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The Mathematics of Congressional Reapportionment: Rounding in Action

Eric P. Choate

The US Constitution awards state representatives “ according to their respective Numbers, ” but is si lent about how this process is done. Rounding makes this problem harder than it seems like it should be, and throughout history, political machi nations have muddied these calculations. This pro vides a great opportunity for secondary teachers to apply computational thinking to a relatively large data set that comes from the American government and history perspective instead of the typical sci ences. Article 1, Section 2, of the Constitution of the United States, requires that members of the House of Representatives shall be apportioned to the states “ according to their respective Numbers. ” It initializes the House with 65 representatives for the original thirteen states and directs an “ Enumeration ” be made within three years of the first meeting of Congress and within every ten years thereafter to reapportion representatives. Nei ther the total number of representatives nor the method of apportionment is specified, and so the first Congress had to decide how to do this. Since their first attempt led to the first presidential veto of an act of Congress, this problem must be harder than computing simple proportions. Why? The answer is, of course, that since each state ’ s share of representatives must be a positive integer, we must round the proportions. We can illustrate the issue with a simple fictitious problem in Table Introduction

1. A fake country splits ten representatives among three states with 10.2, 10, and 9.8 million people. Their quotas —the exact non - integer number of seats each state should get—is between 3.2 and 3.4, so each state rounds down to three representa tives. Reasonable, yes, but it has an important problem, it only awards nine representatives, not ten. Or, if the total increases to twenty, now each state rounds up to seven so that one too many seats are awarded. How do we fix this? Since the first

census, Congress has used four different methods to correct the sum. All four start with the same quotas with their differences lying in how they round. They can lead to different apportionments, but sometimes the size of the House was chosen precisely because that value showed the same re sults for two different methods. You may notice that here we started with the sim plest possible scenario illustrating the problem. We can do this one in our heads. This is sufficient to illustrate how the methods are implemented. How ever, to reproduce the apportionment based on the 2020 Census, we need to apply the method to 50 Table 1: Simple apportionment of a fake country with three states

Virginia Mathematics Teacher vol. 48, no. 1

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