Quebec Education Program Progression of Learning — Grade 12

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12-1.11.b.

Squares and square roots

12-1.11.c.

Numbers in exponential notation (integral exponent)

12-1.11.d.

Numbers in scientific notation

12-1.11.e.

Cubes and cube roots

12-1.11.f.

Numbers in exponential notation (fractional exponents)

12-1.11.g.

Numbers using radicals or rational exponents

12-1.11.h.

Numbers in logarithmic notation using the equivalence (log)(a)x = n <=> a^n = x, if necessary

12-1.15.b.

Numbers expressed in different ways (fractional, decimal, exponential [integral exponent], percentage, square root, scientific notation)

12-10.A.3.

Establishes relationships between units of mass

12-10.B.1.

Chooses the appropriate unit of time for the context

12-10.B.3.

Establishes relationships between units of time: second, minute, hour, day, daily cycle, weekly cycle, yearly cycle

12-10.B.4.

Distinguishes between duration and position in time

12-10.C.2.

Estimates and determines the degree measure of angles

12-10.C.3.

Describes the characteristics of different types of angles: complementary, supplementary, adjacent, vertically opposite, alternate interior, alternate exterior and corresponding

12-10.C.4.

Determines measures of angles using the properties of the following angles: complementary, supplementary, vertically opposite, alternate interior, alternate exterior and corresponding

12-10.C.5.

Finds unknown measurements using the properties of figures and relations

12-10.C.5.a.

Measures of angles in a triangle

12-10.C.5.b.

Degree measures of central angles and arcs

12-10.C.7.

Determines the correspondence between degrees and radians

12-10.C.8.

Justifies statements using definitions or properties associated with angles and their measures

12-10.D.1.

Chooses the appropriate unit of length for the context

12-10.D.4.

Constructs relations that can be used to calculate the perimeter or circumference of figures

12-10.D.5.

Finds the following unknown measurements, using properties of figures and relations

12-10.D.5.a.

Perimeter of plane figures

12-10.D.5.b.

A segment in a plane figure, circumference, radius, diameter, length of an arc, a segment resulting from an isometry or a similarity transformation

12-10.D.5.c.

Segments in a solid resulting from an isometry or a similarity transformation

12-10.D.5.d.

Segments or perimeters resulting from equivalent figures

12-10.D.6.

Justifies statements concerning measures of length

12-10.E.4.

Constructs relations that can be used to calculate the area of plane figures: quadrilateral, triangle, circle (sectors)

12-10.E.5.

Uses relations that can be used to calculate the area of a right cone and a sphere

12-10.E.6.

Finds unknown measurements, using properties of figures and relations

12-10.E.6.a.

Area of circles and sectors

12-10.E.6.b.

Area of figures that can be split into circles (sectors), triangles or quadrilaterals

12-10.E.6.c.

Lateral or total area of right prisms, right cylinders and right pyramids

12-10.E.6.d.

Lateral or total area of solids that can be split into right prisms, right cylinders or right pyramids

12-10.E.6.f.

Area of figures resulting from a similarity transformation

12-10.E.6.g.

Area of a sphere, lateral or total area of right cones and decomposable solids

12-10.E.7.

Justifies statements concerning measures of area

12-10.F.4.

Establishes relationships between

12-10.F.4.b.

Measures of capacity

12-10.F.5.

Constructs relations that can be used to calculate volumes: right cylinders, right pyramids, right cones and spheres

12-10.F.6.

Finds unknown measurements using properties of figures and relations

12-10.F.6.a.

Volume of right prisms, right cylinders, right pyramids, right cones and spheres

12-10.F.6.b.

Volume of solids that can be split into right prisms, right cylinders, right pyramids, right cones and spheres

12-10.F.6.c.

Volume solids resulting from an isometry or a similarity transformation

12-10.F.6.d.

Volume of equivalent solids

12-10.F.7.

Justifies statements concerning measures of volume or capacity

12-10.G.1.

Determines, through exploration or deduction, different metric relations associated with plane figures

12-10.G.2.

Finds unknown measurements in various situations

12-10.G.2.a.

In a right triangle rectangle using

12-10.G.2.a.i.

Pythagorean relation

12-10.G.2.a.iii.

Trigonometric ratios: sine, cosine, tangent

12-10.G.2.c.

In a circle: measure of arcs, chords, inscribed angles, interior angles and exterior angles

12-10.G.4.

