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COURSE OUTLINE

Course Title: Advanced Functions
Course Code: MHF4U
Grade: 12
Course Type: University Preparation
Credit Value: 1
Prerequisite: Functions, MCR3U or Mathematics for College Technology, MCT4U
Curriculum Policy Document: Mathematics, The Ontario Curriculum, Grades 11 and 12, 2007 (Revised)
Department: Mathematics
Course Developer: Ken Stewart
Development Date: 2007
Course Revised by: -
Revision Date: -

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.

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.	6 hours
Unit 2	Exponent Laws This unit begins with a review of the rules associated with exponents. Exponential functions, examples and applications of them are the first new topic in this unit followed by logarithmic functions, identities and applications.	26 hours
Unit 3	Trigonometry This unit examines the meaning and application of radian measure. Students will 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 involving trigonometric equations and prove trigonometric identities.	26 hours
Unit 4	Trigonometry Functions and Graphs 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.	25 hours
Unit 5	Polynomial Functions In this unit students learn to identify and describe some key features of polynomial functions and to make connections between the numeric, graphical, and algebraic representations of polynomial functions. The concepts of identifying and describing some key features of the graphs of rational functions are also presented. Finally students will represent rational functions graphically; solve problems involving polynomial and simple rational equations graphically and algebraically so they can demonstrate an understanding of how to solve polynomial and simple rational inequalities.	25 hours
	Final Evaluation The final assessment task is a proctored three hour exam worth 30% of the student’s final mark.	3 hours
	Total	110 hours

Teaching / Learning Strategies:

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	Ideal Problem Solving	Multimedia Productions
Logical Mathematical Intelligence	Graphing Applications	Problem Posing
Model Analysis		Self-Assessments

Assessment and Evaluation Strategies of Student Performance:

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. 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
Problem Solving	Assessment	Peer/teacher	Marking scheme
Research	Assessment	Peer/teacher	Anecdotal records
Problem Solving	Evaluation	Teacher	Marking scheme
Graphing	Evaluation	Teacher	Checklist
Investigations	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.

Overall Expectations - MHF4U

EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Overall Expectations
MEL.01	demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions;
MEL.02	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;
MEL.03	solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications.
TRIGONOMETRIC FUNCTIONS
Overall Expectations
MTF.01	demonstrate an understanding of the meaning and application of radian measure;
MTF.02	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;
MTF.03	solve problems involving trigonometric equations and prove trigonometric identities.
POLYNOMIAL AND RATIONAL FUNCTIONS
Overall Expectations
MPR.01	identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of polynomial functions;
MPR.02	identify and describe some key features of the graphs of rational functions, and represent rational functions graphically;
MPR.03	solve problems involving polynomial and simple rational equations graphically and algebraically;
MPR.04	demonstrate an understanding of solving polynomial and simple rational inequalities.
CHARACTERISTICS OF FUNCTIONS
Overall Expectations
MCF.01	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;
MCF.02	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;
MCF.03	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.

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 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 the student's reports and will consist of well-formulated multiple choice questions. These will be evaluated using a checklist.

The report card will focus on two distinct but related aspects of student achievement; the achievement of curriculum expectations and the development of learning skills. The report card will contain separate sections for the reporting of these two aspects.

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%	Level R	Insufficient achievement of curriculum expectations. A credit will not be granted.

