1. Dropping Lowest Grades What score(s) should be dropped to maximize a students grade? Dustin M. Weege Concordia College 2008 Secondary Mathematics Education Not always what you think it to be

2. The Plan Natural ideas for dropping grades ▫Flaws Mention of other possible methods for determining the best scores to drop Optimal Drop Function

3. If the teacher is basing the final grade on the student’s raw score Drop the lowest score earned ▫Notice the percentages Table 1: Alan’s quiz scores Quiz12345 Score262436 Possible81240424 Percentage2550607525 “Natural” ideas for dropping grades

4. Also Consider: Drop the lowest score earned ▫Notice the percentages Table 1: Alan’s quiz scores Quiz12345 Score462426 Possible81240424 Percentage50 605025 “Natural” ideas for dropping grades

5. Drop the lowest percentage ▫What should be dropped?  Drop 3 (lowest percentage)?  Total percentage of  Drop 2?  Total percentage of Table 2: Beth’s quiz scores Quiz123 Score80201 Possible100 20 Percentage80205 “Natural” ideas for dropping grades

6. What do we know? Highest percentage will always be in the optimal retained set. ▫Reason: if S has grades that are less than the largest percentage, then the average will be less than the largest percentage. The reverse is not necessarily true ▫The grade with the smallest score does not necessarily appear in the optimal deletion set ▫Ex. Beth Table 2: Beth’s quiz scores Quiz123 Score80201 Possible100 20 Percentage80205

7. Dropping more than one grade Conflict arises depending on the number of scores dropped ▫Remove 1 score:  quiz 4 would be best to drop  63.4% ▫Remove 2 scores  quizzes 2 & 3 would be best to drop  74.6% Table 3: Carl’s quiz scores Quiz1234 Score10042143 Possible100915538 Percentage10046258

8. How can we find the best set? Brute Force ▫Try all possibilities ▫Flaw: Can take too long especially with large quantities of scores Greedy Algorithm ▫Do the best in each situation ▫Flaw: ex. Carl  Drop 4 & 3  Yields a total score of 100+42=142 out of 191  74.3%  Drop 2 & 3  Yields a total score of 100+3=103 out of 138  74.6% ▫Compare : (2&4  73.5%; 1&4  38.4%) Table 3: Carl’s quiz scores Quiz1234 Score10042143 Possible100915538 Percentage10046258

9. Can’t we come up with something? You know it. Leave it up to a MIT student and a Professor who received his BA in Mathematics at U of M- Duluth Jonathan M. Kane Not actually Daniel M. Kane, but this showed up in a Google Images search

10. Terminology - also found on handout k  K – Total number of assignments; s.t. K  j – quiz # (1,2,3,…k) s.t. k>0 m j – earned points on quiz n j – possible points on quiz r – score(s) dropped/”deletion set” “optimal deletion set” – set of quiz(zes) dropped in order to yield the highest possible grade k-r – Number of scores counted/“retained” S – retained grades (*note: S  K) S best – Optimal retained set q – the average score in S q best – best possible value for q “Optimal Drop function” -

11. Optimal Drop Function q – Defined as: q best – q is defined s.t. the S score is maximized Define for every j: ▫By substitution we get: Equation 1 Equation 2 Equation 3

12. Optimal Drop Function Notice: q is the average score in S is a linear, decreasing function iff

13. Optimal Drop Function q is the average score in S is a linear, decreasing function is also a linear, decreasing function ▫Example from Carl iff Table 3: Carl’s quiz scores Quiz1234 Score10042143 Possible100915538 Percentage10046258

14. Optimal Drop Function F(q) is the max of the sum of linear, decreasing functions, ▫F(q) must be a piecewise, linear, decreasing function Equation 4

15. Optimal Drop Function F(q) is the max of the sum of linear, decreasing functions, ▫F(q) must be a piecewise, linear, decreasing function Equation 4

16. Optimal Drop Function Now, find the rational number q, so that  Recall Equation 4

17. Optimal Drop Function Next, we need to find the line that yields the highest possible q value s.t.. ▫From this we are able to determine ▫  composed of the top k-r f j (q) values.

18. Optimal Drop Function Tasks: ▫Evaluate each for each j ▫Identify the k-r largest from the values ▫S is the set of j values from the largest k-r values ▫Calculate

19. Optimal Drop Function – Ex. Carl Drop 2 scores: Possibilities for S: 1&21&31&4 2&32&43&4 Estimate: q to be.75 Table 3: Carl’s quiz scores Quiz1234 Score10042143 Possible100915538 Percentage10046258

20. Relevance Determining the best set of scores to drop Understanding Computer gradebooks Determine cuts that are necessary to be made in a company based on several assessments

21. Sources 1.Daniel M. Kane and Jonathan M. Kane Dropping Lowest Grades Mathematics Magazine, (2006) 79 (June) pp. 181- 189.Dropping Lowest Grades 2.Daniel Kane's Homepage http://web.mit.edu/dankane/www/http://web.mit.edu/dankane/www/ 3.Jonathan Kane Home Page http://faculty.mcs.uww.edu/kanej/kane.htm http://faculty.mcs.uww.edu/kanej/kane.htm 4.http://www.ams.org/news/home-news-2007.htmlhttp://www.ams.org/news/home-news-2007.html 5.Daniel Biebighauser’s brain