The concepts included are limits, derivatives, antiderivatives and definite integrals. These concepts will be applied to solve problems of rates of change, maximum and minimum, curve sketching and areas. The classes of functions used to develop these concepts and applications are: polynomial, rational, trigonometric, exponential, and logarithmic.
MATH 103Introductory Algebra for Arts and Social Science0.50 units
An introduction to applications of algebra to business, the behavioural sciences, and the social sciences. Topics will be chosen from linear equations, systems of linear equations, linear inequalities, functions, set theory, permutations and combinations, binomial theorem, probability theory.
Prerequisites
Enrolled in a program offered by the Faculties of Arts, Environment, or Health
Antirequisites
Complete all of the following
Not completed nor concurrently enrolled in any of the following:
MATH 104Introductory Calculus for Arts and Social Science0.50 units
An introduction to applications of calculus in business, the behavioural sciences, and the social sciences. The models studied will involve polynomial, rational, exponential, and logarithmic functions. The major concepts introduced to solve problems are rate of change, optimization, growth and decay, and integration.
Prerequisites
Enrolled in a program offered by the Faculties of Arts, Environment, Health, or Science
Antirequisites
Not completed nor concurrently enrolled in any of the following:
Vectors in 2- and 3-space and their geometry. Linear equations, matrices, and determinants. Introduction to vector spaces. Eigenvalues and diagonalization. Applications. Complex numbers.
Prerequisites
Complete all of the following
Must have completed the following: 4U Calculus and Vectors
Functions: review of polynomials, exponential, logarithmic, trigonometric. Operations on functions, curve sketching. Trigonometric identities, inverse functions. Derivatives, rules of differentiation. Mean Value Theorem, Newton's Method. Indeterminate forms and L'Hopital's rule, applications. Integrals, approximations, Riemann definite integral, Fundamental Theorems. Applications of the integral.
Prerequisites
Complete all of the following
Must have completed the following: 4U Calculus and Vectors
Functions of engineering importance; review of polynomial, exponential, and logarithmic functions; trigonometric functions and identities. Inverse functions (logarithmic and trigonometric). Limits and continuity. Derivatives, rules of differentiation; derivatives of elementary functions. Applications of the derivative, max-min problems, Mean Value Theorem. Antiderivatives, the Riemann definite integral, Fundamental Theorems. Methods of integration, approximation, applications, improper integrals.
Prerequisites
Complete all of the following
Must have completed the following: 4U Calculus and Vectors
Methods of integration: by parts, trigonometric substitutions, partial fractions; engineering applications, approximation of integrals, improper integrals. Linear and separable first order differential equations, applications. Parametric curves and polar coordinates, arc length and area. Infinite sequences and series, convergence tests, power series and applications. Taylor polynomials and series, Taylor's Remainder Theorem, applications.
Elementary approximation methods: interpolation; Taylor polynomials and remainder; Newton's method, Landau order symbol, applications. Infinite series: Taylor series and Taylor's Remainder Theorem, geometric series, convergence test, power series, applications. Functions of several variables: partial derivatives, linear approximation and differential, gradient and directional derivative, optimization and Lagrange multipliers. Vector-valued functions: parametric representation of curves, tangent and normal vectors, line integrals and applications.
MATH 124Calculus and Vector Algebra for Kinesiology0.50 units
Review of trigonometry and basic algebra. Introduction to vectors in 2- and 3-space: sums, addition, dot products, cross products and angles between vectors. Solving linear systems in two and three variables. Functions of a real variable: powers, rational functions, trigonometric, exponential and logarithmic functions, their properties. Intuitive discussion of limits and continuity. Derivatives of elementary functions, derivative rules; applications to curve sketching, optimization. Relationships between distance, velocity, and acceleration. The definite integral, antiderivatives, the Fundamental Theorem of Calculus; change of variable and integration by parts; applications to area, centre of mass.
Prerequisites
Complete all of the following
Must have completed the following: 4U Advanced Functions
Functions of a real variable: powers, rational functions, trigonometric, exponential and logarithmic functions, their properties and inverses. Intuitive discussion of limits and continuity. Definition and interpretation of the derivative, derivatives of elementary functions, derivative rules and applications. Riemann sums and other approximations to the definite integral. Fundamental theorems and antiderivatives; change of variable. Applications to area, rates, average value.
