Advanced Functions
Grade 12, University Preparation (MHF4U)

 

Course Title : Advanced Functions, Grade 12, University Preparation (MHF4U)
Course Name : Advanced Functions
Course Code : MHF4U
Grade : 12
Course Type : University Preparation
Credit Value : 1.0
Prerequisite : Functions, Grade 11, University Preparation or Mathematics for College Technology for College Technology
Curriculum Policy Document: Mathematics, The Ontario Curriculum, Grades 11 and 12, 2010 (Revised)
Course Developer: USCA Academy
Department: Mathematics
Development Date: August 2021
Most Recent Revision Date: August 2021

Course Description

 

This course extends students’ experience with functions. Students will investigate the properties of polynomial, rational, logarithmic, and trigonometric functions; develop techniques for combining functions; broaden their understanding of rates of change; and develop facility in applying these concepts and skills. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended both for students taking the Calculus and Vectors course as a prerequisite for a university program and for those wishing to consolidate their understanding of mathematics before proceeding to any one of a variety of university programs.

Overall Curriculum Expectations

A1

demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions;

A2

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;

A3

solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications.

B1

demonstrate an understanding of the meaning and application of radian measure;

B2

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;

B3

solve problems involving trigonometric equations and prove trigonometric identities.

C1

identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of polynomial functions;

C2

identify and describe some key features of the graphs of rational functions, and represent rational functions graphically;

C3

solve problems involving polynomial and simple rational equations graphically and algebraically;

C4

demonstrate an understanding of solving polynomial and simple rational inequalities.

D1

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;

D2

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;

D3

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.

Outline of Course Content

Unit Titles and Descriptions Time and Sequence
Unit 1 Basic Skills Review

Many situations can be modeled graphically. Interpreting those graphs is something that requires you to become familiar with all of the aspects of these graphs. Students will recall a few basic facts of a distance time graph. In previous math courses, students saw some transformations and studied their effects on a given graph. These will also be reviewed. Finally, the concepts of function notation, relation, range, domain, and function notation will all be reviewed.

10 hours
Unit 2 Polynomial Functions

In this unit, students learn to identify and describe characteristics of polynomial functions including key features of their graphs and the relationship between finite differences and equations. The connection between equations and graphs of polynomial functions will also be examined. Equation-solving skills and graphing skills are combined to solve polynomial equations and inequalities. The relationship between the Remainder Theorem and the Factor Theorem is identified. Techniques for factoring polynomial functions of degree greater than two are examined and also applied to determine the roots of polynomial equations. Families of polynomial functions are analyzed. Finally, inequalities are solved graphically using technology, and algebraic methods for solving factorable polynomial inequalities.

20 hours
Unit 3 Rational Functions

In this Unit, students learn to analyse properties of those rational functions created by taking the reciprocal of linear functions and quadratic functions. The equations and key features of the graphs of these rational functions are analyzed. Different forms of rational functions are explored, and solved using a variety of methods, such as algebraically and using technology. Finally, connections between real-world situations and rational functions are explored through problem solving.

12 hours
Unit 4 Trigonometric Functions

In this unit, students his unit begins by studying trigonometry concepts and then applies these concepts to analyze trigonometric functions. The graphs of the sine, cosine, and tangent functions are analyzed and their key features are identified. Transformations of these graphs will also be examined. Finally, trigonometric equations are solved by combining factoring techniques with knowledge of trigonometric ratios of special angles.

24 hours
Unit 5 Exponential and Logarithmic Functions

In this unit students study the exponential function and its inverse, including writing equations to fit data and graphing inverse functions. Then, logarithms and transformations of logarithmic functions are explored. The power law of logarithms is examined, including solving problems, evaluating logarithms, and graphing logarithmic functions. Problems and applications connecting logarithms and the physical sciences will be solved. Techniques to solve exponential equations are investigated and applied. The Product and Quotient Laws of logarithms are studied and techniques to solve logarithmic equations are demonstrated. Finally, mathematical modeling with exponential and logarithmic equations is examined to solve problems.