Proves trigonometric identities by using algebraic properties, definitions (sine, cosine, tangent, cosecant, secant, cotangent), Pythagorean identities, and the properties of periodicity and symmetry

12-10.G.5.

Justifies statements concerning

12-10.G.5.a.

Pythagorean relation

12-10.G.5.b.

Metric or trigonometric relations

12-10.H.1.

Defines a vector: magnitude (length or norm), direction, sense

12-10.H.2.

Represents a vector graphically (arrow in a plane or pair in a Cartesian plane)

12-10.H.3.

Identifies properties of vectors

12-10.H.4.

Performs operations on vectors

12-10.H.4.a.

Determination of the resultant or projection of a vector

12-10.H.4.c.

Multiplication of a vector by a scalar

12-10.H.4.d.

Scalar product of two vectors

12-10.H.4.e.

Linear combination of vectors

12-10.H.5.

Justifies statements using properties associated with vectors

12-10.H.6.

Analyzes and models situations using vectors (e.g. displacements, forces, speeds or velocities)

12-11.A.1.

Locates objects/numbers on an axis, based on the types of numbers studied

12-11.A.2.

Locates points in a Cartesian plane, based on the types of numbers studied (x- and y-coordinates of a point)

12-11.B.1.

Uses the concept of change to

12-11.B.1.a.

Calculate the distance between two points

12-11.B.1.b.

Determine the coordinates of a point of division using a given ratio (including the coordinates of a midpoint)

12-11.B.1.c.

Calculate and interpret a slope

12-11.B.2.

Determines the relative position of two straight lines using their respective slope (intersecting at one point, perpendicular, non-intersecting parallel or coincident)

12-11.B.3.

Models, with or without technological tools, a situation involving

12-11.B.3.a.

Straight lines: graphically and algebraically

12-11.B.3.b.

A half-plane: graphically and algebraically

12-11.B.3.c.

Parallel lines and perpendicular lines

12-11.B.4.

Determines the equation of a line using the slope and a point or using two points

12-11.D.3.

Analyzes and models situations using conics: describing the elements of a conic: radius, axes, directrix, vertices, foci, asymptotes, regions; graphing a conic and its internal and external region; constructing the rule of a conic based on its definition; finding the rule (standard form) of a conic and its internal and external region; validating and interpreting the solution, if necessary

12-11.D.3.a.

Parabola centred at the origin and resulting from a translation

12-11.D.3.b.

Circle, ellipse and hyperbola centred at the origin

12-11.D.3.c.

Circle, ellipse and hyperbola resulting from a translation

12-11.D.4.

Determines the coordinates of points of intersection between

12-11.D.4.a.

A line and a conic

12-11.E.1.

Establishes the relationship between trigonometric ratios and the standard unit circle (trigonometric ratios and lines)

12-11.E.2.

Determines the coordinates of points associated with significant angles using metric relations in right triangles (Pythagorean relation, properties of angles: 30 degrees, 45degrees, 60degrees)

12-11.E.3.

Analyzes and uses periodicity and symmetry to determine coordinates of points associated with significant angles in the standard unit circle

12-11.E.4.

Proves Pythagorean identities

12-12.A.1.

Describes the basic elements of graph theory: degree distance, path, circuit

12-12.A.3.

Constructs graphs: directed graphs, weighted graphs, coloured graphs, trees

12-12.B.1.

Determines elements of a situation associated with vertices and edges

12-12.B.2.

Represents a situation using a graph

12-12.B.3.

Compares graphs, if necessary

12-13.2.b.

Compares and interprets different voting procedures and their results

12-3.14.a.

Integral exponents (rational base) and fractional exponents

12-3.14.b.

Powers of bases (change of base), exponents, radicals (nth root), using their properties

12-3.14.c.i.

Definition and change of base

Absolute values

12-3.4.b.

Uses, in different contexts, the properties of divisibility: 2, 3, 4, 5 and 10

12-3.7.a.

Numbers written in decimal notation, using rules of signs

12-3.7.c.

Numbers written in fractional notation

12-4.1.a.

A certain percentage of a number

12-4.1.b.

The value corresponding to 100 per cent

12-4.5.a.

Ratios and rates qualitatively (equivalent rates and ratios, unit rate)

12-4.5.b.

Ratios and rates quantitatively (equivalent rates and ratios, unit rate)

12-5.A.1.

