MCV4U – Calculus and Vectors Grade 12

Course Type: University Preparation

Credit: 1.0

Ontario Curriculum: Grade 12 Mathematics – Calculus & Vectors

MCV4U Prerequisite: The Advanced Functions course MHF4U must be taken prior to or concurrently with Calculus and Vectors MCV4U.

NCAA Approved Course

MCV4U – Calculus and Vectors Grade 12 course builds on students’ previous experience with functions and their developing understanding of rates of change. Students in MCV4U 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 MCV4U 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.

MCV4U Course Outline

Unit 1 – Rate of Change
Students will 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. Students will learn about limit properties and continuous and
discontinuous functions.
Unit 2 – Derivatives and their Applications
Students will graph the derivatives of polynomial functions and make connections between the numeric,
graphical, and algebraic representations of a function and its derivative. Students will verify graphically and
algebraically the rules for determining derivatives. They will further apply these rules to determine the
derivatives of polynomial, rational and radical functions, and simple combinations of functions. Students
will solve related problems.
Unit 3- Curve Sketching and Optimization
Students will 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. Students will 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.
Unit 4 – Derivatives of Exponential and Trigonometric Functions
Students will graph the derivatives of sinusoidal, and exponential functions, and make connections between
the numeric, graphical, and algebraic representations of a function and its derivative. Students will verify
graphically and algebraically the rules for determining derivatives and apply these rules to determine the
derivatives of sinusoidal and exponential functions, and simple combinations of functions. In addition, they
will solve related problems.
Unit 5 – Geometry and Algebra of Vectors
Students will demonstrate an understanding of vectors in two-space and three-space by representing them
algebraically and geometrically and by recognizing their applications in real world scenarios. In addition,
students will 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. Students will 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. In addition, students will represent lines and planes using scalar, vector, and parametric
equations, and solve problems involving distances and intersections of lines and planes.

A. Rate of Change
By the end of this course, students will:
● 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
By the end of this course, students will:
● 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
By the end of this course, students will:
● 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.

In this course, students will experience the following activities.

General:

Presentations with embedded videos are utilized to outline concepts, explain theory with the use of
examples and practice questions, and incorporate multi-media opportunities for students to learn more
(e.g. online simulations, quizzes, etc.).

Reflection is an opportunity for students to look back at concepts and theories with new eyes, to relate
theory to practice, and to align learning with their own values and beliefs.

Discussions with the instructor are facilitated through video conferencing, discussing the concepts and
skills being studied. This enables two-way communication between the student and the instructor, to share ideas and ask questions in dialogue. This also helps to build a relationship between the student and
instructor.

Instructor demonstrations (research skills, etc.) are opportunities for the instructor to lead a student
through a concept or skill through video conferencing, videos, or emailing with the student.

Discussion forums are an opportunity for students to summarize and share their ideas and perspectives
with their peers, which deepens understanding through expression. It also provides an opportunity for
peer-to-peer feedback.

Practical extension and application of knowledge are integrated throughout the course. The goal is to help
students make connections between what they learn in the classroom and how they understand and relate
to the world around them and their own lives. Learning becomes a dynamic opportunity for students to be
more aware that their learning is all around them and enable them to create more meaning in their lives.

Individual activities/assignments assessments are completed individually at a student’s own pace and are
intended to expand and consolidate the learning in each lesson. Individual activities allow the teacher to
accommodate interests and needs and to assess the progress of individual students. For this reason,
students are encouraged to discuss IEPs (Individual Education Plans) with their teacher and to ask to modify
assessments if they have a unique interest that they feel could be pursued in the assessment. The teacher
plays an important role in supporting these activities by providing ongoing feedback to students, both orally
and in writing.

STEM:

Virtual simulations are interactive websites that provide students with an opportunity to ask questions,
explore hypotheses, relate variables, examine relationships, and make connections between theory and
application in a safe environment that promotes intellectual risk taking and curiosity.

Virtual labs are interactive websites that provide students with an opportunity to follow a procedure to test
hypotheses using scientific apparatus, gather and record observations, analyze observations using formula
and relevant theory/concepts, and then formulate conclusions that relate hypotheses to analysis.

