Functions and Applications, Grade 11, University / College Preparation
Course Title: Functions and Applications, Grade 11, University / College Preparation (MCF3M)
Course Name: Functions and Applications
Course Code: MCF3M
Course Type: University / College Preparation
Credit Value: 1.0
Prerequisite: MPM2D, Principles of Mathematics, Grade 10, Academic or MFM2P, Foundations of Mathematics, Grade 10, Applied
Curriculum Policy Document: Mathematics, The Ontario Curriculum, Grades 11 and 12, 2007 (Revised)
Course Developer: Virtual High School
Department Head: André Bergeron, B.Sc., B.Ed., OCT
Development Date: 2004
Most Recent Revision Date: 2012
Tuition Fee (CAD): $579
This course introduces basic features of the function by extending students' experiences with quadratic relations. It focuses on quadratic, trigonometric, and exponential functions and their use in modeling real-world situations. Students will represent functions numerically, graphically, and algebraically; simplify expressions; solve equations; and solve problems relating to applications. Students will reason mathematically and communicate their thinking as they solve multi-step problems.
Resources required by the student:
Note: This course is entirely online and does not require or rely on any textbook.
- A scanner, smart phone camera, or similar device to digitize handwritten or hand-drawn work,
- A non-programmable, non-graphing, scientific calculator.
- Spreadsheet software (e.g. Microsoft ExcelTM, Mac NumbersTM, or equivalent)
Overall Curriculum Expectations
|A. Quadratic Functions|
|A1||expand and simplify quadratic expressions, solve quadratic equations, and relate the roots of a quadratic equation to the corresponding graph;|
|A2||demonstrate an understanding of functions, and make connections between the numeric, graphical, and algebraic representations of quadratic functions;|
|A3||solve problems involving quadratic functions, including problems arising from real-world applications.|
|B. Exponential Functions|
|B1||simplify and evaluate numerical expressions involving exponents, and make connections between the numeric, graphical, and algebraic representations of exponential functions;|
|B2||identify and represent exponential functions, and solve problems involving exponential functions, including problems arising from real-world applications;|
|B3||demonstrate an understanding of compound interest and annuities, and solve related problems.|
|C. Trigonometric Functions|
|C1||solve problems involving trigonometry in acute triangles using the sine law and the cosine law, including problems arising from real-world applications;|
|C2||demonstrate an understanding of periodic relationships and the sine function, and make connections between the numeric, graphical, and algebraic representations of sine functions;|
|C3||identify and represent sine functions, and solve problems involving sine functions, including problems arising from real-world applications.|
Teaching and Learning Strategies:
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 following mathematical processes are used throughout the course as strategies for teaching and learning the concepts presented:
- Problem solving: This course scaffolds learning by providing students with opportunities to review and activate prior knowledge (e.g. reviewing concepts related to quadratic equations from prior mathematics courses), and build off of this knowledge to acquire new skills. The course guides students toward recognizing opportunities to apply knowledge they have gained to solve problems.
- Selecting Tools and Computational Strategies: This course models the use of spreadsheet software and a TVM solver for personal finance to familiarize students with available software and resources which will allow them to simplify calculations in order to better and more accurately manage money.
- Connecting: The connects the concepts taught to real-world applications, such as simple harmonic motion and sound or light waves.
- Representing: Through the use of examples, practice problems, and solution videos, the course models various ways to demonstrate understanding, poses questions that require students to use different representations as they are working at each level of conceptual development - concrete, visual or symbolic, and allows individual students the time they need to solidify their understanding at each conceptual stage.
- Self-Assessment: Through the use of interactive activities (e.g. multiple choice quizzes, and drag-and-drop activities) students receive instantaneous feedback and are able to self-assess their understanding of concepts.
Assessment, Evaluation and Reporting Strategies of Student Performance:
Our theory of assessment and evaluation follows the Ministry of Education's Growing Success document, and it is our firm belief that doing so is in the best interests of students. We seek to design assessment in such a way as to make it possible to gather and show evidence of learning in a variety of ways to gradually release responsibility to the students, and to give multiple and varied opportunities to reflect on learning and receive detailed feedback.
