These work samples were drawn from two different classrooms although the
task given to the students was the same:
The length of a string that is 160 centimeters long
is cut into three pieces. The second piece is 40% longer than the first
and the third is 20% shorter than the first. How long is each piece?
These samples of student work were
produced under the following conditions: 
 alone 
in a group 
in class 
 as homework 
with teacher feedback 
with peer feedback 
timed 
opportunity for revision 
Sample 1 shows a standard approach that a student of algebra might take.
The student appropriately used algebra to represent the first, second,
and third piece of string.
Sample 2 shows a different approach. Here the student
used an understanding of percent, decimal, and fraction concepts, as well
as proportional reasoning, to achieve a solution.
These work samples illustrate standardsetting
performances for the following parts of the standards:

a 
Number and Operation
Concepts: Consistently and accurately add, subtract, multiply, and
divide rational numbers. 
b 
Number and Operation
Concepts: Use the inverse operation to determine unknown quantities
in equations. 
c 
Number and Operation
Concepts: Consistently and accurately apply and convert the different
kinds and forms of rational numbers. 
e 
Number and Operation
Concepts: Interpret percent as part of 100 as a means of comparing
quantities of different sizes. 
b 
Function and Algebra
Concepts: Represent relationships. 
d 
Function and Algebra
Concepts: Find solutions for unknown quantities in simple equations. 
a 
Problem Solving
and Mathematical Reasoning: Formulation. 
b 
Problem Solving
and Mathematical Reasoning: Implementation. 
c 
Problem Solving
and Mathematical Reasoning: Conclusion. 
b 
Mathematical Communication:
Organize work, explain facets of a solution orally and in writing,
label drawings, and use other techniques to make meaning clear to
the audience. 
Sample 1


a
Number and Operation Concepts: The student
consistently and accurately adds, subtracts, multiplies and divides
rational numbers using appropriate methods…with the different
kinds and forms of rational numbers…written as decimals,
as percents, or as proper, improper, or mixed fractions.
Sample
1 shows accurate multiplication and division of simple decimals.
This is evidenced here as part of the solution to the equation.
The student used different methods to complete these multiplication
problems.
In Sample 2, the student made a sketch of the string divided into
10, 14, and 8 pieces respectively and showed that the total length
of the three sections equals 160 cm. This shows understanding
of “10” as a useful common divisor. The student used
10 as the denominator to express the percent increase and percent
decrease of the second and third pieces as fractions. Although
the student’s diagram does not show marks in the third section,
this does not make the diagram incomplete. The student clearly
stated, “3rd section 8 pieces.”
b
Number and Operation Concepts: The student
uses the inverse operation to determine unknown quantities in
equations.
The student
in Sample 1 used the inverse operation to eliminate the 3.20 coefficient
of the variable x.
c
Number and Operation Concepts: The student
consistently and accurately applies and converts the different
kinds and forms of rational numbers.
The work in Sample 2 demonstrates the student’s understanding
of the equivalency of these numbers as percents, decimals, and
fractions. Based on this, the student divided the first piece
of string into 10 equal parts and represented the percent increase
and percent decrease of the second and third pieces of string
as “”
and “”
respectively.
e
Number and Operation Concepts: The student
interprets percent as part of 100 as a means of comparing quantities
of different sizes.
The student
in Sample 1 correctly represented the three pieces of string in
relation to 100%. The student assigned a value of 140% to the
second piece of string and 80% to the third one.
The student
in Sample 2 demonstrated a competency in the use of percent increase
and percent decrease of a quantity.
b
Function and Algebra Concepts: The student
represents relationships...
The student
in Sample 1 gave the length of each piece of string as a function
of x, the length of the first piece.
d
Function and Algebra Concepts: The student
finds solutions for unknown quantities in simple equations.
The student
in Sample 1 correctly solved for x in this equation.

a
Problem Solving and Mathematical Reasoning:
Formulation. The student participates in the formulation of problems;
that is, given the basic statement of a problem situation, the
student extracts pertinent information from situations and figures
out what additional information is needed.
The student
in Sample 1 accurately showed variables with representations for
each piece of string.
b
Problem Solving and Mathematical Reasoning:
Implementation. The student makes the basic choices involved in
planning and carrying out a solution; that is, the student…
• invokes problem solving strategies, such as
illustrating with sense–making sketches to clarify situations.
• solves for unknown…quantities
using algebra.
The student
in Sample 1 set up the equation correctly and used algebraic methods
to solve it.
c
Problem Solving and Mathematical Reasoning:
Conclusion. The student provides closure to the solution process
through summary statements.
The student
in Sample 1 correctly solved each equation in the legend and presented
a summary of the results.
140% of 50 = 70 and
80% of 50 = 40.
The last statement, “Answer: The first piece of string is
50 cm…” shows complete understanding of the problem.
The student in Sample 2 recognized the proportional relationship
between the total length of the string (160 cm) and the number
of sections made on the string (32 sections). The statement, “Since
the total length is 160 cm, therefor [sic] it’s 5 cm each
piece 160÷32 = 5,” demonstrates understanding of proportional
relationships.
Sample 2
b
Mathematical Communication: The student
uses the language of mathematics and communicates about mathematics;
that is, the student: organizes work…labels drawings and
uses other techniques to make meaning clear to the audience.
In Sample 2 the student displayed the work in an
organized manner, explained the different facets of the solution,
and labeled the drawings to make the meaning clear to the audience.
There are some errors in these samples of student
work (e.g., sentence structure in the first sentence of Sample
1 and spelling (“therefor” instead of “therefore”
in Sample 2). These errors do not detract from the quality of
the mathematics.

