Course Syllabi

Mathematics Course Syllabi

Since this is a liberal arts mathematics course whose purpose is to expose students to a variety of mathematical topics that are of current interest, not all topics listed below can be covered in a one-semester course. Typically, three of the major topics listed below are the focus of a course. Topics are chosen by the individual instructor. Each learning outcome corresponds with one major topic.

Sets

  • Basic concepts
  • Venn diagrams and subsets
  • Set operations
  • Infinite sets, cardinal numbers, countability and uncountability

Logic

  • Statements
  • Truth tables/Equivalent statements
  • Euler diagrams
  • Analysis of arguments
  • Applications: Computer circuits

Numeration and Mathematical Systems, Number Theory

  • Historical numeration systems
  • Arithmetic in the Hindu-Arabic system
  • Converting between number bases
  • Clock arithmetic and modular systems
  • Other finite mathematical systems
  • Groups
  • Prime and composite numbers
  • Greatest common divisor and least common multiple
  • Perfect numbers
  • Sequences
  • Fibonacci numbers
  • Golden ratio
  • Solved and unsolved problems in number theory

Fundamental Counting Principle

  • Permutations
  • Combinations
  • Fundamental Counting Principle
  • Applications to Probability

Voting and Apportionment

  • Plurality Method
  • Pairwise Comparison Method
  • Borda Method
  • Condercet Criterion
  • Benefits and shortcomings of each method
  • Applications

Graph Theory

  • Walks, paths and circuits
  • Colorability
  • Euler Circuits
  • Hamiltonian Circuits
  • Trees

Personal Financial Management

  • Time value of money
  • Consumer credit
  • Truth in lending
  • Costs and advantages of home ownership
  • Financial investments

Learning Outcomes

  • The student will have demonstrated the knowledge and use of applications of set concepts.
  • The student will have demonstrated the knowledge and use of applications of basic logic.
  • The student will have demonstrated an understanding of mathematical systems and their properties.
  • The student will have demonstrated the comprehension of the Fundamental Counting Principle as applied to the area of probability and its applications.
  • The student will demonstrate knowledge of the major voting and apportionment methods.
  • The student will have demonstrated an understanding of basic graph theory.
  • The student will have demonstrated knowledge of the fundamental principles and real-world applications of personal financial management.

Fundamental Concepts

  • Development of the real number system
  • Polynomial and rational expressions
  • Coordinate system
  • Equations

Functions

  • Notation, evaluation, domain and range
  • Linear and quadratic functions
  • Applications

Systems of Measurement

  • Metric system
  • Apothecaries' system
  • Household system
  • Oral medications
  • Parenteral medications: intramuscular medications
  • Intravenous medications: infusion rates, length of time, piggyback medications

Exponential and Logarithmic Functions

  • Definition of exponents
  • Exponential functions, graphs
  • Logarithms: Properties and graphs
  • Applications

Trigonometric Functions

  • Definitions
  • General angle: Special angles
  • Basic identities
  • Graphs
  • Applications

Learning Outcomes

  • The student will have demonstrated an understanding of mathematical systems and their properties.
  • The student will have demonstrated an understanding of mathematical calculations in the metric system.
  • The student will have demonstrated an understanding of exponential, logarithmic and trigonometric functions.

Prerequisite Skills

  • Time Tables (up to 10x10) without calculator, Categories of Real Numbers
  • Comparators, Signed Numbers, Absolute Value, Fractions
  • Evaluating a Numeric Exponent
  • Joining a Zoom meeting, navigating Blackboard

Module 1: Algebra Essentials

  • Real number properties
  • Order of operations, simplifying/evaluating expressions
  • Exponents and (opt) scientific notation
  • Radicals and rational exponents
  • Complex numbers
    • Operations
    • (opt) Powers of i

Module 2: Polynomial and Rational Expressions

  • Polynomial operations: addition, subtraction, multiplications, division
  • Factoring polynomials
  • Rational expressions and operations

Module 3: Linear Equations

  • Simple: 5 Step Method
  • Solving In Terms Of
  • Applications: Weighted Average, Modeling
  • Rational: Proportion Method
  • Rational: Constant Denominators
  • Rational: Variable Denominators (monomial and binomial)

Module 4: Quadratic Equations

  • Square Root Method
  • Factoring
  • (opt) Completing the Square
  • Quadratic Formula

Module 5: Absolute Value Equations & Inequalities

  • Absolute Value Equations
  • Interval Notation
  • Simple and Compound Linear Inequalities
  • Absolute Value Inequalities

