AS & A2 Maths Scheme of Work

Transcription

1 AS & A Maths Scheme of Work Date 1 st September 014 Topic & page ref in Heinemann Live Text Core 1: p1-14 Ch1. ALGEBRA & FUNCTIONS 1.1 Simplifying terms by collecting like terms 1. Rules of Indices 1.3 Expanding an expression 1.4 Factorising expressions 1.5 Factorising quadratic expressions 1.6 Rules of Indices for all R 1.7 Surds (p10) 1.8 Rationalising the denominator Notes Teachers are free to use whatever resources they wish but must adhere to the timings of the SOW. It is suggested that in class use the LiveText CD ROM to go through examples. Skills developed & Examples students should be able to answer the end of each section Homework Resources Students should learn the squares from 1 to 16 ; cube numbers from 1 3 to 6 3 which will help them solve fractional indices problems. Examples: Fractional Indices: if 81 ½ is 9 then 5 ½ is.? Simplify a 5 a 3 ; m 4 m ; (p ) 5 ; (xy ) 3 ; solve n =16; solve 3 x-1 = 7 Simplify: Rationalise the denominator; ; ; Expand Factorise and solve : x + 8x + 15 = 0; x + 7x + 6 = 0 ; 4x -1 = 0; And worth at this stage pointing out how to find the roots (or solutions) and the critical values which can be used to sketch a curve of the fn. Staff should try to explicitly differentiate homework to meet the needs of all learners. Heinemann C1 Live Text on CD to use in lessons to support explanations. Tarsias : Manipulating Surds Standards Unit N11 Surds Standards Unit N1 using indices as PDF. These can be used as homework to stretch all students. There are also TESTS that can be flashed in lessons on the IWB. September Core 1: p15 6 Ch. QUDARATIC FUNCTIONS.1 plotting graphs of quadratic functions. solving quadratic equations by factorisation.3 competing the square.4 solving quadratic equations by competing the square.5 solving quadratic equations by using the formula.6 sketching graphs of quadratic functions Examples: Know and learn the quadratic formula: Solve by completing the square: x + 8x + 15 = 0; x 1x + 7 = 0 By completing the square, find the minimum value of x 4x 9. Show that the line y = x 4 is a tangent to the circle with equation x + y = 8 Extension: Reproduce the proof of the Quadratic formula Standards Unit C1 Linking the properties & forms of Quadratic Functions September Core 1: p7 40 Ch3. EQUATIONS & INEQUALITIES 3.1 Simultaneous equations by Examples: Solve x 4y 7 and x + y = 16 by elimination and substitution What about: 3x + y = 10 and x + xy + y = 17

2 elimination 3. Simultaneous equations by substitution 3.3 Simultaneous equations with 1 linear & 1 quadratic 3.4 solving linear inequalities 3.5 solving quadratic inequalities September Core 1: p41 68 Ch4. SKETCHING CURVES 4.1 sketching graphs of cubic functions 4. interpreting graphs of cubic functions 4.3 sketching the reciprocal function 4.4 using intersections points of graphs to solve equations 4.5 The effect of f(x+a), f(x-a) and f(x) + a 4.6 The effect of af(x), -f(x) and f(-x) 4.7 performing transformations on the sketches of curves Solve for x: (a) 5x x 16 (b) x 5 Solve and sketch x + 8x ; x - 10x Examples: Know the graphs of: y = x; y = x ; y = x 3 ; y = y = x ; Understand and sketch transformations of any given graph, inc. f(x+a), f(x-a), f(ax), af(x), -f(x) and f(-x) say, for f(x) = x Extension Questions: October Core 1: p73 90 Ch5. COORDINATE GEOMETRY IN THE (x, y) PLANE 5.1 The equation of a straight line 5. The gradient of the straight line 5.3 y y 1 = m(x x 1) 5.4 the formula for finding the equation of a straight line 5.5 Parallel and perpendicular lines To know that: The equation of a straight line can be written as y = mx + c, where m is the gradient and c is the intercept with the vertical axis. Lines are parallel if they have the same gradient. Two lines are perpendicular if the product of their gradients is -1. If the gradient of a line is m, then the gradient of a perpendicular line is 1 m The gradient of a line passing through the points 1 1, 1 and, is y x y x y y. x x 1 The equation of the straight line with gradient m that passes through the point x y is y y m( x x ). 1 1 The distance between the points with coordinates x1, y1 and x, y is x x1 y y1. The midpoint of the line joining the points x1 x y1 y. x1, y1 and x, y is, Example: Find the equation of the perpendicular bisector of the line joining the points 1, 1 Condensed 1 page notes with questions for coordinate Geometry

