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The task
Students were given the following assignment:

In the clues below, each variable represents a different digit 0-9. Determine the value of each variable.

g + g + g = d
j + e = j
= d
b + g = d
f - b = c
i / h = a (h > a)
a x c = a

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:
b Number and Operation Concepts: Use and understand the inverse relationships between multiplication and division, addition and subtraction.
d Number and Operation Concepts: Be familiar with characteristics of numbers and with properties of operations.
d Function and Algebra Concepts: Find solutions for unknown quantities in simple equations.
b Problem Solving and Mathematical Reasoning: Implementation.
c Problem Solving and Mathematical Reasoning: Conclusion.
d Problem Solving and Mathematical Reasoning: Mathematical reasoning.

What the work shows
The task is a logic puzzle. The work shows strength in mathematical reasoning. Strong knowledge of properties of numbers and operations is particularly evident.

b Number and Operation Concepts: The student uses and understands the inverse relationships between addition and subtraction, multiplication and division…; uses the inverse operation to determine unknown quantities in equations.
The student used transitivity and inverse operations to determine that g=3. A possible solution was inadvertently lost when the student decided to “divide each side by g,” because division by zero is undefined. She should have been aware, though, that g=0 also satisfies =3g. As in the second step, this possibility would have been dismissed because “e is already 0.”

d Number and Operation Concepts: The student is familiar with characteristics of numbers…and with properties of operations….
The student recognized the additive identity, 0.
The student recognized the multiplicative identity, 1. She realized the effect of multiplication by zero. The latter was dismissed as a possibility because zero was used elsewhere.

d Function and Algebra Concepts: The student finds solutions for unknown quantities in simple equations.

b Problem Solving and Mathematical Reasoning: Implementation. The student…
• invokes problem solving strategies…

The student paused to review what she had done and what remained. Understanding and solving the remaining clues was easier after eliminating the digits already used once.
• solves for unknown or undecided quantities….

c Problem Solving and Mathematical Reasoning: Conclusion. The student…verifies… results with respect to the original problem situation….


d Problem Solving and Mathematical Reasoning: Mathematical reasoning. The student…makes justified, logical statements.

The “process of elimination” applies.