Module 6: Rectangular Coordinate System & Equations of Lines

  • Points and Lines in the Plane
  • (opt) Midpoint Formula
  • (opt) Distance Formula
  • Slope
  • x- and y-intercepts
  • Graphing Linear Equations
  • Finding Linear Equations
  • Parallel and Perpendicular Lines

Module 7: Systems of Equations

  • (opt) Graphical Solutions
  • Substitution
  • Elimination

Module 8: Graphing Quadratic Equations

  • Vertex
  • Direction of Concavity
  • Axis of Symmetry
  • Standard and Vertex Forms
  • Applications: Parabolic Modeling

Module 9: Functions & Graphs

  • Definitions and Properties of Functions
  • Notation and Algebra of Functions
  • Domain and Range
  • Average Rate of Change
  • Piecewise Functions
  • Transformations
    • Key Functions
    • Vertical and Horizontal Shifts
    • Reflections (x-axis)

Textbook: Essentials of Mathematics for Elementary Teachers, By Musser, Burger, Peterson.

Prerequisite: Placement by the Mathematics Department

Problem Solving

  • Polya’s Four Step Process
  • Strategies for Problem Solving
  • Look for a Pattern
  • Make a List
  • Solve a Simpler Problem
  • Draw a Diagram
  • Use a Variable
  • Use Properties of Numbers

Whole Numbers & Operations

  • Whole Numbers & Numeration
  • Operations with Whole Numbers

Number Theory

  • Primes & Composite Numbers
  • Tests for Divisibility
  • Greatest Common Factor
  • Least Common Multiple

Rational Numbers

  • Set of Rational Numbers (Fractions)Concept of a Fraction
  • Representations of Fractions
  • Equivalence and Ordering
  • Operations with Fractions
  • Decimals
  • Place Value
  • Operations with Decimals
  • Ratio and Proportion
  • Percent

Integers

  • Representations of Integers (including negative numbers)
  • Operations with Integers

Learning Outcomes

  • The student will have demonstrated a comprehension of problem solving strategies.
  • The student will have demonstrated a basic understanding of the set of integers and mathematical operations using them.
  • The student will have demonstrated a basic understanding of the set of rational numbers and their applications.

Geometry

  • Defining and Exploring Lines and Angles
  • Modeling Phenomena with Angles
  • Exploring Interior and Exterior Angles in a Triangle
  • Problem Solving with Angles
  • Defining and Exploring Circles and Spheres
  • Classifying and Constructing Triangles, Quadrilaterals, and Other Polygons

Measurement

  • Exploring the Concept of Length
  • Comparing Length, Area, Volume, and Dimension
  • Understanding Measurement Error and Precision
  • Making Sense of Unit Conversions

Area of Shapes

  • Defining and Exploring Area
  • Finding and Justifying Areas of Rectangles
  • Exploring Different Strategies for Finding Areas of Triangles
  • Exploring Principles of Area
  • Finding and Justifying Areas of Parallelograms and Other Polygons
  • Problem Solving with Areas
  • Exploring Area and Circumference of Circles
  • Approximating Areas of Irregular Shapes
  • Exploring Relationships Between Perimeter and Area
  • Exploring the Pythagorean Theorem

Solid Shapes

  • Making and Analyzing Polyhedra and other solids
  • Exploring Nets and Surface Area of Solids
  • Exploring and Explaining Volumes of Solid Shapes
  • Problem Solving with Volume

Transformation Geometry

  • Using Dynamic Geometry Software
  • Exploring Reflections, Translations, and Rotations
  • Exploring Symmetry
  • Making Sense of Congruence
  • Exploring Criterion for Triangle Congruence
  • Compass Constructions
  • Making Sense of Similarity
  • Exploring Dilations
  • Exploring Relationships between Areas, Volumes and Similarity

Learning Outcomes

  • The student will develop a deeper knowledge of measurement and geometry and how these concepts can be applied in problem solving.
  • The student will develop the ability to clearly communicate and explain mathematics content be able to explain various procedures and justify formulas, both orally and in writing.
  • The student will develop the ability to engage in mathematics problem solving and analyze alternative ways of solving problems.

