Guide to Planning Functions and Applications, Grade 11, University/College Preparation (MCF3M)

Transcription

1 Guide to Planning Functions and Applications, Grade 11, University/College Preparation (MCF3M) Targeted Implementation and Planning Supports for Revised Mathematics

2 This is intended to provide an overview of the curriculum revisions. Teachers are required to refer to the curriculum documents for program planning. 000 Functions and Applications 006 Functions and Applications Strand Sub-Strand Strand Sub-Strand Financial Applications of Sequences and Series Solving Problems Involving Arithmetic and Geometric Sequences and Series Solving Problems Involving Compound Interest and Annuities (some expectations incorporated into Exponential Functions) Solving Problems Involving Financial Decision Making Trigonometric Functions Trigonometric Functions Solving Problems Involving the Sine Law and the Cosine Law in Applying the Sine Law and the Cosine Law in Acute Triangles Oblique Triangles Understanding the Meaning and Application of Radian Measure Investigating the Relationships Between the Graphs and the Connecting Graphs and Equations of Sine Functions Equations of Sinusoidal Functions Solving Problems Involving Models of Sinusoidal Functions Solving Problems Involving Sine Functions Tools for Operating and Communicating with Functions Manipulating Polynomials, Rational Expressions, and Exponential Expressions Understanding Inverses and Transformations and Using Function Notation Communicating Mathematical Reasoning (some expectations incorporated into Quadratic Functions and Exponential Functions) Quadratic Functions Solving Quadratic Equations Investigating Properties of Quadratic Functions Solving Problems Involving Quadratic Functions Exponential Functions Solving Financial Problems Involving Exponential Functions Properties of Exponential Functions Solving Problems Involving Exponential Functions Functions and Applications, Grade 11, University/College Preparation (MCF3M) 1

3 Introduction Beginning in September 006, teachers are required to implement mathematics programs in accordance with the curriculum document for their grade level. In the Grade 11, University/College Preparation Course (MCF3M) there are three strands: Quadratic Functions Exponential Functions Trigonometric Functions The sub-groupings within each strand reflect particular aspects of knowledge and skills that are addressed within it. As a result of the revisions made as part of the Curriculum Review process, the curriculum for all Grades 1 through 11 has changed in three ways. Some expectations have been: reworded or revised for clarity removed added Expectations that have been reworded for clarity generally call for teachers to continue using instructional strategies and resources chosen for implementation of the 000 curriculum. Rewording and inclusion of examples and sample problems allow teachers to verify that their programs reflect the intended scope and emphasis in connection with the content in these expectations. To ensure that their understanding is unchanged by the rewording, teachers should read these expectations found in the second column of the clarification chart in this guide. Expectations that have been revised for clarity generally identify the depth and breadth of treatment intended for the grade level. Teachers are expected to choose instructional activities and to assess and evaluate students within these parameters. Some resources that are currently used for the Functions and Applications, Grade 11, University/College Preparation course include inappropriate content for the revised program, and must be carefully reviewed before using. Further clarification of the intent of an expectation has been achieved through inclusion of examples and/or a sample problem. These illustrate the kind of skill level, depth of learning, and complexity that the expectation entails. When an expectation has been removed from the Functions and Applications, Grade 11, University/College Preparation course, it may have: - moved to an earlier grade level - moved to a later grade level - been deleted from the curriculum altogether - been incorporated into the mathematical process expectations When an expectation has been added to the Functions and Applications, Grade 11, University/College Preparation course, it may be that: - content has been moved from an earlier grade level - content has been moved from a later grade level - new content has been added to the overall Grade 1 11 program, with some of that content placed in the Functions and Applications Grade 11, University/College Preparation Course. For example, throughout the grades, increased emphasis is placed on the concepts of multiple representations and inverse operations. In the revisions chart provided in this guide, these categories are used to identify changes to the mathematics curriculum: Remove Add Functions and Applications, Grade 11, University/College Preparation (MCF3M)

