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UAF DEPARTMENT OF MATHEMATICS AND STATISTICS
MATH 103X CONCEPTS AND CONTEMPORARY APPLICATIONS OF MATHEMATICS
COURSE SYLLABUS GUIDELINES
FALL 2014-FALL 2015
Syllabi must clearly satisfy university requirements, available at :
http://www.uaf.edu/uafgov/faculty-senate/curriculum/course-degree-procedures-/uafsyllabus-requirements/
Course Description: Applications of mathematics in modern society. Topics may include
voting systems, probability and statistics and applications of graph theory in management
science; uses of probability and statistics in industry, government and science; and
applications of geometry to engineering and astronomy. Problem solving
emphasized. Effective From Fall 2014 until Fall 2015
Prerequisites: Placement into Math 103x determined by the UAF Math Placement Policy as
stated in the UAF Catalog. Enforcement of prerequisites is expected, though instructors have
permission to override this requirement when appropriate.
Textbook: Excursions in Modern Mathematics 8th ed by Tannenbaum ISBN: ???
Students should have the option of purchasing the e-book for this course through
MyMathLab.
Software: MyMathLab (optional) For MyMathLab Help (click here)
If you do not plan to use MyMathLab but would like to have the resources available to your
students who have access (click here for access information).
Drop Date: Please note that the University Drop Date Sept 19 2014 deadline will be strictly
enforced.
Withdrawal Date: Please note that the University Withdrawal Date Oct 31, 2014 deadline
will be strictly enforced.
Instructor Availability: All Instructors will have regular, posted office hours (or online
meetings) during the week.
Grading Policy: The final grade in this course will be determined as follows:
Written Assessed Work
At most 30%
Online Assessed Work
At most 15%
At least 2 Midterm Exams
At least 40% total
Comprehensive Final Exam
At least 20%
Instructors should provide written feedback to students approximately weekly throughout the
semester. This can be through humanly-graded assignments or email correspondence. Students
must have a mechanism for estimating their current grade in the course.
To avoid confusion for the student, the syllabus should state: You need to earn a grade of C- or
better in this course to receive core credit.
The syllabus must include a grading scale in some form. One example is below. Plus/minus
grading is at the discretion of the instructor, but must be stated explicitly.
A
90-100
B
80-89
C
70-79
D
60-69
F
0-59
Exams: All exams for this course are closed book. All exams should be paper-pencil exams,
written and graded by a faculty member. Midterm and final exams from previous semesters
should not be reused by an instructor. Limited reuse of edited problems is acceptable. An exam
can have no more than 10% of points from multiple choice questions.
The final exam will be a comprehensive exam. Students will not be allowed to take the exam
early or late unless there is written verifiable proof of the reason for missing the exam (e.g., a
doctor’s note, police report, court notice, etc., clearly stating the date AND time of the
conflicting circumstances). In the event the final exam is not taken, under rare circumstances
where the student has a legitimate reason for missing the exam, a makeup exam will be
administered.
The comprehensive final exam should contain problems of the types listed below. (IS THIS LAST
SENTENCE CORRECT – that is: do you really mean there should be one of these type problems
on the final??)
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Finding the winner of an election using the four methods
Identify winning coalitions of a weighted voting system
Calculate the Banzhaf power index
Calculate the Shapley-Shubik power index
Determine basics of a graph (vertices, edges, connected, path, circuit, etc.)
Find Euler paths and/ or circuits
Find a minimum-cost Hamiltonian Circuit
Understand the Brute Force, Nearest Neighbor and Cheapest-Link Algorithms
Find a minimal spanning tree of a graph
Find mean, median, and quartiles of a data set
Create a box plot of a data set
Find the standard deviation of a data set
Compute the probability of an event
Find the probability space for a random phenomena
Find probability using permutations and/or combinations
Calculate the mean and standard deviation of a given statistic
Find the mean and standard deviation of a normal curve
Use the 68-95-99.7 rule to calculate normal probabilities
Sample Course Schedule:
Week 1: Introduction and begin Chapter 1(this is always a short week Thursday and Friday)
Week 2: Chapter 1
Week 3: Chapter 1
Week 4: Chapter 2
Week 5: Chapter 2
Week 6: Midterm
Week 7: Chapter 5
Week 8: Chapter 6
Week 9: Chapter 6
Week 10: Chapter 7
Week 11: Midterm
Week 12: Chapter 15
Week 13: Chapter 16
Week 14: Chapter 16
Week 15: Chapter 17
Week 16: Final Exam
Course Content and approximate timing:
Mathematics of Social Choice (Chapters 1&2 – 9 hours)
The Mathematics of Voting
Analyze and interpret a preference schedule.
Rearrange a preference schedule to accommodate the elimination of one or more alternatives.
Explain the difference between majority rule and the plurality method.
List and discuss the four voting methods.
Determine the winner from a preference schedule using each of the 4 voting methods.
Explain what is meant by a Condorcet Candidate.
Describe the process of insincere voting.
List three factors that can affect the outcome of an election.
List and discuss the four fairness criteria. Understand how a method “violates” a fairness
criterion.
