The student analyzed the locker scenario successfully, studying
the factors (“only after a long while of experimenting”)
of natural numbers and how those divisors could correspond to the
opening and closing of lockers. The student carefully explained
the correspondence to the reader and attempted to justify the work,
going beyond the level required by the task prompt.
In attempting to generalize the locker results for larger, then
arbitrary, numbers of students and lockers, the student created
a large table of values and inferred from them the desired relationship.
Troubled in his attempt to write a symbolic rule for the number
of lockers that would remain open (a more significant result than
that which was sought by the prompt), he overcame the problem by
developing his own rendition of the classic greatestinteger function.
This work sample shows strength across several standards, most
thoroughly in relation to
and . The
student took an apparently unfamiliar situation (becoming more difficult
as “the locker problem” continues its spread across middle
grade classrooms) and determined a mathematical context and approach
that yielded a solution. He used appropriate representations, organization,
and reasoning both to attain and to explain his solution.
Observe that the prompt asks “Which lockers will remain open?”
rather than “How many lockers?” which is the moreinvolved
question the student answered in the work that follows. The “Extra!”
section asks for some generalization, but it still does not pose
the question of “How many?”
d
Number and Operation Concepts: The student
is familiar with characteristics of numbers (e.g., divisibility,
prime factorization)….
The student analyzed the parity of the numbers of factors and how
that determines a locker’s final position, explaining why locker
numbers with even numbers of factors end up closed and why those
with odd numbers of factors remain open. The student went on to
determine that it is only the perfect squares that have odd numbers
of factors. Here (“…because they have the same number
of factors as their square root and another factor—itself”),
though, there is a flaw in the logic. It is true that squares of
prime numbers have just one more factor than their square roots,
and it seems the student saw this in 4, 9, and 25 (,
and
) and
made too strong a claim. Observe that he did recognize that 16 ()
gives a different result and he attempted to account for this (“8
is also here to be multiplied by 2”). However, instead of concluding
that extra factors exist for squares of composite numbers, he only
recognized that they exist for squares of square numbers.
A fundamental
idea underlying the locker problem—that square numbers have
an odd number of factors, whereas other natural numbers have even
numbers of factors—is just an extension of a common occurrence
in classrooms when factorization is taught. “Pairing”
of factors is often done by teachers and students to check for missing
numbers in a factorization. For example, the factors of 63 are 1,
3, 7, 9, 21, and 63; they can be paired as 1 x 63, 3 x 21, and 7
x 9, with all pairs giving product 63. Perfect squares, though,
have a “twist”: the factors of 36 are 1, 2, 3, 4, 6, 9,
12, 18, 36; 1 x 36, 2 x 18, 3 x 12, and 4 x 9 are all paired, leaving
6 by itself (
= 36).
a
Function and Algebra Concepts: The student
discovers, describes, and generalizes patterns…and represents
them with variables and expressions.
c
Function and Algebra Concepts: The student
analyzes tables…to determine functional relationships.
The student
created his own version of the “greatest integer function,”
often symbolized [x], which returns the largest integer less than
or equal to the real number x. Presumably unacquainted with this
function, he created it for himself with his rule of computing the
square root and subtracting the remainder, .
Each variable is defined and explained well.
The work demonstrates understanding of concepts of irrational numbers
that surpasses the demand of ,
which focuses on proficiency with the rationals in the middle school.
Notice that the student recognized that, except for square numbers,
the written decimal forms of his square roots are only approximations
of the numbers’ actual values (“the square root is 22.36068”).

b
Problem Solving and Mathematical Reasoning:
Implementation. The student…invokes problem solving strategies,
such as…organizing information in a table….
The student
noticed early in the problem the need to examine the numbers of
factors of the natural numbers and to think about the implications
of the factorization on whether or not the corresponding locker
would be open or closed in the end. This is astute, particularly
so early in the problem. More commonly, students notice experimentally
that those lockers whose numbers are perfect squares (of natural
numbers) remain open when all is done. It is only on reflection
(asking “Why?”) that they reason and realize the relationship
between the perfect squares and their odd numbers of factors.
c
Problem Solving and Mathematical Reasoning:
Conclusion. The student…generalizes solutions and strategies
to new problem situations.
The student
tackled the related question of “How many open lockers?”
This is no more complicated for the 100 locker scenario, but it
is considerably more involved for a generalization to n lockers.
d
Problem Solving and Mathematical Reasoning:
Mathematical reasoning. The student…
• formulates conjectures and argues why they must be or seem
true…
• makes justified, logical statements.
f
Mathematical Skills and Tools: The student
uses equations, formulas, and simple algebraic notation appropriately.
g
Mathematical Skills and Tools: The student
reads and organizes data on charts….
h
Mathematical Skills and Tools: The
student uses…calculators,…as appropriate, to achieve solutions.
a
Mathematical Communication: The student uses
mathematical language and representations with appropriate accuracy,
including numerical tables….
b
Mathematical Communication: The student organizes
work, explains facets of a solution…in writing, labels drawings,
and uses other techniques to make meaning clear to the audience.
The answer
to the task’s primary question (“Which lockers will remain
open?”) deserves more prominent placement so as not to be overlooked
by the reader. Despite this minor point, the entire piece of work
is clear and well organized.
c
Mathematical Communication: The student uses
mathematical language to make complex situations easier to understand.
The student
used examples to clarify confusion that might arise from the function
he created. Observe that he provided examples to illustrate both
of the cited cases, with remainder and without remainder. Of course,
the latter could be subsumed in the former by saying, “with
remainder of zero.”
No apology
necessary!
