b
Geometry and Measurement Concepts: The student
works with…three dimensional figures and their properties,
including…cubes….
d
Geometry and Measurement Concepts: The student
visualizes objects…in space….
j
Geometry and Measurement Concepts: The student
investigates geometric patterns, including sequences of growing
shapes.
Throughout the responses, the students worked with the structure
of large cubes built up from smaller cubes, visualizing them in
terms of the small cubes that are visible and those that are hidden,
and representing the visible and hidden cubes in large cubes of
various sizes.
a
Function and Algebra Concepts: The student
models given situations with formulas…, and interprets given
formulas…in terms of situations.
b
Function and Algebra Concepts: The student…uses basic types
of functions…[including] cube….
h
Function and Algebra Concepts: The student…uses
and manipulates expressions involving variables…in work with
formulas….
i
Function and Algebra Concepts: The student
represents functional relationships in formulas [and] tables….
o
Function and Algebra Concepts: The student
uses functions to analyze patterns and represent their structure.
Here the
student began to formulate the generalization asked for in Question
3 of the task. The variable “x” was chosen to represent
the number of small cubes making up each dimension of the large
cube. (The variable “n” would be more in keeping with
standard practice.) The total number of small cubes in a large “x
by x by x” cube is given as x³, while the number of hidden
cubes is given as (x1)³. The latter fact was based on the
empirical observation “I noticed that the number of hidden
cubes was the same number of cubes in the next size cube.”
Expressing
the number of visible cubes is harder than expressing the number
of hidden cubes. The student expressed the number of visible cubes
directly by summing the number of cubes on the three visible faces
(and making sure not to count the same cube more than once):
 top face: x2
 front face: x (x1) = x2  x
 side face: (x1) (x1) = x2  2x + 1
Summing these gives the total number of visible cubes: 3x²
 3x + 1.
The same result could have been obtained a little more easily
by subtracting the number of hidden cubes (which is (x1)³)
from the total number of cubes (which is x³).
Here the
student formulated the generalization asked for in Question 3 of
the task. The variable “N” is chosen to represent the
number of small cubes making up each dimension of the large cube,
and the total number of small cubes that cannot be seen in a large
(N+1) (N+1) (N+1) cube is given as
(N) (N) (N). This formulation of the generalization would have been
more compact if exponent notation N³ had been used. The student
implicitly used this generalization to fill in the rows of the table.

Sample 3
Sample 3 Translation
Here the
student formulated very concisely the generalization asked for in
Question 3 of the task. The variable “n” is chosen to
represent the number of small cubes making up each dimension of
the large cube.
The student
used this generalization (rather than counting directly) to fill
in the data for side lengths 3, 4, and 5.
It is
not clear how the student arrived at the cubic function given in
the response. Is it an observation that the numerical entries in
the “total cubes” column of the table are all perfect
cubes:
1 = 1³, 8 = 2³, 27 = 3³, etc.? Or is it the geometrical
insight that an n by n by n large cube has n³ small cubes in
it? The difference between these two possible ways of seeing the
cubic pattern is the difference between:
(i) data analysis (get numerical data from
the geometrical situation case by case, then forget the situation
and analyze the data numerically); and
(ii) “structural analysis” (directly
analyze the geometric structure of the situation).
c
Problem Solving and Mathematical Reasoning:
Conclusion. The student formulates generalizations of the results
obtained.
The student
went beyond the prompt’s request for an explanation and provided
the generalization.
e
Mathematical Skills and Tools: The student
makes and uses rough sketches, schematic diagrams…to enhance
a solution.
The student
made effective use of diagrams as a way of illustrating and hence
visualizing the structure of the large cubes. The small cubes in
the diagrams are numbered systematically, indicating an organized
process of using the diagrams to reveal the pattern.
c
Mathematical Communication: The student organizes
work and presents mathematical procedures and results clearly, systematically,
succinctly, and correctly.
In Sample 1, the diagrams are connected and interpreted with explanatory
text. In Sample 2, the text is clear, concrete, and illustrated
with examples.
In sample
3, the conciseness is particularly
effective. 