Project
Adding Volumes of
Solid Figures
Objectives To find the volumes of solid figures composed of
two
non-overlapping rectangular prisms.
t
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Practice
EM Facts
Workshop
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Family
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Doing the Project
Recommended Use During or after Unit 9
Assessment
Management
Common
Core State
Standards
Interactive
Teacher’s
Lesson Guide
Curriculum
Focal Points
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP8
Content Standards
5.MD.3a, 5.MD.3b, 5.MD.4, 5.MD.5a, 5.MD.5b, 5.MD.5c
Key Concepts and Skills
• Use unit cubes to calculate volume. [Measurement and Reference Frames Goal 2]
• Use a formula to calculate the volumes of rectangular prisms. [Measurement and Reference Frames Goal 2]
• Explore properties of solid figures. [Geometry Goal 2]
Materials
Math Journal 2, Activity Sheet 8
Math Masters, p. 410
transparent tape
per group of students: about 100 centimeter
cubes
• Write number sentences with variables to model volume problems. [Patterns, Functions, and Algebra Goal 2]
Key Activities
Students use centimeter cubes and formulas to explore the volumes of solid figures
composed of two rectangular prisms.
Key Vocabulary
unit cube
Extending the Project
Ex
Students find the volumes of rectangular prisms to find the total approximate volume
of Willis Tower.
Materials
Math Masters, p. 410A
Project 9
Adding Volumes of Solid Figures
990A
Student Page
Date
Time
1 Doing the Projects
TAB
Box A
TAB
Rectangular Prism Patterns
▶ Exploring the Volume
TAB
TAB
of Solid Figures
(Math Journal 2, Activity Sheet 8)
Provide each group of students with about 100 centimeter cubes.
In each group, students should display the two open boxes that
they constructed in Lesson 9-8 from Activity Sheet 8. If the
constructed boxes are no longer available, distribute copies of
Activity Sheet 8 as needed.
Box B
TAB
Have students tape together Boxes A and B to form one solid
figure. Specify that they should put the prisms together so that
two faces are together and edges line up where possible. Note that
students may tape the prisms together in different ways. The figure
below is one example.
TAB
TAB
SMALL-GROUP
ACTIVITY
TAB
Math Journal 2, Activity Sheet 8
AS5-IBC1__EMCS_S_MJ2_G5_AS_576434.indd AS11
3/8/11 12:38 PM
Ask: Suppose each box has a lid so that each is a rectangular
prism. Is the figure formed by the two rectangular prisms also a
rectangular prism? Explain. Sample answer: No. A rectangular
prism is formed by six faces that are rectangles. The new figure
that is formed has more than six rectangular faces. Mention that
a three-dimensional shape that is not of a specific type is simply
called a solid figure.
Ask: How could you use the centimeter cubes to find the volume
of the solid figure you formed? Sample answer: Fill each box with
centimeter cubes. Add or count the cubes used in all. What is the
volume of the solid figure? 69 cubic centimeters
990B Project 9
Adding Volumes of Solid Figures
Project Master
Ask students to attach the two prisms in different ways (such as
one above the other or one next to the other) and compare the
volume of each new solid that has been formed. Students should
recognize that the volume is the same no matter how the prisms
are attached. Ask a volunteer to explain why this is so. Sample
answer: The two prisms have the same volumes no matter how
they are put together, so the sum of the two volumes is the same.
Name
PROJECT
9
Date
Time
Building a Solid Figure to Find Volume
For Problems 1–3, do the following:
a. Use centimeter cubes to build each rectangular prism.
b. Find the volume of each rectangular prism.
c. Find the volume of the solid figure formed by the two rectangular prisms.
1.
Length
l
Width
w
Height
h
Volume V
(cubic units)
Rectangular Prism A
1
2
3
Rectangular Prism B
2
3
4
6
24
Solid Figure Formed
30
by Prisms A and B
Explain that in this case we can consider each dimension of a
centimeter cube as being 1 unit long. Mention that a cube with side
lengths of 1 unit is called a unit cube. Explain that in general,
when a solid figure can be packed without gaps and overlaps using
n unit cubes, the volume of the solid figure is n cubic units.
▶ Building a Solid Figure
to Find Volume
SMALL-GROUP
ACTIVITY
2.
Length
l
Width
w
Height
h
Volume V
(cubic units)
Rectangular Prism C
5
3
2
Rectangular Prism D
2
3
5
30
30
Solid Figure Formed
60
by Prisms C and D
3.
Length
l
Width
w
Height
h
Volume V
(cubic units)
Rectangular Prism E
12
3
1
Rectangular Prism F
3
3
3
36
27
Solid Figure Formed
63
by Prisms E and F
4.
Explain how to find the volume of a solid figure made from two rectangular
prisms, one with dimensions 3 cm by 4 cm by 5 cm and one with dimensions
2 cm by 5 cm by 4 cm.
Sample answer: I would use the formula V = l ∗ w ∗ h
to find the volume of each rectangular prism and then add
the volumes. So, the total volume is 3 ∗ 4 ∗ 5 + 2 ∗ 5 ∗ 4 =
60 + 40, or 100 cm3.
(Math Masters, p. 410)
Have students use the centimeter cubes to build the two
rectangular prisms in each of Problems 1–3 on Math Masters,
page 410. Students find the volume of each prism and then find
the volume of the solid figure formed by the two prisms. Encourage
students to use a formula (either V = B ∗ h or V = l ∗ w ∗ h) to
find the volumes of the rectangular prisms. Students should use
the cubes to check their results.
