Dicing Cheese
You have a large rectangular block of cheese. You know its volume, V in cubic centimeters. Using a special cheese dicing machine, you cut the whole block up into small cubes, all exactly the same size.
When you spread these small cubes out one layer thick, with no spaces in between, they completely fill a flat, rectangular tray. You know the area, A of the tray in square centimeters.

1. In terms of V and A, find the length of the side of one of these small cubes.
2. In terms of V and A, find how many cubes were made.

This task about area and volume may appear to be rather simple. It involves only basic, rectangular shapes, and the only formulas needed are the most elementary ones (formulas for the volume of a cube and the area of a face of a cube). Yet the task deeply probes students’ conceptual understanding of area and volume. Anyone who has merely memorized formulas will make little headway here.

 Circumstances of performance This sample of student work was produced under the following conditions: - alone in a group - in class as homework with teacher feedback with peer feedback - timed opportunity for revision - with manipulatives with calculator

The task asks the student to express the number n of cubes and their side length l in terms of the total volume V and the total area A they cover. One way to proceed is to write down these observations about the volume V and the area A:

1. Since the volume of one small cube is l³, the total volume is V = nl³.
2. Since the area of a face of one cube is l², the total area they cover is A = nl².

Eliminating n from these two equations allows us to express l in terms of V and A, while eliminating l allows us to express n in terms of V and A.

It is interesting that there are many approaches quite different from this one that students use to solve this problem. For example, looking at the cubes spread out on the tray as a rectangular solid, its volume can be written as V = l A. This gives the length l immediately in terms of V and A as l = . Another method uses “dimensional analysis” to argue that l = (perhaps with a dimensionless constant) is the only possible formula for l in terms of V and A that has the right units (the right “dimensions”). “Dimensional analysis” is a technique that keeps track of the “dimensions” of quantities. For example, volume has the dimensions L³ (where L stands for length), area has the dimensions L², and speed has the dimensions (where T stands for time). These dimensions can be operated on algebraically. Hence, a volume divided by an area has the dimensions = L = length.

 This work sample illustrates a standard-setting performance for the following parts of the standards: a Geometry and Measurement Concepts: Model situations geometrically to formulate and solve problems. b Geometry and Measurement Concepts: Work with three dimensional figures and their properties. e Geometry and Measurement Concepts: Know and use formulas for area, surface area, and volume. k Geometry and Measurement Concepts: Work with geometric measures of length, area, and volume. a Function and Algebra Concepts: Model given situations with formulas. h Function and Algebra Concepts: Define, use, and manipulate expressions involving variables. a Problem Solving and Mathematical Reasoning: Formulation. b Problem Solving and Mathematical Reasoning: Implementation. c Problem Solving and Mathematical Reasoning: Conclusion. a Mathematical Skills and Tools: Carry out symbol manipulations effectively. c Mathematical Communication: Organize work and present mathematical procedures and results clearly, systematically, succinctly, and correctly.

One feature of the task that needs comment is the fact that specific numbers are not given for V and A. The task is designed to assess students’ abilities to deal with the abstractness of this “numberless” formulation. Assigning specific numbers would make the task somewhat easier. For example, in another version of the task that was used with other students, the specific values A = 9,000 square centimeters and V = 5,400 cubic centimeters were given. This made it easier for the students to create a concrete picture of the situation, and hence easier to get started.

What the work shows
 a Geometry and Measurement Concepts: The student models situations geometrically to formulate and solve problems. To start off both questions, the student used these facts about the whole mass of cheese: - The number of small cubes is equal to the total volume V divided by the volume of one cube. - The number of small cubes is equal to the total area A covered divided by the face area of one cube. b Geometry and Measurement Concepts: The student works with…three dimensional figures and their properties, including…cubes…. k Geometry and Measurement Concepts: The student works with geometric measures of length, area, volume…. This is evident throughout the student work. e Geometry and Measurement Concepts: The student knows [and] uses…formulas for…area, surface area, and volume of many types of figures. To continue, the student used the area and volume formulas for a cube of side length l: volume = l³ area of a face = l² a Function and Algebra Concepts: The student models given situations with formulas…. h Function and Algebra Concepts: The student defines [and] uses…variables…in work with formulas…. Throughout, the response uses relevant formulas for area and volume and their interrelation in the derivation of formulas for the side length l and the number of cubes n. h Function and Algebra Concepts: The student defines, uses, and manipulates expressions involving variables…in work with formulas…[and] equations…. The student substituted the result obtained for Question 1, namely l = into the equations of Question 2. The second result is obtained by manipulation and substitution: # of cubes = a Problem Solving and Mathematical Reasoning: Formulation. The student…asks and answers a series of appropriate questions in pursuit of a solution and does so with minimal “scaffolding” in the form of detailed guiding questions. b Problem Solving and Mathematical Reasoning: Implementation. The student…selects appropriate mathematical concepts and techniques from different areas of mathematics and applies them to the solution of the problem…. c Problem Solving and Mathematical Reasoning: Conclusion. The student…concludes a solution process with a useful summary of results…. The response shows the formulation and implementation of an approach to a difficult and non-routine problem, and clearly indicates the results of this approach. There are two independent derivations of the second result, one starting with the area A and the other starting with the volume V. a Mathematical Skills and Tools: The student carries out…symbol manipulations effectively…. c Mathematical Communication: The student organizes work and presents mathematical procedures and results clearly, systematically, succinctly, and correctly. Although the response is brief, it is easy to follow and to the point.