High School Block:

Foundations of Mathematics 110

Foundations of Mathematics 110 provides foundations to more complex mathematical reasoning and includes the study of philosophical or logical aspects of mathematics. Learners will create mathematical statements and communicate their validity using reasoning skills; think critically and apply knowledge to solve geometric problems; connect algebraic equations and their visual representations on a graph using linear and quadratic functions; develop a personal approach to finances using number sense; and explore how technologies can support mathematical endeavours. Learners will enact and apply prior knowledge of factoring, linear graphs, trigonometric ratios, and properties of parallel lines. This course develops pathways to further studies in mathematics and/or preparatory skills for calculus.

Topics include numerical and logical reasoning; angles and triangles; sine and cosine laws; systems of linear inequalities; quadratic functions; renting and buying; and investment portfolios.

Geometry, Measurement, and Finance 10; Number, Relations, and Functions 10 are recommended prior to enrolling in Foundations of Mathematics 110.

For Educators: Foundations of Mathematics 110 Companion Document

CONTEXTS AND CONCEPTS

Synthesis

Strategies

  • Using algorithms, mental procedures, technology/tools, and other strategies
  • Determining appropriate units of measure and precision
  • Using the most efficient strategies
  • Determining the reasonableness of the answer and explaining thinking
  • Verifying solutions with substitution
  • Simplifying
  • Comparing

Processes

  • Deriving mathematical rules and algorithms
  • Using calculators
  • Modeling and using simulations
  • Extrapolation and interpolation
  • Deductive and inductive reasoning
  • Linear optimization/programming
  • Working with vertexes, axis of symmetry, maximums and minimums, domain, range, roots, and intercepts
  • Factoring and the quadratic formula
  • Working with appreciation, depreciation, leasing, and renting concepts
  • Working with bond, investment certificate, and share concepts
  • Working with budget, fixed and variable expenses concepts

Fluency

  • Conversion between formats, representations, and equivalents of numbers
  • Ways to present information/data
  • Similarity
  • Scale
  • Working with oblique triangles

Communication

  • Using variable terms and constant terms
  • Using coefficients, exponents, powers, and bases
  • Working with inequalities
  • Polynomial expressions
  • Using degrees of terms and degrees of polynomials
  • Using titles, labels for axis, values
  • Conjectures and counterexamples
  • Two-column proof formats
  • Introducing terms constraint and feasible region
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GRADE 11

Strand: Relations and Functions

Big Idea: Algebraic and Graphical Reasoning

Skill Descriptor: Solve problems involving systems of linear inequalities in two variables.

Global Competencies: CTPS

Achievement Indicators:

  • Graph linear inequalities by justifying elements of graphing
  • Model systems of linear inequalities in two variables
  • Graph systems of linear inequalities
  • Determine the solution region in systems of linear inequalities
  • Verify the solution of systems of linear inequalities
  • Analyze the solution region of systems of linear inequalities
  • Solve linear optimization problems

Skill Descriptor: Determine the vertex, intercepts, domain and range, axis of symmetry, of quadratic functions.

Global Competencies: CTPS

Achievement Indicators:

  • Determine the intercepts of graphs of quadratic functions
  • Determine the roots of quadratic equations by factoring
  • Determine the roots of quadratic equations by applying the quadratic formula
  • Represent graphs of quadratic functions that have zero, one and two 𝑥-intercepts
  • Express quadratic equations in factored form, using the zeros of a corresponding function or the 𝑥-intercepts of the graphs
  • Determine the coordinates of the vertex of graphs of quadratic functions
  • Determine the equation of the axis of symmetry of graphs of quadratic functions, given the 𝑥-intercepts of the graphs
  • Determine if the 𝑦-coordinate of vertexes are maximums or minimums
  • Determine the domain and range of quadratic functions
  • Draw the graphs of quadratic functions
  • Solve applied mathematics problems that involve characteristics of quadratic functions

Strand: Shape and Space

Big Idea: Geometry

Skill Descriptor: Derive proofs that involve the properties of angles and triangles.

Global Competencies: CTPS

Achievement Indicators:

  • Generalize relationships between pairs of angles formed by transversals and parallel lines using inductive reasoning
  • Construct parallel lines and transversals
  • Prove properties of angles formed by transversals and parallel lines in triangles using deductive reasoning
  • Prove relationships between sides and/or angles using congruent triangle properties using deductive reasoning
  • Generalize a rule for the relationship between the sum of the interior angles and the number of sides (n) in polygons using inductive reasoning
  • Identify and correct errors in given proofs of properties involving angles and/or congruent triangles

Skill Descriptor: Solve problems using the properties of angles and triangles.

Global Competencies: CTPS

Achievement Indicators:

  • Apply angle properties to determine if two lines are parallel
  • Analyze parallel lines and a transversal to solve or verify angle measurements

Skill Descriptor: Solve problems involving the cosine law and the sine law.

Global Competencies: CM, CTPS

Achievement Indicators:

  • Represent problems involving the cosine law or sine law
  • Explain the steps in given proofs of the sine law or cosine law
  • Apply the cosine law and sine law
  • Recognize and apply the ambiguous case of the sine law

Strand: Logical Reasoning

Big Idea: Conjectures

Skill Descriptor: Analyze and prove conjectures using logical reasoning.

Global Competencies: CM ,CTPS

Achievement Indicators:

  • Create and justify conjectures by observing patterns
  • Determine and justify if given arguments are valid
  • Create counterexamples to prove conjectures false
  • Compare and contrast inductive and deductive reasoning using examples
  • Prove algebraic and number relationships
  • Prove conjectures using deductive reasoning
  • Identify errors in given proofs

Strand: Number

Big Idea: Financial Applications

Skill Descriptor: Analyze costs and benefits of renting, leasing and buying.

Global Competencies: CM, CTPS, ICE, SASM

Achievement Indicators:

  • Identify and describe examples of assets that appreciate or depreciate
  • Compare, renting, leasing, and buying and justify personal choice given specific situations
  • Manipulate formulae involving renting, leasing or buying to solve problems
  • Complete cost-and-benefit analysis using technology

Skill Descriptor: Analyze investment portfolios.

Global Competencies: CTPS, ICE, SASM

Achievement Indicators:

  • Compare and contrast given investment portfolios
  • Determine the total value of investments when there are regular contributions to the principal using technology
  • Graph and compare the total value of investments with and without regular contributions
  • Apply the Rule of 72 to solve investment problems
  • Determine possible investment strategies to achieve financial goals using technology
  • Compare and contrast long-term and short-term investment options
  • Solve investment problems

Skill Descriptor: Solve problems that involve personal budgets.

Global Competencies: CM, CTPS, ICE, SASM

Achievement Indicators:

  • Identify income and expenses that should be included in personal budgets
  • Explain considerations that must be made when developing personal budgets
  • Create personal budgets based on given income and expense data to achieve personal goals
  • Analyze unexpected events related to personal budgets
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