Advanced Mathematics Support Programme OCR Year 2 Pure Core Suggested Scheme of Work ( )

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1 OCR Year 2 Pure Core Suggested Scheme of Work ( ) This template shows how Integral Resources and FMSP FM videos can be used to support Further Mathematics students and teachers. This template is for the compulsory Core Pure component of Year 2 Further Mathematics you will also need to deliver optional elements alongside this. This content taken with the AS material makes up 50% of the OCR A Level Further Mathematics content. It forms approximately ⅔ of the content examined in A level papers Y540 & Y541 (Pure Core). Integral Resources FMSP FM videos Teacher access to the Integral Resources (integralmaths.org/2017/) for Further Mathematics is available free of charge to all schools/colleges that register with the Further Mathematics Support Programme: furthermaths.org.uk. This will include access to the FM videos. A single student login will also be included so that teachers can give students direct access to the FM videos. Integral Resources FMSP FM video+ Individual student access to the full range of Integral Resources and the FM videos for Further Mathematics is available at a cost of 30 per student or via a full school/college subscription to Integral. Teachers will get access to the management system so they can monitor their students' progress: furthermaths.org.uk/fmvideos. Integral Resources include a wide range of resources for both teacher and student use in learning and assessment. Interactive resources and ideas for using technology are featured throughout. Sample resources are available via: integralmaths.org/2017/. FM videos are available for individual components of AS and A level Further Mathematics. There will be around 4-5 videos of 5-10 minutes in length for each section in Integral. The intention of these videos is that they are sufficient to introduce students to the concepts so that they can learn the material by working through appropriate examples. FM videos are ideal for schools/colleges teaching Further Mathematics with small groups and/or limited time allocation. They are also useful to support less experienced teachers of Further Mathematics. See furthermaths.org.uk/fm-videos. Scheduling will depend on circumstances, but the template breaks the study into topic sections. Each section corresponds to one set of videos and may be allocated approximately equal time this would equate to approximately one week of teaching time for a single teacher delivering the complete course. Further information on scheduling can be found at furthermaths.org.uk/offering-fm. FMSP Area Coordinators will be able to offer additional guidance if needed: furthermaths.org.uk/regions.

2 OCR Year 2 Pure Core Suggested Scheme of Work ( ) Date Topic Specification statements Integral Resources Exercises & Assessment Integral Resources Series and Induction. use formulae for the sums of integers, squares and cubes and use these to sum related series. use the method of differences to find the sum of a (finite or infinite) series. Be able to construct inductive proofs of a more demanding nature, including conjecture followed by proof. Series and induction / Series and induction 1: Summing series Series and induction / Series and induction 2: Further series and induction Section test S1 Exercise level 3 Section test S2 Exercise level 3 FM videos Notes Other resources 1.1 (Recap) Sigma notation 1.2 The standard formula for the sum to n terms of r 1.3 The standard formula for the sum to n terms of r² and r³ 1.4 Using the standard formulae to find combined sums 1.5 The method of differences Formulae will be given, but learners may be asked to prove them. Including the use of partial fractions. This topic may be tested using any relevant content including sums of series. Note that there are no further FM videos for induction as this topic was covered extensively in the Year1/AS course. Vectors: The equation of a plane. use the equation of a plane in cartesian and vector form. Be able to find the angle between two planes Vectors / Vectors 1: The equation of a plane Series and induction / Series and induction: Topic Assessment Section test V1 4.1 The equation of a plane in parametric form 4.2 The equation of a plane using the normal to the plane 4.3 The equation of a plane in Cartesian form 4.4 The angle between two planes Learners should know and be able to use the forms: ax + by + cz = d r = a + λb + μc (r a). n = 0 r. n = p Includes being able to convert from one form to another Vectors: Lines and Planes. Be able to find the angle between a line and a plane. Vectors / Vectors 2: Lines and plane 4.5 The intersection of a line and a plane

