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MATH 0018 - The Nature of Mathematics

http://catalog.sierracollege.edu/course-outlines/math-0018/

Catalog Description DESCRIPTION IS HERE: Prerequisite: Completion of Intermediate Algebra or appropriate placement Hours: 54 lecture Description: Introduces students to the art and application of mathematics in the world around them. Topics include mathematical modeling, voting and apportionment, and mathematical reasoning with applications chosen from a variety of disciplines. Not recommended for students entering elementary school teaching or business. (CSU, UC-with unit limitation) Units 3 Lecture-Discussion 54 Laboratory By Arrangement Contact Hours 54 Outside of Class Hours Course Student Learning Outcomes Solve college level math problems from a variety of different areas. Utilize linear, quadratic, exponential, and logarithmic equations, systems of equations, and their graphs to analyze mathematical applications from various disciplines. Analyze given information and implement strategies for solving problems involving mathematical and logical reasoning. Use mathematical modeling as a problem solving tool in other disciplines and contexts. Logically present clear, complete, accurate, and sufficiently detailed solutions to communicate reasoning and demonstrate the method of solving problems. Course Content Outline I. Mathematical Modeling A. Applications of linear and quadratic functions and graphs, using tools such as regression lines, optimization, and linear programming B. Exponential and logarithmic function applications such as growth and decay problems, logistic equations, business and financial applications, and resource analysis C. Modeling with other mathematical tools and algorithms: applications such as symmetry, tilings, fair division, group theory, graph theory, and networks II. Voting and Apportionment A. Apportionment Methods B. Voting systems 1. Mathematics of Voting systems 2. Weighted voting systems III. Mathematical Reasoning: Development of mathematical reasoning through study of topics such as numeric and geometric patterns, sequences, probability and chance, and combinatorics IV. Other Topics from Higher Mathematics A. Modular arithmetic and cryptology B. Topics from pure mathematics such as logic, set theory, game theory, non-Euclidean and fractal geometry, and chaos theory Course Objectives Course Objectives 1. Solve college level math problems from a variety of different mathematical subject areas, especially topics not usually covered in a traditional mathematics course. 2. Analyze given information and develop strategies for solving problems involving mathematical and logical reasoning. 3. Recognize and apply the concepts of mathematics as a problem-solving tool in other disciplines and contexts. 4. Utilize linear, quadratic, exponential, and logarithmic equations, systems of equations, and their graphs to analyze mathematical applications from various disciplines. 5. Compare and contrast apportionment methods and voting systems, using an appropriate level of mathematics to support any conclusions. Methods of Evaluation Objective Examinations Problem Solving Examinations Projects Reading Assignments 1. Read selections in the textbook or online concerning the Fibonacci sequence. One example is at http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html Be prepared to discuss in class where we find Fibonacci numbers and why they occur in nature so frequently. 2. Read an article about how the Best Picture Oscar winner is chosen, and compare and contrast this method to one of the voting methods studied in class. How does this method hold up the Arrow's Fairness Criteria? Be prepared to discuss in class. 3. Research examples of how the Golden Ratio has been used by humans to creatively (for example in art or architecture). Show understanding of where and how exactly this mathematical ratio is incorporated into the chosen examples and be prepared to discuss in class. Writing, Problem Solving or Performance 1. Create a weighted voting system with 4 members in which 1 person has veto power. Calculate the Banzhaf Power Index for the system. Compare this system to a voting system with 5 members in which one person equals one vote. Calculate the Banzhaf Power Index for this system and use it in your discussion. 2. Use the Division Algorithm to show that the remainder when a number n is divided by m is equal to the position n would be on a mod m clock. 3. Public Key Encryption: Using the 2 public numbers 7 and 143, encode the following string of numbers: "2 83 3 61 38". 4. Write about the relationships between the Fibonacci sequence and the Golden ratio. How are a Fibonacci spiral and a Golden spiral different? 5. Create a population sequence model for a given species. Write the model in infinite list notation, using a recursive formula, and using an explicit formula. Other (Term projects, research papers, portfolios, etc.) 1. Create a graph model of a residential neighborhood from Google Maps for a mail carrier route. After determining the degree of each vertex, Eulerize the graph and find a route for the mail carrier to deliver the mail to all the residents. 2. Create something that utilizes Fibonacci numbers and/or the Golden Ratio in multiple ways. Write an explanation showing how these numbers were incorporated and demonstrate understanding of the concepts. Methods of Instruction Lecture/Discussion Distance Learning Other materials and-or supplies required of students that contribute to the cost of the course.