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states, splitting 435 representatives. We cannot do these calculations in our heads or even with pencil, paper, and a calculator, not because they are of a harder type, but because the number of calcula tions grows so rapidly with the number of states. We must think about this problem algorithmically to instruct a computer to do these calculations for us. I have taught this material in a college class on the mathematics of democracy, and it motivates students to learn a programming language like Matlab or R. This may not be the best approach for high school classes. Our focus here is to do this algorithmic thinking in a spreadsheet, such as Ex cel or the sample Google Sheet referenced in this article, which I encourage you to copy from https:// tinyurl.com/y3artfkk. Formulas without the ability to apply them to real - world data are not helpful. We begin with a historical summary of the four different methods Congress has used to reappor tion the House of Representatives, including dis cussions of what motivated changes when they happened and an explanation of why we chose the current system that does not round like we teach our elementary school students to round. We con clude with suggested classroom activities or as signments in which students compare the different methods, see how they differ from each other, and apply the methods to various hypothetical scenari os based on populations of the states and our choice for the number of representatives. A note on resources: The best mathematical re source for information on apportionment is Fair Representation (Balinski & Young, 2001). The first half is written for a less technical audience and takes more of a narrative approach to the histo ry of the methods used by Congress to reapportion itself, whereas the second half takes a more theo retical approach, proving theorems about how vari ous apportionment methods work or perhaps do not work as we would hope. Also of note, is The American Census: A Social History (Anderson, 2015), which focuses more on the history of the census but also discusses how it has been used in reapportionment. Additional presentations of this problem are in Bradberry (1992), Caulfield (2010), Malkevitch (2002a, 2002b), and Swenson (2022).

1790: The First Apportionment

Counting a nation of nearly four million people in 1790 was a daunting task. The census date was August 2, 1790, but the US marshals tasked with this duty were not able to report their results until 1792. Vermont and Kentucky became states during the course of this tabulation, and so they were also included as states. The enumeration counted the entire free population plus three - fifths of the en slaved population, as it would until the passage of the 14th Amendment in 1868. The total enumera tion was P = 3,615,924, and state - by - state results are given on the page titled “1790 Census ” of the Google Sheet. Next, Congress had to decide N , the size of the House of Representatives, and then , the number of representatives for state i. There are just three constitutional constraints: 1) Each state gets at least one representative, 2) Seats are awarded proportionally with the enumeration, and 3) There are at most 30,000 people per repre sentative, which caps the size of the House. In 1790, the largest value of N that kept P/N greater than 30,000 was 120. Secretary of the Treasury Alexander Hamilton proposed using that value and the following method to round the quotas so that the total was correct.

The Method of Hamilton

1. Compute each state ’ s quota

, where

is the enumeration of state i.

2. Round each quota down so that state i is awarded seats, where [] is the floor or greatest integer function. The sum of the is now less than or equal to the total N .

3. Compute each state ’ s remainder which would be between 0 and 1.

,

4. Rank the states from largest remainder to smallest, which establishes the priority for awarding the remaining seats.

For

example,

Virginia ’ s

quota

was

so that _______ with the remainder _______ . This was the second largest remainder after New Jersey.

Virginia Mathematics Teacher vol. 48, no. 1

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