## Grades, *s, and ,s |

**Bill Zwicker**

Union College

October 16, 2006

7:00 pm

Bailey Hall 312

Refreshments will be served

Grades, *s, and ,s You are a professor, and you are planning to teach a course. There will be 3 tests, each graded on a scale from 1 to 5, with 5 as the highest possible grade and no fractional grades allowed. You want each test to be equally important in deciding a student’s final course grade, for which there are only two possibilities:PassandFail. Finally, you don’t want the system to be “perverse” – if you realize you made an error, and as a result youraiseone of Sara’s test grades, her final course grade should never switch from Pass to Fail as a result. Here are a few possible grading systems:How many other possibilities can you think of?

- Everyone passes, regardless of their test grades
^{(1)}.

- Everyone fails, regardless of their grades.

- You pass if and only if your average test score is 3 or higher.

- You pass if and only if your median test score is 3 or higher.

The table below gives the number of possible Pass-Fail grading systems when there are n tests, each graded on a scale from 1 to j. (We were using n = 3, j = 5, above.)

What interesting patterns can you find in the table?We’ll extend a standard method in finite combinatorics to confirm what may be the most surprising of these patterns. If time permits, we’ll explain the connections between these grading systems and the theory of voting with abstention.

^{(1)}This Professor is probably not in the Mathematics department.

j/n 2 3 4 5 6 7 2 4 5 6 7 8 9 3 8 16 32 64 128 256 4 16 66 352 2431 21760 252586 5 32 352 9304 683464 161960220 6 64 2431 683464 7 128 21760 161960220 8 252586

For additional information, send e-mail to math@union.edu or call (518) 388-6246.

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