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Calculus and Vectors
MCV4U
Course Overview
Department
Mathematics
Course Title
Calculus and Vectors
Grade
12
Course Type
University Preparation - Online Delivery
Course Code
MCV4U
Credit Value
1.0
Prerequisite
MHF4U, Advanced Functions, Grade 12, University Preparation (may be taken concurrently)
Developers
IMG Education
Revision Date
September 2019
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.
Course Resources
This course is entirely online and does not require any additional resources such as a textbook.<br>Students will be required to use:
Students will be required to use:
A scanner, smartphone camera, or similar device to digitize handwritten or hand-drawn work.
A non-programmable, non-graphing, scientific calculator.
Spreadsheet software (e.g. Google Sheets or equivalent).
Overall Course Expectations
Demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions;
Identify and describe some key features of the graphs of logarithmic functions, make connections among the numeric, graphical, and algebraic representations of logarithmic functions, and solve related problems graphically;
Solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications.
Demonstrate an understanding of the meaning and application of radian measure;
Make connections between trigonometric ratios and the graphical and algebraic representations of the corresponding trigonometric functions and between trigonometric functions and their reciprocals, and use these connections to solve problems;
Solve problems involving trigonometric equations and prove trigonometric identities.
Identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of polynomial functions;
Identify and describe some key features of the graphs of rational functions, and represent rational functions graphically;
Solve problems involving polynomial and simple rational equations graphically and algebraically;
Demonstrate an understanding of solving polynomial and simple rational inequalities.
Demonstrate an understanding of average and instantaneous rate of change, and determine, numerically and graphically, and interpret the average rate of change of a function over a given interval and the instantaneous rate of change of a function at a given point;
Determine functions that result from the addition, subtraction, multiplication, and division of two functions and from the composition of two functions, describe some properties of the resulting functions, and solve related problems;
Compare the characteristics of functions, and solve problems by modelling and reasoning with functions, including problems with solutions that are not accessible by standard algebraic techniques.
Course Content Outline
-Rate of Change & Limits: different ways to manipulate limit so it can be evaluated
-Continuity of Functions: piecewise functions; types of continuity; testing continuity using limits
-Derivative Rules: power, product, quotient, chain
-Displacement, Velocity, Acceleration: relate and interpret graphs; algebraic relationships using Rate of Change; derivatives
-Extreme Values & Optimization: identifying maximum/minimum using 1st derivative; various optimization scenarios (for example: optimization of distance, optimization of time, optimization of money)
-Original Function: intervals of increase/decrease; asymptotes (vertical, horizontal, oblique); use limits to classify behaviour
-First Derivative: critical points (can also be used for intervals if increase/decrease; identifies one point of inflection (where m=0)
-Second Derivative: points of inflection; concavity (can use to classify critical points as max/min)
-Interval Table(s): use to organize entire solution before sketching
-Exponential Functions: natural number, e; derivative of natural & general exponential function
-Logarithmic Functions: relation between logarithm and exponential functions; using relationship to solve equations; derivatives of logarithmic functions
-Trigonometric Functions: derivatives of sine, cosine, tangent, reciprocals
-Optimization & Curve Sketching
-Vector & Vector Operations: add, subtract, scalar multiplication
-Vectors in R2 and R3: graphical representation; algebraic representations
-Combining Vectors: spanning sets
-Force Vectors, Velocity Vectors: vector components; resultant and equilibrant vectors
-Algebraic Operations with Vectors: dot product, cross product; perpendicular vectors; angles between vectors; scalar & vector projections
