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:< /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

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:<12/span>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.