3 (3, ) and (5, -6). Example: Find the point of intersection of the lines: x + y = 3 and y = 3x 1. Extension Question: October Core 1: p Ch6. ARITHMETIC SEQUNECES 6.1 Introduction to Sequences 6. the nth term 6.3 recurrence relationships 6.4 Arithmetic sequences 6.5 Arithmetic series 6.6. the sum to n of an arithmetic series 6.7 The sigma notation Students usu. struggle with the notion of Un better to start with simple sequences and explain how to find the nth term (like at KS3) Formula for the nth term and the sum of a series will be given If numbers ascend in 3 s, that s the 3 x table = 3n. Then find the number before the 1 st term (=5), so, nth term is 3n+5 nth term in sequence 8, 11, 14, 17,...,...,... Standards Unit N13 Analysing sequences October Core 1: p11 13 Ch7. DIFFERENTIATION 7.1 derivative of f(x) 7. gradient of 7.3 gradients of simple functions 7.4 gradients of functions with power 7.5. re-writing expressions to make them easier to differentiate Rate of change of a function at a point 7.8 Equations of Tangents and Normals Find : 4 y = x 5x ; y = 7x 1x 5 ; y = 1 5x 6 x 4 9 ; y = 1 x x NB: A turning point occurs where the gradient is zero, i.e. where 0. And you can also use the nd derivative to decide whether a turning point is a maximum or a minimum: If d y 0 then it is a minimum dx ; dy 0 then it isa maximum. dx Equation of a tangent Standards Unit C exploring functions involving fractional and negative powers of x Standards Unit C3 matching functions & derivatives Standards Unit C4 differentiating & Integrating fractional and negative powers of x tells you the gradient of a curve. The gradient m of a tangent line at the point x1, y 1 can be found from. The equation of the tangent is then y y1 m( x x1). Standards Unit C5 Finding stationary points of cubic functions Perpendicular lines

4 Suppose lines have gradients m & m. These lines are perpendicular if m m, i.e. 1. m m 1 1 Equation of a normal To find the equation of a normal at the point x y : 1, Find the gradient from then find the gradient m of the normal using 1 m dy dx and the equation of the normal is y y1 m( x x1) Example: Find the equation of the normal to the graph y = x(x + 1) (x ) at x = -1. October OCTOBER HALF TERM October Core 1: p Ch8. INTEGRATION 8.1 Integrating 8. Integrating simple expressions 8.3 using the 8.4 Simplifying before integrating 8.5 Finding c Rule: Increase the power by 1 and divide by the new power. Integration is the reverse of differentiation. Example: Find y if dy x 6x dx and y = 4 when x = 3 (answer: 3 x y 3x x 16) 3 Questions: 1. v (3x 4x ) dx. If v = 3 when x = 0, find v as a function of x. Hence calculate the value of v when x = 1. Standards Unit C4 differentiating & Integrating fractional and negative powers of x November/ December 3 rd November START CORE Core : p1 17 Ch1. ALGEBRA & FUNCTIONS 1.1 Simplifying algebraic fractions 1. Dividing a polynomial 1.3 factorising a polynomial 1.4 Using the remainder theorem 6 hour teacher on their own teaching C 4 hour teacher start the Applied module Factor Theorem: (x a) is a factor of a polynomial f(x) if f(a) = 0. Remainder Theorem: The remainder when a polynomial f(x) is divided by (x a) is f(a). Extended version of the factor theorem: (ax + b) is a factor of a polynomial f(x) if b f 0 a 3 1. g(x) = x 3x 13x 15. (a) Show that g(-5) = 0 and g(3) = 0. (b) Hence factorise g(x). (c) Sketch the graph of y = g(x). (d) Write down the full set of values of x for which g(x) > 0. Heinemann C Live Text on CD to use in lessons to support explanations. Tarsias Condensed 1 page notes with questions for factor theorem Standards Unit A11 factorising cubics Extenstion Question:

5 December Core : p18 37 Ch: THE SINE & COSINE RULE.1 Sine rule for missing sides. Sine rule for unknown angles.3 Solutions for a missing angles.4 Cosine rule to find unknown sides.5 Cosine rule to find missing angles.6 Sine, Cosine & Pythagoras.7 Area of a triangle Core : p38 50 Ch3: EXPONENTIALS & LOGARITHMS 3.1 The function y = a x 3. writing expressions as a logarithm 3.3 calculating using log to base Laws of Logs 3.5 solving a x = b 3.6 changing the base Know the 3 trig ratios using: S O H C A H T O A Only the cosine rule formula will be provided in the formula book. Know the Area of a triangle is A = 1 sin ab C Know sin x tan x and cos x Extension questions: sin xcos x 1 Standards Unit A13 simplifying Log expressions

6 December CHRISTMAS HOLIDAYS Revise for the C1 MOCK Exam in January Look into 1-Day revision sessions at UCL/Imperial College January 015 C1 MOCK Exam (internal) C1 Solomon Paper?? Solomon Paper TBC at dept. meeting closer to the time This will take place during lesson time. Y1 will have a Mock week later in the year. January Core : p51 7 Ch4. COORDINATE GEOMETRY IN THE (x, y) PLANE 4.1 The mid-point of a line 4. Distance between two points 4.3 The equations of a circle The equation of a circle centre (a, b) with radius r is ( x a) ( y b) r. Example: Find the centre and the radius of the circle with equation x x y x Extension Question: January Core : p76 86 Ch5. THE BINOMIAL EXPANSION 5.1 Pascal s Triangle 5. Combinations and Factorial Notation 5.3 Using ( ) in the binomial expansion 5.4 Expanding ( ) ( ) Note that in C n e.g. 1: Find the expansion of 4 3x y. e.g. : Find the first 4 terms in the expansion 10 a 3b. 4 4 e.g. 3: Find the coefficient of xy in the expansion of E.g. Find the non-zero value of b if the coefficient of is equal to the coefficient of x y 5 x in the expansion of 8 x in the expansion of 6 bx. b x Condensed 1 page notes with questions for Binomial theorem January/ February Core : p Ch6: RADIAN MEASURE 6.1 Using radians to measure angles 6. The length of an arc 6.3 The area of a sector 6.4 The area of a segment Know that 360 o = π radians Why are there 360 o in a circle? What is 1 radian? Convert rads into degrees Convert 150 o into radians Prove the length of an arc is l = rθ Show that the area of a sector is A = Show that the area of a segment in a circle is A = ( )

