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.
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 standardsetting
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. 

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.
