MCV4U - Grade 12 Calculus and Vectors
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MCV4U - Grade 12 Calculus and Vectors
| Course Code: | MCV4U |
| Course Type: | University Preparation |
| Credit Value: | 1.0 |
| Prerequisite: | Advanced Functions, Grade 12, University Preparation |
Course Description for MCV4U Grade 12 Calculus and Vectors Online Course
Grade 12 Calculus & Vectors (MCV4U) takes your understanding of mathematics to new heights by building on your knowledge of functions and deepening your grasp of rates of change. In this course, you’ll tackle problems involving vectors, lines, and planes in three-dimensional space using both geometric and algebraic approaches. You’ll also expand your understanding of rates of change by exploring derivatives of polynomial, sinusoidal, exponential, rational, and radical functions. These skills will be applied to model real-world relationships, giving you practical tools for complex problem-solving.
This course is perfect for students aiming for careers in science, engineering, economics, or business, particularly those planning to take university-level courses in calculus, linear algebra, or physics. Along the way, you’ll refine the advanced mathematical processes essential for success in these fields.
Overview of Units And Timelines For Grade 12 Calculus and Vectors MCV4U
Here’s the suggested sequence for delivering course units, along with the recommended hours needed to complete each one. For a detailed breakdown of the specific expectations and activities included in each unit, refer to the Unit Overviews provided in the MCV4U course profile.
Unit
Titles and Descriptions
Time and Sequence
Unit 1
Concepts of Calculus
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. The indeterminate form of a limit involving factoring, rationalization, change of variables and one-sided limits are all included in the exercises undertaken next in this unit. To further investigate the concept of a limit, the unit briefly looks at the relationship between a secant line and a tangent line to a curve. To this point in the course students have been given a fixed point and have been asked to find the tangent slope at that value, in this section of the unit students will determine a tangent slope function similar to what they had done with a secant slope function. Sketching the graph of a derivative function is the final skill and topic.
15 hours
Unit 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. The next section is dedicated to finding the derivative of relations that cannot be written explicitly in terms of one variable. Next students will simply apply the rules they have already developed to find higher order derivatives. As students saw earlier, if given a position function, they can find the associated velocity function by determining the derivative of the position function. They can also take the second derivative of the position function and create a rate of change of velocity function that is more commonly referred to as the acceleration function which is where this unit ends.
16 hours
Unit 3
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 detail 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. The key features of a properly sketched curve are all reviewed separately before putting them all together into a full sketch of a curve.
6 hours
Unit 4
Derivative Applications and Related Rates
A variety of types of problems exist in this unit and are generally grouped into the following categories: Pythagorean Theorem Problems (these include ladder and intersection problems), Volume Problems (these usually involve a 3-D shape being filled or emptied), Trough Problems, Shadow problems and General Rate Problems. During this unit students will look at each of these types of problems individually.
8 hours
Unit 5
Derivative of Exponents and Log Functions-Exponential Functions
This unit begins with examples and exercises involving exponential and logarithmic functions using Euler’s number (e). But as students have already seen, many other bases exist for exponential and logarithmic functions. Students will now look at how they can use their established rules to find the derivatives of such functions. The next topic should be familiar as the steps involved in sketching a curve that contains an exponential or logarithmic function are identical to those taken in the curve sketching unit studied earlier in the course. Because the derivatives of some functions cannot be determined using the rules established so far in the course, students will need to use a technique called logarithmic differentiation which is introduced next.
6 hours
Unit 6
Trig Differentiation and Application
A brief trigonometry review kicks off this unit. Then students turn their attention to special angles and the CAST rule which has been developed to identify which of the basic trigonometric ratios is positive and negative in the four quadrants. Students will then solve trigonometry equations using the CAST rule to locate other solutions. Two fundamental trigonometric limits are investigated for the concepts of trigonometric calculus to be fully understood. The unit ends, as in all other units in the course, with an assignment and a unit quiz.
8 hours
Unit 7
Vectors
There are four main topics pursued in this initial unit of the course. These topics are: an introduction to vectors and scalars, vector properties, vector operations and plane figure properties. Students will tell the difference between a scalar and vector quantity, they will represent vectors as directed line segments and perform the operations of addition, subtraction, and scalar multiplication on geometric vectors with and without dynamic geometry software. Students will conclude the first half of the unit by proving some properties of plane figures, using vector methods and by modeling and solving problems involving force and velocity. Next students learn to represent vectors as directed line segments and to perform the operations of addition, subtraction, and scalar multiplication on geometric vectors with and without dynamic geometry software. The final topic involves students in proving some properties of plane figures using vector methods.