4 Curriculum Revisions that may Require Adjustments to Program NOTE: Teacher notes are in orange. Italics indicate revision. Strand: Financial Applications of Sequences and Series (renamed Exponential Functions with two sub strands deleted) Overall Expectations Solving Problems Involving Arithmetic and Geometric Sequences and Series (deleted) Solving Problems Involving Compound Interest and Annuities (renamed Solving Financial Problems Involving Exponential Functions) Solving Problems Involving Financial Decision Making (renamed Solving Financial Problems Involving Exponential Functions) Reworded demonstrate an understanding of compound interest and annuities, and solve related problems. (from renamed strand: Exponential Functions) Revised - determine, through investigation, that compound interest is an example of exponential growth; - solve problems, using a scientific calculator, that involve the calculation of the amount, A (also referred to as future value, FV), and the principal, P (also referred to as present value, PV), using the compound interest formula in the form n n A= P( 1+ i) [or FV = PV ( 1+ i) ], with a scientific calculator; - explain the meaning of the term annuity, through investigation of numerical and graphical representations using technology. - compare, using a table of values and graphs, the simple and compound interest earned for a given principal (i.e., investment) and a fixed interest rate over time; - solve problems, using technology, that involve the amount, the present value, and the regular payment of an ordinary annuity in situations where the compounding period and the payment period are the same. Functions and Applications, Grade 11, University/College Preparation (MCF3M) 3

5 Strand: Tools for Operating and Communicating with Functions (renamed Exponential Functions) Note: Expectations from this strand have been incorporated into two new strands: Quadratic Functions and Exponential Functions. For clarity, changes have been separated into two charts corresponding to the two new strands. Sub-groupings Reworded Revised Overall Expectations simplify and evaluate numerical expressions involving exponents, and make connections between the numeric, graphical, and algebraic representations of exponential functions. Manipulating Polynomials, Rational Expressions, and Exponential Expressions (renamed Connecting Graphs and Equations of Exponential Functions) Understanding Inverses and Transformations and Using Function Notation (renamed Solving Problems Involving Exponential Functions) Communicating Mathematical Reasoning (incorporated into the Process Expectations) - determine, through investigations using a variety of tools and strategies, the value of a power with a m n rational exponent (i.e., x where x > 0 and m and n are integers); - determine, through investigation, the exponent rules for multiplying and dividing numerical expressions involving exponents and the exponent rule for simplifying numerical expressions involving a power of a power, and use the rules to simplify numerical expressions containing integer exponents. - determine, through investigation, and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals, and asymptotes for exponential functions represented in a variety of ways (from Manipulating Polynomials, Rational Expressions, and Exponential Expressions). - collect data that can be modelled as an exponential function, through investigation with and without technology, from primary sources using a variety of tools or from secondary sources, and graph the data. Functions and Applications, Grade 11, University/College Preparation (MCF3M) 4

6 Strand: Tools for Operating and Communicating with Functions (renamed Quadratic Functions) Reworded Revised Overall Expectations demonstrate an understanding of functions, and make connections between the numeric, graphical, and algebraic representations of quadratic functions. Manipulating Polynomials, Rational Expressions, Exponential Expressions (split into two new sub strands: Solving Quadratic Equations and Connecting Graphs and Equations of Quadratic Functions) Understanding Inverses and Transformations and Using Function Notation (deleted) Communicating Mathematical Reasoning (This sub-group has been incorporated into the Process Expectations) Strand: Trigonometric Functions Solving Quadratic Equations - determine the real roots of a variety of quadratic equations and describe the advantages and disadvantages of each strategy (i.e., graphing; factoring; using the quadratic formula). Solving Quadratic Equations - relate the real roots of a quadratic equation to the x- intercept(s) of the corresponding graph, and connect the number of real roots to the value of the discriminant. Connecting Graphs and Equations of Quadratic Functions - describe the information that can be obtained by inspecting the standard form f ( x) = ax + bx+ c, the vertex form f ( x) = a( x h) + k, and the factored form f ( x) = a( x r)( x s) of a quadratic function. Reworded Revised Overall Expectations solve problems involving trigonometry in acute triangles using the sine law and the cosine law, including problems arising from real-world applications; demonstrate an understanding of periodic relationships and the sine function, and make connections between the numeric, graphical, and algebraic representations of sine functions; identify and represent sine functions, and solve problems involving sine functions, including those arising from real-world applications. Functions and Applications, Grade 11, University/College Preparation (MCF3M) 5