Growing Success articulates the vision the Ministry has for the purpose and structure of assessment and evaluation techniques. There are seven fundamental principles that ensure best practices and procedures of assessment and evaluation by Virtual High School teachers. VHS assessments and evaluations,
- are fair, transparent, and equitable for all students;
- support all students, including those with special education needs, those who are learning the language of instruction (English or French), and those who are First Nation, Métis, or Inuit;
- are carefully planned to relate to the curriculum expectations and learning goals and, as much as possible, to the interests, learning styles and preferences, needs, and experiences of all students;
- are communicated clearly to students and parents at the beginning of the course and at other points throughout the school year or course;
- are ongoing, varied in nature, and administered over a period of time to provide multiple opportunities for students to demonstrate the full range of their learning;
- provide ongoing descriptive feedback that is clear, specific, meaningful, and timely to support improved learning and achievement;
- develop students’ self-assessment skills to enable them to assess their own learning, set specific goals, and plan next steps for their learning.
For a full explanation, please refer to Growing Success.
The Final Grade:
The evaluation for this course is based on the student's achievement of curriculum expectations and the demonstrated skills required for effective learning. The final 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 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 final evaluations administered at the end of the course. The final assessment may be a final exam, a final project, or a combination of both an exam and a project.
The Report Card:
Student achievement will be communicated formally to students via an official report card. Report cards are issued at the midterm point in the course, as well as upon completion of the course. Each report card will focus on two distinct, but related aspects of student achievement. First, the achievement of curriculum expectations is reported as a percentage grade. Additionally, the course median is reported as a percentage. The teacher will also provide written comments concerning the student's strengths, areas for improvement, and next steps. Second, the learning skills are reported as a letter grade, representing one of four levels of accomplishment. The report card also indicates whether an OSSD credit has been earned. Upon completion of a course, VHS will send a copy of the report card back to the student's home school (if in Ontario) where the course will be added to the ongoing list of courses on the student's Ontario Student Transcript. The report card will also be sent to the student's home address.
Program Planning Considerations:
Teachers who are planning a program in this subject will make an effort to take into account considerations for program planning that align with the Ontario Ministry of Education policy and initiatives in a number of important areas.
Planning Programs for Students with Special Education Needs
Virtual High School is committed to ensuring 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.
Virtual High School pays particular attention to the following beliefs: (1) all students can succeed, (2) each student has his or her own unique patterns of learning, (3) successful instructional practices are founded in evidence-based research, tempered by experience, (4) an open and accessible learning environment with differentiated instruction are effective and interconnected means of meeting the learning or productivity needs of any group of students, (5) classroom teachers are the key educators for a student's literacy and numeracy development, (6) classroom teachers need the support of the larger community to create a learning environment that supports students with special education needs, and finally, (7) fairness is not sameness.
The provision of special education programs and services for students at Virtual High School rests within a legal framework The Education Act and the regulations related to it set out the legal responsibilities pertaining to special education. They provide comprehensive procedures for the identification of exceptional pupils, for the placement of those pupils in educational settings where the special education programs and services appropriate to their needs can be delivered, and for the review of the identification of exceptional pupils and their placement.
If the student requires either accommodations, then Virtual High School will take into account these needs of exceptional students as they are set out in the students' existing Individual Education Plan. The online courses offer a vast array of opportunities for students with special educations needs to acquire the knowledge and skills required for our evolving society. Students who use alternative techniques for communication may find a venue to use these special skills in these courses. There are a number of technical and learning aids that can assist in meeting the needs of exceptional students as set out in their Individual Education Plan. In the process of taking their online course, students may use a personal amplification system, tele-typewriter (via Bell relay service), an oral or a sign-language interpreter, a scribe, specialized computer programs, time extensions, ability to change font size, oral readers, etc.
Accommodations (instructional, environmental or assessment) allow the student with special education needs access to the curriculum without changes to the course curriculum expectations. VHS will develop a Personal Education Plan (PEP) based on the student’s existing Individual Education Plan.
Program Considerations for English Language Learners
This Virtual High School online course provide a number of strategies to address the needs of ESL/ELD students. This online course must be flexible in order to accommodate the needs of students who require instruction in English as a second language or English literacy development. The Virtual High School 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. Virtual High 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. On a larger scale, well written content will aid ESL students in mastering not only the content of this course, but as well, the English language and all of its idiosyncrasies. Virtual High School has created course content to enrich the student's learning experience. Many occupations in Canada require employees with capabilities in the English language. Enabling students to learn English language skills will contribute to their success in the larger world.