-Applications of Vector Operations: work (dot product); torque (cross product); area of parallelogram
-Representations of Lines & Planes; lines in R2: slope ­intercept, Cartesian, vector; lines in R3: vector, parametric, symmetric; planes in R3: vector, parametric, Cartesian
-Intersection of Lines & Planes: geometric explanations and interpretations; algebraic techniques for solving – interpretation of solution(s)
-Distances between Points, Lines, Planes: manipulate given information to produce a geometry that works with given equation(s)
Total
110 hrs
Teaching/Learning Strategies
A variety of teaching and learning strategies are used, including but not limited to:
Asynchronous video lessons, explaining and modeling key concepts
Asynchronous Power Points with key concepts, applications and relevant examples
Synchronous sessions with the instructor
Inquiry and problem solving process using a prescribed series of steps
Oral and written reports and presentations
Open ended discussion questions
Reading responses focusing on critical thinking and analysis
Research skills focused upon using secondary sources to support ideas
Written responses, often scaffolded with graphic organizers
Mind maps: organize societal, environmental, and economic consideration of concepts discussed throughout the course
Case Study Analysis: students will consider data from experiments or research to answer analysis/thinking questions
Brainstorming: group generation of ideas expressed without criticism or analysis
Collaborative and cooperative learning: small group learning providing high levels of student engagement and interdependence
Structured discussion and debate: allows for detailed analysis
Formal presentation: collaborative approach to constructing presentations
Student Led Discussion: students will design open-ended critical thinking questions to discuss with their classmates and facilitate discussion
Use success criteria to set goals for learning
Develop and use checklists to reflect on skills used and those to improve
Specific questions and discussion will guided self-reflection skills for reflecting on on-going learning, process, interpretation and evaluation
Assessment and Evaluation
The primary purpose of assessment and evaluation is to improve student learning. Assessment for learning, as learning and of learning as outlined in the Ministry document Growing Success will be the focus in this course.
Assessment will be ongoing and is intended to provide students with feedback for improvement, as intended in using for and as assessment procedures.
Examples of Assessment For/As Learning
Graphic organizers
Teacher feedback
Peer feedback
Self-reflection
Practice with feedback
Sticky note activities
Placemat activities
Jigsaw
Practice quizzes
Success Criteria
Exit tickets
Goal setting
Reading/writing logs
Examples of Assessment Of Learning
Quizzes/Tests
Projects
Differentiated Products (e.g. newscast, graphic organizers, outlines)
Oral reports
Presentations
Media production
Technology products
Evaluation will be summative and will occur at key points during the term (70 percent) and at/near the end of the course (30 percent). Evaluation will provide information about student achievement based on student performance in the four areas of Knowledge/Understanding, Thinking/Inquiry, Communication and Application. In determining a final grade, the teacher will especially consider the student’s most recent consistent level of achievement.
Products including projects, portfolios, essays, reports, written assignments, in-class (online) assignments, and rich performance tasks such as presentations, seminars, independent research, exhibitions, recitals, skills demonstration, role-playing, and work samples.
Observations including instructional question and answer sessions for review of previous material, questions and answers, journals/learning logs, group and independent work, tests and quizzes, and discussions.
Conversations, including, student-teacher interviews and conferences, informal feedback and review, and discussions to review self and peer evaluations.
The final evaluation is comprised of an exam, independent student project, or a combination of both, that is worth 30 percent of the final mark. It will that evaluate student mastery of the curriculum in these areas: knowledge/understanding, thinking/inquiry, application and communication.
KTAC Total % of Course
Knowledge 25%
Thinking 25%
Communication 25%
Application 25%
100%