7 February Core : p Ch7: GEOMETRIC SEQUENCES & SERIES 7.1 Geometric sequences 7. geometric progression & the nth term 7.3 Using a G.P to solve problems 7.4 Sum of a G.P 7.5 Sum to infinity of a geometric series If 3, x and 9 are the first three terms of a geometric sequence, find x and the value of the 4 th term. What is the first term in the GP 3, 6, 1, 4 to exceed 1 million? Show that the general term for the sum of a GP is Show that the sum to infinity of a GP is Find ( ) ( ) ( ) or ( ) ( ) Standards Unit N13 Analysing sequences February February/ March FEBRUARY HALF TERM Core : p Ch8. GRAPHS OF TRIGONOMETRICAL FUNCTIONS 8.1 Sin, Cos & Tan functions 8. Values of trig functions in all 4 quadrants 8.3 Exact values & surds for trig functions 8.4 Graphs of Sin θ, Cos θ & Tan θ 8.5 simple transformations of Sin θ, Cos θ & Tan θ Standards Unit A1 matching activities & probing questions as PDF ask me or someone for it March Core : p Ch9. DIFFERENTIATION 9.1 Increasing & decreasing functions 9. Stationary points 9.3 Using turning points to solve problems Students need to be able to confidently find areas, surface areas & volumes of various D & 3D shapes inc. circles & arcs Condensed 1 page notes with questions Standards Unit C

8 March Core : p Ch10. TRIGONOMETRICAL IDENTTIS AND SIMPLE EQUATIONS Know and use ; Sketch the graphs of: and show coordinates of intersection with the axes 10.1 Simple Trigonometric identities 10. Solving simple Trig equations March Year 1 MOCK week C1 MOCK exam (Hall) We will assess C during April along with mocks for the Applied modules (D1, M1 and S1) which will take place during lesson time. March/April EASTER HOLIDAYS Students to continue with their revision into the Easter Holidays April Core : p continued Ch10. TRIGONOMETRICAL IDENTITIES AND SIMPLE EQUATIONS 10.3 Solving Equations of the form: Sin (nθ + a), Cos(nθ + a) & Tan(nθ + a) 10.4 Solving quadratic Trigonometrical equations Example: (a) Solve the equation sin x = ⅓ in the interval 0 x 540 (b) The height of the water above mean tide level in a harbour t hours after midnight is h metres, given by the equation h1.8sin(30 t90). Use your answers to part (i) to find three times on the same day when the water is 0.6m above mean tide level. Extension Questions: Condensed 1 page notes with questions April Core : p Ch11. INTEGRATION 11.1 Simple Definite integration 11. Area under a curve 11.3 Area under a curve that gives negative values 11.4 Area between a line & a curve 11.5 The trapezium rule Definite Integration Example: 4 Find: ( x 1)( x )d x. 1 Evaluate: 3, ( x 1) dx 0 1, x 1 dx 7 3. x 11 ( x )( x 5) dx Finding areas Integration can be used to find the area underneath a curve. Example 1: Find the area beneath the curve 4. 1 y3x 5 between the lines x = and x = Tarsias :

9 x y NB: Areas beneath the x-axis are negative. You need to calculate areas above and below the axes separately. Example : The diagram shows the curve y = x(x 3). Find the shaded area (answer: ) y x - To find the area between curves you can use the formula: Area= (top curve - bottom curve)dx Extension Questions:

10 C1 & C Deadline Week Core 1 & Core REVISION & CATCH-UP WEEK Teachers to aim to complete all teaching by this week to allow time for past paper practice, revision & last minute intervention. Comprehensive notes are for C from the 1-day revision day at UCL C & M1 Notes from May 013 Lectures at UCL on Fronter April C1 & C MOCK EXAMS Teachers to conduct these during lesson time or do a HOME-Mock to save lesson time. Papers to use will be discussed nearer the time. The papers will be printed for you. May REVISION & INTERVENTION PAST PAPERS Year 1 study leave starts May 6 th May 30 th May MAY HALF TERM May/June EXTERNAL AS EXAMS Exams for C1, C, S1 & M1 Students should do about 15-0 past papers for every modules they will be sitting in the summer this could be a combination of actual and Solomon papers Jun 015 START OF THE NEW TIMETABLE_ START C3 SOW June Core 3: p1 11 Ch1. ALGEBRAIC FRACTIONS 1.1 Simplify algebraic fractions by NB: Both 4hr &6hr teachers to teach C3 until October Halfterm

