Course Description
This course broadens students’ knowledge of functions. Students will examine the properties of polynomial, rational, logarithmic, and trigonometric functions, as well as create ways for combining them, increase their understanding of rates of change, and improve their ability to apply these concepts and skills. Students will also improve their understanding of the mathematical procedures required for senior mathematics success. This course is designed for students who are taking Calculus and Vectors as a prerequisite for a university degree, as well as those who want to brush up on their math skills before enrolling in one of the many university programs.
- Course Code: MHF4U
- Department: Mathematics
- Course Type:
- Credit Value:1.0
- Prerequisite:MCR3U, Functions, Grade 11, University or MCT4C, Mathematics for College Technology, Grade 12, College
- Curriculum Policy Document:Mathematics, The Ontario Curriculum, Grades 11 and 12, 2007
Course Outline
Advanced Functions MHF4U – Total Hours (110 Hours)
- Unit One: Characteristics of Functions (26 Hours) In this unit, students will recall a few basic facts of a distance time graph. They will review transformations and their effects on a given graph. The concepts of function notation, relation, range, domain, and function notation will all be reviewed. Finally, strategies will be employed to aid in the understanding of rates of change.
- Unit Two: Polynomial and Rational Functions (27 Hours) In this unit the students will investigate polynomial and rational functions. They will extend their knowledge about linear and quadratic functions to include cubic, quartic and quintic functions. Students will explore their graphs and characteristics, also distinguish polynomial functions from sinusoidal and exponential functions, and compare and contrast the graphs of various polynomial functions with the graphs of other types of functions. Students will also determine, through investigation with and without technology, key features (i.e., vertical and horizontal asymptotes, domain and range, intercepts, positive/negative intervals, increasing/decreasing intervals) of the graphs of rational functions that are the reciprocals of linear and quadratic functions, and make connections between the algebraic and graphical representations of these rational functions. Students will solve problems involving applications of simple rational functions and equations and explain the difference between the solution to an equation in one variable and the solution to an inequality in one variable, also demonstrate that given solutions satisfy an inequality and determine solutions to simple rational inequalities in one variable by graphing the corresponding functions, using graphing technology, and identifying intervals for which x satisfies the inequalities.
- Unit Three: Trigonometric Functions (27 Hours) In this unit, students recognize the radian as an alternative unit to the degree for angle measurement, define the radian measure of an angle as the length of the arc that subtends this angle at the centre of a unit circle, and develop and apply the relationship between radian and degree measure. Students will also sketch the graphs of f(x) = sinx and f(x) = cosx for angle measures expressed in radians, and determine and describe some key properties (e.g., period of 2 π , amplitude of 1) in terms of radians. Students represent a sinusoidal function with an equation, given its graph or its properties, with angles expressed in radians. Students recognize that trigonometric identities are equations that are true for every value in the domain, prove trigonometric identities through the application of reasoning skills, using a variety of relationships, and verify identities using technology.
- Unit Four : Exponential and Logarithmic Functions (27 Hours) In this unit, students make connections between related logarithmic and exponential equations and between the laws of exponents and the laws of logarithms, verify the laws of logarithms with or without technology, and use the laws of logarithms to simplify and evaluate numerical expressions. Students will also solve problems involving exponential and logarithmic equations algebraically, including problems arising from real-world applications.
- Final Exam (3 Hours) The final assessment task is a proctored three hours exam worth 30% of the student’s final mark.
Resources required by the students
Note: This course is entirely online and does not require or rely on any textbook.
A scanner, smartphone camera, or similar device to upload handwritten or hand-drawn work
MHF4U Advanced Functions, Grade 12, University Preparation
Overall Expectations
By the end of this course, students will:
- 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.
Strategies for Assessment and Evaluation of Student Performance
There are three forms of assessment that will be used throughout this course:
- Assessment for learning: will directly influence student learning by reinforcing the connections between assessment and instruction, and provide ongoing feedback to the student. Assessment for learning occurs as part of the daily teaching process and helps teachers form a clear picture of the needs of the students because students are encouraged to be more active in their learning and associated assessment. Teachers gather this information to shape their teaching environment.
