AS/A level subject criteria for mathematics: consultation draft

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Annabella Blake
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1 : consulttion drft For first teching from September 2012 April 2009 QCA/09/4176

2 Contents 1 Introduction 3 2 Aims nd lerning outcomes 3 3 Subject content 4 4 Key skills 13 5 Assessment objectives 13 6 Scheme of ssessment 14 7 Performnce descriptions 15 Appendix 16 2

3 1 Introduction 1.1 These subject criteri set out the knowledge, understnding, skills nd ssessment objectives common to ll dvnced subsidiry (AS) nd dvnced (A) level specifictions in mthemtics. They provide the frmework within which the wrding body cretes the detil of the specifiction. Subject criteri re intended to: help ensure consistent nd comprble stndrds in the sme subject cross the wrding bodies define the reltionship between the AS nd A level specifictions, with the AS s subset of the A level ensure tht the rigour of A level is mintined help higher eduction institutions nd employers know wht hs been studied nd ssessed. Any specifiction tht contins significnt elements of the subject mthemtics must be consistent with the relevnt prts of these subject criteri. 2 Aims nd lerning outcomes Aims 2.1 AS nd A level specifictions in mthemtics should encourge students to: develop their understnding of mthemtics nd mthemticl processes in wy tht promotes confidence nd fosters enjoyment develop n understnding of coherence nd progression in mthemtics use technology s tool for exploring mthemtics, modelling with mthemtics nd solving mthemticl problems develop wreness of the historicl nd culturl roots of mthemtics develop n wreness of the relevnce of mthemtics to other fields of study, to the world of work nd to society in generl 3

4 tke incresing responsibility for their own lerning nd the evlution of their own mthemticl development be well prepred for progression to further study nd to employment. Lerning outcomes 2.2 AS nd A level specifictions in mthemtics should enble students to: reson logiclly nd recognise incorrect resoning, to generlise nd to construct mthemticl proofs pply rnge of mthemticl skills nd techniques to both structured nd unstructured problems develop their understnding of how different res of mthemtics re connected recognise tht sitution my be represented mthemticlly by mthemticl model nd understnd the model s reltionship with the corresponding rel world problem refine nd improve mthemticl models communicte mthemticl ides effectively nd use mthemtics s n effective mens of communiction red nd comprehend mthemticl rguments nd informtion concerning mthemtics nd its pplictions use technology nd recognise when such use my be inpproprite, nd be wre of limittions. 3 Subject content 3.1 Mthemtics is rigorous nd coherent discipline with rich historicl nd culturl heritge. While there is progression of mteril through ll levels t which the subject is studied, there is lso the possibility to develop bredth nd depth within mny spects. The criteri build on the knowledge, understnding nd skills studied in higher tier GCSE mthemtics. The core content for AS is subset of the core content for A level. Progression in the subject my extend in number of wys from AS nd A level, into the complementry study of further mthemtics or onto relted courses in higher eduction. 4

5 Knowledge, understnding nd skills 3.2 AS nd A level specifictions in mthemtics should require: construction nd presenttion of mthemticl rguments, including proofs, through pproprite use of logicl deduction nd precise sttements involving correct use of symbols nd pproprite connecting lnguge b correct understnding nd use of mthemticl lnguge, nottion nd grmmr these re listed in the Appendix. These requirements should pervde the core content mteril set out below. 3.3 Core content mteril for AS nd A level exmintions in mthemtics is listed below. AS core content is listed in the second column, with A2 core content in the right-hnd column Algebr nd functions AS core content A2 core content () Lws of indices for ll rtionl exponents. (b) Use surds nd pi in exct clcultions. (c) Qudrtic functions nd their grphs. The discriminnt of qudrtic function. Completing the squre. Solution of qudrtic equtions. (d) Simultneous equtions: nlyticl solution by substitution, eg of one liner nd one qudrtic eqution. (e) Solution of liner nd qudrtic inequlities. (f) Algebric mnipultion of polynomils, including expnding brckets nd collecting like terms, fctoristion nd simple lgebric division; use of the Fctor Theorem. Simplifiction of rtionl expressions including fctorising nd cncelling, nd lgebric division. 5

