Dr. John Maharry

Textbook Excursions in Modern Mathematics

- Describe an appropriate sample space of a random experiment.
- Apply the multiplication rule, permutations, and combinations to counting problems.
- Understand the concept of a probability assignment.
- Identify independent events and their properties.
- Use the language of odds in describing probabilities of events.

Web Links

- Video on Expected Value
- Another one from KhanAcademy
**Chapter 1: The Mathematics of Voting: The Paradoxes of Democracy**

- Construct and interpret a preference schedule for an election involving preference ballots.
- Implement the plurality, Borda count, plurality-with-elimination, and pairwise comparisons vote counting methods.
- Rank candidates using recursive and extended methods.
- Identify fairness criteria as they pertain to voting methods.
- Understand the significance of Arrows' impossibility theorem.

Web Links

- http://en.wikipedia.org/wiki/Marquis_de_Condorcet
- http://www.cut-the-knot.org/Curriculum/SocialScience/SocialChoice.shtml
- Results of 2000 Presidential Election http://www.infoplease.com/ipa/A0876793.html
- http://en.wikipedia.org/wiki/Jean-Charles_de_Borda
- http://en.wikipedia.org/wiki/Voting_system
- http://en.wikipedia.org/wiki/Monotonicity_criterion
- http://en.wikipedia.org/wiki/Arrow's_impossibility_theorem
- http://en.wikipedia.org/wiki/Kenneth_Arrow

- State the basic apportionment problem.
- Implement the methods of Hamilton, Jefferson, Adams, and Webster to solve apportionment problems.
- State the quota rule and determine when it is satisfied.
- Identify paradoxes when they occur.
- Understand the significance of Balinski and Young's impossibility theorem.

Web links

- http://www.cut-the-knot.org/Curriculum/SocialScience/AHamilton.shtml
- http://en.wikipedia.org/wiki/Article_One_of_the_United_States_Constitution#Clause_3:_Apportionment_of_Representatives_and_Taxes
- http://en.wikipedia.org/wiki/Alabama_paradox#Alabama_Paradox
- http://www.cut-the-knot.org/Curriculum/SocialScience/Webster.shtml

- Identify and model Euler circuit and Euler path problems.
- Understand the meaning of basic graph terminology.
- Classify which graphs have Euler circuits or paths using Euler's circuit theorems.
- Implement Fleury's algorithm to find an Euler circuit or path when it exists.
- Eulerize and semi-eulerize graphs when necessary.
- Recognize an optimal eulerization (semi-eulerization) of a graph.

Web Links

- Cool Web Applet that makes a graph out of the structure of any Web Page http://www.aharef.info/static/htmlgraph/
- Flash Applet to explore Euler Circuits http://www.flashandmath.com/mathlets/discrete/graphtheory/euler.html

- Identify and model Hamilton circuit and Hamilton path problems.
- Recognize complete graphs and state the number of Hamilton circuits that they have.
- Identify traveling-salesman problems and the difficulties faced in solving them.
- Implement brute-force, nearest-neighbor, repeated nearest-neighbor, and cheapest-link algorithms to find approximate solutions to traveling-salesman problems.
- Recognize the difference between efficient and inefficient algorithms.
- Recognize the difference between optimal and approximate algorithms.

Web Links

- http://wps.prenhall.com/esm_tannenbaum_excursions_5/14/3687/943975.cw/index.html
- http://www.flashandmath.com/mathlets/discrete/graphtheory/graph2.html
- http://en.wikipedia.org/wiki/W._T._Tutte
- http://www-e.uni-magdeburg.de/mertens/TSP/TSP.html
- Find a Hamilton Circuit in the following Graphs http://www.flashandmath.com/mathlets/discrete/graphtheory/graph2.html
- Find a Hamilton Circuit or Euler Circuit in several graphs (deletes edges as you use them) http://www.cut-the-knot.org/Curriculum/Combinatorics/GraphPractice.shtml
- Several different approximation algorithms that run on random sets of vertices http://web.telia.com/~u85905224/tsp/TSP.htm.
- Another complicated approximate solution algorithm http://www.sund.de/netze/applets/som/som2/index.htm
- Nearest-Neighbor and Insertion Algorithms that show one step at a time http://www-e.uni-magdeburg.de/mertens/TSP/TSP.html
- Applet to run Brute Force Nearest Neighbor and Cheapest Link (Greedy) http://www.wiley.com/college/mat/gilbert139343/java/java09_s.html

- Identify and use a graph to model minimum network problems.
- Classify which graphs are trees.
- Implement Kruskal's algorithm to find a minimal spanning tree.
- Understand Torricelli's construction for finding a Steiner point.
- Recognize when the shortest network connecting three points uses a Steiner point.
- Understand basic properties of the shortest network connecting a set of (more than three) points.

- Generate the Fibonacci sequence and identify some of its properties.
- Identify relationships between the Fibonacci sequence and the golden ratio.
- Define a gnomon and understand the concept of similarity.
- Recognize gnomonic growth in nature.

- Understand how a transition rule models population growth.
- Recognize linear, exponential, and logistic growth models.
- Apply linear, exponential, and logistic growth models to solve population growth problems.
- Differentiate between recursive and explicit models of population growth.
- Apply the general compounding formula to answer financial questions.
- State and apply the arithmetic and geometric sum formulas in their appropriate contexts.

- Represent a weighted voting system using a mathematical model.
- Calculate the Banzhaf and Shapley-Shubik power distribution in a weighted voting system.

- State the fair-division problem and identify assumptions used in developing solution methods.
- Recognize 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.

- Understand and use digraph terminology.
- Schedule a project on N processors using the priority-list model.
- Apply the backflow algorithm to find the critical path of a project.
- Implement the decreasing-time and critical-path algorithms.
- Recognize optimal schedules and the difficulties faced in finding them.

- Describe the basic rigid motions of the plane and state their properties.
- Classify the possible symmetries of any finite two-dimensional shape or object.
- Classify the possible symmetries of a border pattern.

- Explain the process by which fractals such as the Koch snowflake and the Sierpinski Gasket are constructed.
- Recognize self-similarity (or symmetry of scale) and its relevance.
- Describe how random processes can create fractals such as the Sierpinski Gasket.
- Explain the process by which the Mandelbrot set is constructed.

- Identify whether a given survey or poll is biased.
- List and discuss the quality of several sampling methods.
- Identify components of a well-constructed clinical study.
- Define key terminology in the data collection process.
- Estimate the size of a population using the capture-recapture method.

- Interpret and produce an effective graphical summary of a data set.
- Identify various types of numerical variables.
- Interpret and produce numerical summaries of data including percentiles and five-number summaries.
- Describe the spread of a data set using range, interquartile range, and standard deviation.

- Identify and describe an approximately normal distribution.
- State properties of a normal distribution.
- Understand a data set in terms of standardized data values.
- State the 68-95-99.7 rule.
- Apply the honest and dishonest-coin principles to understand the concept of a confidence interval.