Mathematics: Advanced Level—Algebraic

Goal Statement

The goals for Advanced Algebraic Mathematics are (1) to provide students with sufficient mathematical knowledge for academic, career, and technical programs whose admission requirements include Math 11 equivalence and (2) to prepare students to enter Provincial Level mathematics courses.

Learning Outcomes

It is expected that learners will use a scientific calculator to evaluate complex expressions with emphasis on using special keys to perform a variety of functions. The use of a graphing calculator or other technology is optional.

1. Basic Algebraic Skills Review

Note: A review of the following basic algebraic skills is suggested but not required. It is expected that learners will be able to:

• perform operations with real numbers including absolute value and exponential notation
• simplify expressions using rules for order of operations including nested parentheses and properties of exponents
• translate common language into algebraic expressions
• evaluate algebraic expressions by substitution

2. Solving Linear Equations and Inequalities

It is expected that learners will be able to:

• solve first degree/linear equations in one variable
• manipulate simple formulas to isolate a specified variable
• solve and graph linear inequalities in one variable
• write set-builder and/or interval notation for the solution set or graph of an inequality
• use linear equations, formulas and linear inequalities to solve applied problems
• find the union (disjunction) and intersection (conjunction) of sets
• solve and graph compound inequalities
• solve absolute value equations

3. Graphing Relations and Functions

It is expected that learners will be able to:

• write linear equations in slope-intercept form
• graph linear equations using a table of values
• graph linear equations using the y-intercept and slope and using x- and y-intercepts
• graph horizontal and vertical lines
• find the slope of a line given two points on the line
• find the equation of a line given graphic data: the slope and y-intercept, the slope and one point, or two points on the line
• determine whether a pair of lines is parallel, perpendicular or neither
• find the equation of a line parallel or perpendicular to a given line and through a given point
• use the definition of function and the vertical line test to distinguish between functions and non-functions
• use and interpret function notation to evaluate functions for given x-values and find x-values for given function values
• determine the domain and range of a function
• use a table of values to graph linear functions and non-linear functions such as quadratic, cubic, square root, reciprocal, and absolute value functions

Optional Outcomes:

• graph exponential functions
• analyze functions to determine line of symmetry, vertices, asymptotes, and intercepts
• understand and demonstrate transformations in graphs resulting from the following changes in the defining equation: translation, reflection, dilation
• use a graphing calculator or other appropriate technology to graph equations
• identify an appropriate graph for a given relation
• develop a model function from a given graph or set of data
• perform linear regression using a graphing calculator to fit a linear function to data
• graph linear inequalities in two variables

4. Systems of Linear Equations and Inequalities

It is expected that learners will be able to:

• solve systems of linear equations in two variables by graphing, substitution and elimination methods
• determine if a system of equations will have one, infinite, or no solution(s)
• use systems of linear equations to solve applied problems

Optional Outcomes:

• solve systems of linear equations in three variables and applied problems using such systems
• graph the solution for a system of linear inequalities in two variables
• use a graphing calculator or other appropriate technology to solve systems of linear equations and inequalities

5. Polynomial Expressions, Equations and Functions

It is expected that learners will be able to:

• identify the degree, terms and coefficients of a polynomial
• distinguish between monomials, binomials, trinomials, and other polynomials
• add, subtract, multiply polynomials
• divide polynomials by monomials
• factor polynomials using an appropriate strategy or a combination of techniques: common factors, difference of squares, difference and sum of cubes, perfect square trinomials, trial/error, or grouping
• solve polynomial equations using the principle of zero products
• solve applied problems using polynomial equations/ functions

Optional Outcomes:

• divide polynomials by binomials using long division
• divide polynomials and binomials using synthetic division

6. Variation, Rational Expressions, and Equations

It is expected that learners will be able to:

• identify situations and find values for which a rational expression will be undefined
• simplify rational expressions
• add, subtract, multiply and divide rational expressions
• solve rational equations
• manipulate formulas involving rational expressions to isolate a specified variable
• solve applied problems that can be modeled with rational equations
• simplify complex rational expressions
• express variations in the form of equations (direct, inverse, joint, combined)
• solve problems involving direct, inverse, joint and combined variation

7. Radical Expressions and Equations

It is expected that learners will be able to:

• identify situations and find values for which a radical expression will be undefined
• write radicals as powers with rational exponents and vice versa
• use rational exponents to simplify radical expressions
• simplify, add, subtract, multiply and divide radical expressions (numeric or algebraic)
• rationalize denominators containing radicals (including the use of conjugates)
• solve equations involving radical expressions or powers with rational exponents and check for extraneous roots
• manipulate formulas involving powers and square roots to isolate a specified variable
• solve applied problems which can be modeled by radical equations, and determine if solutions are reasonable given the context of the problem

Optional Outcomes:

• identify imaginary and complex numbers and express them in standard form
• add, subtract, multiply, and divide complex numbers

8. Quadratic Equations and Functions

It is expected that learners will be able to:

• solve quadratic equations by factoring, principle of square roots, completing the square and the quadratic formula
• use the discriminant to identify the number and type of solutions of a quadratic equation
• write a quadratic equation given its solutions
• solve rational and radical equations reducible to a quadratic pattern and check that answers are reasonable
• solve selected polynomial equations that can be factored simplifying to linear and/or quadratic factors
• graph quadratic functions of the form f(x) = a(x -h)² + k and demonstrate translations, reflections and stretching/shrinking resulting from changes in the function equation
• find the vertex, line of symmetry, minimum or maximum values, x– and y-intercepts, domain and range, given the function f(x) = a(x -h)² + k
• rewrite f(x) = ax² + bx + c as f(x) = a(x -h)² + k by completing the square
• solve problems that can be modeled using quadratic equations such as maximum and minimum problems

Optional Outcomes:

• solve quadratic equations having complex number solutions
• use a graphing calculator or other appropriate technology to graph and solve quadratic equations
• solve quadratic inequalities by graphing
• solve polynomial and rational inequalities algebraically

9. Trigonometry

It is expected that learners will be able to:

• label the sides of a right triangle with respect to a given angle
• determine sine, cosine, and tangent ratios of an angle in a right triangle using the side lengths
• use a scientific calculator to find the trigonometric value for a given angle and to find an angle given its trigonometric value
• solve right triangles and applied problems using the basic trigonometric ratios, the Pythagorean theorem, and sum of the angles (180°)
• use the Law of Sines and the Law of Cosines to solve non-right (oblique) triangles and applied problems

Optional Outcomes:

• use ^1/2 bcsinA to find the area of a triangle
• determine the quadrant for positive and negative angles in standard position
• identify coterminal angles
• determine primary trigonometric function values for angles in standard position
• identify reference angles
• evaluate primary trigonometric functions for any angle in a variety of conditions
• solve trigonometric equations involving the primary functions over a specific domain
• use the trigonometric definitions to deduce unknown trigonometric values from given values

10. Optional Topics

Learners may wish to complete either A or B but these outcomes are not required.

A. Geometry

• recall the properties of parallel lines, similar and congruent figures, polygons, angle relationships, angle measurements, and basic compass and straightedge construction
• demonstrate an understanding of the following properties of a circle:
  • the perpendicular bisector of a chord passes through the centre of the circle
  • the line joining the midpoint of a chord to the centre is perpendicular to the chord
  • the line through the centre, perpendicular to a chord, bisects the chord
  • central angles containing equal chords or arcs are equal (the converse is also true)
  • inscribed angles containing the same or equal chords (on the same side of chord) or arcs are equal
  • an inscribed angle equals half the central angle containing the same or equal chords (on the same side of chord) or arcs are equal
  • an inscribed angle in a semicircle measures 90°
  • opposite angles of a cyclic (inscribed) quadrilateral are supplementary
  • a tangent is perpendicular to the radius at the point of contact (the converse is also true)
  • tangents from an external point are equal
  • the angle between a chord and tangent equals the inscribed angle of the opposite side of the chord (the converse is also true)
• demonstrate and clearly communicate deductive reasoning in the solution of applied problems

B. Data Analysis

• explain the uses and misuses of statistics
• demonstrate an understanding of mean, median, mode, range, quartiles, percentiles, standard deviation, the normal curve, z-scores, sampling error and confidence intervals
• graphically present data in the form of frequency tables, line graphs, bar graphs, and stem and leaf plots
• design and conduct a statistics project, analyze the data, and communicate the outcomes