The following table provides a summary description of achievement in each percentage grade range and corresponding level of achievement:

Percentage Grade Range

Achievement Level

Summary Description

95-100%

87-94%

80-86%

Level 4+

Level 4

Level 4-

A very high to outstanding level of achievement. Achievement is above the provincial standard

77-79%

73-76%

70-72%

Level 3+

Level 3

Level 3-

A high level of achievement. Achievement is at the provincial standard.

67-69%

63-66%

60-62%

Level 2+

Level 2

Level 2-

A moderate level of achievement. Achievement is below, but approaching, the provincial standard.

57-59%

53-56%

50-52%

Level 1+

Level 1

Level 1-

A passable level of achievement. Achievement is below the provincial standard

Below 50%:  Insufficient achievement of curriculum expectations. A credit will not be granted.

 

Level 3 (70-79%) is the provincial standard. Teachers and parents can be confident that students who are achieving at level 3 are well prepared for work in the next grade or the next course.

 

Student Achievement Chart - Levels of Achievement

The achievement chart also identifies four levels of achievement, defined as follows:

 

Level 1 (50-59%) represents achievement that falls much below the provincial standard.The student demonstrates the specified knowledge and skills with limited effectiveness. Students must work at significantly improving learning in specific areas, as necessary, if they are to be successful in the next grade/course.

 

Level 2 (60-69%) represents achievement that approaches the provincial standard. The student demonstrates the specified knowledge and skills with some effectiveness. Students performing at this level need to work on identified learning gaps to ensure future success.

 

Level 3 (70-79%) represents the provincial standard for achievement. The student demonstrates the specified knowledge and skills with considerable effectiveness. Parents of students achieving at level 3 can be confident that their children will be prepared for work in subsequent grades/courses.

 

Level 4 (80% +) identifies achievement that surpasses the provincial standard.The student demonstrates the specified knowledge and skills with a high degree of effectiveness.However, achievement at level 4 does not mean that the student has achieved expectations beyond those specified for the grade/course.

 

Knowledge and Understanding

  • knowledge of content
  • understanding of content

Thinking

  • use of planning skills
  • use of processing skills
  • use of critical/creative thinking processes, skills, and strategies

Communication

  • expression and organization of ideas and information in oral, visual, and/or written forms
  • communication for different audiences and purposes in oral, visual, and/or  written forms
  • use of conventions, vocabulary, and terminology of the discipline in oral, visual, and/or written forms

Application

  • application of knowledge and skills in familiar contexts
  • transfer of knowledge and skills to new contexts
  • making connections within and between various contexts

 

Learning Skills

The development of learning skills and work habits is an integral part of a student’s learning. To the extent possible, however, the evaluation of learning skills and work habits, apart from any that may be included as part of a curriculum expectation in a subject or course, will not be considered in the determination of a student’s grades.

Learning Skills and Work Habits

Sample Behaviours

The Student:

Responsibility

  • fulfils responsibilities and commitments within the learning environment;
  • completes and submits class work, homework,and assignments according to agreed-upon timelines;
  • takes responsibility for and manages own behaviour.

Organization

  • devises and follows a plan and process for completing work and tasks;
  • establishes priorities and manages time to complete tasks and achieve goals;
  • identifies,gathers,evaluates,and uses information, technology, and resources to complete tasks.

Independent Work

  • independently monitors, assesses, and revises plans to complete tasks and meet goals;
  • uses class time appropriately to complete tasks;
  • follows instructions with minimal supervision.

Collaboration

  • accepts various roles and an equitable share of work in a group;
  • responds positively to the ideas,opinions,values,and traditions of others;
  • builds healthy peer-to-peer relationships through personal and media-assisted interactions;
  • works with others to resolve conflicts and build consensus to achieve group goals;
  • shares information, resources, and expertise and promotes critical thinking to solve problems and make decisions.

Initiative

  • looks for and acts on new ideas and opportunities for learning;
  • demonstrates the capacity for innovation and a willingness to take risks;
  • demonstrates curiosity and interest in learning;
  • approaches new tasks with a positive attitude;
  • recognizes and advocates appropriately for the rights of self and others.

Self-regulation

  • sets own individual goals and monitors progress towards achieving them;
  • seeks clarification or assistance when needed;
  • assesses and reflects critically on own strengths,needs,and interests;
  • identifies learning opportunities,choices,and strategies to meet personal needs and achieve goals;
  • perseveres and makes an effort when responding to challenges.
Program Planning Considerations

At Freedom High School we believe in every students’ ability to succeed and we are committed to enabling and empowering each student to reach his or her full potential. We recognize that every student has unique interests, abilities, and goals.   