6 (g) Grphs of functions; sketching curves defined by simple equtions. Geometricl interprettion of lgebric solution of equtions. Use of intersection points of grphs of functions to solve equtions. (h) Definition of function. Domin nd rnge of functions. Composition of functions. Inverse functions nd their grphs. (i) The modulus function. (j) The effect of simple trnsformtions on the grph of y = f( x) s represented by y = f( x), y = f( x) +, y = f( x+ ), y = f( x). Combintions of these trnsformtions. (k) Rtionl functions. Prtil frctions (denomintors not more complicted thn repeted liner terms) Coordinte geometry in the (x, y) plne AS core content A2 core content () Eqution of stright line, including the forms y y = m( x x ) x + by + c = nd Conditions for two stright lines to be prllel or perpendiculr to ech other. Coordintes of mid-point. (b) Coordinte geometry of the circle using the eqution of circle in the form ( x ) + ( y b) = r, nd including use 6

7 of the following circle properties: the ngle in semicircle is right ngle the perpendiculr from the centre to chord bisects the chord the perpendiculrity of rdius nd tngent. (c) Prmetric equtions of curves nd conversion between Crtesin nd prmetric forms Sequences nd series AS core content A2 core content () Sequences, including rithmetic nd geometric sequences, those given by formul for the nth term nd those generted by simple reltion of the form xn+ = 1 f( xn). (b) (c) Arithmetic series, including the use of Σ nottion. Geometric series; the sum to infinity of convergent geometric series, including the use of r < 1 nd the use of Σ nottion. (d) Binomil expnsion of (1 + x) n for positive integer n. (e) Binomil series for ny rtionl n. 7

8 3.3.4 Trigonometry AS core content A2 core content () The sine nd cosine rules, nd the re of 1 tringle in the form bsinc. 2 (b) (c) Rdin mesure, including use for rc length nd re of sector. Sine, cosine nd tngent functions. Their grphs, symmetries nd periodicity. (d) Knowledge of secnt, cosecnt nd cotngent, nd of rcsin, rccos nd rctn. Their reltionships to sine, cosine nd tngent. Understnding of their grphs nd pproprite restricted domins. (e) Knowledge nd use of 2 2 sin θ + cos θ = 1. sinθ tnθ = nd cosθ 2 2 Knowledge nd use of sec θ = 1+ tn θ 2 2 nd cosec θ = 1+ cot θ. (f) Knowledge nd use of double ngle formule; use of formule for sin( A ± B ) ), cos ( A ± B) nd tn( A ± B), nd of expressions for cosθ + bsinθ in the equivlent forms rcos( θ ± ) or rsin( θ ± ). (g) Solution of simple trigonometric equtions in given intervl. Solution of trigonometric equtions in given intervl Exponentils nd logrithms AS core content A2 core content 8

9 () x,, nd its grph. The exponentil function (e) nd its y= > 0 grph. (b) Lws of logrithms: ( > 0 ): log x + log y log ( xy), x log x log y log, y The logrithmic function (ln) nd its grph; logrithmic nd exponentil functions s the inverse of one nother. klog x log ( x ). k (c) The solution of equtions of the form x = b ( b, > 0). (d) Exponentil growth nd decy Differentition AS core content A2 core content () The tngent to the grph of y = f( x) s limit of chords; the grdient of the tngent s limit. The terminology of derivtives nd their interprettion s rte of chnge; second-order derivtives. Differentition of simple functions from first principles. (b) Differentition of nd differences. n x, nd relted sums Differentition of e x, ln x, sin x, cos x, tn x nd their sums nd differences. (c) Applictions of differentition to grdients, tngents nd normls, mxim nd minim nd sttionry points, incresing nd decresing functions. 9

