Course Title: Calculus and Vectors, Grade 12 University Preparation (MCV4U)
Prerequisite: MHF4U — Advanced Functions, Grade 12 University Preparation (must be taken prior to or concurrently with MCV4U).
Hours: 110 hours of scheduled instruction · Credit: 1.0
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.
Mathematical Process Expectations
Throughout this course, students will apply seven mathematical processes:
- Problem Solving — develop, select, apply, compare, and adapt a variety of problem-solving strategies.
- Reasoning and Proving — develop and apply reasoning skills to make mathematical conjectures, assess conjectures, and justify conclusions.
- Reflecting — demonstrate that they are reflecting on and monitoring their thinking.
- Selecting Tools and Computational Strategies — select and use a variety of concrete, visual, and electronic learning tools and appropriate strategies.
- Connecting — make connections among mathematical concepts and procedures, and relate them to other contexts.
- Representing — create a variety of representations of mathematical ideas (numeric, geometric, algebraic, graphical).
- Communicating — communicate mathematical thinking orally, visually, and in writing using clear and precise mathematical vocabulary.
Strands and Overall Expectations
Strand A — Rate of Change
By the end of this course, students will:
- A1. 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;
- A2. 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;
- A3. 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.
Strand B — Derivatives and Their Applications
By the end of this course, students will:
- B1. 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;
- B2. 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.
Strand C — Geometry and Algebra of Vectors
By the end of this course, students will:
- C1. demonstrate an understanding of vectors in two-space and three-space by representing them algebraically and geometrically and by recognizing their applications;
- C2. 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;
- C3. 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;
- C4. represent lines and planes using scalar, vector, and parametric equations, and solve problems involving distances and intersections.
Specific Expectations (Selected)
A1. Investigating Instantaneous Rate of Change
- A1.1 describe examples of real-world applications of rates of change, represented in a variety of ways
- A1.2 describe connections between average rate of change as the slope of a secant on the graph of a function, and instantaneous rate of change as the slope of the tangent
- A1.3 make connections, with or without graphing technology, between an approximate value of the instantaneous rate of change at a given point on the graph of a smooth function and average rates of change over intervals containing the point
- A1.4 recognize that the slope of a secant on a graph of a function approaches the slope of the tangent at a given point as the interval over which the secant is taken approaches zero
- A1.5 make connections between the slope of a secant on the graph of a function (e.g., \( f(x) = x^2 \)) and the average rate of change of the function over an interval, and between the slope of the tangent to a point on the graph of a function and the instantaneous rate of change of the function at that point
- A1.6 determine the limit of a polynomial, a rational, or an exponential function
- A1.7 determine instantaneous rates of change using limits
A2. Investigating the Concept of the Derivative Function
- A2.1 determine numerically and graphically the intervals over which the instantaneous rate of change is positive, negative, or zero for a function whose graph is given
- A2.2 generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function
- A2.3 make connections between the graphs of a function and its derivative function
- A2.4 define the derivative function \( f'(x) \) as \( \displaystyle f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h} \)
- A2.5 determine the derivatives of polynomial functions by simplifying the algebraic expression \( \frac{f(x+h)-f(x)}{h} \) and then taking the limit
A3. Investigating and Applying Properties of Derivatives
- A3.1 verify the power rule for functions of the form \( f(x)=x^n \) (n positive integer)
- A3.2 verify the constant, constant multiple, sum, and difference rules graphically and numerically
- A3.3 determine algebraically the derivatives of polynomial functions, and use these derivatives to determine the instantaneous rate of change at a point and to determine point(s) at which a given rate of change occurs
- A3.4 verify that the power rule applies to functions of the form \( f(x)=x^n \), where n is a rational number
- A3.5 verify algebraically the chain rule using monomial functions, and the product rule using polynomial functions
- A3.6 solve problems, using the product and chain rules, involving the derivatives of polynomial functions, sinusoidal functions, exponential functions, rational functions, radical functions, and other simple combinations
B1. Connecting Graphs and Equations of Functions and Their Derivatives
- B1.1 sketch the graph of a derivative function, given the graph of a function that is continuous over an interval
- B1.2 recognize the second derivative as the rate of change of the rate of change
- B1.3 determine algebraically the equation of the second derivative \( f''(x) \) of a polynomial or simple rational function \( f(x) \), and make connections, through investigation using technology, between the key features of the graph of the function and those of the first and second derivatives
- B1.4 describe key features of a polynomial function, given information about its first and/or second derivatives
- B1.5 sketch the graph of a polynomial function, given its equation, by using a variety of strategies
- B1.6 sketch the graph of a function, given the equation of its derivative
B2. Solving Problems Using Mathematical Models and Derivatives
- B2.1 make connections between the concept of motion and the concept of the derivative
- B2.2 make connections between the graphical or algebraic representations of derivatives and real-world applications
- B2.3 solve problems, using the derivative, that involve instantaneous rates of change, including problems arising from real-world applications
- B2.4 solve optimization problems involving polynomial, simple rational, and exponential functions drawn from a variety of applications
- B2.5 solve problems arising from real-world applications by applying a mathematical model and the concepts and procedures associated with the derivative to determine mathematical results, and interpret and communicate the results
