MATHEMATICS LEARNING AREA. Methods Units 1 and 2 Course Outline. Week Content Sadler Reference Trigonometry

Transcription

1 MATHEMATICS LEARNING AREA Methods Units 1 and 2 Course Outline Text: Sadler Methods and 2 Week Content Sadler Reference Trigonometry Cosine and Sine rules Week 1 Trigonometry Week 2 Radian Measure Radian Measure Week Review sine, cosine and tangent as ratios of side lengths in right-angled triangles Understand the unit circle definition of cos, sin and tan and periodicity using degrees Examine the relationship between the angle of inclination of a line and the gradient of that line Cosine and Sine rules Establish and use the cosine and sine rules, including consideration of the ambiguous case and the 1 Formula Area= bcsin A for the area of a triangle 2 Circular measure and radian measure Define and use radian measure and understand its relationship with degree measure Circular measure and radian measure Define and use radian measure and understand its relationship with degree measure Calculate the lengths of arcs and areas of sectors and segments in circles Understand the unit circle definition of cos, sin and tan and periodicity using radians Recognise the exact values of cos, sin and tan at integer multiples of 6 and 4 Chapter 1 Chapter 1 Chapter 2 Chapter 2 Response 1 1

2 Function Week 4 Functions Understand the concept of a function as a mapping between sets and as a rule or a formula that Defines one variable quantity in terms of another Use function notation; determine domain and range; recognise independent and dependent Variables Understand the concept of the graph of a function Chapter 3 Linear Functions Week 5 Quadratic Functions Week 6 Quadratic Equations Week 7 Linear relationships Determine the coordinates of the mid-point between two points Determine an end-point given the other end-point and the mid-point Examine examples of direct proportion and linearly related variables Recognise features of the graph y = mx + c, including its linear nature, its intercepts and its slope or gradient Determine the equation of a straight line given sufficient information; including parallel and perpendicular lines Solve linear equations, including those with algebraic fractions and variables on both sides Quadratic relationships Examine examples of quadratically related variables Recognise features of the graphs of y = x 2, y = a(x-b) 2 +c, and y = a(x-b)(x-c) including their parabolic nature, turning points, axes of symmetry and intercepts Determine turning points and zeros of quadratics and understand the role of the discriminate Recognise features of the graph of the general quadratic y=ax 2 + bx + c Quadratic relationships Solve quadratic equations, including the use of the quadratic formula and completing the square Determine the equation of a quadratic given sufficient information Chapter 4 Chapter 5 Investigation 1 Chapter 6 2

3 Polynomial and other Functions Week 8-9 Inverse proportion Examine examples of inverse proportion Recognise features and determine equations of the graphs of their hyperbolic shapes and their asymptotes 1 y and x y a, including x b Chapter 7 Powers and polynomials Recognise features of the graphs of y = x n for n N, n=-1 and n= 2 1, including shape, and Behaviour as x and x Identify the coefficients and the degree of a polynomial Expand the quadratic and cubic polynomials from factors Recognise features and determine equations of the graphs of y=x 3, y=a(x-b) 3 +c and y=k(x-a)(x-b)(x-c), including shape, intercepts and behaviour as x and x Factorise cubic polynomials in cases where a linear factor is easily obtained Solve cubic equations using technology and algebraically in cases where a linear factor is easily obtained Functions Examine translations and the graphs of y=f(x)+a and y=f(x-b) Examine dilations and the graphs of y=cf(x) and y=f(dx) Graphs of relations Recognise features and determine equations of the graphs of ( x a) ( y b) r x y r and, including their circular shapes, their centre and their radii Recognise features of the graph y x, including its parabolic shape and its axis of symmetry 3

4 Trigonometric Functions Week 10 Sets and Probability Week Trigonometric functions Recognise the graphs of y = sin x, y = cos x, and y = tan x on extended domains Examine amplitude changes and the graphs of y = asin x and y = acos x Examine period changes and the graphs of y = sin bx, y = cos bx and y = tan bx Examine phase changes and the graphs of y = sin(x-c), y = cos(x-c) and y = tan(x-c) Examine the relationships sin(x+ 2 )= cos x and cos(x- 2 ) = sin x Prove and apply the angle sum and difference identities Identify contexts suitable for modelling by trigonometric functions and use them to solve Practical problems Solve equations involving trigonometric functions using technology, and algebraically in simple Cases Language of events and sets Review the concepts and language of outcomes, sample spaces, and events, as sets of outcomes Use set language and notation for events, including: a. A (or A ) for the complement of an event A b. A B and A B for the intersection and union of events A and B respectively c. A B C and A B C for the intersection and union of the three events A,B and C respectively d. recognise mutually exclusive events Use everyday occurrences to illustrate set descriptions and representations of events and set Operations Review of the fundamentals of probability Review the probability as a measure of likelihood of occurrence of an event Review the probability scale: 0 P ( A) 1 for each event A, with P(A) = 0 if A is an impossibility and P(A) = 1 if A is a certainty Review the rules: P( A) 1 P( A) and P( A B) A) B) A B) Use relative frequencies obtained from data as estimates of probabilities Chapter 8 Response 2 Chapter 9 Investigation 2 4