Prerequisites
Complete 1 of the following
Must have completed the following:
MATH104 - Introductory Calculus for Arts and Social Science (0.50)
Must have completed the following: 4U Calculus and Vectors
Antirequisites
Complete all of the following
Not completed nor concurrently enrolled in any of the following:
Transforming and evaluating integrals; application to volumes and arc length; improper integrals. Separable and linear first order differential equations and applications. Introduction to sequences. Convergence of series; Taylor polynomials, Taylor's Remainder theorem, Taylor series and applications. Parametric/vector representation of curves; particle motion and arc length. Polar coordinates in the plane.
An introduction to the language of mathematics and proof techniques through a study of the basic algebraic systems of mathematics: the integers, the integers modulo n, the rational numbers, the real numbers, the complex numbers and polynomials.
Prerequisites
Complete all of the following
Must have completed at least 1 of the following: 4U Calculus and Vectors, 4U Mathematics of Data Management
MATH 136Linear Algebra 1 for Honours Mathematics0.50 units
Systems of linear equations, matrix algebra, elementary matrices, computational issues. Real n-space, vector spaces and subspaces, basis and dimension, rank of a matrix, linear transformations, and matrix representations. Determinants, eigenvalues and diagonalization, applications.
MATH 137Calculus 1 for Honours Mathematics0.50 units
Absolute values and inequalities. Sequences and their limits. Introduction to series. Limits of functions and continuity. The Intermediate Value theorem and approximate solutions to equations. Derivatives, linear approximation, and Newton's method. The Mean Value theorem and error bounds. Applications of the Mean Value theorem, Taylor polynomials and Taylor's theorem, Big-O notation. Suitable topics are illustrated using computer software.
Prerequisites
Must have completed the following: 4U Calculus and Vectors
Antirequisites
Not completed nor concurrently enrolled in any of the following:
MATH 138Calculus 2 for Honours Mathematics0.50 units
Introduction to the Riemann integral and approximations. Antiderivatives and the fundamental theorem of calculus. Change of variables, methods of integration. Applications of the integral. Improper integrals. Linear and separable differential equations and applications. Tests for convergence for series. Binomial series, functions defined as power series and Taylor series. Vector (parametric) curves in R2. Suitable topics are illustrated using computer software.
MATH 199Mathematical Discovery and Invention0.50 units
A course in problem solving in which intriguing and difficult problems are solved. Problems are taken mainly from the elementary parts of applied mathematics, computer science, statistics and actuarial science, pure mathematics, and combinatorics and optimization. Material relevant to the problems is taught in depth.
MATH 207Calculus 3 (Non-Specialist Level)0.50 units
Multivariable functions and partial derivatives. Gradients. Optimization including Lagrange multipliers. Polar coordinates. Multiple integrals. Surface integrals on spheres and cylinders. Introduction to Fourier Series.
MATH 213Signals, Systems, and Differential Equations0.50 units
Laplace transform methods for: solving linear ordinary differential equations, classical signals, and systems. Transfer functions, poles, and zeros; system stability. Frequency response of linear systems and its log-scale representation (Bode plot). Fourier series. Applications in areas of interest for software engineers and computer scientists. Brief introduction to Fourier transforms in the context of signals and systems.
Systems of linear equations; their representation with matrices and vectors; their generalization to linear transformations on abstract vector spaces; and the description of these linear transformations through quantitative characteristics such as the determinant, the characteristic polynomial, eigenvalues and eigenvectors, the rank, and singular values.
MATH 217Calculus 3 for Chemical Engineering0.50 units
Curves and surfaces in R3. Multivariable functions, partial derivatives, the chain rule, gradients. Optimization, Lagrange Multipliers. Double and triple integrals, change of variable. Vector fields, divergence and curl. Vector integral calculus: Green's theorem, the Divergence theorem and Stokes' theorem. Applications in engineering are emphasized.
MATH 218Differential Equations for Engineers0.50 units
First order equations, second order linear equations with constant coefficients, series solutions, the Laplace transform method, systems of linear differential equations. Applications in engineering are emphasized.
