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The official Manitoba Pre-Calculus Mathematics, Grade 12 (40S) curriculum
Manitoba defines Pre-Calculus Mathematics, Grade 12 (40S) by strands and outcomes. MapleMind teaches the same curriculum reorganized for one-skill-at-a-time tutoring — the table shows exactly where every official strand lands, and the ministry's own wording is quoted under each unit below.
Official source Manitoba's official curriculumRead it on the government site — edu.gov.mb.ca ↗| Official strand | Outcomes | Where MapleMind teaches it |
|---|---|---|
| Strand T | 6 | Trigonometry |
| Strand R | 14 | Relations and Functions |
| Strand P | 4 | Permutations, Combinations and Binomial Theorem |
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How MapleMind teaches Pre-Calculus Mathematics, Grade 12 (40S) — every unit, lesson, and skill
Every skill below runs as a short session: a plain-words lesson, a worked example, solving it together, then a five-question skill check that earns up to three stars. Guided Mode keeps it teaching instead of answer-handing — turning it off needs a parent's password.
Unit 1TrigonometryOfficial strand · Strand T
Angles in standard position in radians, the unit circle, the six trigonometric ratios, trigonometric functions and their graphs, trigonometric equations, and proving identities.
Angles in Standard Position
- Sketching and converting angles12P.T.1 — Sketch, in standard position, an angle given in degrees or radians, and convert an angle measure between degrees and radians in exact and approximate form.
- Coterminal angles and arc length12P.T.1 — Determine coterminal angles and their general form in degrees or radians, and explain and apply the relationship between the radian measure of an angle and the arc length on a circle of radius r.
The Unit Circle and Trigonometric Ratios
- The equation of the unit circle12P.T.2 — Derive the equation of the unit circle from the Pythagorean theorem, describe the six trigonometric ratios using a point on the terminal arm, and generalize the equation of a circle centred at the origin with radius r.
- Exact trigonometric ratios12P.T.3 — Determine, using the unit circle or a reference triangle, the exact value of a trigonometric ratio for angles that are multiples of 30, 45, 60, or 90 degrees (or their radian equivalents), and explain the strategy.
- Solving with trigonometric ratios12P.T.3 — Determine approximate trigonometric ratios with technology, find the angles in a domain given a ratio, and determine exact ratios from the coordinates of a point on the terminal arm, then solve a problem using trigonometric ratios.
Trigonometric Functions and Graphs
- Graphs of sine, cosine, and tangent12P.T.4 — Sketch the graphs of y = sin x, y = cos x, and y = tan x and determine their characteristics — amplitude, asymptotes, domain, period, range, and zeros.
- Transforming trigonometric graphs12P.T.4 — Determine how a, b, c, and d affect the graphs of sine and cosine, sketch y = a sin b(x - c) + d using transformations, write the equation from a graph, and model a context with a trigonometric function.
- Solving trigonometric equations12P.T.5 — Solve first- and second-degree trigonometric equations algebraically over a restricted domain in degrees and radians, stating exact solutions where possible and verifying them.
- General solutions and identities in equations12P.T.5 — Determine the general solution of a trigonometric equation, relate it to the zeros of the corresponding function, and use identities to simplify and solve a trigonometric equation.
Trigonometric Identities
- Proving with reciprocal, quotient, and Pythagorean identities12P.T.6 — Distinguish a trigonometric identity from an equation, determine non-permissible values, and prove identities algebraically using reciprocal, quotient, and Pythagorean identities.
- Sum, difference, and double-angle identities12P.T.6 — Use the sum, difference, and double-angle identities (restricted to sine, cosine, and tangent) to prove identities and to determine the exact value of a trigonometric ratio.
The official wording — 6 outcomes in this unit
- 12P.T.1
Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
- 12P.T.2
Develop and apply the equation of the unit circle.
- 12P.T.3
Solve problems, using the six trigonometric ratios for angles expressed in radians and degrees.
- 12P.T.4
Graph and analyze the trigonometric functions sine, cosine, and tangent to solve problems.
- 12P.T.5
Solve, algebraically and graphically, first- and second-degree trigonometric equations with the domain expressed in degrees and radians.
