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: 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).
 * 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?

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

 * 1) 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.


 * 1) 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).
 * 2) Find the apportionment as described in problem #1 (use M = 160) under Hamilton's method.
 * 3) Find the apportionment as described in problem #1 under Adams's method.

Test Your Mettle Quiz

 * 1) 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.
 * 2) 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.