Course outline Mathematics: Methods ATAR Year 11

Transcription

1 Course outline Mathematics: Methods ATAR Year 11 Unit 1 Sequential In Unit 1 students will be provided with opportunities to: underst the concepts techniques in algebra, functions, graphs, trigonometric functions, counting probability solve problems using algebra, functions, graphs, trigonometric functions, counting probability apply reasoning skills in the context of algebra, functions, graphs, trigonometric functions, counting probability interpret evaluate mathematical information ascertain the reasonableness of solutions to problems communicate their arguments strategies when solving problems. Semester 1 is based on a 13 week teaching block 1 week review block. Time Allocation Placement Week 1 (2 hours) Weeks 1 2 (5 hours) Topics Textbook Reference Key teaching points Semester 1 (Unit 1 plus intro of Unit 2) Topic 1: Lines linear relationships ( ) Functions graphs coordinates of mid-points end-point direct proportion linearly related variables Ch 1 (excuding ) Topic 1: Functions graphs Ch 1 (excuding ) Weeks 2 4 (7 hours) Topic 1: Functions graphs Ch features of the graph of y mx c equations of a straight lines given sufficient information; including parallel perpendicular lines solve linear equations, including those with algebraic fractions variables on both sides Quadratic relationships ( ) examine examples of quadratically related variables 2 2 features of the graphs of y x, y a( x b) c y a( x b)(x c) including their parabolic nature, turning points, axes of symmetry intercepts solve quadratic equations, including the use of quadratic formula completing the square equation of a quadratic, turning points, zeros, discriminant graph of the general quadratic Inverse Proportion ( ) examples of inverse proportion 2 y ax bx c equations of the graphs of,, including their hyperbolic shapes their asymptotes. Powers polynomials ( ) recognise features of the graphs of Identify coefficients the degree of a polynomial exp quadratic cubic polynomials from factors shape, behaviour as features equations of the graphs of ; shape, intercepts behaviour as factorise cubic polynomials (in cases where a linear factor is easily obtained) QBC Mathematics Course Outline: Yr 11 Methods 1 2

2 Weeks 4 6 (8 hours) Weeks 6 8 (5 hours) Weeks 8 11 (10 hours) Topic 1: Functions graphs Ch 4 Topic 2: Trigonometric Functions Ch 3 Topic 2: Trigonometric Functions Ch 6 solve cubic equations using technology, algebraically in cases where a linear factor is easily obtained Graphs relations ( ) features equations of the graphs of their circular shapes, centres radii graph of, shape axis of symmetry Functions ( ) the concept of a function as a mapping as a rule or a formula that defines one variable quantity in terms of another use function notation; determine domain range; recognise independent dependent variables the graph of a function translations the graphs of dilations the graphs of distinction between functions relations the vertical line test Sine cosine rules ( ) right-angled triangles trig. ratios unit circle definition of cos ɵ, sin ɵ tan ɵ periodicity using degrees angle of inclination of a line the gradient of that line establish use the cosine sine rules, including consideration of the ambiguous case the formula Area 1 bc sin A for the area of triangle. Circular measure radian measure ( ) use radian measure degree measure calculate lengths of arcs areas of sectors segments in circles Trigonometric functions ( ) underst the unit circle definition of sin ɵ, cos ɵ tan ɵ periodicity using radians recognise the exact values of sin ɵ, cos ɵ tan ɵ at integer multiples of 6 4 recognise the graphs of y sin x, y cos x y tan x on extended domains examine amplitude changes the graphs of y asin x y acos x examine period changes the graphs of y sinbx, y cosbx y tan bx examine phase changes the graphs of y sin(x - c), y cos(x - c) y tan(x - c) examine the relationships sin x cos x Week (4 hours) Counting probability Ch 2 prove apply the angle sum difference identities identify contexts suitable for modelling by trigonometric functions use them to solve practical problems solve equations involving trigonometric functions using technology, algebraically in simple cases Combinations ( ) underst the notion of a combination as a set of r objects taken from a set of n distinct objects use the notation the formula for the number of combinations of r objects taken from a set of n distinct objects exp x y n for small positive integers n QBC Mathematics Course Outline: Yr 11 Methods 1 2

3 recognise the numbers as binomial coefficients (as coefficients in the expansion of x y n use Pascal s triangle its properties Weeks (4 hours) Weeks (4 hours) Counting probability Ch 2 Counting probability Ch 2 Language of events sets ( ) review the concepts language of outcomes, sample spaces, events, as sets of outcomes use set language notation for events, including: a. for the complement of an event A b. for the intersection union of events A B respectively c. for the intersection union of the three events A, B C respectively d. recognise mutually exclusive events. use everyday occurrences to illustrate set descriptions representations of events set operations Review of the fundamentals of probability ( ) review probability as a measure of the likelihood of occurrence of an event review the probability scale: for each event A with P(A) 0 if A is an impossibility P(A) 1 if A is a certainty review the rules: P(A) 1 - P(A) Week 15 use relative frequencies obtained from data as estimates of probabilities Revision/end of Unit 1 assessment Semester 2 is based on a 17 week teaching block 2 weeks review block. Unit 2 sequential In Unit 2 students will be provided with opportunities to: underst the concepts techniques used in algebra, sequences series, functions, graphs calculus solve problems in algebra, sequences series, functions, graphs calculus apply reasoning skills in algebra, sequences series, functions, graphs calculus interpret evaluate mathematical statistical information ascertain the reasonableness of solutions to problems communicate arguments strategies when solving problems. Time Allocation Placement Weeks (6 hours) Topics Counting Probability Key teaching points Semester 2 (Unit 2 plus review of Unit 1) Conditional probability independence ( ) underst the notion of a conditional probability recognise use language that indicates conditionality Ch 5 QBC Mathematics Course Outline: Yr 11 Methods 1 2