3 Vectors: Finding distances. Matrices and simultaneous. Be able to find the intersection of a line and a plane. Be able to find the distance between two parallel lines and the shortest distance between two skew lines. Be able to find the shortest distance between a point and a line. Be able to find the shortest distance between a point and a plane. Be able to determine, for two or three linear simultaneous where no unique solution exists, whether the have an infinite set of solutions (the are consistent) or Further vectors / Further vectors 1: Finding distances Matrices / Matrices 1: Matrices and simultaneous Section test V2 4.6 The angle between a line and a plane Vectors / Vectors: Topic assessment Section test FV1 5.1 The perpendicular distance from a point to a plane 5.2 The perpendicular distance from a point to a line in 2D 5.3 The perpendicular distance from a point to a line in 3D 5.4 The perpendicular distance between two parallel lines 5.5 The perpendicular distance between skew lines For skew lines, the formula D = (b a).n n where a and b are position vectors of points on each line and n is a mutual perpendicular to both lines, will be given. Either n will be given, or it must be established from given information including by use of the vector product. The formula D = ax 1+by 1 c a 2 +b 2 where the coordinates of the point are (x 1, y 1 ) and the equation of the line is given by ax + by = c, will be given. The formula D = b.n p n where b is the position vector of the point and the equation of the plane is given by r. n = p, will be given. Further vectors / Further vectors: Topic assessment Section test M1 Exercise level Solving two simultaneous linear 5.2 Three simultaneous (unique solution) 5.3 Three simultaneous Finding the solution set in the infinite case is excluded.

4 no solutions (the are inconsistent). Be able to interpret the solution or failure of solution of three simultaneous linear in terms of the geometrical arrangement of three planes. (no unique solution) Learners should know and be able to identify the different ways in which two or three distinct planes can intersect in 3-D space, including cases where two or three of the planes are parallel. Learners should understand and be able to apply the geometric significance of the singularity of a matrix in relation to the solution(s) or non-existence of them. Finding the line of intersection of two or more planes is excluded. Polar Coordinates. Further Calculus 1: Improper integrals. use polar coordinates Be able to convert between polar and cartesian coordinates. Be able to sketch polar curves, with r given as a function of θ Be able to find the area enclosed by a polar curve. Be able to evaluate improper integrals where either the integrand is undefined at a value in the range of integration or where the range of integration is infinite. / Polar coordinates / Polar coordinates 1: Polar curves Polar coordinates / Polar coordinates 2: Finding areas Further calculus / Further calculus 1: Improper integrals Matrices / Matrices: Topic assessment Exercise level Polar coordinates Identify significant 1.2 Sketching polar features of polar Section test P1 curves curves such as Exercise level Converting symmetry, and least between forms and greatest values of r. Exercise level 1 Section test P2 Exercise level Polar curves: finding areas Includes use of trigonometric. Using 1 2 r2 dθ Polar coordinates / Polar coordinates: Topic assessment 1.1 Integrals with Section test FC1 infinite limits (convergent) 1.2 Integrals with infinite limits (divergent) 1.3 Integrals with the function undefined at a limit 1.4 Integrals that cross a value where

5 Further calculus 2: Inverse trigonometric. Understand the definitions of inverse trigonometric. Be able to differentiate and integrate inverse trigonometric. Further calculus / Further calculus 2: Inverse trigonometric Section test FC2 Exercise level 3 the function is undefined 2.1 Differentiating arcsin x 2.2 Differentiating arctan x 2.3 Integrating using arcsin x 2.4 Integrating using arctan x Further calculus 3: Further integration. Be able to integrate using partial fractions. Recognise integrals of of the form 1 1 a 2 x 2 and a 2 +x2 and be able to integrate related by using trigonometric substitutions. Further calculus / Further calculus 3: Further integration Section test FC3 Exercise level Partial fractions recap 3.2 Partial fractions: denominator (ax + b)(cx 2 + d) 3.3 Integrating: denominator (ax + b)(cx + d) 3.4 Integrating: denominator (ax + b) (cx + d) Integrating: denominator (ax + b)(cx 2 + d) 4.1 Further integration using arcsinx 4.2 Further integration using arctanx 4.3 Substituting x=sinu 4.4 Substituting x=tanu 4.5 Substituting x=a sin u or x=a tan u

