Volume Unit 2
U N I T
Structuring Cubic Volumes: Nets of Buildings
2
Mathematical Concepts
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We call the space occupied by a 2-dimensional figure an area.
We call the space occupied by a 3-dimensional structure a volume.
The measure of the volume of a structure is the ratio of the volume of
the structure to the volume of a unit. Practically, this is established by
counting the number of units that tile the volume.
Nets are 2-dimensional representations of the surface of 3-D structures.
Equivalence is a relation indicating that two different mathematical
objects are the same in some respect.
Proofs are convincing explanations that answer why.
Contents
Mathematical Concepts
1
Unit Overview
Unit Overview
1
The teacher demonstrates the net of a cubic apartment from the previous
lesson and challenges students to generate alternative nets that will have
the same result. During whole class conversation, students establish the
grounds for considering which nets are equivalent (e.g., nets that can be
made congruent by an isometry or composition of isometries are
equivalent). Following establishment of these grounds, students consider
collectively which nets were generated most often, and visualize these
data using a dot plot or some other display. The teacher asks students how
they will know if they have generated all possible nets of the cube. The
teacher guides a discussion of a convincing explanation that there are 11
possible different nets, if isometries are used to establish equivalence.
Materials & Preparation
2
Mathematical Background
3
Instruction
4
Students’ Way of Thinking
6
Formative Assessment
8
Formative Assessment
Record
10
Students then view nets of four different apartment buildings and for each
net, predict the building’s surface area and volume. Then students fold the
nets to constitute each building and compare their predictions with counts
of faces and cubes.
Nets of Buildings Worksheet
11
1 Area Unit
Materials and Preparation
Volume Unit 2
Prepare
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Square grid paper so that students can outline different nets.
If available, square Polydron® pieces, 6 per student.
Scissors so that students can cut out and fold each net.
Copies of the nets worksheet, one for each student.
Copies of the 14-square net worksheet, two for each student.
14 square Polydron® or Magformer® pieces for public
demonstration of how to fold the squares to form two different
prisms.
Academic Vocabulary
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Area
Volume
Net
Unit
Proof
Equivalence
2 Mathematical Concepts
Unit Overview
Materials and Preparation
Mathematical Background
Instruction
Students’ Ways of Thinking
Formative Assessment
Formative Assessment Record
Nets of Buildings Worksheet
Area Unit
Mathematical Background
Volume Unit 2
Surface Area
The area of the two dimensional regions of the surface of a three
dimensional structure is called a surface area.
Volume
The space enclosed or occupied by a three dimensional structure is called
a volume.
Volume Measure
Volume measure is a ratio of the space enclosed by a three dimensional
structure and a three-dimensional unit of measure. Units of volume
measure, typically cubes, tile the volume. Hence, counting cubes,
including fractional cubic units, measures volume. For example, a
!
!
structure with a measure of 50 ! cubic units has a volume that is 50 ! times
that of the volume of the cubic unit.
Net
A net is a flat connected 2-dimensional representation of the surface of a
3-dimensional structure that can be folded to constitute the surface of the
structure.
Equivalence
Equivalence is a judgment that two different mathematical objects are
nevertheless the same in some way. For nets, it is common to consider
nets that can be related by isometries (slides, reflections, turns, or glides)
as equivalent (see Students’ Ways of Thinking).
Proof
A proof is a convincing explanation for why a result is true. In this
instance, students try to establish how many different nets of a cube are
possible. One way to proceed is to systematically generate all possible nets
with a column of 4 vertical faces, with three vertical faces, and with two
vertical faces (see Students’ Ways of Thinking).
Prism
A prism is a polyhedron with a polygonal base and a translated copy of
that base not in the same plane. N other parallelogram faces, all of which
are identical, join corresponding sides of the two bases.
3 Mathematical Concepts
Unit Overview
Materials and Preparation
Mathematical Background
Instruction
Students’ Ways of Thinking
Formative Assessment
Formative Assessment Record
Nets of Buildings Worksheet
Area Unit
Instruction
Volume Unit 2
Whole Group
Students view a Polydron® or paper model of a cubic “apartment.”
The teacher introduces students to a net representation by showing how
tracing each face of the cube results in a two-dimensional representation
of the faces and the edges of the cube. The teacher demonstrates how
folding the net results in a cube.
Individual
With Polydron® squares or square grid paper and scissors, each
student generates one net of the cube. The net can be the same as that of
the teacher or different than that of the teacher.
Whole Group
The teacher selects nets produced by students to compare nets to
decide about which nets can be considered different. The purpose of
the conversation is to develop criteria for considering whether or not two
different nets can be considered as equivalent or as truly different.
Teacher note. Students usually decide that nets that can be made
congruent via reflection, rotation, or translation are the same. One visible
instance is to contrast a T net, consisting of 6 squares arranged in the form
!
of the letter T, with a T net that is ! turn rotated, and to ask students to
consider whether or not they should be treated as distinct or as the same.
Partners
Partners work together to produce as many distinct nets as they can.
They make paper cut outs of every net that they generate that is distinct.
4 Mathematical Concepts
Unit Overview
Materials and Preparation
Mathematical Background
Instruction
Students’ Ways of Thinking
Formative Assessment
Formative Assessment Record
Nets of Buildings Worksheet
Area Unit
Instruction
Volume Unit 2
Whole Group
The teacher invites a student to post a net, and all students who have
generated the same net also post theirs, either in a row or in a column, to
produce a visual display of the class results. The teacher guides discussion
about what the class notices about the display and considers why some
nets seem to be more frequently generated than others. Following this
conversation, the teacher asks students whether or not they have
discovered all the possible nets of a cube. How could they convince
someone else that there were no others that could be found?