Describes, using his/her own words and mathematical language, numerical patterns

12-5.A.2.

Describes, using his/her own words and mathematical language, series of numbers and family of operations

12-5.A.3.

Adds new terms to a series when the first three terms or more are given

12-5.A.4.

Describes the role of components of algebraic expressions:

12-5.A.4.d.

Coefficient, degree, term, constant term, like terms

12-5.A.5.

Constructs an algebraic expression using a register (type) of representation

12-5.A.7.

Recognizes or constructs equivalent algebraic expressions

12-5.A.8.

Recognizes or constructs

12-5.A.8.a.

Equalities and equations

12-5.B.1.

Calculates the numeric value of an algebraic expression

12-5.B.2.

Performs the following operations on algebraic expressions, with or without objects or diagrams: addition and subtraction, multiplication and division by a constant, multiplication of first-degree monomials

12-5.B.3.

Factors out the common factor in numerical expressions (distributive property of multiplication over addition or subtraction)

12-5.B.4.a.

Algebraic expressions of degree less than 3

12-5.B.4.b.

Algebraic expressions

12-5.B.5.a.

Algebraic expressions by a monomial

12-5.B.5.b.

A polynomial by a binomial (with or without a remainder)

12-5.B.5.c.

A polynomial by another polynomial (with or without a remainder)

12-5.B.6.

Factors polynomials by

12-5.B.6.a.

Finding the common factor

12-5.B.6.b.

Factoring by grouping (polynomials including decomposable second-degree trinomials)

12-5.B.6.c.

Completing the square (factoring and switching from one type of notation to another)

12-5.B.6.d.

Using formulas for trinomials of the form ax^2 + bx + c: x1 = (-b + or - square root (b^2-4ac)/2a)

12-5.B.6.e.

Substituting second-degree algebraic identities (perfect square trinomial and difference of two squares)

12-5.B.7.

Manipulates rational expressions

12-5.C.1.

Recognizes whether a situation can be translated by

An inequality

12-5.C.10.

Solves first-degree inequalities in one variable

12-5.C.11.

Solves the following types of equations or an inequalities in one variable

12-5.C.11.b.

Exponential, logarithmic or square root, using the properties of exponents, logarithms and radicals

Absolute value

12-5.C.11.e.

First-degree trigonometric involving a sine, cosine or tangent expression

12-5.C.11.f.

Trigonometric that can be expressed as a sine, cosine or tangent function

12-5.C.12.

Solves a second-degree equation in two variables

12-5.C.13.

Validates a solution, with or without technological tools, by substitution

12-5.C.14.

Solves an inequality graphically and checks the feasible region of a

12-5.C.14.a.

First-degree inequality in two variables

12-5.C.14.b.

Second-degree inequality in two variables

12-5.C.2.

Recognizes or constructs

12-5.C.2.a.

Relations or formulas

12-5.C.2.b.

Inequality relations and first-degree inequalities in one variable

12-5.C.3.

Manipulates relations or formulas (e.g. isolates an element)

12-5.C.4.

Represents a situation using

12-5.C.4.a.

A first-degree equation with one unknown

12-5.C.4.b.

A first-degree inequality with a variable

12-5.C.5.a.

An equation using another register (type) of representation, if necessary

12-5.C.5.b.

An inequality using another register (type) of representation, if necessary

12-5.C.6.

Determines the missing term in an equation (relations between operations) :1 a + b = , a + = c, + b = c, a b = , a = c, b = c, a b = , a = c, b = c, a b = , a = c, b = c

12-5.C.7.

Transforms arithmetic equalities and equations to maintain equivalence (properties and rules for transforming equalities) and justifies the steps followed, if necessary

12-5.C.8.

Transforms inequalities to maintain equivalence (properties and rules for transforming inequalities) and justifies the steps followed, if necessary

12-5.C.9.

Uses different methods to solve first-degree equations with one unknown of the form ax + b = cx + d : trial and error, drawings, arithmetic methods (inverse or equivalent operations), algebraic methods (balancing equations or hidden terms)

12-5.D.1.

Determines whether a situation may be translated by a system of

12-5.D.2.

Translates a situation algebraically or graphically using a system of

Solves a system

12-5.D.3.a.

Of first-degree equations in two variables of the form y = ax + b by using tables of values, graphically or algebraically (by comparison), with or without technological tools

12-5.D.3.b.

Of first-degree equations in two variables

12-5.D.3.c.

