Math 116

Dr. John Maharry

Textbook Excursions in Modern Mathematics

Chapter 15: Chances, Probability, and Odds: Measuring Uncertainty

• 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
• 
  

Chapter 4: The Mathematics of Apportionment: Making the Rounds

• 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

Chapter 6: The Traveling-Salesman Problem: Hamilton Joins the Circuit

• 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.

13-year old kids uses Fibonacci Numbers to build a better solar panel : http://whatsnext.blogs.cnn.com/2011/11/18/meet-a-13-year-old-solar-panel-developer/?hpt=hp_c1

• 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.

Topics we don't cover

Chapter 2: Weighted Voting Systems: The Power Game

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

Chapter 3: Fair Division: The Mathematics of Sharing

• 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.

Chapter 8: The Mathematics of Scheduling: Directed Graphs and Critical Paths

• 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.

Chapter 11: Symmetry: Mirror, Mirror, off the Wall...

• 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.

Chapter 12: The Geometry of Fractal Shapes: Fractally Speaking

• 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.

Chapter 13: Collecting Statistical Data: Censuses, Surveys, and Clinical Studies

• 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.

Chapter 14: Descriptive Statistics: Graphing and Summarizing Data

• 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.

Chapter 16: Normal Distributions: Everything Is Back to Normal (Almost)

• 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.