular Prisms
PARTNER
ACTIVITY
15–30 Min
Example:
(Math Masters, p. 389)
To explore the concept of volume, have students use 24 centimeter
cubes to build as many rectangular prisms as possible, each with
a different base. The area of the base must be greater than 1 cm2,
and the height of the prism must be greater than 1 cm.
Have students create a table on an Exit Slip (Math Masters,
page 389) to organize their work. The table should include the
area of the base, the height, and the volume of each prism they
make. Remind students to include the units. (See margin.)
ENRICHMENT
Exploring Penticubes
Area of Base
(sq cm)
Height
(cm)
Volume
(cu cm)
2
12
24
12
2
24
3
8
24
8
3
24
4
6
24
6
4
24
PARTNER
ACTIVITY
15–30 Min
(Math Masters, p. 389)
To investigate volume and surface area, have
students build penticubes, which are 3-dimensional
figures with a volume of 5 cubic units. They are
constructed from 5 cubes connected by at least one face. There
are 29 possible penticubes.
Have students compare the surface areas of the penticubes and
describe on an Exit Slip (Math Masters, page 389) anything they
notice about the figures with similar surface areas. Expect
responses such as the following:
All but two of the penticubes have a surface area of 22 square
units. The penticubes that are circled to the right have a
surface area of 20 square units.
There are 29 possible penticubes.
If a cube touches only 1 other cube face, then the surface area of
that individual cube is 5 square units. If a cube touches two faces,
then the surface area of that individual cube is 4 square units.
The more faces the cubes touch, the smaller the surface area
is. For example, in the circled penticubes, 4 of the cubes touch
2 faces each, thereby creating a smaller surface area.
ELL SUPPORT
Building a Math Word Bank
SMALL-GROUP
ACTIVITY
5–15 Min
(Differentiation Handbook)
To provide language support for volume, have students use the
Word Bank Template found in the Differentiation Handbook. Ask
students to write the terms volume and cubic units, draw pictures
relating to each term, and write other related words. See the
Differentiation Handbook for more information.
Lesson 11 4
871