Composed of a first-degree equation in two variables and a second-degree equation in two variables

12-5.D.3.e.

Involving various functional models (mostly graphical solutions)

Solves a system

12-5.D.4.a.

Of first-degree inequalities in two variables

12-5.D.4.b.

Involving various functional models (mostly graphical solutions)

12-5.D.5.

Validates the solution, with or without technological tools

12-5.D.6.

Interprets the solution or makes decisions if necessary, depending on the context

12-5.E.1.

Analyzes a situation to be optimized

12-5.E.1.a.

Mathematizing the situation using a system of first-degree inequalities in two variables

12-5.E.1.b.

Drawing a bounded or unbounded polygon of constraints to represent the situation

12-5.E.1.c.

Determining the coordinates of the vertices of the bounded polygon (feasible region)

12-5.E.1.d.

Recognizing and defining the function to be optimized

12-5.E.2.

Optimizes a situation by taking into account different constraints and makes decisions with respect to this situation

12-5.E.2.a.

Determining the best solution(s) for a particular situation, given a set of possibilities

12-5.E.2.b.

Validating and interpreting the optimal solution, depending on the context

12-5.E.2.c.

Justifying the solution(s) chosen

12-5.E.2.d.

Changing certain conditions associated with the situation to provide a more optimal solution, if necessary

12-6.A.1.

Identifies patterns in various situations and in various forms

12-6.A.2.

Analyzes situations using different registers (types) of representation

12-6.A.3.

Represents a situation generally using a graph

12-6.A.5.

Recognizes relations, functions and inverses

12-6.A.6.

Describes, in the functions under study, the role of

12-6.A.6.a.

Multiplicative parameters

12-6.A.7.

Performs operations on functions (including composition)

12-6.B.1-9.a.

Polynomial functions of degree 0 or 1

12-6.B.1-9.b.

Second-degree polynomial functions

12-6.B.1-9.b.ii.

f(x) = (bx)^2 or f(x) = a(bx)^2

12-6.B.1-9.b.iii.

f(x) = ax^2+ bx + c, f(x) = a(b(x - h))^2 + k, f(x) = a(x - x1)(x - x2)

12-6.B.1-9.c.

Square root functions

12-6.B.1-9.c.i.

f(x) = a(root of (bx))

12-6.B.1-9.c.ii.

f(x) = a(root of (b(x-h)) + k

12-6.B.1-9.d.

Rational functions

12-6.B.1-9.d.i.

f(x) = k/x or xy = k where k is an element of the set of positive rational numbers.

12-6.B.1-9.d.ii.

f(x) = a(1/(b(x-h)) + k and f(x) = ((ax+b)/(cx+d))

12-6.B.1-9.e.

Exponential functions

12-6.B.1-9.f.

Logarithmic functions

f(x) = a logc bx

12-6.B.1-9.k.i.

Modelling periodic occurences (e.g. natural phenomena such as tides or sound, medical or electrical phenomena)

12-6.B.1-9.k.ii.

Sinusoidal : f(x) = a sin b(x - h) + k, f(x) = a cos b(x - h) + k

12-6.B.1-9.k.iii.

Tangent : f(x) = a tan b(x - h) + k

12-6.B.1.

Models a situation verbally, algebraically, graphically, using a table of values or a scatter plot

12-6.B.2.

Finds the rule of a function or its inverse, depending on the context

12-6.B.3.

Represents and interprets the inverse

12-6.B.4.

Interprets parameters (multiplicative or additive) and describes the effect of changing their value, if necessary

12-6.B.5.

Describes the properties of real functions: domain, range, interval within which the function is increasing or decreasing, sign, extrema, x-intercept and y-intercept

12-6.B.6.

Determines values or data by solving equations and inequalities

12-6.B.8.

Compares situations or graphical representations

12-6.B.9.

Makes decisions, if necessary, depending on the context

12-7.A.11.

Recognizes certain, probable, impossible, simple, complementary, compatible, incompatible, dependent, independents events

12-7.A.12.

Distinguishes between mutually exclusive and non-mutually exclusive, and between dependent and independent events

12-7.A.13.

Uses fractions, decimals or percentages to quantify a probability

12-7.A.14.

Recognizes that a probability is always between 0 and 1

12-7.A.9.

Enumerates the possible outcomes of a random experiment using

12-7.A.9.a.

Tables, tree diagram

12-7.A.9.b.