Every course at Freedom High School is grounded in the principles of Universal Design and allows for a flexible learning environment that can accommodate individual learning differences. Our mission is to provide education that is not limited by physical, financial, or social factors. Instruction as well as assessment is differentiated and adapted to support every student’s needs and ensure each student's’ success. The use of technology supports students by allowing each learner to learn at his or her own speed.  Educators pay particular attention to the following beliefs:

  • all children can succeed,
  • each child has his or her own unique patterns of learning,
  • successful instructional practices are founded on evidence-based research, tempered by experience,
  • universal design and differentiated instruction are effective and interconnected means of meeting the learning or productivity needs of any group of children,
  • classroom educators are the key educators for a child's literacy and numeracy development,
  • classroom educators need the support of the larger community to create a learning environment that supports children with special education needs, and
  • fairness is not sameness.

Based on legislation in Ontario (Education Act, Ontario Human Rights Code, Charter of Rights and Freedoms and the Special Education in Ontario Policy and Resource Guide, 2017), Freedom High School accommodates students requiring course accommodations.

Students needing instructional, environment, or assessment accommodations in a course of study without change to the curriculum expectations are eligible to earn a course credit through the program offerings at Freedom High School.

Freedom High School supports English Language Learners and English as a Second Language learners both through dedicated ESL/ELL courses as well as within credit courses.  Our language learning courses help learners acquire a basic level of English language proficiency and prepare for credit courses. Additionally, each credit course employs best practices and strategies to support ESL and ELL learners.   

The ELL and ESL policy at Freedom High School is based on The Ontario Curriculum Grades 9-12: English as a Second Language and English Literacy Development, 2007 policy and best practices to support English Language Learners. Freedom High School offers support for English Language Learners in a variety of accommodations related to:

  • instructional strategies, including but not limited to:
    • bilingual resources,
    • cloze activities,
    • visual cues,
    • cooperative learning and peer tutoring,
    • guided reading and writing;
  • learning resources, including but not limited to:
    • graphic organizers,
    • use of visual materials,
    • simplified texts,
    • bilingual dictionaries,
    • jigsaw and learning games,
    • role play activities;
  • assessment accommodations (timing/scheduling, setting, presentation, response), including but not limited to:
    • extra time or scheduling accommodations,   
    • use of exemplars,
    • key word lists / translation on assessments,
    • oral responses,
    • scaffolded written responses.

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.

Students use critical-thinking skills in when they assess, analyse, and/or evaluate the impact of something and when they form an opinion and support that opinion with a rationale. In order to think critically, students need to ask themselves effective questions in order to: interpret information; analyse situations; detect bias in their sources; determine why a source might express a particular bias; examine the opinions, perspectives, and values of various groups and individuals; look for implied meaning; and use the information gathered to form a personal opinion or stance, or a personal plan of action with regard to making a difference.

Students approach critical thinking in various ways. Some students find it helpful to discuss their thinking, asking questions and exploring ideas. Other students may take time to observe a situation or consider a text carefully before commenting; they may prefer not to ask questions or express their thoughts orally while they are thinking.

Freedom High School courses support critical thinking by giving students opportunities to interpret information, analyse situations, and detect biases.

Literacy, mathematical literacy, and inquiry/research skills are critical to students’ success in all subjects of the curriculum and in all areas of their lives.

Literacy is defined as the ability to use language and images in rich and varied forms to read, write, listen, view, represent, and think critically about ideas. It involves the capacity to access, manage, and evaluate information; to think imaginatively and analytically; and to communicate thoughts and ideas effectively. The activities and tasks that students undertake in all Freedom High School courses involve oral, written, and visual communication skills. For example, students use language to record their observations, to describe their inquiries in both informal and formal contexts, and to present their findings in presentations and reports in oral, written, graphic, and multimedia forms. 