Helping students become environmentally responsible is a role assumed by Virtual High School. 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. Good curriculum design following the resource document - The Ontario Curriculum, Grades 9-12: Environmental Education, Scope and Sequence of Expectations, 2011, will assist Virtual High School staff to weave environmental education in and out of the online course content. 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.
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 Virtual High School 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 of the Virtual High School community.
The most effective way to enable all students to learn about healthy and respectful relationships is through the school curriculum. Virtual High School teachers can promote this learning in a variety of ways. For example, they can help students develop and practise the skills they need for building healthy relationships by giving them opportunities to apply critical-thinking and problem solving strategies and to address issues through group discussions, role play, case study analysis, and other means. Virtual High School can also have a positive influence on students by modelling the behaviours, values, and skills that are needed to develop and sustain healthy relationships, and by taking advantage of “teachable moments” to address immediate relationship issues that may arise among students.
At Virtual High School, all staff strive to create a climate of cooperation, collaboration, respect, and open-mindedness. These attitudes and attributes enable our students to develop an awareness of the complexity of a range of issues. Moreover, in examining issues from multiple perspectives, students develop not only an understanding of various positions on these issues but also a respect for different points of view. Virtual High School students will hopefully develop empathy as they analyse events and issues from the perspectives of people all over the world. These attitudes and attributes provide a foundation on which students can develop their own identity, explore interconnectedness with others, and form and maintain healthy relationships.
Equity and Inclusive Education
The Virtual High School equity and inclusive education strategy focuses on respecting diversity, promoting inclusive education, and identifying and eliminating discriminatory biases, systemic barriers, and power dynamics that limit the ability of students to learn, grow, and contribute to society. Antidiscrimination education continues to be an important and integral component of this strategy.
In an environment based on the principles of inclusive education, all students, parents, caregivers, and other members of the school community - regardless of ancestry, culture, ethnicity, sex, physical or intellectual ability, race, religion, gender identity, sexual orientation, socio-economic status, or other similar factors - are welcomed, included, treated fairly, and respected. Diversity is valued, and all members of the Virtual High School community feel safe, comfortable, and accepted. Every student is supported and inspired to succeed in a culture of high expectations for learning. In an inclusive education system, all students see themselves reflected in the curriculum, their physical surroundings, and the broader environment, so that they can feel engaged in and empowered by their learning experiences.
Virtual High School can give students a variety of opportunities to learn about diversity and diverse perspectives. By drawing attention to the contributions of women, the perspectives of various ethno-cultural, religious, and racial communities, and the beliefs and practices of First Nations, Mêtis, and Inuit peoples, teachers enable Virtual High School students from a wide range of backgrounds to see themselves reflected in the curriculum. It is essential that learning activities and materials used to support the curriculum reflect the multicultural nature of society that is Canada. In addition, Virtual High School differentiates the instruction and assessment strategies to take into account the background and experiences, as well as the interests, aptitudes, and learning needs, of all students.
Financial Literacy Education
Financial literacy may be defined as having the knowledge and skills needed to make responsible economic and financial decisions with competence and confidence. Since making financial decisions has become an increasingly complex task in the modern world, students need to have knowledge in various areas and a wide range of skills in order to make informed decisions about financial matters. Students need to be aware of risks that accompany various financial choices. They need to develop an understanding of world economic forces as well as ways in which they themselves can respond to those influences and make informed choices. Virtual High School considers it essential that financial literacy be considered an important attribute of a well-educated population. 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 and other issues. The goal is to help students acquire the knowledge and skills that will enable them to understand and respond to complex issues regarding their own personal finances and the finances of their families, 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. The Ministry of Education and Virtual High School are working to embed financial literacy expectations and opportunities in all courses as appropriate, as part of the ongoing curriculum review process.