- 12P.T.6
Prove trigonometric identities, using Q Q reciprocal identities Q Q quotient identities Q Q Pythagorean identities
Unit 2Relations and FunctionsOfficial strand · Strand R
Operations and compositions of functions, the full family of transformations, inverses, logarithms and their laws, and the analysis of exponential, logarithmic, polynomial, radical, and rational functions.
Operations and Compositions of Functions
- Operations on functions12P.R.1 — Sketch and write the equation of a function that is the sum, difference, product, or quotient of two functions, and determine its domain and range.
- Compositions of functions12P.R.1 — Evaluate and determine the equation of a composite function of the forms f(g(x)), g(f(x)), or f(f(x)), state any restrictions, and write a function as a composition of two or more functions.
Transformations of Functions
- Translations of graphs12P.R.2 — Compare graphs of y - k = f(x) and y = f(x - h) to y = f(x), generalize the effects of h and k, sketch translated graphs, and write the equation of a translated function.
- Compressions and stretches12P.R.3 — Compare graphs of y = af(x) and y = f(bx) to y = f(x), generalize the effects of a and b, sketch stretched graphs, and write the equation of a compression or stretch.
- Combining transformations12P.R.4 — Apply translations, compressions, and stretches together, sketching y - k = af(b(x - h)) and writing the equation of a fully transformed function from its graph.
- Reflections of graphs12P.R.5 — Sketch reflections of y = f(x) through the x-axis, the y-axis, and the line y = x, generalize the coordinate rules, and write the equation of a reflected function.
- Inverses of relations12P.R.6 — Sketch the inverse of a relation using the line y = x or the swap (x, y) to (y, x), determine whether an inverse is a function, restrict a domain so the inverse is a function, and find the inverse of a linear or quadratic relation.
Logarithms
- Understanding logarithms12P.R.7 — Explain the relationship between logarithms and exponents, convert between logarithmic and exponential form, and determine or estimate the exact value of a logarithm.
- Laws of logarithms12P.R.8 — Develop, prove, and apply the product, quotient, and power laws of logarithms to write equivalent logarithmic expressions.
Exponential and Logarithmic Functions
- Graphing exponential functions12P.R.9 — Sketch and analyze an exponential function of the form y = a to the x (a greater than 0), identifying its domain, range, horizontal asymptote, and intercepts, and apply transformations.
- Graphing logarithmic functions12P.R.9 — Sketch and analyze a logarithmic function of the form y = log base b of x (b greater than 1), identifying its domain, range, vertical asymptote, and intercepts, and show graphically that it is the inverse of the exponential function.
- Solving exponential and logarithmic equations12P.R.10 — Solve exponential equations where bases are or are not powers of one another, solve logarithmic equations, verify solutions, and explain why a logarithmic solution may be extraneous.
- Applications of exponential and logarithmic equations12P.R.10 — Solve problems involving exponential growth or decay, the application of exponential equations to loans, mortgages, or investments, and logarithmic scales such as the Richter or pH scale.
Polynomial, Radical, and Rational Functions
- Long and synthetic division of polynomials12P.R.11 — Divide a polynomial expression by a binomial of the form x - a using long division or synthetic division, and explain how the two methods are related.
- The remainder and factor theorems12P.R.11 — Explain the remainder theorem, relate the linear factors of a polynomial to its zeros, and apply the factor theorem to express a polynomial as a product of factors.
- Graphing polynomial functions12P.R.12 — Sketch and analyze a polynomial function (degree five or less), relating its zeros, the roots of its equation, and the x-intercepts, and explaining the role of degree, leading coefficient, and multiplicity.
- Radical functions12P.R.13 — Sketch and analyze a radical function involving one radical using a table of values and transformations, state its domain and range, and relate its roots to the x-intercepts.
- Rational functions12P.R.14 — Graph and analyze a rational function with monomial, binomial, or trinomial numerator and denominator, describing its behaviour near non-permissible values and distinguishing asymptotes from holes.
The official wording — 14 outcomes in this unit
- 12P.R.1
Demonstrate an understanding of operations on, and compositions of, functions.
- 12P.R.2
Demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of functions and their related equations.