10 (d) (e) Differentition using the product rule, the quotient rule, the chin rule Differentition of simple functions defined implicitly or prmetriclly Integrtion AS core content A2 core content () Indefinite integrtion nd differentition s inverse processes. (b) (c) Integrtion of n x, n 1. Simple pproximtion of re under curve. Integrtion of e x, 1, sin x, cos x. x Interprettion of the definite integrl s the re under curve. Evlution of definite integrls. (d) Evlution of volumes of revolution. (e) Simple cses of integrtion by substitution nd integrtion by prts. These methods s the inverse processes of the chin nd product rules, respectively. (f) Simple cses of integrtion using prtil frctions. (g) Formtion nd nlyticl solution of simple first-order differentil equtions with seprble vribles. 10

11 3.3.8 Numericl methods AS core content A2 core content () Loction of roots of f( x) = 0 by considering chnges of sign of f( x ), in n intervl of x in which f( x ) is continuous. (b) (c) Approximte solution of equtions using itertive methods, including recurrence reltions of the form xn+ = 1 f( xn). Numericl integrtion of functions Vectors AS core content () (b) (c) (d) (e) A2 core content Vectors in two nd three dimensions. Mgnitude of vector. Algebric opertions of vector ddition nd multipliction by sclrs, nd their geometricl interprettions. Position vectors. The distnce between two points. Vector equtions of lines. The sclr product. Its use for clculting the ngle between two lines Applictions The AS pplictions develop from the pure mthemtics content. The references given below re to the relted pure mthemtics sections from which the pplictions cn be developed. 11

12 AS core content bsic ides of mthemticl modelling, including business, mechnicl nd sttisticl contexts A2 core content further development of mthemticl modelling () Optimistion: liner progrmming Grphs nd networks Simple lgorithmic procedures 3.3.6, 3.3.1e, Primm s nd Kruskl s lgorithms, minimum spnning tree, Dijkstr s lgorithm (b) (c) Kinemtics: speed, distnce, ccelertion, velocity time grphs Constnt ccelertion, including verticl motion under grvity Vrible ccelertion (polynomil functions of time) 3.3.1g, 3.3.2, 3.3.6, Forces: force s vector (mgnitude, direction), resolution of forces in 2-D, drwing force digrms, tringle of forces, inclined plne, resistnce, equilibrium/dynmic movement (limited to predicting whether the object will move nd, if so, in which direction) Newton s lws Friction Tension nd thrust connected Prticles (not compound pulleys) Projectiles s n exmple of motion in 2-D where displcement is function of time (d) Working within the sttisticl problemsolving process: interpret, explin nd evlute dt using pproprite digrms nd summry sttistics (including stndrd devition nd use of Σ nottion) Simple continuous rndom vribles Norml distribution Estimtion of popultion prmeters Distribution of the smple men 12

13 (e) Probbility: smple spce, combined events, conditionl probbilities, rndom vrible Modelling with probbility distributions (geometric nd binomil) Centrl limit theorem for the smple men Inference 3.3.3c,d 4 Key skills 4.1 AS nd A level specifictions in mthemtics should provide opportunities for developing nd generting evidence for ssessing relevnt key skills from the list below. Where pproprite, these opportunities should be directly cross-referenced, t specified level(s), to the key skills stndrds, which my be found on the QCA website ( Appliction of number Communiction Improving own lerning nd performnce Informtion nd communiction technology Problem-solving Working with others 5 Assessment objectives 5.1 All cndidtes must be required to meet the following ssessment objectives. The ssessment objectives re to be weighted in ll specifictions s indicted in the following tble. 13

14 Assessment objectives Weighting AS level A2 level AO1 Recll, select nd use mthemticl fcts, concepts nd techniques AO2 Select nd pply mthemtics to model rel nd bstrct situtions. Solve structured nd unstructured problems in both fmilir nd less fmilir contexts AO3 Interpret informtion nd results, nd evlute the effectiveness of ny model or strtegy used AO4 Use precise sttements, logicl deduction nd inference to construct nd test the vlidity of mthemticl rguments Scheme of ssessment 6.1 A level specifictions in mthemtics will consist of two units t AS nd two units t A2. Synoptic ssessment 6.2 Synoptic ssessment in mthemtics should tke plce cross the A2 units nd should encourge cndidtes to: demonstrte understnding of the connections between different elements of the subject solve problems tht require cndidtes to bring different spects of mthemtics together solve substntil unstructured problems tht require extended resoning. 6.3 All AS nd A level specifictions in mthemtics must explicitly refer to the importnce of cndidtes using cler, precise nd pproprite mthemticl lnguge. These references must drw ttention to the relevnt demnds of ssessment objective AO AS nd A level specifictions in mthemtics must: 14

