AS & A2 Maths Date 1st September 2014 Topic & page ref in Heinemann Live Text Core 1: p1-14 Ch1. ALGEBRA & FUNCTIONS 1.1 Simplifying terms by collecting like terms 1.2 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 Scheme of Work 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. 2014 – 2016 Skills developed & Examples students should be able to answer the end of each section Students should learn the squares from 12 to 162; cube numbers from 13 to 63 which will help them solve fractional indices problems. Examples: Fractional Indices: if 81½ is 9 then 25½ is ….? Simplify a5 a3; m4 m2; (p2)5; (2xy2)3; solve 2n =16; solve 32x-1 = 27 Simplify: √ Rationalise the denominator; Homework Resources 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 available: Manipulating Surds Standards Unit N11 – Surds √ √ ;; √ √ Standards Unit N12 – using indices √ √ Expand √√ Factorise and solve : x2 + 8x + 15 = 0; 2x2 + 7x + 6 = 0 ; 4x2 -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. September September Core 1: p15 – 26 Ch2. QUDARATIC FUNCTIONS Know and learn the quadratic formula: Core 1: p27 – 40 Ch3. EQUATIONS & INEQUALITIES Examples: √ Standards Unit C1 – Linking the properties & forms of Quadratic Functions Solve by completing the square: x2 + 8x + 15 = 0; 2x2 – 12x + 7 = 0 By completing the square, find the minimum value of x2 4x 9. Show that the line y = x – 4 is a tangent to the circle with equation x2 + y2 = 8 Extension: Reproduce the proof of the Quadratic formula √ Solve x 4y 7 and x + 2y = 16 by elimination and substitution What about: 3x + y = 10 and x2 + 2xy + 2y2 = 17 3.1 Simultaneous equations by These can be used as homework to stretch all students. There are also TESTS that can be flashed in lessons on the IWB. Solomon worksheets Examples: 2.1 plotting graphs of quadratic functions 2.2 solving quadratic equations by factorisation 2.3 competing the square 2.4 solving quadratic equations by competing the square 2.5 solving quadratic equations by using the formula 2.6 sketching graphs of quadratic functions Solomon worksheets available as PDF. Solomon worksheets September elimination 3.2 Simultaneous equations by substitution 3.3 Simultaneous equations with 1 linear & 1 quadratic 3.4 solving linear inequalities 3.5 solving quadratic inequalities Core 1: p41 – 68 Ch4. SKETCHING CURVES Solve for x: (a) 5x 2 x 16 (b) x2 25 4.1 sketching graphs of cubic functions 4.2 interpreting graphs of cubic functions 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) = x2 Solve and sketch x2 + 8x + 15 0; x2 - 10x + 21 0 Examples: Solomon worksheets Know the graphs of: y = x; y = x2; y = x3; y = y = 2x ; Extension Questions: 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 October Core 1: p73 – 90 Ch5. COORDINATE GEOMETRY IN THE (x, y) PLANE 5.1 The equation of a straight line 5.2 The gradient of the straight line 5.3 y – y1 = m(x – x1) 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 The gradient of a line passing through the points y y . x1 , y1 and x2 , y2 is 2 1 m x2 x1 The equation of the straight line with gradient m that passes through the point x1 , y1 is y y1 m( x x1 ) . The distance between the points with coordinates 2 2 x1 , y1 and x2 , y2 is x2 x1 y2 y1 . The midpoint of the line joining the points x x2 y1 y2 . x , y and x , y is 1 , 1 1 2 2 2 2 Example: Find the equation of the perpendicular bisector of the line joining the points Solomon worksheets Condensed 1 page notes available with questions for coordinate Geometry (3, 2) and (5, -6). Example: Find the point of intersection of the lines: 2x + y = 3 and y = 3x – 1. Extension Question: October Core 1: p91 – 111 Ch6. ARITHMETIC SEQUNECES 6.1 Introduction to Sequences 6.2 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 October Core 1: p112 – 132 Ch7. DIFFERENTIATION 7.1 derivative of f(x) 7.2 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 7.6 7.7 Rate of change of a function at a point 7.8 Equations of Tangents and Normals 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 Solomon worksheets If numbers ascend in 3’s, that’s the 3 x table = 3n. Then find the number before the 1st term (=5), so, nth term is 3n+5 nth term in sequence 8, 11, 14, 17, ..., ..., ... Standards Unit N13 – Analysing sequences Solomon worksheets Find : y = 5x6 1 x4 9 ; y = 2 x 2 x 2 NB: A turning point occurs where the gradient is zero, i.e. where 0. y = 2 x4 5x2 ; y = 7 x2 12 x 5 ; 1 2 And you can also use the 2nd derivative to decide whether a turning point is a maximum or a minimum: If d2y 0 then it is a minimum ; dx 2 d2 y 0 then it is a maximum. dx 2 Equation of a tangent tells you the gradient of a curve. The gradient m of a tangent line at the point x1 , y1 can be found from . The equation of the tangent is then y y1 m( x x1) . Perpendicular lines Standards Unit C2 – 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 Standards Unit C5 – Finding stationary points of cubic functions Suppose 2 lines have gradients m1 & m2 . These lines are perpendicular if m1 m2 1 , i.e. m 1 . 2 m1 Equation of a normal To find the equation of a normal at the point x1 , y1 : Find the gradient from then find the gradient m of the normal using m and the equation of the normal is y y1 m( x x1 ) 1 dy dx Example: Find the equation of the normal to the graph y = x(x + 1) (x – 2) at x = -1. October OCTOBER HALF TERM October Core 1: p133 – 142 Ch8. INTEGRATION 8.1 Integrating 8.2 Integrating simple expressions 8.3 using the ∫ 8.4 Simplifying before integrating 8.5 Finding ‘c’ November/ December START CORE 2 3rd November Core 2: p1 – 17 Ch1. ALGEBRA & FUNCTIONS 1.1 1.2 1.3 1.4 Simplifying algebraic fractions Dividing a polynomial factorising a polynomial Using the remainder theorem Solomon worksheets Rule: Increase the power by 1 and divide by the new power. Integration is the reverse of differentiation. Example: 3 Find y if dy x 2 6 x 2 and y = 4 when x = 3 (answer: y x 3x 2 2 x 16 ) dx 3 Questions: 1. v (3x 2 4 x 2)dx . Standards Unit C4 – differentiating & Integrating fractional and negative powers of x If v = 3 when x = 0, find v as a function of x. Hence calculate the value of v when x = 1. 6 hour teacher on their own teaching C2 – 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 1. g(x) = x3 3x2 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 C2 Live Text on CD to use in lessons to support explanations. Solomon worksheets available Tarsias available Condensed 1 page notes available with questions for factor theorem Standards Unit A11 – factorising cubics Extenstion Question: December Core 2: p18 – 37 Ch2: THE SINE & COSINE RULE Know the 3 trig ratios using: SOH CAH TOA 2.1 Sine rule for missing sides 2.2 Sine rule for unknown angles 2.3 Solutions for a missing angles 2.4 Cosine rule to find unknown sides 2.5 Cosine rule to find missing angles 2.6 Sine, Cosine & Pythagoras 2.7 Area of a triangle Only the cosine rule formula will be provided in the formula book. Core 2: p38 – 50 Ch3: EXPONENTIALS & LOGARITHMS Extension questions: 3.1 The function y = ax 3.2 writing expressions as a logarithm 3.3 calculating using log to base 10 3.4 Laws of Logs 3.5 solving ax = b 3.6 changing the base Solomon worksheets available Know the Area of a triangle is A = 1 ab sin C 2 Know tan x sin x and sin 2 x cos2 x 1 cos x Solomon worksheets available Standards Unit A13 – simplifying Log expressions December CHRISTMAS HOLIDAYS Revise for the C1 MOCK Exam in January Look into 1-Day revision sessions at UCL/Imperial College January 2015 C1 MOCK Exam (internal) C1 Solomon Paper ?? Solomon Paper – TBC at dept. meeting closer to the time This will take place during lesson time. Y12 will have a Mock week later in the year. January Core 2: p51 – 72 January Ch4. COORDINATE GEOMETRY IN THE (x, y) PLANE The equation of a circle centre (a, b) with radius r is ( x a)2 ( y b)2 r 2 . Example: Find the centre and the radius of the circle with equation x2 2 x y 2 6 x 6 0 4.1 The mid-point of a line 4.2 Distance between two points 4.3 The equations of a circle Extension Question: Core 2: p76 – 86 e.g. 1: Find the expansion of 3x y 4 . e.g. 2: Find the first 4 terms in the expansion 2a 3b 10 . 