Calculus and Vectors, Grade 12, University Preparation
| Course Title | Calculus and Vectors, Grade 12, University Preparation |
| Course Code | MCV4U |
| Grade | 12 |
| Course Type | University Preparation |
| Credit Value | 1.0 |
| Prerequisite | Advanced Functions, Grade 12, taken prior to or concurrently |
| Co requisite | None |
| Department | Mathematics |
| Name of the Department Head | Colleen McMillan (B.Sc.,B.Ed.,OCT) |
| Course Developer or Teacher | Colleen McMillan (B.Sc.,B.Ed.,OCT) |
| Curriculum Policy Document | Mathematics, The Ontario Curriculum, Grades 11 & 12, 2018 (Revised) |
| Course development date | May 2020 |
| Most recent revision date | June 2026 |
| Course reviser | Colleen McMillan (B.Sc.,B.Ed.,OCT) |
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.
By the end of this course, students will develop the following skills in these different areas:
| 1. Rate of Change | |
| 1.1 | 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; |
| 1.2 | graph the derivatives of polynomials, exponential and sinusoidal functions, and make connections between the numeric, graphical and algebraic representations of a function and its derivative; |
| 1.3 | verify graphically and algebraically the rules for determining derivatives; apply these rules to determine the derivatives of polynomial, exponential, sinusoidal functions, rational and radical functions, as well as simple combination of functions; and solve related problems |
| 2. Derivatives and their Applications | |
| 2.1 | 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; |
| 2.2 | solve problems, including optimization types, 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. |
| 3. Geometry and Algebra of Vectors | |
| 3.1 | demonstrate an understanding of vectors in 2-space and 3-space by representing them algebraically and geometrically and by recognizing their application; |
| 3.2 | perform operations on vectors in 2- and 3-space, and use the properties of these operations to solve problems, including those arising from real-world applications; |
| 3.3 | distinguish between the geometric representations of a single linear equation or a system of 2 linear equations in 2- and 3- space, and determine different geometric configurations of lines and planes in 3-space; |
| 3.4 | represent lines and planes using scalar, vector and parametric equations; and solve problems involving distances and intersections. |
| Time Allocated | Online/Offline Component | Learning Expectations | |
|---|---|---|---|
| 1. Limits and First Principles | |||
A variety of mathematical operations with functions are needed in order to do the calculus of this course. This unit begins with students developing a better understanding of these essential concepts. Students will then deal with rates of change problems and the limit concept. While the concept of a limit involves getting close to a value but never getting to the value, often the limit of a function can be determined by substituting the value of interest for the variable in the function. Students will work with several examples of this concept and learn techniques to evaluate limits of indeterminate form. These basic ideas will be extended and expanded to be able to distinguish between average and instantaneous rates of change to help students solve problems arising in real-world applications. | 12 hours | (6 hrs online / 6 hrs offline) | A1.1-1.6, A2.2-2.4 |
| 2. Derivatives | |||
The concept of a derivative is, in essence, a way of creating a short cut to determine the tangent line slope function that would normally require the concept of a limit. Once patterns are seen from the evaluation of limits, rules can be established to simplify what must be done to determine this slope function. This unit begins by examining those rules including: the power rule, the product rule, the quotient rule and the chain rule followed by a study of the derivatives of composite functions. Students will learn to take derivatives of polynomial, rational and radical functions. The skills obtained in this unit and the next will be the foundation for the following units of curve sketching and applications.
| 13 hours | (5 hrs online / 8 hrs offline) | A2.1-2.3, A3.1-3.5 |
| 3. Derivatives Part 2 | |||
Derivative rules are further established for exponential, logarithmic and | 10 hours | (4 hrs online / 6 hrs offline) | A2.4-2.8 |
| 4. Derivatives and Curve Sketching | |||
In previous math courses, functions were graphed by developing a table of values and smooth sketching between the values generated. This technique often hides key details of the graph and produces a dramatically incorrect picture of the function. These missing pieces of the puzzle can be found by the techniques of calculus learned thus far in this course. Students will use the first derivative test to determine maximum and minimum values and intervals ofincrease and decrease of functions; second derivative test to determine intervals of concavity. The key features of a properly sketched curve are all determined separately before putting them all together into a full sketch of a curve. | 12 hours | (5 hrs online / 7 hrs offline) | B1.1-1.5, B2.3 |
| 5. Applications (Optimization and Related Rates) | |||
Derivatives will be used to determine rates of change in real life applications such as displacement, velocity and acceleration; population growth, volume and flow rates, etc. Optimization and related rates problems from a variety of applications will be solved from algebraic models. | 13 hours | (5 hrs online / 8 hrs offline) | B2.1-2.5 |
| 6. Geometric and Algebraic Representation of Vectors | |||
Vectors are introduced as quantities with magnitude and direction and the distinction between scalar and vector quantities are defined. Students will learn to represent vectors geometrically and algebraically in 2-space and 3-space; and use both representations to perform addition, subtraction and scalar multiplication of vectors and determine associative and distributive properties. | 11 hours | (5 hrs online / 6 hrs offline) | C1.1-1.4, C2.1-2.3 |
| 7. Dot & Cross Product | |||
Applications involving work and torque are used to introduce and lend context to the dot and cross products of Cartesian vectors. The vector and scalar projections of Cartesian vectors are written in terms of the dot product. The properties of vector products are investigated and proven. These vector products will be revisited to predict characteristics of the solutions of systems of lines and planes in the intersections of lines and planes. | 12 hours | (5 hrs online / 7 hrs offline) | C1.5, C2.3-2.8 |
