Calculus and Vectors
Grade 12, University Preparation (MCV4U)
|Course Title :||Calculus and Vectors, Grade 12, University Preparation (MCV4U)|
|Course Name :||Calculus and Vectors|
|Course Code :||MCV4U|
|Course Type :||University Preparation|
|Credit Value :||1.0|
|Prerequisite :||Advanced Functions, Grade 12, University Preparation|
|Curriculum Policy Document:||Mathematics, The Ontario Curriculum, Grades 11 and 12, 2010 (Revised)|
|Course Developer:||USCA Academy|
|Development Date:||August 2021|
|Most Recent Revision Date:||August 2021|
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 modeling 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
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; verify graphically and algebraically the rules for determining derivatives
A3 determine the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions, and simple combinations of functions; and solve related problems.
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.
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.
Outline of Course Content
|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.
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.
|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.
|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.
|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.
|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.
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.
|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.
|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.
The final assessment task is a three hour exam worth 30% of the student’s final mark.
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.
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.
|Classroom discussion Self-evaluation Peer assessment||Classroom discussion Small group discussion||Presentations of research Debates|
|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)
Poster presentations Tests
In Class Presentations
Some assessment and evaluation strategies include:
|Self-Assessment Quizzes||Diagnostic||Self/Teacher||Marking scheme|
|Problem Solving||Diagnostic||Self/Peer/Teacher||Marking scheme|
|Graphing Application||Diagnostic||Self||Anecdotal records|
|Teacher/Student Conferencing||Assessment||Self/Teacher||Anecdotal records|
|Problem Solving||Assessment||Teacher||Marking scheme|
|Problem Solving||Evaluation||Teacher||Marking scheme|
|Unit Tests||Evaluation||Teacher||Marking scheme|
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:
70% 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.
30% 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.
Textbook Nelson, Calculus and Vectors 12, 2008
- Graphing calculator
- Various internet websites
For the teachers who are planning a program in mathematics must take into account several important areas. The areas of concern to all teachers that are outlined in the policy document of Ontario Ministry of Education, include the following:
- teaching approaches
- types of secondary school courses
- education for exceptional students
- the role of technology in the curriculum
- English as a second language (ESL) and English literacy development (ELD)
- career education
- cooperative education and other workplace experiences
- health and safety in mathematics
It is important to ensure that all students, especially those with special education needs, are provided with the learning opportunities and supports they require to gain the knowledge, skills, and confidence needed to succeed in a rapidly changing society. The context of special education and the provision of special education programs and services for exceptional students in Ontario are constantly evolving. Provisions included in the Canadian Charter of Rights and Freedoms and the Ontario Human Rights Code have driven some of these changes. Others have resulted from the evolution and sharing of best practices related to the teaching and assessment of students with special educational needs. Accommodations (instructional, environmental or assessment) allow the student with special education needs access to the curriculum without changes to the course curriculum expectations.
Environmental education teaches students about how the planet's physical and biological systems work, and how we can create a more sustainable future. Good curriculum design following the resource document. This ensures that the student will have opportunities to acquire the knowledge, skills, perspectives and practices needed to become an environmentally literate citizen. The online course should provide opportunities for each student to address environmental issues in their home, in their local community, or even at the global level.
USCA helps students to become environmentally responsible. The first goal is to promote learning about environmental issues and solutions. The second goal is to engage students in practicing and promoting environmental stewardship in their community. The third goal stresses the importance of the education system providing leadership by implementing and promoting responsible environmental practices so that all stakeholders become dedicated to living more sustainably. Environmental education teaches students about how the planet's physical and biological systems work, and how we can create a more sustainable future.
USCA provides a number of strategies to address the needs of ESL/ELD students to accommodate the needs of students who require instruction in English as a second language or English literacy development. Our teacher considers it to be his or her responsibility to help students develop their ability to use the English language properly. Appropriate accommodations affecting the teaching, learning, and evaluation strategies in this course may be made in order to help students gain proficiency in English, since students taking English as a second language at the secondary level have limited time in which to develop this proficiency. School determines the student's level of proficiency in the English Language upon registration. This information is communicated to the teacher of the course following the registration and the teacher then invokes a number of strategies and resources to support the student in the course.
Throughout their secondary school education, students will learn about the educational and career opportunities that are available to them; explore and evaluate a variety of those opportunities; relate what they learn in their courses to potential careers in a variety of fields; and learn to make appropriate educational and career choices. The skills, knowledge and creativity that students acquire through this course are essential for a wide range of careers. Being able to express oneself in a clear concise manner without ambiguity in a second language, would be an overall intention of this course, as it helps students prepare for success in their working lives.