6 Maclaurin Series. Be able to find the Maclaurin series of a function, including the general term. Know that a Maclaurin series may converge only for a restricted set of values of x. Be able to recognise and use the Maclaurin series of standard : e x, ln (1 + x), sin x, cos x and (1 + x) n and based on these. Maclaurin series / Maclaurin series 1: Using Maclaurin series Further calculus / Further calculus: Topic assessment Exercise level 1 The interval of validity should be Section test MS1 understood. Exercise level Polynomial approximations 2.2 The general Maclaurin series 2.3 Maclaurin series using standard 2.4 Maclaurin series of composite Proof of the interval of validity and the use of non-real values of x are excluded. Maclaurin series / Maclaurin series: Topic assessment Hyperbolic 1: Introduction. Hyperbolic 2: Inverse hyperbolic. Understand the definitions of hyperbolic, know their domains and ranges and be able to sketch their graphs. Understand and use the identity cosh 2 x sinh 2 x = 1 Be able to differentiate and integrate hyperbolic. use the definitions of the inverse hyperbolic and know their domains and ranges. Be able to derive and use the logarithmic forms of the inverse hyperbolic. MEI_FM_Pure / Hyperbolic / Hyperbolic 1: Introducing hyperbolic Hyperbolic / Hyperbolic 2: The inverse hyperbolic function Exercise level 1 Section test H1 Exercise level 3 Exercise level 1 Section test H2 Exercise level Definitions and graphs of hyperbolic 1.2 Solving simple 1.3 Hyperbolic function identities 1.4 Differentiating and integrating hyperbolic 2.1 Inverse hyperbolic 2.2 Differentiating inverse hyperbolic 2.3 Integration using inverse hyperbolic Learners may be asked to derive or use other identities, but no prior knowledge of them is assumed. Prior knowledge of other identities is excluded. arsinh x and artanh x can take any values but arcosh x 0 arsinhx = ln (x + x 2 + 1)

7 Recognise integrals of of the form 1 1 x 2 +a 2 and and be able to x 2 a2 integrate related by using substitutions. 2.4 Integration using hyperbolic substitutions arcoshx = ln (x + x 2 1) x 1 artannhx = ln (1+x ) 2 1 x 1 < x < 1 Hyperbolic / Hyperbolic : Topic assessment 1 Complex numbers 1a: de Moivre's theorem. Complex numbers 1b: de Moivre's theorem roots of. Understand and use de Moivre s theorem. Know that every non-zero complex number has n distinct nth roots, and that on an Argand diagram these are the vertices of a regular n-gon. Be able to represent complex roots of unity on an Argand diagram. Be able to explain why the sum of all the nth roots is zero. Be able to apply complex numbers to geometrical problems. OCR_FM_Pure / Complex numbers / Complex numbers 1: de Moivre's theorem (part) Complex numbers / Complex numbers 1: de Moivre's theorem (part) Exercise level 1 Q1-4 Q1-2 Exercise level 1 Q5-7 Q3-5 Section test C1 Exercise level De Moivre's theorem 4.2 Powers of complex numbers 4.3 Powers of complex numbers in x + iy form 4.4 Using de Moivre's theorem 5.1 Solving z n = The nth roots of unity 5.3 The nth roots of any complex number 5.4 Solving Answers may be asked for in either cartesian or modulus-argument form. e.g. To locate the roots of unity on an Argand diagram or to prove results about sums of roots of unity. Complex numbers 2: Applications of Be able to apply de Moivre s theorem to finding multiple Complex numbers / Complex numbers 2: Exercise level 1 Section test C2 4.5 Finding multiple angle identities using de Moivre's theorem Express trigonometrical ratios of multiple angles in terms of

8 de Moivre's theorem. angle formulae and to summing suitable series. Understand the definition re iθ = cos θ + i sin θ and hence the form z = re iθ Know that the distinct nth roots of re iθ are: r 1 θ + 2kπ n [cos ( ) n θ + 2kπ + sin ( )] n for k = 0,1... n 1. Understand the effect of multiplication by a complex number on an Argand diagram. Applications of de Moivre's theorem Exercise level The exponential form: z= re iθ 6.2 Using the exponential form of a complex number 6.3 Expressing sin θ and cos θ in terms of complex numbers 6.4 Using the exponential forms of sin θ and cos θ 6.5 Summing series using de Moivre s Theorem powers of trigonometrical ratios of the fundamental angle. e.g. sin 3θ = 3 sin θ 4sin 3 θ Use expressions for sin θ and cos θ in terms of e iθ or equivalent relationships. e.g. sin θ = eiθ e iθ 2 Express powers of sin θ and cos θ in terms of series of trigonometric ratios of multiples of the fundamental angle. e.g. sin 5 θ = 1 (10 sin θ 5 sin 3θ + sin 5θ) 16 Multiplication by re iθ corresponds to enlargement with scale factor r with rotation through θ about the origin. e.g. multiplication by i corresponds to a rotation of π about the origin. 2 Complex numbers / Complex numbers: Topic assessment Applications of Integration: Volumes of revolution and mean values. Be able to derive formulae for and calculate volumes of solids of revolution. evaluate the mean value of a function. Applications of integration 5.1 Introduction 5.2 Rotation about the x-axis 5.3 Rotation about the y-axis 5.4 Curves defined parametrically 5.5 Mean of a function To include solids generated using either coordinate axis as the axis of rotation, and the volume of a solid formed by rotation of a region between two curves. This includes curves defined parametrically. The mean value of f(x) on the interval [a, b] 1 is b f(x)dx b a a