11 cancelling common factors 1. Multiply and divide algebraic fractions. 1.3 Add and subtract algebraic fractions 1.4 Dividing algebraic factions and the remainder theorem. Core 3: p1 30 Ch. FUNCTIONS 1.1 Mapping diagrams and graphs of operations. 1. Functions & Function notation 1.3 Range, Mapping diagrams, graphs & definitions of functions 1.4 Using composite functions 1.5 Finding &using inverse functions July Core 3: p31 44 Ch3. THE EXPONENTIAL & LOG FUNCTIONS IV has matching activities/tarsias & extension problems 3.1 Introducing exponential functions of the form y = a x 3. Graphs of exponential functions and modelling using y = a x 3.3 Using e x and the inverse of the exponential function log ex Sketch the functions ax, a > 0, ex, lnx and and their graphs. 3 rd July 1 st Sept 014 SUMMER HOLIDAYS 1 st Sept 014 C3: Review Chapters 1-3 ( week) Sept 014 C3: p45 57 Ch4 NUMERICAL METHODS 4.1 finding approximate roots of f(x) = 0 graphically 4. using iterative & algebraic methods to find approximate roots of f(x) = 0 Before you teach Ch5 familiarise yourself with Autograph - speak with BMM on how to use this software

12 September C3: p63 8 Ch5 TRANSFORMING GRAPHS OF FUNCTIONS 5.1 Sketching graphs of the modulus function ( ) 5. Sketching graphs of the function ( ) 5.3 solving equations involving a modulus 5.4 applying a combinations of transformations to sketch curves 5.5 sketching transformations & labelling the coordinates of a given point Autograph Standards Unit A1

13 October 014 C3: p Ch6 TRIGONOMETRY 6.1 The functions secant θ, cosecant θ and cotangent θ 6. The graphs of secant θ, cosecant θ and cotangent θ 6.3 simplifying expressions, proving identities & solving equations using sec θ, cosec θ and cot θ 6.4 using identities 6.5 using inverse trigonometrical functions and their graphs October 014 C3: p Ch7 FURTHER TRIGONOMETRIC IDENTITIES & THEIR APPLICATIONS 7.1 using additional trigonometrical formulae 7. using double angle trigonometrical formulae 7.3 solving equations and proving identities using double angle formulae 7.4 using the form in solving trigonometrical problems 7.5 the factor formulae October 014 November 014 OCTOBER HALF TERM Core3: p Ch8 DIFFERENTIATION 8.1 Differentiating using the chain rule 8. Differentiating using the product rule 8.3 Differentiating using the quotient rule 8.4 Differentiating the exponential function 8.5 finding the differential of the logarithmic function 8.6 Differentiating sin x 8.7 Differentiating cos x 8.8 Differentiating tan x 8.9 Differentiating further trigonometric functions 8.10 Differentiating functions formed by combining trigonometrical, exponential, logarithmic & C3 PAST PAPER BOOKLETS DISTRIBUTED for students to revise from over the Christmas break papers including full solutions we will use Solomon Papers A-L

14 polynomial functions December 014 December 014 START TEACHING CORE 4 Core 4: p1 9 Ch1. PARTIAL FRACTIONS Must start C4 before Christmas to allow you time for revision & past papers of C3 & C4 in April & May 1.1 Adding & subtracting algebraic fractions 1. Partial fractions with two linear factors in the denominator 1.3 Partial fractions with three or more linear factors in the denominator 1.4 Partial fractions with repeated linear factors in the denominator 1.5 Improper fractions into partial fractions December 014 CHRISTMAS HOLIDAYS January 015 Core 4: p10 Ch. COORDINATE GEOMETRY IN THE (x, y) PLANE 1.6 Parametric equations used to define the coordinates of a point 1.7 Using parametric equations in coordinate geometry 1.8 Converting parametric equations into Cartesian equations 1.9 Finding the area under a curve given by parametric equations Standards Unit A14 Exploring equations in parametric form Core 4: p3 35 Ch3. THE BINOMIAL EXPANSION 3.1 The binomial expansion for a positive integral index 3. using the binomial expansion to expand ( ) 3.3 using Partial fractions with the binomial expansion

