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4.4
Painted Cubes
Goals
• Explore the “Painted Cube” situation, which has
edge length 10 cm and then have them generalize
it to one with edge length of n cm.
Let the class work in groups of three to four.
linear, quadratic, and cubic functions
• Compare linear, quadratic, and exponential
functions
When a painted cube with edge length n is
separated into n3 small cubes, how many of these
cubes will have paint on three faces? Two faces?
One face? No face? To answer this question,
students can consider cubes with smaller
dimensions and then look for patterns in the
data to make generalizations.
Launch 4.4
Explore 4.4
If students are having difficulty getting started
suggest that they study the sketches and build
cubes like these to find the answers to these
painted cube questions. Having a 10 3 10 3 10
base 10 block and a large supply of 1 cm cubes for
each group to examine while they work might be
helpful.
Have one or more groups put their data table
on a transparency to use in the summary.
Suggested Questions
You can use a Rubik’s™ Cube to launch the
problem.
Note: Having a Rubik’s Cube, a base ten block,
or some other large cube to show students while
you discuss it would be helpful. Also having small
unit cubes or sugar cubes around so students can
build some of the smaller cubes of length 2, 3, or 4
will be very helpful.
• Where are the unpainted cubes located? (They
Suggested Questions Hold up a large cube:
• Where are the cubes that have only one face
• How many faces, edges and corners does a
cube have? (A cube has six faces, 12 edges
and 8 corners.)
change from cube to cube? (No.)
• Suppose we paint a cube with edge length of
have? (cube)
• What are the dimensions of this cube? (Its
dimensions are two less the original cube.)
painted? (They are on the faces of the cube
but not along the edges.)
• Within one face of the large cube, what shape
is formed by the set of cubes with only one
face painted? (They form a square.)
• How many faces are there? (6)
• Where are the cubes that have two faces
painted? (They form the edges of the cube
but not the corners.)
• How many edges are there? (12)
• Where are the cubes that have three faces
painted? (They form the corners of the cube.)
• How many corners are there? (8)
In this problem you will explore these
questions for cubes of edge length 1 to 6 cm and
then students will generalize for edge lengths of
10 cm or n cm. Note: you could leave the problem
more open by posing the problem with a cube of
Investigation 4
What Is a Quadratic Function? 137
4
10 cm and then separate it into 1,000 small
centimeter cubes. How many of the smaller
cubes will have paint on three faces? Two
faces? One face? No faces? (Let students
make some conjectures. Some might notice
that the cube on a corner will have three
faces painted.)
• What shape does this set of unpainted cubes
I N V E S T I G AT I O N
• Do the number of faces, edges and corners
are on the inside of the large cube; they are
not visible from the outside.)
Suggested Questions
• What pattern in the differences would suggest
a linear relationship?
• What pattern in the differences would suggest
a quadratic relationship?
Summarize 4.4
Cubes painted on one face
During the discussion give students an
opportunity to verbalize the patterns of change
they have found in their tables. This will help
them write symbolic statements for Question C.
Put up a transparency of a table of data that
the students collected to refer to during the
summary. Ask students to study the data they
collected. Possible student responses are provided.
Suggested Questions
• What kind of relationship—linear, quadratic,
exponential, or other—do you observe in the
plotted graph or table of data for the number
of cubes with paint on 3 faces? (This one
seems to be linear since it is always 8.)
Cubes painted on two faces
• What kind of relationship—linear, quadratic,
exponential, or other—do you observe in the
plotted graph or table of data for the number
of cubes with paint on 2 faces? (This one
seems to be linear since they seem to be in a
straight line. Also, the table seems to be
increasing by 12 each time.)
• What kind of relationship—linear, quadratic,
exponential, or other—do you observe in the
plotted graph or table of data for the number
of cubes with paint on one face? (The
relationship seems like it might be quadratic
because the second difference is constant.)
Cubes painted on three faces
If students are having trouble seeing patterns
of change for Question C in the tables, ask them
to look at the patterns of change.
138 Frogs, Fleas, and Painted Cubes
• What kind of relationship—linear, quadratic,
exponential, or other—do you observe in the
plotted graph or table of data for the number
of cubes with paint on 0 faces? [The pattern
of change indicates this is not linear, nor
quadratic, nor exponential. In fact the
relationship is cubic, P = (n 2 2)3, though
students will not necessarily see this without
some discussion.]
• Let’s look closer at the pattern of change in the
data for zero faces.
