DRAFT:
Advanced Functions, Grade 12, University Preparation
Adapted by Walter Whiteley from the October 2005 draft, following up the Ministry Task Force Recommendations:
This course builds on students’ experience with functions and their developing understanding of rate of change. Students will investigate and apply the properties of polynomial, exponential, and logarithmic functions; broaden their understanding of the mathematics associated with rates of change; and develop facility with the concepts and skills needed for the study of calculus in a variety of university courses. Students will continue to develop the mathematical processes necessary for success in senior mathematics.
Prerequisite: Functions, Grade 11, University Preparation
General process expectations for all grade 9-12 mathematics courses
PROBLEM SOLVING / develop, select, apply, compare and adapt a variety of problem-solving strategies as they pose and solve problems and conduct investigations, to help deepen their mathematical understanding;REASONING AND
PROVING / develop and apply reasoning skills (e.g., generalization through inductive reasoning, generalization through inductive reasoning, use of counter-examples, construction of proofs) to make mathematical conjectures, assess conjectures, and justify conclusions, and plan and construct organized mathematical arguments;
REFLECTING / demonstrate that they are reflecting on and monitoring their thinking to help clarify their understanding as they complete an investigation or solve a problem (e.g., by assessing the effectiveness of strategies and processes used, by proposing alternative approaches, by judging the reasonableness of results, by verifying solutions);
SELECTING TOOLS AND
COMPUTATIONAL
STRATEGIES / select and use a variety of concrete, visual, and electronic learning tools and appropriate computational strategies to investigate mathematical ideas and to solve problems;
CONNECTING / make connections among mathematical concepts and procedures, and relate mathematical ideas to situations or phenomena drawn from other contexts (e.g., other curriculum areas, daily life, current events, art and culture, sports);
REPRESENTING / create a variety of representations of mathematical ideas (e.g., numeric, geometric, algebraic, graphical, pictorial representations; onscreen dynamic representations), connect and compare them, and select and apply the appropriate representations to solve problems;
COMMUNICATING / communicate mathematical thinking orally, visually, and in writing, using precise mathematical vocabulary and a variety of appropriate representations, and observing mathematical conventions.
Strand: Polynomial and Rational Functions
Overall Expectations
1 determine the characteristics of a polynomial function of various degrees given its graph and make connections to its algebraic representation;
2 determine the characteristics of a polynomial function and rational function of various degrees, given its algebraic representation and make connections to its graph;
Strand: Trigonometric, Exponential and Logarithmic Functions
Overall Expectations
21 extend an understanding of exponential functions and the related logarithmic functions to solve problems involving exponential growth and decay;
22 extend an understanding of trigonometric functions using radian measure and solve related problems;
23 consolidate their understanding of the characteristics of functions by considering compound functions and the composition of functions as well as reasoning about general functions.
Strand: Rates of Change
Overall Expectations
48 determine and interpret the average and instantaneous rates of change of given functions in multiple representations (numerical, graphical, algebraic)
49 demonstrate an understanding of the relationship between the shape of a graph and the rate of change of the dependent variable.
50 investigate and interpret the rates of change of functions drawn from the natural and social sciences;
NOTE: There will be no explicit use of limits, or of techniques for derivatives. The rates of change will have some overlap with the Grade 12 Calculus and Vectors course, for which this is a prerequisite.