Modified Filling and Wrapping 2.3
Domain:
Geometry
Big Idea (Cluster):
Solve real-world and mathematical problems involving area, surface area, and volume.
Common Core Standards:
6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes
of the appropriate unit fraction lengths, and show that the volume is the same as would be found by
multiplying the edge lengths of the prism. Apply formulas V=lwh and V = Bh to find volumes to solve realworld and mathematical problems
Mathematical Practice(s):
MP 1: Make sense of problems and persevere in solving them
MP 2: Reason abstractly and quantitatively
MP 3: Construct viable arguments and critique the reasoning of others
MP 5: Use appropriate tools strategically
MP 6: Attend to precision
MP 7: Look for and make sure of structure
MP 8: Look for and express regularity in repeated reasoning
Content Objectives:
Language Objectives:
Students will be able to understand that prisms can
be filled with identical layers which lead to volume
formula.
Record data in a table
Students will be able to develop a formula for
finding the volume of any rectangular prism.
Share strategy with partner, group, or class
Write a strategy for finding volume
Listen to a classmate’s strategy and respond with your
Students will be able to apply formula of the volume ideas
of a rectangular prism with fractional edge lengths.
Vocabulary:
Base
Cube
Cubic Units
Dimensions
Formula
Height
Prior Knowledge: Concepts students need to know
Layers
Length
Rectangular Prism
Volume
Width
Beginning in 2014-2015, Grade 5 students will find the
volume of a right rectangular prism with whole
number side lengths. They will have applied the
formula V=lwh and V = Bh in the context of solving
real world and mathematical problems. They will also
have a basic understanding of volume as layering of a
three-dimensional base shape repeatedly.
Until 2014-2015, you need to teach both whole
number and rational edge lengths as this will not be
prior knowledge for Grade 6 students.
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Modified Filling and Wrapping 2.3
Questions to Develop Mathematical Thinking:
Common Misconceptions:

Students may struggle to see the area of the base face
is the same numerical value as the volume of the base
layer when understanding the volume formula of a
rectangular prism can be either V= lxwxh or V = Bh.
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What measures of the figure are involved in
solving for total number of cubes, volume?
In an arrangement of cubes, what dimensions
make up the base layer?
In an arrangement of cubes, what dimension
makes up the identical layers?
Why is the number of cubes in the bottom layer
equal to the area of the base face?
Provide students the opportunity to explore the
difference between base face area being square units
and base layer being cubic units. Base layer is cubic
units because the volume of the base layer is found by
multiplying length times width times a height of 1
which leads to the same numerical value as area.
ASSESSMENT:
Observe student work and listen to student discussion for:
 Connections between length times width and the single layer
 Connections between height and the stacked identical layers
 Connections between single layer, identical layer, and volume formula
 Accurate calculation and labeling of volume
Use problem C as a formative assessment. Have students share out their strategies for solving volume of a
rectangular prism. Have students connect the model of one of the boxes to their volume strategy.
Problem D could be used as an exit ticket.
MATERIALS:

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Up to 80 centimeter or inch cubes per group to build the boxes if needed
Copies of “Modified Filling and Wrapping Investigation 2.3” student explore
Copies of “Modified Filling and Wrapping Investigation 2.3” labsheet
INSTRUCTIONAL PLAN:
Launch: (5-10 minutes)
From the Modified Investigation 2.1, have students describe what each arrangement for a specific set of
blocks had in common. Students will likely describe each arrangement was in the shape of a rectangular
prism and had the same amount of blocks. Ask students what measurement the block represents versus the
face on the blocks. Students should connect the faces relate to area and the block relates to volume.
After discussion, have students read the “Did you Know?” on page 22 to make connection between surface
area and volume notations. Hand out modified lesson and begin discussion on ready-made boxes. For a realworld connection, you might share this CNN article about a degree in packaging.
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Modified Filling and Wrapping 2.3
Explore: (20-30 minutes)
Have students read the task individually and share out to the class or partner student actions we, teacher and
student, should see and hear during the task. For example, we should see all students building models,
counting layers, and recording measures. We should hear all students counting accurately, checking their
work, discussing strategies, and explaining their reasoning.
The teacher notes for the “Explore” section are the same as the CMP2 Filling and Wrapping Teacher’s Guide
for Investigation 2.3 on page 47-48. The only difference is the connection to surface area has been removed
from this modified lesson and students are provided with more opportunities to connect the base layer and
identical stacked layers to a strategy for solving volume of any rectangular prism. Within Common Core
Standards for Mathematics, students should be flexible with understanding and applying both volume
formulas, V=lwh and V = Bh.
When I observe students:

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Listen for connections between length times width and the single layer (MP 2 and MP 7)
When students struggle to make the above connection, have students look at one of their box models
and ask students which dimensions make up the base layer both numerically and in general?
Listen for connections between height and the stacked identical layers (MP 2 and MP 7)
When students struggle to make the above connection, have students look at one of their box models
and ask students which dimension represents the stacked layers both numerically and in general?
Listen for connections between single layer, identical layer, and volume formula (MP 2, MP 7, and
MP 8)
Have students share their strategy for finding volume and describe how their strategy is represented in
the model of a rectangular prism stacked with unit cubes (MP 3 and MP 5)
When students struggle to identify V=Bh as an alternate volume formula, have students make
observations from the table. Ask students how to more generally describe the single layer than length
times width. Where else have we used length times width? Where is length times width represented
within volume?
Look at student work for accurate calculation and labeling of volume (MP 6)
Questions to Develop Mathematical Thinking as you observe:
 What measures of the figure are involved in solving for total number of cubes, volume?
 In an arrangement of cubes, what dimensions make up the base layer?
 In an arrangement of cubes, what dimension makes up the identical layers?
 Why is the number of cubes in the bottom layer equal to the area of the base face?
Solutions:
A. See table at end of lesson plan.
B. See student work. Students may describe the number of the cubes in each layer are length times
width. The identical layers are the same as height. The total number of cubes is the multiplication of
all three dimensions.
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Modified Filling and Wrapping 2.3
C. You can find volume by multiplying length times width times height. If students do not identify area
of base times height. Question students to make a more generalized strategy that could be applied to
other prisms.
D. The company did advertise accurately since 5 x 2 1/3 x 2 2/5 is 28 cm3. I applied the volume strategy
by multiplying length times width times height for volume.
E. ATC could buy Box W or X for the 18 or 24 block sets. The 30 set blocks would fit in Box Y or Z
with a lot of extra room since they hold 80 blocks and you only need 30 to fit inside.
Summarize:
The teacher notes for the “Summarize” section are the same as the CMP2 Filling and Wrapping Teacher’s
Guide for Investigation 2.3 on page 47-48. The only difference is the focus on both volume formulas. Within
Common Core Standards for Mathematics, students should be flexible with understanding and applying both
volume formulas, V=lwh and V = Bh with whole, decimal, and fractional edge lengths.
Focus summary on investigation problems B and C. Have students describe their connections between the
measures on the table and their general formula/strategy for finding volume of a rectangular prism. While
students share out strategies, ask students, “Do you agree or disagree with the provided strategy?” You may
also ask, “Did anyone find another way to solve for volume?”
In addition, ask the following question to probe students’ understanding and possible misconception:
 Why is the number of cubes in the bottom layer equal to the area of the base? This is a common
misconception since area is two-dimensional and the bottom layer is three-dimensional. Ask students the
height of the single layer and what happens to a calculation when you multiply by 1. This is an
opportunity to talk about the Property of Identity.
Feedback for lesson improvement:
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