Table of COntents
Introduction – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 3
Format of Books – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 4
Suggestions for Use – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 7
Annotated Answer Key and Extension Activities – – – – – – – – – – – 9
Reproducible Tool Set – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – 183
ISBN 978-0-8454-8875-1
Copyright © 2016 The Continental Press, Inc.
Excepting the designated reproducible blackline masters, no part of this publication may be reproduced
in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without
the prior written permission of the publisher. All rights reserved. Printed in the United States of America.
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34 Volume of Irregular Figures
Pages
274 and 275
Objective
To find the volume of figures made up of more than
one rectangular prism
PAR
T
1 Introduction
Introduction
Tell students that they can use the volume formula for
a rectangular prism to find the volume of figures made
up of more than one rectangular prism. The volume
for the entire figure is the sum of the volumes of the
rectangular prisms that make it up. Work through the
sample problem as a class. Find the volume of each
part of the building and then add to find the total
volume of the building.
Think About It
Students should recognize how they can use the given
information. If they know the volume of the irregular
figure and the volume of one part of it, they can
subtract the volume of the part from the volume of
the whole to find the volume of the other part.
V 5 l 3 w 3 h or V 5 B 3 h
1,000 cubic inches
6
6 3 1,000
6,000 in. 3
Indiana Academic Standard
2
4 yards
1 yard
3 yards
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FL Ind Math ATE G5_U6.indd 157
5.M.6 Find volumes of solid figures composed of
two non-overlapping right rectangular prisms by
adding the volumes of the non-overlapping parts,
applying this technique to solve real-world problems
and other mathematical problems.
UNIT 6
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LESSON
34 Volume of Irregular Figures
PAGES
276 AND 277
PAR
T
2 Focused
Instruction
Focused
Instruction
Students will work through two problems to find
the volumes of irregular figures. The first figure is
composed of 6 congruent cubes. Students will find
the volume of one cube and then multiply it by 6 to
find the total volume. The second figure is made up
of two rectangular prisms. Students must identify the
dimensions of the prisms and use the volume formula
to find the volume of each prism. They can then find
the volume of the entire figure by adding.
4 square yards
12 cubic yards
1 yard
3 yards
1 yard
3 cubic yards
Add them together.
15 cubic yards
Conclude the Focused Instruction section by having
students find the volumes of two irregular figures.
270 cu cm
2,250 cu ft
PAR
T
Practice
Guided
Practice
3 Guided
Students should complete the Guided Practice section
on their own. Offer assistance as needed, pointing out
the reminder and hint boxes along the right side of
the page.
Connections to Process Standards
for Mathematics
• Make sense of problems and persevere in
solving them.
• Model with mathematics.
• Use appropriate tools strategically.
• Attend to precision.
• Look for and make use of structure.
158
UNIT 6
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66
The volume of each part is found by V 5 l 3 w 3 h. The main
part of the house is 40 2 18 5 22 feet long, so V 5 22 3 15 3
30 5 9,900 ft 3. The volume of the garage is V 5 18 3 15 3
10 5 2,700 ft 3. The total volume of the house is 9,900 1
2,700 5 12,600 ft 3.
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PAGES
278 AND 279
LESSON
34 Volume of Irregular Figures
PAR
T
Practice
I ndependent
Practice
4 Independent
Answer Rationales
DOK 2
5.M.6
No, the volume of the studio is 12 3 12 3 12 5 1,728 cubic feet,
and the tool room is 6 3 6 3 9 5 324 cubic feet. The total
volume is 1,728 1 324 5 2,052 cubic feet, which is less than
the required 2,100 cubic feet.
DOK 3
5.M.6
The larger box has sides of 8 inches, and the smaller box has
sides of 6 inches.
The volume of the larger part is 512 cubic inches, because the
sides of the cube are each 8 inches. The volume of the smaller
part is 216 cubic inches, because the sides of the cube are each
6 inches. The volume of the box is 512 1 216 5 728 cubic inches.
DOK 2
5.M.6
30
DOK 2
5.M.6
8,200
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1
Find the volume of the studio and of the tool room
to determine if the combined volume is at least
2,100 cubic feet. The volume of the studio is 12 3
12 3 12 5 1,728 cubic feet. The volume of the tool
room is 6 3 6 3 9 5 324 cubic feet. The sum of
the two volumes is 1,728 1 324 5 2,052 cubic feet,
which is less than 2,100 cubic feet.