• When we look at the differences between
y-values in tables where x changes by 1 each
time, what tells us that a relationship is linear?
(First differences are constant.)
• What tells that a relationship is quadratic?
(The second differences are constant.)
If you asked some groups to make tables of
differences, have these students put up their tables
of differences or demonstrate finding second
differences. (Figures 2 and 3)
• Explain that once constant differences of 0 are
reached it is pointless to go further. (Figure 4)
In this situation, the third differences are
constant.
• If we continued to take differences in each of
these cases, what would happen?
Figure 2
Figure 3
Figure 4
2
0
3
12
4
24
5
36
6
48
Edge Length of
Large Cube
Cubes Painted
on 1 Face
2
0
3
6
4
24
5
54
6
96
Edge Length of
Large Cube
Cubes Painted
on 0 Faces
2
0
3
1
4
8
5
27
6
64
First
Differences
Second
Differences
12
0
12
0
12
0
12
First
Differences
6
18
30
42
First
Differences
Second
Differences
12
12
12
Second
Differences
1
7
19
37
6
12
18
Investigation 4
Third
Differences
0
0
Third
Differences
6
6
4
Cubes Painted
on 2 Faces
I N V E S T I G AT I O N
Edge Length of
Large Cube
What Is a Quadratic Function? 139
Suggested Question
• We call (n 2 2)3 a cubic expression. What
• Let’s use this information to work backward
and find the number of cubes painted on
0 faces for an edge length of 7 centimeters.
(Figure 5)
• What kind of equation do you think describes
this relationship? (Students may not guess
“cubic” yet.)
To help students write the equations, go back
to some of the questions from the Explore. For
example, to write an expression for the number of
cubes with no faces painted:
Suggested Questions
• Where are the unpainted cubes located? (They
are on the inside of the large cube; they are
not visible from the outside.)
• What shape does this set of unpainted cubes
have? (cube)
• What are the dimensions of this cube? (Its
dimensions are two less than the original
cube.)
would you conjecture about the first, second,
or third differences for a cubic relationship?
(First and second differences are not
constant. Third differences are constant.)
Repeat these questions from the Explore for
the pattern of cubes on the faces [6(n - 2)2], on
the edges [12(n - 2)], and on the corners (8).
Use your calculator to make sketches of the
graphs of these relationships.
• Describe the shapes of the graphs.
• How many different kinds of relationships did
we explore in this problem? How can we
recognize each relationship from its table,
graph and equation? (For cubics they may
only be able to say something about the
equation containing a variable whose highest
exponent is 3. They may also say something
about the third difference.)
You can end the summary by going back to the
cube (10 3 10 3 10) and answering the original
questions.
• So how can we write this in symbols?
[(n - 2)(n - 2)(n - 2) or (n - 2)3]
Figure 5
Edge Length of
Large Cube
Cubes Painted
on 0 Faces
2
0
3
1
4
8
5
27
6
64
7
125
140 Frogs, Fleas, and Painted Cubes
First
Differences
1
7
19
37
61
Second
Differences
6
12
18
24
Third
Differences
6
6
6
At a Glance
4.4
Painted Cubes
PACING 1 day
Mathematical Goals
• Explore the “Painted Cube” situation, which has linear, quadratic, and
cubic functions
• Compare linear, quadratic, and exponential functions
Launch
Having a Rubik’s Cube, a base ten block, or some other large cube to show
students would be helpful. Hold up a large cube:
• How many faces, edges, and corners does a cube have?
• Do the number of faces, edges, and corners change from cube to cube?
• Suppose we paint a cube with edge length of 10 cm and then separate it
Materials
•
Base ten thousands
blocks (optional;
1 per group)
•
Rubik’s cube or other
large cube (optional)
into 1,000 small centimeter cubes. How many of the smaller cubes will
have paint on three faces? Two faces? One face? No faces?
In this problem you will explore these questions for cubes of edge length
1 to 6 cm and then generalize for edge lengths of 10 cm or n cm. Let the
class work in groups of three to four.
Explore
Materials
Suggest students study the sketches and build cubes like these to find the
answers to these painted cube questions. To get started ask:
•
Centimeter or other
unit cubes (in four
colors or with
colored dot stickers,
or sugar cubes and
colored markers)
•
Transparency 4.4
• Where are the unpainted cubes located? What shape does this set of
unpainted cubes have? What are the dimensions of this cube? Where are
the cubes that have only one face painted?
• Within one face of the large cube, what shape is formed by the set of
cubes with only one face painted?