Elementary Functions

  • Functions: definition, notation and graphs
  • Elementary functions and transformations
  • Linear functions and straight lines
  • Quadratic functions: algebraic solutions
  • Applications

Additional Elementary Functions

  • Exponential Functions
    • Definitions and graphs
    • Properties
    • Base “e” exponential functions
    • Growth and decay applications
  • Logarithmic Functions
    • Definitions and Graphs
    • Properties
    • Applications

Mathematics of Finance

  • Simple interest
  • Compound interest

Systems of Linear Equations

  • Linear systems in two variables: algebraic and graphical solutions
  • Augmented matrices
  • Gauss-Jordan Elimination
  • Matrices: definitions and basic operations
  • Inverse of a square matrix
  • Matrix equations and linear systems 

Linear Inequalities and Linear Programming

  • Systems of linear inequalities in two variables
  • Linear programming- a geometric approach 

Learning Outcomes

  • The student will have demonstrated understanding of elementary functions.
  • The student will have demonstrated knowledge in solving systems of linear equations and matrix operations.
  • The student will have demonstrated the ability to apply acquired knowledge to solving business application problems.   

Brief Review of Algebra

  • Exponents
  • Factoring
  • Quadratic equation

Limits and Derivatives

  • Function concepts
  • Limits and continuity
  • Definition of the derivative
  • Derivative formulas
  • Business applications

Applications of the Derivative

  • Maxima and minima
  • First and second derivative tests
  • Business applications

Logarithmic and Exponentional Functions

  • Definitions and properties
  • Differentiation
  • Applications

Integration

  • Indefinite integrals
  • Definite integrals
  • Applications

Learning Outcomes

  • The student will demonstrate an understanding of linear, polynomial and rational functions and their applications.
  • The student will understand the concept of limits and continuity.
  • The student will understand the concept of the derivative.

Ratio and Proportional Relationships

  • Define ratios using the multiple-batches perspective, modeling with a ratio table and a double number line.
  • Define ratios using the variable-parts perspective, modeling with a ratio table and strip diagram.
  • Solve and explain proportion problems using a variety of methods and elementary models such as strip diagrams, ratio tables, and double or triple number lines.
  • Identify unit rates, explain their meaning, and apply them to solving problems
  • Justify why methods of solving proportions are valid.

Algebra

  • Write numerical expressions to match picture models/designs used in elementary classrooms.
  • Write expressions for quantities and explain why they are formulated the way they are.
  • Generate and recognize equivalent and non-equivalent expressions
  • Solve equations and explain the reasoning about numbers, operations, and expressions
  • Solve equations algebraically, and describe how the method can be viewed in terms of a pan balance, when appropriate.
  • Interpret solutions to linear equations and inequalities.
  • Provide examples of equations in one variable that have no solution and those that have infinitely many solutions.
  • Explain how to solve word problems with the aid of models: strip diagrams and algebraically.
  • Solve problems involving repeated patterns and explain your reasoning.
  • Solve problems involving arithmetic sequences of numbers that correspond to sequences of figures, and explain equations for arithmetic sequences.
  • Identify relationships between the corresponding terms of two numerical patterns (e.g., finding a rule for a function table).
  • Differentiate between dependent and independent variables.
  • Sketch graphs of functions to match a written description.
  • When given a scenario that gives rise to a linear function, connect details of the situation to the table, graph, and equation.
  • Contrast patterns of change in linear and other types of functions.

Statistics

  • Identify and formulate statistical questions, distinguish from other questions.
  • Solving problems involving measures of center (mean, median, mode) and range.
  • Explain the concept of mean as “making even” or “leveling out” while demonstrating with elementary manipulatives.
  • Recognize which measure of center best describes a set of data
  • Determine how changes in data affect measures of center or range.
  • Describe a set of data (e.g., overall patterns, outliers).
  • Describe decision-making around representing and interpreting data presented in various forms.
  • Interpret and critique various displays of data (e.g., box plots, histograms, scatter plots).
  • Identify, construct, and complete appropriate graphs that represent given data (e.g., circle graphs, bar graphs, line graphs, histograms, scatter plots, double bar graphs, double line graphs, box plots, and line plots/dot plots).

Probability

  • Explain how principles of probability are applied to determine probabilities in simple cases.
  • Make sense of empirical probability and theoretical probability.
  • Apply principles of multiplication to counting outcomes of multistage experiments.

Learning Outcomes

  • Teacher candidate will develop a deep conceptual knowledge of mathematical concepts underlying the algebra, statistics and probability taught in K-8, and be able to apply this knowledge to solve a wide variety of problems.
  • Teacher candidate will develop the ability to problem solve and explore phenomenon related to algebra, statistics and probability through inquiry, as well as analyze the viability of multiple ways of solving problems.
  • Teacher candidate will be able to explain why procedures and algorithms related to algebra, statistics, and probability work with supporting math drawings and manipulatives, communicating both orally and in writing.

This course will emphasize the use of computer software for the analysis of data and the performance of statistical tests.