- Assessment as learning: is the use of a task or an activity to allow students the opportunity to use assessment to further their own learning. Self and peer assessments allow students to reflect on their own learning and identify areas of strength and need. These tasks offer students the chance to set their own personal goals and advocate for their own learning.
- Assessment of learning: will occur at or near the end of a period of learning; this summary is used to make judgements about the quality of student learning using established criteria, to assign a value to represent that quality and to communicate information about achievement to students and parents.
Teaching and Learning Strategies
This course’s overarching goal is to enable students to use mathematics’ language competently, confidently, and flexibly. A number of instructional methodologies are used to give learning opportunities to meet a variety of learning styles, interests, and ability levels. The teaching and learning methodologies will be built on the following mathematical processes:
- Communicating: This course provides students with several opportunities to convey their knowledge, both orally and in writing. Students will use discussion boards to discuss what they’ve learned, produce reports that tie what they’ve studied to real-world applications, and make presentations to demonstrate comprehension of some ideas.
- Problem solving: By building on existing knowledge and abilities, this course scaffolds student learning. Students will have the opportunity to study earlier ideas and will be confronted with challenges that will require them to apply their skills in creative ways to solve problems in real-world situations.
- Reflecting: The reflective process is modeled in this course. The course shows proper communication using some examples and practice exercises to explain intermediate phases and reflect on solutions to see if they make sense in the given context.
- Selecting Tools and Computational Strategies: This course demonstrates how to utilize graphing tools to solve problems and introduce students to technologies that can make problem solving easier and more precise.
- Connecting: Students will apply the concepts learned in class to real-world situations. Through posed puzzles, investigations, and enrichment activities, students will be able to connect past concepts to new topics.
- Self-Assessment: Students receive immediate feedback and can self-assess their grasp of ideas through the use of interactive activities (e.g. multiple choice quizzes, hangman, puzzle activities).
- Feedback: Through constant communication with the instructor, timely feedback and encouragement was given.
The Final Grade
| Percentage of Final Mark | Categories of Mark Breakdown |
| 70% | Assessments of Learning Tasks Throughout the Term |
| 30% | Final Written Examination And/ Or RST |
A student’s final grade is reflective of their most recent and most consistent level of achievement.
The balance of the weighting of the categories of the achievement chart throughout the course is:
| Total | Knowledge | Inquiry | Communication | Application |
| 100% | 25% | 25% | 25% | 25% |
Cheating and Plagiarism
Forest Green Academy International commits to having policies for assessments that minimize the risk of cheating. We also commit to begin each course with refresher learning on academic integrity.
In the event of incidences of academic dishonesty, the student, Academic Director (and, in the case of students under 18, their parents) will be notified of the occurrence, of the consequence, and of the potential consequences of subsequent incidents.
Improper Citation
Grades 11 and 12
First Instance: A warning and an opportunity to redo the piece.
Subsequent Instance: An opportunity to redo the piece to a maximum grade of 75%.
Unaccredited Paraphrasing
Grade 11 and 12
First Instance: An opportunity to redo the piece to a maximum grade of 75%.
Subsequent Instance: An opportunity to redo the piece to a maximum grade of 50%.
Unaccredited Verbatim
Grade 11 and 12
First Instance: An opportunity to redo the piece to a maximum grade of 50%.
Subsequent Instance: A grade of zero. No opportunity to resubmit.
Full Plagiarism
Grade 11 and 12
First Instance: A grade of zero. No opportunity to resubmit.
Subsequent Instance: A grade zero. No opportunity to resubmit.
Instructional Approaches
Teachers will use a variety of instructional strategies to help students become independent, strategic and successful learners. The key to student success is effective, accessible instruction. When planning this course of instruction, the teacher will identify the main concept and skills of the course, consider the context in which students will apply their learning and determine the students’ learning goals. The instructional program for this course will be well planned and will support students in reaching their optimal level of challenge for learning, while directly teaching the skills that are required for success.
Understanding student strengths and needs will enable the teacher to plan effective instruction and meaningful assessments. Throughout this course the teacher will continually observe and assess the students’ readiness to learn, their interests, and their preferred learning styles and individual learning needs.
