Course of Study

Course of Study for Advanced Functions DE (MHF4Ud)

Private School	Lakefield College School Lakefield ON K0L 2H0 (705) 652-3324
Department	Sciences
Course Developers	John Braeckman
Development Date	1 Sep 2009
Course Revisers	John Braeckman,Julia McGill,Dayna Taylor
Revision Date	8 Jun 2015
Course	Advanced Functions DE Grade 12 University type
Ministry Course Code	MHF4U
School Course Code	MHF4Ud
Credit Value	1.00 Ontario Course credit
Ministry Curriculum	The Ontario Curriculum Grades 11 and 12: Mathematics - 2007 (revised)
Prerequisite/Co-requisite	MCR3U

Course Description/Rationale:
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:

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.

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

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;

solve problems involving trigonometric equations and prove trigonometric identities.

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

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

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

demonstrate an understanding of solving polynomial and simple rational inequalities.

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;

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;

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:

Teaching & Learning Strategies:
Socratic Lessons, Guided Investigations (with technology and without), On-line Group Discussions, Lecture, Problem Sets

Strategies for Assessment & Evaluation of Student Performance:
Assessment For Learning Tools (Teacher-Assessments)

homework assessments
question/answer periods
on-line group discussions (peer-to-peer and student-teacher)

Assessment As Learning Tools (Self-Assessments)

notebook review
homework checks (by student)
checklists
assignment corrections and feedback

Assessment Of Learning (Evaluation)

Assignments
On-line group discussions
Final Exam

Considerations for Program Planning:

Individual student needs will be addressed through the use of adaptive software, and through implementing the concept of universal curriculum design through which each student will be able to meet expectations and demonstrate growth and mastery through a variety of methods geared to his/her interests and abilities.

Term Work: 70% (Includes those tools listed in the "Assessment/Evaluation Strategies" section above)
Summative Evaluation 30% (1 Written Exam)

A student's overall term mark (Term Work and Summative Evaluation) will be based on the following breakdown: Knowledge 30%
Inquiry: 20%
Communication: 20%
Application: 30%

Learning Skills

Growing Success

http://www.edu.gov.on.ca/eng/policyfunding/growSuccess.pdf

Considerations for Program Planning:
Special Educational Needs:

Ample extension material is provided.
Identified students will receive extra time on tests.
Full notes are available online for all students.
Video lessons provide opportunity for students to control the pace of learning.

English Language Learners:

Full notes are available online for all students.
Video lessons provide opportunity for students to control the pace of learning.
Word problems are modified to address language differences

Information and Communications Technology:

Computer technology is readily employed in the form of graphing calculators, CAS systems, and geometry modelling software

Environmental Education:
Characteristics of Functions

recognize real-world applications of combinations of functions (e.g., the motion of a damped pendulum can be represented by a function that is the product of a trigonometric function and an exponential function; the frequencies of tones associated with the numbers on a telephone involve the addition of two trigonometric functions), and solve related problems graphically

Sample problem: The rate at which a contaminant leaves a storm sewer and enters a lake depends on two factors: the concentration of the contaminant in the water from the sewer and the rate at which the water leaves the sewer. Both of these factors vary with time. The concentration of the contaminant, in kilograms per cubic metre of water, is given by c(t) = t2, where t is in seconds. The rate at which water leaves the sewer, in cubic metres per second, is given by w(t) = 1/t4 + 10. Determine the time at which the contaminant leaves the sewer and enters the lake at the maximum rate. Note: The sample problem above illustrates how the learning context can be related to the environment − in this case, to water pollution.

Financial Literacy:
Characteristics of Functions

determine the composition of two functions [i.e., f(g(x))] numerically (i.e., by using a table of values) and graphically, with technology, for functions represented in a variety of ways (e.g., function machines, graphs, equations), and interpret the composition of two functions in real-world applications

Sample problem: For a car travelling at a constant speed, the distance driven, d kilometres, is represented by d(t) = 80t, where t is the time in hours. The cost of gasoline, in dollars, for the drive is represented by C(d) = 0.09d. Determine numerically and interpret C(d(5)), and describe the relationship represented by C(d(t)).

solve problems involving the composition of two functions, including problems arising from real-world applications

Sample problem: The speed of a car, v kilometres per hour, at a time of t hours is represented by v(t) = 40 + 3t + t2. The rate of gasoline consumption of the car, c litres per kilometre, at a speed of v kilometres per hour is represented by c(v) = (v/500 - 0.1)2 + 0.15. Determine algebraically c(v(t)), the rate of gasoline consumption as a function of time. Determine, using technology, the time when the car is running most economically during a four-hour trip.

Resources:

Erdman, W., et al. (2008) Advanced Functions 12. McGraw-Hill Ryerson

TI - 83+ Graphing Calculator and/or online graphing software

Laptop with the following applications: MS Excel, Graphing programs (Fathom, Geometer's Sketchpad, Graph) and MS Word with Mathtype or Equation Editor