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 Code: MHF4U

Course Name: Advanced Functions Grade 12, University Preparation

Department: Mathematics

Hours: 110

Credit Value: 1.0

Pre-requisites: Functions 11, Grade 11, MCR3U; or Mathematics for College Technology, MCT4C

Curriculum Policy Documents: The Ontario Curriculum, Mathematics: Grades 11 and 12. 2007.

Growing Success: Assessment, Evaluation, and Reporting in Ontario School. 2010.

Development Date: August 2019

Developed By: Kelda Wells

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Overall Curriculum Expectations

  1. demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions.
  2. 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.
  3. solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications.

  1. demonstrate an understanding of the meaning and application of radian measure.
  2. make connections between trigonometric ratios and the graphical and algebraic representations the corresponding trigonometric functions and between trigonometric functions and their and use these connections to solve problems.
  3. solve problems involving trigonometric equations and prove trigonometric identities.

  1. identify and describe some key features of polynomial functions, and make connections between numeric, graphical, and algebraic representations of polynomial functions.
  2. identify and describe some key features of the graphs of rational functions, and represent functions graphically.
  3. solve problems involving polynomial and simple rational equations graphically and algebraically
  4. demonstrate an understanding of solving polynomial and simple rational inequalities

Overall Expectations
By the end of this course, students will be able to:
  1. 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.
  2. 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.
  3. compare the characteristics of functions, and solve problems by modeling and reasoning with functions, including problems with solutions that are no accessible by standard algebraic techniques.
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Course Content

Unit Unit Title Approx. Duration
Unit 1 Polynomial Functions 21 hours
Unit 2 Polynomial Equations and Inequalities 21 hours
Unit 3 Rational Functions, Equations, and Inequalities 21 hours
Unit 4 Trigonometric Functions 22 hours
Unit 5 Exponential and Logarithmic Functions 22 hours
Final Examination 3 hours
TOTAL 110 hours
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Evaluation Breakdown

Evaluation Type Percentage of Final Grade
Test 1 12%
Test 2 12%
Test 3 12%
Test 4 12%
Test 5 12%
Cumulative Project 10%
Final Exam 30%
Total 100%

Achievement Chart

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
80-100% Level 4 A very high to outstanding level of achievement. Achievement is above the provincial standard.
70-79% Level 3 A high level of achievement. Achievement is at the provincial standard.
60-69% Level 2 A moderate level of achievement. Achievement is below, but approaching the provincial standard.
50-59% 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.

Program Planning Considerations

This course will include a variety of instructional strategies to help students become independent, strategic and successful learners.  The key to student success is effective, accessible instruction in order to empower students to become lifelong learners. Students will be required to identify the main concepts and skills of the course, consider the context in which they will apply their learning and work toward meeting the stated learning goals.
During this course, the teacher will provide multiple opportunities for students to apply their knowledge and skills and consolidate and reflect upon their learning. It is the student's responsibility to make the most of each learning opportunity provided and grow in their understanding and application of learning skills and strategies that will ensure success in this course and in life beyond.

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:
  • Modelling of expectations
  • Reference to and use of online ESL and subject-specific and dictionaries
  • Concrete examples and materials whenever possible
  • Use of a variety of learning resources including visual material and cues, , graphic organizers and visual
  • Materials that reflect cultural diversity
  • Pre-writing strategies
  • Previewing course readings / texts

When at all possible, this course will include opportunities to integrate environmental education into the material studied, encouraging the students to explore a range of environmental concerns using issue-based analysis and some of the following strategies:
  • Community Connections
  • Environmental Perspective
  • Simulation
  • Problem Solving
  • Surveys
  • Co-operative Learning
  • Inquiry-based learning
  • Cross-curricular connections
When appropriate, the students will have opportunities to analyse the environmental impact of the concepts being studied, such as the impact of Canadian and international political policies, Canadian and international laws, cultural and social differences, human rights legislation and the protection and stewardship of the Earth.

Every student is entitled to learn in a safe, respectful and caring environment, free from violence, discrimination and harassment. Antidiscrimination education encourages all students to:
  • think critically about themselves and others in the world around them in order to promote fairness, healthy relationships, and active responsible citizenship.
  • work to high standards, as it affirms the worth of all students, and helps students strengthen their sense of identity and develop a positive self-image.
  • value and show respect for diversity.
The course content will use the following strategies to help create a healthy learning environment for all students.
  • Developing a learning environment where all students feel safe
  • Promotion of diversity and inclusivity in the classroom
  • Getting students involved within their school community
  • Making community connections
  • Peer tutoring
  • Role playing
  • Group discussion
  • Case Study analysis

Classes taught within the Ontario curriculum seek to create an atmosphere of equity and inclusion based 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.
The Ontario curriculum entrusts educators to create an environment based on the principles of inclusive education, where 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.
During this course, the teacher will give students a variety of opportunities to learn about diversity and diverse perspectives.

