Course Overview & Instructional Approach
Pre-Calculus 12 explores the patterns, relationships, and structures that connect many areas of mathematics. Students investigate how functions can be represented, transformed, analyzed, and used to describe real-world situations.
The course develops algebraic reasoning, mathematical communication, and problem-solving skills while preparing students for future studies in mathematics, science, engineering, technology, business, and other quantitative disciplines.
Throughout the course, students are encouraged to look for connections between concepts and develop a deeper understanding of the mathematical relationships that unite seemingly different topics.
Course Structure
The course is organized into three major units.
Polynomials & Transformations
The first unit develops the foundational language of functions and transformations.
Students investigate how functions behave, how graphs can be transformed, and how algebraic representations relate to graphical and numerical representations.
Topics include:
- transformations of functions
- polynomial functions
- factoring techniques
- equations and inequalities
- rational expressions and equations
- function analysis
Students learn to identify key characteristics of functions, make connections between multiple representations, and develop confidence working with increasingly complex algebraic relationships.
Exponentials & Logarithms
This unit begins with patterns in geometric growth and develops the ideas that lead naturally to exponential and logarithmic functions.
Topics include:
- geometric sequences
- geometric series
- exponent laws
- exponential functions
- compound interest
- growth and decay models
- logarithmic functions
- logarithmic laws
- logarithmic equations
- applications and modelling
A recurring theme throughout this unit is the relationship between functions and their inverses. Students learn to move fluently between symbolic, graphical, numerical, and contextual representations.
Trigonometric Functions & Proofs
The final unit treats trigonometry primarily as the study of periodic functions and mathematical relationships.
Students investigate:
- the unit circle
- exact values
- solving trigonometric equations
- graphing sine and cosine functions
- transformations of trigonometric functions
- modelling periodic behavior
- trigonometric identities
- proof and verification
Particular emphasis is placed on recognizing patterns, making connections, and developing algebraic reasoning. This unit serves as a culmination of many of the skills developed throughout the course.
Learning Through Problem Solving
Assignments are an important part of the learning experience in this course.
Questions are carefully selected and organized to support conceptual understanding, develop procedural fluency, and strengthen problem-solving skills. Concepts are introduced progressively, allowing students to build confidence before applying ideas in more complex situations.
Assignments are regularly reviewed and refined to improve clarity, strengthen connections between topics, and support student learning throughout the course.
Thinking Like a Mathematician
Students are encouraged to approach mathematics as an exploration of patterns, relationships, and structure.
Throughout the course, students learn to:
- identify patterns
- analyze relationships
- make conjectures
- justify conclusions
- communicate reasoning
- recognize structure in unfamiliar situations
Students are encouraged to approach unfamiliar problems by looking for patterns, identifying relationships, and connecting new ideas to previous learning.
These habits become increasingly valuable as students progress into more advanced mathematics and related disciplines.
Mathematical Modelling
Mathematical modelling is a recurring theme throughout the course.
Students regularly learn to:
- build models
- analyze models
- solve models
- interpret solutions
Modelling provides opportunities for students to apply mathematical ideas in meaningful contexts and develop a deeper understanding of how mathematics can be used to describe relationships and make predictions.
Examples include:
- financial growth
- periodic phenomena
- population models
- optimization contexts
- real-world function analysis
Students learn that developing an appropriate mathematical model is often an important part of solving a problem.
Mathematical Communication
Students are expected to communicate mathematics clearly and precisely.
This includes:
- showing complete solutions
- using appropriate notation
- defining variables
- organizing work logically
- explaining reasoning
- verifying solutions when appropriate
A correct answer without supporting reasoning is often considered incomplete.
Communication is treated as an essential mathematical skill rather than an optional addition.
Technology
Technology is used to support mathematical understanding, exploration, and communication.
Students regularly use:
- graphing calculators
- Desmos
- spreadsheets when appropriate
Technology provides opportunities to visualize mathematical relationships, explore patterns, verify results, and investigate concepts from multiple perspectives.
Assessment Philosophy
Assessment is intended to support learning and provide students with opportunities to demonstrate understanding throughout the course.
Students receive regular opportunities to:
- practice new skills
- receive feedback
- revisit challenging concepts
- demonstrate growth over time
Assessment tasks include:
- assignments
- modelling activities
- investigations
- cumulative review opportunities
- tests
Both procedural fluency and conceptual understanding are valued throughout the course.
Preparing Students for Future Mathematics
The ultimate goal of this course extends beyond any particular unit or assessment.
Students who successfully complete this course should leave with:
- strong algebraic fluency
- confidence working with functions
- experience solving unfamiliar problems
- effective mathematical communication skills
- an appreciation for mathematical structure and patterns
Whether students continue into calculus, science, engineering, technology, business, or other fields, these skills provide a strong foundation for future learning.
Final Thoughts
Mathematics is a discipline built upon patterns, relationships, and logical reasoning.
The purpose of this course is to help students develop the ability to recognize structure, communicate ideas clearly, solve problems effectively, and approach new challenges with confidence.
Success in Pre-Calculus 12 comes from developing understanding, persistence, and mathematical reasoning that can be applied both within mathematics and beyond the classroom.