HIGHLINE SCHOOL DISTRICT
MATH OLYMPIAD		April 7, 2001
1/2	The Great Escape Problem Scoring Guide

Correct Answer

Points	Look for the following:
4	Correct answer, 84 coins, is given.
0	Answer is not correct

Strategy

Points	Look for the following:
4	Chose a valid strategy.(See discussion of valid strategies at the end of this rubric) Applied the strategy correctly and completely
3	Chose a valid strategy Applied the strategy correctly but with small omissions
2	Chose a valid strategy Strategy was partially or incompletely developed
0	No attempt at response or inappropriate strategy

Problem Understanding

Points	Look for the following:
4	Shows understanding of the concept of fractional parts Understood that each fractional reduction was on remaining coins
2	Understood concept of fractional parts Either did not understand that each fractional reduction was on remaining coins or computed remaining fractional parts incorrectly
0	No attempt at response.

Communication

Points	Look for the following:
4	Described strategy and solution clearly and completely, step-by-step Correct presentation of solution, with correct labels, terminology, and symbols
3	Described strategy correctly One step may not be described, requiring you to guess that it was done
2	Describe a strategy in some fashion, but discussion is incomplete. Steps are missing (they jumped to the answer)
0	No attempt at response.

Reasonable Result

Points	Look for the following:
4	Reworked problem using a different strategy
3	Used their answer to run through each of the calculations forward to arrive at the final number of coins, 7.
2	Made some statement about checking their answer
0	No attempt to demonstrate reasonableness.

Valid strategies:

1. Visual representation: Drew a rectangle similar to the one below representing the transactions as they occur: Where the columns are thirds of the coins and the rows are fourths. The columns and rows are shaded or otherwise annotated to show the fraction of the total coins that are given to each guard (1,2,3,or 4):

   1	2	3
   1	2	4
   1	2	4
   1	2

   So 1/12 of the original is left after all the guards have taken their coins, so 1/12 of the original is 7, therefore, the original number of coins is 12*7 = 84
   Note: a pie chart or other similar representation works also.
2. Guess and Check: Here is a typical guess-and-check scenario:
   Guess that Spavadio had 75 coins.
   Gave 1/3 to the first guard, leaving 50 coins.
   Gave 1/2 to the second guard leaving 25 coins.
   Gave 1/4 to the third guard...whoops! Can't take 1/4 of 25
   New guess: 84 coins.
   Gave 1/3 to first guard, leaving 56 coins.
   Gave 1/2 to second guard, leaving 28 coins.
   Gave 1/4 to third guard, leaving 21 coins.
   Gave 2/3 to fourth guard, leaving 7 coins.It worked!
3. Work backwards:
   
   1. Gave 2/3 to last guard and had 7 remaining, so he must have had 21 coins before he got to the last guard. (2/3 of 21 is 14,21-14 = 7)
   2. Gave 1/4 to third guard, so he must have had 28 coins before he got to the third guard (1/4 of 28 is 7, 28-7 = 21)
   3. Gave 1/2 to second guard, so he must have had 56 coins before he got to the second guard(1/2 of 56 = 28,56-28=28)
   4. Gave 1/3 to first guard, so he must have had 84 coins before he got to the first guard (1/3 of 84 = 28, 84-28=56)
4. Use equations: The above working backwards strategy can be used with equations to produce the results at each backward step:
   1. Let X be number of coins after 3rd guard. So:
      (1/3)X = 7, so X = 21;(After giving 2/3, 1/3 X is left)
   2. Let Y be the number of coins after 2nd guard. So:
      (3/4)Y= 21, so Y = 28; (after giving 1/4, 3/4 Y is left)
   3. Let Z be the number of coins after the 1st guard. So:
      (1/2)Z = 28, so Z = 56; (after giving 1/2, 1/2 Z is left)
   4. Let C be the number of coins before the 1st guard. So:
      (2/3)C = 56, so C = 84, the correct answer.