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The task
The student is called on to analyze a scenario in which school lockers are alternately opened and closed in a particular manner. The task calls for evidence from , as the student determines how to describe the situation mathematically and achieve a solution. That solution should provide the student ample opportunity to show prowess in , since this problem lends itself to features that illustrate this standard—appropriate mathematical language, well-organized work and ideas, clarity, assorted mathematical representations, etc.

As part of the problem conclusion, if not earlier in the student’s work, the reflective question “Why?” should come to mind upon discovery that only those lockers whose numbers are perfect squares of natural numbers are open in the end. The student who investigates the reason for this will realize the relationship between the factors that divide those perfect squares and the procedure by which lockers were opened and closed. This will demonstrate considerable achievement under .

Finally, the “Extra!” section of the prompt leads students to generalize and “find a rule for any number,” providing an opportunity to show evidence under .


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
This work sample illustrates a standard-setting
performance for the following parts of the standards:
d Number and Operation Concepts: Be familiar with characteristics of numbers.
a Function and Algebra Concepts: Discover, describe, and generalize patterns.
c Function and Algebra Concepts: Analyze tables to determine functional relationships.
b Problem Solving and Mathematical Reasoning: Implementation.
c Problem Solving and Mathematical Reasoning: Conclusion.
d Problem Solving and Mathematical Reasoning: Mathematical reasoning.
f Mathematical Skills and Tools: Use equations, formulas, and simple algebraic notation appropriately.
g Mathematical Skills and Tools: Read and organize data on charts.
h Mathematical Skills and Tools: Use calculators, as appropriate, to achieve solutions.
a Mathematical Communication: Use mathematical language and representations with appropriate accuracy.
b Mathematical Communication: Organize work, explain a solution in writing, and use other techniques to make meaning clear to the audience.
c Mathematical Communication: Use mathematical language to make complex situations easier to understand.

What the work shows

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 greatest-integer 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 more-involved 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!