Social Justice Mathematics Project 4: Gerrymandering
Overview
This project will explore the idea of election fairness through the concept of Gerrymandering. Gerrymandering is a means of unfairly drawing district boundaries that favor one party over another.There are legal concepts and mathematical concepts that can be considered in determining the fairness of district boundaries.
Objective:
Students will:
- Explore shapes that create fair voting districts.
- Calculate compactness as a measure for voting fairness.
- Discuss the implications of using compactness to measure voting fairness.
Social Justice Mathematics Project 4 "Gerrymandering": Exploring the Information Introduction
This project will explore the idea of election fairness through the concept of Gerrymandering. Gerrymandering is a means of unfairly drawing district boundaries that favor one party over another.There are legal concepts and mathematical concepts that can be considered in determining the fairness of district boundaries.
Objective:
Students will:
- Explore shapes that create fair voting districts.
- Calculate compactness as a measure for voting fairness.
- Discuss the implications of using compactness to measure voting fairness.
Social Justice Mathematics Project 4 "Gerrymandering": Exploring Politics and Math
Under the terms of the Constitution, state legislatures are entitled to drawing congressional districts, and take the sole role of doing so in most states. Some states have opted to have independent commissions draw their districts, while others have advisory commissions, though the final decision is still made by the legislature.In states with only one representative (Alaska, Delaware, Montana, North Dakota, South Dakota, Vermont, Wyoming), it’s easy: the whole state is the one and only district (called an “at-large” district). In the other states, there are many legal restrictions. The easiest to describe are as follows:
• Districts must be (roughly) the same size in population within each state. (A violation of this is called malapportionment.)
• Districts must be contiguous: A person must be able to walk between any two points within the district while remaining in the district.
• Districts must be compact though there is no satisfactory definition of this.
• Districts must respect communities of interest such as neighborhoods, minority communities, etc.
• Districts must not be drawn with racial concerns as the “predominant factor”.
Gerrymandering is the act of purposefully drawing district lines to favor one political group/party over others. • It is named for Massachusetts Governor Elbridge Gerry, who in 1812 approved a map for state senate districts which contained one oddly shaped district, believed to be drawn to favor his Democratic-Republican Party. • Its shape was likened to a monster and a salamander by commentators, resulting in the portmanteau “Gerry-mander”.
But are strange shapes necessarily bad? Consider Illinois’s Fourth Congressional District, which is frequently lambasted for its peculiar “earmuffs” shape. This odd shape doesn’t necessarily demonstrate bad intentions. In this case, the district was “gerrymandered” so as to connect two majority Hispanic parts of Chicago, thereby providing a common voice to this demographic. So it’s not unprecedented to sacrifice shape in favor of a more substantive ideal.
Social Justice Mathematics Project 4 "Gerrymandering": Exploring the Information
Introduction:
Compactness in the redistricting setting is a way to describe shapes that might make for more fair districts for vorting purposes. The basic idea is that the shapes of electoral districts should not be too stretched out nor should the boundaries look like undulating or jagged. There are two measures that we will use to determine whether a certain district (consisting of the area of a shape that is formed by a closed loop boudnary) represents a fair "shape". For the first measure, Polsby-Popper givces a quantification of the jaggedness of a planar shapes boundaries. Basically, if it is too jagged, it is not fair. For the second measure, Reock quantifies the dispersion or oblongness of a shape. Thsi measure essentially claims that a circle is the most fair shape.
Both measures are based on something called the isoperimetric inequality which states that:
\(L^2\ge4\pi(A)\)
Where A is the area enclosed by a planar curve of length L. The only shape for which equality exists between the left and right side is the circle. In that case L= Circumference = C. So since,
\(C=2 \pi r\)
and
\(A = \pi r^2\)
we can see that:
\(L^2 = C^2 = (2 \pi r)^2= 4 \pi^2 r^2 = 4 \pi \cdot \pi r^2 = 4\pi \cdot A\)
Finally, note that the Polsby-Popper measure is the ratio:
\((4 \pi A) \over L^2\)
Answer the following question/s:
1. Suppose each dot is a voter. Notice that the red voters win the majority in the first two districts and the blue voters win the majority in the last three districts. The Polsby-Popper Measure for the first districting option is done for you. Compute the Polsby Popper Measure for the last two districting shapes and determine which of the political parties (blue or red) wins each district.
Districting Option #1
Each district has a length of 10 and a width of 1. Therefore L=Perimeter=22 units. The area of each column/district is 10. Therefore the Polsby-Popper measure is:
\({4 \pi A \over L^2} = {4 \pi 10 \over 22^2} = {40 \pi \over 484} = {125.7 \over 484} = 0.26 \)
Your turn:
Districting Option #2.
Districting Option #3.
Social Justice Mathematics Project 4 "Gerrymandering": Looking at the Concepts More Deeply
These concepts are relevant because since 1967, states which are apportioned more than one representative have been required to be divided into districts (i.e., physical regions which partition the state), each of which must hold its own election for a representative using the plurality method. We will consider more unusual shapes within the rectangular whole.
1. First, compute the Polsby-Popper Measure for each of the following districts, and determine who wins each district (Stars vs. Blanks).
2. Here is a set of questions which starts with the same set-up as the first examples, but it is up to you to create districts with the following criteria:
- Draw 5 districts of 10 people in which the districting results in 3 blue and 2 red districts?
- Draw 5 districts of 10 people in which the districting results in 5 blue districts?
- Draw 5 districts of 10 people in which the districting results in 2 blue and 3 red districts?
3. Finally, consider the following "state" in which there are 36 people, 20 which are blue and 16 of which are "red".
- Can you draw 4 districts of 9 people which result in 4 "blue" districts?
- Can you draw 4 districts of 9 people which result in 1 blue, and 3 red districts?