24 hours
Unit 6 Characteristics of Functions

This unit develops students understanding of average and instantaneous rate of change, both numerically and graphically, and how to 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. Students will be taught how to determine functions that result from the addition, subtraction, multiplication, and division of two functions and from the composition of two functions, to describe some properties of the resulting functions, and to solve related problems. The unit helps students discover how to compare the characteristics of functions, and solve problems by modeling and reasoning with functions, including problems with solutions that are not accessible by standard algebraic techniques.

24 hours
Final Evaluation

The final assessment task is a three-hour exam worth 30% of the student’s final mark.

3 hours
Total 110 hours

Since the over-riding aim of this course is to help students use language 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. These include:

Guided Exploration Problem Solving Graphing
Visuals Direct Instruction Independent Reading
Independent Study Cooperative Learning Multimedia Productions
Logical Mathematical Intelligence Graphing Applications Problem Posing
Model Analysis Group discussion Self-Assessments

Assessment is a systematic process of collecting information or evidence about student learning. Evaluation is the judgment we make about the assessments of student learning based on established criteria. The purpose of assessment is to improve student learning. This means that judgments of student performance must be criterion-referenced so that feedback can be given that includes clearly expressed next steps for improvement. Tools of varying complexity are used by the teacher to facilitate this.

The assessment will be based on the following processes that take place in the classroom:

Assessment FOR Learning Assessment AS Learning Assessment OF Learning

During this process the teacher seeks information from the students in order to decide where the learners are and where they need to go.

During this process the teacher fosters the capacity of the students and establishes individual goals for success with each one of them.

During this process the teacher reports student’s results in accordance to established criteria to inform how well students are learning.

Conversation Conversation Conversation
Classroom discussion Self-evaluation Peer assessment Classroom discussion small group discussion Post-lab conferences Presentations of research Debates
Observation Observation Observation
Drama workshops (taking direction) Steps in problem solving Group discussions Presentations Group Presentations
Student Products Student Products Student Products
Reflection journals (to be kept throughout the duration of the course)
Check Lists
Success Criteria
Practice sheets
Socrative quizzes
Projects
Poster presentations Tests
In Class Presentations

For the more complex evaluations, the criteria are incorporated into a rubric where levels of performance for each criterion are stated in language that can be understood by students.

Strategy Purpose Who Assessment Tool
Self-Assessment Quizzes Diagnostic Self/Teacher Marking scheme
Problem Solving Diagnostic Self/Peer/Teacher Marking scheme
Graphing Application Diagnostic Self Anecdotal records
Homework check Diagnostic Self/Teacher Checklist
Teacher/Student Conferencing Assessment Self/Teacher Anecdotal records
Problem Solving Assessment Peer/teacher Marking scheme
Investigations Assessment Self/Teacher Checklist
Problem Solving Evaluation Teacher Marking scheme
Graphing Evaluation Teacher Checklist
Unit Tests Evaluation Teacher Marking scheme
Final Exam Evaluation Teacher Checklist

Assessment is embedded within the instructional process throughout each unit rather than being an isolated event at the end. Often, the learning and assessment tasks are the same, with formative assessment provided throughout the unit. In every case, the desired demonstration of learning is articulated clearly and the learning activity is planned to make that demonstration possible. This process of beginning with the end in mind helps to keep focus on the expectations of the course as stated in the course guideline. The evaluations are expressed as a percentage based upon the levels of achievement.



The evaluation of this course is based on the four Ministry of Education achievement categories of knowledge and understanding (25%), thinking (25%), communication (25%), and application (25%). The evaluation for this course is based on the student's achievement of curriculum expectations and the demonstrated skills required for effective learning.

