Curriculum Overview · Ontario Ministry of Education

📋 MCR3U Curriculum Overview

Functions, Grade 11 University Preparation

Aligned with The Ontario Curriculum, Grades 11 and 12: Mathematics (2007 Revised) · Growing Success (2010)

1. Course Identification

Course Code
MCR3U
Course Title
Functions
Grade / Type
11 / University (U)
Credit Value
1.0
Scheduled Hours
110
Prerequisite
MPM2D Principles of Mathematics, Grade 10 Academic
Curriculum Document
2007 Revised
Policy Framework
Growing Success (2010)

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. MCR3U is the prerequisite for both MHF4U (Advanced Functions) and MCV4U (Calculus and Vectors), and for many university-bound mathematics, science, business, and engineering programs.

2. Strands & Expectations

Strand A — Characteristics of Functions

Overall Expectations: demonstrate an understanding of functions, their representations, and their inverses, and make connections between the algebraic and graphical representations of functions using transformations; 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; demonstrate an understanding of equivalence as it relates to simplifying polynomial, radical, and rational expressions.

Key Specific Expectations

  • A1.1 Explain the meaning of the term function; distinguish a function from a relation that is not a function using ordered pairs, mappings, tables of values, graphs, and the vertical-line test.
  • A1.2 Represent linear and quadratic functions using function notation; substitute into and evaluate functions; explain the meaning of \( f(a) \).
  • A1.3 Determine the domain and range of linear, quadratic, and absolute-value functions; use set-builder and interval notation.
  • A1.4 Determine, through investigation, the relationship between a function and its inverse, including the role of the line \( y = x \), and the conditions under which the inverse is also a function.
  • A1.5 Determine the inverse of a linear or quadratic function from its equation.
  • A1.6 Determine, through investigation, the roles of the parameters \( a, k, d, c \) in functions of the form \( y = a f(k(x - d)) + c \) for parent functions \( f(x) = x^2, x, |x|, \tfrac{1}{x}, \sqrt{x} \).
  • A2.1 Determine the maximum or minimum value of a quadratic function by completing the square or by using the formula \( x = -\tfrac{b}{2a} \).
  • A2.2 Solve problems involving quadratic functions arising from real-world applications, including projectile motion, area, and revenue.
  • A2.3 Determine, through investigation, the connections between the real roots of a quadratic equation and the x-intercepts of the corresponding quadratic function; interpret the discriminant.
  • A3.1 Simplify polynomial expressions by expanding, factoring, and combining like terms.
  • A3.2 Verify, through investigation, that equivalent algebraic expressions evaluate to the same value for any input.

Strand B — Exponential Functions

Overall Expectations: evaluate powers with rational exponents, simplify expressions containing exponents, and describe properties of exponential functions represented in a variety of ways; make connections between the numeric, graphical, and algebraic representations of exponential functions; identify and represent exponential functions, and solve problems involving exponential functions, including problems arising from real-world applications.

Key Specific Expectations

  • B1.1 Graph, with and without technology, an exponential relation given its equation in the form \( y = a^x \) (\( a > 0, a \neq 1 \)); identify key properties (domain, range, intercepts, asymptote, growth/decay).
  • B1.2 Evaluate, with and without technology, numerical expressions containing integer and rational exponents and rational bases.
  • B1.3 Simplify algebraic expressions containing integer and rational exponents using the laws of exponents.
  • B2.1 Distinguish exponential functions from linear and quadratic functions by making comparisons in tables of values and on graphs; recognise the constant ratio between successive y-values.
  • B2.3 Determine, through investigation, the roles of \( a, k, d, c \) in functions of the form \( y = a\,b^{k(x-d)} + c \).
  • B3.1 Collect data that can be modelled by an exponential function; investigate the time-doubling and time-halving forms.
  • B3.2 Identify exponential functions in real-world applications (population growth, radioactive decay, compound interest); pose and solve related problems.
  • B3.3 Solve simple exponential equations using a same-base strategy.

Strand C — Discrete Functions

Overall Expectations: demonstrate an understanding of recursive sequences, represent recursive sequences in a variety of ways, and make connections to Pascal's triangle; demonstrate an understanding of the relationships involved in arithmetic and geometric sequences and series, and solve related problems; make connections between sequences, series, and financial applications, and solve problems involving compound interest and ordinary annuities.

Key Specific Expectations

  • C1.1 Make connections between sequences and discrete functions; represent sequences using function notation \( t(n) \) or \( t_n \).
  • C1.2 Determine the terms of a sequence given its general term, or generate a sequence recursively.
  • C1.3 Determine, through investigation, recursive patterns in Pascal's triangle and connect to sequences.
  • C2.1 Identify sequences as arithmetic or geometric, or neither; determine \( a, d, r \) and the general term.
  • C2.2 Determine the formula for the general term of an arithmetic sequence: \( t_n = a + (n-1)d \).
  • C2.3 Determine the formula for the general term of a geometric sequence: \( t_n = a r^{n-1} \).
  • C2.4 Determine the sum of the first \( n \) terms: \( S_n = \tfrac{n}{2}(2a+(n-1)d) \) for arithmetic series and \( S_n = \tfrac{a(r^n-1)}{r-1} \) for geometric series.
  • C2.5 Solve problems involving arithmetic and geometric sequences and series.
  • C3.1 Solve problems, using a scientific calculator, that involve compound interest: \( A=P(1+i)^n \).
  • C3.2 Determine the future value and present value of an ordinary simple annuity using \( FV=R\,\dfrac{(1+i)^n-1}{i} \) and \( PV=R\,\dfrac{1-(1+i)^{-n}}{i} \).
  • C3.3 Solve real-world problems involving annuities and amortisation tables (mortgages, RESPs, retirement planning).