5.1 Pascal’s Triangle 5.2 Combinations and Factorial Notation January/ February 2 E.g. Find the non-zero value of b if the coefficient of is equal to the coefficient of ( ) Core 2: p87 – 101 Ch6: RADIAN MEASURE 6.1 Using radians to measure angles 6.2 The length of an arc 6.3 The area of a sector 6.4 The area of a segment Solomon worksheets available e.g. 3: Find the coefficient of x 4 y 4 in the expansion of 1 x 3y 5.3 Using ( ) in the binomial expansion ) 5.4 Expanding ( ∑ Condensed 1 page notes available with questions for Binomial theorem Note that in C2 n Ch5. THE BINOMIAL EXPANSION Know that radians 360o = 2π x5 Solomon worksheets available x 2 8 in the expansion of b 2 x 6 in the expansion of 2 bx 8 . Solomon worksheets available Why are there 360o in a circle? What is 1 radian? Convert rads into degrees Convert 150o 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 = ( ) February Core 2: p102 – 118 If 3, x and 9 are the first three terms of a geometric sequence, find x and the value of the 4th term. What is the first term in the GP 3, 6, 12, 24 … to exceed 1 million? Ch7: GEOMETRIC SEQUENCES & SERIES Show that the general term for the sum of a GP is 7.1 Geometric sequences 7.2 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 February February/ March ( ( ) ) or ( ( ) ) Solomon worksheets available Standards Unit N13 – Analysing sequences Show that the sum to infinity of a GP is ) Find ∑ ( FEBRUARY HALF TERM Core 2: p119 – 137 Standards Unit A12 – matching activities & probing questions – available as PDF – ask me or someone for it Ch8. GRAPHS OF TRIGONOMETRICAL FUNCTIONS 8.1 Sin, Cos & Tan functions 8.2 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, Solomon worksheets available Cos θ & Tan θ March Core 2: p141 – 153 Ch9. DIFFERENTIATION 9.1 Increasing & decreasing functions 9.2 Stationary points 9.3 Using turning points to solve problems Students need to be able to confidently find areas, surface areas & volumes of various 2D & 3D shapes inc. circles & arcs Condensed 1 page notes available with questions Standards Unit C2 – Solomon worksheets available March Core 2: p154 – 170 Know and use Sketch the graphs of: Ch10. TRIGONOMETRICAL IDENTTIS AND SIMPLE EQUATIONS Solomon worksheets available ; and show coordinates of intersection with the axes 10.1 Simple Trigonometric identities 10.2 Solving simple Trig equations March Year 12 MOCK week March/April EASTER HOLIDAYS April Core 2: p154 – 170 continued… Ch10. TRIGONOMETRICAL IDENTITIES AND SIMPLE EQUATIONS 10.3 Solving Equations of the form: Sin (nθ + a), Cos(nθ + a) & Tan(nθ + a) C1 MOCK exam (Hall) We will assess C2 during April along with mocks for the Applied modules (D1, M1 and S1) which will take place during lesson time. Students to continue with their revision into the Easter Holidays 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 h 1.8sin(30t 90) . 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 available with questions Solomon worksheets available 10.4 Solving quadratic Trigonometrical equations April Core 2: p171 – 192 Ch11. INTEGRATION 2 Tarsias available: Definite Integration Example: Find: 4 (2 x 1)( x 2)dx . 1 11.1 Simple Definite integration 11.2 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 Evaluate: , 3 1( x 1)dx 0 x 1 2 1 dx 2 , 7 x 11 ( x 2)( x 5) dx . 3 Finding areas Integration can be used to find the area underneath a curve. Example 1: Find the area beneath the curve y 3x2 5 between the lines x = 2 and x = 4. Solomon worksheets available y 50 40 30 20 10 -1 1 2 3 4 5 x NB: Areas beneath the x-axis are negative. You need to calculate areas above and below the axes separately. Example 2: The diagram shows the curve y = x(x – 3). Find the shaded area (answer: 1 56 4 12 6 13 ) y 10 8 6 4 2 -1 1 2 3 4 5 x -2 To find the area between 2 curves you can use the formula: Area= (top curve - bottom curve)dx Extension Questions: C1 & C2 Deadline Week Core 1 & Core 2 April C1 & C2 MOCK EXAMS May REVISION & INTERVENTION 26th May – 30th May MAY HALF TERM May/June EXTERNAL ‘AS’ EXAMS Jun 2015 START OF THE NEW TIMETABLE_ START C3 SOW June Core 3: p1 – 11 Ch1. ALGEBRAIC FRACTIONS Teachers to aim to complete all teaching by this week to allow time for past paper practice, revision & last minute intervention. Comprehensive notes are available for C2 from the 1-day revision day at UCL REVISION & CATCH-UP WEEK 1.1 Simplify algebraic fractions by C2 & M1 Notes from May 2013 Lectures at UCL on Fronter 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. PAST PAPERS Year 12 study leave starts May Students should do about 15-20 past papers for every modules they will be sitting in the summer – this could be a combination of ‘actual’ and Solomon papers Exams for C1, C2, S1 & M1 NB: Both 4hr &6hr teachers to teach C3 until October Halfterm Solomon worksheets available cancelling common factors 1.2 Multiply and divide algebraic fractions. 