MCB4U: Advanced Functions and Introductory Calculus, Grade 12, University Preparation
This course builds on students' experience with functions and introduces the basic concepts and skills of calculus. Students will investigate and apply the properties of polynomial, exponential, and logarithmic functions; broaden their understanding of the mathematics associated with rates of change; and develop facility with the concepts and skills of differential calculus as applied to polynomial, rational, exponential, and logarithmic functions. Students will apply these skills to problem solving in a range of applications.
This Advanced Functions and Introductory Calculus course provides students with background skills in functions as a foundation for the study of the elementary principles of differential calculus. Calculus is a powerful problem-solving tool used in disciplines ranging from physics and chemistry to economics and physiology. The intent in this introductory course is to prepare students for university calculus courses and the wide variety of programs (engineering, social sciences, economics, etc.) that uses the skills of calculus. This course prepares students for applying and using calculus in their future studies.
John von Neumann said: "The Calculus was the first achievement of modern mathematics, and it is difficult to overestimate its importance." To master calculus, students need sufficient background skills in function manipulation. In previous courses, students examined quadratic, basic polynomial, and rational functions. Algebraic manipulation and identifying connections between the algebraic and graphical representations of these functions continue to be important skills.
Completion of MCB4U prepares students to enter the diverse range of university programs that require calculus as an admission requirement; the contextual examples and activities should be drawn from a variety of areas. The diverse nature of the courses that students take in their post secondary education necessitates a balance of emphasis between building solid foundations and providing sufficient opportunities to engage in investigations and problems that involve a rigorous level of mathematics. This balance requires students to consistently demonstrate the ability to:
The recommended sequence begins with Unit 1: Investigating the Graphs of Polynomial Functions and Unit 2: Underlying Concepts of Calculus, which extend from the Grade 11 courses and introduce elementary calculus. A more advanced treatment of polynomials and the derivative are considered in Units 3 and 4. Units 2 and 4 create a solid foundation for the problem solving that takes place in the last three units. Before students begin applying their derivative skills, however, they are introduced to logarithmic and exponential functions (Unit 5); this knowledge is needed for the next two units on curve sketching and rates of change. In Unit 6, the graphical understanding of the derivative is considered with an algebraic focus. In Unit 7, a variety of problems are investigated to analyze models of many types of functions using calculus techniques. The final summative unit brings together the concepts and skills across all three strands.
- practice and consolidate skills;
- deal with applications that address a broad scope of scenarios;
- reflect on and summarize new learning;
- investigate and construct mathematical concepts independently;
- conjecture and, through inquiry, test a hypothesis;
- generate multiple types of solutions to complex problems which may cross strands, require the use of appropriate technology, and require abstract thinking;
- expand the depth of their inquiry in order to solve higher-order problems.
MGA4U: Geometry and Discrete Mathematics, Grade 12, University Preparation
This course enables students to broaden mathematical knowledge and skills related to abstract mathematical topics and to the solving of complex problems. Students will solve problems involving geometric and Cartesian vectors, and intersections of lines and planes in three-space. They also develop an understanding of proof, using deductive, algebraic, vector, and indirect methods. Students will solve problems involving counting techniques and prove results using mathematical induction.
This course is designed for students planning to study university programs that are highly focused on mathematics, including engineering, pure mathematics, computer science, and the physical sciences. Contextual examples and activities should thus be drawn largely from these fields. Because of the academic demands of these programs, the expectations of this course require students to consistently demonstrate the ability to:
This course is comprised of three strands: Proof and Problem Solving, Geometry, and Discrete Mathematics. These strands have been divided into a total of six units. Unit 1 - Deductive Geometry, encompasses the Proof and Problem Solving strand, in which students apply deductive, algebraic, and vector methods to demonstrate and prove properties of plane figures. The rigors of this curriculum require students to expand the depth of their understanding of the comprehensiveness of problem solving techniques. Independent work skills are stressed in this unit. The Geometry strand is divided into three units. In Unit 2 - Vectors and Unit 3 - Vector Applications, students investigate, manipulate, and apply geometric vectors. In Unit 4 - Intersections of Lines and Planes, students determine equations of lines and planes, solve systems of equations using matrices, and determine intersections of lines and planes. In the Discrete Mathematics strand, students solve problems involving counting techniques. Students use mathematical induction to prove the binomial theorem and the formulas for the sums of series. The Discrete Mathematics strand is also divided into two units. Unit 5 - Mathematical Induction and Combinatorics introduces sequences and series, mathematical induction, and counting techniques. Students further explore these concepts and apply their learning in Unit 6 - Application of Counting Techniques. A final summative unit contains activities designed
- research, investigate, and construct mathematical concepts independently;
- conjecture and, through inquiry, test a hypothesis in a variety of ways including using technology;
- generate multiple types to solutions to complex problems which may cross strands and require abstract thinking;
- analyze and design proofs from multiple perspectives.
MDM4U:Mathematics of Data Management, Grade 12, University Preparation
This course broadens students' understanding of mathematics as it relates to managing information and focuses on a culminating project throughout the course. Students will apply methods for organizing and analyzing large amounts of information; apply counting techniques, probability, and statistics in modelling and solving problems; and carry out a culminating project that integrates the expectations of the course and encourages perseverance and independence. Successful completion of MDM4U prepares students for any undergraduate course in probability and statistics. Such courses are typically a requirement for students in their second year of most four-year undergraduate programs in both the sciences and humanities. In particular, students planning to pursue university programs in business, social sciences, or the humanities will find this course of relevance.