Networks, tables, diagrams, Venn diagrams

12-7.B.1.

Represents an event using different registers (types of representation))

12-7.B.10.

Chooses and applies the concept of odds/chance (odds for and odds against) or probability, depending on the context

12-7.B.11.

Determines the odds for or odds against

12-7.B.12.

Interprets and makes decisions with respect to the odds obtained

12-7.B.13.

Calculates mathematical expectation

12-7.B.15.

Interprets the resulting mathematical expectation and makes appropriate decisions

12-7.B.4.

Calculates the probability of an event

12-7.B.5.

Calculates the probability of outcomes of random experiments related to situations involving arrangements, permutations or combinations

12-7.B.6.

Associates the type of probability to a situation: experimental, theoretical, subjective

12-7.B.8.

Calculates conditional probabilities

12-7.B.9.

Interprets probabilities and makes appropriate decisions

12-8.A.1.

Conducts a survey or a census

12-8.A.1.a.

Formulates questions for a survey

12-8.A.1.d.

Collects, describes and organizes data (classifies or categorizes) using tables

12-8.A.10.

Calculates and interprets an arithmetic mean

12-8.A.11.

Determines and interprets

12-8.A.11.a.

Measures of central tendency: mode, median, weighted mean

12-8.A.11.b.

Measures of dispersion:

12-8.A.11.b.iv.

Standard deviation

12-8.A.11.c.

Measures of position:

12-8.A.12.

Chooses the appropriate statistical measures for a given situation

12-8.A.2.

Recognizes possible sources of bias

12-8.A.3.

Interprets data presented in a table or a bar graph, a pictograph, a broken-line graph or a circle graph

12-8.A.4.

Distinguishes different types of statistical variables: qualitative, discrete or continuous quantitative

12-8.A.5.

Chooses appropriate register(s) (types) of representation to organize, interpret and present data

12-8.A.8.

Understands and calculates the arithmetic mean

12-8.A.9.

Describes the concept of arithmetic mean (levelling or balance point)

12-8.B.1.

Compares experimental and theoretical data

12-8.B.10.

Compares two-variable distributions

12-8.B.3.

Associates the most appropriate functional model with a scatter plot :

12-8.B.3.a.

First-degree polynomial function

12-8.B.9.

Interpolates or extrapolates values using

12-8.B.9.a.

A regression line

12-8.B.9.b.

The functional model best suited to the situation

12-9.A.10.

Justifies statements using definitions or properties of plane figures

12-9.A.5.

Recognizes and names regular convex polygons

12-9.A.6.

Decomposes plane figures into circles (sectors), triangles or quadrilaterals

12-9.A.7.

Describes circles and sectors

12-9.A.8.

Recognizes and draws main segments and lines

12-9.A.8.a.

Diagonal, altitude, median, perpendicular bisector, bisector, apothem, radius, diameter, chord

Leg, hypotenuse

12-9.A.9.

Identifies the properties of plane figures using geometric transformations and constructions

12-9.B.4.

Describes solids:

12-9.B.4.a.

Vertex, edge, base, face

12-9.B.4.b.

Altitude, apothem, lateral face

12-9.B.5.

Tests Euler's relation on convex polyhedrons

12-9.B.7.

Represents three-dimensional figures in the plane, using different procedures:

12-9.B.7.b.

Projection and perspective (e.g. orthogonal projections [different views], parallel projections [cavalier and axonometric perspectives] or central projections [with one or two vanishing points])

12-9.C.2.

Identifies properties and invariants resulting from geometric constructions and transformations

12-9.C.4.

Constructs the image of a figure under a translation, rotation and reflection

12-9.C.5.

Recognizes dilatation with a positive scale factor

12-9.C.6.

Constructs the image of a figure under a dilatation with a positive scale factor

12-9.D.2.

Recognizes congruent or similar figures

12-9.D.4.

Determines the properties and invariants of congruent or similar figures

12-9.D.5.

Determines the minimum conditions required to conclude that triangles are congruent or similar

12-9.D.6.

Demonstrates the congruence or similarity between triangles or finds unknown measurements using minimum conditions

12-9.D.7.

Recognizes equivalent figures (plane figures or solids)

12-9.D.8.

Justifies statements using definitions or properties of congruent, similar or equivalent figures, depending on the cycle and year

12-xxxx

Note: Statements 1 to 9 apply to the functions listed below.