Mathematics, Secondary 4 (Science)

Defines the concept absolute value in context (e.g. difference between two numbers, distance between two points)

I can define absolute value as a distance, write its definition by cases, and use |a − b| as the distance between two numbers in a scientific situation.

Identifies the coefficient, degree, terms, constant term and like terms of an algebraic expression

I can name the terms, coefficients, degree, constant term and like terms of a polynomial, and justify why like terms combine.

Interprets parameters (multiplicative or additive) and describes the effect of changing their values

I can interpret each parameter of a rule in its scientific situation and describe, and justify in general, the effect of changing its value.

Multiplies algebraic expressions

I can multiply polynomials of any degree, organize the partial products, and check an expansion with a value of the variable.

Analyzes second-degree polynomial functions of the form f(x) = ax² + bx + c, f(x) = a(b(x – h))² + k or f(x) = a(x – x₁)(x – x₂)

I can analyze a second-degree function in general, standard and factored form, switch between the forms, and find its rule from its vertex or its zeros and one other point.

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

I can divide a polynomial by another polynomial, write the result with its remainder, and verify it with P = D × Q + R.

Validates the solution, with or without technological tools

I can validate the solution of a system of equations by substitution, by a graph or a table made with technology, and by the meaning of the situation.

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

I can construct the algebraic expression of a quantity from words, a table of values, a graph or a figure, and prove that two constructions of the same quantity are equivalent.

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

I can factor polynomials by grouping, including second-degree trinomials that decompose, and explain why the method works.

Factors by completing the square (switching from one type of notation to another)

I can complete the square to rewrite ax² + bx + c in standard form, and use it to factor and to switch between the forms of a second-degree expression.

Factors trinomials of the form ax² + bx + c using formulas

I can factor ax² + bx + c with the formulas x₁, x₂ = (−b ± √(b² − 4ac)) ÷ 2a and ax² + bx + c = a(x − x₁)(x − x₂), and decide from the discriminant whether it factors.

Factors by substituting second-degree algebraic identities (perfect square trinomial and difference of two squares)

I can recognize and factor perfect square trinomials and differences of two squares, and prove the identities they come from.

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

I can interpret the solution of a system in a scientific situation, say what it means and what it does not, and draw a justified general conclusion.

Describes the role of multiplicative parameters in the functions under study

I can describe the role of the multiplicative parameters a and b in the functions under study, recognize them in a rule, and predict how each one changes the graph.

Describes the role of additive parameters in the functions under study

I can describe the role of the additive parameters h and k in the functions under study, recognize them in a rule, and use them to place a graph.

Determines values or data by solving equations and inequalities

I can find unknown values in a scientific situation by solving first- and second-degree equations and inequalities, and state the solution set with its reasons.

Recognizes or constructs equivalent algebraic expressions

I can recognize and construct equivalent algebraic expressions, and prove an equivalence with algebraic properties rather than with examples.

Interpolates and extrapolates data, if applicable

I can interpolate and extrapolate with a function that models experimental data, and state when an extrapolation can be trusted.

Solves exponential, logarithmic and square root equations and inequalities using the properties of exponents, logarithms and radicals

I can solve exponential, logarithmic and square-root equations and inequalities with the properties of exponents, logarithms and radicals, and justify every solution I keep.

Solves a second-degree equation in two variables

I can solve a second-degree equation in two variables for one variable when the other is known, describe its solution set as the points of a curve, and justify the number of solutions.

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

I can validate a solution of an equation or an inequality by substituting it into the original statement, with or without technology, and explain why validation catches false solutions.

Analyzes a second-degree inequality in two variables

I can graph the solution region of a second-degree inequality in two variables, check whether points belong to it, and justify which side of the parabola is shaded.

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

I can interpret the solutions of an equation or an inequality in a scientific situation, keep those the situation allows, and state a general conclusion.

Solves systems of first-degree equations in two variables

I can solve a system of two first-degree equations by comparison, substitution or elimination, choose the most efficient method, and prove when the system has no solution or infinitely many.

Solves systems of second-degree equations in relation to conics, changing variables if applicable

I can solve a system made of second-degree equations, such as a line and a circle or two parabolas, by substitution or a change of variable, and interpret its 0, 1, 2 or more solutions.

Represents and interprets the inverse of a function

I can represent the inverse of a function in a table, a graph and a rule, interpret it in a scientific situation, and decide when the inverse is itself a function.

Analyzes a situation to be optimized using a system of first-degree inequalities

I can translate the constraints of a situation into a system of first-degree inequalities, draw the bounded or unbounded polygon of constraints, compute its vertices exactly and define the function to be optimized.