The acquisition and development of literacy skills is promoted in all Freedom High School courses and builds on, reinforces, and enhances mathematical literacy. For example, clear, concise communication often involves the use of diagrams, charts, tables, graphs, and graphic text.

Inquiry is at the heart of learning in all courses offered at Freedom High School. Students are encouraged to develop their ability to ask questions and to explore a variety of possible answers to those questions. As they advance through the grades, they acquire the skills to locate relevant information from a variety of sources, such as books, newspapers, dictionaries, encyclopedias, interviews, videos, and the Internet. The questioning they practised in the early grades becomes more sophisticated as they learn that all sources of information have a particular point of view and that the recipient of the information has a responsibility to evaluate it, determine its validity and relevance, and use it in appropriate ways. The ability to locate, question, and validate information allows a student to become an independent, lifelong learner.

Acting Today, Shaping Tomorrow: A Policy Framework for Environmental Education in Ontario Schools, 2009 states that schools “have a vital role to play in preparing our young people to take their place as informed, engaged, and empowered citizens who will be pivotal in shaping the future of our communities, our province, our country and our global environment.”

The three goals of environmental education are organized around the themes of teaching and learning, student engagement and community connections, and environmental leadership. The first goal is to promote learning about environmental issues and solutions. The second goal is to engage learners in practising and promoting environmental stewardship, both at school and in the community. The third goal stresses the importance of providing leadership by implementing and promoting responsible environmental practices throughout the education system so that staff, parents, community members, and children become dedicated to living more sustainably.

The environmental education goals outlined in Acting Today, Shaping Tomorrow  have been embedded in each course of study offered by Freedom High School.  Instead of learning about environmental topics in one course, students learn about sustainability, stewardship, and the Earth’s biological and physical systems through our interdisciplinary approach to teaching and learning.

At Freedom High School, we pride ourselves on our commitment to equity, inclusivity and diversity, where each person, regardless of his or her ancestry, culture, ethnicity, sex, physical or intellectual ability, race, religion, gender identity, sexual orientation, socio-economic status, is welcomed and valued as an integral member of our community.  Freedom High School seeks to provide a safe, inclusive, and supportive environment where all our stakeholders, including students, teachers, staff, parents and guardians, and all community members feel included, respected, and treated fairly.  

In addition to creating an equitable and inclusive environment, Freedom High School also draws attention to the contributions and perspectives of First Nations, Metis, and Inuit peoples.  All our courses are developed with the Ontario First Nations, Metis, and Inuit Education Policy Framework in mind and our curriculum reflects the diversity of Canadian society.

Financial literacy may be defined as “having the knowledge and skills needed to make responsible economic and financial decisions with competence and confidence”. Freedom High School believes that financial literacy is an important component in developing well-rounded individuals and setting them up for future success. In addition to acquiring knowledge in such specific areas as saving, spending, borrowing, and investing, students need to develop skills in problem solving, inquiry, decision making, critical thinking, and critical literacy related to financial issues.

The goal at Freedom High School is to help students acquire the knowledge, skills and attitudes that will enable them to understand and respond to complex issues regarding their personal finances, as well as to develop an understanding of local and global effects of world economic forces and the social, environmental, and ethical implications of their own choices as consumers. Using The Ontario Curriculum Grades 9-12: Financial Literacy Scope and Sequence of Expectations, 2016 resource guide, Freedom High School endeavours to integrate financial literacy learning opportunities into all of our courses as appropriate to that subject.

The school library can help to build and transform students’ knowledge to support lifelong learning in our information- and knowledge-based society. The school library program supports student success across the language curriculum by encouraging students to read widely, teaching them to read for understanding and enjoyment, and helping them to improve their research skills and to use information gathered through research effectively. Freedom High School supports students by providing access to a multitude of online resources such as image and video libraries, digital libraries and databases, articles, journals, and more. FHS teachers also promote the respect of intellectual property rights for all literature and media.

