Title
Scheduling Math Classes (Jo Ann Ellerbrock)
Goals
(1) Students will explore the use of quota ratios in determining course section allocations.
(2) Students will apply different divisor methods of apportionment (Hamilton, Jefferson, Webster, Hill, Adams)
(2) Students will explore the impact of small changes in a ratio on the outcomes of a situation.
(3) Students will work in small groups to arrive at a consensual methods of determining allocations and be able to explain, and justify, their choice to the rest of the class.
Abstract
This activity explores using four different divisor methods to a high school scheduling apportionment situation. Students are also asked to determine which of several methods is best, and to justify their decision to their peers.
Problem Statement
Remind students of the effects of class size on the learning environment. This activity provides an opportunity for students to explore the allocation of school resources in a limited situation.
Instructor Suggestions
(1) Discuss with students their feelings about class size in a school. Include a discussion about the factors that drive class size.
(2) Discuss quotas and the methods of apportionment listed below. This is a good opportunity to predict the effects of each method on the outcome and discuss the circumstances under which each method would be preferable.
Jefferson Method: Rounds all fractions down to zero and takes only the integer parts as the apportionment.
Hamilton Method: Assigns the integer part of each quota and then adds 1 to the quotas with the largest fractional parts. The number of 1's so added is enough to reach the desired size.
Webster Method: Rounds fractions greater than 0.5 upward and fractions less than 0.5 downward.
Hill Method: Computes the geometric mean of the integers directly above and below the quota and rounds up if the quota exceeds the geometric mean, down if it doesn't.
Adams Method: Rounds all positive fractions upward to the next integer.
(3) Distribute the "Scheduling Math Classes" activity sheet (attached), and have students compute quotas, apportionment, and class size individually for Part A. (This could be a homework assignment.)
(4) Have students form groups small groups and come to consensus and present the method they feel would be the most fair for students attending Downtown Utopian High for Part B.
(5) Have students compute quotas, apportionment, and class size for each method for "Opening Day" enrollments (Part C) and discuss if their chosen method is still the "most fair".
Materials
"Scheduling Math Classes" worksheet, pencils, erasers
Time
This activity will take one to three class periods depending on how much background skills the students already have on the divisor methods involved and how much discussion there is on class sizes.
Mathematics Concepts
Discrete Mathematics Concepts
Divisor Methods (Hamilton, Jefferson, Webster, Hill, Adams), Apportionment,
Related Mathematics Concepts
Ratios, Proportions, Rounding, Fractional Parts, Arithmetic Mean, Geometric Mean, Alabama paradox, quota monotonicity (paradox), Number Sense, Computation
NCTM Standards Addressed
Problem Solving, Communication, Reasoning, Connections, Discrete Mathematics
Colorado Model Content Standards Addressed
Algebraic Techniques (2), Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity could be integrated into the study of proportions in an Algebra or Geometry class or into the study of decimals and fractions in a Pre-Algebra or General Math class.
It also provides a nice bridge to a discussion on legislative apportionment in Social Studies Classes
Further Investigation
Apportionment methods could be redefined to take account of various minimum or maximum conditions.
Variations/Comments
This exercise is modified from a problem found in For All Practical Purposes: Introduction to Contemporary Mathematics, Second Edition, a project of the Consortium for Mathematics and Its Applications (COMAP), 57 Bedford Street, Suite 210, Lexington, MA 02173.
References/Resources
Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete mathematics through applications. New York: W.H. Freeman and Company
Kenney, M.J., & Hirsch, C.R. (Eds.). (1991). Discrete mathematics across the curriculum, K-12. Reston, VA: National Council of Teachers of Mathematics.
Steen, Lynn A. (Ed.). (1991). For All Practical Purposes. New York: W.H. Freeman and Company