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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 12
Lesson 12: From Unit Cubes to the Formulas for Volume
Student Outcomes
Students extend the volume formula for a right rectangular prism to the formula π = Area of base β height.
They understand that any face can be the base.
Lesson Notes
This lesson is a continuation of the ideas in Lesson 11 and the lessons in Module 5 of Grade 5 (Topics A and B).
The word face, though referenced in the last lesson, should be taught to students who may not know this meaning of it.
A student-friendly definition and i l lustration can be posted on the wall (along with definitions of edge(s) and vertex/
vertices). Here is a l ink to a useful i l lustration: http://www.11plusforparents .co.uk/Maths/shape8.html .
Classwork
Example 1 (10 minutes)
Look at the rectangular prisms in the first example. Write a numerical expression for the volume of each
rectangular prism.
Answers provided below.
What do these expressions have in common?
They have the same dimensions for the lengths and widths.
What do these dimensions represent?
The area of the bases of the rectangular prisms.
Rewrite each of the numerical expressions to show what they have in common.
Answers provided below.
If we know volume for a rectangular prism as length times width times height, what is another formula for
volume that we could use based on these examples?
Area of the base times the height.
What is the area of the base of each of the rectangular prisms?
π΄ = π π€; π΄ = (15 ππ. ) (112
ππ. ); π΄ = 2212
in2
How would we use the area of the base to determine the volumes? (Think about the unit cubes we have been
using. The area of the base would be the first layer of unit cubes. What would the height represent?)
We would multiply the area of the base times the height. The height would represent how many layers
of cubes it would take to fill up the rectangular prism. Sample answers are below.
How do the volumes of the first and second rectangular prisms compare? The first and third?
The volume of the second prism is twice that of the first because the height is doubled. The volume of
the third prism is three times that of the first because the height is tripled.
Scaffolding:
You may want to use unit
cubes to help students visualize
the layers in this problem.
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MP.8
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 12
Example 1
a. Write a numerical expression for the volume of each of the rectangular prisms above.
(ππ ππ. )(πππ
ππ. )(π ππ.) (ππ ππ. ) (πππ
ππ. ) (π ππ.) (ππ ππ. ) (πππ
ππ. ) (π ππ.)
b. What do all of these expressions have in common? What do they represent?
All of the expressions have (ππ ππ. )(πππ
ππ. ). This is the area of the base.
c. Rewrite the numerical expressions to show what they have in common.
(ππππ
πππ) (π ππ. ) (ππππ
πππ) (π ππ.) (ππππ
πππ) (π ππ.)
d. If we know volume for a rectangular prism as length times width times height, what is another formula for
volume that we could use based on these examples?
(Area of the base)(height), or area of the base times height.
e. What is the area of the base for all of the rectangular prisms?
(ππ ππ. ) (ππ
π ππ. ) = ππ
π
π πππ
f. Determine the volume of each rectangular prism using either method.
(ππ ππ. ) (πππ
ππ. ) (π ππ. ) = ππππ
πππ or (ππππ
πππ) (π ππ. ) = ππππ
πππ
(ππ ππ. ) (πππ
ππ. ) (π ππ. ) = πππ πππ or (ππππ
πππ) (π ππ. ) = πππ πππ
(ππ ππ. ) (πππ
ππ. ) (π ππ. ) = πππππ
πππ or (ππππ
πππ) (π ππ. ) = πππππ
πππ
g. How do the volumes of the first and second rectangular prisms compare? The volumes of the first and third?
πππ πππ = ππππ
πππ Γ π ππ.; πππππ
πππ = ππππ
πππ Γ π ππ.
The volume of the second prism is twice that of the first because the height is doubled. The volume of the
third prism is three times as much as the first because the height is triple the first prismβs height.
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 12
What do you think would happen to the volume if we turn this prism on its side so
that a different face is the base? (Have students calculate the area of the base
times the height for this new prism. To help students visualize what is happening
with this rotation, you could use a textbook or a stack of index cards and discuss
how this prism is similar and/or different to the rectangular prisms in part (a).)
Answers will vary. Some students may see that the volume will be the
same no matter which face is the base.
Area of the base = (3 ππ. ) (11
2 ππ. )
Area of the base = 4.5 ππ2
Volume =Area of the base Γ height
Volume = (412
ππ2) (15 ππ. )
Volume = 6712
ππ3
How does this volume compare with the volume you calculated using the other
face as the base?
The volumes in both solutions are the same.
What other expressions could we use to determine the volume of the prism?
Answers will vary. Some possible variations are included below.
15 in. Γ 11
2 in. Γ 3 in.
15 in. Γ 3 in. Γ 11
2 in.
3 in. Γ 15 in. Γ 112
in.
45 in2 Γ 112
in.
We notice that 3 in. Γ 15 in. Γ 11
2 in. and 45 in2 Γ 1
1
2 in. are equivalent and both represent the volume. How
do they communicate different information?
The first expression (3 in. Γ 15 in. Γ 112
in.) shows that the volume is the product of three edge lengths.
The second (45 in2 Γ 11
2 in.) shows that the volume is the product of the area of the base times the
height.
