
WASHINGTON
STATE MATHEMATICS COUNCIL 2000
MIDDLE SCHOOL MATH OLYMPIAD Session I PROBLEM SOLVING 
5678 
Scoring
Guidelines: The Tiles Game
CORRECT ANSWER [Scores
of 0, 2, 3, and 4 are possible.]
Points 
Look
for the following: 
4 
Grades
7 and 8: 27/216,
1/8, .125, or 12.5% Grades
5 and 6: 12/64,
3/16, .1875, or 18.75% 
3 
·
If answer is in fractional form, denominator and numerator
are both correct in first form of answer, but simplified form is incorrect. ·
If answer is in decimal or percent form, original
(fraction) form of answer is correct, but conversion to decimal or percentage
is incorrect. 
2 
·
Numerator or
denominator used to identify answer is incorrect, leading to incorrect
fraction, decimal, or percent form of answer. (One of the two – numerator or
denominator  is/was originally
correct.) 
0 
·
Answer is not correct (and would not have been with
correct calculation). 
PROBLEM
UNDERSTANDING [Scores of 0, 2, 3, and 4 are possible.]
Students were asked: Do you
understand what the problem is asking?
What did you notice about patterns?
How did that help you solve the problem?
Points 
Look
for the following: 
4 
Grades
7 and 8: ·
Discovers there are 6 x 6 x 6 or 216 permutations (i.e.,
shows strong understanding that there is a need to discover all possible permutations of 3 tiles). ·
Discovers that there are 27 permutations resulting in a
sum of 7 or that there are 6 combinations resulting in a sum of 7 (i.e.,
shows strong understanding of need to find the permutations that will sum to
7, whether this is done by combinations first or by listing permutations). ·
Shows evidence of approaching the problem in an orderly
way Grades
5 and 6: ·
Discovers there are 4 x 4 x 4 or 64 permutations (i.e.,
shows strong understanding that there is a need to discover all possible permutations of 3 tiles). ·
Discovers that there are 12 permutations resulting in a
sum of 5 or that there are 3 combinations resulting in a sum of 5 (i.e.,
shows strong understanding of need to find the permutations that will sum to
5, whether this is done by combinations first or by listing permutations). ·
Shows evidence of approaching the problem in an orderly
way. 
3 
Grades
7 and 8: ·
Discovers there are 6 x 6 x 6 or 216 permutations (i.e.,
shows understanding of the need to discover all possible permutations of 3
tile numbers, even if method used is not completely valid). ·
Shows understanding of the need to determine the number of
these permutations that result in a sum of 7, even if method for doing so is
not completely valid. ·
Shows evidence of approaching the problem in an orderly
way. Grades
5 and 6: ·
Discovers there are 4 x 4 x 4 or 64 permutations (i.e.,
shows understanding of the need to discover all possible permutation of 3
tile number, even if method used is not completely valid). ·
Shows understanding of the need to determine the number of
these permutations that result in a sum of 5, even if method for doing so is
not completely valid. ·
Shows evidence of approaching the problem in an orderly
way. 
2 
·
Shows understanding that the problem has two parts
(numerator & denominator). ·
Shows some evidence of approaching – or attempting to
approach  the problem in an orderly way. 
0 
·
No evidence that an attempt was made to understand the
problem and/or to communicate this understanding. 
STRATEGY
Students were
asked: What is your strategy for solving
the problem (e.g., table, list, etc.)?
Is your strategy valid? Is it
carried out completely?
Points 
Look
for the following: 
4 
·
A valid strategy is used: makes a table, or uses logic and
calculations. Strategy is completely carried through. 
3 
·
A valid strategy is used, but may not be completely
carried through or may not be used for completely valid reasons (making
strategy not 100% valid). 
2 
·
A strategy was applied, but reasoning is confused or not
logical. 
1 
·
An attempt was made to use a strategy, but strategy was
poor (very incomplete). 
0 
·
No evidence of a strategy 
COMMUNICATION
Students were
asked: Is your reasoning about the
problem and about your strategies clearly communicated  using words,
pictures, tables, symbols?
Points 
Look
for the following: 
4 
·
Described problem understanding, strategy, and solution
clearly and completely, stepbystep, with all steps included. ·
Used appropriate labels, terminology, and symbols. 
3 
·
Described problem understanding, strategy, and solution,
but steps may be missing from the explanation. The evaluator is required to infer that intermediate steps were
performed correctly. ·
Used appropriate labels, terminology, and symbols (with
perhaps very minor errors). 
2 
·
Strategy was described, but steps may be missing or out of
order. Understanding
of problem may or may not be directly communicated (but should be indirectly
communicated). 
1 
·
An attempt was made to explain strategy, but reasoning is
confusing. Understanding
of problem is poorly communicated, directly or indirectly. 
0 
·
No sentences or phrases were provided to explain strategy,
understanding, solution. 
Students were asked: How did you
check that your answer was reasonable?
Points 
Look
for the following: 
4 
There is evidence that the result was checked. 
2 
A claim was made that the answer was checked for reasonableness, but there is no substantiating evidence of this check. 
0 
There is no evidence of or claim about a check for reasonableness. 