# Quebec Education Program Progression of Learning — Grade 11

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#### 11-1.11.b.

Squares and square roots

#### 11-1.11.c.

Numbers in exponential notation (integral exponent)

#### 11-1.11.d.

Numbers in scientific notation

#### 11-1.11.e.

Cubes and cube roots

#### 11-1.11.f.

Numbers in exponential notation (fractional exponents)

#### 11-1.11.g.

Numbers using radicals or rational exponents

#### 11-1.11.h.

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

#### 11-1.15.b.

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

#### 11-10.A.3.

Establishes relationships between units of mass

#### 11-10.B.1.

Chooses the appropriate unit of time for the context

#### 11-10.B.3.

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

#### 11-10.B.4.

Distinguishes between duration and position in time

#### 11-10.C.2.

Estimates and determines the degree measure of angles

#### 11-10.C.3.

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

#### 11-10.C.4.

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

#### 11-10.C.5.

Finds unknown measurements using the properties of figures and relations

#### 11-10.C.5.a.

Measures of angles in a triangle

#### 11-10.C.5.b.

Degree measures of central angles and arcs

#### 11-10.C.7.

Determines the correspondence between degrees and radians

#### 11-10.C.8.

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

#### 11-10.D.1.

Chooses the appropriate unit of length for the context

#### 11-10.D.4.

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

#### 11-10.D.5.

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

#### 11-10.D.5.a.

Perimeter of plane figures

#### 11-10.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

#### 11-10.D.5.c.

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

#### 11-10.D.5.d.

Segments or perimeters resulting from equivalent figures

#### 11-10.D.6.

Justifies statements concerning measures of length

#### 11-10.E.4.

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

#### 11-10.E.5.

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

#### 11-10.E.6.

Finds unknown measurements, using properties of figures and relations

#### 11-10.E.6.a.

Area of circles and sectors

#### 11-10.E.6.b.

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

#### 11-10.E.6.c.

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

#### 11-10.E.6.d.

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

#### 11-10.E.6.f.

Area of figures resulting from a similarity transformation

#### 11-10.E.6.g.

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

#### 11-10.E.7.

Justifies statements concerning measures of area

#### 11-10.F.4.

Establishes relationships between

#### 11-10.F.5.

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

#### 11-10.F.6.

Finds unknown measurements using properties of figures and relations

#### 11-10.F.6.a.

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

#### 11-10.F.6.b.

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

#### 11-10.F.6.c.

Volume solids resulting from an isometry or a similarity transformation

#### 11-10.F.6.d.

Volume of equivalent solids

#### 11-10.F.7.

Justifies statements concerning measures of volume or capacity

#### 11-10.G.1.

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

#### 11-10.G.2.

Finds unknown measurements in various situations

#### 11-10.G.2.a.

In a right triangle rectangle using

#### 11-10.G.2.a.iii.

Trigonometric ratios: sine, cosine, tangent

#### 11-10.G.2.c.

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

#### 11-10.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

#### 11-10.G.5.

Justifies statements concerning

#### 11-10.G.5.b.

Metric or trigonometric relations

#### 11-10.H.1.

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

#### 11-10.H.2.

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

#### 11-10.H.3.

Identifies properties of vectors

#### 11-10.H.4.

Performs operations on vectors

#### 11-10.H.4.a.

Determination of the resultant or projection of a vector

#### 11-10.H.4.c.

Multiplication of a vector by a scalar

#### 11-10.H.4.d.

Scalar product of two vectors

#### 11-10.H.4.e.

Linear combination of vectors

#### 11-10.H.5.

Justifies statements using properties associated with vectors

#### 11-10.H.6.

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

#### 11-11.A.1.

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

#### 11-11.A.2.

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

#### 11-11.B.1.

Uses the concept of change to

#### 11-11.B.1.a.

Calculate the distance between two points

#### 11-11.B.1.b.

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

#### 11-11.B.1.c.

Calculate and interpret a slope

#### 11-11.B.2.

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

#### 11-11.B.3.

Models, with or without technological tools, a situation involving

#### 11-11.B.3.a.

Straight lines: graphically and algebraically

#### 11-11.B.3.b.

A half-plane: graphically and algebraically

#### 11-11.B.3.c.

Parallel lines and perpendicular lines

#### 11-11.B.4.

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

#### 11-11.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

#### 11-11.D.3.a.

Parabola centred at the origin and resulting from a translation

#### 11-11.D.3.b.

Circle, ellipse and hyperbola centred at the origin

#### 11-11.D.3.c.

Circle, ellipse and hyperbola resulting from a translation

#### 11-11.D.4.

Determines the coordinates of points of intersection between

#### 11-11.E.1.

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

#### 11-11.E.2.

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

#### 11-11.E.3.

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

#### 11-11.E.4.

Proves Pythagorean identities

#### 11-12.A.1.

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

#### 11-12.A.3.

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