Teachers will use differentiated instructional approaches such as:
- adjusting the method or pace of instruction
- using a variety of resources
- allowing a wide choice of topics
- adjusting the learning environment
- scaffolding instruction
During this course, the teacher will provide multiple opportunities for students to apply their knowledge and skills and consolidate and reflect upon their learning.
Planning the Program for Students with Special Educational Needs
The teacher in this course is the key educator of students with special education needs. The teacher has a responsibility to help all students learn, and will work collaboratively with the guidance counselor, where appropriate, to achieve this goal. In planning this course, the teacher will pay particular attention to the following guidelines:
- All students have the ability to succeed
- Each student 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 students
- Online teachers are the key educators for a student’s literacy and numeracy development
- Online teachers need the support of the larger school community to create a learning environment that supports students with special education needs
- Fairness is not sameness
The teacher will use the following strategies:
- Extra time on tests and extended deadlines for major assessments
- Complete tasks or present information in ways that cater to individual learning styles
- Variety of teaching and learning strategies
- Scaffolding
- Break down (chunk) assignments
- A computer for assessments and exams
- Formula sheets, memory aids
- oral and written instructions
- Cue cards during instruction and Assessments
- Graphic organizers
- Specific strategies to enhance recall
- Non-verbal cues and reminders to remain focused
- Oral testing
- Allow for sufficient response time
- Experiential learning experiences so that students can make connections between curriculum and real-world examples
- Conferencing
- Prompting students through lessons and assessments
- Refocusing strategies
- Periodic breaks
Planning the Program for Students with English as a Second Language
In planning this course for students with linguistic backgrounds other than English, the teacher will create a safe, supportive, and welcoming environment that nurtures the students’ self-confidence while they are receiving course instruction. Most English language learners who have developed oral proficiency in everyday English will nevertheless require instructional scaffolding to meet curriculum expectations. The teacher will adapt the instructional program in order to facilitate the success of these students in their classes. Appropriate adaptations and strategies for this course will include:
- Body language and non-verbal communication
- Model expectations
- Subject-specific dictionary
- Cooperative learning
- Concrete examples and materials
- Avoid idioms
- Bilingual Dictionaries
- Buddy system
- Peer tutors
- Allow sufficient response time
- Graphic organizers
- Scaffolding
- Story maps
- Conferencing
- Pre-writing strategies
- Literature circle
- Journal
- Previewing course readings / texts
- Materials that reflect cultural diversity
- Free voluntary reading
- Guided Reading
- Guided Writing
- Think Aloud
- Whole-Class Response
- Editing checklist
Supporting First Nations, Métis and Inuit Students
We will promote active and engaged citizenship, which includes greater awareness of the distinct place and role of Indigenous (First Nation, Métis, and Inuit) peoples in our shared heritage and in the future in Ontario.
We will:
- increase the focus in school strategic planning to promote the voluntary, confidential self-identification of First Nation, Métis, and Inuit students as a means to enhance the success and well-being of Aboriginal students and to help close the achievement gap
- continue to identify and share practices and resources to help improve First Nation, Métis, and Inuit student achievement and close the achievement gap
- increase the training in our schools to respond to the learning and cultural needs of First Nation, Métis, and Inuit students
- provide quality programs, services, and resources at our schools to support First Nation, Métis, and Inuit student
- provide quality programs, services, and resources at our schools who support First Nation, Métis, and Inuit students to help create learning opportunities that support improved academic achievement and identify building
- provide curriculum links that facilitates learning about contemporary and traditional First National, Métis, and Inuit cultures, histories, and perspectives among all students
- develop awareness among teachers of the learning styles of First Nation, Métis, and Inuit students and employ instructional methods designed to enhance the learning of all First Nation, Métis, and Inuit students
- implement targeted learning strategies for effective oral communication and mastery of reading and writing
- implement strategies for developing critical and creative thinking
- provide access to a variety of accurate and reliable Aboriginal resources such as periodicals, books, software, and resources in other media, including materials in the main Aboriginal languages in schools with First Nation, Métis, and Inuit students
- provide a supportive and safe environment for all First Nation, Métis, and Inuit students
The Role of Information and Communication Technology
ICT tools will be integrated into this course for whole-class instruction and for the design of curriculum units that contain varied approaches to learning in order to meet diverse needs and interests of the students in this class. At the beginning of this class, all students will 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. ICT used in this course will include:
- Websites
- Online libraries
- Archives
- Public records
- YouTube
- Curriculum Digital Resources
- Widgets
- Online Graphing Calculator
- Cell phones
- iPads
- DVDs
- Digital Camera
- Edsby
- G Suite
- Office 365
- Gizmos
- Labster
- Gradeslam
- Mathspace
- Mathletics
- Screencastify
Environmental Education Connections
Although there are no specific environmental connections in this course, teachers will develop an environmental understanding fostered through the learning context (e.g., problems and examples related to environmental issues such as climate change, habitat destruction, population growth, energy conservation, and waste management).
