Getting Your Fair Share -- Jelly beans, student groups, and Alexander Hamilton

Aaron Montgomery, Central Washington University, Ellensburg, WA
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This material is replicated on a number of sites as part of the SERC Pedagogic Service Project

Summary

In the Spreadsheets Across the Curriculum Module, students develop a spreadsheet that will apportion money to student organizations based on a method designed by Alexander Hamilton. In end of module assignments, the students will address other apportionment problems (distributing jelly beans) resulting from fractional remainders from division. Students will learn about percentages and ratios. They will also learn that there is no such thing as fair apportionment.

Learning Goals

Students will In the process, students will

Context for Use

This material could be used in a class that includes either a section on apportionment or percentage. This module contains only basic mathematical operations and could be used in a course emphasizing basic mathematics. However, this module does assume some familiarity with Excel and should probably not be used as an introduction into Excel.

This module is the first of a pair of modules (the second under development) that focus on various methods of apportionment. Once the second module is complete, a link to that module will be available here.

Description and Teaching Materials

SSAC2007.JF1075.AM1.1 (PowerPoint 186kB Sep1 07)
This module is a PowerPoint presentation with embedded spreadsheets. Students work though the presentation, answering questions and developing the spreadsheet. If the embedded spreadsheets are not visible, save the PowerPoint file to a disk and open it from there.

This PowerPoint file is the student version of the module. An instructor version is available by request. The instructor version includes the completed spreadsheet. Send your request to Len Vacher (vacher@usf.edu) by filling out and submitting the Instructor Module Request Form.

Teaching Notes and Tips

The module can be used with a focus on political science (and apportionment in general) or it can be used in a historical context. I found historical census data for the original 13 colonies at Deane Merrill's website on August 30, 2007. Let me know if this link goes dead or if you find contradictory information. You could also discuss the issues surrounding the original apportionment of representatives presented in the US Constitution and the first presidential veto.

A "quota" method of apportionment is a method that always assigns to each claimant either their upper or lower quota. Hamilton's method is a quota method of apportionment whereas Jefferson's, Webster's and the Hill-Huntington Methods are what are known as "divisor" methods. The divisor methods are covered in the next module.

An alternative quota method is Lownde's Method. Instead of giving the excess to those claimants with the largest fractional part, Lownde's Method gives the excess to those claimants with the largest "relative fractional part" where the relative fractional part of a number is obtained by dividing the fractional part by the integer part of that number. You may ask students to discuss whether Lownde's Method gives an advantage to larger or smaller states (Lownde's was a South Carolina representative which may provide a clue here).

Students are expected to have a familiarity with Excel formulas. Although some guidelines in using the LARGE() and IF() functions are included with the module.

Assessment

End-of-module questions are included inside the module and there is a pre- and post-test available in the instructor's version.

References and Resources

Apportionment is a classic problem. Further examples of apportionment paradoxes can be found in two articles (Apportionment Apportionment II) at the AMS website (one example in the questions is based on the second article). Searching the web for apportionment paradoxes and you will find many sites with information. One that includes java applets is The Constitution and Paradoxes

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