5 Conditional probability and independence Week Understand the notion of a conditional probability and recognise and use language that indicates Conditionality Use the notation P( A B) and the formula P( A B) A B) P( B) Understand the notion of independence of an event A from an event B, as defined by P( A B) A) Establish and use the formula P( A B) A) P( B) for independent events A and B, and recognise the symmetry of independence Use relative frequencies obtained from data as estimates of conditional probabilities and as indications of possible independence of events Combinations Understand the notion of a combination as a set of r objects taken from a set of n distinct objects n n n! Use the notation and the formula = for the number of combinations of r r r r!( n r)! Objects taken from a set of n distinct objects Expand (x+y) n for small positive integers n n Recognise the numbers as binomial coefficients (as coefficients in the expansion (x+y) n ) r Use Pascal s triangle and its properties Chapter 10 Response 3 Week Revision and Examinations Indices and Index Laws Week 19 Indices and index laws Review indices (including fractional and negative indices) and the index laws Use radicals and convert to and from fractional indices Chapter 1 5

6 Indices and Index Laws Exponential Functions Week 20 Exponential Functions Week 21 Sequences Week Indices and index laws Understand and use scientific notation and significant figure Exponential functions Establish and use the algebraic properties of exponential functions Recognise the qualitative features of the graph of y = a x (a > 0), including asymptotes, and of its translations (y = a x + b and y = a x-c ) Exponential functions Identify contexts suitable for modelling by exponential functions and use them to solve practical Problems Solve equations involving exponential functions using technology, and algebraically in simple Cases Arithmetic sequences Recognise and use the recursive definition of an arithmetic sequence: t n 1 = tn + d Develop and use the formula tn = t1 + (n 1)d for the general term of an arithmetic sequence and recognise its linear nature Use arithmetic sequences in contexts involving discrete linear growth or decay, such as simple interest Geometric sequences Recognise and use the recursive definition of a geometric sequence: t t r n Develop and use the formula t n t1r for the general term of a geometric sequence and recognise its exponential nature Understand the limiting behaviour as n of the terms tn in a geometric sequence and its dependence on the value of the common ratio r n 1 n Chapter 1 Chapter 2 Chapter 2 Chapter 3 6

7 Series Week Rates of Change Week Arithmetic Series Establish and use the formula for the sum of the first n terms of an arithmetic sequence Geometric Series n r Establish and use the formula S n t1 for the sum of the first n terms of a geometric r 1 Sequence Use geometric sequences in contexts involving geometric growth or decay, such as compound interest Rates of change f ( x h) f ( x) Interpret the difference quotient as the average rate of change of a function f h Use the Leibniz notation δx and δy for changes or increments in the variables x and y y f ( x h) f ( x) Use the notation for the difference quotient x h where y = f(x) f ( x h) f ( x) y Interpret the ratios and h x as the slope or gradient of a chord or secant of the graph y = f(x) The concept of the derivative Examine the behaviour of the difference quotient introduction to the concept of a limit Define the derivative f (x) as lim h 0 dy Use the Leibniz notation for the derivative: = dx dy = f (x) where y = f(x) dx 7 f ( x h) f ( x) h as h 0 as an informal f ( x h) f ( x) h y lim x 0 and the correspondence x Chapter 4 Response 4 Investigation 3 Chapter 5

8 Applications of Differentiation Week Antidifferentiation Week 32 Rectilinear Motion Week 33 Week Interpret the derivative as the instantaneous rate of change Interpret the derivative as the slope or gradient of a tangent line of the graph of y = f(x) Computation of derivatives Estimate numerically the value of a derivative for simple power functions Examine examples of variable rates of change of non-linear functions d n n Establish the formula ( x ) nx for non-negative integers n expanding (x+h) n or by dx n n factorising ( x h) x Properties of derivatives Understand the concept of the derivative as a function Identify and use linearity properties of the derivative Calculate derivatives of polynomials Applications of derivatives Determine instantaneous rate of change Determine the slope of a tangent and the equation of a tangent Sketch curves associated with simple polynomials, determine stationary points, and local and global maxima and minima, and examine behaviour as x and x Solve optimisation problems arising in a variety of contexts involving polynomials on finite interval domains Anti-derivatives Calculate anti-derivatives of polynomial functions Rectilinear motion Construct and interpret position-time graphs with velocity as the slope of the tangent Recognise velocity as the first derivative of displacement with respect to time Revision and Examinations 8 Chapter 6 Chapter 7 Chapter 8 Response 5

9 Assessment Record Mathematics Methods Year 11- and 2 Assessment Date Administered Response 1 () Week 3 Response 2 () Week 10 Response 3 () Week 15 Content Cosine and Sine Rules, Circular Measure and Radian Measure Linear, Functions, Quadratic Functions, Polynomial and other Functions and Trigonometric Functions Sets and Probability, Combinations My Raw Mark (%) YEAR Weighting 4% 6% 6% YEAR Value (Raw mark weighting) Total weighting of each assessment type 40% Response 4 () Week 25 Index Laws, Exponential Functions, Sequences and Series Response 5 () Week 33 Rates of Change, Applications of Differentiation, Anti-differentiation, Rectilinear Motion Investigation 1 Week 6 Functions 12% 12% % Investigation 2 Week 24 Sequences % 20% Investigation 3 Week 29 Rates of Change % Semester 1 EXAM Week 17 Semester 2 EXAM Week 35 and 2 15% 25% 40% 9

10 Refer to College Diary ASSESSMENT PROCEDURE 10