Teachers provide varied opportunities for students to learn about ethical issues and to explore the role of ethics in decision making.  The teacher will support and encourage students to make ethical judgements when evaluating evidence and positions on various issues, and when drawing their own conclusions about issues, advancements, and global events.
During this course the teacher will provide support and oversight to students throughout the inquiry process, ensuring that students engaged in inquiry-based activities are aware of potential ethical concerns and address them in a respectful manner. If students are conducting surveys and/or interviews, teachers will supervise student activities to ensure that they respect the dignity, privacy, and confidentiality of their participants.
The teacher will thoroughly address the issue of plagiarism with students.  Students will be reminded of the ethical issues surrounding plagiarism, and the consequences of plagiarism will be clearly discussed before students engage in an inquiry.
The following strategies will be used to develop students’ understanding of ethics:
  • Making community connections
  • Peer tutoring
  • Role playing
  • Group discussions
  • Case study analysis
  • Simulation
  • Problem solving
  • Cross-curricular connections
  • Media connections
  • Surveys and interviews
  • Model ethical behavior
  • Explore ethical standards
  • Explore ethical concerns
  • Inclusive practices
  • Foster positive relationships with others
  • Assist students in developing an understanding of ethical judgments
  • Assist students in understanding confidentiality standards

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. Financial literacy education can provide the preparation Ontario students need to make informed decisions and choices in a complex and fast-changing financial world.
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 through a variety of activities in problem solving, inquiry, decision making, critical thinking, and critical literacy related to financial issues. Students will come to understand the social, environmental, and ethical implications of their own choices as consumers.  Strategies that will be used will include:
  • Community connections
  • Simulation
  • Problem Solving
  • Inquiry-based learning
  • Cross-curricular connections
  • Issue-based analysis
  • Critical literacy skills
  • Setting financial goals
  • Developing intra-personal skills

Literacy involves a range of critical-thinking skills and is essential for cross-curricular learning. It is the responsibility of all of our teachers to explicitly teach literacy and inquiry skills. 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.  Students will be exposed to literacy skills relating to oral, written, and visual communication.  The following skills will be developed in this course:
  • Reading, interpreting, and analysing various texts, including diaries, letters, government legislation and policy documents, interviews, speeches, information from non-governmental organizations, news stories, and fiction and non-fiction books
  • Extracting information
  • Analysing various types of maps and digital representations, including charts, diagrams, pictures, etc.
  • Using appropriate and correct terminology, including that related to the concepts of disciplinary thinking
  • Making community connections
  • Peer tutoring
  • Role playing
  • Group discussions
  • Case study analysis
  • Simulation
  • Problem solving
  • Cross-curricular connections
  • Media connections
  • Kinesthetic opportunities
  • Foster use of proper terminology
  • Inquiry and research skills
  • Help students to develop a language for literacy, inquiry and numeracy skills
  • Assist students with developing communication skills in areas of literacy, inquiry and numeracy

Central to successful education is the focus on experiential learning.  Planned learning experiences in the community, including job shadowing and job twinning, field trips, work experience, hands-on experiences and cooperative education, 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.  Planned learning experiences in this course will include:
  • Experiential Learning: Library visits, Guest Speaker, Role Playing, mock trials, writing anthology, author visits, writers reading, theatre workshops

As part of every course, students must be made aware that health and safety in the classroom are the responsibility of all participants – at home, at school, and in the workplace. Teachers will model safe practices at all times and communicate safety requirements to students.  Health and safety issues not usually associated with a particular course may be important when the learning involves field trips and field studies. Teachers will preview and plan these activities carefully to ensure students’ health and safety.

Although Sophio Academy does not have an official school library, students are encouraged to use e-books, local libraries, and archives to develop important research and inquiry skills.

Sophio Academy maintains and expects a high standard of academic honesty from all of its students as a vital and essential part of their character growth, academic practice and in preparation for their post-secondary studies. Students are expected to properly reference and cite the work of others in their course work and assignments using the proper recognized MLA formatting. As plagiarism is a serious offence that can have significant consequences, Sophio Academy is committed to instilling the values of honesty and hard work within our students which involves the proper acknowledgement of other's ideas and work.
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