Introduction to Statistics

  • Nature of statistical data
  • Data collection and sampling 

Descriptive Statistics

  • Frequency distributions, graphical representations of data
  • Mean, median, mode
  • Measures of variation: Variance, standard deviation, Chebyshev’s Theorem (optional), Empirical Rule
  • Measures of position: Percentiles, quartiles, the box plot. 

Probability

  • Basic concepts and rules
  • Permutations, combinations, fundamental counting principle
  • Conditional probability
  • Probability distributions: expected value and standard deviation
  • Binomial distribution
  • Normal distribution
  • Poisson distribution (optional)
  • Geometric distribution (optional) 

Sampling Distributions and Confidence Intervals

  • Methods and Errors in Sampling
  • Central Limit Theorem
  • Confidence intervals for population means
  • Confidence intervals for population proportions
  • Determining the sample size
  • Normal approximation to binomial distribution (optional)
  • Chi Square distribution (optional) 

Hypothesis Testing

  • Type I and II errors
  • Hypothesis testing for the means – large and small samples
  • Hypothesis testing for population proportions
  • Critical value and p-value methods 

Correlation and Regression

  • Equation of regression line
  • Correlation and goodness of fit

Learning Outcomes

  • The student will demonstrate an understanding of different ways to collect, organize and describe data sets.
  • The student will demonstrate an understanding of the basic concepts of probability and be able to apply them to study probability distributions.
  • The student will be able to use sample statistics to make inferences about a population parameter.

Graphing Calculator: TI-82 or higher

Review of Fundamental Concepts

  • Real number system
  • Equations
  • Coordinate systems in two systems; the straight line & circle

Functions

  • Definition and properties of functions
  • Graphs of functions: zeros, intercepts, asymptotes
  • Special functions: polynomial, rational
  • Composite and inverse functions

Trigonometric Functions

  • Radian and degree measure of angles; arc length
  • Trigonometric functions of angles
    • Definition; cofunctions; coterminal
    • General angle
    • Basic identities: reciprocals, quotient & Pythagorean
    • Special angles & unit circle
    • Graphs of sine/cosine curves
    • Inverse trigonometric functions
    • Applications

Analytic Trigonometry

  • Trigonometric identities and equations
  • Sum/Difference formulas
  • Multiple angle formulas
  • Law of sines/Law of cosines
  • Applications

Exponential and Logarithmic Functions

  • Review exponents and their properties and laws 
  • Definition and properties
  • Exponential and logarithmic equations
  • Applications

Learning Outcomes

  • The student will demonstrate an understanding of polynomials and rational functions.
  • The student will demonstrate an ability to solve logarithmic and exponential equations.
  • The student will demonstrate an understanding of trigonometric functions, identities and applications.

Review of Functions

  • Definition and notation; Domain and range
  • Linear functions, Graphing

Limits and Continuity

  • Limits: intuitive, analytical and graphical approach
  • Limits at Infinity
  • Definition of continuity, Intermediate Value Theorem
  • Limits and continuity of the trigonometric functions

Differentiation

  • Tangent line to a curve, instantaneous rate of change
  • Limit definition of the derivative
  • Techniques of differentiation
  • Derivatives of the trigonometric functions
  • Chain rule; Implicit differentiation
  • Differentials and the Tangent Line approximation

Application of the Derivative

  • Related rates
  • Relative extrema
  • Applications of the derivative to curve sketching
  • Applications of maxima and minima
  • Rolle's Theorem and the Mean Value Theorem
  • Rectilinear motion; velocity and acceleration
  • Newton's Method (optional)

Logarithmic and Exponential Functions

  • Integration and differentiation involving logarithmic and exponential functions

Integration

  • Antiderivatives and the indefinite integral
  • Area and the definite integral
  • The Fundamental Theorem of Calculus
  • Change of variables and the Substitution Method
  • The Mean Value Theorem for Integrals

Learning Outcomes

  • The student will understand the concept of limits and continuity.
  • The student will understand the concept of the derivative.
  • The student will understand the concept of the integral.

