Gerrymandering Math

Can math save us from dark depression?

Facilitator(s): Solange and Eric
Date of Meeting: November 9, 2016
Problem: pdf · url · url2
Further Reading: pdf

Among the thoughts racing through our minds on election night was the realization that we had decided to have a CAMI meeting on the next day. What were we thinking? Solange and I spoke before the meeting. We briefly considered scrapping our plans to explore gerrymandering math and do something to get our minds off the election, but eventually decided that we should take the opportunity to talk with other teachers about this moment.

Solange started the meeting by asking the group to talk in pairs about how they were feeling about the election, and what ideas they had for addressing it in their classrooms. She shared the electoral and popular vote numbers:

as of 3pm, 11/9 Popular Vote Electoral Vote
Donald Trump 59,460,467 279
Hillary Clinton 59,670,969 228

A teacher who watched the election results come in with an evening class said that the consensus of his students was that we were in trouble (not the words they used) either way, but his morning class the next day talked in specifics about the way they felt in real trouble now. Another teacher spoke about the trauma among students and teachers at her program. Many people were in tears. Students asked if they would be deported, if their marriages to a citizen would be invalidated, if ESOL classes would be ended. One group mentioned that the popular vote and the electoral vote were not proportional.

I started the mathematical part of the meeting by asking participants to draw a 5×10 grid on graph paper, then divide it into 5 equal sections. Straight lines weren’t required. Here are a couple examples:

gerrymandering1   gerrymandering2

I then asked the group to color the first 2 columns of their boxes pink and the last 3 columns blue. I asked them to imagine that this box of 50 squares is a state broken up into different districts, each which elects a representative to Congress. There are 10 voters in each district and 5 representatives total for the state. A blue or pink representative will win each district, depending on the majority color in that district.*

(*I didn’t actually explain it this way, having not thought clearly enough about it. And there may be a better way to explain the setup.)

What would the 5 representatives be for this way of dividing?


Blue wins all 5 districts 6 blue votes : 4 red votes.

Does Blue win all the votes no matter how you divide the different districts?

We came up with some interesting results, but I’ll let you answer this for yourself.

We then looked at a puzzle from


Stephanie looked at this puzzle and made the following argument:

  • There are 5 districts and 9 blue voters.
  • To win the majority in a district, you need 3 voters from the blue party.
  • To win the majority overall, you need to win 3 districts.
  • Therefore, if you can distribute 3 blue voters to 3 different districts, you can win the majority.

Maritza used this information to divide the region into 5 districts with the following breakdown:

 blue  red
 3 2
 3 2
 3 2
 0 5
 0 5
total of 9 voters total of 16 voters

We successfully gerrymandered the puzzle! Then, we derived the following principles of gerrymandering:

  • Concentrate the opposition’s votes in districts you’re willing to lose. Dilute the majority party’s votes by concentrating them in separate districts.
  • Determine the minimum majority in each district.
  • Split your party’s votes into multiples of that minimum majority and break the districts in a way that divides these multiples into districts they control.

We then looked at an example of gerrymandering in the United States.


I shared that after doing a little reading on mathematics and gerrymandering (most of which I didn’t fully understand), I noticed that area and perimeter are often used to calculate indices of gerrymandering. Simplistically, the ratio of a district’s perimeter to its area would let you know how gerrymandered the district is.

We then looked at recent results of gerrymandering in the House of Representatives. The first thing we noticed was the difference between proportion of votes cast and the representation by seats. My understanding is that this is the result of Republicans being in power when the redistricting happened after the 2010 census.


A draft lesson for gerrymandering math is available as a Google Slide Show.

The last half hour of the meeting was devoted to, who shared a free group code at the NCTM Conference in Philadelphia, valid through December. Solange shared a few interesting lessons:

  • Ballot Boxing – about the popular and electoral vote
  • Leonardo Numbers – about bees and the Fibonacci sequence
  • Sweet Tooth – about Halloween candy, graphing and the phenomenon of diminishing returns

I’m not sure electoral math is making me feel any better about the future of this country, but I love my CAMI people!


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