7 Solving Problems Involving the Sine Law and the Cosine Law in Oblique Triangles (renamed Applying the Sine Law and the Cosine Law in Acute Triangles) Understanding the Meaning and Application of Radian Measure (deleted) Investigating the Relationships between the Graphs and the Equations of Sinusoidal Functions (renamed: Connecting Graphs and Equations of Sine Functions) Solving Problems Involving Models of Sinusoidal Functions (renamed Solving Problems Involving the Sine Function) Reworded - sketch the graph of f ( x) = sinx for angle measures expressed in degrees, and determine and describe its key properties (i.e., cycle, domain, range, intercepts, amplitude, period, maximum and minimum values, increasing/decreasing intervals); - determine, through investigation using technology, and describe the roles of the parameters a, c, and d in functions in the form f ( x) = asinx, f ( x) = sinx+ c, and f ( x) = sin( x d) in terms of transformations on the graph of f ( x) = sinx with angles expressed in degrees (i.e., translations; reflections in the x-axes; vertical sketches and compressions). - pose and solve problems based on applications involving a sine function by using a given graph or a graph generated with technology from its equation. Revised - solve problems including those that arise from realworld applications, by determining the measures of the sides and angles of right triangles using the primary trigonometric ratios; - solve problems involving two right triangles in two dimensions; - solve problems that require the use of the sine law or the cosine law in acute triangles, including problems arising from real-world applications. - describe key properties of periodic functions arising from real-world applications, given a numerical or graphical representation; - make connections between the sine ratio and the sine function by graphing the relationship between angles from 0º to 360º and the corresponding sine ratios, with or without technology, defining this relationship as the function f ( x) = sinx, and explaining why it is a function; - make connections, through investigation with technology, between changes in a real-world situation that can be modelled using a periodic function and transformations of the corresponding graph. (Formerly under Solving Problems Involving Models of Sinusoidal Functions.); - sketch graphs of f ( x) = asinx, f ( x) = sinx+ c, and f ( x) = sin( x d) by applying transformations to the graph of f ( x) = sinx, and state the domain and range of the transformed functions. - identify sine functions, including those that arise from real-world applications involving periodic phenomena, given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range. Functions and Applications, Grade 11, University/College Preparation (MCF3M) 6

8 Curriculum Revisions that Require Change to Program Strand: Financial Applications of Sequences and Series (mostly deleted some expectations have been incorporated into other strands) Overall Expectations Solving Problems Involving Arithmetic and Geometric Sequences and Series Solving Problems Involving Compound Interest and Annuities Solving Problems Involving Financial Decision Making Remove solve problems involving arithmetic and geometric sequences and series; solve problems involving financial decision making, using spreadsheets or other appropriate technology. - write terms of a sequence, given the formula for the nth term; - determine a formula for the nth term of a given sequence; - identify sequences as arithmetic or geometric, or neither; - determine the value of any term in an arithmetic or a geometric sequence, using the formula for the nth term of the sequence; - determine the sum of the terms of an arithmetic or a geometric series, using appropriate formulas and techniques. - derive the formulas for compound interest and present value, the amount of an ordinary annuity, and the present value of an ordinary annuity, using the formulas for the nth term of a geometric sequence and the sum of the first n terms of a geometric series; - demonstrate an understanding of the relationships between simple interest, arithmetic sequences, and linear growth. - describe the manner in which interest is calculated on a mortgage (i.e., compounded semi-annually but calculated monthly) and compare this with the method of interest compounded monthly and calculated monthly; - generate amortization tables for mortgages, using spreadsheets or other appropriate software; - analyse the effects of changing the conditions of a mortgage; - communicate the solutions to problems and the findings of investigations with clarity and justification. Add Functions and Applications, Grade 11, University/College Preparation (MCF3M) 7