Review

  • Fundamental Theorem of Calculus, Definite Integral
  • Area Under Curve, Substitution Method

Applications of Integration

  • Area between two curves
  • Volume of a Solid of Revolution - Disk and Shell methods
  • Arc length
  • Area of a surface of revolution (optional)

Techniques of Integration

  • Inverse trigonometric functions and their derivatives
  • Integration by parts
  • Trigonometric integrals
  • Trigonometric substitutions
  • Method of partial fractions
  • Numerical integration (Trapezoidal Estimate, Simpson's Rule) (optional)

Indeterminate Forms and Improper Integrals

  • Indeterminate forms, L'Hopital's Rule
  • Improper integrals

Infinite Series

  • Sequences
  • Convergent and divergent series
  • Integral test and p-series
  • Direct Comparison and Limit Comparison Test
  • Ratio and Root Test
  • Power series - radius and interval of convergence
  • Representing functions as power series
  • Maclaurin and Taylor series

Parametric Equations and Polar Coordinates

  • Parametric equations
  • Tangent lines and arc length
  • Rectangular and polar coordinates
  • Graphs of polar equations
  • Area in polar coordinates

Learning Outcomes

  • The student will demonstrate an understanding of integration techniques.
  • The student will demonstrate an understanding of calculus involving solids of revolution and the arc length of a curve.
  • The student will understand the concept of infinite series.

Solid "Analytic Geometry"

  • Coordinates: the distance formula
  • Vectors in space
  • Equation of a line in space
  • Equation of a plane, Angles
  • Distance from a point to a plane
  • Spheres and cylinders
  • Quadric surfaces
  • Other coordinate systems: cylindrical and spherical

Partial Differentiation

  • Limits and continuity of functions of several variables
  • Partial derivatives
  • The total differential; applications
  • The Chain Rule; applications
  • Second and higher order derivatives
  • Directional derivatives and gradients
  • Maxima and minima

Multiple Integration

  • Definition of the integral
  • Double and triple integrals
  • Interpretation as area and volume
  • Double integral in polar coordinates
  • Volumes by double integrals in cylindrical coordinates
  • Triple integrals in cylindrical and spherical coordinates
  • Change of variables

Vector Analysis

  • Vector fields
  • Divergence and curl of a vector field
  • Line integrals
  • Conservative vector fields and independence of path
  • Green's Theorem
  • Surfaces
  • Surface integrals
  • Divergence Theorem and applications
  • Stokes' Theorem and applications

Learning Outcomes

  • The student will understand the basic geometry of three-dimensional space.
  • The student will understand differentiation of functions of several variables.
  • The student will understand integration of functions of several variables.

Introduction to Differential Equations

  • Modeling via Differential Equations: Population Growth
    • Unlimited Population Growth
    • Logistic Population Models
    • Predator - Prey Systems
  • Analytic, Qualitative, and Numerical Approaches

First Order Differential Equations

  • Analytic Techniques
    • Separation of Variables
    • Linear Differential Equations
    • Change of Variables (optional)
  • Qualitative Techniques: Slope fields
  • Numerical Techniques: Euler's Method
  • Extstence and Uniqueness of Solutions
  • Bifurcations

First Order Systems

  • Modeling via Systems
  • The Geometry of Systems
  • Analytic Methods for Special Systems
  • Euler's Method for Systems

Linear Systems

  • Properties of Linear Systems and the Linearity Principle
  • Straight Line Solutions
  • Phase Planes for Linear Systems with Real Eigenvalues
  • Complex Eigenvalues
  • Special Cases: Repeated and Zero Eigenvalues
  • Second order Linear Equations
  • The Trace-Determinant Plane

Forcing and Resonance

  • Forced Harmonic Oscillators
  • Sinusoidal Forcing
  • Undamped Forcing and Resonance
  • Amplitude and Phase of the Steady State
  • The Tacoma Narrows Bridge (optional)

Laplace Transform

  • Laplace Transforms
  • Discontinuous Functions
  • Second Order Equations
  • Delta Functions and Impulse Forcing
  • Convolutions

Numerical Methods (optional as a substitution for parts of Chapter V and Chapter VI)

  • Numerical Error in Euler's Method
  • Improving Euler's Method
  • The Runge-Kutta Method

Learning Outcomes

  • The student will demonstrate an understanding of finding solutions to linear first order differential equations explicitly.
  • The student will demonstrate an understanding of solving higher order linear equations explicitly.
  • The student will understand power series methods and approximation methods to solve linear equations and matrix methods to solve linear systems.

