Euclidean Geometry • Equitable Distributing • Linear Programming • Set Theory • Nonstandard Analysis • Advice • Topology • Number Theory • Computation of Time (Previous | Next)

In life, there is often distribution unfairness. Although the following is not new, equitable distributing is to be demonstrated by examples that occur daily at school:

1. How do we distribute work-results equitably?

2. How do we channel groups equitably through the cafeteria?

3. How do we form equitably groups in physical instruction?

To 1.: We first put down the work results alphabetically, arranged according to the names of the present pupils (in trays, on the table or the like). The names are or will be put down alphabetically in a spread-sheet analysis (except the missing pupils). Behind each name, a random number is produced. The names are arranged according to the random number. We call the pupils according to this order (possibly in groups) so that they can receive their work results.

To 2.: At the beginning, we form groups with members that would like to dine together or are to do this. These groups or their names are arranged as described under 1. Then for each group (the available output time) * group size / diners is computed. The second group goes into the cafeteria if the time determined for the first group is up. The third group follows if additionally the time for the second group has elapsed and so on. If necessary, the points in time have to be computed rounded. If singular groups have time only in certain intervals, they are distributed first on the time intervals. Then the procedure continues with the remainder time for the groups left.

To 3.: At the beginning, groups of same capacity are formed and the individual names are written in a firm sequence one below the other and serially numbered due to an assigned group number. If group members are missing, we filter them first by noting e.g. the entry "missing" behind their name and selecting the empty entries afterwards.

The group size should be a multiple of the number of groups that can be formed. The sequence of the names should be firm, for reasons of the probability. Changes of the capacity are considered in changed groups. Behind each name, the numbering is divided by the number of groups. If the numbering does not begin with 0, a number must be subtracted before accordingly. The decimal places of the received values are truncated. This can be attained also by a rounding of the values reduced by 1/2. In a further column, now random numbers are registered.

Thereupon the three columns are arranged according to the values determined first, then additionally according to the random numbers. In a fourth column, beginning from above, the numbers from 1 to number of groups are registered. After the last value, it is again continued with 1 up to the number of groups and so on. Now the numbers are sorted again according to names. Finally, we can announce the group assignments in alphabetical order.

If the number of pupils to be divided is not a multiple of the number of groups, we should form proxies from the same group for each name. The namebearer who have the smallest time absent by sitting out are intended to sit out. They represent their deputies (e.g., if there are several runs in an instruction unit). The time absent is credited accordingly.

A simple procedure with a dice (instead of a computer) functions as follows: The pupils are set up ordered according to their (self-determined or over-directed) capacity on a line. The one with the highest capacity is assigned to one of the groups by throwing the dice. The next one is determined under the remaining groups by the dice and so on. If each group has a member, it is continued as with the one with the highest capacity now with the one having the henceforth highest capacity.

Without considerable aids, we can form groups also as follows: We request all pupils to move at random each time in a circle after their capacity. We prevent couple formation by the request "A and B apart to the circle line!". Everyone remains at her or his place after the command "stop!". We now request one of the pupils to leave her or his place and to move on an approximate circle line around the remaining pupils or around a group each time.

After a further command "stop!", we divide the pupils by imaginary straight lines into groups. First, the intersection is built with the circle line of the group the straight line forms through the break point of the one who stands on the circle line and the respective centre of the circle. The groups are defined by parallel straight lines that go parallel to the straight line through the respective intersection and the respective centre of the circle.

These procedures are more equitable than the judging selection by individual pupils. Three capacities are sufficient according to experience.

© 2006-2007 by Boris Haase

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