Directional derivative and the chain rule for multivariable functions. Optimization, Lagrange multipliers. Double and triple integrals on simple domains; transformations and Jacobians; change of variable in multiple integrals. Vector fields, divergence and curl. Vector integral calculus: Line and surface integrals, Green's Theorem, Stokes' Theorem, Gauss' Theorem, conservative vector fields.
MATH 228Differential Equations for Physics and Chemistry0.50 units
First-order equations, second-order linear equations with constant coefficients, series solutions and special functions, the Laplace transform method. Applications in physics and chemistry are emphasized.
MATH 235Linear Algebra 2 for Honours Mathematics0.50 units
Orthogonal and unitary matrices and transformations. Orthogonal projections, Gram-Schmidt procedure, best approximations, least-squares. Inner products, angles and orthogonality, orthogonal diagonalization, singular value decomposition, applications.
MATH 237Calculus 3 for Honours Mathematics0.50 units
Calculus of functions of several variables. Limits, continuity, differentiability, the chain rule. The gradient vector and the directional derivative. Taylor's formula. Optimization problems. Mappings and the Jacobian. Multiple integrals in various co-ordinate systems.
Topology of real n-dimensional space: completeness, closed and open sets, connectivity, compact sets, continuity, uniform continuity. Differential calculus on multivariable functions: partial differentiability, differentiability, chain rule, Taylor polynomials, extreme value problems. Riemann integration: Jordan content, integrability criteria, Fubini's theorem, change of variables. Local properties of continuously differentiable functions: open mapping theorem, inverse function theorem, implicit function theorem.
Prerequisites
Complete all of the following
Must have completed the following:
MATH146 - Linear Algebra 1 (Advanced Level) (0.50)
MATH 600Introduction to Mathematical Software for Teachers0.25 units
This course exposes students to the technical tools that professional mathematicians use. The software presented in the course will enhance each student's communication, presentation, visualization, and problem-solving skills. The course will also take a brief look at the history of mathematical communication and its impact on the development of the subject.
This course will explore the basic properties of probability focusing on both discrete and continuous random variables. Topics include: Laws of probability, discrete and continuous random variables, probability distributions, mean, variance, generating functions, Markov chains, problem solving, history of probability.
Corequisites
Completed or concurrently enrolled in:
MATH600 - Introduction to Mathematical Software for Teachers (0.25)
MATH692 - Reading, Writing and Discovering Proofs (0.25)
This course discusses some of the mathematical and scientific aspects of empirical, or data-based, problem solving. Topics will include methods for the design of experiments and surveys, and the analysis of data using statistical models. Examples will illustrate the application of these methods to data-based problems in science, health, business and industry.
This course explores the foundations of linear algebra and some of its applications. An emphasis will be placed on the development of mathematical thinking and the importance of proof in mathematical teaching. Topics include: matrices, linear mappings, vector spaces, determinants, diagonalization, inner products, the Fundamental Theorem of Linear Algebra, and the method of least squares.
Corequisites
Completed or concurrently enrolled in:
MATH600 - Introduction to Mathematical Software for Teachers (0.25)
MATH692 - Reading, Writing and Discovering Proofs (0.25)
This course explores the many fascinating properties of the natural numbers. Topics include: the Euclidean algorithm, congruences and modular arithmetic, primitive roots and quadratic residues, sums of squares, multiplicative functions, continued fractions and Diophantine equations.
Corequisites
Completed or concurrently enrolled in:
MATH600 - Introduction to Mathematical Software for Teachers (0.25)
MATH692 - Reading, Writing and Discovering Proofs (0.25)
This course explores the intersection between mathematics and computer science by examining general methods for solving problems efficiently. We will study a variety of algorithm design techniques for problems in diverse application areas and use mathematical models to compare algorithms. Students will be introduced to "big ideas" from computer science, such as the generalization and reuse of previous solutions, and the notion of limits to the power of computation.
