# MCR3U（11年级函数）

Course name: Functions

Course code: MCR3U

Course Type: Pre-University

Credit value: 1.0

Pilot course: MPM2D

Course Description

This course introduces the mathematical concept of the function by extending students' experiences with linear and quadratic relations. Students will investigate properties of discrete and continuous functions, including trigonometric and exponential functions; represent functions numerically, algebraically, and graphically; solve problems involving applications of functions; investigate inverse functions; and develop facility in determining equivalent algebraic expressions. Students will reason mathematically and communicate their thinking as they solve multi-step problems.

 Unit Titles and Descriptions Exploring FunctionsStudents will explore the concepts of relations and functions in this unit, their representations, their inverses, and how to make connections between the algebraic and graphical representations of functions using transformations. Students will learn how to apply and determine the different transformations that can be applied to functions using transformational parameters, along with learning to apply function and set notations, and graphing methods consistently. Characteristics of FunctionsStudents will explore properties of functions, in particular by determining the zeros and the maximum or minimum of a quadratic function, and solve problems involving quadratic functions, including problems arising from real-world applications. By the end of the unit, students will be able to demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical, and rational expressions. Exponential FunctionsThis unit will explore several topics including evaluating powers with rational exponents, simplifying expressions containing exponents, and describing properties of exponential functions represented in a variety of ways. The emphasis will be on modelling and problem solving using these concepts. Discrete FunctionsThe unit begins with an exploration of recursive sequences and how to represent them in a variety of ways. Making connections to Pascal's triangle, demonstrating understanding of the relationships involved in arithmetic and geometric sequences and series, and solving related problems involving compound interest and ordinary annuities will form the rest of the unit. TrigonometryThis unit concentrates students' attention on determining the values of the trigonometric ratios for angles less than 360° proving simple trigonometric identities and solving problems using the primary trigonometric ratios. The sine law and the cosine law are developed. Students will learn to demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numeric, graphical, and algebraic representations of sinusoidal functions while solving problems involving sinusoidal functions, including problems arising from real-world applications Trigonometric Functions and graphsStudents will investigate the relationship between the graphs and the equations of sinusoidal functions sketching and describing the graphs and describing their periodic properties. Final Assessment ExamThis course includes a two and a half hour final exam and is worth 30% of your final grade.

Overall Curriculum Expectations

 A. Characteristics of Functions A1 demonstrate an understanding of functions, their representations, and their inverses, and make connections between the algebraic and graphical representations of functions using transformations; A2 determine the zeros and the maximum or minimum of a quadratic function, and solve problems involving quadratic functions, including problems arising from real-world applications; A3 demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical, and rational expressions. B. Exponential Functions B1 evaluate powers with rational exponents, simplify expressions containing exponents, and describe properties of exponential functions represented in a variety of ways; B2 make connections between the numeric, graphical, and algebraic representations of exponential functions; B3 identify and represent exponential functions, and solve problems involving exponential functions, including problems arising from real-world applications. C. Discrete Functions C1 demonstrate an understanding of recursive sequences, represent recursive sequences in a variety of ways, and make connections to Pascal's triangle; C2 demonstrate an understanding of the relationships involved in arithmetic and geometric sequences and series, and solve related problems; C3 make connections between sequences, series, and financial applications, and solve problems involving compound interest and ordinary annuities. D. Trigonometric Functions D1 determine the values of the trigonometric ratios for angles less than 360º; prove simple trigonometric identities; and solve problems using the primary trigonometric ratios, the sine law, and the cosine law; D2 demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numeric, graphical, and algebraic representations of sinusoidal functions; D3 identify and represent sinusoidal functions, and solve problems involving sinusoidal functions, including problems arising from real-world applications.

Teaching and Learning Strategies:

The over-riding aim of this course is to help students use the language of mathematics skillfully, confidently and flexibly, a wide variety of instructional strategies are used to provide learning opportunities to accommodate a variety of learning styles, interests, and ability levels.The following mathematical processes are used throughout the course as strategies for teaching and learning the concepts presented.

Problem Solving: This course scaffolds learning by providing students with opportunities to review and activate prior knowledge (e.g. reviewing concepts related to trigonometry from prior mathematics courses), and build off of this knowledge to acquire new skills. The course guides students toward recognizing opportunities to apply knowledge they have gained to solve problems.

Selecting Tools and Computational Strategies: This course models the use of spreadsheet software and a TVM solver for personal finance to familiarize students with available software and resources which will allow them to simplify calculations in order to better and more accurately manage money.

Connecting: This course connects the concepts taught to real-world applications, such as simple harmonic motion and sound or light waves.

Representing: Through the use of examples, practice problems, and solution videos, the course models various ways to demonstrate understanding, poses questions that require students to use different representations as they are working at each level of conceptual development - concrete, visual or symbolic, and allows individual students the time they need to solidify their understanding at each conceptual stage.

Self-Assessment: Through the use of interactive activities (e.g. multiple choice quizzes, and drag-and-drop activities) students receive instantaneous feedback and are able to self-assess their understanding of concepts.