| Name of School | Department | Course Developer | Course Dev Date | Course Revisor / Revision Date |
| Advanced Knowledge Academy | Mathematics | Junhyung Park | Nov 1, 2025 | Nov 1, 2025 |
| Course Title & Grade | Ministry Course Code | Course Type | Credit Value | Developed from |
| Calculus and Vectors, Grade 12 | MCV4U | University Preparation | 1.0 | The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007 (revised) |
| Prerequisite | Advanced Functions, Grade 12 University (MHF4U) |
Welcome to Grade 12 Calculus and Vectors (MCV4U).
Textbook and Homework Link Here
Course Description
This course builds on students’ previous experience with functions and their developing understanding of rates of change. Students will solve problems involving geometric and algebraic representations of vectors and representations of lines and planes in three-dimensional space; broaden their understanding of rates of change to include the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions; and apply these concepts and skills to the modelling of real-world relationships. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended for students who choose to pursue careers in fields such as science, engineering, economics, and some areas of business, including those students who will be required to take a university-level calculus, linear algebra, or physics course.
Overall Curriculum Expectations
All courses in mathematics, Grade 1-12 have both common “Mathematical Process Expectations” and “Overall Expectations”, which are specific for each course.
The 7 mathematical process expectations are to be integrated into student learning in all areas of the course. Throughout this course, students will show:
Problem Solving
– Develop, select, apply, compare, and adapt a variety of problem-solving strategies as they pose and solve problems and conduct investigations, to help deepen their mathematical understanding;
Reasoning and Proving
– Develop and apply reasoning skills (e.g., use of inductive reasoning, deductive reasoning, and counter-examples; construction of proofs) to make mathematical conjectures, assess conjectures, and justify conclusions, and plan and construct organized mathematical arguments;
Reflecting
– Demonstrate that they are reflecting on and monitoring their thinking to help clarify their understanding as they complete an investigation or solve a problem (e.g., by assessing the effectiveness of strategies and processes used, by proposing alternative approaches, by judging the reasonableness of results, by verifying solutions);
Selecting Tools and Computational Strategies
– Select and use a variety of concrete, visual, and electronic learning tools and appropriate computational strategies to investigate mathematical ideas and to solve problems; Connecting
– Make connections among mathematical concepts and procedures, and relate mathematical ideas to situations or phenomena drawn from other contexts (e.g., other curriculum areas, daily life, current events, art and culture, sports);
Representing
– Create a variety of representations of mathematical ideas (e.g., numeric, geometric, algebraic, graphical, pictorial representations; onscreen dynamic representations), connect and compare them, and select and apply the appropriate representations to solve problems;
Communicating
– Communicate mathematical thinking orally, visually, and in writing, using precise mathematical vocabulary and a variety of appropriate representations, and observing mathematical conventions.
A. Rate of Change
- Demonstrate an understanding of rate of change by making connections between average rate of change over an interval and instantaneous rate of change at a point, using the slopes of secants and tangents and the concept of the limit.
- Graph the derivatives of polynomial, sinusoidal, and exponential functions, and make connections between the numeric, graphical, and algebraic representations of a function and its derivative.
- Verify graphically and algebraically the rules for determining derivatives; apply these rules to determine the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions, and simple combinations of functions; and solve related problems.
B. Derivatives and Their Applications
- Make connections, graphically and algebraically, between the key features of a function and its first and second derivatives, and use the connections in curve sketching
- Solve problems, including optimization problems, that require the use of the concepts and procedures associated with the derivative, including problems arising from real-world applications and involving the development of mathematical models.
C. Geometry and Algebra of Vectors
- Demonstrate an understanding of vectors in two-space and three-space by representing them algebraically and geometrically and by recognizing their applications
- Perform operations on vectors in two-space and three-space, and use the properties of these operations to solve problems, including those arising from real-world applications.
- Distinguish between the geometric representations of a single linear equation or a system of two linear equations in two-space and three-space, and determine different geometric configurations of lines and planes in three-space.
- Represent lines and planes using scalar, vector, and parametric equations, and solve problems involving distances and intersections.
