Standards - Mathematics

2019 - Mathematics Course of Study Document

Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.26
Explain and apply mathematical aspects of fair division, with respect to classic problems of apportionment, cake cutting, and estate division. Include applications in other contexts and modern situations.

Unpacked Content

Knowledge

Students know:

How to divide.

Skills

Students are able to:

Explain and use ways to divide objects and argue how their proposed method could be considered fair.

Understanding

Students understand that:

In some cases of division, fairness can be achieved through agreement on method by parties involved.

Vocabulary

Fair division
Continuous division
Discrete division

Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.27
Identify and apply historic methods of apportionment for voting districts including Hamilton, Jefferson, Adams, Webster, and Huntington-Hill. Identify issues of fairness and paradoxes that may result from methods.

COS Examples
Examples: the Alabama paradox, population paradox

Unpacked Content

Knowledge

Students know:

Apportionment is a method of dividing based on population.

Skills

Students are able to:

Calculate an ideal ratio for diving objects based on populations.
Use the ideal ratio to determine exact district quotas and apply a variety of apportionment methods to determine apportionment of discrete objects such as representative seats.
Use a variety of apportionment methods to adjust the divisor and calculate subsequent quotas.
Identify specific examples of paradoxes or violation of the quota rule.

Understanding

Students understand that:

Some methods result in unfair paradoxes and may favor larger or smaller districts.
Apportionment methods are used to divide discrete objects when they cannot be divided exactly proportional to the populations.

Vocabulary

Ideal ratio
Quota
Hamilton method
Jefferson method
Adams method
Arithmetic mean
Geometric mean
Webster method
Huntington-Hill method
Population paradox
Alabama paradox
Quota rule

Mathematics (2019) Grade(s): 09-12 - Applications of Finite Mathematics

MA19.FM.28
Use spreadsheets to examine apportionment methods in large problems.

COS Examples
Example: apportion the 435 seats in the U.S. House of Representatives using historically applied methods

Unpacked Content

Knowledge

Students know:

How to apply various methods of apportionment with smaller data sets.

Skills

Students are able to:

Determine and use formulas for spreadsheet cells for determining parts of the apportionment process including calculating quotas, truncated quotas, mean and geometric mean of upper and lower quotas, and final apportionments using various methods (Hill, Webster, Hamilton, etc.)

Understanding

Students understand that:

Apportionment methods can be applied efficiently on large data sets using spreadsheets.

Vocabulary

Truncate
Quota
Arithmetic mean
Geometric mean

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