15 January 015 Core 4: p36 50 Ch4. DIFFERENTIATION 4.1 Differentiating functions given parametrically 4. Differentiating relations which are implicit 4.3 Differentiating the function 4.4 Differentiating rates of change 4.5 Simple differential equations February FEBRUARY HALF TERM 015 February Core 4: p51 86 Ch5. VECTORS 5.1 Vector Definitions and Vector Diagrams 5. Vector arithmetic and the unit vector 5.3 using vectors to describe points in or 3 dimensions 5.4 Cartesian components of a vector in D 5.5 Cartesian components of a vector in 3D 5.6 Extending D vector results to 3D 5.7 The scalar product 5.8 The vector equation of a straight line 5.9 Intersecting straight line vectors equations 5.10 The angle between two straight lines March/April Core 4: p87 18 Ch6. INTEGRATION 6.1 Integrating standard functions 6. Integrating using the reverse chain rule 6.3 using trigonometric identities in integration 6.4 using partial fractions to integrate expressions 6.5 using standard patterns to integrate expressions 6.6 Integration by substitution C4 PAST PAPER BOOKLETS DISTRIBUTED for students to revise from over the half-term break papers including full solutions use Edexcel

16 6.7 Integration by parts 6.8 Numerical integration 6.9 Integration to find areas and volumes 6.10 using integration to solve differential equations 6.11 Differential equations in context Mid April 015 Core 4: REVISION EASTER HOLIDAYS April 015 C3, C4 + APPLIED MODULES REVISION PAST PAPERS PAST PAPER BOOKLETS April 015 C3, C4 + APPLIED MODULES REVISION PAST PAPERS PAST PAPER BOOKLETS NOTES FOR THE TEACHER DEADLINE AS Teachers must aim to complete teaching by end of March 015 to leave sufficient time for exam prep & past paper revision A Teachers must aim to complete teaching by mid-april 015 to leave sufficient time for exam prep & past paper revision MAIN RESOURCE Teachers will use the LiveText for all modules. Students will buy these themselves and bring to each lesson. Additional resources are from MEP, click HERE HOMEWORK

17 A variety of tasks can be set ranging from short Q&A to extended pieces of investigation work. When you set homework you MUST mark it and record it. You should also ask students to make summary notes of each chapter as independent study. Fronter has been loaded with a wealth of homework practice which students should be directed to by you. Students are expected to spend as much time outside lessons as in them i.e. about 5 hours on maths outside lessons each week. Most of this time will be spent on homework set by the teacher. = I am confident with what I am doing (able) set Mixed exercise/review exam style questions = I am ok with this but could do with a little more practice (so-so) set questions from normal exercises focussing on end of exercise questions = I am struggling with this topic/subject (weak) set usual exercises for extra practice (Ex 1A, 1B etc.) FMSP REVISION COURSES Payment to be collected before the publication of revision dates. Places to be allocated on a first come first served basis. Deposits to be collected by front office and must NOT be handled by the Maths department. G&T PROVISION Pure Investigations and Pure what if & why problems for the most able from The Centre for Teaching Maths (Plymouth University) covering C1-C4 RULES FOR CLOSING THE GAP: Know your students; Plan effectively; Enthuse & Inspire; Engage & Guide; Feedback appropriately & Evaluate together. ASSESSMENT: What about short tests in class? Teachers should simply get students to do questions straight from the book to avoid printing costs maybe do a couple of carefully chosen questions each month to assess student retention of prior learning or maybe flash a select few questions on the IWB Alternatively, the Integral website from FMSP has lots of End of chapter assessments speak to Mr Mani about these