Systems of Linear Equations

  • Introduction
  • Gaussian Elimination and Gauss-Jordan Elimination

Matrices

  • Operations and properties
  • Inverse of a matrix
  • Elementary matrices

Determinants

  • The determinant of a matrix
  • Evaluation of a determinant using elementary operations
  • Properties

Vector Spaces

  • Vectors in Rn
  • Vector spaces
  • Subspaces of vector spaces
  • Spanning sets and linear independence
  • Basis and dimension
  • Rank of a matrix and systems of linear equations
  • Change of basis

Inner Product Spaces

  • Length and dot product in Rn
  • Inner product spaces
  • Orthonormal bases: Gram-Schmidt Process
  • Mathematical models and Least Square Analysis

Linear Transformations

  • Introduction
  • Kernel and range of a linear transformation
  • Matrices for linear transformations
  • Transition matrices and similarity

Eigenvalues and Eigenvectors

  • Introduction
  • Diagonalization
  • Symmetric matrices and orthogonal diagonalization

Learning Outcomes

  • The student will demonstrate an understanding of elementary row operations and their applications.
  • The student will demonstrate an understanding of determinants, eigenvalues and eigenvectors.
  • The student will understand the basic theory of vector spaces involving linear maps between them, kernel and range, basis and dimension, and the construction of an orthonormal basis.

Topics vary each semester

Numeration Systems

  • Babylonian, Egyptian, Mayan, Chinese Numeration Systems
  • Calculations in different systems
  • Ambiguity & Use of Zero
  • Development of Base 10 systems including Hindu-Arabic Numerals

Zero

  • Earliest uses of zero
  • Controversy & paradoxes around zero
  • Operations with zero; Indeterminate forms
  • Limits approaching zero
  • Infinitesimals; Development of Calculus

Infinity

  • Philosophical conceptions of infinity
  • Controversy & paradoxes around infinity
  • Limits at infinity & infinite limits
  • Indeterminate Forms; Connections to Calculus

Selected Topics (varies by Instructor):

  • Development of Deductive Logic throughout history, (Aristotle, Chrysippus, Wittgenstein, Russell, Whitehead, Post)
  • Development of Non-Euclidean Geometry (Euclid, Saccheri, Bolyai, Lobachevsky, Reimann, Klein, Beltrami)
  • Mathematical Biographies – what does it mean to think like a mathematician?
  • History of mathematics within a particular culture or region

Techniques of Proof

  • Statements
  • Mathematical induction
  • Proof by contradiction

Elementary Set Theory

  • Sets and subsets
  • Combining sets

Relations and Functions

  • Definition and basic properties
  • Injective and surjective functions
  • Composition and invertible functions
  • Equivalence relations

The Integers

  • Axioms and basic properties
  • Greatest common divisor, Euclidean Algorithm
  • Prime numbers, unique factorization
  • Congruences

Infinite Sets

  • Countable sets
  • Uncountable sets, Cantor's Theorem
  • Collections of sets

Learning Outcomes

  • The student will demonstrate a knowledge of set theory and logic.
  • The student will be able to construct mathematical proofs.
  • The student will be able to demonstrate an understanding of functions and proofs involving relations and functions.

Euclidean Geometry

  • Axioms of Euclidean Geometry
  • Properties of triangles
  • Pythagorean Theorem
  • Isometries/congruence/similarity
  • Medians, centroids

Hyperbolic Geometry

  • Models- Poincare Upper Half Plane Model and The Poincare Disk
  • Isometries
  • Parallel and Ultraparallel lines
  • Metric/Lengths
  • Triangles

Spherical Geometry

  • Triangles
  • Metric/Lengths

Learning Outcomes

  • The student will demonstrate an understanding of Euclidean Geometry
  • The student will demonstrate an understanding of Hyperbolic Geometry
  • The student will demonstrate an understanding of Spherical Geometry

Topological Spaces

  • Open, Closed, and Clopen Sets
  • Basis for a Topology
  • Elementary Topologies
  • Limit Points

Transformations

  • Continuous Functions
  • Homeomorphism
  • Preservation of Properties

Connectivity and Compactness

  • Connected Spaces
  • Subspaces of the Real Line
  • Compactness
  • Compact subsets of the Real Line
  • Other forms of compactness

Algebraic Topology

  • Homotopy of Paths
  • The Fundamental Group
  • Covering Maps
  • Retractions

Learning Outcomes

  1. The student will demonstrate an understanding of various topological spaces.
  2. The student will demonstrate an understanding of continuous functions and their applications to homeomorphisms and preservations of topological properties.
  3. The student will demonstrate an understanding of connectivity and compactness, especially as it applies to the real line.

