Course Overview & Instructional Approach
Pre-Calculus 11 develops the algebraic reasoning and mathematical fluency students need to work with increasingly complex mathematical relationships. Students extend their understanding of numbers, expressions, equations, functions, and trigonometry while learning to recognize the structures and patterns that connect these areas of mathematics.
A major theme throughout the course is the movement from arithmetic toward more abstract algebraic thinking. Students learn not only how to carry out mathematical procedures, but also how to select appropriate strategies, explain their reasoning, connect different representations, and determine whether solutions are reasonable.
The course provides an important foundation for Pre-Calculus 12 and future studies in mathematics, science, engineering, technology, business, and other quantitative disciplines.
Course Structure
The BC curriculum identifies the mathematical content and curricular competencies students are expected to develop but does not prescribe a particular unit sequence. This course organizes that content into two major threads, Expressions & Equations.
Introductory Skills
The course begins by strengthening the algebraic skills that students will use throughout Pre-Calculus 11. Students are expected to be familiar with, if not fluent in:
- the distributive property
- the exponent laws
- expanding and factoring polynomials
- solving linear equations
- substitution and function notation
- graphing and interpreting basic functions
The goal of this review is not simply to repeat previous mathematics, but to establish the fluency students need before working with more complex algebraic relationships.
Expressions
This unit focuses on how mathematical relationships can be represented, simplified, and understood in different forms.
Students extend their understanding of the real number system, exponents, radicals, polynomial factoring, and rational expressions, with an emphasis on recognizing equivalent forms and choosing useful representations.
They also explore quadratic functions through their equations and graphs, making connections between standard, factored, and vertex form and the information each representation provides.
Trigonometry is introduced through angles in standard position, the unit circle, and trigonometric ratios, helping students connect geometric relationships with numerical and algebraic representations. Financial mathematics provides another application of this thinking as students interpret expressions describing compound growth, investments, loans, and regular payments.
A central question is: How can I represent this relationship in a form that helps me understand it?
Across the unit, students are asked to recognize structure, move between representations, and explain how different forms describe the same underlying mathematical relationship.
Equations
This unit builds on students’ fluency with expressions by focusing on how mathematical relationships can be used to determine unknown quantities.
Students solve radical, rational, and quadratic equations and inequalities, using a range of algebraic and graphical strategies. Particular attention is given to choosing an appropriate method, recognizing restrictions, checking solutions, and identifying extraneous results.
Quadratic equations are connected directly to quadratic functions, allowing students to interpret solutions as zeros, intercepts, maximum or minimum values, and features of mathematical models.
In trigonometry, students use trigonometric relationships, the sine law, and the cosine law to determine unknown sides and angles and solve indirect-measurement problems. Financial mathematics extends the same reasoning to decisions involving interest, investments, loans, payments, buying, and leasing.
Rather than treating each type of equation as a separate procedure, students are encouraged to identify the information they know, determine the relationship connecting those quantities, and select a strategy that will reveal the unknown.
A central question is: What relationship connects what I know to what I am trying to find?
Across the unit, students are expected to select appropriate strategies, carry out procedures accurately, interpret solutions in context, determine whether results are reasonable, and communicate their reasoning clearly.
Learning Through Problem Solving
Assignments are an important part of the learning experience in this course.
Questions are carefully selected and organized to develop procedural fluency while also strengthening conceptual understanding and problem-solving skills.
Concepts are introduced progressively so that students can first develop familiarity with new mathematical structures before applying them in more complex or unfamiliar situations.
Students regularly encounter questions that require them to:
- recognize mathematical structure
- select an appropriate strategy
- connect new ideas to previous learning
- interpret mathematical representations
- identify restrictions
- determine whether an answer is reasonable
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 rather than as a collection of unrelated procedures.
Throughout the course, students learn to:
- identify patterns and analyze relationships
- make conjectures and generalize mathematical ideas
- compare strategies and justify conclusions
- communicate reasoning
- recognize structure in unfamiliar situations
An important question throughout Pre-Calculus 11 is:
What do I already know that can help me understand this new problem?
For example, students examine how operations with rational numbers extend to rational expressions, how exponent laws connect to radicals, and how strategies used to solve linear equations can be extended to quadratic, radical, and rational equations.
These connections help students develop mathematical flexibility rather than relying on a separate memorized procedure for every type of problem.
Mathematical Modelling
Mathematical modelling is used throughout the course to connect abstract mathematics with situations in which mathematics can help describe relationships or support decisions. Students learn that creating an appropriate mathematical model is often as important as carrying out the calculations that follow.
Students learn to:
- identify relevant quantities
- build mathematical relationships
- select appropriate representations
- solve mathematical models
- interpret solutions
- evaluate whether results are reasonable
Examples include quadratic relationships, indirect measurement, financial growth, investments and loans, geometric relationships, and problems involving domain and range restrictions.
Mathematical Communication
Students are expected to communicate mathematics clearly and precisely.
This includes:
- showing complete solutions
- using appropriate notation
- defining variables when necessary
- organizing work logically
- explaining reasoning
- identifying restrictions
- verifying solutions
- interpreting answers in context
A correct numerical answer without sufficient supporting reasoning may not provide complete evidence of mathematical understanding. Communication is treated as an essential mathematical skill rather than an optional addition.
Technology
Technology is used to support mathematical understanding, exploration, modelling, and communication. Students are expected to understand the mathematics underlying technological solutions and to determine whether the results produced by technology are reasonable.
Students may use graphing calculators, Desmos, and spreadsheets
Technology allows students to visualize relationships, investigate patterns, compare algebraic and graphical representations, test conjectures, and verify results.
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
- apply previous learning
- demonstrate growth over time
Assessment tasks may include, problem-solving activities, modelling tasks, graphical investigations, cumulative review opportunities, and tests.
Both procedural fluency and conceptual understanding are valued throughout the course. Students should increasingly be able not only to carry out mathematical procedures accurately, but also to select strategies, explain their reasoning, connect representations, identify errors, and apply mathematics in unfamiliar situations.
Preparing Students for Future Mathematics
Pre-Calculus 11 provides an important transition between the mathematics students encounter in earlier grades and the more abstract study of functions in Pre-Calculus 12.
Students who successfully complete the course should leave with:
- stronger algebraic fluency
- confidence manipulating expressions and equations
- experience working with multiple representations
- effective mathematical communication skills
- experience applying mathematics in unfamiliar contexts
These skills provide the foundation for the study of polynomial, rational, exponential, logarithmic, and trigonometric functions in Pre-Calculus 12.
Final Thoughts
Mathematics becomes increasingly powerful when students begin to recognize that seemingly different ideas are connected.
The purpose of Pre-Calculus 11 is to help students recognize these structures, develop confidence working with algebra, communicate mathematical ideas clearly, and apply their understanding to new problems.
Success in Pre-Calculus 11 comes from developing mathematical fluency, reasoning, persistence, and the ability to make connections between ideas.