A variety of teaching and learning strategies will be used throughout the course to engage students in the learning process. Strategies will be used which will build on students’ prior knowledge and to assess where students are in their understanding of mathematical concepts enabling them to learn the curriculum. Differentiated teaching/learning strategies will be used within units of study to support the learning needs of students in the class. Students are encouraged to share learning strategies with their teacher which helps them improve their learning. Examples of teaching and learning strategies which will be used throughout the course include, but are not limited to:
- Direct Instruction (Teacher led and discussion based)
- Class Activities (Teach facilitation)
- Experiential Learning
- Worksheets/Surveys
- Teacher Modeling
- Text-based Modeling
- Use of Computers/Internet
- 1:1 Conferencing Teacher & Student
- Independent Work
- Project Based Learning
- Use of Visual Representations
- Problem Solving Exercises
- Teacher Directed Learning
- Use of Computer Software/Technology
- Investigative Inquiry
The primary purpose of assessment and evaluation is to improve student learning and to help students assume responsibility for their learning. Mid-semester and final marks are determined through evaluations or Assessments of Learning which typically occur towards the end of a unit and end of semester. During the learning process information about a student’s learning is gathered and used by the teacher and student to inform decisions that affect goal-setting and teaching in the classroom. The data gathered as Assessment as Learning and Assessment for Learning do not carry a mark weight but do play a crucial role in student success as they help inform the teacher and the student about each student’s progress. All types of assessments allow teachers to provide descriptive feedback that is clear, specific, meaningful, and timely to support improved learning and achievement.
Grading
Throughout this course, you will be given multiple opportunities to practice and reinforce the concepts you are learning. While these formative tasks do not contribute to your mark in the course, they will assist you in preparing for summative assessments which will count towards your final grade.
The final grade is based on performance in 3 areas: Products, Observations and conversations.
- Unit 1: Rate of Change and Derivatives
- Unit 2: Curve Sketching/Optimization
- MID-SEMESTER POINT
- Unit 3: Trigonometric & Exponential Derivatives
- Unit 4: Geometric & Algebraic Vectors
- Unit 5: Lines and Planes
- Final Exam
Advanced Knowledge Academy adheres to the assessment & evaluation practices as outlined in the Ministry of Education’s Growing Success documents. The final mark for this course will be calculated according to the following breakdown:
| 30% (Summative + Exam) | |
| Term Work | 70% |
| Final Exam | 30% |
Learning Skills (responsibility, organisation; independent work, collaboration, initiative, self-regulation) will also be assessed separately from the achievement of the learning expectations and will be reported on the provincial report card.
| Unit 1: Rate of Change and Derivatives | 30 hours |
| Unit 2: Curve Sketching/Optimization | 25 hours |
| Unit 3: Trigonometric & Exponential Derivatives | 20 hours |
| Unit 4: Geometric and Algebraic Vectors | 20 hours |
| Unit 5: Lines and Planes | 15 hours |
| Total: | 110 hours |
| Assessment Tools | ||
| Marking Schemes, Rubrics, Checklists, Rating Scales | ||
| Assessment for Learning | Assessment as Learning | Assessment of Learning |
| Quizzes Presentations Project Drafts Demonstrations – Conferencing Questioning Independent Study Assignment Proofing | Problem Solving (process focused) Work Sheets Direct Instruction Quizzes Self/Peer Assessment Tests Presentations | Projects Demonstrations – Questioning Independent Study Assignments Problem Solving (Process Focused) |
Considerations for Program Planning
In an asynchronous online mathematics course, instructional approaches are guided by Ministry and board policy, centering on universal design, differentiated instruction, and clear learning objectives. Lessons are structured to activate prior knowledge, scaffold complex concepts, and encourage students to articulate their reasoning through a variety of media, including written explanations, visual organizers, and interactive math tools. Teachers model problem-solving strategies using pre-recorded videos, simulations, and annotated examples, ensuring students have access to multiple representations such as numerical, algebraic, graphical, and contextual formats. The course provides opportunities for inquiry-based learning, practice with immediate feedback, collaboration through discussion forums, and reflection on mathematical thinking. These approaches support the development of conceptual understanding, procedural fluency, and real-world application skills for diverse learners.