12 hours
Unit 8
Vector Applications
Cartesian vectors are represented in two-space and three- space as ordered pairs and triples, respectively. The addition, subtraction, and scalar multiplication of Cartesian vectors are all investigated in this unit. 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.
16 hours
Unit 9
Intersection of Lines and Planes
This unit begins with students determining the vector, parametric and symmetric equations of lines in R2 and R3. Students will go on to determine the vector, parametric, symmetric and scalar equations of planes in 3-space. The intersections of lines in 3-space and the intersections of a line and a plane in 3-space are then taught. Students will learn to determine the intersections of two or three planes by setting up and solving a system of linear equations in three unknowns. Students will interpret a system of two linear equations in two unknowns geometrically, and relate the geometrical properties to the type of solution set the system of equations possesses. Solving problems involving the intersections of lines and planes, and presenting the solutions with clarity and justification forms the next challenge. As work with matrices continues students will define the terms related to matrices while adding, subtracting, and multiplying them. Students will solve systems of linear equations involving up to three unknowns, using row reduction of matrices, with and without the aid of technology and interpreting row reduction of matrices as the creation of new linear systems equivalent to the original constitute the final two new topics of this important unit.
16 hours
Unit 10
Final Evaluation
The final assessment task is a three hour exam worth 30% of the student’s final mark.
3 hours
Total
110 hours
Calculus and Vectors Grade 12
Throughout this course students will:
Problem solve: by developing, selecting, applying, and adapting a variety of problem-solving strategies
Reason and prove: by developing and applying reasoning skills to make mathematical conjectures, assess conjectures, and justify conclusions, plan and construct mathematical arguments;
Reflect: by monitoring their thinking to help clarify understanding as they complete an investigation or problem;
Select tools and computational strategies: by selecting and using a variety of concrete, visual, and electronic learning tools and computational strategies;
Connect: by relating mathematical ideas to situations or phenomena drawn from other contexts;
Represent: by making representations (e.g. Numeric, geometric, algebraic, graphical, pictorial and onscreen);
Communicate: by thinking orally, visually and in writing using precise mathematical vocabulary and conventions. Teachers will employ guided exploration, visuals, model analysis, direct instruction, problem posing and self-assessment to enable these student strategies.
Calculus and Vectors Grade 12
Assessment is a systematic process of collecting information or evidence about a student’s progress towards meeting the learning expectations. Assessment is embedded in the instructional activities throughout a unit. The expectations for the assessment tasks are clearly articulated and the learning activity is planned to make that demonstration possible. This process of beginning with the end in mind helps to keep focus on the expectations of the course. The purpose of assessment is to gather the data or evidence and to provide meaningful feedback to the student about how to improve or sustain the performance in the course. Scaled criteria designed as rubrics are often used to help the student to recognize their level of achievement and to provide guidance on how to achieve the next level. Although assessment information can be gathered from a number of sources (the student himself, the student’s course mates, the teacher), evaluation is the responsibility of only the teacher. For evaluation is the process of making a judgment about the assessment information and determining the percentage grade or level.
