Objective
To guide the development and use of a formula for
finding the volume of a rectangular prism.
1
materials
Teaching the Lesson
Key Activities
Students solve cube-stacking problems and use the results to derive a formula for the
volume of a rectangular prism.
Math Journal 2, pp. 298, 300–302
Study Link 11 4
Key Concepts and Skills
centimeter cubes
centimeter ruler
• Find the area of the base and the surface area of a rectangular prism.
slate
[Measurement and Reference Frames Goal 2]
• Count unit cubes and use a formula to find the volume of a rectangular prism.
[Measurement and Reference Frames Goal 2]
• Solve problems involving spatial visualization. [Geometry Goal 3]
• Describe a rule for a pattern and use the rule to solve problems.
[Patterns, Functions, and Algebra Goal 1]
• Write number models with parentheses. [Patterns, Functions, and Algebra Goal 3]
Key Vocabulary rectangular prism • volume • formula
Ongoing Assessment: Informing Instruction See page 874.
Ongoing Assessment: Recognizing Student Achievement Use journal page 302.
[Measurement and Reference Frames Goal 2]
2
Ongoing Learning & Practice
Students play Chances Are to practice using probability language to describe the
likelihood of an event.
Students practice and maintain skills through Math Boxes and Study Link activities.
materials
Math Journal 2, p. 299
Student Reference Book, pp. 236 and 237
Study Link Master (Math Masters, p. 331)
Game Master (Math Masters, p. 464)
Chances Are Event and Probability Cards
(Math Masters, pp. 462, 463, 465, and 466)
3
materials
Differentiation Options
READINESS
Students use interlocking cubes to
build cube stacks and solve spatial
visualization problems.
ENRICHMENT
Students estimate the volume of a sheet of
notebook paper.
Teaching Master (Math Masters, p. 332)
Teaching Aid Master (Math Masters, p. 388
or 389)
interlocking cubes; sheet of notebook
paper; scissors; stick-on notes
Technology
Assessment Management System
Journal page 302, Problems 1, 2, and 5
See the iTLG.
872
Unit 11 3-D Shapes, Weight, Volume, and Capacity
Getting Started
Mental Math and Reflexes
Math Message
Pose mental addition problems. Suggestions:
Complete journal page 298.
16 ⫹ 4 ⫽ 20
25 ⫹ 5 ⫽ 30
11 ⫹ 9 ⫽ 20
37 ⫹ 3 ⫽ 40
18 ⫹ 19 ⫽ 37
49 ⫹ 17 ⫽ 66
48 ⫹ 16 ⫽ 64
32 ⫹ 18 ⫽ 50
253 ⫹ 29 ⫽ 282
76 ⫹ 149 ⫽ 225
126 ⫹ 259 ⫽ 385
644 ⫹ 39 ⫽ 683
Study Link 11 4 Follow-Up
䉬
Working in small groups, have students describe the
items in their home that had volumes equal to about
1
ᎏᎏ of, the same as, and 2 times the volume of the open box. Ask
2
students to use centimeter cubes to determine the volume of the
box. 96 cm3 Ask: Suppose the box had a lid. What would be the
surface area of the closed box? 136 cm2
1 Teaching the Lesson
䉴 Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Math Journal 2, p. 298)
Review the answers on journal page 298. Discuss the use of
variables to stand for quantities such as length and width. Two
formulas that students are likely to give are A ⫽ l ⴱ w and
A ⫽ b ⴱ h. To support English language learners, write the two
formulas on the board.
Tell students that in this lesson they will apply their knowledge
of area formulas to develop a formula for finding the volume
of a rectangular prism.
䉴 Solving Cube-Stacking Problems
Student Page
PARTNER
ACTIVITY
(Math Journal 2, pp. 300 and 301)
Date
Time
LESSON
Area of a Rectangle
11 5
134
1. Write a formula for the area of a rectangle. In your formula, use A for area. Use l and
w for length and width, or b and h for base and height.
Fill in the column for Box 1 with the class. You might wish to use
the following prompts:
●
l w, or A
Number model:
How many layers of cubes are needed to fill the box? 5
How can you tell? There are 5 cubes in the stack.
