Grade 12: Mathematics

Here is a comprehensive contents outline for Grade 12 Mathematics, aligned with the British Columbia (BC) curriculum in Canada.

The BC Grade 12 math curriculum offers several distinct pathways to cater to students’ different interests, career goals, and post-secondary plans. The three main courses are:

1. Pre-Calculus 12: For students intending to study STEM (Science, Technology, Engineering, and Mathematics) fields in university. It focuses on abstract reasoning, algebraic skills, and advanced function analysis.

2. Calculus 12: An introductory course for students who have completed Pre-Calculus 12 and wish to begin their study of calculus.

3. Foundations of Mathematics 12: For students intending to pursue programs in the humanities, social sciences, or arts, or for those entering trades or direct entry into the workforce. It focuses on logical reasoning, financial mathematics, and statistical analysis.

The following outlines break down the content for these distinct pathways.

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Pathway 1: Pre-Calculus 12

Big Ideas of the Course:

• The concept of a function is a unifying theme throughout mathematics.

• Algebra allows us to generalize relationships through abstract analysis.

• Transformations of functions are powerful tools for visualizing and understanding mathematical relationships.

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Unit 1: Function Transformations

• Key Concepts:

  • Graphing and analyzing functions under combinations of vertical and horizontal translations, reflections, and stretches.

  • Mapping notation: $y=af(b(x-p))+q$

  • Applying transformations to parent functions (e.g., y=x2,y=x,y=∣x∣,y=1xy=x2,y=x​,y=∣x∣,y=x1​).

Unit 2: Radical Functions

• Key Concepts:

  • Analyzing and graphing functions involving square roots, y=ab(x−p)+qy=ab(x−p)​+q.

  • Solving radical equations algebraically and identifying extraneous roots.

  • Determining the domain and range of radical functions.

Unit 3: Polynomial Functions

• Key Concepts:

  • Characteristics of polynomial functions (degree, leading coefficient, end behaviour).

  • Analyzing and graphing polynomial functions (cubic, quartic).

  • Solving polynomial equations of degree greater than two.

  • The Remainder Theorem and The Factor Theorem.

Unit 4: Trigonometric Functions and Graphs

• Key Concepts:

  • Graphing and analyzing the sine and cosine functions, $y=a\sin (b(x-c))+d$ and $y=a\cos (b(x-c))+d$.

  • Understanding and applying the effects of transformations on trigonometric graphs (amplitude, period, phase shift, vertical displacement).

  • Solving trigonometric equations using graphs and algebra, with attention to the general solution.

Unit 5: Logarithmic and Exponential Functions

• Key Concepts:

  • The relationship between logarithms and exponents: y=log⁡bxy=logb​x is equivalent to by=xby=x.

  • Graphing and analyzing exponential and logarithmic functions.

  • The Laws of Logarithms and their application to simplifying and solving equations.

  • Solving exponential and logarithmic equations (e.g., population growth, radioactive decay).

Unit 6: Rational Functions

• Key Concepts:

  • Graphing and analyzing rational functions of the form y=f(x)g(x)y=g(x)f(x)​, where $f(x)$ and $g(x)$ are polynomials.

  • Identifying key features: vertical asymptotes, horizontal asymptotes, points of discontinuity (holes), and intercepts.

  • Solving rational equations.

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Pathway 2: Calculus 12

Big Ideas of the Course:

• The concept of a limit is foundational to calculus.

• Derivatives are functions that represent an instantaneous rate of change.

• Integrals represent a limit of a sum and can be used to find net change.

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Unit 1: Limits and Continuity

• Key Concepts:

  • Estimating limits from graphs and tables of values.

  • Evaluating limits using direct substitution, factoring, and rationalization.

  • Understanding the concept of continuity and identifying points of discontinuity.

Unit 2: Derivatives

• Key Concepts:

  • The Derivative as a Limit: Understanding the derivative as the limit of the difference quotient: f′(x)=lim⁡h→0f(x+h)−f(x)hf′(x)=limh→0​hf(x+h)−f(x)​.

  • Derivative Rules: Power rule, product rule, quotient rule, and chain rule for polynomial, rational, radical, and simple trigonometric functions (sine and cosine).

  • Applications of Derivatives:

    • Finding the equation of a tangent line.

    • Determining instantaneous rates of change.

    • Analyzing motion (position, velocity, acceleration).

Unit 3: Applications of Derivatives

• Key Concepts:

  • Curve Sketching: Using the first derivative test to find intervals of increase/decrease and critical points. Using the second derivative test to find points of inflection and intervals of concavity.

  • Optimization Problems: Solving applied max/min problems (e.g., maximizing area, minimizing cost).

Unit 4: Integrals

• Key Concepts:

  • Anti-differentiation: Finding indefinite integrals of polynomial, simple trigonometric, and exponential functions.

  • The Definite Integral: Understanding the definite integral as a limit of Riemann sums and as net area under a curve.

  • The Fundamental Theorem of Calculus: Connecting derivatives and integrals.

  • Applications of Integrals: Calculating areas and total change.

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Pathway 3: Foundations of Mathematics 12

Big Ideas of the Course:

• Logical reasoning helps us make sense of the world and is used in developing mathematical ideas.

• Statistical analysis allows us to explore, understand, and communicate data.

• Financial literacy promotes understanding of financial decisions and their effects.

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Unit 1: Financial Mathematics

• Key Concepts:

  • Compound interest, annuities, and mortgages.

  • Analyzing and comparing credit options.

  • Creating and analyzing a budget.

Unit 2: Logical Reasoning and Set Theory

• Key Concepts:

  • inductive and deductive reasoning.

  • Set theory (Venn diagrams, unions, intersections, complements).

  • Solving problems using logical reasoning and set theory.

Unit 3: Probability

• Key Concepts:

  • Conditional probability and independent events.

  • Permutations and combinations.

  • Solving problems involving probability.

Unit 4: Relations and Functions

• Key Concepts:

  • Representing relations and functions in different ways.

  • Analyzing and modeling with linear, quadratic, and exponential functions in contextual situations.

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Curricular Competencies (The “Doing” of Mathematics)

Throughout all courses, students will be expected to develop these skills:

• Reasoning and Analyzing

• Understanding and Solving

• Communicating and Representing

• Connecting and Reflecting

This outline provides a solid foundation for what a student can expect to learn in BC’s Grade 12 math courses. The choice of pathway is critical and should be based on a student’s future academic and career plans.