Discuss “Arrow’s Impossibility Theorem.”
Weighted Voting Systems
Interpret the symbolic notation for a weighted voting system by identifying the quota and the
weight of each
voter.
Identify the winning coalitions in a given weighted voting system.
Determine whether a weighted voting system has a dictator, any dummies, or voters with veto
power.
Calculate the Banzhaf power index for a given weighted voting system.
List the possible permutations (sequential coalitions) for a three- or four-voter weighted voting
system.
Calculate the Shapley-Shubik index for a three voter weighted voting system.
Management Science (Chapters 5,6,&7 - 12 hours)
Euler Circuits
Determine by observation if a graph is connected, and determine the degree of each vertex.
Construct graphs that model real world situations.
Define an Euler circuit, and determine whether a graph contains an Euler circuit.
Find an Euler circuit and identity the circuit by numbering the edges.
If a graph does not contain an Euler circuit, “eulerize” the graph by duplicating a minimum
number of edges.
Identify the types of problems whose solutions involve Euler circuits.
Hamilton Circuits
Give the definition of a Hamilton circuit.
Explain the difference between an Euler circuit and a Hamilton circuit.
Identify a given application as being an Euler circuit problem or a Hamilton circuit problem.
Explain what is meant by a complete graph on N vertices.
Calculate N! for a given value of N.
Calculate the number of Hamilton circuits and the number of edges in a complete graph with a
given number of vertices.
Define the term algorithm.
Explain the brute force method for finding the minimum-cost Hamilton circuit.
Find an approximate solution to the TSP by applying the nearest-neighbor and cheapest-link
algorithms.
The Mathematics of Networks
Explain the difference between a graph and a tree.
Explain what is required for a graph to be a tree.
Identify which types of applications are solved by using Euler circuits, Hamilton circuits, or
minimum spanning trees.
Find a minimum-cost spanning tree by applying Kruskal’s algorithm.
Statistics (Chapters 15,16,&17 - 12 hours)
Descriptive Statistics
Calculate the median, mean, and 1st and 3rd quartiles of a set of data.
Calculate the range and inter-quartile range of a given data set
List the five-number summary for a given data set and construct a box plot.
Find the standard deviation for a small data set.
Explain in your own words the meaning of standard deviation.
Explain the difference between a bar graph and a histogram.
Construct a bar graph or histogram for a small data set.
Find the mean, median, and quartiles of data represented by a bar graph or frequency table.
Analyze a pie chart.
Chances and Probability
Describe the sample space for a given random phenomena.
Explain what is meant by the probability of an outcome.
List the two laws of probability.
Apply the laws of probability to determine the validity of a probability space.
Identify which probability law is not satisfied for a given illegitimate probability space.
Compute the probability of an event when the probability space of the experiment is given.
Write the probability space for a given random phenomena.
Identify the different situations in which permutations or combinations are used.
Normal Distributions
Define statistical inference.
Explain the difference between a parameter and a statistic.
Identify both the parameter and the statistic in a simple inferential setting.
Explain what a random variable is.
Explain the difference between the honest and dishonest-coin principles.
Using an appropriate formula, calculate the mean and standard deviation of a given statistic.
Discuss the effect of an increased sample size on the statistic's sampling error.
Explain the difference between the population mean and the sample mean.
Describe a normal curve.
Locate the mean and standard deviation from a graph of a normal curve.
Explain the 68-95-99.7 rule and apply it to compute normal probabilities.
Give the mean and standard deviation of a normally distributed data set, and compute the
percent of the population that falls within a given interval.
1 Section from the following – 5 hours
Mathematics of Sharing (Ch3)
Mathematics of Apportionment (Ch4)
Mathematics of Scheduling (Ch8)
Growth in Nature (Ch9)
Financial Mathematics (Ch10)
Censuses, Surveys, Polls and Studies (Ch14)
Topics for Assessment:
For the final exam students should be able to:
 Differentiate between the methods for finding a winner of an election and be able to
use the method to determine a winner.(Plurality, Borda Count, Plurality with elimination
and pairwise comparisons)
 Understand the mechanics behind a weighted voting system
 Find the power distribution of players using both the Banzhaf and Shapely-Shubik Power
indices
 Find edges and vertices of a graph
 Find paths, circuits, Euler paths and Euler circuits
 Find Hamiltonian paths or circuits using different algorithms (Brute forces, nearest
neighbor, repetitive nearest neighbor, or cheapest link)
 Find a spanning tree of a graph and be able to apply Kruskal’s Algorithm
 Find quartiles, range, interquartile range, mean, median and standard deviation for a
data set
 Describe a sample space for a random phenomena
 List outcomes and probabilities for an event
 Find probabilities using permutations or combinations
 Find the mean and standard deviation for a normal curve
 Use the 68-95-99.7 rule to calculate normal probabilities
Criteria for Assessment:
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Students are able to master problem solving skills
Students learn to manipulate abstract symbols
Students learn a broad spectrum of mathematical applications
 Basic statistics
 Graph theory and its applications
 Probability
 Social choice and voting systems