Chabot College
Fall 2012
Course Outline for Mathematics 47
MATHEMATICS FOR LIBERAL ARTS
Catalog Description:
47 – Mathematics for Liberal Arts
3 units
An introduction to a variety of mathematical concepts for students interested in liberal arts. Focus is on
using mathematics to help make informed decisions. Applications include voting practices, apportionment
and personal finance. Prerequisite: Mathematics 54, 54L, 55, 55L, or 55B or equivalent (completed with a
grade of “C” or higher) or an appropriate skill level demonstrated through the mathematics assessment
process. 3 hours.
[Typical contact hours: lecture 52.5]
Prerequisite Skills:
Before entering the course the student should be able to:
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perform basic operations on complex numbers;
solve quadratic equations by factoring, completing the square, and quadratic formula;
find complex roots of a quadratic equation;
sketch the graphs of functions and relations:
a. algebraic, including polynomial and rational
b. logarithmic
c. exponential
d. circles;
find and sketch inverse functions;
perform function composition;
solve exponential and logarithmic equations;
apply the concepts of logarithmic and exponential functions;
solve systems of linear equations in three unknowns using elimination and substitution;
apply the properties of and perform operations with radicals;
apply the properties of and perform operations with rational exponents;
solve equations and inequalities involving absolute values;
solve equations involving radicals;
graph linear inequalities in two variables;
find the distance between two points;
find the midpoint of a line segment.
Expected Outcomes for Students:
Upon completion of the course the student should be able to:
1. apply a given voting method to determine the election result when given a description of the voting
method and the preferences of a small population of voters;
2. explain how the fairness criterion is violated when given the outcome of a voting method that violates
one of the fairness criteria:
3. prepare an argument for or against changing from majority voting to another voting method;
4. determine the critical voters in a winning coalition given a weighted voting system;
5. apply a given apportionment method to determine the apportionment when given the relevant
information about the distribution of the population and the total number of representatives;
6. explain the paradox or violation of the quota rule when given an outcome of an apportionment method
having a paradox or violation and describe how it leads to controversy;
7. calculate future value using the principles of simple interest;
8. calculate the interest rate per compounding period, future value, present value, and effective annual
rate using the principles of compound interest;
Chabot College
Course Outline for Mathematics 47, page 2
Fall 2012
9. compare the future value for simple interest and compound interest, including different compounding
periods;
10. identify and explain the meaning of the finance charge, average daily balance, and balance subject to
finance charge when given a credit card statement;
11. determine the monthly payment and the APR when given a consumer loan charging add-on simple
interest;
12. find a particular term in the Fibonacci sequence using the recursion formula and Binet’s formula;
13. discuss the occurrences of the Fibonacci numbers in art, architecture and nature;
14. compare the properties of sequences that have the same recursive formula as a Fibonacci sequence
but different start values;
15. explain the relationship between the Fibonacci sequence and the number phi;
16. explain the relationship between the number phi and the golden rectangle;
17. explain the relationship between the phi and the quadratic equation x 2-x-1=0;
18. determine if a rectangle is a golden rectangle;
19. classify a statement as simple or compound;
20. identify a connective used in a compound statement;
21. translate an English statement into symbolic notation;
22. construct truth tables;
23. classify a statement as a tautology, self-contradiction, or neither;
24. identify logically equivalent statements;
25. determine and write the negation, converse, inverse and contrapositive of a statement;
26. determine the validity of an argument;
27. convert a numeral from one representation scheme to another;
28. perform calculations using numerals in various bases; and
29. discuss the advantages and disadvantages of a given base for computations done by human or
computer;
30. determine whether two graphs are isomorphic when given the diagrams of the two graphs;
31. diagram a connected graph, determine the degree of each vertex and determine whether the graph
contains an Euler path or circuit when given the description of a connected graph;
32. apply an algorithm to find an Euler path or circuit in a connected graph;
33. determine whether a sequence is a Hamilton circuit when given a graph and a sequence vertices; and
34. solve the traveling salesperson problem when given a small weighted graph, using a) the brute force
algorithm and b) the nearest neighbor algorithm.
Course Content:
1. Voting Methods
a. Borda count
b. Plurality with elimination
c. Pairwise comparison
d. Preference ballot
e. Approval voting
2. Fairness Criteria for Voting Methods
a. Majority criterion
b. Condorcet’s criterion
c. Independence-of-irrelevant alternatives criterion
d. Monotonicity criterion
3. Arrow’s Impossibility Theorem
4. Weighted Voting Systems
a. Weights and Quotas
b. Coalitions
c. Banzhaf Power Index
5. Apportionment methods
a. Standard divisors and quotas
b. Modified divisors and quotas
c. Hamilton’s method
Chabot College
Course Outline for Mathematics 47, page 3
Fall 2012
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d. Jefferson’s method
e. Adam’s method
f. Webster’s method
g. Huntington-Hill method
Paradoxes and Violations
a. Population paradox
b. Alabama paradox
c. New-states paradox
d. The quota rule
e. Absolute and relative unfairness
Simple Interest
a. Future value
b. Present value
Compound Interest
a. Future value
b. Present value
c. Effective Annual Rate
Credit Card Statements
a. Average daily balance
b. Finance charge
c. Balance subject to finance charge
Consumer loan charging add-on simple interest
a. Monthly payment
b. APR
Annuities
a. Sum of geometric series
b. Future value
c. Sinking fund
Amortization
a. Present value
b. Monthly payments
c. Loan payoff amount
Fibonacci Numbers
a. Definition
i. Recursive
ii. Binet’s formula
b. Properties of the sequence
c. Phi
d. Golden rectangle
Logic
a. Simple and compound statements
b. Connectives
c. Symbolic Notation
d. Statements
i. Tautology
ii. Self-contradiction statements
iii. Negation
iv. Converse
v. Inverse
vi. Contrapositive
e. Validity of an Argument
i. Truth tables
ii. Common argument forms
iii. Euler circles
Numeral Representation Schemes
a. Additive (Roman numerals)
Chabot College
Course Outline for Mathematics 47, page 4
Fall 2012
b. Multiplicative (Chinese)
c. Positional
i. Binary
ii. Decimal
d. Elemental (Prime factorization)
16. Graphs
a. Isomorphic
b. Connected
c. Paths and circuits
i. Euler
ii. Hamilton
Methods of Presentations:
1. Informal lectures
2. Class discussion of problems, solutions and student’s questions
3. Audio-visual materials
Assignments and Methods of Evaluating Student Progress:
1. Typical Assignments
a. Exercises from the textbook such as the following: Draw a graph that has a Hamilton circuit but no
Euler circuit. Specify the Hamilton circuit, and explain why the graph has no Euler circuit.
b. Determine whether the following argument is valid or invalid. If Hurricane Gustave hit that grove of
trees, then the trees are devastated. The trees are not devastated. Therefore, Hurricane Gustave
did not hit that grove of trees.
1. Methods of Evaluating Student Progress
a. Homework
b. Quizzes
c. Class participation
d. Examinations
e. Final examination
Textbook(s) (Typical):
Mathematical Ideas by Miller/Heeren/Hornsby, 12th edition, Addison Wesley, 2012
Mathematics All Around by Pirnot, 4th edition, Addison Wesley, 2010
Special Student Materials:
Scientific calculator
Created by Stubblebine, Traugott, Wah