1. No category

Dear Family,
Content
Overview
In this unit, we are reviewing the volume of rectangular prisms
with whole number edge lengths, and analyzing the difference
between surface area and volume along with the units used to
measure them.
The surface area of a solid figure is the total area of all its faces.
Volume is the measure of the space that a three-dimensional figure
occupies. The solid figure below is made of centimeter cubes.
l
l is the length.
w is the width.
h is the height.
B is the area of the base.
h
w
B
Surface Area
SA = (2 × 10) + (2 × 2) +
(2 × 5) = 34
Surface Area = 34 cm2
Volume
V = lwh or V = Bh
V = 5 × 2 × 1 or V = 10 × 1
Volume = 10 cm3
Surface area is measured in square units.
Volume is measured in cubic units.
This unit also introduces the volume of rectangular prisms with
fractional edge lengths.
1
V = Bh = 12 • 2__
= 30,
2
Volume = 30 unit3
3
1
1
V = lwh = 6__
• 2__
• 3 = 48__
,
4
2
2
3
Volume = 48__
unit3
4
If you need practice materials or if you have any questions, please
call or write to me.
Sincerely,
Your child’s teacher
Unit 6 addresses the following standards from the Common Core State Standards for Mathematics with
California Additions: 6.G.1, 6.G.2, 6.G.4, 6.EE.2, 6.EE.2c, and all Mathematical Practices.
UNIT 6 LESSON 1
What Is Volume?
147
Estimada familia:
Un vistazo
general al
contenido
En esta unidad, repasaremos cómo obtener el volumen
de prismas rectangulares cuyos lados tienen longitudes
expresadas en números enteros. También analizaremos la
diferencia entre el área total y el volumen y examinaremos
las unidades de medida usadas.
El área total de un cuerpo geométrico es la suma del área de
todas sus caras. El volumen es la medida del espacio que ocupa
una figura tridimensional.
l
h
a
Ab
Área total
At = (2 × 10) + (2 × 2) +
(2 × 5) = 34
Área total = 34 cm2
l es el largo
a es el ancho
h es la altura
Ab es el área de la base
Volumen
V = lah ó V = Abh
V = 5 × 2 × 1 ó V = 10 × 1
Volumen = 10 cm3
El área total se mide en unidades cuadradas.
El volumen se mide en unidades cúbicas.
En esta unidad también se presenta el volumen de prismas
rectangulares cuyos lados tienen longitudes expresadas en
fracciones.
1
V = Abh = 12 • 2__
= 30,
2
Volumen = 30 unidades3
3
1
1
V = lah = 6__
• 2__
• 3 = 48__
,
4
2
2
3
Volumen = 48__
unidades3
4
Si necesita material para practicar o si tiene preguntas, por favor
comuníquese conmigo.
Atentamente,
El maestro de su hijo
En la Unidad 6 se aplican los siguientes estándares auxiliares, contenidos en los Estándares estatales comunes de
matemáticas con adiciones para California: 6.G.1, 6.G.2, 6.G.4, 6.EE.2, 6.EE.2c, y todos los de prácticas matemáticas.
148
UNIT 6 LESSON 1
What Is Volume?
6–1
► Nets for Cubic Units and a Rectangular Prism
Cut out the nets and form the solid figure.
Unit Cube
1 unit
1 unit
1 unit
1 in.
1 in.
1 in.
1 cm
1 cm 1 cm
UNIT 6 LESSON 1
What Is Volume?
149
6–1
► Nets for Cubic Units and a Rectangular Prism (continued)
150
UNIT 6 LESSON 1
What Is Volume?
6–1
Content Standards 6.G.1, 6.G.4, 6.EE.2, 6.EE.2c
Mathematical Practices MP.3, MP.6, MP.7
► Cubic Units
The volume of a solid figure is the amount of space
occupied by the figure. Volume is measured in cubic units.
Vocabulary
volume
unit cube
centimeter cube
inch cube
cubic unit (unit3)
cubic centimeter (cm3)
cubic inch (in.3)
1. How can you measure the amount of space each of
these rectangular prisms takes up? How much space is
inside each of the rectangular prisms? How many unit
cubes does it take to fill the rectangular prism?
A unit cube is a cube with each edge 1 unit long. The
volume of a unit cube is 1 cubic unit. It can be written
1 cubic unit or 1 unit3.
V=
1 unit3
1 unit
1 unit
1 unit
2. Label the length, width, and height of the
centimeter cube on the right. Write the volume
of a centimeter cube in two ways.
Write the volume of the cube in two ways.
3. inch cube
UNIT 6 LESSON 1
4. meter cube
5. foot cube
6. yard cube
What Is Volume?
151
152
UNIT 6 LESSON 1
What Is Volume?
6–2
► Nets for Part of a Unit Cube
Cut out the nets and form the solid figures.
A
C
B
UNIT 6 LESSON 2
Fractional Unit Cubes
153
► Nets for Part of a Unit Cube (continued)
154
UNIT 6 LESSON 2
Fractional Unit Cubes
6–3
► Prism Layers
Cut out the nets and form the solid figures.