Corequisites
Completed or concurrently enrolled in:
MATH600 - Introduction to Mathematical Software for Teachers (0.25)
MATH692 - Reading, Writing and Discovering Proofs (0.25)
MATH 642Introduction to Computer Science: A Mathematical Perspective0.50 units
The connections between mathematics and computer science are varied and deep. This course will introduce students to foundational ideas in computer science and their relationship to foundational ideas in mathematics through the use of a functional programming language designed for education. No prior experience with programming is required, though students with exposure to popular programming languages will also benefit from this alternate approach.
Corequisites
Completed or concurrently enrolled in:
MATH600 - Introduction to Mathematical Software for Teachers (0.25)
MATH692 - Reading, Writing and Discovering Proofs (0.25)
This course will use mathematics to study the foundations of problem solving and computation. A decision problem (answering "yes" or "no") can be characterized as a set of strings of characters encoding inputs that yield "yes" answers, and a computer can be characterized as a simple mathematical machine model that processes strings of characters. By studying properties of classes of problems of increasing complexity, we will be able to establish relationships among classes, membership in classes, and hard problems for classes. The course will culminate in the examination of classes and models that correspond to the power of computers, and the demonstration that there exist problems that cannot be solved.
Corequisites
Completed or concurrently enrolled in:
MATH600 - Introduction to Mathematical Software for Teachers (0.25)
MATH692 - Reading, Writing and Discovering Proofs (0.25)
This course will explore the foundations of differential calculus, the role of rigour in mathematics, and the use of sophisticated mathematical software. Topics include: A brief primer on logic and proof, axiom of choice and other ideas from set theory, convergence of sequences and the various forms of the completeness axiom for R, detailed study of limits, continuity and the Intermediate Value Theorem, fundamentals of differentiation and the importance of linear approximation, role of the Mean Value Theorem, the nature and existence of extrema, Taylor's Theorem and polynomial approximation, MAPLE as a tool for discovery.
Corequisites
Completed or concurrently enrolled in:
MATH600 - Introduction to Mathematical Software for Teachers (0.25)
MATH692 - Reading, Writing and Discovering Proofs (0.25)
This course explores the foundations of integral calculus and the use of series in approximating the basic functions of mathematics. Topics include: Understanding the Riemann Integral and its flaws, the idea of Lebesque, the geometric meaning of the Riemann-Stieltjes integral, the Fundamental Theorem of Calculus, numerical integration, numerical series, uniform convergence of functions and the extraordinary nature of power series, Fourier Series.
MATH 650Mathematical Modeling with Differential Equations0.50 units
Solving and interpreting differential equations motivated by a variety of systems from the physical and social sciences. Analytical solutions of standard linear and non-linear equations of first and second order; phase portrait analysis; linearization of non-linear systems in the plane. Numerical and graphical solutions using mathematical software.
Corequisites
Completed or concurrently enrolled in:
MATH600 - Introduction to Mathematical Software for Teachers (0.25)
MATH692 - Reading, Writing and Discovering Proofs (0.25)
MATH 661Problem Solving and Proof in Geometry0.50 units
This course explores the Euclidean geometry of triangles and circles from elementary to advanced settings. The course also briefly discusses three-dimensional geometry. An emphasis is placed on proof, on problem solving, on problem creation in a geometric context, and on aspects of teaching geometry. Where possible, problems that combine multiple areas of mathematics are used.
Corequisites
Completed or concurrently enrolled in:
MATH600 - Introduction to Mathematical Software for Teachers (0.25)
MATH692 - Reading, Writing and Discovering Proofs (0.25)
MATH 674Special Topics in Mathematical Connections0.25 units
This course is intended to give the student insight to an important area of mathematics and how it connects with problems in the real world. Each topic consists of one six-week module. The emphasis will be on how the mathematics is used in a real world context.
Corequisites
Completed or concurrently enrolled in:
MATH600 - Introduction to Mathematical Software for Teachers (0.25)
MATH692 - Reading, Writing and Discovering Proofs (0.25)
We explore the who, where, when and why of some of the most important ideas in mathematics. Topics include: William T. Tutte and Decryption, Euclid and the Delian Problem, Archimedes and his estimate of Pi, Al Khwarizmi and Islamic mathematics, Durer and the Renaissance, Descartes and Analytic Geometry, and Kepler and Planetary Motion.
