# UTPA STEM/CBI Courses/The Mathematics of Apportionments

Course Title: Contemporary Math
Lecture Topic: The Mathematics of Apportionment
Instructor: Nam Nguyen
Institution: University of Texas-Pan American

## Backwards Design

Course Objectives
* Primary Objectives- By the next class period students will be able to:
** Understand the Basic concepts of Apportionment
** What is an apportionment problem?
** What is the nature of the problem?
** What are the mathematical issues we must deal with?
** Why should we care?
* Sub Objectives- The objectives will require that students be able to:
** Understand Apportionment problems
** Hamilton's method and the Quota Rule
** Jefferson's method's
** Adams's method
* Difficulties- Students may have difficulty:
** Being able to find the modified divisor D and modified quota.
** Using various methods for the purpose of comparison
* Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
** Determining the numbers of seats each state would have in the House of Representatives (look at the last 10 years census and now)
Model of Knowledge
* Concept Map
** Understand the Standard Divisor
** Understand the Standard Quota
** Understand the Upper and Lower quota
** The Conventional Rounding of standard quota
* Content Priorities
** Enduring Understanding
*** Apportion-We are dividing and assigning things, and we are doing this on a proportional basic and in planned organization fashion.
      * Hamilton's method:
          1) Calculate each state's standard quota .
          2) Give each state its Lower Quota.
          3) Give surplus seats ( one at a time) to the states with the largest residues(fractional parts) until there are no more surplus seats.
      * Jefferson's method:
          1) Find a "Suitable" divisor D.
          2) Using D as the divisor, compute each state's modified quota.
          3) Each state is apportioned its modified lower quota.
      * Adams's method:
          1) Find a "Suitable" divisor D.
          2) Using D as the divisor, compute each state's modified quota.
          3) Each state is apportioned its modified upper quota.
* Important to Do and Know
** The "State" is the term we will use to describe the parties having a stake in the apportionment.
** The "Seat" is the term that describes the set of M identical, indivisible objects that are being divided among the N states.
** The "Population" is a set of N positive numbers that are used as a basic for the apportionment of the seat to the states.
** The Standard Divisor (SD) this is the ratio of population to seats.
** The standard quotas of a state is the exact fractional number of seats that the state would get if fractional seats were allowed.
** Upper and Lower quotas : the lower quota(L)-the standard quota rounded down; the upper quota(U)-the standard quota rounded up.
* Worth Being Familiar with
** Quota rule : No state should be apportioned a number of seats smaller than its lower quota or larger than its upper quota.
** Understand and how to compute the Standard Divisor
** Understand and how to compute the Standard Quota
Assessment of Learning
* Formative Assessment
** In Class (groups)
*** Read problem carefully, understand what is given and what being asked for.
** Homework (individual)
*** assign homework after the lecture
* Summative Assessment
** Exam
** Group project presentation

## Legacy Cycle

OBJECTIVE
By the next class period, students will be able to:
* Understand the Basic concepts of apportionment and how to compute by using various method
* Understand the Basic concepts of apportionment
* What is an apportionment problem?
* What is the nature of the problem?
* What are the mathematical issues we must deal with?
* Why should we care?
The objectives will require that students be able to:
* What is an apportionment problem?
* What is the nature of the problem?
* What are the mathematical issues we must deal with?
* Why should we care?
THE CHALLENGE The U.S constitution requires a national census once every ten years. Census data is also used to guide local decision makers on where to build new roads, hospitals, transportation needs, child-care and senior centers, schools, and more. Your goal is to determine how many seats each state would have in the House of Representatives by using various methods (note: use the census 2000 versus 2010).
GENERATE IDEAS
* The instructor will lead students by giving some different examples of Jefferson's and Adams's method.
* The instructor will lead students by showing how to compute the Standard Divisor and Standard quota.
MULTIPLE PERSPECTIVES Provide a short video clip about the House of Representatives.
** The instructor will ask group members for comments and critique.
** The instructor will give comments to the students and present the complete solution.
** Assign more examples.
RESEARCH & REVISE
* Give introduction of the Apportionment
* Handout worksheets and walk students through the use of each method
TEST YOUR METTLE
* Form base-groups or groups and give the result of their finding and conclusion.
GO PUBLIC
* Students will be able to answer similar questions presented at the beginning of the lesson.
* Students would be asked to turn in a brief write up.
* Students will have an exam covering these topics.

## Pre-Lesson Quiz

* The Bandana Republic is a small country consisting of four states: A population 3,310,000, B population 2,670,000, C population 1,330,000, and D population 690,000. Suppose that there are M=160 seats to be apportioned among the four states based on their respective populations.
       a) Find the standard divisor.
       b) Find each state's standard standard quota.
* According to the 2000 U.S. Census 7.43% of the U.S. population lived in Texas. Compute Texas Standard Quota in 2000 (hint: there are 435 seats in the House of Representatives).
* Find the apportionment as described in problem #1 (use M = 160) under Hamilton's method.
* Find the apportionment as described in problem #1 under Adams's method.

## Test Your Mettle Quiz

* Suppose that there were 11 candy bars in the box. Given that Bob did homework for a total of 54 minutes, Peter did homework for a total of 243 minutes, and Ron did homework for a total of 703 minutes, apportion the 11 candy bars among the children using Hamilton's method.
* Suppose that before mom hands out the candy bars, the children decide to spend a "little" extra time on homework. Bob puts in an extra 12 minutes (for a total of 56 minutes ), Peter an extra 12 minutes ( for a total of 255 minutes), and Ron an extra 86 minutes ( for a total of 789 minutes). Using these new totals, apportion the 11 candy bars among the children using Hamilton's method.