9 Strand: Quadratic Functions (new strand) Remove Add Overall Expectations expand and simplify quadratic expressions, solve quadratic equations, and relate the roots of a quadratic equation to the corresponding graph; solve problems involving quadratic functions, including those arising from real-world applications. Solving Quadratic Equations - pose and solve problems involving quadratic relations arising from real-life applications and represented by tables of values and graphs; - represent situations using quadratic expressions in one variable, and expand and simplify quadratic expressions in one variable; - factor quadratic expressions in one variable, including those for which a 1, differences of squares, and perfect square trinomials, by selecting and applying an appropriate strategy; - solve quadratic equations by selecting and applying a factoring strategy; - determine, through investigation, and describe the connection between the factors used in solving a quadratic equation and the x-intercepts of the corresponding quadratic relation; - explore the algebraic development of the quadratic formula, and apply the formula to solve quadratic equations, with technology. Functions and Applications, Grade 11, University/College Preparation (MCF3M) 8

10 Connecting Graphs and Equations of Quadratic Functions Remove Add - substitute into and evaluate linear and quadratic functions represented using function notation, including functions arising from real-world applications; - explain the meanings of the terms domain and range, through investigation using numeric, graphical, and algebraic representations of linear and quadratic functions, and describe the domain and range of a function appropriately; - explain any restrictions on the domain and the range of a quadratic function in contexts arising from real-life applications; - determine, through investigation using technology, and describe the roles of a, h, and k in quadratic functions of the form f ( x) = a( x h) + k in terms of transformations on the graph of f ( x) = x (i.e., translations; reflections in the x-axis; vertical stretches and compressions); - sketch graphs of gx ( ) = ax ( h) + k by applying one or more transformations to the graph of f ( x) = x ; - express the equation of a quadratic function in the standard form f ( x) = ax + bx+ c, given the vertex form f ( x) = a( x h) + k, and verify, using graphing technology, that these forms are equivalent representations; - express the equation of a quadratic function in the vertex form f ( x) = a( x h) + k, given the standard form f ( x) = ax + bx+ c, by completing the square, including cases where b a is a simple rational number, and verify, using graphing technology, that these forms are equivalent representations; - sketch graphs of quadratic functions in the factored form f ( x) = a( x r)( x s) by using the x-intercepts to determine the vertex; - sketch the graph of a quadratic function whose equation is given in the standard form f ( x) = ax + bx+ c by using a suitable strategy, and identify the key features of the graph. Functions and Applications, Grade 11, University/College Preparation (MCF3M) 9

11 Solving Problems Involving Quadratic Functions Remove Add - solve problems arising from real-world applications, given the algebraic representation of a quadratic function; - collect data that can be modelled as a quadratic function, through investigation with and without technology from primary sources using a variety of tools, or from secondary sources and graph the data; - determine, through investigation using a variety of strategies, the equation of the quadratic function that best models a suitable data set graphed on a scatter plot, and compare this equation to the equation of a curve of best fit generated with technology. Strand: Trigonometric Functions Sub-groupings Remove Add Overall Expectations demonstrate an understanding of the meaning and application of radian measure. Solving Problems Involving the Sine Law and the Cosine Law in Oblique Triangles (renamed Applying the Sine Law and the Cosine Law in Acute Triangles) Understanding the Meaning and Application of Radian Measure - determine the sine, cosine, and tangent of angles greater than 90, using a suitable technique, and determine two angles that correspond to a given single trigonometric function value. - define the term radian measure; - describe the relationship between radian measure and degree measure; - represent, in applications, radian measure in exact form as an expression involving π and in approximate form as a real number - determine the exact values of the sine, cosine, and tangent of the special angles 0, 6 π, 4 π, 3 π, π and their multiples less than or equal to π ; - prove simple identities, using the Pythagorean identity, sin x + cos x = 1, and the quotient sin x relation, tan x = ; cos x - solve linear and quadratic trigonometric equations on the interval 0 x π ; - demonstrate facility in the use of radian measure in solving equations and in graphing. - verify, through investigation using technology, the sine law and the cosine law; - describe conditions that guide when it is appropriate to use the sine law or the cosine law, and use these laws to calculate sides and angles in acute triangles. Functions and Applications, Grade 11, University/College Preparation (MCF3M) 10