9 First order Second order 1: Homogeneous Understand how to introduce and define variables to describe a given situation in mathematical terms. Be able to relate 1st and 2nd order derivatives to verbal descriptions and so formulate. Know the language of kinematics, and the relationships between the various variables. Use in modelling in kinematics and in other contexts. Know the difference between a general solution and a particular solution. Be able to find both general and particular solutions. Recognise where the integrating factor method is appropriate. Be able to find an integrating factor and understand its significance in the solution of an equation. Be able to solve an equation using an integrating factor and find both general and particular solutions. Be able to solve of the form y + ay + by = 0 by using the auxiliary equation. OCR_FM_Pure / First order 1.1 Modelling with 1.2 General and particular solutions 1.3 Separation of variables 1.4 Applying the separation of variables method 1.5 Integrating factor: multiplying by a factor to give a perfect derivative 1.6 Finding integrating factors 1.1 Auxiliary equation method for 1st order DEs The will not be restricted to those which candidates can solve analytically. Sufficient information will be given about contexts which may be unfamiliar. v = dx dt = x a = dv dv = v dt dx = x Equations which can be rearranged into the form dy + P(x)y = Q(x) dx Integrating factor I(x) = e P(x)dx E.g. a particular solution through a given point. a and b are constants Understand the term complementary Function.

10 Second order 2: Nonhomogeneous Understand and use the relationship between different cases of the solution and the nature of the roots of the auxiliary equation. Be able to solve of the form y + ay + by = f(x) by solving the homogeneous case and adding a particular integral to the complimentary function. Be able to find particular integrals in simple cases. Understand the relationship between different cases of the solution and the nature of the roots of the auxiliary equation. 1.2 Auxiliary equation method for 2nd order DEs 1.3 Finding particular solutions (initial conditions) 1.4 Finding particular solutions (boundary conditions) 1.5 Auxiliary equation with repeated roots 1.6 Auxiliary equation with imaginary roots 1.7 Auxiliary equation with complex roots 2.1 Finding the complementary function 2.2 Using a trial function to find a particular integral 2.3 Particular integrals: polynomials 2.4 Particular integrals: trigonometric 2.5 Particular integrals: exponential Discriminant > 0 Discriminant = 0 Discriminant < 0 Cases where f(x) is a polynomial, trigonometric or exponential function. Includes cases where the form of the complementary function affects the form required for the particular integral. Includes cases where the form of the particular integral is given.

11 Second order 3: Modelling with 2nd order Be able to solve the equation for simple harmonic motion x = ω 2 x and be able to relate the solution to the motion. Be able to model damped oscillations using 2nd order. Be able to interpret the solutions of modelling damped oscillations in words and graphically. 2.6 Particular integrals: special cases 3.1 Simple Harmonic Motion 3.2 Damped oscillations 3.3 Damped oscillations: plotting/interpreting solutions Includes use of the formulae x = A cos ωt + B sin ωt or x = R sin( ωt + ɸ) in modelling situations. Learners may quote these formulae without proof when not asked to derive it or to solve the SHM equation. The damping will be described as over-, critical or under- according to whether the roots of the auxiliary equation are real distinct, equal or complex. Amplitude refers to the local maximum distance from the equilibrium position. The amplitude decreases with time. Analyse and interpret model situations with one independent variable and two dependent variables which lead to coupled 1st order simultaneous linear and find the solution. 3.4 Solving simultaneous 3.5 Finding particular solutions 3.6 Simultaneous : plotting/interpreting solutions Analyse and interpret model situations with one independent variable and two dependent variables which lead to coupled 1st order simultaneous linear and find the solution. e.g. Predator-prey models, continuous population models, industrial processes. Includes solution by eliminating one variable to produce a single second order equation.