Healthy Relationships in the Learning Environment
At Forest Green Academy International, every student is entitled to learn in a safe, respectful and caring environment, free from violence and harassment. The teacher will create a safe and supportive environment in the class by cultivating positive relationships between students and between the teacher and their students.
Equity and Inclusive Education in the Learning Environment
At Forest Green Academy International we embrace multiculturalism, human rights and diversity as fundamental values. Bullying, hate propaganda and cyber bullying, racism, religious intolerance, homophobia and gender-based violence are still evident in our communities and, unfortunately, in our schools. At Emerald Valley Academy International we address the needs of a rapidly changing and increasingly complex society by ensuring that our policies evolve with changing societal needs.
Ethics in the Learning Environment
At Forest Green Academy International teachers provide varied opportunities for students to learn about ethical issues and to explore the role of ethics in decision making.
Financial Literacy Connections
There is a growing recognition that the education system has a vital role to play in preparing young people to take their place as informed, engaged, and knowledgeable citizens in the global economy.
Because making informed decisions about economic and financial matters has become an increasingly complex undertaking in the modern world, where appropriate, the teacher will give students the opportunity to build knowledge and skills.
Literacy, Mathematical Literacy, and Inquiry Skills
At Forest Green Academy International, it is the responsibility of all of our teachers to explicitly teach literacy and inquiry skills.
Cooperative Education
Cooperative education programs allow students to earn secondary school credits while completing a work placement in the community. These programs compliment students’ academic programs and are valuable for all students, whatever their post-secondary destination.
Cooperative education courses may be earned using this course as one of the related courses.
Experiential Learning
Central to the philosophy at Forest Green Academy International is the focus on experiential learning. Planned learning experiences in the community, including job shadowing, field trips, and hands-on experiences will provide our students with opportunities to see the relevance of their classroom learning in a work setting, make connections between school and work, and explore a career of interest as they plan their pathways through secondary school and make postsecondary plans.
Health and Safety in the Learning Environment
As part of every course, students must be made aware that health and safety in their learning environment are the responsibility of all participants – at home, at school, and in the workplace. Teachers will model safe practices at all times when communicating with students online.
The Role of the School Library
Although Forest Green Academy International does not have an official school library, students are encouraged to use e-books, local libraries, GALE resource archives and Curriculum Video Digital resources to develop important research and inquiry skills.
Promotion of Careers
The knowledge and skills students acquire in this course will be useful in helping students recognize the value of their education and applications to the world outside of school and identify possible careers, essential skills and work habits required to succeed. Students will learn how to connect their learning in asking questions and finding answers to employable skills.
During this course the teacher will:
- Ensure that all students develop the knowledge and skills they need to make informed education and career/life choices
- Provide a learning environment and online school-wide opportunities for this learning
- Engage parents and the broader community in the development, implementation, and evaluation of the program to support students in their learning
- Use the four-step inquiry process linked to the four areas of learning:
- Knowing yourself – Who am I ?
- Exploring opportunities – What are my opportunities?
- Making decisions and setting goals – Who do I want to become?
- Achieving goals and making transitions – What is my plan for achieving my goals?
The teacher will support students in this course in education and career/life planning by providing them with learning opportunities, filtered through the lens of the four inquiry questions, that allow them to apply subject-specific knowledge and skills to work-related situations; explore subject-related education and career/life options, and become competent, self-directed planners.