Achievement Chart: Mathematics, Grades 9-12

Categories	50-59% (Level 1)	60-69% (Level 2)	70-79% (Level 3)	80-100% (Level 4)
Knowledge and Understanding - Subject-specific content acquired in each course (knowledge), and the comprehension of its meaning and significance (understanding)
	The student:
Knowledge of content (e.g., facts, terms, procedural skills, use of tools)	demonstrates limited knowledge of content	demonstrates some knowledge of content	demonstrates considerable knowledge of content	demonstrates thorough knowledge of content
Understanding of matematical concepts	demonstrates limited understanding of content	demonstrates some understanding of content	demonstrates considerable understanding of content	demonstrates thorough and insightful understanding of content
Thinking - The use of critical and creative thinking skills and/or processes
	The student:
Use of planning skills -understanding the problem (e.g., formulating and interpreting the problem, making conjectures) -making a plan for problem solving	uses planning skills with limited effectiveness	uses planning skills with moderate effectiveness	uses planning skills with considerable effectiveness	uses planning skills with a high degree of effectiveness
Use of processing skills -carrying out a plan (e.g., collecting data, questioning, testing, revising, modelling, solving, inferring, forming conclusions) -looking back at the solution (e.g., evaluating reasonableness, making convincing arguments, reasoning, justifying, proving, reflecting)	uses processing skills with limited effectiveness	uses processing skills with some effectiveness	uses processing skills with considerable effectiveness	uses processing skills with a high degree of effectiveness
Use of critical/creative thinking processess (e.g., problem solving, inquiry)	uses critical / creative thinking processes with limited effectiveness	uses critical / creative thinking processes with some effectiveness	uses critical / creative thinking processes with considerable effectiveness	uses critical / creative thinking processes with a high degree of effectiveness
Communication - The conveying of meaning through various forms
	The student:
Expression and organization of ideas and mathematical thinking (e.g., clarity of expression, logical organization), using oral, visual, and written forms (e.g., pictorial, graphic, dynamic, numeric, algebraic forms; concrete materials)	expresses and organizes mathematical thinking with limited effectiveness	expresses and organizes mathematical thinking with some effectiveness	expresses and organizes mathematical thinking with considerable effectiveness	expresses and organizes mathematical thinking with a high degree of effectiveness
Communication for different audiences (e.g., peers and teachers) and purposes (e.g., to present data, justify a solution, express a mathematical argument) in oral, visual, and written forms	communicates for different audiences and purposes with limited effectiveness	communicates for different audiences and purposes with some effectiveness	communicates for different audiences and purposes with considerable effectiveness	communicates for different audiences and purposes with a high degree of effectiveness
Use of conventions, vocabulary, and terminology of the discipline (e.g., terms, symbols) in oral, visual, and written forms	uses conventions, vocabulary, and terminology of the discipline with limited effectiveness	uses conventions, vocabulary, and terminology of the discipline with some effectiveness	uses conventions, vocabulary, and terminology of the discipline with considerable effectiveness	uses conventions, vocabulary, and terminology of the discipline with a high degree of effectiveness
Application - The use of knowledge and skills to make connections within and between various contexts
	The student:
Application of knowledge and skills in familiar contexts	applies knowledge and skills in familiar contexts with limited effectiveness	applies knowledge and skills in familiar contexts with some effectiveness	applies knowledge and skills in familiar contexts with considerable effectiveness	applies knowledge and skills in familiar contexts with a high degree of effectiveness
Transfer of knowledge and skills to new contexts	transfers knowledge and skills to new contexts with limited effectiveness	transfers knowledge and skills to new contexts with some effectiveness	transfers knowledge and skills to new contexts with considerable effectiveness	transfers knowledge and skills to new contexts with a high degree of effectiveness
Making connections within and between various contexts (e.g., connections between concepts, representations, and forms within mathematics; connections involving use of prior knowledge and experience; connections between mathematics, other disciplines, and the real world))	makes connections within and between various contexts with limited effectiveness	makes connections within and between various contexts with some effectiveness	makes connections within and between various contexts with considerable effectiveness	makes connections within and between various contexts with a high degree of effectiveness

Potential Resources:

• MHF4U online course of study
• visuals
• videos
• graphing calculator
• various internet websites

Program Planning Considerations for Mathematics:

Teachers who are planning a program in Mathematics must take into account considerations in a number of important areas. Essential information that pertains to all disciplines is provided in the companion piece to this document, The Ontario Curriculum, Grades 9 to 12: Program Planning and Assessment, 2000. The areas of concern to all teachers that are outlined there include the following:

• 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

Considerations relating to the areas listed above that have particular relevance for program planning in Mathematics are noted here.

Education for Exceptional Students. In planning courses in Mathematics, teachers should take into account the needs of exceptional students as set out in their Individual Education Plan. All Mathematics courses reflect the real world very closely, which offers a vast array of opportunities for exceptional students. Students who use alternative techniques for communication may find a venue for their talents as they go about researching the nature of their world.

The Role of Technology in the Curriculum. Information technology is considered a learning tool that must be accessed by Mathematics students when the situation is appropriate. As a result, students will develop transferable skills through their experience with word processing, internet research, presentation software, and equation editors as would be expected in any environment. 

English As a Second Language and English Literacy Development (ESL/ELD). This Mathematics course can provide a wide range of options to address the needs of ESL/ELD students. Assessment and evaluation exercises will help ESL students in mastering the English language and all of its idiosyncrasies. In addition, since all occupations require employees with a wide range of English skills and abilities, many students will learn how the operation of their own physical world can contribute to their success in their social world.

Career Education. Mathematics definitely helps prepare students for employment in a huge number of diverse areas - Engineering, Science, Business, etc. 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, solve problems, make connections between this Mathematics course and the larger world, etc., would be an overall intention of this Mathematics course, as it helps students prepare for success in their working lives.

Cooperative Education and Other Workplace Experiences. 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 of Mathematics should maintain links with community-based workers to ensure that students have access to hands-on experiences that will reinforce the knowledge they have gained in school.

Health and Safety. The Mathematics program provides the reading and analytical skills for the student to be able to explore the variety of concepts relating to health and safety in the workplace. Teachers who provide support for students in workplace learning placements need to assess placements for safety and ensure that students can read and understand the importance of issues relating to health and safety in the workplace.