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UTPA STEM/CBI Courses/The Mathematics of Apportionments

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Course Title: Contemporary Math

Lecture Topic: The Mathematics of Apportionment

Instructor: Nam Nguyen

Institution: University of Texas-Pan American

Backwards Design

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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

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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

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  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

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  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.
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