| 8. Equations of Lines and Planes | |||
Lines and planes are represented by vector, parametric and Cartesian equations in 2-space and 3-space. Students will recognize the geometric and algebraic representations of a normal to a plane; learn to create equations of lines and planes and to convert between the different forms. | 12 hours | (6 hrs online / 6 hrs offline) | C3.1-3.3, C4.1-4.3,4.5, 4.6 |
| 9. Relationships between Points Lines and Planes | |||
Planes, lines and points can intersect in a multitude of ways. Students will use algebraic techniques to find the intersections and relationships and interpret the results geometrically. Students will find the intersection of two and three planes by setting up a system of linear equations. Matrices and row reduction will be introduced to solve linear systems. | 12 hours | (5 hrs online / 7 hrs offline) | C4.3, 4.4,4.7 |
| 10. Final Examination | |||
This is a proctored exam worth 25% of the final grade. | 3 hours | (3 hrs online) | — |
| Total | 110 Hours | ||
| Online | Offline |
|---|---|
| Communicating with the instructor | Reading materials of the course |
| Contributing to forums | Completing assignments |
| Watching instructional videos | Completing practice activities |
| Watching additional resource videos | Reviewing for tests and exams |
| Practicing through online quizzes | |
| Completing online timed exam |
A variety of assessment and evaluation methods, strategies and tools are required as appropriate to the expectation being assessed. These include diagnostic, formative and summative within the course and within each unit.
Assessment FOR Learning and Assessment AS Learning is obtained through a variety of means, including the following:
- Ongoing descriptive feedback, including descriptive feedback on students’ plans for their venture
- Self-assessment
- Peer assessment
- Student/Teacher communication on a regular basis to:
- verbalize observations
- ask questions
- clarify understanding
Evidence of student achievement (assessment of learning) is collected through ongoing observations of most consistent work, with consideration given to most recent work from various sources.
Assessment and evaluation in this course will be based on the provincial curriculum expectations. Students will be provided with numerous and varied opportunities to demonstrate the full extent of their achievement. Categories of assessment and breakdowns are as follows:
- Knowledge – 30%
- Thinking Inquiry – 25%
- Application – 25%
- Communication – 20%
A final grade will be determined as follows:
- Term Work – 65%
- Final Examination – 25%
- Attendance and Participation – 10%
| Students enrolled in this course through CHS’s Hybrid (H) program will participate in synchronous learning via live online teaching sessions, online support material, and student-to-student discussions organized throughout the semester. Conversely, students taking this course as part of CHS’s Self-Paced (S) program will learn asynchronously through recorded video lessons, presentations, online support material, and simulations. While self-paced students have up to one year to complete the course, they are encouraged to finish within five months.
A variety of strategies will be used in the online delivery of this course. Instructional strategies will include but are not limited to:
Learning goals and success criteria are listed at the beginning of each lesson. The success criteria are used to develop the assessment tools in this course, including rubrics. The over-riding aim of this course is to help students use the language of mathematics skillfully, confidently and flexibly. A wide variety of instructional strategies are used to provide learning opportunities to accommodate a variety of learning styles, interests, and ability levels. The following mathematical processes are used throughout the course as strategies for teaching and learning the concepts presented. |
|
| Problem Solving | Course scaffolds learning by providing students with opportunities to review and activate prior knowledge (e.g. reviewing concepts related to rates of change) and build off of this knowledge to acquire new skills. The course guides students toward recognizing opportunities to apply knowledge they have gained to solve problems. |
| Selecting Tools and Computational Strategies | Course models the use of graphing software to familiarize students with available software and resources which will allow them to simplify calculations in order to better understand the characteristics of functions. |
| Connecting | Course uses 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 | Through the use of examples, practice problems, and solution videos, the course models various ways to demonstrate understanding, poses questions that require students to use different representations as they are working at each level of conceptual development – concrete, visual or symbolic, and allows individual students the time they need to solidify their understanding at each conceptual stage. |
| Self-Assessment | Through the use of interactive activities (e.g. whiteboard group work, online quizzes and simulations) students receive instantaneous feedback and are able to self-assess their understanding of concepts. |
Students with special needs and English Language Learners will be provided with accommodation, including additional time, assistive technology and scribe where available. Teachers who are planning a program in this subject make an effort to take into account considerations for program planning that align with the Ontario Ministry of Education policy and initiatives in a number of important areas.
Learning Skills listed below are key to student success. Learning Skills are assessed independently of achievement and are determined through observation and participation. A check list and student-teacher communication will be used to determine the level in each category.
1. Responsibility
2. Organization
3. Independent Work
4. Collaboration
5. Initiative
6. Self-Regulation
- Calculator
- Graph paper
- Internet access
- Presentation and video lessons
- Activities and Assignments
- On-Line resources