• Where are the cubes that have two faces painted? How many edges are
there? Where are the cubes that have three faces painted? How many
corners are there?
If students are having trouble seeing patterns of change for Question C
in the tables, ask them to look at the patterns of change.
• What pattern in the differences would suggest a linear relationship? A
quadratic relationship?
Summarize
Give students an opportunity to verbalize the patterns of change they have
found in their tables.
Materials
•
Student notebooks
• What kind of relationship—linear, quadratic, exponential, or other—do
you observe in the plotted graph or table of data for the number of
cubes with paint on 3 faces (on 2 faces, on 1 face, on 0 faces)?
continued on next page
Investigation 4
What Is a Quadratic Function? 141
Summarize
continued
• When we look at the differences between y-values in tables where
x changes by 1 each time, what tells us that a relationship is linear?
Quadratic? If we continued to take differences in each of these cases,
what would happen?
If you asked some groups to make tables of differences, have these
students put up their tables of differences or demonstrate finding
second differences. Explain that once constant differences of 0 are
reached it is pointless to go further.
• Notice that when third differences are constant, the expanded form of
the equation contains the variable raised to the third power. Or, it is
the product of three terms that contain the variable x, like x ? x ? x
or x3, or (x 2 2)(x 2 2) (x 2 2). We call such a relationship a cubic
relationship.
To help students write the equations, go back to some of the
questions from the Explore.
ACE Assignment Guide
for Problem 4.4
used as a factor 3 times, x ? x ? x or x3. The
relationship is not linear, quadratic, or
exponential.
Core 27–30
Other Connections 41–50; Extensions 56, 57;
unassigned choices from previous problems
Adapted For suggestions about adapting
Exercise 4 and otherACE exercises, see the
CMP Special Needs Handbook.
Connecting to Prior Units 41, 43–45: Covering and
Surrounding
Answers to Problem 4.4
2. The number of cubes painted on three
faces is 8 regardless of the edge length of
the large cube. The number of cubes
painted on two faces is the edge length of
the large cube minus 2 and then multiplied
by 12. The number of cubes painted on one
face is the edge length of the large cube
minus 2, squared, and then multiplied by 6.
The number of cubes painted on zero faces
is the edge length of the large cube minus 2
then raised to the third power.
A. 1. and 2. (Figure 6)
B. 1. The number of centimeter cubes in a large
cube is the edge length of the large cube
Figure 6
Painted Faces of a Centimeter Cube
Edge Length of
Large Cube
Number of
cm Cubes
Number of cm Cubes Painted on
3 Faces 2 Faces 1 Face 0 Faces
2
8
8
0
0
0
3
27
8
12
6
1
4
64
8
24
24
8
5
125
8
36
54
27
6
216
8
48
96
64
142 Frogs, Fleas, and Painted Cubes
2.
80
60
40
20
0
1
3
5
7
Edge Length (cm)
x
9
Edge Length vs. Cubes Painted on 1 Face
600
300 y
200
100
0
1
3
5
7
Edge Length (cm)
x
9
400
200
0
1
3
5
7
Edge Length (cm)
x
9
8
300
200
100
0
1
3
5
7
Edge Length (cm)
x
9
6
4
2
x
0
2
4
6
8
Edge Length (cm)
Investigation 4
What Is a Quadratic Function? 143
4
0
We can predict that (edge, 3 faces) and
(edge, 2 faces) will be linear because the
table indicates first differences that are
constant. We can predict (edge, 1 face)
will be quadratic because the second
differences are constant.
I N V E S T I G AT I O N
Number of Cubes
Edge Length vs. Cubes Painted on 3 Faces
10 y
Edge Length vs. Unpainted Cubes
400 y
Number of Cubes
Number of Cubes
Edge Length vs. Total Number of Cubes
800 y
Number of Cubes
C. 1. For the equations below; let n be the
number of cubes and O represent edge
length of the large cube in cm.
For the relationship between edge length of
the large cube and total number of cubes,
n = O3.
For three faces painted: n = 8
For two faces painted: n = 12(O - 2) or
n = 12O - 24
For one face painted: n = 6(O - 2)2 or
n = 6O2 - 24O + 24
For zero faces painted: n = (O - 2)3
Edge Length vs. Cubes Painted on 2 Faces
100 y
Number of Cubes
3. Three faces: the relationship is linear
Two faces: the relationship is linear
One face: the relationship is quadratic
Zero faces: Not linear, quadratic, nor
exponential
144 Frogs, Fleas, and Painted Cubes