Teacher note. If no student suggests it, then suggest to students that they
create a system for making all the possible nets with a column of 4 squares.
Partners
Partners work together to generate all possible nets of a cube using the
system suggested by the teacher or by a peer.
Teacher note. Prompt students if necessary to consider a column of 3
squares as well as a column with 4 squares. See Students’ Ways of
Thinking.
Whole Group
The teacher leads a whole class conversation, calling upon different
partners to demonstrate how they found all possible nets with a column of
4 faces, with a column of 3 faces, and with a column of 2 faces (See
Students’ Ways of Thinking). The teacher labels the process of generating
a compelling explanation a proof, emphasizing how the class now knows
that there are only 11 possible nets, given the grounds of equivalence
established by the class.
5 Mathematical Concepts
Unit Overview
Materials and Preparation
Mathematical Background
Instruction
Students’ Ways of Thinking
Formative Assessment
Formative Assessment Record
Nets of Buildings Worksheet
Area Unit
Students’ Ways of Thinking
Volume Unit 2
STUDENTS’
WAYS OF
THINKING
This depicts a proof generated by a third grade class for 11 possible nets of a cube.
Note that some nets originally proposed as different are re-considered as equivalent.
Having considered all possible configurations by exhaustive search of columns with
4, 3 and 2 squares (called backbones by the children), there can no other possibilities.
6 Mathematical Concepts
Unit Overview
Materials and Preparation
Mathematical Background
Instruction
Students’ Ways of Thinking
Formative Assessment
Formative Assessment Record
Nets of Buildings Worksheet
Area Unit
Instruction
Volume Unit 2
Individual
Each student receives a copy of the worksheet with nets of five
apartment buildings. For each net, the student predicts the surface area
for each building and how many apartments are in each building. Then
each student cuts out and folds each net, comparing their predictions to the
counts of windows and apartments evident in each building.
Whole Group
The teacher guides a discussion, focusing on the nets that students found
most challenging to predict. The folded nets are depicted below.
A
B
C
D
7 E
Mathematical Concepts
Unit Overview
Materials and Preparation
Mathematical Background
Instruction
Students’ Ways of Thinking
Formative Assessment
Formative Assessment Record
Nets of Buildings Worksheet
Area Unit
Instruction
Volume Unit 2
Optional Investigation of a 14-square net
Individual
Each student receives two copies of the 14-square net and is challenged to
see if the same net can be folded in different ways to create two different
structures.
Teacher note. The 14-square net can be folded into two different prisms.
One solution has a convex base (all interior angles less than 180 degrees).
The other solution has a concave base in the shape of an L.
Whole Group
Students share solutions.
8 Mathematical Concepts
Unit Overview
Materials and Preparation
Mathematical Background
Instruction
Students’ Ways of Thinking
Formative Assessment
Formative Assessment Record
Nets of Buildings Worksheet
Area Unit
Formative Assessment
Volume Unit 2
Formative Assessment
The formative assessment is a design task with multiple possible solutions.
After students design and fold their nets into structures, select a few
different designs to emphasize different solutions to the same challenge.
9 Mathematical Concepts
Unit Overview
Materials and Preparation
Mathematical Background
Instruction
Students’ Ways of Thinking
Formative Assessment
Formative Assessment Record
Nets of Buildings Worksheet
Area Unit
Formative Assessment
Volume Unit 2
NAME_______________________________
Using the square grid paper, design a net of your own apartment building
that will occupy a volume of at least 8 apartments (cubes). Then fold it to
show that it really has 8 (or more) apartments. Compare the surface area to
the volume.
10 Mathematical Concepts
Unit Overview
Materials and Preparation
Mathematical Background
Instruction
Students’ Ways of Thinking
Formative Assessment
Formative Assessment Record
Nets of Buildings Worksheet
Area Unit
Formative Assessment Record
Volume Unit 2
NAME_______________________________________
Indicate the levels of mastery demonstrated by circling those for which there is clear evidence:
Item
Level
Description
Circle highest level of performance
ToVM 3C
Find and compare volumes of right rectangular prisms by counting unit cubes (including “hidden cubes”). Item 1
Design a net of a
structure consisting of
8 or more cubes.
ToVM 3B
Find and compare
volumes of right
rectangular prisms by
counting unit cubes (no
hidden cubes).
ToVM 1C
Differentiate surface
area from volume.
Finds the number of apartments/cubes
with a net that generates hidden units.
11 Finds the number of apartments
with a net that does not generate
any hidden units.
Counts the visible cubic faces as
windows.
Notes
Area Unit
Nets of Building Worksheet 1
Volume Unit 2
Building
A
Building
B
Building
Building
12 Building
C
Mathematical Concepts
Unit Overview
Materials and Preparation
Mathematical Background
Instruction
Students’ Ways of Thinking
Formative Assessment
Formative Assessment Record
Nets of Buildings Worksheet
D
E
Area Unit
Nets of Building Worksheet 2
Volume Unit 2
13 Mathematical Concepts
Unit Overview
Materials and Preparation
Mathematical Background
Instruction
Students’ Ways of Thinking
Formative Assessment
Formative Assessment Record
Nets of Buildings Worksheet