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COURSE OUTLINE

Course Title: Functions
Course Code: MCF3M
Grade: 11
Course Type: University / College Preparation
Credit Value: 1
Prerequisite: MPM2D or MFM2P
Curriculum Policy Document: Mathematics, The Ontario Curriculum, Grades 11 and 12, 2000
Department: Mathematics
Course Developer: Mrs. Uma Gnanaharan
Development Date: October 2004
Course Revised by: -
Revision Date: -

Course Description:

This course introduces some financial applications of mathematics and extends students’ experiences with functions. Students will solve problems in personal inance involving applications of sequences and series; investigate properties and applications of trigonometric functions; develop facility in operating with polynomials, rational expressions, and exponential expressions; develop an understanding of inverses and transformations of functions; and develop facility in using function notation and in communicating mathematical reasoning.

Unit	Titles and Descriptions	Time and Sequence
Unit 1	Financial Applications In this unit students will be involved in solving three kinds of problems. They will solve problems involving arithmetic and geometric sequences and series. They will solve problems involving compound interest and annuities and they will solve problems involving financial decision making, using spreadsheets or other appropriate technology. Throughout the unit students will be taught to communicate mathematical reasoning with precision and clarity.	20 hours
Unit 2	Exploring Functions Several topics will be considered in this unit including solving first-degree inequalities and graphing their solutions on number lines, performing operations with polynomials, investigating quadratic functions, applying the method of completing square to solve max/min problems involving quadratic functions, investigating complex numbers graphically using the graphing calculator, applying properties of complex numbers by adding, subtracting, multiplying and dividing complex numbers in the form a + bi, determine the roots (complex or real) of the quadratic equations by factoring, completing the square and the quadratic formula.	22 hours
Unit 3	Transformations In this unit students will develop definitions for the terms function, and relation and explain the relationship between them. They will use the vertical line test and other methods to determine whether a relation is a function and use the function notation to evaluate functions. Time will be spent identifying the graph of various functions, the properties of the discrete and continuous graphs and the relationship between a function and its inverse. Inverse functions will be represented using function notation and the inverse of a given function or graph will be determined. Transformation of functions will be represented using function notation.	20 hours
Unit 4	Trigonometry In this unit students will solve problems involving the sine and cosine law in oblique triangles.	22 hours
Unit 5	Trigonometry Functions There are two major parts to this unit. One involves investigating the relationship between the graphs and the equations of sinusoidal functions and the other targets solving problems involving models of sinusoidal functions.	20 hours
Unit 6	Review Students will review the major concepts of the course in this review unit before attempting the final examination.	4 hours
	Final Evaluation The final assessment task is a proctored two hour exam worth 30% of the student’s final mark.	2 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:

Ideal Problem Solving	Problem Solving	Graphing
Visuals	Direct Instruction	Independent Reading
Independent Study	Model Analysis	Multimedia Productions
Logical Mathematical Intelligence	Graphing Applications	Problem Posing
	Guided Exploration	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 Activities	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 - MCF3M

QUADRATIC FUNCTIONS
Overall Expectations
MQF.01	expand and simplify quadratic expressions, solve quadratic equations, and relate the roots of a quadratic equation to the corresponding graph;
MQF.02	demonstrate an understanding of functions, and make connections between the numeric, graphical, and algebraic representations of quadratic functions;
MQF.03	solve problems involving quadratic functions, including problems arising from real-world applications.
EXPONENTIAL FUNCTIONS
Overall Expectations
MEF.01	simplify and evaluate numerical expressions involving exponents, and make connections between the numeric, graphical, and algebraic representations of exponential functions;
MEF.02	identify and represent exponential functions, and solve problems involving exponential functions, including problems arising from real-world applications;
MEF.03	demonstrate an understanding of compound interest and annuities, and solve related problems.
TRIGONOMETRIC FUNCTIONS
Overall Expectations
MTF.01	solve problems involving trigonometry in acute triangles using the sine law and the cosine law, including problems arising from real-world applications;
MTF.02	demonstrate an understanding of periodic relationships and the sine function, and make connections between the numeric, graphical, and algebraic representations of sine functions;
MTF.03	identify and represent sine functions, and solve problems involving sine functions, including problems arising from real-world applications.

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

Resources:

• MCF3M online course of study
• visuals
• videos
• graphing calculator
• various internet websites

Resource Materials:

• Addison-Wesley Functions and Relations 11; Robert Alexander, Peter Taylor, Peter J. Harrison, Kevin Maguire, Linda Rajotte, Margaret Sinclair, Kevin Spry, Claire Burnett, Enid Haley, Lesley Haynes, Mei Lin Cheung, Nirmala Nutakki; Pearson Education Canada, 2001
• Harcourt Mathematics 11 Functions/Relations; Ronald Green, Gordon Nicholls, Loraine Wilson, Ronald Dunkley, Enzo Carli; Harcourt Canada
• McGraw-Hill Ryerson Mathematics 11; Barbara Canton, Fred Ferneyhough, Lynda Ferneyhough, Michael Hamilton, George Knill, Louis Lim, John Rodger, Mike Webb, Chris Dearling, Frank Maggio; McGraw-Hill Ryerson, 2001
• Nelson Mathematics 11; David Zimmer, Chris Kirkpatrick, Ralph Montesanto, Dan Charbonneau, Christine Suurtamm, Susanne Trew, Rob McLeish; Nelson Thomson Learning, 2001

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.