Mathematics, Secondary 5 (Science)

Unit 1: Arithmetic

The PDA Arithmetic branch, taught in the Science (Natural Science) sequence register (formal mathematics as the object of study (Quebec Science program: "develop formal proofs... to confirm a truth", properties of mathematical objects, abstract thinking)).

• Writes numbers using radicals or rational exponents

• Estimates the value of the power of an exponential expression with respect to its components: base (between 0 and 1, greater than 1), exponent (positive or negative, integral or fractional)

• Estimates the order of magnitude of a real number using scientific notation

• Manipulates numerical expressions involving powers (change of base), exponents and radicals (nth root), using their properties
• Manipulates numerical expressions involving logarithms
• Manipulates numerical expressions involving absolute values

Unit 2: Algebra

The PDA Algebra branch, taught in the Science (Natural Science) sequence register (formal mathematics as the object of study (Quebec Science program: "develop formal proofs... to confirm a truth", properties of mathematical objects, abstract thinking)).

• Factors trinomials of the form ax² + bx + c using formulas
• Determines values or data by solving equations and inequalities

• Solves rational equations and inequalities in one variable
• Solves absolute value equations and inequalities in one variable
• Solves trigonometric equations that can be expressed as a sine, cosine or tangent function

• Solves inequalities when analyzing a situation to be optimized
• Models a situation verbally, algebraically, graphically, using a table of values or a scatter plot

• Uses inequalities when optimizing a situation
• Optimizes a situation by taking into account different constraints and makes decisions with respect to this situation determining the best solution(s) for a particular situation, given a set of possibilities validating and interpreting the optimal solution, depending on the context justifying the solution(s) chosen changing certain conditions associated with the situation to provide a more optimal solution, if necessary
• Finds the rule of a function or its inverse, depending on the context

• Solves systems of second-degree equations in relation to conics, changing variables if applicable
• Represents and interprets the inverse of a function

• Solves systems of first-degree inequalities in two variables
• Interprets parameters (multiplicative or additive) and describes the effect of changing their values

• Performs operations on functions (including composition)
• Interpolates and extrapolates data, if applicable

• Analyzes square root functions of the form f(x) = a√b(x – h) + k

• Analyzes rational functions of the form f(x) = (ax + b)/(cx + d)

• Analyzes exponential functions of the form f(x) = a·c^(b(x – h)) + k

• Analyzes logarithmic functions of the form f(x) = a log_c b(x – h) + k

• Analyzes piecewise functions

• Analyzes absolute value functions of the form f(x) = a|b(x – h)| + k

• Analyzes sinusoidal functions of the form f(x) = a sin b(x – h) + k or f(x) = a cos b(x – h) + k
• Analyzes tangent functions of the form f(x) = a tan b(x – h) + k

Unit 3: Statistics

The PDA Statistics branch, taught in the Science (Natural Science) sequence register (formal mathematics as the object of study (Quebec Science program: "develop formal proofs... to confirm a truth", properties of mathematical objects, abstract thinking)).

• Interpolates or extrapolates values using the functional model best suited to the situation

Unit 4: Geometry

The PDA Geometry branch, taught in the Science (Natural Science) sequence register (formal mathematics as the object of study (Quebec Science program: "develop formal proofs... to confirm a truth", properties of mathematical objects, abstract thinking)).

• Defines the concept of radian
• Analyzes and models situations using vectors (e.g. displacements, forces, speeds or velocities)

• Determines the correspondence between degrees and radians

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

• Proves trigonometric identities by using algebraic properties, definitions (sine, cosine, tangent, cosecant, secant, cotangent), Pythagorean identities, and the properties of periodicity and symmetry
• Determines the resultant or projection of a vector
• Adds and subtracts vectors
• Multiplies a vector by a scalar
• Calculates the scalar product of two vectors
• Expresses a vector as a linear combination of vectors
• Applies the Chasles relation to vectors

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

• Identifies properties of vectors

• Justifies statements using properties associated with vectors

Unit 5: Analytic Geometry

The PDA Analytic Geometry branch, taught in the Science (Natural Science) sequence register (formal mathematics as the object of study (Quebec Science program: "develop formal proofs... to confirm a truth", properties of mathematical objects, abstract thinking)).

• Calculates and interprets a slope

• Analyzes and models situations involving geometric loci in the Euclidian and Cartesian planes
• Determines the coordinates of points associated with significant angles using metric relations in right triangles (Pythagorean relation, properties of angles: 30°, 45°, 60°)

• Describes and represents a parabola centred at the origin or resulting from a translation
• Describes and represents the circle, ellipse and hyperbola centred at the origin
• Analyzes and uses periodicity and symmetry to determine coordinates of points associated with significant angles in the standard unit circle

• Finds the points of intersection between a line and a conic
• Finds the points of intersection between two conics (a parabola and a conic)
• Proves Pythagorean identities