- 12P.R.3
Demonstrate an understanding of the effects of horizontal and vertical compressions and stretches on the graphs of functions and their related equations.
- 12P.R.4
Apply translations, compressions, and stretches to the graphs and equations of functions.
- 12P.R.5
Demonstrate an understanding of the effects of reflections on the graphs of functions and their related equations, including reflections through the Q Q x-axis Q Q y-axis Q Q line y = x
- 12P.R.6
Demonstrate an understanding of inverses of relations.
- 12P.R.7
Demonstrate an understanding of logarithms.
- 12P.R.8
Demonstrate an understanding of the product, quotient, and power laws of logarithms.
- 12P.R.9
Graph and analyze exponential and logarithmic functions.
- 12P.R.10
Solve problems that involve exponential and logarithmic equations.
- 12P.R.11
Demonstrate an understanding of factoring polynomials of degree greater than 2 (limited to polynomials of degree ≤ 5 with integral coefficients).
- 12P.R.12
Graph and analyze polynomial functions (limited to polynomial functions of degree ≤ 5).
- 12P.R.13
Graph and analyze radical functions (limited to functions involving one radical).
- 12P.R.14
Graph and analyze rational functions (limited to numerators and denominators that are monomials, binomials, or trinomials).
Unit 3Permutations, Combinations and Binomial TheoremOfficial strand · Strand P
The fundamental counting principle, permutations, combinations, and the binomial theorem.
Counting and Arrangements
- The fundamental counting principle12P.P.1 — Count the total number of items in a sample space using lists and tree diagrams, explain why choices multiply rather than add, and solve a counting problem using the fundamental counting principle.
- Permutations12P.P.2 — Determine the number of permutations of n elements taken n or r at a time using factorial and nPr notation, explain why n must be at least r, and account for identical elements.
- Combinations12P.P.3 — Determine the number of combinations of n elements taken r at a time, explain the difference between a permutation and a combination, and explain and use the identity nCr = nC(n - r).
The Binomial Theorem
- Pascal's triangle and binomial patterns12P.P.4 — Explain the patterns in the expansion of a binomial power by multiplying factors, generate rows of Pascal's triangle, and relate its rows to the coefficients and to combinations.
- Expanding with the binomial theorem12P.P.4 — Expand a binomial power using the binomial theorem, and determine a specific term in the expansion.
The official wording — 4 outcomes in this unit
- 12P.P.1
Apply the fundamental counting principle to solve problems.
- 12P.P.2
Determine the number of permutations of n elements taken r at a time to solve problems.
- 12P.P.3
Determine the number of combinations of n different elements taken r at a time to solve problems.
- 12P.P.4
Expand powers of a binomial in a variety of ways, including using the binomial theorem (restricted to exponents that are natural numbers).
Printable workbook · A keepsake of the year
A Pre-Calculus Mathematics, Grade 12 (40S) workbook worth keeping
Built from the same official curriculum as this page. Before and after each skill above, your student colours in how sure they feel — so the two of you can see, on one page, what's clicking and what needs another look. It's a quiet way to follow how the year is really going.
By June it's full of their own handwriting: units worked through, confidence grown, a mid-year check-in, notes from parent-teacher night, and a certificate at the end. Less a worksheet, more a record of the year worth keeping on the shelf.
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Yes. MapleMind's AI tutor covers all 34 skills in Manitoba's Pre-Calculus Mathematics, Grade 12 (40S) — you pick the exact skill, and the tutor teaches it step by step: a short lesson, a worked example, solving together, then a skill check to show it stuck.
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Yes. Every skill in this course maps to an official outcome code from Manitoba's Grade 12 Mathematics curriculum, and the ministry's own wording is quoted under each unit on this page — with the official government source linked so you can check it yourself.
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Yes — 14 languages, including French. Both the app and the tutor's explanations switch to the language you choose.
Where can I see the official Manitoba curriculum for Pre-Calculus Mathematics, Grade 12 (40S)?
The official source is linked on this page — Manitoba's official curriculum. The outline here follows it: every MapleMind skill carries its official outcome code, and the ministry's own wording is quoted under each unit.
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