The Math of Elections

The materials on this page accompany the presentation Percents, Voting, and the Government, which was a featured session at NCTM Phoenix on Oct 27, 2016, and a regular session at NCTM Philadelphia on Nov 1, 2016. The following files can be used to explore apportionment and voting:

Elections Hands-On Activity (PDF)

The Hands-On Activity can be used with students in a non-digital environment.

  • The first table (How Many Votes?) provides simple numbers for calculating the apportionment using the Hamilton and Jefferson methods.
  • The second table (Who Wins?) can be used to determine the winner in elections using different types of voting. (The candidates are flavors of Ben ‘n Jerry’s ice cream.) Note that the PowerPoint slides also contain information about the winner in a run-off election, even though that’s not included in the hands-on activity.
  • The second table (Now, Who Wins?) is meant to show some strange results. Use it to compare Hamilton, Jefferson, and the Equal Proportions methods, both as is as well as adding an additional state with a population of 19 people.
  • The Just for Fun puzzle is, well, just for fun. (I’m an amazing prognosticator. Without seeing which numbers you chose, I can predict the sum of your four numbers.)

Election Apportionment Spreadsheet (XLSX)

The spreadsheet can be used for exploration with the various apportionment methods. It is recommended that students be given access to the spreadsheet so they can explore on their own. (That is, the spreadsheet should not be used just for a lecture about apportionment. Students don’t learn as much that way!)

  • The sheets with _Session at the end of the name are to be used in conjunction with slides 12-15 of the PPT file.
  • The sheets with _HOA at the end of the name correspond to the Hands-On Activity.
  • The sheets names just Hamilton and Jefferson can be used to show paradoxes and differences between the two methods. In particular, if you unhide rows 10-15 on the Jefferson sheet, the differences between the two methods is explicitly shown.
  • The two EqProp sheets can be used to show the Method of Equal Proportions, one with a small set of data, the other with the full set of data from the United States.
  • The Raw Data sheet shows the electoral votes of every state, and this sheet can be used to answer the question, “What percentage of the population is needed to elect a President?” The Bar Graph sheet will show a bar graph of the states electoral power, pulled from column E of the Raw Data sheet.

Percents, Voting, and the Government (PPT)

This is a PDF version of the slides used during the presentation.

  • Slides 3-9 present some background information.
  • Slides 12-30 are for apportionment methods.
  • Slides 31-39 are for voting methods.
  • Slides 40-43 are for additional background information.
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