12 Sub-groupings Remove Add Investigating the Relationships - sketch the graphs of simple sinusoidal functions Between the Graphs and the - write the equation of a sinusoidal function, given Equations of Sinusoidal Functions its graph and given its properties; (renamed Connecting Graphs and Equations of Sine Functions) Solving Problems Involving Models of Sinusoidal Functions (renamed Solving Problems Involving Sine Functions) - sketch the graph of y = tan x; identify the period, domain, and range of the function; and explain the occurrence of asymptotes. - explain the relationship between the properties of a sinusoidal function and the parameters of its equation, within the context of an application, and over a restricted domain. - collect data that can be modelled as a sine functions, through investigation with and without technology, from primary sources using a variety of tools or from secondary sources and graph the data; - predict, by extrapolating, the future behaviour of a relationship modelled using a numeric or graphical representation of a periodic function. Functions and Applications, Grade 11, University/College Preparation (MCF3M) 11

13 Strand: Tools for Operating & Communication with Functions (mostly deleted some expectations have been incorporated into quadratic Functions and Exponential Functions) Manipulating Polynomials, Rational Expressions, and Exponential Expressions Understanding Inverses and Transformations and Using Function Notation Remove - solve first-degree inequalities and represent the solutions on number lines; - identify the structure of the complex number system and express complex numbers in the form a + bi, where i = 1 - explain the relationship between a function and its inverse (i.e., symmetry of their graphs in the line y = x; the interchange of x and y in the equation of the function; the interchanges of the domain and range), using examples drawn from linear and quadratic functions, and from the functions f (x) = x and f (x) = 1 x ; Add Communicating Mathematical Reasoning - represent inverse functions, using function notation, where appropriate; - represent transformations of the functions defined by f(x) = x, f(x) = x, f(x) = x, f(x) = sin x, and f(x) = cos x, using function notation; - describe, by interpreting function notation, the relationship between the graph of a function and its image under one or more transformations; - state the domain and range of transformations of the functions defined by f(x) = x, f(x) = x, f(x) = x, f(x) = sin x, and f(x) = cos x. - explain mathematical processes, methods of solution, and concepts clearly to others; - present problems and their solutions to a group, and answer questions about the problems and the solutions; - communicate solutions to problems and to findings of investigations clearly and concisely, orally and in writing, using an effective integration of essay and mathematical forms; - demonstrate the correct use of mathematical language, symbols, visuals, and conventions; - use graphing technology effectively. Functions and Applications, Grade 11, University/College Preparation (MCF3M) 1

14 Strand: Exponential Functions (new strand) Remove Add Overall Expectations identify and represent exponential functions, and solve problems involving exponential functions, including those arising from real-world applications. Connecting Graphs and Equations of Exponential Functions Solving Problems Involving Exponential Functions) Solving Financial problems Involving Exponential Functions - evaluate, with and without technology, numerical expressions containing integer and rational exponents and rational bases; - graph, with and without technology, an exponential x relation, given its equation in the form y = a (a > 0, a 1), define this relation as the function x f ( x) = a, and explain why it is a function; - distinguish exponential functions from linear and quadratic functions by making comparisons in a variety of ways within the same context when possible. - identify exponential functions, including those that arise from real-world applications involving growth and decay, given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range; - solve problems using given graphs or equations of exponential functions arising from a variety of real-world applications by interpreting the graphs or by substituting values for the exponent into the equations. - solve problems, using a TVM Solver in a graphing calculator or on a website, that involve the calculation of the interest rate per compounding period, i, or the number of compounding periods, n, in the compound interest n formula A = P( 1+ i) [or FV PV ( 1 i) = + ]. n Functions and Applications, Grade 11, University/College Preparation (MCF3M) 13