Strand D — Trigonometric Functions

Overall Expectations: determine the values of the trigonometric ratios for angles less than 360°; prove simple trigonometric identities; solve problems using the primary trigonometric ratios, the sine law, and the cosine law; demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numeric, graphical, and algebraic representations of sinusoidal functions; identify and represent sinusoidal functions, and solve problems involving sinusoidal functions, including problems arising from real-world applications.

Key Specific Expectations

  • D1.1 Determine the exact values of the primary and reciprocal trigonometric ratios for the special angles 0°, 30°, 45°, 60°, 90°, and their multiples up to 360°.
  • D1.2 Determine the values of the trig ratios for angles between 0° and 360° using the unit circle and the CAST rule; recognise the role of the reference angle.
  • D1.3 Determine the measure of two angles between 0° and 360° given a primary trig ratio (i.e., recognise the dual solutions in different quadrants).
  • D1.4 Prove simple trigonometric identities using the quotient identity \( \tan x = \tfrac{\sin x}{\cos x} \) and the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \).
  • D1.5 Solve problems requiring the sine law (including the ambiguous SSA case) and the cosine law in 2-D and 3-D.
  • D2.1 Describe key properties of periodic functions: cycle, period, amplitude, equation of axis (midline).
  • D2.2 Predict, by extrapolating, the future behaviour of a periodic relationship by extending the graph in both directions.
  • D2.3 Sketch the graphs of \( y = \sin x \) and \( y = \cos x \) for \( -360° \le x \le 360° \); state period, amplitude, range, intercepts, maxima/minima.
  • D2.4 Determine, through investigation, the roles of \( a, k, d, c \) in \( y = a\sin(k(x-d))+c \) and \( y = a\cos(k(x-d))+c \).
  • D3.1 Pose and solve real-world problems modelled by sinusoidal functions: tides, Ferris wheels, hours of daylight, biorhythms, sound waves.
  • D3.2 Determine the equation of a sinusoidal function from a description of an oscillation or from a graph.

3. Mathematical Processes

The seven mathematical processes are integrated into every strand of MCR3U. Students engage in:

Problem SolvingDevelop, select, apply, compare, and adapt strategies to investigate situations and solve problems.
Reasoning & ProvingDevelop and apply reasoning skills to make and investigate conjectures (e.g., predicting the next term of a sequence) and to construct/defend arguments.
ReflectingDemonstrate that they are reflecting on and monitoring their thinking to clarify understanding (e.g., checking the reasonableness of a sinusoidal model).
Selecting Tools & StrategiesUse a variety of concrete, visual, and electronic tools (Desmos, GeoGebra, scientific calculator, spreadsheet) and computational strategies.
ConnectingMake connections among mathematical concepts and procedures, and relate ideas across strands (e.g., geometric series → annuities; transformations → sinusoids).
RepresentingCreate a variety of representations (numeric, geometric, algebraic, graphical) and translate between them.
CommunicatingCommunicate mathematical thinking orally, visually, and in writing using appropriate vocabulary, notation, and conventions.

4. Achievement Chart

Per Growing Success (2010), each of the four categories carries equal weight (25%) within both term work and the final evaluation.

Knowledge & Understanding
25%

Subject-specific content acquired (knowledge) and the comprehension of its meaning and significance (understanding). Includes definitions, properties, exact values of trig ratios, and standard computations.

Thinking & Inquiry
25%

Use of critical and creative thinking skills and processes — planning, processing, making connections, justifying conclusions in novel and multi-step problems (e.g. ambiguous-case sine law, sinusoidal modelling).

Communication
25%

Conveying meaning through various forms — clarity of expression, use of mathematical conventions, vocabulary, terminology, and labelled diagrams/graphs.

Application
25%

Use of knowledge and skills to make connections within and between contexts — real-world modelling: projectile motion, compound interest, annuities, periodic phenomena.

Levels of achievement: Level 4 (80–100%, exceeds standard) · Level 3 (70–79%, provincial standard) · Level 2 (60–69%, approaching) · Level 1 (50–59%, limited) · R (below 50%, insufficient).

5. Evaluation Policy

Final mark = 70% Term Work + 30% Final Evaluation, per Growing Success (2010).

ComponentWeightDescription
Unit Tests (Ch 1–8, Assessment OF Learning)45%Eight chapter unit tests covering all four achievement categories, 60 marks each.
Quizzes & Diagnostics (Assessment FOR/AS Learning)10%Formative checks; lowest score dropped per term.
Performance Tasks & Investigations10%Modelling, investigation, communication of process (e.g. annuity-design task, sinusoidal modelling task).
Culminating Performance Task5%Cross-strand application due before the final exam window.
Subtotal — Term Work70%Reported on the Provincial Report Card
Final Exam (Ch 1–8 comprehensive, 3 hours, 100 marks)30%Equally weighted across K/U, Thinking, Communication, Application (25 marks each).
Final Course Mark100%One credit toward the OSSD; eligible university prerequisite for MHF4U and MCV4U.

6. Chapter ↔ Strand Mapping

ChapterTitlePrimary StrandSpecific Expectations Addressed
Ch 1Introduction to FunctionsAA1.1–1.6
Ch 2Quadratic Functions & EquationsAA1.5, A2.1–2.3, A3.1–3.2
Ch 3Exponential FunctionsBB1.1–1.3, B2.1, B2.3
Ch 4Solving Exp. Equations & ApplicationsBB3.1–3.3
Ch 5Sequences & SeriesCC1.1–1.3, C2.1–2.5
Ch 6Financial MathematicsCC3.1–3.3
Ch 7Trig Ratios & Sine/Cosine LawsDD1.1–1.5
Ch 8Sinusoidal FunctionsDD2.1–2.4, D3.1–3.2
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