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

I can model a scientific situation with a first-degree, second-degree or step function, moving between words, a rule, a table, a graph and a scatter plot, and justify the choice of model.

Analyzes step functions

I can analyze a step function: write it by pieces, draw its graph with correct open and closed dots, state its domain and range, and compare the greatest-integer, truncation and rounding functions.

Analyzes greatest-integer functions of the form f(x) = a[b(x – h)] + k

I can graph and analyze f(x) = a[b(x − h)] + k, explaining from its parameters the width and height of its steps, the side of the closed dots and its properties.

Represents data using a scatter plot or a double-entry (two-variable) distribution table

I can represent two-variable data in a scatter plot with suitable scales and in a double-entry table with chosen classes, and say what each representation shows best.

Models two-variable data using a first-degree polynomial function

I can model two-variable data with a first-degree function when the scatter plot shows a linear trend, find its rule and interpret its parameters in the experiment's units.

Describes and interprets the relationship between two variables, if any

I can describe the relationship between two variables from a scatter plot, by its direction, its form and its strength, and interpret it in the situation without claiming a cause.

Gives a qualitative assessment of a linear correlation

I can give a qualitative assessment of a linear correlation, its direction and its strength in words, by judging how closely the points follow a straight line.

Approximates and interprets the linear correlation coefficient

I can approximate the linear correlation coefficient with the rectangle method, compute it with technology, and interpret its sign and its size.

Represents a regression line algebraically or graphically

I can represent a regression line graphically and algebraically, by the Mayer method, the median-median method or technology, and use its equation.

Interpolates or extrapolates values using a regression line

I can use a regression line to interpolate and extrapolate values, and state the conditions under which each estimate is reliable.

Compares two-variable distributions

I can compare two-variable distributions by the direction, form and strength of their relationships and by the rates of their regression lines, and draw a justified conclusion.

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

I can state and justify the minimum conditions for congruent triangles (SSS, SAS, ASA) and for similar triangles (AA, SAS, SSS), and show why other combinations are not enough.

Determines measures of segments or perimeters in equivalent figures

I can find unknown segments and perimeters of figures equivalent to a given figure, and prove which equivalent figure has the least perimeter.

Finds unknown measurements using metric or trigonometric relations

I can find unknown lengths in triangles by choosing the right metric or trigonometric relation, and justify each step with the relation used.

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

I can write a proof that two triangles are congruent or similar using minimum conditions, and use the correspondence to find unknown measurements.

Determines the area of equivalent figures

I can determine the area of figures equivalent to a given figure, find their unknown dimensions, and prove equivalences between figures.

Determines the volume of equivalent solids

I can determine the volume of solids equivalent to a given solid, find their unknown dimensions, and show that equivalent solids can differ in total area.

Recognizes equivalent figures (plane figures or solids)

I can recognize equivalent plane figures and equivalent solids by computing or by reasoning, and distinguish equivalence from congruence and similarity.

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

I can explore a metric relation in plane figures, state it as a conjecture, and prove it in general with similar triangles, the Pythagorean relation or areas.

Finds unknown measurements using the trigonometric ratios sine, cosine and tangent

I can find unknown sides and angles of right triangles with the sine, cosine and tangent ratios, and prove relations between the ratios.

Finds unknown measurements using the cosine law

I can prove the cosine law, use it to find sides and angles of any triangle, and choose between the cosine law and the sine law.

Calculates the area of a triangle given the measure of an angle and the lengths of two sides or given the measures of two angles and the length of one side

I can calculate the area of a triangle from two sides and the included angle or from two angles and a side, and prove the formula A = ½ab sin C.

Calculates the distance between two points

I can calculate the distance between two points of the Cartesian plane and use distances to prove properties of figures.

Calculates and interprets a slope

I can calculate the slope of a segment or a line, interpret it as a rate of change in the situation's units, and use slopes to prove that points are collinear.

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

I can determine whether two lines meet at one point, are perpendicular, are parallel and distinct, or are coincident, from their slopes and y-intercepts, and justify the conclusion.

Represents straight lines graphically and algebraically

I can represent a line graphically and algebraically in functional, general and symmetric form, convert between the forms, and justify each conversion.

Represents a half-plane graphically and algebraically

I can represent a half-plane by an inequality and a graph, decide its boundary and its side, and prove which side satisfies the inequality.

Represents parallel lines and perpendicular lines

I can represent parallel and perpendicular lines graphically and algebraically, and prove the slope conditions that characterize them.

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

I can determine the equation of a line from its slope and a point or from two points, check it, and interpret its parameters.

Determines the equation of a line parallel or perpendicular to another

I can determine the equation of a line parallel or perpendicular to a given line through a point, and use it to find the distance from a point to a line.