Environmental issues and financial contexts are authentically integrated into math tasks, challenging students to analyze real-world data, model change, and make informed decisions. Topics such as exponential growth and decay, compound interest, and resource allocation support both environmental stewardship and the development of valuable financial literacy skills.
Technology is at the core of asynchronous mathematics instruction, functioning as both the primary mode of content delivery and a catalyst for mathematical exploration. Students use digital graphing software, dynamic algebra systems, video lessons, and interactive modules to investigate functions and solve complex problems. Communication and collaboration are fostered through online forums, cloud-based assignments, and digital portfolios, allowing students to share ideas and receive feedback asynchronously. Teachers provide digital resources and support safe, responsible technology use, guiding students in evaluating sources, safeguarding privacy, and practicing digital citizenship in all coursework interactions.
The course is intentionally designed to ensure accessibility for all, with particular supports for students with special education needs and English language learners. Accommodations, such as adaptive technologies, flexible timelines, and multi-modal instructions, are provided to facilitate equitable participation. Instruction is scaffolded to develop mathematical vocabulary and reasoning, with frequent opportunities for students to demonstrate understanding in formats that draw on their strengths—such as oral explanations, graphical responses, and collaborative digital projects.
Curriculum materials reflect diverse perspectives, cultural backgrounds, and identities, in alignment with equity and inclusive education policies. Problems and examples are selected with careful attention to representation, drawing on Indigenous and global mathematical traditions
where appropriate. The course fosters a climate of respect and encourages students to relate mathematical concepts to their lived experiences, advancing equity and social justice.
Students explore the relevance of mathematics to a wide array of careers, with assignments and projects that simulate workplace scenarios and problem-solving processes. Cooperative education principles are embodied through virtual group work, the use of discussion boards for peer feedback, and opportunities for authentic, project-based learning experiences that develop transferable skills.
Assessment is ongoing and varied, focusing on both process and product, and includes diagnostic, formative, and summative evaluations in accordance with Growing Success. Clear guidelines on academic honesty are communicated at the outset of the course: students are expected to submit original work, and incidents of plagiarism are addressed according to school policy, with an emphasis on restorative approaches and learning from errors. Policies regarding late assignments balance the importance of meeting deadlines with recognition of individual circumstances; extensions are available for valid reasons, but persistent lateness may result in academic consequences such as reduced achievement levels, as outlined in Growing Success. Students are provided with feedback and opportunities to reflect on their progress and develop self-regulation skills in the independent online environment.
The course encourages digital wellness, with expectations for respectful online conduct, guidance on healthy screen time practices, and resources to support mental health. Well-being is prioritized through clear communication channels for academic and personal support, promoting a safe and inclusive online space for learning.
By integrating these considerations, the course ensures alignment with Ministry and board initiatives while fostering a rigorous, inclusive, and supportive online mathematics learning environment.
