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, step-by-step, 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.

REASONABLE RESULT

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.