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The task
The context of this task is a science fair with three attending middle schools of different populations. The student must consider the numbers of students as fractions and percents of the total. The student also must appropriately divide among the schools both the area in which the fair will be conducted and the cost of the fair.

Circumstance 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 is a Released Task from the New Standards reference examination. Students were expected to need no more than fifteen minutes to solve it.

This work sample illustrates a standard-setting
performance for the following parts of the standards:
a Number and Operation Concepts: Add, subtract, multiply, and divide rational numbers.
c Number and Operation Concepts: Apply and convert the different kinds and forms of rational numbers.
f Number and Operation Concepts: Reason proportionally to solve problems involving equivalent fractions.
c Problem Solving and Mathematical Reasoning: Conclusion.
a Mathematical Skills and Tools: Compute accurately with arithmetic operations on rational numbers.

What the work shows

a Number and Operations Concepts: The student…adds, subtracts, multiplies, and divides rational numbers….
The student made appropriate computations and accurately divided the cost of the science fair proportionally, according to the approximate populations given at the beginning of the task.

c Number and Operations Concepts: The student…applies and converts the different kinds and forms of rational numbers.
The work makes clear the relationship between fractions and quotients, that “ = 3÷10,” etc. He converted the fractions into percentages.

f Number and Operations Concepts: The student…reasons proportionally to solve problems involving equivalent fractions….
The rectangle is not divided precisely in 5:3:2 ratio, but it is very nearly so. The partitions are drawn sufficiently for the given task. The school populations were approximations anyway (“about 1,000 students,” etc.).
In part two, the student exhibited understanding of some fundamental concepts of fractions. He used fractions to express a relationship where the whole is not a single item but a set (of 2,000 students). He determined the whole given the parts. He used equivalent fractions and simplified here, as appropriate.
The student accurately divided the cost of the science fair proportionally, according to the approximate populations given at the beginning of the task.

c Problem Solving and Mathematical Reasoning: Conclusion. The student verifies and interprets results with respect to the original problem situation….
The “check” that the three percentages computed must add up to 100%, the total, is more convincing evidence of understanding of percent than the actual rote computations performed by the student to obtain 50%, 30%, and 20%.
The student checked that the charges computed for each school indeed add up to the $300 cost of the fair.

a Mathematical Skills and Tools: The student computes accurately with arithmetic operations on rational numbers.