Increasing reliance on computers, telecommunication networks, and information technologies in society and the workplace makes it essential for students to become computer literate and to develop “information literacy” skills. Information literacy is the ability to access, select, gather, critically evaluate, create, and communicate information, and to use the information obtained to solve problems and make decisions. In preparation for further education, employment, citizenship, and lifelong learning, students must be capable of deriving meaning from information by using a wide variety of information literacy skills.

Freedom High School uses technology in courses, where appropriate, to ensure that students acquire the knowledge, skills, and attitudes that will allow them to use computer and information technology safely, effectively, confidently, and ethically throughout their academic careers and lives.

Freedom High School courses integrate current events and issues within the curriculum expectations and do not treat them as separate topics. The integration of current events and issues into the curriculum helps students make connections between what they are learning in class and past and present-day local, national, and global events, developments, and issues. Examining current events helps students analyse controversial issues, understand diverse perspectives, develop informed opinions, and build a deeper understanding of the world in which they live. In addition, investigating current events will stimulate students’ interest in and curiosity about the world around them.

The guidance and career education program plays a central role in secondary schools by providing students with the tools they need for success in school, in the workplace, and in their daily lives. Consulting the Ontario Ministry of Education Guidance and Career Education Curriculum documents (The Ontario Curriculum Grades 9 and 10: Guidance and Career Education, 2006, The Ontario Curriculum Grade 11 and 12: Guidance and Career Education, 2006, and The Ontario Curriculum Grades 9 to 12: Course Descriptions and Prerequisites, 2011, and other policy and resource documents), Freedom High School aims to support students in their program planning. Freedom High School provides guidance to students in the form of course selection and academic counselling to ensure that students are taking the appropriate courses to fulfill their academic goals. Students will be encouraged to keep a journal throughout their courses and record their strengths and interests as it relates to future career plans. Freedom High School also assists students in applying for university and/or college by communication student achievement directly to post-secondary institutions.  

Freedom High School sees cooperative education and workplace experience as an important aspect of education.  We believe that these experiences will enrich a student's understanding of curriculum through real-life learning and increase a student's knowledge of employment opportunities.  Although Freedom High School does not currently offer cooperative education programs, the school supports students who wish to apply for a cooperative or workplace experience and connects students with Ministry of Education policies and links.

Freedom High School courses integrate current events and issues within the curriculum expectations and do not treat them as separate topics. The integration of current events and issues into the curriculum helps students make connections between what they are learning in class and past and present-day local, national, and global events, developments, and issues. Examining current events helps students analyse controversial issues, understand diverse perspectives, develop informed opinions, and build a deeper understanding of the world in which they live. In addition, investigating current events will stimulate students’ interest in and curiosity about the world around them.

Freedom High School aims to provide and safe, healthy and caring learning environment for all its stakeholders, where all students, staff, and parents or guardians are and feel safe, included and accepted.  Freedom High School provides its stakeholders with freedom from violence and harassment, ensuring that each person feels safe and supported. The learning environment, instructional materials, and teaching and assessment strategies reflect the diversity of all learners. To ensure a safe and healthy environment, Freedom High School complies with all provincial and federal health and safety legislation. The Ontario Curriculum Grades 9-12: Health and Safety Scope and Sequence of Expectations, 2017

Freedom High School courses provide 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 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. If students are conducting surveys and/or interviews, teachers must supervise their activities to ensure that they respect the dignity, privacy, and confidentiality of their participants. Teachers should ensure that they thoroughly address the issue of plagiarism with students. In a digital world in which we have 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 the more “blatant” forms of plagiarism, but also more nuanced instances that can occur. Students often struggle to find a balance between writing in their own voice and acknowledging the work of others in the field. Merely telling students not to plagiarize, and admonishing those who do, is not enough. The skill of writing in one’s own voice, while appropriately acknowledging the work of others, must be explicitly taught to all students. Using accepted forms of documentation to acknowledge sources is a specific expectation within the inquiry and skill development strand for each course.