Cube-n-ometry Unit
Inspired by: MathScience Innovation Center
(Permission Granted to Modify)
Modified for: Wake County’s 7th Grade Math Classes
Part 1: Questions to Ponder

As you view the next few slides, ask
yourself the following questions:
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What can be seen in the pictures?
How were the pictures taken?
Where would you need to be to take the
pictures?
Why are these pictures important to the
specified professions?
2
What would you take a picture
of if you were an…

Architect?
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Aerial view
of Middle
Creek High
School
Buildings,
land,
streets,
etc.
3
What would you take a picture
of if you were a…
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Cartographer
or map maker?
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Aerial view of
Raleigh, NC
Views of
buildings,
streets,
roads, etc.
4
What would you take a picture
of if you were a…
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Geologist?

Cross
sectional
view of
Earth’s
main
layers
5
What would you take a picture
of if you were a…

Doctor?
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X-rays of humans
(surgeons and
dentists) or even
of animals
(veterinarians)
6
Questions to Answer

So, let’s answer the questions now:

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What could be seen in the pictures?
How were the pictures taken?
Where would you need to be to take the
pictures?
Why are these pictures important to the
specified professions (map-maper,
architect, scientist, doctor, etc)
7
Introduction to
Visualizing Figures

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As you can see, POINTS OF VIEW are
extremely important in all careers and
professions.
In this unit – “Cube-n-ometry” – we’re
going to:

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Strengthen our ability to visualize threedimensional objects, and
Explore methods for representing cubic
structures.
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So…Let’s Explore Cubes

What makes a cube from the
beginning…
Square

Cube
That is, what shape makes up a cube
and how many are there in a cube?
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A cube is made up of six (6) squares!
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Exploring Dimensions
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What differences can you see in the two
figures?
Threedimensional
Two dimensional
Square
Cube
The major difference between two-dimensional and
three-dimensional objects is that three-dimensional
objects have depth/width.
10
3-Dimensional Versus
2-Dimensional Figures

Let’s explore solids and faces…
Face
Solid
A face is the TWODIMENSIONAL
surface of a solid.
A cube is made up
of 6 identical
faces, all of which
are squares.
Face
A solid is a THREEDIMENSIONAL figure
(such as a CUBE).
THREEDIMENSIONAL
figures have length,
width, and depth.
11
Exploring Cubic Arrangements

Today we are going to explore possible
arrangements of:
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3 Cubes,
4 Cubes, and
Upon each task, we’ll create
ISOMETRIC DRAWINGS of our 3D
figures using isometric dot paper.
12
Part 2: Isometric Drawings
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What’s an ISOMETRIC DRAWING?
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An isometric drawing is a perspective
drawing of a three-dimensional figure.
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Hence, isometric drawings show the 3RD
dimension – “width” or “depth.”
ISOMETRIC DRAWINGS are drawn from
the CORNER of a three-dimensional figure
and can be placed onto isometric dot
paper.
13
Example of Isometric Dot Paper
14
Arrangements of 3 Cubes
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3 Cubes are arranged so that at least one face of
each cube must meet the face of another cube.
You cannot have a second story on a house,
unless there is a first story below it- it needs
support!
Rotations of the same arrangement do not count
as different houses. If you have to PICK UP or
DROP an arrangement, it is considered a
different house.
15
Arrangements of 3 Cubes
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How many arrangements of 3 cubes are there?
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There are FOUR arrangements of 3 cubes.
16
Create Isometric Drawings of
Your Cubic Arrangements
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Using isometric dot paper, record each
arrangement by creating an ISOMETRIC
DRAWING of the cubic arrangement.
17
Create Isometric Drawings of
Your Cubic Arrangements
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Arrangements of 4 Cubes
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New scenario!
Four cubes are arranged so that at least one
face of each cube must meet the face of
another cube.
How many arrangements of 4 cubes are
possible?
19
Arrangements of 4 Cubes
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How many arrangements of 4 cubes are
there?
A few examples are…
20
Create Isometric Drawings of
Your Cubic Arrangements
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How Many Arrangements
of 4 Cubes Are There?
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There are FIFTEEN arrangements of 4 cubes!
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Part 3: Orthogonal Drawings
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What’s an ORTHOGONAL
DRAWING?
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An orthogonal drawing is a twodimensional sketch of a three-dimensional
figures’ face.
Orthogonal drawings are the FRONT, TOP,
and SIDE views of a three-dimensional
figure.
23
Using 3D Figures to Generate
Orthogonal Drawings
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Views of a threedimensional block figure
can be broken down
into three views:
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Front View
Top View
Side View
These views are referred to as
ORTHOGONAL DRAWINGS
 Orthogonal drawings are twodimensional representations of
3-D figures
24
Construct Building 1
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Build this three dimensional block
figure.
What does it look like from the:
25
Watch the Views Light
Up on Building 1
Orthogonal Views of Building 1
27
Construct Building 2

If you were to
snap a picture of
this object what
do you think it
would look like
from the:
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Front?
Top?
Side?
28
Orthogonal Views of Building 2
29
Building 3
• Build this block figure and draw the front, top, and
side views using your grid paper.
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Orthogonal Views of Building 3
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Building 4
• Build this block figure and draw the front, top, and
side views using your grid paper.
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Orthogonal Views of Building 4
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Part 4: Building 3D Figures
Using Orthogonal Drawings
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PowerPoint Presentation – Building 3D
figures using orthogonal views
Designing Spaces Section 3: “Seeing all
Possibilities”
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Drawing 3-D Views Packet
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Construct Homes from Orthogonal Views
Figures 1-6 (Classroom Strategies)
Orthogonal to 3-D to Isometric
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4 Buildings
34
Building 3D Figures Using
Orthogonal Drawings
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Now. . .let’s do the opposite!
Given the orthogonal drawings front, top, and side views - can you
build the three-dimensional block
figures using the least number of
cubes possible.
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Hint: Once you’ve constructed the 3-D
figure, see if one or more cubes can be
removed to generate the same
orthogonal drawings or views.
35
Views of Building 1
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Given the views below, can you
build the three-dimensional block
figure?
Front
Top
Side
36
Picture of Building 1
Good Job!
37
Views of Building 2

Given the views below, can you
build the three-dimensional block
figure?
Front
Top
Side
38
Picture of Building 2
39
Number of Cubes
for Building 2 (Option 1)
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How many cubes are needed to construct
this building?
Front

Top
Side
8 Cubes
40
Least Number of Cubes
for Building 2 (Option 2)

What is the least number of cubes needed
to construct this building?
Front

Top
Side
Hint: Remove cubes to obtain the same
orthogonal views.
41
Picture of Building 2 with
the Least Number of Cubes
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ONLY 6 Cubes are needed!
42
Views of Building 3
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Given the views below, can you
build the three-dimensional block
figure?
43
Picture of Building 3
44
Least Number of Cubes
for Building 3
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What is the least number of cubes needed
to construct this building?
10 cubes minus 1 unnecessary cube = 9 cubes!
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