UNIT 6 LESSON 3
Compose Rectangular Prisms with Fractional Edge Lengths
155
► Prism Layers (continued)
156
UNIT 6 LESSON 3
Compose Rectangular Prisms with Fractional Edge Lengths
Unit 6
1. Select True or False for each statement.
1a. Surface area and volume both measure the
amount of space occupied by a solid figure.
True
False
1b. Surface area and volume are measured
using the same unit.
True
False
1c. Surface area and volume are both
attributes of solid figures.
True
False
2. Fill in the bubble next to the unit used to measure volume.
square units
cubic units
linear units
Explain your answer choice in words or with a labeled sketch.
3. The volume of a rectangular pyramid can be found
using the formula V = Bh.
Part A: Use the tiles to show a different formula you can use to find the
volume of a rectangular prism. You will not need to use every tile.
w
h
B
l
V=
Part B: Explain how the two formulas are related, and why you
can use either formula to find the volume of a rectangular prism.
4. The equation for the volume of the square prism
is V = Mt. Label the prism.
UNIT 6 TEST
157
Unit 6
5. Choose one value from each column to make
the volume of the rectangular prism true.
V = 24 ft3
2 ft
w
3
5
9
10
l
l
1
2
3
4
w
6. Use mental math to find the volume.
V=
10 in.
5 in.
5 in.
7. The volume of a prism can be found
by packing it with unit cubes of the
appropriate edge lengths.
Part A
1 ft
5 ft
4
1
ft
4
What is the longest edge length that would be
appropriate for the unit cubes? Explain your answer.
edge lengths
Part B
If the longest dimension of the prism was 1_1 feet, would the
8
length you chose in Part A be sensible? Explain.
158
UNIT 6 TEST
Unit 6
8. Cecilia used layers to find the volume of the prism below.
Part A Explain how to find the volume using layers.
Part B Use layers to find the volume of the prism.
1
4 2 units
1
6 2 units
4 units
V=
cubic units
9. Find the volume. Show your work.
3
ft
4
2
ft
3
1
ft
2
V=
10. The dimensions on the tiles below are values for l and h.
1
in., identify those expressions that
For a prism with w = 2__
2
will produce a volume of 270 in.3 and those that will produce
a volume of 300 in.3
9 in. • 12 in.
1
2__
in. • 48 in.
2
1
16 in. • 7__
in.
2
15 in. • 8 in.
Volume: 270 in.3
UNIT 6 TEST
1
24 in. • 4__
in.
2
1
in.
2
2
6 in. • 18 in.
20 in. •
2
5__
in.
5
h
l
Volume: 300 in.3
159
Unit 6
11. Choose one value from each column to make
the volume of the rectangular prism true.
w
3
V = 24 4 ft3
h
1
1__
4
3
3
1__
4
4
2
5
1
2__
4
6
h
2
3
ft
4
w
Select Yes or No to indicate if the equation represents
the volume for the prism with sides e.
12a. V = e • e • e
Yes
No
12b. V = 3e
Yes
No
12c. V = e3
Yes
No
13. Substitute the dimensions shown below into the volume
formula to find the volume of the prism.
Show your work.
1
yd
3
3 yd
1
4 yd
2
V=
160
UNIT 6 TEST
e
e
e
Unit 6
14. Anthony labeled this rectangular prism.
Part A
t
Select the equations that represent the volume
of the prism. Mark all that apply.
A
V= b+ b+ t
C
V = b2t
B
V= b• b•t
D
V = b2 + t
b
b
Part B
Show your work. Find the volume of the prism
if b = 3 cm and t = 7.5 cm.
V=
15. Write an equation for the volume of the
prism using the given variables.
W
V=
c
16. A gift box has a volume of 199 in.3 The area of the base of
3
in.2 What is the height?
the box is 49__
4
h=
1
17. A small pool in the shape of a rectangular prism has a length of 6__
ft,
2
a width of 5 ft and a height of 24 in. Jeb says the volume of the pool
is 780 ft3.
Is Jeb correct? If he is, show how he may have found the volume.
If not, explain what he did wrong and give the correct volume.
V=
UNIT 6 TEST
161
Unit 6
18. Choose Yes or No to indicate if the given
dimensions represent a rectangular prism
with a volume of 96 cubic inches.
18a. 2 in. • 6 in. • 8 in.
Yes
No
18b. 2 in. • 4 in. • 12 in.
Yes
No
18c. 4 in. • 12 in. • 4 in.
Yes
No
18d. 3 in. • 4 in. • 9 in.
Yes
No
18e. 4 in. • 4 in. • 6 in.
Yes
No
19. What is the volume of a rectangular prism
with a base area of 52 square inches and a
height of 14 inches? Show your work.
V=
20. The volume of a rectangular prism is 6,037.5 cubic inches.
The area of the base of the prism is 525 square inches.
What is the height of the prism? Show your work.
h=
21. John wants to place a layer of mulch on top of his
12 ft by 24 ft vegetable garden. If a 1.5 ft3
bag of mulch costs $4, how much will it cost
to place a 3-inch layer of mulch on the garden?
Show your work.
$
162
UNIT 6 TEST