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.
bNumber 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.”
dNumber 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.
dFunction and Algebra Concepts: The student
finds solutions for unknown quantities in simple equations.
bProblem 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….
cProblem Solving and Mathematical Reasoning:
Conclusion. The student…verifies… results with respect
to the original problem situation….
dProblem Solving and Mathematical Reasoning:
Mathematical reasoning. The student…makes justified, logical
statements. The “process
of elimination” applies.
= d
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
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.
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.
The “process
of elimination” applies.