The percentage grade represents the quality of the student's overall achievement of the expectations for the course and reflects the corresponding level of achievement as described in the achievement chart for the discipline.
A credit is granted and recorded for this course if the student's grade is 50% or higher. The final grade for this course will be determined as follows:

70% of the grade will be based upon evaluations conducted throughout the course. This portion of the grade will reflect the student's most consistent level of achievement throughout the course, although special consideration will be given to more recent evidence of achievement.

30% of the grade will be based on a final exam administered at the end of the course. The exam will contain a summary of information from the course and will consist of well-formulated multiple-choice questions. These will be evaluated using a checklist.

Textbook

  • McGraw-Hill Ryerson, Advanced Functions 12, 2009 Potential Resources graphing calculator various internet websites

For the teachers who are planning a program in mathematics must take into account several important areas. The areas of concern to all teachers that are outlined in the policy document of Ontario Ministry of Education, include the following:

teaching approaches

types of secondary school courses

education for exceptional students

the role of technology in the curriculum

English as a second language (ESL) and English literacy development (ELD)

career education

cooperative education and other workplace experiences

health and safety in mathematics

It is important to ensure that all students, especially those with special education needs, are provided with the learning opportunities and supports they require to gain the knowledge, skills, and confidence needed to succeed in a rapidly changing society. The context of special education and the provision of special education programs and services for exceptional students in Ontario are constantly evolving. Provisions included in the Canadian Charter of Rights and Freedoms and the Ontario Human Rights Code have driven some of these changes. Others have resulted from the evolution and sharing of best practices related to the teaching and assessment of students with special educational needs. Accommodations (instructional, environmental or assessment) allow the student with special education needs access to the curriculum without changes to the course curriculum expectations.

Environmental education teaches students about how the planet's physical and biological systems work, and how we can create a more sustainable future. Good curriculum design following the resource document. This ensures that the student will have opportunities to acquire the knowledge, skills, perspectives and practices needed to become an environmentally literate citizen. The online course should provide opportunities for each student to address environmental issues in their home, in their local community, or even at the global level.

USCA helps students to become environmentally responsible. The first goal is to promote learning about environmental issues and solutions. The second goal is to engage students in practicing and promoting environmental stewardship in their community. The third goal stresses the importance of the education system providing leadership by implementing and promoting responsible environmental practices so that all stakeholders become dedicated to living more sustainably. Environmental education teaches students about how the planet's physical and biological systems work, and how we can create a more sustainable future.

USCA provides a number of strategies to address the needs of ESL/ELD students to accommodate the needs of students who require instruction in English as a second language or English literacy development. Our teacher considers it to be his or her responsibility to help students develop their ability to use the English language properly. Appropriate accommodations affecting the teaching, learning, and evaluation strategies in this course may be made in order to help students gain proficiency in English, since students taking English as a second language at the secondary level have limited time in which to develop this proficiency. School determines the student's level of proficiency in the English Language upon registration. This information is communicated to the teacher of the course following the registration and the teacher then invokes a number of strategies and resources to support the student in the course.

Throughout their secondary school education, students will learn about the educational and career opportunities that are available to them; explore and evaluate a variety of those opportunities; relate what they learn in their courses to potential careers in a variety of fields; and learn to make appropriate educational and career choices. The skills, knowledge and creativity that students acquire through this course are essential for a wide range of careers. Being able to express oneself in a clear concise manner without ambiguity in a second language, would be an overall intention of this course, as it helps students prepare for success in their working lives.

By applying the skills they have developed, students will readily connect their classroom learning to real-life activities in the world in which they live. Cooperative education and other workplace experiences will broaden their knowledge of employment opportunities in a wide range of fields. In addition, students will increase their understanding of workplace practices and the nature of the employer-employee relationship. Teachers should maintain links with community-based businesses to ensure that students have access to hands-on experiences that will reinforce the knowledge they have gained in school.

