Math 5 --- Contemporary Mathematics
Catalog Description
A study of mathematical concepts that include inductive and deductive reasoning, mathematical modeling and analysis of linear and exponential functions, geometric symmetries, geometry of fractals, sequences and series, dynamics of population growth, statistics, mathematics of finance and management science, mathematics of methods of voting, fair division, and problem-solving techniques that include a variety of practical problems. This course is designed for liberal arts students.
Special notes or advisories: Graphing calculators are required
Prerequisites
Math 120 (or equivalent) with "C" or better or appropriate score on assessment exam
Describe representative skills without which the student would be highly unlikely to succeed: Ability to solve linear and exponential equations analytically and graphically. Ability to use technology in the study of the course topics. Ability to verbally express mathematical concepts as applied to the course topics./p>
Course Learning Outcomes
What should the student be able to do as a result of taking this course? State some of the objectives in terms of specific, measurable student accomplishments.
- Accurately communicate mathematical ideas using correct mathematical notation, graphs, and vocabulary.
- Demonstrate appropriate use of the graphing calculator to explore mathematical concepts and verify their quantitative conclusions.
- Solve problems and applications demonstrating the skills required for college-level mathematics.
- Demonstrate the characteristics of an effective learner, such as note-taking, critical reading, communication through writing, verbal discussions, presentations, etc.
- Explain the concepts of mathematics of social choice, statistics, growth, symmetry, finance, and management science and use the concepts to solve problems in these fields.
- Perform symbolic manipulations and use technology that will support success in the outcomes.
Course Content
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Themes: What themes, if any, are threaded throughout the learning experiences in this course?
- Critical thinking.
- Problem solving.
- Symbol manipulation.
- Use of technology.
- Communication.
- Application to real world situations.
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Concepts: What concepts do students need to understand to demonstrate course outcomes?
- A multiple-step problem-solving process.
- The presentation of mathematical solutions in a logical, coherent structure, including the use of writing skills, grammar, illustrations, and punctuation.
- The use of the graphing calculator as a problem-solving tool.
- The relationship between mathematical concepts to real-world problems.
- Mathematical concepts to include definitions, properties, graphs, and their application to the problem-solving process.
- The recognition that proper symbolic manipulation is an important tool in multiple problem-solving situations.
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Issues: What primary issues or problems, if any, must students understand to achieve course outcomes.
- The role of the student in becoming a successful learner.
- The importance of writing/presenting mathematics using correct notation and grammar.
- The limitations of technology.
- The influence of mathematics in the liberal arts, social and environmental issues, finance, and data analysis.
- The necessity to read unfamiliar mathematics using the text and other resources.
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Skills: What skills must students master to demonstrate course outcomes?
- Use a calculator to: graph linear and exponential functions, enter data and find appropriate mathematical models, determine regression equations.
- Topics for study will be selected from the following:
- The Mathematics of Voting:
- Interpret preference ballots and construct a preference schedule.
- Compare the results of counting votes using the different methods of plurality, Borda count, plurality-with-elimination, and pairwise comparisons.
- Apply recursive and extended methods to rank candidates.
- Identify fairness criteria and violations of these criteria.
- Explain Arrows' Impossibility Theorem.
- Identify the paradoxes of voting methods.
- The Mathematics of Fair Division:
- Identify assumptions used in developing solutions to fair-division problems.
- Compare the differences between continuous and discrete fair-division problems.
- Apply the divider-chooser, lone-divider, lone-chooser, and last-diminisher methods to continuous fair-division problems.
- Apply the method of sealed bids and the method of markers to discrete fair-division problems.
- The Mathematics of Apportionment:
- State the basic apportionment problem.
- Solve apportionment problems using the historic methods of Hamilton, Jefferson, Adams, and Webster.
- Apply the presently used Huntington-Hill Method to apportionment problems.
- Determine violations of the quota rule.
- Identify paradoxes.
- Explain Balinski and Young's Impossibility Theorem.
- The Mathematics of Spiral Growth:
- Define and generate the Fibonacci sequence.
- Identify occurences of Fibonacci numbers in nature and art.
- Compare the Fibonacci sequence to features of the golden ratio.
- Define a gnomon and identify similar figures.
- The Mathematics of Symmetry:
- State the basic rigid motions of the plane.
- Describe the properties of reflections, rotations, translations, and glide reflections.
- Define symmetry as a rigid motion.
- Classify the symmetries of two-dimensional shapes.
- The Mathematics of Fractal Shapes:
- Identify the iterative processes that produce the Koch snowflake and the Sierpenski Gasket.
- Construct examples of fractal shapes.
- Explain symmetry of scale.
- Describe random processes used to create fractals.
- Explain the construction of the Mandelbrot set.
- The Mathematics of Population Growth:
- Differentiate between examples of linear and exponential growth.
- Explain a transition rule that models population growth.
- Solve population growth problems.
- Compare recursive and explicit models of population growth.
- Apply the general compounding formula to answer questions of economic and environmental concern.
- State and apply the formulas for arithmetic and geometric series.
- The Mathematics of Descriptive Statistics:
- Summarize a data set to produce tables, graphs, or equations.
- Interpret and produce percentiles, mean, median, mode, and standard deviation.
- Describe the spread of the data using range and standard deviation.
- Identify an approximately normal distribution.
- State the properties of a normal disbtribution.
- Interpret standardized data values.
- Explain the 68-95-99.7 rule.
- Calculate probabilities.
- The Mathematics of Management Science:
- Identify and model Hamilton circuit and path problems.
- Recognize traveling-salesman problems and the difficulties.
- Identify algorithms.
- Model minimum network problems.
- Classify which graphs are trees.
- Recognize the basic properties of the shortest network connecting a set of points.
- Analyze the mathematics of scheduling using digraph terminology.
- The Mathematics of Voting:
Representative Learning Activities
What will the students be doing (i.e., Listening to lectures, participating in discussions and/or group activities, attending a field trip, etc.)? Relate the activities directly to the Course Learning Outcomes.
- Listening to lectures.
- Participating in group activities or assignments.
- Participating in in-class assignments or discussions.
- Completing daily homework assignments.
- Completing online activities on the computer.
- Reading the textbook and other printed resources.
- Creating posters.
Assessment Tasks
How will the student show evidence of achieving the Course Learning Outcomes? Indicate which assessments (if any) are required for all sections.
- Representative assessment tasks:
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- Writing assignments to develop communication of mathematical concepts.
- Quizzes and tests.
- Group projects or other in-class activities.
- Portfolios.
- Individual projects, posters, or presentations.
- Homework.
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Required assessments for all sections – to include but not limited to:
- At least two proctored, closed-book examinations. An individual final project or a comprehensive, proctored, closed-book final exam.
Examples of Appropriate Texts or Other Readings
- Author: Stewart; Title: Calculus, Early Transcendentals, Sixth Edition; Date: 2007