2
Part A To find the dimensions of the larger cube,
recognize that the sides of a cube are all equal, so
the dimensions are 8 3 8 3 8. The area of the base
of the smaller cube is 36 square inches. The base is
a square, so the sides of the square are the same.
The sides of the square are 6 inches since 6 3 6 5
36. A cube has dimensions that are all the same, so
the dimensions of the smaller cube are 6 3 6 3 6.
Part B The volume of the prop can be found by
finding the volume of each cube and then finding
the sum. The volume of the larger cube is V 5 l 3
w 3 h, so 8 3 8 3 8 5 512 cubic inches. The
volume of the smaller cube is V 5 l 3 w 3 h, so
6 3 6 3 6 5 216 cubic inches. The sum of the
volumes is 512 1 216 5 728 cubic inches.
3
The bottom tier of the pyramid has dimensions of
4 3 4 3 1. Each tier has a length and width that
is 1 inch smaller than the tier below it, so the next
tier is 3 3 3 3 1. The third tier is 2 3 2 3 1, and
the fourth tier is 1 3 1 3 1. Find the volume by
multiplying the dimensions of each tier and adding
the products: 4 3 4 3 1 5 16; 3 3 3 3 1 5 9;
2 3 2 3 1 5 4; 1 3 1 3 1 5 1. The sum of the
volumes is 16 1 9 1 4 1 1 5 30 cubic inches.
4
To find the volume of the birdhouse, find the
volume of each part of the birdhouse. The bottom
portion of the birdhouse has one dimension in
yards, so convert the dimension to inches: 1 yard 5
3 feet; 1 foot 5 12 inches; so 1 yard 5 36 inches.
Multiply the dimensions of the bottom part of the
birdhouse: 36 3 10 3 20 5 7,200 cubic inches. The
volume of the top part of the birdhouse is 10 3
10 3 10 5 1,000 cubic inches. The total volume of
the birdhouse is 7,200 1 1,000 5 8,200 cubic inches.
UNIT 6
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LESSON
34 Volume of Irregular Figures
PAGES
280 AND 281
5
Part A To find the volume of the steps, think of
each tier as an individual prism. The dimensions of
each prism are as follows: bottom prism 5 30 3
30 3 8; middle prism 5 20 3 30 3 8; top prism 5
10 3 30 3 8. To find the total volume, find the
volume of each tier and then add to find the sum of
the volumes.
Part B Find the volume of each prism and then
add the products to find the total volume.
Bottom prism 5 30 3 30 3 8 5 7,200 cubic inches
Middle prism 5 20 3 30 3 8 5 4,800 cubic inches
Top prism 5 10 3 30 3 8 5 2,400 cubic inches
The total volume of the stairs is 7,200 1 4,800 1
2,400 5 14,400 cubic inches.
6
A cube is a rectangular prism with edges that are all
the same length. The small cube has a side length
of 5 yards, so its volume is 5 3 5 3 5 5 125 cubic
yards. The side length of the large cube is 2 yards
longer than the small cube, so its side length is 2 1
5 5 7 yards. The volume of the large cube is 7 3
7 3 7 5 343 cubic yards. The total volume of the
cubes is 125 1 343 5 468 cubic yards.
DOK 2
5.M.6
I separated the steps into rectangular prisms. The bottom
prism is 30 in. by 30 in. by 8 in. The second prism is 20 in. by
30 in. by 8 in. The top prism is 10 in. by 30 in. by 8 in. I used the
volume formula V 5 l 3 w 3 h to find the volume of each prism.
30 3 30 3 8 5 7,200
20 3 30 3 8 5 4,800
10 3 30 3 8 5 2,400
7,200 1 4,800 1 2,400 5 14,400
14,400
DOK 2
5.M.6
468
Extension Activity
Have students bring in or provide boxes shaped
like rectangular prisms. Give each student a box
and then have the students get in groups of two or
three. The students should find the volume of their
box and then create an irregular figure with all the
boxes in the group. Finally, they should calculate the
volume of the entire figure. Have groups share their
figure and its volume with the class.
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UNIT 6
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