Literacy, Mathematical Literacy, and Inquiry Skills
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. Literacy includes critical thinking and reasoning to solve problems and make decisions related to issues of fairness, equity, and social justice. Literacy connects individuals and communities and is an essential tool for personal growth and active participation in a cohesive, democratic society. Literacy involves a range of critical-thinking skills and is essential for learning across the curriculum. Literacy instruction takes different forms of emphasis in different subjects, but in all subjects, literacy needs to be explicitly taught. 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.
Many of the activities and tasks that students undertake in the Virtual High School courses involve the literacy skills relating to oral, written, and visual communication. For example, they develop literacy skills by reading, interpreting, and analysing various texts. In addition, they develop the skills needed to construct, extract information from, and analyse various types information presented in a variety of media forms. In all Virtual High School courses, students are required to use appropriate and correct terminology, including that related to the concepts of disciplinary thinking, and are encouraged to use language with care and precision in order to communicate effectively.
Inquiry and research are at the heart of learning in all subject areas at Virtual 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 print and electronic sources. The questioning they practiced 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.
Critical Thinking and Critical Literacy
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 Virtual High School courses when they assess, analyse, and/or evaluate the impact of something and when they form an opinion about something and support that opinion with a rationale. In order to think critically, students need to examine the opinions and values of others, detect bias, 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, including many First Nations, Mêtis, and Inuit 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.
The development of these critical-thinking skills is supported in every course at Virtual High School. As students work to achieve the curriculum expectations in their particular course, students frequently need to identify the possible implications of choices. As they gather information from a variety of sources, they need to be able to interpret what they are listening to, reading, or viewing; to look for instances of bias; and to determine why a source might express a particular bias.
The Role of the School Library
The school library program in many schools 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. Virtual High School teachers assist students in accessing a variety of online resources and collections (e.g., professional articles, image galleries, videos, databases). Teachers at Virtual High School will also guide students through the concept of ownership of work and the importance of copyright in all forms of media.
The Role of Information and Communications Technology
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.
The Ontario Skills Passport: Making Learning Relevant and Building Skills
The Ontario Skills Passport (OSP) is a free, bilingual, web-based resource that provides teachers and students with clear descriptions of the "Essential Skills" and work habits important in work, learning, and life. Virtual High School can engage students by using OSP tools and resources to show how what they learn in class can be applied in the workplace and in everyday life. For further information on the Ontario Skills Passport, including the Essential Skills and work habits, visit http://www.skills.edu.gov.on.ca.
Education and Career/Life Planning
As online students progress through online courses, teachers are available to help the student prepare for employment in a number of diverse areas. With the help of teachers, students will learn to set and achieve goals and will gain experience in making meaningful decisions concerning career choices. The skills, knowledge and creativity that students acquire through this online course are essential for a wide range of careers. 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 framework of the program is a four-step inquiry process based on four questions linked to four areas of learning: (1) knowing yourself - Who am I?; (2) exploring opportunities - What are my opportunities?; (3) making decisions and setting goals - Who do I want to become?; and (4) achieving goals and making transitions - What is my plan for achieving my goals?
Cooperative Education and Other Forms of Experiential Learning
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. Virtual High School will try to help students link to Ministry programs to ensure that students have information concerning programs and opportunities.
Planning Program Pathways and Programs Leading to a Specialist High Skills Major
Virtual High School courses are well suited for inclusion in Specialist High Skills Majors (SHSMs) or in programs designed to provide pathways to particular apprenticeship, college, university, or workplace destinations. In some SHSM programs, courses at Virtual High School can be bundled with other courses to provide the academic knowledge and skills important to particular economic sectors and required for success in the workplace and postsecondary education, including apprenticeship training.
Health and Safety
In order to provide a suitable learning environment for the Virtual High School staff and students, it is critical that classroom practice and the learning environment complies with relevant federal, provincial, and municipal health and safety legislation and by-laws, including, but not limited to, the Workplace Safety and Insurance Act, the Workplace Hazardous Materials Information System (WHMIS), the Food and Drug Act, the Health Protection and Promotion Act, the Ontario Building Code, and the Occupational Health and Safety Act (OHSA). The OHSA requires all schools to provide a safe and productive learning and work environment for both students and employees.
Virtual 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 Virtual High School 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 at Virtual High School 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. 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 Virtual High School courses. Using accepted forms of documentation to acknowledge sources is a specific expectation within the inquiry and skill development strand for each course.