1.3 Add and subtract algebraic fractions 1.4 Dividing algebraic factions and the remainder theorem. Core 3: p12 – 30 Ch2. FUNCTIONS Solomon worksheets available 1.1 Mapping diagrams and graphs of operations. 1.2 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 3.1 3.2 3.3 Introducing exponential functions of the form y = ax Graphs of exponential functions and modelling using y = ax Using ex and the inverse of the exponential function logex 23rd July – 1st Sept 2014 SUMMER HOLIDAYS 1st Sept 2014 C3: Review Chapters 1-3 ( week) Sept 2014 C3: p45 – 57 Ch4 NUMERICAL METHODS 4.1 finding approximate roots of f(x) = 0 graphically 4.2 using iterative & algebraic methods to find approximate roots of f(x) = 0 IV has matching activities/tarsias & extension problems Solomon worksheets available Sketch the functions ax, a > 0, ex, lnx and and their graphs. Before you teach Ch5 familiarise yourself with Autograph - speak with BMM on how to use this software Solomon worksheets available September C3: p63 – 82 Ch5 TRANSFORMING GRAPHS OF FUNCTIONS Autograph 5.1 Sketching graphs of the modulus | ( )| function 5.2 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 Solomon worksheets available Standards Unit A12 October 2014 C3: p83 – 105 Ch6 TRIGONOMETRY Solomon worksheets available 6.1 The functions secant θ, cosecant θ and cotangent θ 6.2 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 2014 C3: p106 –131 Ch7 FURTHER TRIGONOMETRIC IDENTITIES & THEIR APPLICATIONS Solomon worksheets available 7.1 using additional trigonometrical formulae 7.2 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 2014 OCTOBER HALF TERM November 2014 Core3: p132 – 151 Ch8 DIFFERENTIATION 8.1 Differentiating using the chain rule 8.2 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 Solomon worksheets available polynomial functions December 2014 START TEACHING CORE 4 December 2014 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 Solomon worksheets available 1.1 Adding & subtracting algebraic fractions 1.2 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 2014 January 2015 CHRISTMAS HOLIDAYS Core 4: p10 – 22 Ch2. 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 Core 4: p23 – 35 Ch3. THE BINOMIAL EXPANSION 3.1 The binomial expansion for a positive integral index 3.2 using the binomial expansion to )2 expand ( 3.3 using Partial fractions with the binomial expansion Solomon worksheets available Standards Unit A14 – Exploring equations in parametric form Solomon worksheets available January 2015 Core 4: p36 – 50 Ch4. DIFFERENTIATION Solomon worksheets available 4.1 Differentiating functions given parametrically 4.2 Differentiating relations which are implicit 4.3 Differentiating the function 4.4 Differentiating rates of change 4.5 Simple differential equations February 2015 FEBRUARY HALF TERM February Core 4: p51 – 86 Ch5. VECTORS Solomon worksheets available 5.1 Vector Definitions and Vector Diagrams 5.2 Vector arithmetic and the unit vector 5.3 using vectors to describe points in 2 or 3 dimensions 5.4 Cartesian components of a vector in 2D 5.5 Cartesian components of a vector in 3D 5.6 Extending 2D 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 – 128 Ch6. INTEGRATION 6.1 Integrating standard functions 6.2 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 Solomon worksheets available 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 2015 Core 4: REVISION EASTER HOLIDAYS April 2015 C3, C4 + APPLIED MODULES REVISION PAST PAPERS PAST PAPER BOOKLETS April 2015 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 2015 to leave sufficient time for exam prep & past paper revision A2 Teachers must aim to complete teaching by mid-April 2015 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 available from MEP, click HERE HOMEWORK 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 available 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