Example 2 (5 minutes)
Example 2
The base of a rectangular prism has an area of ππ
π in2. The height of the prism is π
π
π in. Determine the volume of the
rectangular prism.
π½ = π¨πππ ππ πππ ππππ Γ ππππππ
π½ = (ππ
π πππ) (π
π
π ππ. )
π½ = (ππ
π πππ) (
π
πππ. )
π½ =ππ
π πππ
ππ π’π§.
π π’π§. π
π
π π’π§.
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 12
Do we need to know the length and the width to find the volume of the rectangular prism?
No, if we know the area of the base, we do not need the length and width. The length and width is
used to calculate the area, and we are already given the area.
Exercises 1β6 (20 minutes)
The cards are printed out and used as stations or hung on the classroom walls so that students can move from question
to question. Copies of the questions can be found at the end of the lesson. Multiple copies of each question can be
printed so that a small number of students visit each question at a time. Students should spend about three minutes at
each station where they will show their work by first writing a numerical expression, and then use the expression to
calculate the volume of the rectangular prism described. They will use the rest of the time to discuss the answers , and
the teacher can answer any questions students have about the lesson.
1. Draw a sketch of the figure. Then calculate the
volume.
Rectangular Prism
Area of the base = 438
ft2
Height = 21
2 ft.
π = π΄πππ ππ πππ π Γ βπππβπ‘
π = (43
8ππ‘2) (2
1
2ππ‘)
π = (35
8 ππ‘2) (
5
2ππ‘)
π =175
16 ππ‘3
2. Draw a sketch of the figure. Write the length,
width, and height in feet. Then calculate the
volume.
Rectangular Prism
Length is 212
times as long as the height.
Width is 3
4 as long as the height.
Height = 3 ft.
πΏππππ‘β = 3 ππ‘ Γ 21
2=
15
2ππ‘
ππππ‘β = 3 ππ‘ Γ3
4 =
9
4ππ‘
π = π π€ β
π = (15
2ππ‘) (
9
4 ππ‘) (3 ππ‘)
π =405
8 ππ‘3
3. Write two different expressions to represent the
volume and explain what each one represents.
Answers will vary. Some possible solutions include
(42
3 π) (
1
3 π) (1
1
8 π) and (
14
9 π2) (1
1
8 π)
The first expression shows the volume as a product of
the three edge lengths. The second expression shows
the volume as a product of a base (423
π) (13
π)
times the height.
π
π
π π¦
ππ
π π¦
π
π π¦
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 12
4. Calculate the volume.
π = π΄πππ ππ πππ π Γ βπππβπ‘
π = (4
3ππ‘2) (
3
10ππ‘. )
π =12
30ππ‘3
π =2
5ππ‘3
5. Calculate the volume.
ππππ’ππ = π΄πππ ππ πππ π Γ βπππβπ‘
ππππ’ππ = (131
2 ππ2) (1
1
3 ππ. )
ππππ’ππ =108
6 ππ3
ππππ’ππ = 18 ππ3
6. Challenge!
Determine the volume of a rectangular prism
whose length and width are in a ratio of 3: 1. The
width and height are in a ratio of 2: 3. The length
of the rectangular prism is 5 ft.
πΏππππ‘β = 5 ππ‘.
ππππ‘β = 5 ππ‘.Γ· 3 =5
3ππ‘.
π»πππβπ‘ =5
3ππ‘.Γ
3
2=
5
2ππ‘.
π = π π€ β
π = (5 ππ‘. ) (5
3ππ‘. ) (
5
2 ππ‘. )
π =125
6 ππ‘3
ππ«ππ = πππ
π π’π§π
ππ
π π’π§.
ππ«ππ =π
π πππ
π
ππππ.
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 12
Extension (3 minutes)
Extension
A company is creating a rectangular prism that must have a volume of π ft3. The company also knows that the area of the
base must be ππ
π ft2. How can you use what you learned today about volume to determine the height of the rectangular
prism?
I know that the volume can be calculated by multiplying the area of the base times the height. So, if I needed the height
instead, I would do the opposite. I would divide the volume by the area of the base to determine the height.
π½ = π¨πππ ππ ππππ Γ ππππππ
π πππ = (ππ
π πππ) (π)
π πππ Γ· ππ
π πππ = ππππππ
ππ
π ππ. = ππππππ
Closing (2 minutes)
How is the formula π = π β π€ β β related to the formula π = Area of the base β height?
When we multiply the length and width of the rectangular prism, we are actually finding the area of the
base. Therefore, the two formulas both determine the volume of the rectangular prism.
Exit Ticket (5 minutes)
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 12
Name Date
Lesson 12: From Unit Cubes to the Formulas for Volume
Exit Ticket
1. Determine the volume of the rectangular prism in two different ways.
2. The area of the base of a rectangular prism is 12 cm2 and the height is 313
cm. Determine the volume of the
rectangular prism.
3
4ft.
3
4 ft.
3
8ft.
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 12
π
πππ.
π
π ππ.
π
πππ.
