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.]

 Points Look for the following: 4 Grades 7 and 8:< /span> .         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 completelyvalid). .         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