15 explicitly include ll the mteril in the relevnt Knowledge, understnding nd skills section of the criteri specifictions re permitted to elborte on detils in different wys, but ll specifictions must show very high degree of consistency nd comprbility in ddressing the mteril in sections 3.2 nd 3.3; for both AS nd A level, the Knowledge, understnding nd skills must ttrct totl credit for the qulifiction permit the use of grphing clcultors in ll units indicte the mthemticl nottion tht will be used this is listed in the Appendix. 6.5 Applictions should collectively hve weighting of per cent. Qulity of written communiction 6.6 AS nd A level specifictions will be required to ssess the cndidtes qulity of written communiction in ccordnce with the guidnce document produced by QCA. 7 Performnce descriptions To be dded Comment [AC1]: TBA 15

16 Appendix 1 Set Nottion is n element of is not n element of { x 1, x 2, } the set with elements x1, x 2, {x: } n( A) the set of ll x such tht the number of elements in set A the empty set the universl set A the complement of the set A N Z Z the set of nturl numbers, {1, 2, 3, } the set of integers, {0, ±1, ±2, ±3, } the set of positive integers, {1, 2, 3, } Z n the set of integers modulo n, {0, 1, 2,, n 1} Q the set of rtionl numbers, p : p Z, q Z q Q the set of positive rtionl numbers, { x Q : x> 0} Q set of positive rtionl numbers nd zero, { x Q : x 0} R the set of rel numbers R the set of positive rel numbers, { x R : x> 0} R the set of positive rel numbers nd zero, { x R : x 0} + C the set of complex numbers ( x, y) the ordered pir x, y A B the Crtesin product of sets A nd B, i.e. A B= {(, b): A, b B} is subset of is proper subset of union intersection 16

17 [, ] b the closed intervl { x R : x b} b the intervl { x R : x< b} [, ) (, b ] the intervl { x R : < x b} b the open intervl { x R : < x< b} (, ) yrx y is relted to x by the reltion R y~ x y is equivlent to x, in the context of some equivlence reltion 2 Miscellneous Symbols = is equl to is not equl to is identicl to or is congruent to is pproximtely equl to is isomorphic to is proportionl to < is less thn is less thn or equl to, is not greter thn > is greter thn is greter thn or equl to, is not less thn p q infinity p nd q p q p or q (or both) ~ p not p p q p implies q (if p then q ) p q p is implied by q (if q then p ) p q p implies nd is implied by q ( p is equivlent to q ) there exists for ll 3 Opertions + b plus b 17

18 b minus b b, b, b. multiplied by b b,, / b b divided by b n i i= n n i 1 2 i= 1 n the positive squre root of n! the modulus of n fctoril n r n! the binomil coefficient for r!( n r)! nn ( 1)...( n r+ 1) or for n Q r! or n C r n Z + 4 Functions f( x) the vlue of the function f t x f:a B f is function under which ech element of set A hs n imge in set B the function f mps the element x to the element y f 1 gf the inverse function of the function f the composite function of f nd g which is defined by gf ( x) = g(f(x)) or g º f lim f( x) x the limit of f( x ) s x tends to Δ x, δ x n increment of x dy dx the derivtive of y with respect to x d n dx y n f( x), f ( x),, ( n) f ( x ) the n th derivtive of y with respect to x the first, second,..., n th derivtives of f( x ) with respect to x 18