4 use the notation the formula Weeks (10 hours) Week (6 hours) Week (9 hours) Week (9 hours) Topic 4: Exponential functions Ch 11 Topic 4: Arithmetic geometric sequences series Ch 8 Topic 4: Arithmetic geometric sequences series Ch 8 Topic 5: Introduction to differential calculus ch 7 underst the notion of independence of an event A from an event B, as defined by establish use the formula for independent events A B, recognise the symmetry of independence use relative frequencies obtained from data as estimates of conditional probabilities as indications of possible independence of events Indices the index laws ( ) review indices (including fractional negative indices) the index laws use radicals convert to from fractional indices underst use scientific notation significant figures Exponential functions ( ) establish use the algebraic properties of exponential functions recognise the qualitative features of the graph of, including asymptotes, of its translations identify contexts suitable for modelling by exponential functions use them to solve practical problems solve equations involving exponential functions using technology, algebraically in simple cases Arithmetic sequences ( ) recognise use the recursive definition of an arithmetic sequence: = + develop use the formula = + ( 1) for the general term of an arithmetic sequence recognise its linear nature use arithmetic sequences in contexts involving discrete linear growth or decay, such as simple interest establish use the formula for the sum of the first terms of an arithmetic sequence Geometric sequences ( ) recognise use the recursive definition of a geometric sequence: develop use the formula for the general term of a geometric sequence recognise its exponential nature underst the limiting behaviour as n of the terms tn in a geometric sequence its dependence on the value of the common ratio r establish use the formula for the sum of the first n terms of a geometric sequence use geometric sequences in contexts involving geometric growth or decay, such as compound interest Rates of change the concept of the derivative ( ) interpret the difference quotient as the average rate of change of a function f use the Leibniz notation variables x y for changes or increments in the use the notation for the difference quotient QBC Mathematics Course Outline: Yr 11 Methods 1 2

5 interpret the ratios as the slope or gradient of a chord or secant of the graph of y = f(x) examine the behaviour of the difference quotient as an informal introduction to the concept of a limit define the derivative use the Leibniz notation for the derivative: the Week (9 hours) Week (12 hours) Topic 5: Introduction to differential Calculus Ch 9 Ch 10 Introduction to differential Calculus Ch 12 correspondence 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 properties of derivatives ( ) estimate numerically the value of a derivative for simple power functions examine examples of variable rates of change of non-linear functions establish the formula for non-negative integers n exping (x + h) n or by factorising (x + h) n x n underst the concept of the derivative as a function identify use linearity properties of the derivative calculate derivatives of polynomials Applications of derivatives Anti-derivatives ( ) determine instantaneous rates of change determine the slope of a tangent the equation of the tangent construct interpret position-time graphs with velocity as the slope of the tangent recognise velocity as the first derivative of displacement with respect to time sketch curves associated with simple polynomials, determine stationary points, local global maxima minima, examine behaviour as x x solve optimisation problems arising in a variety of contexts involving polynomials on finite interval domains calculate anti-derivatives of polynomial functions Week Revision/end of course assessment Week 35 SEM 2 Exam Revision/end of Unit 2 assessment QBC Mathematics Course Outline: Yr 11 Methods 1 2

6 Assessment outline Mathematics: Methods ATAR Year 11 Units 1 2 combined Assessment type Response 40% 6% Periodic (fortnightly) 4% Term 1 Week 6 5% Term 1 Week % Term 2 Week 5 6% Term 3 Week 2 6.5% Term 3 Week 5 7% Term 3 Week 10 Investigation 20% 3% Term 1 Week 3 3.5% Term 1 Week 7 4% Term 2 Week 3 4.5% Term 2 Week 10 5% Term 3 Week 9 Examination 40% 15% Semester 1 Week 16 Total 100% 100% 25% Semester 2 Week 30 Task 1: Short skills/concept tests (10 15 minutes) Task 3: Test 1 Linear quadratic relationships Task 5: Test 2 Inverse proportions, powers/polynomials, graphs/relations functions (plus revision of concepts from previous tasks) Task 7: Test 3 Shape measurement (plus revision of concepts from previous tasks) Task 10: Test 4 Uni-variate data (plus revision of concepts from previous tasks) Task 11: Test 5 Comparing data trigonometric applications (plus revision of concepts from previous tasks) Task 13: Test 6 Linear equations graphs (plus revision of concepts from previous tasks) Task 14: Test 7 Linear equations graphs (plus revision of concepts from previous tasks) Task 2: Investigation 1 Preparation activity with in-class validation: Investment strategies/personal budgets Task 4: Investigation 2 In-class investigation: Storing using data in matrices Task 6: Investigation 3 Preparation activity with in-class validation: the swimming pool project Task 9: Investigation 4 Preparation application activity with in-class validation: Statistical investigation Task 12: Investigation 5 Preparation activity with in-class validation: Algebra applications Examination Task 8: Semester 1 examination Section One: calculator-free (35%), Section Two: calculator-assumed (65%). Question selection from Unit 1 content knowledge, skills processes Examination Task 14: Semester 2 examination section One: calculator-free (35%), Section Two: calculator-assumed (65%). Question selection from Unit 1 Unit 2 content knowledge, skills processes QBC Mathematics Course Outline: Yr 11 Methods 1 2