Introduction

  • What is a Graph?
  • The Degree of a Vertex
  • Isomorphic Graphs
  • Subgraphs
  • Degree Sequences
  • Connected Graphs
  • Cut-Vertices and Bridges
  • Special Graphs
  • Digraphs

Trees

  • Properties of Trees
  • Rooted Trees
  • Depth-First Search Algorithm
  • Beadth-First Search Algorithm
  • The Minimum Spanning Tree Problem

Paths and Distances in Graphs

  • Distance in Graphs
  • Distance in Weighted Graphs
  • The Center and Median of a Graph

Eulerian Graphs

  • Characterizing Eulerian Graphs
  • The Chinese Postman Problem

Hamiltonian Graphs

  • Characterizing Hamiltonian Graphs
  • The Traveling Salesman Problem

Planar Graphs

  • Properties of Planar Graphs
  • A Planarity-Testing Algorithm
  • The Crossing Number and Thickness of a Graph

Coloring Graphs

  • Vertex Colorings
  • Edge Colorings
  • The Four Color Problem

Additional Topics depending on time and student interest

  • Networks
  • Matchings and Factorizations
  • Digraphs
  • Extremal Graph Theory
  • Going Deeper into Topics 1 - 7

Learning Outcomes

  • The student will understand and be able to apply the basic definitions of graph theory.
  • The student will understand the concepts of connectivity, distances, and traversability.
  • The student will understand the concepts of planarity and coloring.

Introduction to Probability

  • Counting methods
  • Conditional probability and independent vs. dependent events
  • Bayes’ Theorem

Discrete Probability Distributions

  • Discrete random variables
  • Expected value and variance of discrete random variables
  • Binomial distribution
  • Hypergeometric distribution
  • Negative binomial distribution
  • Poisson distribution

Continuous Probability Distributions

  • Continuous random variables
  • Expected value and variance of continuous random variables
  • Uniform distribution
  • Exponential distribution
  • Normal distribution
  • Central Limit Theorem
  • Normal approximations of discrete distributions

Bivariate Distributions

  • Discrete and continuous bivariate distributions
  • Correlation coefficient
  • Conditional bivariate distributions
  • Bivariate normal distribution

Functions of Random Variables

  • Functions of one random variable
  • Functions and transformations of two random variables
  • Independent random variables
  • Moment generating functions

Learning Outcomes

  1. The student will demonstrate an understanding of probabilities and how to compute them.
  2. The student will demonstrate an understanding of probability distributions.
  3. The student will demonstrate an understanding of expected value and variance

Confidence Intervals

  • Confidence intervals for means
  • Confidence intervals for difference of means
  • Confidence intervals for proportions and difference of proportions
  • Confidence intervals for standard deviations and ratio of standard deviations
  • Distribution-free confidence intervals for percentiles

Hypothesis Testing

  • Hypothesis testing for means
  • Hypothesis testing for difference of means
  • Hypothesis testing for proportions and difference of proportions
  • Hypothesis testing for standard deviations and ratio of standard deviations
  • Wilcoxon Tests
  • Neyman-Pearson Lemma/best critical regions
  • Power of a statistical test
  • Likelihood ratio tests / maximum likelihood

Regression

  • Linear Regression
  • Confidence intervals and prediction intervals
  • Hypothesis testing for regression
  • Multiple linear regression
  • Polynomial regression

Goodness of Fit

  • Chi-Square goodness of fit
  • Testing for independence using contingency tables
  • ANOVA – one way
  • ANOVA – two way

Learning Outcomes

  1. The student will demonstrate an understanding of survey sampling and parameter estimation.
  2. The student will demonstrate an understanding of hypothesis testing.
  3. The student will demonstrate an understanding of descriptive statistics

Solutions of Equations in One Variable

  • Bisection Method
  • Fixed Point Iteration Method
  • Newton-Raphson Method
  • Secant Method
  • Linear and Quadratic Convergence of Newton’s Method

Direct Solutions of Linear Systems

  • Gaussian Elimination Method
  • Partial Pivoting
  • Scaling
  • Matrix Inverse
  • LU – Factorization
  • Matrix Norms
  • Jacobi and Gauss-Seidel Iterative Method (Optional)

Interpolation Theory

  • Lagrange Interpolation
  • Newton’s Divided Differences
  • Hermite Polynomials
  • Cubic Splines

Numerical Integration

  • Simpson’s Rule and Trapezoidal Rule
  • Composite Simpson’s and Trapezoidal Rules
  • Newton Cotes Formulas
  • Error estimates for Simpson’s and Trapezoidal Rules
  • Gaussian Quadrature

Non-linear Systems of Equations

  • Multivariate Newton’s Method
  • Broyden’s Method (Quasi Newton’s Method)

Approximation theory of Functions

  • Method of Least Squares
  • Trigonometric Approximation
  • Fast Fourier Transforms

Learning Outcomes

  • The student will demonstrate an understanding of numerical root-finding techniques.
  • The student will demonstrate an understanding of the theory of Interpolation of functions and Numerical Integration.
  • The student will demonstrate an understanding of the use of computer programming or software packages in Numerical Analysis.