19 y dx the indefinite integrl of y with respect to x b y dx the definite integrl of x = nd x = b y with respect to x between the limits V x the prtil derivtive of V with respect to x the first, second,... derivtives of x with respect to t 5 Exponentil nd Logrithmic Functions e bse of nturl logrithms x e, exp x exponentil function of x log x logrithm to the bse of x ln x, log e x nturl logrithm of x lg x, log 10 x logrithm of x to bse 10 6 Trigonometric nd Hyperbolic Functions sin, cos, tn cosec, sec, cot sin -1, cos -1, tn -1 cosec -1, sec -1, cot -1, rcsin, crcos, rctn, rccosec, rcsec, rccot sinh, cosh, tnh cosech, sech, coth sinh -1, cosh -1, tnh -1 cosech -1, sech -1, coth -1 } } } } the trigonometric functions the inverse trigonometric functions the hyperbolic functions the inverse hyperbolic functions 7 Complex Numbers i squre root of 1 z complex number, z = x+ i y = r(cosθ + isin θ ) Re z the rel prt of z, Re z = x Im z the imginry prt of z, Im z = y 19

20 z the modulus of z, 2 2 z = x + y rg z the rgument of z, rg z =θ, π < θ π z * the complex conjugte of z, x i y 8 Mtrices M 1 M M T mtrix M the inverse of the mtrix M the trnspose of the mtrix M de t M or M the determinnt of the squre mtrix M 9 Vectors the vector â i, the vector represented in mgnitude nd direction by the directed line segment AB unit vector in the direction of j, k unit vectors in the directions of the Crtesin coordinte xes, the mgnitude of the mgnitude of.b the sclr product of nd b b the vector product of nd b 10 Probbility nd Sttistics A, B, C, etc. events A B A B P( A) union of the events A nd B intersection of the events A nd B probbility of the event A A complement of the event P( A B) probbility of the event A conditionl on the event B X, Y, R, etc. rndom vribles A x, y, r, etc. vlues of the rndom vribles X, Y, R etc 20

21 x, x, observtions 1 2, frequencies with which the observtions x x f1, f2 1, 2, occur p( x) probbility function P( X = x) of the discrete rndom vrible X p1, p2, probbilities of the vlues x 1, x 2, of the discrete rndom vrible X f( x), g( x ), F( x), G( x ), the vlue of the probbility density function of continuous rndom vrible X the vlue of the (cumultive) distribution function P(X x) continuous rndom vrible X E( X ) expecttion of the rndom vrible X E(g( X )) expecttion of g( X ) of Vr( X ) vrince of the rndom vrible X G( t) probbility generting function for rndom vrible which tkes the vlues 0, 1, 2, B( n, p) binomil distribution with prmeters n nd p 2 N(μ, σ ) norml distribution with men μ nd vrince σ μ popultion men 2 σ popultion vrince σ x, m smple men popultion stndrd devition s, ˆ σ φ unbised estimte of popultion vrince from smple, s = ( xi x) n 1 probbility density function of the stndrdised norml vrible with distribution N(0, 1) Φ corresponding cumultive distribution function ρ product moment correltion coefficient for popultion r product moment correltion coefficient for smple Co v(x, Y ) covrince of X nd Y 21

22 The regultory uthorities wish to mke their publictions widely ccessible. Plese contct us if you hve ny specific ccessibility requirements. First published in Crown copyright 2009 Council for the Curriculum, Exmintions nd Assessment 2009 Qulifictions nd Curriculum Authority 2009 Reproduction, storge or trnsltion, in ny form or by ny mens, of this publiction is prohibited without prior written permission of the publisher, unless within the terms of the Copyright Licensing Agency. Excerpts my be reproduced for the purpose of reserch, privte study, criticism or review, or by eductionl institutions solely for eduction purposes, without permission, provided full cknowledgement is given. Printed in Gret Britin. The Qulifictions nd Curriculum Authority is n exempt chrity under Schedule 2 of the Chrities Act Qulifictions nd Curriculum Authority 83 Piccdilly London W1J 8QA

6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS

6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS 6. CONCEPTS FOR ADVANCED MATHEMATICS, C (475) AS Objectives To introduce students to number of topics which re fundmentl to the dvnced study of mthemtics. Assessment Emintion (7 mrks) 1 hour 30 minutes.