Chabot College
Fall 2013
Course Outline for Mathematics 47
MATHEMATICS FOR LIBERAL ARTS
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Catalog Description:
MTH 47 - Mathematics for Liberal Arts
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3.00 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: MTH 54 (completed with a grade of "C" or higher) or , MTH 54L (completed with a grade of
"C" or higher) or , MTH 55 (completed with a grade of "C" or higher) or , MTH 55L (completed with a grade
of "C" or higher) or , MTH 55B (completed with a grade of "C" or higher) or an equivalent course or an
appropriate skill level demonstrated through the mathematics assessment process.
Units
Contact Hours
Week
Term
3.00
Lecture
Laboratory
Clinical
Total
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3.00
3.00
0
0.00
3.00
52.50
0
0.00
52.50
Prerequisite Skills:
Before entry into this course, the student should be able to:
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describe data using concepts of frequency and measures of central tendency;
identify functions, find domain and range, and use function notation in the context of real data;
identify the slope of a line using that it is parallel to another line;
find average rates of change;
graph and find the equations of linear functions in the context of real data;
solve problems involving direct and inverse proportionality;
find linear models for data;
find linear system models for data and interpret solutions to these linear systems;
perform operations using the properties of rational exponents;
graph exponential functions and interpret real growth and decay situations and data with
exponential functions;
solve exponential equations using logarithms;
analyze real situations and data by using exponential functions with base e and natural logarithmic
functions;
find inverse functions and compose functions in the context of real data;
graph quadratic, power, and logarithmic functions;
analyze real situations and data using quadratic functions;
choose an appropriate model for a realistic situation given a choice of mathematical models.
solve quadratic equations by factoring, completing the square, and quadratic formula;
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;
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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.
sketch the graphs of functions and relations:
a. logarithmic
b. exponential
c. 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;
solve quadratic equations by factoring, completing the square, and quadratic formula;
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 this course, the student should be able to:
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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;
explain how the fairness criterion is violated when given the outcome of a voting method that
violates one of the fairness criteria:
prepare an argument for or against changing from majority voting to another voting method;
determine the critical voters in a winning coalition given a weighted voting system;
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;
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;
compare the future value for simple interest and compound interest, including different
compounding periods;
determine which method of computing financial charges minimizes the total financial charges on a
particular loan and/or credit card;
observe patterns and form conjectures about properties of Fibonacci-like sequences;
construct truth tables;
write the negation, converse, inverse and contrapositive of a statement;
determine the validity of a logical argument;
apply modular arithmetic to solve application problems;
discuss the advantages and disadvantages of a given base for computations done by human or
computer;
15. explain the advantage in a positional numeration system of using a larger base over a smaller
base;
16. determine whether two graphs are isomorphic when given the diagrams of the two graphs;
17. 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;
18. apply an algorithm to find an Euler path or circuit in a connected graph;
19. determine whether a sequence is a Hamilton circuit when given a graph and a sequence vertices;
and
20. solve the traveling salesperson problem when given a small weighted graph, using a) the brute
force algorithm and b) the nearest neighbor algorithm.
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Course Content:
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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
D. Jefferson's method
E. Adam's method
F. Webster's method
G. Huntington-Hill method
6. Paradoxes and Violations
A. Population paradox
B. Alabama paradox
C. New-states paradox
D. The quota rule
E. Absolute and relative unfairness
7. Simple Interest
A. Future value
B. Present value
8. Compound Interest
A. Future value
B. Present value
C. Effective Annual Rate
9. Credit Card Statements
A. Average daily balance
B. Finance charge
C. Balance subject to finance charge
10. Consumer loan charging add-on simple interest
A. Monthly payment
B. APR
11. Annuities
A. Sum of geometric series
B. Future value
C. Sinking fund
12. Amortization
A. Present value
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Methods of Presentation
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B. Monthly payments
C. Loan payoff amount
Fibonacci Numbers
A. Definition
a. Recursive
b. Binet's formula
B. Properties of the sequence
C. Phi
D. Fibonacci like sequences
Logic
A. Simple and compound statements
B. Connectives
C. Symbolic Notation
D. Statements
a. Tautology
b. Self-contradiction statements
c. Negation
d. Converse
e. Inverse
f. Contrapositive
E. Validity of an Argument
a. Truth tables
b. Common argument forms
c. Euler diagrams
Numeral Representation Schemes
A. Additive (Roman numerals)
B. Multiplicatives (Chinese)
C. Positional
a. Binary
b. Decimal
D. Elemental (prime factorization
Graphs
A. Isomorphic
B. Connected
C. Paths and circuits
a. Euler
b. Hamilton
Lecture/Discussion
Problem Solving
Presentation of audio-visual materials
Group Activities
Assignments and Methods of Evaluating Student Progress
1. Typical Assignments
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Exercises from the textbook such as the following: Make a conjecture about the next
equation in the list and then verify using proof by induction. 1 x 1 = 1; 1 x 3 = 3; 2 x 4 = 8;
3 x 7 = 21; 5 x11 = 55
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.
Exercises from the textbook such as the following: Steven Booth finds that whether he
sorts his White Sox ticket stubs into piles of 10, piles of 15, or piles of 20, there are always
2 left over. What is the least number of stubs he could have, assuming he has more than
2 stubs?
2. Methods of Evaluating Student Progress
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Homework
Quizzes
Class Participation
Exams/Tests
Final Examination
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Textbooks (Typical):
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Miller/Heeren/Hornsby (2012). Mathematical Ideas (12th/e). Addison-Wesley.
Pirnot (2010). Mathematics All Around (4th/e). Addison-Wesley.
Special Student Materials
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Scientific calculator