The task
Students were given the task displayed here.

This task requires an interesting combination of geometry (spatial visualization) and algebra (expressing the general relationship symbolically).

Sample 2 was completed in Haitian-Creole in a bilingual classroom. The translation was provided by the Haitian-Creole calibration team.

Sample 3 was completed in Chinese in a bilingual classroom. The translation was provided by the Chinese calibration team.

Mathematics required by the task
The idea at the heart of the task is the following fact about three-dimensional geometry: A large cube which is made up as an “n by n by n” assembly of small, identical cubes contains exactly n³ small cubes. The task statement illustrates a diagram of such large cubes for the case n = 3 and n = 4.

It is necessary to visualize this situation spatially to appreciate another important fact. The small cubes that are hidden from view in a diagram of a large n by n by n cube actually form a large (n-1) by (n-1) by (n-1) cube. This means that there is a total of (n-1)³ small cubes that are hidden from view.

Finally, an algebraic representation seems essential to express the generalization asked for in Question 3. For example, the number of visible cubes in a large n by n by n cube can be expressed as the total number of cubes minus the number of invisible cubes:

total # of cubes - # of hidden cubes = # of visible cubes
n³ - (n-1)³ = 3n² - 3n + 1

(These expressions make sense if and only if n is a whole number.)

The student work illustrates another way in which the visible cubes are counted directly.

Sample 1

These work samples illustrate standard-setting performances for the following parts of the standards:
b	Geometry and Measurement Concepts: Work with three dimensional figures and their properties.
d	Geometry and Measurement Concepts: Visualize objects in space.
j	Geometry and Measurement Concepts: Investigate geometric patterns.
a	Function and Algebra Concepts: Model given situations with formulas.
b	Function and Algebra Concepts: Use basic types of functions.
h	Function and Algebra Concepts: Use and manipulate expressions involving variables.
i	Function and Algebra Concepts: Represent functional relationships.
o	Function and Algebra Concepts: Use functions to analyze patterns and represent their structure.
c	Problem solving and Mathematical Reasoning: Conclusion.
e	Mathematical Skills and Tools: Make and use rough sketches or schematic diagrams to enhance a solution.
c	Mathematical Communication: Organize work and present mathematical procedures and results clearly, systematically, succinctly, and correctly.

Sample 2

Sample 2 Translation

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 (x-1)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 (x-1) = x2 - x
- side face: (x-1) (x-1) = 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 (x-1)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.