Chabot College Fall 2013 Course Outline for Mathematics 47 Catalog Description: MATHEMATICS FOR LIBERAL ARTS MTH 47 - Mathematics for Liberal Arts 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 3.00 52.50 Laboratory 0 0 Clinical 0.00 0.00 Total 3.00 3.00 52.50 Prerequisite Skills: Before entry into this course, the student should be able to: 1. describe data using concepts of frequency and measures of central tendency; 2. identify functions, find domain and range, and use function notation in the context of real data; 3. identify the slope of a line using that it is parallel to another line; 4. find average rates of change; 5. graph and find the equations of linear functions in the context of real data; 6. solve problems involving direct and inverse proportionality; 7. find linear models for data; 8. find linear system models for data and interpret solutions to these linear systems; 9. perform operations using the properties of rational exponents; 10. graph exponential functions and interpret real growth and decay situations and data with exponential functions; 11. solve exponential equations using logarithms; 12. analyze real situations and data by using exponential functions with base e and natural logarithmic functions; 13. find inverse functions and compose functions in the context of real data; 14. graph quadratic, power, and logarithmic functions; 15. analyze real situations and data using quadratic functions; 16. choose an appropriate model for a realistic situation given a choice of mathematical models. 17. solve quadratic equations by factoring, completing the square, and quadratic formula; 18. sketch the graphs of functions and relations: a. algebraic, including polynomial and rational b. logarithmic c. exponential d. circles; 19. find and sketch inverse functions; 20. perform function composition; 21. solve exponential and logarithmic equations;
22. apply the concepts of logarithmic and exponential functions; 23. solve systems of linear equations in three unknowns using elimination and substitution; 24. apply the properties of and perform operations with radicals; 25. apply the properties of and perform operations with rational exponents; 26. solve equations and inequalities involving absolute values; 27. solve equations involving radicals; 28. graph linear inequalities in two variables; 29. find the distance between two points; 30. find the midpoint of a line segment. 31. sketch the graphs of functions and relations: a. logarithmic b. exponential c. circles; 32. find and sketch inverse functions; 33. perform function composition; 34. solve exponential and logarithmic equations; 35. apply the concepts of logarithmic and exponential functions; 36. solve systems of linear equations in three unknowns using elimination and substitution; 37. solve quadratic equations by factoring, completing the square, and quadratic formula; 38. sketch the graphs of functions and relations: a. algebraic, including polynomial and rational b. logarithmic c. exponential d. circles; 39. find and sketch inverse functions; 40. perform function composition; 41. solve exponential and logarithmic equations; 42. apply the concepts of logarithmic and exponential functions; 43. solve systems of linear equations in three unknowns using elimination and substitution; 44. apply the properties of and perform operations with radicals; 45. apply the properties of and perform operations with rational exponents; 46. solve equations and inequalities involving absolute values; 47. solve equations involving radicals; 48. graph linear inequalities in two variables; 49. find the distance between two points; 50. find the midpoint of a line segment. Expected Outcomes for Students: Upon completion of this 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. compare the future value for simple interest and compound interest, including different compounding periods; 8. determine which method of computing financial charges minimizes the total financial charges on a particular loan and/or credit card; 9. observe patterns and form conjectures about properties of Fibonacci-like sequences; 10. construct truth tables; 11. write the negation, converse, inverse and contrapositive of a statement; 12. determine the validity of a logical argument; 13. apply modular arithmetic to solve application problems; 14. 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. 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 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
B. Monthly payments C. Loan payoff amount 13. Fibonacci Numbers A. Definition a. Recursive b. Binet's formula B. Properties of the sequence C. Phi D. Fibonacci like sequences 14. 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 15. Numeral Representation Schemes A. Additive (Roman numerals) B. Multiplicatives (Chinese) C. Positional a. Binary b. Decimal D. Elemental (prime factorization 16. Graphs A. Isomorphic B. Connected C. Paths and circuits a. Euler b. Hamilton Methods of Presentation 1. Lecture/Discussion 2. Problem Solving 3. Presentation of audio-visual materials 4. Group Activities Assignments and Methods of Evaluating Student Progress 1. Typical Assignments A. 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 B. 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. C. 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 A. Homework B. Quizzes C. Class Participation D. Exams/Tests E. Final Examination
Textbooks (Typical): 1. Miller/Heeren/Hornsby (2012). Mathematical Ideas (12th/e). Addison-Wesley. 2. Pirnot (2010). Mathematics All Around (4th/e). Addison-Wesley. Special Student Materials 1. Scientific calculator