Appendix A: Principles of Math 11 Prescribed Learning Outcomes

The organizers for Principles of Math 11 are as follows:

Problem Solving
Number (Number Operations)
Patterns and Relations (Patterns)
Patterns and Relations (Variables and Equations)
Patterns and Relations (Relations and Functions)
Shape and Space (Measurement)
Shape and Space (3-D Objects and 2-D Shapes)

It is expected that students will use a variety of methods to solve real-life, practical, technical, and theoretical problems

It is expected that students will:

• solve problems that involve a specific content area such as geometry, algebra, trigonometry, statistics, probability
• solve problems that involve more than one content area
• solve problems that involve mathematics within other disciplines
• analyse problems and identify the significant elements
• develop specific skills in selecting and using an appropriate problem-solving strategy or combination of strategies chosen from, but not restricted to, the following:
  • guess and check
  • look for a pattern
  • make a systematic list
  • make and use a drawing or model
  • eliminate possibilities
  • work backward
  • simplify the original problem
  • develop alternative original approaches
  • analyse keywords
• demonstrate the ability to work individually and co-operatively to solve problems
• determine that their solutions are correct and reasonable
• clearly explain the solution to a problem and justify the processes used to solve it
• use appropriate technology to assist in problem solving

Number (Number Operations)

It is expected that students will solve consumer problems, using arithmetic operations.

It is expected that students will:

• solve consumer problems, including:
  • wages earned in various situations
  • property taxation
  • exchange rates
  • unit prices
• reconcile financial statements including:
  • cheque books with bank statements
  • cash register tallies with daily receipts
• solve budget problems, using graphs and tables to communicate solutions
• solve investment and credit problems involving simple and compound interest

It is expected that students will apply the principles of mathematical reasoning to solve problems and to justify solutions.

It is expected that students will:

• differentiate between inductive and deductive reasoning
• explain and apply connecting words, such as and, or, and not, to solve problems
• use examples and counterexamples to analyse conjectures
• distinguish between an if–then proposition, its converse and its contrapositive
• prove assertions in a variety of settings, using direct and indirect reasoning

Patterns and Relations (Variables and Equations)

It is expected that students will represent and analyse situations that involve expressions, equations and inequalities.

It is expected that students will:

• graph linear inequalities, in two variables
• solve systems of linear equations, in two variables:
  • algebraically (elimination and substitution)
  • graphically
• solve systems of linear equations, in three variables:
  • algebraically
  • with technology
• solve non-linear equations, using a graphing tool
• solve nonlinear equations:
  • by factoring
  • graphically
• use the Remainder Theorem to evaluate polynomial expressions, the Rational Zeros Theorem, and the Factor Theorem to determine factors of polynomials
• determine the solution to a system of nonlinear equations, using technology as appropriate

Patterns and Relations (Relations and Functions)

It is expected that students will:

• represent and analyse quadratic, polynomal and rational functions, using technology as appropriate
• examine the nature of relations with an emphasis on functions

It is expected that students will:

• determine the following characteristics of the graph of a quadratic function:
  • vertex
  • domain and range
  • axis of symmetry
  • intercepts
• perform operations on functions and compositions of functions
• determine the inverse of a function
• connect algebraic and graphical transformations of quadratic functions, using completing the square as required
• model real-world situations, using quadratic functions
• solve quadratic equations, and relate the solutions to the zeros of a corresponding quadratic function, using:
  • factoring
  • the quadratic formula
  • graphing
• determine the character of the real and non-real roots of a quadratic equation, using:
  • the discriminant in the quadratic formula
  • graphing
• describe, graph, and analyse polynomial and rational functions, using technology
• formulate and apply strategies to solve absolute value equations, radical equations, rational equations, and inequalities

Shape and Space (Measurement)

It is expected that students will:

• solve problems involving triangles, including those found in 3-D and 2-D applications.
• solve coordinate geometry problems involving lines and line segments, and justify the solutions.

It is expected that students will:

• solve problems involving ambiguous case triangles in 3-D and 2-D
• solve problems involving distances between points and lines
• verify and prove assertions in plane geometry, using coordinate geometry

Shape and Space (3-D Objects and 2-D Shapes)

It is expected that students will develop and apply the geometric properties of circles and polygons to solve problems.

It is expected that students will:

• use technology with dynamic geometry software to confirm and apply the following properties:
  • the perpendicular from the center of a circle to a chord bisects the chord
  • the measure of the central angle is equal to twice the measure of the inscribed angle subtended by the same arc
  • the inscribed angles subtended by the same arc are congruent
  • the angle inscribed in a semicircle is a right angle
  • the opposite angles of a cyclic quadrilateral are supplementary
  • a tangent to a circle is perpendicular to the radius at the point of tangency
  • the tangent segments to a circle, from any external point, are congruent
  • the angle between a tangent and a chord is equal to the inscribed angle on the opposite side of the chord
  • the sum of the interior angles of an n-sided polygon is 180(n-2)
• prove the following general properties, using established concepts and theorems:
  • the perpendicular bisector of a chord contains the center of the circle
  • the measure of the central angle is equal to twice the measure of the inscribed angle subtended by the same arc (for the case when the center of the circle is in the interior of the inscribed angle)
  • the inscribed angles subtended by the same arc are congruent
  • the angle inscribed in a semicircle is a right angle
  • the opposite angles of a cyclic quadrilateral are supplementary
  • a tangent to a circle is perpendicular to the radius at the point of tangency
  • the tangent segments to a circle from any external point are congruent
  • the angle between a tangent and a chord is equal to the inscribed angle on the opposite side of the chord
  • the sum of the interior angles of an n-sided polygon is 180(n-2)
• solve problems, using a variety of circle properties, and justify the solution strategy used

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Maintained by: Mathematics Coordinator
Revised: November 28, 2000

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