Corequisites
Completed or concurrently enrolled in:
MATH600 - Introduction to Mathematical Software for Teachers (0.25)
MATH692 - Reading, Writing and Discovering Proofs (0.25)
MATH 681Problem Solving and Mathematical Discovery0.50 units
This course aims to develop the student's mathematical problem solving ability. Common heuristics such as patterning, trying a simpler problem, considering cases, and thinking inductively will be examined. A wide range of challenging problems from various branches of mathematics will provide the medium through which these important principles and broad strategies are experienced.
Corequisites
Completed or concurrently enrolled in:
MATH600 - Introduction to Mathematical Software for Teachers (0.25)
MATH692 - Reading, Writing and Discovering Proofs (0.25)
MATH 682History of Mathematics I: Antiquity0.50 units
This course examines the history of mathematics through the prism of a particular person working on a particular problem in a particular geographical and temporal place. Topics may include: Egyptian and Babylonian mathematics, Chinese mathematics, mesoamerican mathematics, Euclid and the Delian Problem; Archimedes and an estimate of Pi; Apollonius and conic sections; Al Khwarizmi and Islamic mathematics; and the transmission of works of antiquity to Europe.
Corequisites
Completed or concurrently enrolled in:
MATH600 - Introduction to Mathematical Software for Teachers (0.25)
MATH692 - Reading, Writing and Discovering Proofs (0.25)
MATH 683History of Mathematics II: Early Modern Period0.50 units
This course examines the history of mathematics through the prism of a particular person working on a particular problem in a particular geographical and temporal place. Topics may include: Durer and projective geometry; Cardano and the solution of quartics; Fermat and probability; Descartes and analytic geometry; and Kepler and planetary motion.
Corequisites
Completed or concurrently enrolled in:
MATH600 - Introduction to Mathematical Software for Teachers (0.25)
MATH692 - Reading, Writing and Discovering Proofs (0.25)
Following a brief introduction to Peace Studies, the social implications of math and computer science are explored through a lens of peace theory. We examine how math and computer science can influence poverty, democracy, policing, environment, health and more. Participants in this course will articulate how to help their own students see math and computer science as a tool for good or evil. Only offered online.
Corequisites
Completed or concurrently enrolled in:
MATH600 - Introduction to Mathematical Software for Teachers (0.25)
MATH692 - Reading, Writing and Discovering Proofs (0.25)
MATH 690Workshop for Teachers of Mathematics0.25 units
This course provides the opportunity to connect and collaborate with other students, to synthesize key takeaways from other courses, and to bridge the gap between educational theory and practice. Course activities may include problem-solving, discussion, analysis of mathematics education research, presentations, and peer review. The course includes a limited number of synchronous meetings, and may include optional in-person participation in a CEMC conference for mathematics teachers.
Corequisites
Completed or concurrently enrolled in:
MATH600 - Introduction to Mathematical Software for Teachers (0.25)
MATH692 - Reading, Writing and Discovering Proofs (0.25)
MATH 692Reading, Writing and Discovering Proofs0.25 units
Objectives: To develop the vocabulary, techniques and analytical skills associated with reading and writing proofs, and to gain practice in formulating conjectures and discovering proofs. Emphasis will be placed on understanding logical structures, recognition and command over common proof techniques, and precision in language. Topics Include: rules of formal logic, truth tables, role of definitions, implications, sets, existential and universal quantifiers, negation and counter-example, proofs by contradiction, proofs using the contrapositive, proofs of uniqueness and induction.
MATH 699Master of Mathematics for Teachers Capstone0.50 units
The capstone course is designed to give students an opportunity to showcase the knowledge they have gained and to provide a forum for bringing that knowledge into their own classroom. As part of this course students will design a mini-course on an approved subject in mathematics.
MATH 900University Mathematics Teaching Techniques0.25 units
The purpose of this course is to provide an introduction to teaching mathematics at the postsecondary level. Students will learn about effective course design, lecture delivery, assessment design, active learning, and how to handle common situations in the classroom. Multiple opportunities to gain practical teaching experience will be provided.
Prerequisites
Enrolled in a program offered by Faculty of Mathematics