3
27
Area
27
Area
56 m
7 meters
height
●
How many cubes are needed to fill the box? There are 5 layers
with 32 cubes in each layer, and 5 ⴱ 32 ⫽ 160, so 160 cubes
are needed to fill the box.
Have students complete the rest of the problems with partners.
square centimeters
4. Find the length of the base of the rectangle.
2
84 in2
Area
?
56
8
7
8
Number model:
m
84
12
7
length of base
7
in.
Try This
5. Find the area of the rectangle.
6. Find the height of the rectangle.
5 cm
●
9
?
Number model:
How many cubes are needed to cover the bottom of the box? 32
h
3. Find the height of the rectangle.
How many cubes can be placed along the longer side of
the box? 8 Along the shorter side? 4
●
b
2. Draw a rectangle with sides measuring 3 centimeters and 9 centimeters. Find the area.
12 in.
Each problem on journal pages 300 and 301 shows a picture of a
box that is partially filled with cubes. Students find the number of
cubes needed to completely fill each box and record the results in
the table on journal page 300.
A
?
11.3 cm
11.3 5
56.5 cm
Number model:
Area
Area
403 cm2
26 cm
2
56.5
403
15.5
26
Number model:
height
15.5
cm
Math Journal 2, p. 298
Lesson 11 5
䉬
873
Ongoing Assessment: Informing Instruction
Watch for students who try to find the volume by counting only the cubes shown
in the picture. Remind them that the cubes that are shown will help determine the
height of the box and the number of cubes needed to cover the bottom, but that
the volume of the box is the total number of cubes needed to fill the box.
Deriving a Formula for the
PARTNER
ACTIVITY
Volume of a Rectangular Prism
(Math Journal 2, pp. 300 and 301)
Remind students that geometric solids, such as those pictured
on journal pages 300 and 301, are called rectangular prisms.
Review the properties of rectangular prisms. To support English
language learners, write students’ responses on the board.
For example:
NOTE The properties to the right are true
of right rectangular prisms. The oblique
rectangular prism shown below has only two
rectangular faces. In Fourth Grade Everyday
Mathematics, the term rectangular prism will
refer only to a right rectangular prism.
A rectangular prism has 6 rectangular faces, 12 edges,
and 8 corners.
Pairs of opposite faces are congruent.
Any face of a rectangular prism can be designated as the base of
the prism. The height of the prism is the distance between the
base and the face opposite the base.
rectangle
rectangle
Allow students 10 to 15 minutes to complete the journal pages.
Then bring the class together to discuss students’ results.
Oblique rectangular prism
Draw a rectangular prism on the board and label the base and
height, as shown.
Student Page
Date
height
Time
LESSON
11 5
Cube-Stacking Problems
Each picture at the bottom of this page and on the next page shows a box that is partially
filled with cubes. The cubes in each box are the same size. Each box has at least one
stack of cubes that goes to the top.
Your task is to find the total number of cubes needed to completely fill each box.
Record your answers in the table below.
Table of Volumes
Placement of Cubes
Box 1
Box 2
Box 3
Box 4
Box 5
Box 6
Number of cubes
needed to cover
the bottom
32
40
24
16
35
25
Number of cubes
in the tallest stack
(Be sure to count
the bottom cube.)
5
7
4
5
6
5
Total number of
cubes needed to
fill the box
160 280 96
Box 1
base
138
Ask students to look for a pattern in the table on journal page 300.
To find the total number of cubes needed to fill each box, multiply
the number of cubes needed to cover the bottom of the box by the
number of cubes in the tallest stack.
80 210 125
Box 2
300
Math Journal 2, p. 300
874
Unit 11 3-D Shapes, Weight, Volume, and Capacity
Student Page
Date
Then call students’ attention to the following relationships:
Time
LESSON
11 5
The number of cubes needed to cover the bottom of the box
is the same as the number of squares needed to cover the
base—that is, the area of the base of the box.
Cube-Stacking Problems
The number of cubes in the tallest stack is the same as the
height of the box.
continued
138
Box 4
Box 3
Therefore, you can find the volume of a rectangular prism
by multiplying the area of a base by the height of the prism.
