Mathematics, Secondary 5 (Technical and Scientific)

Unit 1: Arithmetic

The PDA Arithmetic branch, taught in the Technical and Scientific sequence register (mathematics tied to technical instruments, measurement and case studies (Quebec TS program: "design, operation or use of certain technical instruments", identifying errors and anomalies)).

• Writes numbers in logarithmic notation using the equivalence log_a x = n ⇔ a^n = x

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

Unit 2: Algebra

The PDA Algebra branch, taught in the Technical and Scientific sequence register (mathematics tied to technical instruments, measurement and case studies (Quebec TS program: "design, operation or use of certain technical instruments", identifying errors and anomalies)).

• Interprets the role of a parameter in an algebraic expression
• Solves systems of first-degree inequalities in two variables
• Solves inequalities involving various functional models (mostly graphical solutions)
• Interprets parameters (multiplicative or additive) and describes the effect of changing their values

• Factors by completing the square (switching from one type of notation to another)
• Manipulates expressions involving additive parameters
• Determines values or data by solving equations and inequalities

• Solves second-degree equations and inequalities in one variable
• Solves exponential, logarithmic and square root equations and inequalities using the properties of exponents, logarithms and radicals
• Solves rational equations and inequalities in one variable
• Solves first-degree trigonometric equations involving a sine, cosine or tangent expression

• Solves a second-degree equation in two variables

• Analyzes a second-degree inequality in two variables

• 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 composed of a first-degree equation and a second-degree equation in two variables
• Solves systems involving various functional models (mostly graphical solutions)
• Represents and interprets the inverse of a function

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

• 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)

• 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 greatest-integer 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: Geometry

The PDA Geometry branch, taught in the Technical and Scientific sequence register (mathematics tied to technical instruments, measurement and case studies (Quebec TS program: "design, operation or use of certain technical instruments", identifying errors and anomalies)).

• Recognizes equivalent figures (plane figures or solids)
• Determines the correspondence between degrees and radians

• Defines the concept of radian
• Determines the area of equivalent figures
• Determines the volume of equivalent solids
• Analyzes and models situations using vectors (e.g. displacements, forces, speeds or velocities)

• Determines measures of segments or perimeters in equivalent figures

• Determines, through exploration or deduction, different metric relations associated with plane figures
• Defines a vector: magnitude (length or norm), direction, sense

• Finds unknown measurements using the cosine law
• Represents a vector graphically (arrow in a plane or pair in a Cartesian plane)

• Determines measures in a circle: arcs, chords, inscribed angles, interior angles and exterior angles

• 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

• Identifies properties of vectors

Unit 4: Analytic Geometry

The PDA Analytic Geometry branch, taught in the Technical and Scientific sequence register (mathematics tied to technical instruments, measurement and case studies (Quebec TS program: "design, operation or use of certain technical instruments", identifying errors and anomalies)).

• Identifies, through observation, the characteristics of geometric transformations in the Cartesian plane: translations, rotations centred at the origin, reflections with respect to the x-axis and y-axis, dilatations centred at the origin, scaling (expansions and contractions)
• Describes, represents and constructs geometric loci in the Euclidian and Cartesian planes, with or without technological tools
• Establishes the relationship between trigonometric ratios and the standard unit circle (trigonometric ratios and lines)

• Defines algebraically the rule for a geometric transformation
• 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°)

• Constructs, in the Cartesian plane, the image of a figure using a transformation rule
• 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
• Describes and represents the circle, ellipse and hyperbola resulting from a translation
• Analyzes and uses periodicity and symmetry to determine coordinates of points associated with significant angles in the standard unit circle

• Anticipates the effect of a geometric transformation on a figure
• Finds the points of intersection between a line and a conic
• Proves Pythagorean identities

Unit 5: Discrete Mathematics

The PDA Discrete Mathematics branch, taught in the Technical and Scientific sequence register (mathematics tied to technical instruments, measurement and case studies (Quebec TS program: "design, operation or use of certain technical instruments", identifying errors and anomalies)).

• Represents, interprets data using matrices

• Performs operations on matrices: addition and subtraction, multiplication with a scalar and with another matrix

• Performs geometric transformations (transformation matrices)

• Solves systems of equations (augmented matrix)