DROPPING LOWEST GRADES MATC Math Club January 26

DROPPING LOWEST GRADES MATC Math Club January 26, 2006 Jonathan Kane, Professor University of Wisconsin-Whitewater

What is dropping lowest grades? • Obtaining the largest raw score • Obtaining the largest mean score

TABLE 1: Alan’s Quiz scores Quiz 1 2 Score 2 6 Possible 8 12 Percentage 25 50 3 24 40 60 4 3 4 75 5 6 24 25 DROP TWO GRADES: Largest raw score Get the largest raw score by dropping Quiz 1 & 4 to give a raw score of 6 + 24 + 6 = 36. NOTE: this drops the grade with the largest percentage.

K = {1, 2, 3, 4, … k} For j K, quiz j has integer maximum possible score. A student earns an integer score. To drop r scores, find is maximized. of size k – r so that the ratio

TABLE 2: Beth’s Quiz Scores Quiz 1 Score 80 Possible 100 Percentage 80 2 20 100 20 3 1 20 5 DROP ONE GRADE • DROP #1: • DROP #2: • DROP #3: (20 + 1) / (100 + 20) = 17. 5% (80 + 1) / (100 + 20) = 67. 5% (80 + 20) / (100 + 100) = 50% The best grade to drop is quiz 2. NOTE: This retains quiz 3 with the lowest percentage and lowest raw score.

TABLE 3: Carl’s Quiz Scores Quiz 1 2 Score 100 42 Possible 100 91 Percentage 100 46 3 14 55 25 4 3 38 8 BEST ONE GRADE TO DROP: Quiz 4 (100+42+14) / (100+91+55) = 63. 4% BEST TWO GRADES TO DROP: Quizzes 2 and 3 (100 + 3) / (100 + 38) = 74. 6%

TABLE 4: Dale’s Quiz Scores Quiz 0 1 2 3 4 5 Score 20+c 21 -b 1 22 -b 2 23 -b 3 24 -b 4 25 -b 5 Possible 40 42 44 46 48 50 Quiz 6 7 8 9 10 Score 26 -b 6 27 -b 7 28 -b 8 29 -b 9 30 -b 10 Possible 52 54 56 58 60 Percetage

Average of the grades quiz 0 and the quizzes in

BETH’S QUIZ GRADES DROP FIVE GRADES • Each bj = 1 • If c = 4, it is best to drop quizzes 1, 2, 3, 4, 5. • If c = 6, it is best to drop quizzes 6, 7, 8, 9, 10. NOTE: A very small change in the problem completely changes the outcome.

BETH’S QUIZ GRADES • C = 11 and each bj = 2 DROP FOUR GRADES • It is best to drop quizzes 1, 2, 3, 4. • DROP FIVE GRADES • It is best to drop quizzes 6, 7, 8, 9, 10. NOTE: The best five grades to drop does not contain any of the best four grades to drop.

COMMON APPROACHES TO FINDING ALGORITHMS • Brute Force • Greedy Algorithm • Dynamic Programming

FINDING THE BEST r OF k GRADES TO DROP BY BRUTE FORCE Consider all the subsets of K. To drop 10 out of 100 grades, you would need to consider 17, 310, 309, 456, 440 subsets. At 1 Million subsets per second, it would take about 200 days to consider them all.

GREEDY ALGORITHM • Take what looks like the best first step. • Then take what looks like the best next step. • Continue until a solution is found. • Maybe it will be the optimal solution.

GREEDY ALGORITHM SHORTEST PATH

DYNAMIC PROGRAMMING • Find the best solution for a small problem. • Increase the size of the problem, and find the best solution for the new size. • Continue increasing the size of the problem in increments until the final size is reached.

DYNAMIC PROGRAMMING KNAPSACK PROBLEM • Your knapsack can hold a weight of 50. • You can take as many of each item as you can fit. • You want to maximize the sum of the benefits.

ITEM TYPE WEIGHT BENEFIT 1 10 5 2 20 12 3 30 20 4 40 24 5 50 30

BEST SOLUTION IF YOU CAN CARRY ONLY A WEIGHT OF 10 1 #1 for a benefit of 5. BEST SOLUTION IF YOU CAN CARRY ONLY A WEIGHT OF 20 2 #1 for a benefit of 10. 1 #2 for a benefit of 12. BEST SOLUTION IF YOU CAN CARRY ONLY A WEIGHT OF 30 1 #1 and best weight 20 for a benefit of 17. 1 #3 for a benefit of 20.

BEST SOLUTION IF YOU CAN CARRY ONLY A WEIGHT OF 40 1 #1 and best weight 30 for a benefit of 25. 1 #2 and best weight 20 for a benefit of 24. 1 #4 for a benefit of 24. BEST SOLUTION IF YOU CAN CARRY ONLY A WEIGHT OF 50 1 #5 for a benefit of 30. 1 #4 and best weight 10 for a benefit of 30. 1 #3 and best weight 20 for a benefit of 32. Best solution: 1 item #2 and 1 item #3.

Greedy algorithms and dynamic programming algorithms will not work for the dropping lowest grades problem because the best solution for n – 1 grades is unrelated to the best solution for n grades.

THE OPTIMAL DROP FUNCTION Want to maximize Rewrite as Define for each j, Want to find of size k – r and q so where q is as large as possible.

Define the Optimal Drop Function F(q) is easy and efficient to calculate. For a given q, just calculate the k fj(q) values and throw away the r smallest.

fj(q) for dropping 2 of Carl’s 4 quiz grades.

FIND ALL INTERSECTOINS There are k lines. Find the intersections of lines. Then calculate F(q) for each q at each intersection point.

BISECTION METHOD • Start with x 1 and x 2 where the function F crosses the axis between x 1 and x 2. • Find x 3 half way in between. Determine on which side of x 3 the function F crosses the axis. • Continue until you are satisfied.

NEWTON’S METHOD? ? Start with any q 1. Find F(q 1). Follow the line segment to q 2 on the axis. Repeat until qbest is found.