Every student is entitled to learn in a safe, caring environment, free from violence and harassment. Students learn and achieve better in such environments. The safe and supportive social environment at UCSA is founded on healthy relationships between all people. Healthy relationships are based on respect, caring, empathy, trust, and dignity, and thrive in an environment in which diversity is honoured and accepted. Healthy relationships do not tolerate abusive, controlling, violent, bullying/harassing, or other inappropriate behaviours. To experience themselves as valued and connected members of an inclusive social environment, students need to be involved in healthy relationships with their peers, teachers, and other members.

Critical thinking is the process of thinking about ideas or situations in order to understand them fully, identify their implications, make a judgement, and/or guide decision making. Critical thinking includes skills such as questioning, predicting, analysing, synthesizing, examining opinions, identifying values and issues, detecting bias, and distinguishing between alternatives. Students who are taught these skills become critical thinkers who can move beyond superficial conclusions to a deeper understanding of the issues they are examining. They are able to engage in an inquiry process in which they explore complex and multifaceted issues, and questions for which there may be no clear-cut answers.

The school library program in USCA can help build and transform students' knowledge in order to support lifelong learning in our information- and knowledge-based society. The school library program of these schools supports student success across the curriculum by encouraging students to read widely, teaching them to examine and read many forms of text for understanding and enjoyment, and helping them improve their research skills and effectively use information gathered through research. USCA teachers assist students in accessing a variety of online resources and collections (e.g., professional articles, image galleries, videos, databases). Teachers at USCA will also guide students through the concept of ownership of work and the importance of copyright in all forms of media.

Information literacy is the ability to access, select, gather, critically evaluate, and create information. Communication literacy refers to the ability to communicate information and to use the information obtained to solve problems and make decisions. Information and communications technologies are utilized by all Virtual High School students when the situation is appropriate within their online course. As a result, students will develop transferable skills through their experience with word processing, internet research, presentation software, and telecommunication tools, as would be expected in any other course or any business environment. Although the Internet is a powerful learning tool, there are potential risks attached to its use. All students must be made aware of issues related to Internet privacy, safety, and responsible use, as well as of the potential for abuse of this technology, particularly when it is used to promote hatred.  

USCA provides varied opportunities for students to learn about ethical issues and to explore the role of ethics in both public and personal decision making. During the inquiry process, students may need to make ethical judgements when evaluating evidence and positions on various issues, and when drawing their own conclusions about issues, developments, and events. Teachers may need to help students in determining appropriate factors to consider when making such judgements. In addition, it is crucial that USCA teachers provide support and supervision to students throughout the inquiry process, ensuring that students engaged in an inquiry are aware of potential ethical concerns and address them in acceptable ways. Teachers will ensure that they thoroughly address the issue of plagiarism with students. In a digital world in which there is easy access to abundant information, it is very easy to copy the words of others and present them as one's own. Students need to be reminded, even at the secondary level, of the ethical issues surrounding plagiarism, and the consequences of plagiarism should be clearly discussed before students engage in an inquiry. It is important to discuss not only dishonest plagiarism but also more negligent plagiarism instances.

 

Unit

Description

Assessments Evaluation Weight

KICA

 

Unit 1

Basic Skills Review Tests, Assignments

11%

25/25/25/25

Unit 2

Polynomial Functions                   

12%

25/25/25/25

Unit 3

Rational Functions

12%

25/25/25/25

Unit 4

Trigonometric Functions    

12%

25/25/25/25

Unit 5

Exponential and Logarithmic Functions        

12%

25/25/25/25

Unit 6

Characteristics of Functions  

12%

25/25/25/25

 

Final Exam                  

30%

25/25/25/25

 

TOTAL

100%

 

The percentage grade represents the quality of the students’ overall achievement of the expectations for the course and reflects the corresponding achievement as described in the achievement charts and will be 70% of the overall course; the final evaluation will be 30% of the overall grade.

 

Percentage of the Mark                  

Categories of Mark Breakdown

70%

Tests (44%)

Assignments (6%)

Student/teacher conference (Observation and Conversation) 10%

30%

Final Exam (30%)    

         

 

 


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