By applying the skills they have developed, students will readily connect their classroom learning to real-life activities in the world in which they live. Cooperative education and other workplace experiences will broaden their knowledge of employment opportunities in a wide range of fields. In addition, students will increase their understanding of workplace practices and the nature of the employer-employee relationship. Teachers should maintain links with community-based businesses to ensure that students have access to hands-on experiences that will reinforce the knowledge they have gained in school.
Every student is entitled to learn in a safe, caring environment, free from violence and harassment. Students learn and achieve better in such environments. The safe and supportive social environment at UCSA is founded on healthy relationships between all people. Healthy relationships are based on respect, caring, empathy, trust, and dignity, and thrive in an environment in which diversity is honoured and accepted. Healthy relationships do not tolerate abusive, controlling, violent, bullying/harassing, or other inappropriate behaviours. To experience themselves as valued and connected members of an inclusive social environment, students need to be involved in healthy relationships with their peers, teachers, and other members.
Critical thinking is the process of thinking about ideas or situations in order to understand them fully, identify their implications, make a judgement, and/or guide decision making. Critical thinking includes skills such as questioning, predicting, analysing, synthesizing, examining opinions, identifying values and issues, detecting bias, and distinguishing between alternatives. Students who are taught these skills become critical thinkers who can move beyond superficial conclusions to a deeper understanding of the issues they are examining. They are able to engage in an inquiry process in which they explore complex and multifaceted issues, and questions for which there may be no clear-cut answers.
The school library program in USCA can help build and transform students' knowledge in order to support lifelong learning in our information- and knowledge-based society. The school library program of these schools supports student success across the curriculum by encouraging students to read widely, teaching them to examine and read many forms of text for understanding and enjoyment, and helping them improve their research skills and effectively use information gathered through research. USCA teachers assist students in accessing a variety of online resources and collections (e.g., professional articles, image galleries, videos, databases). Teachers at USCA will also guide students through the concept of ownership of work and the importance of copyright in all forms of media.
Information literacy is the ability to access, select, gather, critically evaluate, and create information. Communication literacy refers to the ability to communicate information and to use the information obtained to solve problems and make decisions. Information and communications technologies are utilized by all Virtual High School students when the situation is appropriate within their online course. As a result, students will develop transferable skills through their experience with word processing, internet research, presentation software, and telecommunication tools, as would be expected in any other course or any business environment. Although the Internet is a powerful learning tool, there are potential risks attached to its use. All students must be made aware of issues related to Internet privacy, safety, and responsible use, as well as of the potential for abuse of this technology, particularly when it is used to promote hatred.
USCA provides varied opportunities for students to learn about ethical issues and to explore the role of ethics in both public and personal decision making. During the inquiry process, students may need to make ethical judgements when evaluating evidence and positions on various issues, and when drawing their own conclusions about issues, developments, and events. Teachers may need to help students in determining appropriate factors to consider when making such judgements. In addition, it is crucial that USCA teachers provide support and supervision to students throughout the inquiry process, ensuring that students engaged in an inquiry are aware of potential ethical concerns and address them in acceptable ways. Teachers will ensure that they thoroughly address the issue of plagiarism with students. In a digital world in which there is easy access to abundant information, it is very easy to copy the words of others and present them as one's own. Students need to be reminded, even at the secondary level, of the ethical issues surrounding plagiarism, and the consequences of plagiarism should be clearly discussed before students engage in an inquiry. It is important to discuss not only dishonest plagiarism but also more negligent plagiarism instances.
|Unit Number||Description||Assessments Evaluation Weight||KICA|
|Unit 1||Concepts of Calculus Tests, Assignments||(7%)||25/25/25/25|
|Unit 3||Curve Sketching||8%||25/25/25/25|
|Unit 4||Derivative applications and Related Rates||7%||25/25/25/25|
|Unit 5||Derivatives of Exponents and Log Functions||8%||25/25/25/25|
|Unit 6||Trig Differentiation and Application||8%||25/25/25/25|
|Unit 8||Vector Applications||8%||25/25/25/25|
|Unit 9||Intersections of Lines and Planes||8%||25/25/25/25|
|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 course; the final evaluation will be 30% of the overall grade.|
|Percentage of the Mark||Categories of Mark Breakdown|
|30%||Student/teacher conference (Observations and Conversation (10%)
Final Exam (30%)