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
The Plan
• Natural ideas for dropping grades
▫ Flaws
• Mention of other possible methods for
determining the best scores to drop
• Optimal Drop Function
“Natural” ideas for dropping grades
• If the teacher is basing the final grade on the
student’s raw score
• Drop the lowest score earned
Table 1: Alan’s quiz scores
Quiz
1
Score
2
Possible
8
Percentage
25
▫ Notice the percentages
2
6
12
50
3
24
40
60
4
3
4
75
5
6
24
25
“Natural” ideas for dropping grades
Also Consider:
• Drop the lowest score earned
Table 1: Alan’s quiz scores
Quiz
1
Score
4
Possible
8
Percentage
50
▫ Notice the percentages
2
6
12
50
3
24
40
60
4
2
4
50
5
6
24
25
“Natural” ideas for dropping grades
• Drop the lowest percentage
▫ What should be dropped?
Table 2: Beth’s quiz scores
Quiz
1
2
3
Score
80
20
1
Possible
100
100
20
Percentage
80
20
5
 Drop 3 (lowest percentage)?
 Total percentage of (80  20)
 50%
(100

100)
 Drop 2?
 Total percentage of (80  1)
 67.5%
(100  20)
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 Table 2: Beth’s quiz scores
Quiz
1
2
▫ Ex. Beth
Score
80
20
Possible
Percentage
100
80
100
20
3
1
20
5
Dropping more than one grade
• Conflict arises depending on the number of
scores dropped Table 3: Carl’s quiz scores
Quiz
1
Score
100
Possible
100
Percentage 100
2
42
91
46
3
14
55
25
▫ 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%
4
3
38
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
Table 3: Carl’s quiz scores
1
2
3
▫ Do the best in each situation Quiz
Score
100 42
14
▫ Flaw: ex. Carl
Possible
100 91
55
Percentage 100 46
25
 Drop 4 & 3
4
3
38
8
 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%)
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 MDuluth
Not actually Daniel M. Kane, but this
showed up in a Google Images search
Jonathan M. Kane
Terminology •
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also found on handout
k K – Total number of assignments; s.t. K
j – quiz # (1,2,3,…k) s.t. k>0
mj – earned points on quiz
nj – 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)
Sbest – Optimal retained set
q – the average score in S
qbest – best possible value for q
“Optimal Drop function” -
Optimal Drop Function
• q – Defined as:


jS
mj
jS
nj
q
Equation 1
• qbest – q is defined s.t. the S score is maximized
• Define for every j: f j q   m j  qn j
▫ By substitution we get:
 f q   0
jS
j
Equation 2
Equation 3
Optimal Drop Function
• q is the average score in S
• f j q   m j  qn j is a linear,
decreasing function
•  jS f j q  is also a linear,
decreasing function
Table 3: Carl’s quiz scores
Quiz
1
2
Score
100 42
Possible
100 91
Percentage
100 46
▫ Example from Carl
•


jS
mj
n
jS j
 q iff
4
3
38
8
100
50
 f q   0
jS
3
14
55
25
j
0.5
-50
1
Optimal Drop Function


F q   max  f j q  : S  K , S  k  r 
 jS

• F(q) is the max of the sum of
linear, decreasing functions,
Equation 4
100
50
▫ F(q) must be a piecewise,
linear, decreasing function
0.5
-50
1
Optimal Drop Function


F q   max  f j q  : S  K , S  k  r 
 jS

• F(q) is the max of the sum of
linear, decreasing functions,
Equation 4
100
50
▫ F(q) must be a piecewise,
linear, decreasing function
0.5
-50
1
Optimal Drop Function


F q   max  f j q  : S  K , S  k  r 
 jS

• Now, find the rational
number q, so that F q   0
Equation 4
100
50
 Recall


jS
mj
n
jS j
q
0.5
-50
1
Optimal Drop Function
• Next, we need to find the line that yields the
highest possible q value s.t. F q   0.
▫ From this we are able to determine Sbest


▫ F q   max  f j q 
 jS

 composed of the top k-r fj(q) values.
Optimal Drop Function
• Tasks:
▫ Evaluate each f j qbest   m j  qbest n j for each j
▫ Identify the k-r largest from the f j qbest  values
▫ S is the set of j values from the largest k-r
f j qbest  values
▫ Calculate F qbest  

jS
f j qbest 
Optimal Drop Function – Ex. Carl
Drop 2 scores:
• Possibilities for S:
1&2
2&3
1&3
2&4
1&4
3&4
• Estimate: q to be .75
f j .75  m j  .75n j
f1 .75  100  .75(100)  25
f 2 .75  42  .75(91)  26.25
f 3 .75  14  .75(55)  27.25
f 4 .75  3  .75(38)  25.5
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
100
50
0.5
-50
F .75   jS f j .75  0.5
1
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
Sources
1. Daniel M. Kane and Jonathan M. Kane Dropping Lowest
Grades Mathematics Magazine, (2006) 79 (June) pp. 181189.
2. Daniel Kane's Homepage http://web.mit.edu/dankane/www/
3. Jonathan Kane Home Page
http://faculty.mcs.uww.edu/kanej/kane.htm
4. http://www.ams.org/news/home-news-2007.html
5. Daniel Biebighauser’s brain