Since 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 assessment will be based on the following processes that take place in the classroom:
| Assessment FOR Learning | Assessment AS Learning | Assessment OF Learning |
|---|---|---|
During this process the teacher seeks information from the students in order to decide where the learners are and where they need to go. | During this process the teacher fosters the capacity of the students and establishes individual goals for success with each one of them. | During this process the teacher reports student’s results in accordance to established criteria to inform how well students are learning. |
| Conversation | Conversation | Conversation |
| Classroom discussion Self-evaluation Peer assessment | Classroom discussion Small group discussion | Presentations of research Debates |
| Observation | Observation | Observation |
| Drama workshops (taking direction) Steps in problem solving | Group discussions | Presentations Group Presentations |
| Student Products | Student Products | Student Products |
| Reflection journals (to be kept throughout the duration of the course) Check Lists Success Criteria | Practice sheets Socrative quizzes | Projects Poster presentations Tests In Class Presentations |
Some assessment and evaluation strategies include:
| Strategy | Purpose | Who | Assessment Tool |
|---|---|---|---|
| Self-Assessment Quizzes | Diagnostic | Self/Teacher | Marking scheme |
| Problem Solving | Diagnostic | Self/Peer/Teacher | Marking scheme |
| Graphing Application | Diagnostic | Self | Anecdotal records |
| Homework check | Diagnostic | Self/Teacher | Checklist |
| Teacher/Student Conferencing | Assessment | Self/Teacher | Anecdotal records |
| Problem Solving | Assessment | Teacher | Marking scheme |
| Investigations | Assessment | Self/Teacher | Checklist |
| Problem Solving | Evaluation | Teacher | Marking scheme |
| Graphing | Evaluation | Teacher | Checklist |
| Unit Tests | Evaluation | Teacher | Marking scheme |
| Final Exam | Evaluation | Teacher | Checklist |
Calculus and Vectors Grade 12
The evaluation of this course is based on the four Ministry of Education achievement categories of knowledge and understanding (25%), thinking (25%), communication (25%), and application (25%). The evaluation for this course is based on the student’s achievement of curriculum expectations and the demonstrated skills required for effective learning.
The percentage grade represents the quality of the student’s overall achievement of the expectations for the course and reflects the corresponding level of achievement as described in the achievement chart for the discipline.
A credit is granted and recorded for this course if the student’s grade is 50% or higher. The final grade for this course will be determined as follows:
- 80% of the grade will be based upon evaluations conducted throughout the course. This portion of the grade will reflect the student’s most consistent level of achievement throughout the course, although special consideration will be given to more recent evidence of achievement.
- 20% of the grade will be based on a final exam administered at the end of the course. The exam will contain a summary of information from the course and will consist of well−formulated multiple-choice questions. These will be evaluated using a checklist.
| Unit Number | Description | Evaluation Weight | KICA |
|---|---|---|---|
| Unit 1 | Strand 1: Scientific investigation skills and career exploration Strand 2: Forces, Work and Energy | Quiz 3% Total 14% | 25ƒ25ƒ25ƒ25 |
| Unit 2 | Strand 1: Scientific investigation skills and career exploration Strand 3: Energy and Momentum | Quiz 3% Total 14% | 25ƒ25ƒ25ƒ25 |
| Unit 3 | Strand 1: Scientific investigation skills and career exploration Strand 4: Electric, Gravitational and Magnetic Fields | Quiz 3% Total 14% | 25ƒ25ƒ25ƒ25 |
| Unit 4 | Strand 1: Scientific investigation skills and career exploration Strand 5: The Wave Nature of Light | Quiz 3% Total 14% | 25ƒ25ƒ25ƒ25 |
| Unit 5 | Strand 1: Scientific investigation skills and career exploration Strand 6: Revolutions in Modern Physics: Quantum Mechanics and Special Relativity | Quiz 3% Total 14% | 25ƒ25ƒ25ƒ25 |
| Culminating activity | 10% | 25ƒ25ƒ25ƒ25 | |
| Final Exam | 20% | 25ƒ25ƒ25ƒ25 | |
| Total | 100% | ||
| The percentage grade represents the quality of the students’ overall achievement of the expectations for the course and reflects the corresponding achievement as described in the achievement charts and will be 70% of the overall grade for the course; the Final evaluations will be 30% of the overall grade, incorporating a student/teacher conference and final exam. | |||
| Percentage of the Mark | Categories of Mark Breakdown | ||
| 70% | Assignments (25%) Tests (30%) Labs and Quiz (15%) | ||
| 30% | Culminating Activity (5%) and In Class discussion and presentations (Observations and Conversation (5%) Final Exam (20%) | ||
Main Resources: Textbook
Nelson Physics 12 University Preparation © 2012
Frequently Asked Questions (FAQ)
This course focuses on understanding rates of change, solving vector problems in 3D space, and applying calculus concepts to real-world scenarios.
Students must have completed Advanced Functions, Grade 12, University Preparation.
It is designed for students pursuing careers in science, engineering, economics, or business fields requiring university-level calculus, linear algebra, or physics.
Topics include derivatives, vector applications, curve sketching, intersections of lines and planes, and exponential and logarithmic differentiation.
The final grade is 80% coursework (assignments, quizzes, tests) and 20% final exam, emphasizing recent and consistent achievement.