Diagrams are visual representations of scientific ideas and concepts. They provide another perspective to
organize ideas. Visuals are thought to promote cognitive plasticity – meaning, they can help us change our
minds or help us to remember an idea.

Graphics/images are visual representations of ideas/concepts. Visuals are thought to promote cognitive
plasticity – meaning, they can help us change our minds or help us to remember an idea.

Charts are visual representations of scientific ideas and concepts using math that support analysis.

Tables involve organizing information in terms of categories (rows and columns). This helps us to
understand the relationships between ideas and data, as well as highlight trends.

Practice problems provide students with a scenario/problem to solve by applying concepts and skills
learned in a context. This helps students to understand the relevance of their learning.

Design projects are an opportunity for students to put their learning to the test in a real-world scenario, to
address a design problem with a direct connection to people, the environment, economics, etc. Students
collect information, apply problem solving, and use critical thinking to develop practical solutions that
directly address their design problems.

As summarized in Growing Success 2010, the primary purpose of assessment and evaluation is to improve student learning. Information gathered through assessment helps teachers to determine students’ strengths and weaknesses in their achievement of the curriculum expectations in each course. This information also serves to guide teachers in adapting curriculum and instructional approaches to students’ needs and in assessing the overall effectiveness of programs and classroom practices. As part of assessment, teachers provide students with descriptive feedback that guides their efforts towards improvement.

Evaluation refers to the process of judging the quality of student work on the basis of established criteria, and assigning a value to represent that quality. All curriculum expectations must be accounted for in instruction, but evaluation focuses on students’ achievement of the overall expectations. A students’ achievement of the overall expectations is evaluated on the basis of his or her achievement of related specific expectations. Teachers will use their professional judgement to determine which specific expectations should be used to evaluate achievement of overall expectations, and which ones will be covered in instruction and assessment but not necessarily evaluated.

In order to ensure that assessment and evaluation are valid and reliable, and that they lead to the improvement of student learning, teachers must use assessment and evaluation strategies that:

● Address both what students learn and how well they learn;
● Are based both on the categories of knowledge and skills and on the achievement level descriptions given in the achievement chart
● Are varied in nature, administered over a period of time, and designed to provide opportunities for students to demonstrate the full range of their learning;
● Are appropriate for the learning activities used, the purposes of instruction, and the needs and experiences of the students;
● Are fair to all students;
● Accommodate students with special education needs, consistent with the strategies outlined in their Individual Education Plan;
● Accommodate the needs of students who are learning the language of instruction;
● Ensure that each student is given clear directions for improvement;
● Promote students’ ability to assess their own learning and to set specific goals
● Include the use of samples of students’ work that provide evidence of their achievement;
● Are communicated clearly to students and parents at the beginning of the school year and at other appropriate points throughout the school year.

The final grade will be determined as follows:

❑ 70% of the grade will be based on evaluation conducted throughout the course. This
portion of the grade should reflect the student’s most consistent level of achievement
throughout the course, although special consideration will be given to more recent
evidence of achievement.

❑ 30% of the grade will be based on a final evaluation administered at or towards the end of
the course. This evaluation will be based on evidence from one or a combination of the
following: an examination, a performance, and/or another method of evaluation suitable to
the course content. The final evaluation allows the student an opportunity to demonstrate
comprehensive achievement of the overall expectations for the course.

(Growing Success: Assessment, Evaluation and Reporting in Ontario Schools. Ontario
Ministry of Education Publication, 2010 p.41)

All students can succeed. Some students are able, with certain accommodations, to participate in the regular course curriculum and to demonstrate learning independently. Accommodations allow access to the course without any changes to the knowledge and skills the student is expected to demonstrate. The accommodations required to facilitate the student’s learning can be identified by the teacher, but recommendations from a School Board generated Individual Education Plan (IEP) if available can also be consulted. Instruction based on principles of universal design and differentiated instruction focuses on the provision of accommodations to meet the diverse needs of learners.

Examples of accommodations (but not limited to) include:

• Adjustment and or extension of time required to complete assignments or summative tasks
• Providing alternative assignments or summative tasks
• Use of scribes and/or other assistive technologies
• Simplifying the language of instruction

To learn more go to our Individual Education Plan (IEP) page.

To learn more about this course including tests and exams please visit our FAQ page