Exit Ticket Sample Solutions
1. Determine the volume of the rectangular prism.
π½ = π β π β π π½πππππ = π¨πππ ππ ππππ β ππππππ
π½ = (π
π ππ. ) (π
π ππ. )(π
π ππ. ) π½πππππ = ( π
ππ πππ) (π
π ππ. )
π½ =ππ
πππ πππ π½πππππ =
ππ
πππ πππ
2. The area of the base of a rectangular prism is ππ ππ¦π and the height is ππ
π ππ¦. Determine the volume of the
rectangular prism.
π½πππππ = π¨πππ ππ ππππ β ππππππ
π½πππππ = (ππ πππ ) (π π
π ππ)
π½πππππ =πππ
π πππ
π½πππππ = ππ πππ
Problem Set Sample Solutions
1. Determine the volume of the rectangular prism.
π½ = π π π
π½ = (ππ
ππ) (
π
ππ) (
π
ππ)
π½ =ππ
ππ ππ
2. The area of the base of a rectangular prism is ππ
π ft2 and the height is π
π
π ft. Determine the volume of the
rectangular prism.
π½πππππ = π¨πππ ππ ππππ Γ ππππππ
π½πππππ = (ππ
π πππ) (π
π
π ππ. )
π½πππππ = (ππ
π πππ) (
π
π ππ. )
π½πππππ =πππ
ππ πππ
π
π π¦
π
π π¦
ππ
π π¦
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 12
3. The length of a rectangular prism is ππ
π times as long as the width. The height is
π
π of the width. The width is π cm.
Determine the volume.
πΎππ ππ = π ππ
π³πππππ = π ππΓ ππ
π =
ππ
π ππ
π―πππππ = π ππ Γπ
π =
π
π ππ
π½πππππ = π π π
π½πππππ = (ππ
π ππ) (π ππ)(
π
π ππ)
π½πππππ =πππ
π πππ
4.
a. Write numerical expressions in two different ways, and explain what each reveals.
(πππ
π ππ. )(π
π
π ππ. ) (π ππ.) represents the product of three edge lengths. (
ππ
π πππ) (π ππ) represents the
product of the base area times the height. Answers will vary.
b. Determine the volume of the rectangular prism.
(πππ
π ππ. )(π
π
π ππ. ) (π ππ. ) = πππ πππ or (
ππ
π πππ) (π ππ. ) = πππ πππ
5. An aquarium in the shape of a rectangular prism has the following dimensions: length = ππ cm, width = πππ
π cm,
height = πππ
π cm.
a. Write numerical expressions in two different ways, and explain what each reveals.
(ππ ππ)(πππ
π ππ) (ππ
π
π ππ) represents the product of the three edge lengths.
(π,πππ πππ) (πππ
π ππ) represents the base area times the height.
b. Determine the volume of the rectangular prism.
(π,πππ πππ) (πππ
π ππ) = ππ,πππ
π
π πππ
πππ
in.
ππππ
in.
π in.
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 12
6. The area of the base in this rectangular prism is fixed at ππ cm2. That means for the varying heights there will be various volumes.
a. Complete the table of values to determine the various heights and volumes.
Height in
Centimeters
Volume in
Cubic
Centimeters
π ππ
π ππ
π πππ
π πππ
π πππ
π πππ
π πππ
π πππ
b. Write an equation to represent the relationship in the table. Be sure to define the variables used in the
equation.
Let π be the height of the rectangular prism in centimeters.
Let π be the volume of the rectangular prism in cubic centimeters.
πππ = π
c. What is the unit rate for this proportional relationship? What does it mean in this situation?
The unit rate is ππ.
For every centimeter of height, the volume increases by ππ cubic centimeters because the area of the base is
ππ cm2. In order to determine the volume, multiply the height by ππ.
7. The volume of a rectangular prism is ππ.πππ cm3. The height is π. ππ cm.
a. Let π¨ represent the area of the base of the rectangular prism. Write an equation that relates the volume, the area of the base, and the height.
ππ.πππ = π.πππ¨
b. Solve the equation for π¨.
ππ.πππ
π. ππ=
π.πππ¨
π. ππ
π¨ = π.π
The area of the base is π.π cm2.
ππ cm
π cm
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 12
Station A
Make a sketch of the figure. Then calculate the volume.
Rectangular prism:
Area of the base = πππ
ft2
Height = πππ
ft.
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 12
Station B
Make a sketch of the figure. Write the length, the width, and
height in feet. Then calculate the volume.
Rectangular prism:
Length is πππ
times as long as the height.
Width is π
π as long as the height.
Height = π ft.
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 12
Station C
Write two different expressions to represent the volume, and
explain what each expression represents.
ππ
π m
ππ
π m
π
π m
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 12
Station D
ππ«ππ =π
π πππ
π
ππ ππ.
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 12
Station E
ππ«ππ = πππ
π π’π§π
ππ
π π’π§.
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’5 Lesson 12
Station F
Challenge!
Determine the volume of a rectangular prism whose length and
width are in a ratio of π: π. The width and height are in a ratio of
π: π. The length of the rectangular prism is π ft.