Volume of a rectangular prism area of base height
Written with variables, this becomes V B h
Box 5
where V is the volume of the rectangular prism, B is the area
of the base, and h is the height of the prism.
Box 6
Formula for the volume of a rectangular prism:
VBh
B is the area of a base.
Have students record the formula at the bottom of journal
page 301.
height
h is the height from that base.
Volume units are cubic units.
base
301
Links to the Future
Math Journal 2, p. 301
Use of a formula to calculate the volume of a prism is a Grade 5 Goal.
Finding Volume
PARTNER
ACTIVITY
(Math Journal 2, p. 302)
Students find the volume of stacks of centimeter cubes and
calculate the volume of rectangular prisms. Ask students to
explain the strategies they used to solve the problems.
For Example:
Problem 2: A portion of the top layer is missing. Calculate
the volume of a completed rectangular prism and subtract the
missing blocks. Alternatively, determine the volume of the
complete prism as shown and add the partial layer of cubes.
Problem 3: There is one complete layer of cubes with three
identical stacks on top. Add the volume of the bottom layer to
the volume of the three stacks to determine the total volume.
NOTE Traditionally, lowercase letters are
used in formulas to represent length, and
uppercase letters are used to represent area
or volume. For example, b stands for the
length of the base of a polygon, and B stands
for the area of the base of a geometric solid.
Student Page
Date
Time
LESSON
11 5
Cube-Stacking Problems
ELL
K I N E S T H E T I C
T A C T I L E
V I S U A L
2.
45
cm3
3.
Volume 25
cm3
Volume 26
cm3
4.
Have students use centimeter cubes to build the stacks and rectangular
prisms on journal page 302.
A U D I T O R Y
138
1.
Volume Adjusting the Activity
continued
Find the volume of each stack of centimeter cubes.
Volume 40
cm3
5. Choose one of the problems from above. Describe the strategy that you used
to find the volume of the stack of centimeter cubes.
Sample answer: For Problem 4, I found the volume of the
tall part, which is 4 cm2 6 cm 24 cm3, and then added
the volume of the two extra squares to get 26 cm3.
Try This
6.
7.
2 cm
3 cm
10 cm
3 cm
8 cm
2 cm
Number model:
Volume (2 3) 3 18
18
cm3
Number model:
Volume (8 10) 2 160
160
cm3
302
Math Journal 2, p. 302
Lesson 11 5
875
Student Page
Date
Time
LESSON
Math Boxes
11 5
1. What is the total number of cubes needed
Ongoing Assessment:
Recognizing Student Achievement
2. Calculate the volume.
to completely fill the box?
125
35 in.
cubes
3
,
6
to come up?
4
,
6
b. a factor of 20
to come up?
or
1
2
or
2
3
in3
4
1 ft
yd
6 yd 6
972 in. 27 yd
3 mi
15,840 ft 24,640 yd 14 mi
a. 13 ft b. 18 ft 6 in. c.
d.
e.
81
26
77
54 28 b.
62 (15) $38
$114
1 dozen shirts? $228
75 shirts? $1,425
a. 2 shirts?
2 Ongoing Learning & Practice
b. 6 shirts?
88 51 (139)
$63.89 $23.56 $87.45
$71.08 ($85.79) $14.71
e.
[Measurement and Reference Frames Goal 2]
6. If 4 shirts cost $76, what is the cost of
a.
d.
in.
129
5. Add.
c.
138
4. Complete.
fraction of the time would you expect
a. a multiple of 2
(25 35) 35
30,625
30,625
Volume 138
3. When you roll a 6-sided die, about what
Use journal page 302, Problems 1, 2, and 5 to assess students’ ability to find
the volume of stacks of centimeter cubes. Students are making adequate
progress if they are able to build the stacks with actual centimeter cubes or look
at the pictures to find the volume and then describe the strategy used. Some
students may be able to solve Problems 3 and 4, which involve more difficult
arrangements of cubes, and Problems 6 and 7, which involve representations
of rectangular prisms that do not show individual cubes.
35 in.
25 in.