#### 11-12.B.1.

Determines elements of a situation associated with vertices and edges

#### 11-12.B.2.

Represents a situation using a graph

#### 11-12.B.3.

Compares graphs, if necessary

#### 11-13.2.b.

Compares and interprets different voting procedures and their results

#### 11-3.14.a.

Integral exponents (rational base) and fractional exponents

#### 11-3.14.b.

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

#### 11-3.14.c.i.

Definition and change of base

#### 11-3.4.b.

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

#### 11-3.7.a.

Numbers written in decimal notation, using rules of signs

#### 11-3.7.c.

Numbers written in fractional notation

#### 11-4.1.a.

A certain percentage of a number

#### 11-4.1.b.

The value corresponding to 100 per cent

#### 11-4.5.a.

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

#### 11-4.5.b.

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

#### 11-5.A.1.

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

#### 11-5.A.2.

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

#### 11-5.A.3.

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

#### 11-5.A.4.

Describes the role of components of algebraic expressions:

#### 11-5.A.4.d.

Coefficient, degree, term, constant term, like terms

#### 11-5.A.5.

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

#### 11-5.A.7.

Recognizes or constructs equivalent algebraic expressions

#### 11-5.A.8.

Recognizes or constructs

#### 11-5.A.8.a.

Equalities and equations

#### 11-5.B.1.

Calculates the numeric value of an algebraic expression

#### 11-5.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

#### 11-5.B.3.

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

#### 11-5.B.4.a.

Algebraic expressions of degree less than 3

#### 11-5.B.4.b.

Algebraic expressions

#### 11-5.B.5.a.

Algebraic expressions by a monomial

#### 11-5.B.5.b.

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

#### 11-5.B.5.c.

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

#### 11-5.B.6.

Factors polynomials by

#### 11-5.B.6.a.

Finding the common factor

#### 11-5.B.6.b.

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

#### 11-5.B.6.c.

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

#### 11-5.B.6.d.

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

#### 11-5.B.6.e.

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

#### 11-5.B.7.

Manipulates rational expressions

#### 11-5.C.1.

Recognizes whether a situation can be translated by

#### 11-5.C.10.

Solves first-degree inequalities in one variable

#### 11-5.C.11.

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

#### 11-5.C.11.b.

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

#### 11-5.C.11.e.

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

#### 11-5.C.11.f.

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

#### 11-5.C.12.

Solves a second-degree equation in two variables

#### 11-5.C.13.

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

#### 11-5.C.14.

Solves an inequality graphically and checks the feasible region of a

#### 11-5.C.14.a.

First-degree inequality in two variables

#### 11-5.C.14.b.

Second-degree inequality in two variables

#### 11-5.C.2.

Recognizes or constructs

#### 11-5.C.2.a.

Relations or formulas

#### 11-5.C.2.b.

Inequality relations and first-degree inequalities in one variable

#### 11-5.C.3.

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

#### 11-5.C.4.

Represents a situation using

#### 11-5.C.4.a.

A first-degree equation with one unknown

#### 11-5.C.4.b.

A first-degree inequality with a variable

#### 11-5.C.5.a.

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

#### 11-5.C.5.b.

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

#### 11-5.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

#### 11-5.C.7.

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

#### 11-5.C.8.

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

#### 11-5.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)

#### 11-5.D.1.

Determines whether a situation may be translated by a system of

#### 11-5.D.2.

Translates a situation algebraically or graphically using a system of

#### 11-5.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

#### 11-5.D.3.b.

Of first-degree equations in two variables

#### 11-5.D.3.c.

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

#### 11-5.D.3.e.

Involving various functional models (mostly graphical solutions)

#### 11-5.D.4.a.

Of first-degree inequalities in two variables

#### 11-5.D.4.b.

Involving various functional models (mostly graphical solutions)

#### 11-5.D.5.

Validates the solution, with or without technological tools

#### 11-5.D.6.

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

#### 11-5.E.1.

Analyzes a situation to be optimized

#### 11-5.E.1.a.

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

#### 11-5.E.1.b.

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

#### 11-5.E.1.c.

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

#### 11-5.E.1.d.

Recognizing and defining the function to be optimized

#### 11-5.E.2.

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

#### 11-5.E.2.a.

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

#### 11-5.E.2.b.

Validating and interpreting the optimal solution, depending on the context

#### 11-5.E.2.c.

Justifying the solution(s) chosen

#### 11-5.E.2.d.

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

#### 11-6.A.1.

Identifies patterns in various situations and in various forms

#### 11-6.A.2.

Analyzes situations using different registers (types) of representation

#### 11-6.A.3.

Represents a situation generally using a graph

#### 11-6.A.5.

Recognizes relations, functions and inverses

#### 11-6.A.6.

Describes, in the functions under study, the role of

#### 11-6.A.6.a.

Multiplicative parameters

#### 11-6.A.7.

Performs operations on functions (including composition)

