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The task
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?

Circumstances of performance
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

What the work shows
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 standard-setting
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.