C1. Representing Vectors Geometrically and Algebraically
- C1.1 recognize a vector as a quantity with both magnitude and direction; identify, gather, and interpret information about real-world applications of vectors
- C1.2 represent a vector as a directed line segment, with directions expressed in different ways
- C1.3 determine, using trigonometric relationships, the Cartesian representation of a vector in two-space given as a directed line segment, or the representation as a directed line segment given the Cartesian representation
- C1.4 recognize that points and vectors in three-space can be represented using ordered triples
C2. Operating with Vectors
- C2.1 perform the operations of addition, subtraction, and scalar multiplication on vectors represented as directed line segments in two-space, and on vectors represented in Cartesian form
- C2.2 determine, through investigation, properties of the operations of addition, subtraction, and scalar multiplication of vectors
- C2.3 solve problems involving the addition, subtraction, and scalar multiplication of vectors, including problems arising from real-world applications
- C2.4 perform the operation of dot product on two vectors represented as directed line segments and in Cartesian form
- C2.5 determine, through investigation, properties of the dot product
- C2.6 perform the operation of cross product on two vectors represented in Cartesian form in three-space
- C2.7 solve problems involving dot product and cross product, including problems arising from real-world applications such as work and torque
C3 & C4. Geometry of Lines and Planes
- C3.1 recognize that the solution points (x, y) in two-space of a single linear equation in two variables form a line, and that points (x, y, z) in three-space of a single linear equation in three variables form a plane
- C3.2 determine, through investigation with technology, the geometric configurations of lines and planes in three-space
- C4.1 recognize a scalar equation for a line in two-space; determine, through investigation, vector and parametric equations of a line in two-space
- C4.2 represent a line in three-space using the scalar, parametric, vector, and symmetric equations
- C4.3 recognize that a line in three-space cannot be represented by a single scalar equation
- C4.4 determine, through investigation, different geometric configurations of combinations of up to three lines and/or planes in three-space; organize the configurations based on whether they intersect and how they intersect
- C4.5 determine the intersection of two lines, a line and a plane, or two planes, by setting up and solving a system of linear equations
- C4.6 determine the distance from a point to a line in two-space and three-space, and from a point to a plane
Strand-to-Unit Mapping
| Unit | Title | Hours | Strand | Key Expectations |
|---|
| 1 | Introduction to Calculus | 10 | A | A1.1–A1.7, A2.1–A2.5 |
| 2 | Derivatives | 14 | A | A2.4, A3.1–A3.6 |
| 3 | Derivative Applications | 14 | B | B2.1–B2.5 |
| 4 | Curve Sketching | 12 | B | B1.1–B1.6 |
| 5 | Exponential and Trig. Derivatives | 12 | A, B | A3.6, B2.4 |
| 6 | Introduction to Vectors | 12 | C | C1.1–C1.4, C2.1–C2.3 |
| 7 | Vector Applications | 12 | C | C2.4–C2.7 |
| 8 | Lines and Planes | 12 | C | C3.1–C3.2, C4.1–C4.3, C4.6 |
| 9 | Intersections | 12 | C | C4.4–C4.6 |
Achievement Chart (Growing Success, 2010)
Student achievement is evaluated against four equally weighted categories. Each category contributes 25% of the grade.
| Category | Weight | What is assessed |
|---|
| K/U Knowledge & Understanding | 25% | Subject-specific content acquired (knowledge), and the comprehension of its meaning and significance (understanding). |
| T Thinking | 25% | The use of critical and creative thinking skills and/or processes (planning, processing, critical/creative thinking). |
| C Communication | 25% | The conveying of meaning through various forms (oral, visual, written), use of conventions, vocabulary, and audience awareness. |
| A Application | 25% | The use of knowledge and skills to make connections within and between various contexts (transfer, making connections, problem solving). |
Levels of Achievement
| Level | Percentage | Description |
|---|
| 4 | 80–100% | Thorough understanding; high degree of effectiveness |
| 3 | 70–79% | Considerable understanding (provincial standard) |
| 2 | 60–69% | Some understanding |
| 1 | 50–59% | Limited understanding |
| R | Below 50% | Insufficient achievement of curriculum expectations |
Assessment & Evaluation Policy
Final Grade Calculation (Growing Success 2010):
70% Term Work — based on assessment OF learning gathered throughout the course
30% Final Evaluation — a final exam, performance, or other summative method (this course uses a 100-mark final exam)
Assessment Types Used
- Assessment AS Learning — Self-directed practice quizzes after each unit; not graded; unlimited retakes; supports metacognition and self-monitoring.
- Assessment FOR Learning — Mid-unit diagnostics; not graded; teacher provides descriptive feedback (Strengths / Next Steps) to inform next instructional steps.
- Assessment OF Learning — End-of-unit summative tests (60 marks each) and the final exam (100 marks); graded against the Achievement Chart with rubrics aligned to the levels above.
Term Work Distribution (70%)
| Component | Weight (of 70%) |
|---|
| 9 Unit Tests (OF Learning) | 50% |
| Performance Tasks / Investigations | 15% |
| Observations & Conversations (Triangulation) | 5% |
Final Evaluation (30%)
- 3-hour, 100-mark cumulative final exam covering all 9 units
- Distribution: K/U 25 marks, Thinking 25 marks, Communication 25 marks, Application 25 marks
- Mix of multiple-choice, short answer, and extended response items
Considerations for Program Planning
- Instructional Approaches: teaching is differentiated for students with varied learning needs; the mathematical processes are integrated into every unit.
- Use of Technology: Desmos, GeoGebra, and graphing calculators are used for investigation, visualization, and verification (not as substitutes for algebraic skill).
- Cross-Curricular Connections: physics (motion, work, force vectors), economics (marginal analysis), and engineering (optimization).
- Equity and Inclusive Education: instruction reflects diverse contributions to mathematics; problems are drawn from a variety of cultural contexts.
- Health and Safety: not applicable as a primary concern; cyber-safety guidelines apply when using online tools.
Source: The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007 (Revised), Ministry of Education, Ontario; Growing Success: Assessment, Evaluation, and Reporting in Ontario Schools, 2010.