Introduction to Modeling

  • Overview of models (physical, theoretical, logical, mathematical)
  • Mathematical models: deterministic versus stochastic
  • Model building and case studies (graph theoretic and calculus-based tools, optimization)

First-Order Dynamical Systems and Population Dynamics

  • First-order difference equations
  • Equilibria, stability theory, and periodicity
  • Discrete and continuous population models
  • Beverton-Holt/Michaelis-Menten equation
  • Logistic map and chaos

Higher-Order Dynamical Systems

  • Linear systems and solutions
  • Stability of planar systems
  • Competition and cooperation models
  • Lotka-Volterra predator-prey model, host-parasitoid models

Markov Chains, Randomness and Simulation

  • Introduction and examples
  • Properties of Markov chains
  • Classification of Markov chains and long-range behavior
  • Simulation models

Possible Applications to Physical and Social Sciences

  • Finance: compound interest and mortgages
  • Social science: social networks/graph theory, scheduling, decision-making
  • Biology: population dynamics, carbon dating, drug and disease modeling
  • Physics/Astronomy: age of the universe, planetary motion and orbits, Kepler’s laws
  • Computer science: data optimization/mining, simulation

Property & Casualty Reserving

  • Coverages
  • Claim Process
  • Loss development triangle
  • Development technique
  • Expected claims technique
  • BF and Cape Cod methods
  • Estimating unpaid DCC

Property & Casualty Ratemaking

  • Ratios
  • Ratemaking data
  • Credibility
  • Exposure
  • Premium on-level calculation
  • Developing and trending losses
  • Indication – pure premium and LR
  • Traditional risk classification

Life Insurance

  • Traditional life and annuity products
  • Survival models – Makeham’s Law
  • Life tables
  • Valuation of insurance benefits - EPV
  • Life annuities
  • Premium calculation principles

Preliminaries

  • Sets
  • Methods of proof 
  • Mappings and operations 

Groups

  • Subgroups
  • Permutation groups
  • Cyclic groups
  • Direct products

Modular Arithmetic

  • Equivalence and congruence relations 
  • Construction of Z/nZ 

Closer Examination of Groups

  • Elementary properties
  • Cosets and Lagrange's Theorem
  • Isomorphisms
  • The Fundamental Theorem of Finite Abelian Groups
  • Cayley's Theorem
  • Factor groups

Learning Outcomes

  1. The student will demonstrate an understanding of basic group theory.
  2. The student will demonstrate an understanding of Lagrange's Theorem and Noether's Isomorphism Theorems for groups.
  3. The student will demonstrate an understanding of basic facts on rings and fields.

Fundamental Concepts

  • Set Theory
    • Basic notions
    • Relations and functions
  • The Real Number System
    • Ordered fields
    • The rational numbers as an ordered field
    • The real numbers as a complete ordered field
    • Sequences of real numbers and their properties
    • The Cauchy criterion

Functions of a Real Variable

  • Continuity
    • Limits and continuity
    • Continuous functions and sequences
    • Properties of continuous functions
    • Intermediate Value Theorem
    • Theorem of Bolzano-Weierstrass
    • Extreme Vaule Theorem
  • Differentiation
    • Properties of derivatives
    • Rolle's Theorem
    • Mean Value Theorem
  • Integration
    • Review of the Riemann integral

Learning Outcomes

  1. The student will demonstrate a rigorous understanding of the complete ordered field of real numbers.
  2. The student will demonstrate a rigorous understanding of convergence.
  3. The student will demonstrate a rigorous understanding of continuity and differentiability.

The Algebra of Complex Numbers

  • Introduction
  • Complex conjugates and absolute values
  • Geometric representation
  • Roots of complex numbers

Analytic Functions

  • Introduction
  • Elementary functions
  • Complex derivative
  • Properties of power series
  • The Cauchy-Riemann equations
  • Harmonic functions

Complex Integration

  • Introduction
  • Contour integration
  • Cauchy's Theorem
  • Cauchy's integral formula
  • Talor series expansion of an analytic function
  • Maximum modulus principle

Residue Theory

  • Introduction
  • Laurent expansion of an analytic function
  • Classification of isolated singularities
  • Cauchy Residue Theorem
  • Applications of the Residue Theorem to the evaluation of real integrals
  • The Argument Principle and Rouche's Theorem

Learning Outcomes

  1. The student will understand the basic properties of complex numbers and functions.
  2. The student will understand the procedures of mapping by elementary functions.
  3. The student will understand how to integrate complex functions using Cauchy's Residue Theorem.