Number model:
Journal page 302
Problems
1, 2, and 5
c.
d.
47
299
Math Journal 2, p. 299
Playing Chances Are
PARTNER
ACTIVITY
(Student Reference Book, pp. 236 and 237;
Math Masters, pp. 462–466)
Students play Chances Are to practice using probability
language to describe the likelihood of an event. See
Lesson 7-11 for additional information.
Math Boxes 11 5
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 299)
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 11-7. The skill in Problem 6
previews Unit 12 content.
Study Link Master
Name
Date
STUDY LINK
Volume
11 5
1.
137 138
Find the volume of each stack of centimeter cubes.
a.
b.
39
Volume 2.
Time
30
Volume cm3
cm3
Calculate the volume of each rectangular prism.
a.
b.
Writing/Reasoning Have students write a response to the
following: Use probability terms to describe the likelihood
of each of the events in Problem 3. Explain your choice of
language. Sample answer: There is a 50-50 chance of rolling a
multiple of 2 because 2, 4, and 6 are multiples of 2. There are 3
favorable outcomes out of 6 possible outcomes. It is likely that a
factor of 20 will come up. 1, 2, 4, and 5 are factors of 20. There
are 4 favorable outcomes out of 6 possible outcomes.
2 cm
6 cm
5 cm
9.7 cm
3 cm
3 cm
Number model:
3.
(3 º 3) º 6 54
54
Volume (2 º 5) º 9.7 97
97
cm3
What is the total number of cubes needed to completely fill each box?
a.
Practice
29
150
cubes
49
5.
40
84 (55)
7.
16 89 65 16 6.
Study Link 11 5
(Math Masters, p. 331)
cubes
21 (19)
73
Math Masters, p. 331
876
INDEPENDENT
ACTIVITY
b.
150
4.
Number model:
Volume cm3
Unit 11 3-D Shapes, Weight, Volume, and Capacity
Home Connection Students find the volume of stacks
of centimeter cubes, calculate the volume of rectangular
prisms, and determine the number of cubes that are
needed to fill boxes.
Teaching Master
Name
3 Differentiation Options
Date
LESSON
Hidden Cubes
11 5
1.
READINESS
Solving Spatial-Visualization Puzzles
PARTNER
ACTIVITY
To explore the representation of 3-dimensional figures with
2-dimensional drawings, have students use interlocking cubes
to build cube stacks and solve spatial-visualization problems.
The stacks of cubes shown below are called soma cubes and were first designed
in 1936 by Piet Hein, a Danish poet and scientist.
Use interlocking cubes to build the stacks shown below. Use a small stick-on note to
label each stack with the appropriate letter. Then record the number of cubes needed
to build each stack.
A
E
(Math Masters, p. 332)
3
4
INDEPENDENT
ACTIVITY
Estimating the Volume
cubes
B
cubes F
4
4
cubes
cubes
C
G
4
4
cubes
4
D
cubes
cubes
Use the cube stacks that you made above to build each of the figures below.
The figures do not have any hidden holes. Record the number of cubes needed
to build each figure and the cube stacks that you used.
8
2.
cubes
I used the following cube stacks to build the figure:
E and F
12 cubes
3.
ENRICHMENT
Time
I used the following cube stacks to build the figure:
C, G, E or C, G, F
Try This
5–15 Min
of a Sheet of Paper
4.
27 cubes
I used the following cube stacks to build the figure:
(Math Masters, p. 388 or 389)
To apply students’ understanding of volume, have them
estimate the volume of a sheet of notebook paper. In a
Math Log or on an Exit Slip, have students write a brief
report describing their strategy. One possible strategy is
given below.
All of them, A–G
Math Masters, p. 332
1. Cut the sheet of paper into 1-inch squares.
2. Stack the squares into a neat pile. Measure the height of the
1
pile of squares. About 4 -inch high The area of the base of the
pile is 1 square inch.
3. Use the formula:
VBh
1
V 1 4
1
V 4
4. The volume of a sheet of notebook paper is about
0.25 cubic inches.
1-inch squares cut from
a single sheet of paper
1-inch cube
Lesson 11 5
877