Discrete Math
Syllabus
Developed by Matt DeCerbo & Andrew Carucci
Lee County High School
Course Description: Discrete Math introduces students to the mathematics of networks,
social choice, and decision making. The course extends students’ application of matrix
arithmetic and probability. Applications and modeling are central to this course of study.
Appropriate technology, from manipulatives to calculators and application software,
should be used regularly for instruction and assessment.
Textbook
Excursions in Modern Mathematics 5th Edition
Peter Tannenbaum
Grading Policy
 Test: 50% of final average
 Quiz: 20% of final average
 Project: 20% of final average
 HW: 10% of final average
Final Average
st
 1 Quarter Grade: 37.5% of final grade
 2nd Quarter Grade: 37.5% of final grade
 Final Exam Grade: 25% of final grade
All students are required to take an MSL which will account for their final exam at the
end of the semester. This exam will account for 25% of your final average.
Course Objectives
Competency Goal 1: The learner will use matrices and graphs to model relationships and
solve problems.
 Use matrices to model and solve problems
 Display and interpret data
 Write and evaluate matrix expressions to solve problems
 Use graph theory to model relationships and solve problems
Competency Goal 2: The learner will analyze data and apply probability concepts to
solve problems.
 Describe data to solve problems
 Apply and compare methods of data collection
 Apply statistical principles and methods of data collection
 Determine measures of central tendency and spread
 Recognize, define, and use the normal distribution curve
 Interpret graphical displays of data
 Compare distributions of data
 Use theoretical and experimental probability to model and solve problems
 Use addition and multiplication principles
 Calculate and apply permutations and combinations
 Create and use simulations for probability models
 Find expected values and determine fairness
 Identify and use discrete random variables to solve problems
 Apply the binomial theorem
 Model and solve problems involving fair outcomes
 Apportionment
 Election theory
 Voting power
 Fair division
Competency Goal 3: The learner will describe and use recursively-defined relationships
to solve problems.
 Use recursion to model and solve problems
 Find the sum of a finite sequence
 Find the sum of an infinite sequence
 Determine whether a given series converges or diverges
 Write explicit definitions using iterative process, including finite differences and
arithmetic and geometric formulas
 Verify an explicit definition with inductive proof
Course Outline
Chapter 1: The Mathematics of Voting – The Paradoxes of Democracy
 Preference Ballots and Preference Schedules
 The Plurality Method
 The Borda Count Method
 The Plurality with Elimination Method
 The Method of Pairwise Comparisons
 Rankings
Chapter 2: Weighted Voting Systems – The Power Game
 Weighted Voting Systems
 The Banzhaf Power Index
 Applications of the Banzhaf Power Index
 The Shapley-Shubik Power Index
Chapter 3: Fair Division – The Mathematics of Sharing
 Fair-Division Games
 Two Players: The Divider-Chooser Method
 The Lone-Divider Method
 The Lone-Chooser Method
 The Last-Diminisher Method
 The Method of Sealed Bids
 The Method of Markers
Chapter 4: The Mathematics of Apportionment – Making the Rounds
 The Apportionment Problem
 The Mathematics of Apportionment: Basic Concepts
 Hamilton’s Method and the Quota Rule
 The Alabama Paradox
 The Population and New-States Paradox
 Jefferson’s Method
 Adam’s Method
 Webster’s Method
Chapter 5: Euler Circuits – The Circuit Comes to Town
 Routing Problems
 Graphs
 Graph Concepts and Terminology
 Graph Models
 Euler’s Theorems
 Fleury’s Algorithm
 Eulerizing Graphs
Chapter 6: The Traveling-Salesman Problem – Hamilton Joins the Circuit
 Hamilton Circuits and Hamilton Paths
 Complete Graphs
 Traveling-Salesman Problems
 Simple Strategies for Solving TSPs
 The Brute-Force and Nearest-Neighbor Algorithms
 Approximate Algorithms
 The Repetitive Nearest-Neighbor Algorithm
 The Cheapest Link Algorithm
Chapter 7: The Mathematics of Networks – It’s All About Being Connected
 Trees
 Minimum Spanning Trees
 Kruskal’s Algorithm
 The Shortest Distance Between Three Points
 The Shortest Network Linking more than Three Points
Chapter 8: The Mathematics of Scheduling – Directed Graphs and Critical Paths
 The Basic Elements of Scheduling
 Directed Graphs
 The Priority-List Model for Scheduling
 The Decreasing-Time Algorithm
 Critical Paths
 The Critical-Path Algorithm
 Scheduling with Independent Tasks
Chapter 9: Spiral Growth in Nature
 Fibonacci Numbers
 The Equation 𝑥 2 = 𝑥 + 1 and the Golden Ratio
 Gnomons
 Gnomonic Growth
Chapter 10: The Mathematics of Population Growth
 The Dynamics of Population Growth
 The Linear Growth Model
 The Exponential Growth Model
 The Logistics Growth Model
Chapter 11: Symmetry – Mirror, Mirror, Off the Wall…
 Geometric Symmetry
 Rigid Motions
 Reflections
 Rotations
 Translations
 Glide Reflections
 Symmetry Revisited
 Patterns
Chapter 12: Fractal Geometry – Fractally Speaking
 The Koch Snowflake
 The Sierpinski Gasket
 The Chaos Game
 The Twisted Sierpinski Gasket
 Self-Similarity in Art and Literature
 The Mandelbrot Set
Chapter 13: Collecting Statistical Data – Censuses, Surveys, and Clinical Studies
 The Population
 Surveys
 Random Sampling
 Sampling: Terminology and Key Concepts
 Clinical Studies
Chapter 14: Descriptive Statistics
 Graphical Descriptions of Data
 Variables: Quantitative and Qualitative; Continuous and Discrete
 Numerical Summaries of Data
 Measures of Spread
Chapter 15: Chances, Probabilities, and Odds – Measuring Uncertainty
 Random Experiments and Sample Spaces
 Counting: The Multiplication Rule
 Permutations and Combinations
 What is a Probability?
 Probability Spaces
 Probability Spaces with Equally Likely Outcomes
 Odds
Chapter 16: Normal Distributions – Everything is back to Normal (Almost)
 Approximately Normal Distributions of Data
 Normal Curves and Normal Distributions
 Standardizing Normal Data Sets
 The 68-95-99.7 Rule
 Normal Curves as Models of Real-Life Data Sets
 Normal Distributions of Random Events
 Statistical Inference