Math Masters, p. 410
410-410A_EMCS_B_MM_G5_Proj_576973.indd 410
3/30/11 1:33 PM
When students complete Problem 4, they should conclude that you
can find the volume of a solid figure formed by two rectangular
prisms by adding the two volumes. So, finding the volume of a
solid figure that includes more than one rectangular prism is an
additive process. You may want to explain that if a solid figure has
overlapping parts, as in the solid figure below, you add the volumes
of the non-overlapping parts to the volumes of the parts that overlap.
2 Extending the Project
▶ Finding the Volume of Willis
INDEPENDENT
ACTIVITY
Tower in Chicago
(Math Masters, p. 410A)
Mention that at 1,450 feet tall (excluding antennas), Willis Tower
in Chicago (formerly named Sears Tower) is the tallest building
in the United States. When it was built in 1973, it was the tallest
building in the world.
Project 9 990C
Project Master
Name
Date
PROJECT
The building consists of nine square “tubes” constructed in a 3-by-3
arrangement. Each tube measures 75 feet on a side. Ask students to
explain how to find the dimensions of the entire base of Willis Tower.
Sample answer: There are three tubes on a side, and each is 75 feet
long. So each side is 225 feet long, making the dimensions of the
base 225 ft by 225 ft. Ask: What is the area of the base? 225 ∗ 225,
or 50,625 ft2
Time
Finding the Volume of Willis Tower
9
At 1,450 feet tall, Willis Tower in Chicago is the tallest
building in the United States. It is composed of nine
rectangular prisms known as “tubes.” The tubes are built
in a 3-by-3 arrangement. Although the tubes are of various
heights, each one has a square base that measures
75 feet on a side.
1.
What is the area of the base of each tube?
5,625
ft2
The table below shows the approximate heights of the
tubes. Only two of them reach all the way to the top.
2.
What formula could you use to find the volume
of one tube?
3.
4.
Sample answer:
V=B∗h
Willis Tower
Complete the table to find the volume of one tube at each given height. Then
find the total volume of the tubes for each height.
Approximate
Height of
Tube
Number of
Tubes at
this Height
Volume of One
Tube at this
Height (ft3)
Total Volume of
Tubes at this Height
(ft3)
646 ft
3
672 ft
2
1,200 ft
2
1,450 ft
2
3,633,750
3,780,000
6,750,000
8,156,250
10,901,250
7,560,000
13,500,000
16,312,500
Describe what you will do to find the total approximate volume of Willis Tower.
Sample answer: I will find the sum of the total volumes of
the tubes for each height.
5.
The total volume of Willis Tower is about
48,273,750
ft3.
Math Masters, p. 410A
410-410A_EMCS_B_MM_G5_Proj_576973.indd 410A
990D Project 9
3/30/11 1:33 PM
Adding Volumes of Solid Figures
Explain that the nine tubes are of various heights, with some being
the same height. For example, although all nine tubes extend up
through the 49th story, only two tubes extend to the full height
of 110 stories. So to find the volume of Willis Tower, you need to
find the volumes of the different tubes and then add. Of course,
multiplication could be used to find the total volume of tubes with
the same dimensions.
Math Masters, page 410A guides students in finding the
approximate volume of Willis Tower.
Name
PROJECT
9
Date
Time
Building a Solid Figure to Find Volume
For Problems 1–3, do the following:
a. Use centimeter cubes to build each rectangular prism.
b. Find the volume of each rectangular prism.
c. Find the volume of the solid figure formed by the two rectangular prisms.
1.
Length
l
Width
w
Height
h
Rectangular Prism A
1
2
3
Rectangular Prism B
2
3
4
Length
l
Width
w
Height
h
Rectangular Prism C
5
3
2
Rectangular Prism D
2
3
5
Length
l
Width
w
Height
h
Rectangular Prism E
12
3
1
Rectangular Prism F
3
3
3
Volume V
(cubic units)
Solid Figure Formed
by Prisms A and B
2.
Volume V
(cubic units)
Solid Figure Formed
by Prisms C and D
3.
Volume V
(cubic units)
by Prisms E and F
4.
Explain how to find the volume of a solid figure made from two rectangular
prisms, one with dimensions 3 cm by 4 cm by 5 cm and one with dimensions
2 cm by 5 cm by 4 cm.
410
Copyright © Wright Group/McGraw-Hill
Solid Figure Formed
Name
Date
PROJECT
9
Time
Finding the Volume of Willis Tower
At 1,450 feet tall, Willis Tower in Chicago is the tallest
building in the United States. It is composed of nine
rectangular prisms known as “tubes.” The tubes are built
in a 3-by-3 arrangement. Although the tubes are of various
heights, each one has a square base that measures
75 feet on a side.
1.
What is the area of the base of each tube?
ft2
The table below shows the approximate heights of the
tubes. Only two of them reach all the way to the top.
2.
What formula could you use to find the volume
of one tube?
Copyright © Wright Group/McGraw-Hill
3.
Willis Tower
Complete the table to find the volume of one tube at each given height. Then
find the total volume of the tubes for each height.
Approximate
Height of
Tube
Number of
Tubes at
this Height
646 ft
3
672 ft
2
1,200 ft
2
1,450 ft
2
Volume of One
Tube at this
Height (ft3)
Total Volume of
Tubes at this Height
(ft3)
4.
Describe what you will do to find the total approximate volume of Willis Tower.
5.
The total volume of Willis Tower is about
ft3.
410A