Achievement Chart
| Category & Criteria | 50–59% (Level 1) | 60–69% (Level 2) | 70–79% (Level 3) | 80–100% (Level 4) |
|---|---|---|---|---|
| Knowledge and Understanding | ||||
| Knowledge of content (facts, terms, formulas, procedures, algorithms, tools) | Demonstrates limited knowledge of mathematical content and procedures; frequent errors in computation or use of tools | Demonstrates some knowledge of mathematical content and procedures; occasional errors in computation or use of tools | Demonstrates considerable knowledge of mathematical content and procedures; minor errors that do not significantly affect results | Demonstrates thorough knowledge of mathematical content and procedures; accurate computation and effective use of tools |
| Understanding of mathematical concepts (relationships, patterns, representations, properties, theorems) | Demonstrates limited understanding of mathematical concepts and relationships among representations | Demonstrates some understanding of mathematical concepts and relationships among representations | Demonstrates considerable understanding of mathematical concepts and relationships among representations | Demonstrates thorough understanding of mathematical concepts and relationships among multiple representations |
| Thinking | ||||
| Use of planning skills (understanding the problem, formulating, selecting strategies, making conjectures) | Uses planning skills with limited effectiveness; difficulty interpreting problems or selecting appropriate strategies | Uses planning skills with some effectiveness; interprets problems with guidance and selects partially effective strategies | Uses planning skills with considerable effectiveness; interprets problems independently and selects appropriate strategies | Uses planning skills with a high degree of effectiveness; analyses problems insightfully and selects optimal strategies |
| Use of processing skills (carrying out plans, modelling, solving, reasoning, justifying, reflecting, checking) | Uses processing skills with limited effectiveness; frequent errors in execution; reasoning is unclear or incomplete | Uses processing skills with some effectiveness; occasional errors in execution; reasoning is partially developed | Uses processing skills with considerable effectiveness; accurate execution; reasoning is logical and mostly complete | Uses processing skills with a high degree of effectiveness; efficient and accurate execution; reasoning is rigorous and complete |
| Use of critical/creative thinking processes (problem solving, inquiry, mathematical reasoning, metacognition, proving) | Uses critical/creative thinking processes with limited effectiveness; reasoning lacks justification | Uses critical/creative thinking processes with some effectiveness; reasoning includes partial justification | Uses critical/creative thinking processes with considerable effectiveness; reasoning includes clear justification and logical arguments | Uses critical/creative thinking processes with a high degree of effectiveness; reasoning includes rigorous justification and formal proofs |
| Communication | ||||
| Expression and organization of mathematical thinking (clarity, logical organization in oral, visual, written forms) | Expresses and organizes mathematical thinking with limited effectiveness; solutions are difficult to follow | Expresses and organizes mathematical thinking with some effectiveness; solutions are partially coherent | Expresses and organizes mathematical thinking with considerable effectiveness; solutions are logical and easy to follow | Expresses and organizes mathematical thinking with a high degree of effectiveness; solutions are rigorous and well-presented |
| Communication for different audiences and purposes (peers, teachers; presenting data, justifying solutions, expressing arguments) | Communicates for different audiences and purposes with limited effectiveness; explanations lack justification | Communicates for different audiences and purposes with some effectiveness; explanations include partial justification | Communicates for different audiences and purposes with considerable effectiveness; explanations include clear justification | Communicates for different audiences and purposes with a high degree of effectiveness; explanations include rigorous justification |
| Use of conventions, vocabulary, and terminology (symbols, notation, mathematical language) | Uses conventions, vocabulary, and terminology of the discipline with limited effectiveness; frequent errors in notation | Uses conventions, vocabulary, and terminology of the discipline with some effectiveness; occasional errors in notation | Uses conventions, vocabulary, and terminology of the discipline with considerable effectiveness; minor errors in notation | Uses conventions, vocabulary, and terminology of the discipline with a high degree of effectiveness; precise and consistent use of notation |
| Application | ||||
| Application in familiar contexts (routine problems, standard algorithms, known procedures) | Applies knowledge and skills in familiar contexts with limited effectiveness; struggles with routine problems | Applies knowledge and skills in familiar contexts with some effectiveness; solves routine problems with occasional errors | Applies knowledge and skills in familiar contexts with considerable effectiveness; solves routine problems accurately and independently | Applies knowledge and skills in familiar contexts with a high degree of effectiveness; solves routine problems efficiently and extends procedures |
| Transfer to new contexts (novel problems, unfamiliar situations, modified conditions) | Transfers knowledge and skills to new contexts with limited effectiveness; struggles with novel problems | Transfers knowledge and skills to new contexts with some effectiveness; solves novel problems with scaffolding | Transfers knowledge and skills to new contexts with considerable effectiveness; solves novel problems independently | Transfers knowledge and skills to new contexts with a high degree of effectiveness; solves novel problems creatively and adapts strategies |
| Making connections within and between contexts (connections among concepts, representations, disciplines, real-world applications) | Makes connections within and between various contexts with limited effectiveness; sees topics in isolation | Makes connections within and between various contexts with some effectiveness; identifies simple relationships | Makes connections within and between various contexts with considerable effectiveness; explains relationships among concepts and applications | Makes connections within and between various contexts with a high degree of effectiveness; synthesizes concepts across topics and disciplines |