#### 11-6.B.1-9.a.

Polynomial functions of degree 0 or 1

#### 11-6.B.1-9.b.

Second-degree polynomial functions

#### 11-6.B.1-9.b.ii.

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

#### 11-6.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)

#### 11-6.B.1-9.c.ii.

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

#### 11-6.B.1-9.d.i.

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

#### 11-6.B.1-9.d.ii.

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

#### 11-6.B.1-9.k.i.

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

#### 11-6.B.1-9.k.ii.

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

#### 11-6.B.1-9.k.iii.

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

#### 11-6.B.1.

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

#### 11-6.B.2.

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

#### 11-6.B.3.

Represents and interprets the inverse

#### 11-6.B.4.

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

#### 11-6.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

#### 11-6.B.6.

Determines values or data by solving equations and inequalities

#### 11-6.B.8.

Compares situations or graphical representations

#### 11-6.B.9.

Makes decisions, if necessary, depending on the context

#### 11-7.A.11.

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

#### 11-7.A.12.

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

#### 11-7.A.13.

Uses fractions, decimals or percentages to quantify a probability

#### 11-7.A.14.

Recognizes that a probability is always between 0 and 1

#### 11-7.A.9.

Enumerates the possible outcomes of a random experiment using

#### 11-7.A.9.b.

Networks, tables, diagrams, Venn diagrams

#### 11-7.B.1.

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

#### 11-7.B.10.

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

#### 11-7.B.11.

Determines the odds for or odds against

#### 11-7.B.12.

Interprets and makes decisions with respect to the odds obtained

#### 11-7.B.13.

Calculates mathematical expectation

#### 11-7.B.15.

Interprets the resulting mathematical expectation and makes appropriate decisions

#### 11-7.B.4.

Calculates the probability of an event

#### 11-7.B.5.

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

#### 11-7.B.6.

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

#### 11-7.B.8.

Calculates conditional probabilities

#### 11-7.B.9.

Interprets probabilities and makes appropriate decisions

#### 11-8.A.1.

Conducts a survey or a census

#### 11-8.A.1.a.

Formulates questions for a survey

#### 11-8.A.1.d.

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

#### 11-8.A.10.

Calculates and interprets an arithmetic mean

#### 11-8.A.11.

Determines and interprets

#### 11-8.A.11.a.

Measures of central tendency: mode, median, weighted mean

#### 11-8.A.11.b.

Measures of dispersion:

#### 11-8.A.12.

Chooses the appropriate statistical measures for a given situation

#### 11-8.A.2.

Recognizes possible sources of bias

#### 11-8.A.3.

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

#### 11-8.A.4.

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

#### 11-8.A.5.

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

#### 11-8.A.8.

Understands and calculates the arithmetic mean

#### 11-8.A.9.

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

#### 11-8.B.1.

Compares experimental and theoretical data

#### 11-8.B.10.

Compares two-variable distributions

#### 11-8.B.3.

Associates the most appropriate functional model with a scatter plot :

#### 11-8.B.3.a.

First-degree polynomial function

#### 11-8.B.9.

Interpolates or extrapolates values using

#### 11-8.B.9.b.

The functional model best suited to the situation

#### 11-9.A.10.

Justifies statements using definitions or properties of plane figures

#### 11-9.A.5.

Recognizes and names regular convex polygons

#### 11-9.A.6.

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

#### 11-9.A.7.

Describes circles and sectors

#### 11-9.A.8.

Recognizes and draws main segments and lines

#### 11-9.A.8.a.

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

#### 11-9.A.9.

Identifies the properties of plane figures using geometric transformations and constructions

#### 11-9.B.4.a.

Vertex, edge, base, face

#### 11-9.B.4.b.

Altitude, apothem, lateral face

#### 11-9.B.5.

Tests Euler's relation on convex polyhedrons

#### 11-9.B.7.

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

#### 11-9.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])

#### 11-9.C.2.

Identifies properties and invariants resulting from geometric constructions and transformations

#### 11-9.C.4.

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

#### 11-9.C.5.

Recognizes dilatation with a positive scale factor

#### 11-9.C.6.

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

#### 11-9.D.2.

Recognizes congruent or similar figures

#### 11-9.D.4.

Determines the properties and invariants of congruent or similar figures

#### 11-9.D.5.

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

#### 11-9.D.6.

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

#### 11-9.D.7.

Recognizes equivalent figures (plane figures or solids)

#### 11-9.D.8.

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

#### 11-xxxx

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