|
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.
- In terms of V and A, find the length of
the side of one of these small cubes.
- 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.
| 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:
- Since the volume of one small cube
is l³, the total volume is V = nl³.
- 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.
|
|
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.
|
|
