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

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1 Edexcel Year 2 Core Pure Suggested Scheme of Work ( ) This template shows how Integral Resources and AMSP FM videos can be used to support Further Mathematics students and teachers. This template is for the compulsory Core Pure component of Year 2 you will also need to deliver optional elements alongside this. Core Pure makes up 50% of the Edexcel A Level Further Mathematics content. It is examined in A level papers 9FM0/01&02 (Core Pure). Integral Resources FM videos Teacher access to the Integral Resources for Further Mathematics is available free of charge to all schools/colleges that register with the Advanced Mathematics Support Programme: 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 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: FM videos are available for individual components of AS and A level Further Mathematics. There are 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. AMSP Area Coordinators will be able to offer additional guidance if needed: furthermaths.org.uk/regions.

2 Edexcel Year 2 Core Pure Suggested Scheme of Work ( ) Date Topic Specification statements Integral Resources Exercises & Assessment Integral Resources Sequences & Series The method of differences Further calculus 1: Improper integrals Further calculus 2: Inverse trigonometric Understand and use the method of differences for summation of series including use of partial fractions. Evaluate improper integrals where either the integrand is undefined at a value in the range of integration or the range of integration extends to infinity. Differentiate inverse trigonometric. Integrate of the form (a 2 x 2 ) 1 2 and (a 2 x 2 ) 1 Edexcel_FM_Pure / Sequences and series / Sequences and series 1: The method of differences 1: Improper integrals 2: Inverse trigonometric Section test S1 FM videos Notes Prerequisites 3.1: The method of differences Be able to sum series such as 1 by use of r(r+1) partial fractions. Understanding that a fraction can be decomposed into the sum or difference of other fractions. Exercise level 3 / Sequences and series / Sequences and series: Topic assessment For example: Section test FC1 Section test FC2 Exercise level Integrals with infinite limits (convergent) 2.2 Integrals with infinite limits (divergent) 2.3 Integrals with the function undefined at a limit 2.4 Integrals that cross a value where the function is undefined. 3.1 Differentiating arcsin x 3.2 Differentiating arctan x 3.3 Integrating using arcsin x 3.4 Integrating using arctan x e x dx x dx 0 Be able to differentiate expressions such as, arcsin x + x (1 x 2 ) and 1 2 arctanx2 Understanding of the rules of integration for met at AS level. e.g. know how to integrate such as 1/(x + a) and e x. Knowledge of inverse trig ; the chain rule and product and quotient rules for differentiation.

3 Further calculus 3: Further integration Maclaurin series 1: Using Maclaurin series Integrate using partial fractions. Be able to choose trigonometric substitutions to integrate associated. Find the Maclaurin series of a function including the general term Recognise and use the Maclaurin series for e x, ln(1 + x), sinx, cos x and (1 + x) n, and be aware of the range of values of x for which they are valid. 3: Further integration Edexcel_FM_Pure / Maclaurin series / Maclaurin series 1: Using Maclaurin series Section test FC3 Exercise level Partial fractions recap 4.2 Partial fractions: denominator (ax + b)(cx 2 + d) 4.3 Integrating: denominator (ax + b)(cx + d) 4.4 Integrating: denominator (ax + b) (cx + d) Integrating: denominator (ax + b)(cx 2 + d) 5.1 Further integration using arcsinx 5.2 Further integration using arctan x 5.3 Substituting x = sin u 5.4 Substituting x = tan u 5.5 Substituting x = a sin u or x = a tan u Includes quadratic factors ax 2 + c in the denominator. How to find partial fractions with denominators of the form (ax+b)(cx+d) or (ax+b)(cx+d)(ex+f); the technique of completing the square; integration by substitution. : Topic assessment 3.2 Polynomial Proof not required Familiarity with approximations of range of validity. differentiating trig 3.3 The general, Section test M2 Maclaurin series logarithmic 3.4 Maclaurin series and the Exercise level 3 using standard binomial expansion.

4 3.5 Maclaurin series of composite / / Maclaurin series / Maclaurin series: Topic assessment Polar coordinates 1: Polar curves Polar coordinates 2: Finding areas Understand and use polar coordinates and be able to convert between polar and Cartesian coordinates Sketch curves with r given as a function of θ, including use of trigonometric. Find the area enclosed by a polar curve. Find tangents parallel to, or at right angles to, the initial line. / Polar coordinates / Polar coordinates 1: Polar curves / Polar coordinates / Polar coordinates 2: Finding area Section test P1 Exercise level Polar coordinates 1.2 Sketching polar curves 1.3 Converting between forms The sketching of curves such as r = p sec (α θ), r = a, r = 2a cosθ, r = kθ, r = a(1 ± cosθ), r = a(3 + 2 cosθ), r = a cos2θ and r 2 = a 2 cos2θ Use of the formula β 1 2 r2 dθ α for area. 1.4 Polar curves: finding areas 1.5 Parallel Section test P2 /perpendicular tangents to the initial Exercise level 3 line Edexcel_FM_Pure / Polar coordinates / Polar coordinates: Topic assessment Familiarity with the reciprocal trig. Ability to integrate range of trig. Hyperbolic 1: Introducing hyperbolic Understand the definitions of hyperbolic sinh x, cosh x and tanh x, including their domains and ranges, and be able to sketch their graphs. Differentiate and integrate hyperbolic. / Hyperbolic / Hyperbolic 1: Introducing hyperbolic function Section test H1 Exercise level Definitions and graphs of hyperbolic 1.2 Solving simple equations 1.3 Hyperbolic function identities 1.4 Differentiating and integrating hyperbolic For example, differentiate tanh 3x, x sinh2 x, cosh2x x + 1 The chain rule and product and quotient rules for differentiation. Familiarity with exponential and logarithmic, including domain and range. Integration techniques covered in the Further Calculus section.

5 Hyperbolic 2: The inverse hyperbolic Applications of Integration 1: Volumes of revolution and mean values of a function Understand and be able to use the definitions of the inverse hyperbolic and their domains and ranges. Derive and use the logarithmic forms of the inverse hyperbolic. Integrate of the form (x 2 + a 2 ) 1 2 and (x 2 a 2 ) 1 2 and be able to choose substitutions to integrate associated. Derive formulae for and calculate volumes of revolution, where the equation of the curve is given in parametric form Understand and evaluate the mean value of a function. / Hyperbolic / Hyperbolic 2: The inverse hyperbolic Section test H2 Exercise level Inverse hyperbolic 2.2 Differentiating inverse hyperbolic 2.3 Integration using inverse hyperbolic 2.4 Integration using hyperbolic substitutions arsinhx = ln( x + x 2 + 1) arcoshx = ln( x + x 2 1) x 1 artanhx = x ln 2 1 x 1 < x < 1 Inverse. Implicit differentiation. Integration by substitution. / Hyperbolic / Hyperbolic : Topic assessment 6.1 Rotation about Be familiar with the Parametric the x-axis for mean equations and parametric curves value of a function parametric 6.2 Rotation about f(x) as differentiation. the y-axis for b 1 parametric curves b a f(x)dx Integration methods such as a 6.3 Mean of a substitution. function First order equations Use separation of variables to solve first order Des. Find and use an integrating factor to solve equations of form dy dx + p(x)y = q(x) and 1.1 Modelling with equations 1.2 General and particular solutions 1.3 Separation of variables The integrating factor e p(x)dx may be quoted without proof. Sketch members Integration methods from previous sections and A Level.

6 recognise when it is appropriate to do so. Find both general and particular solutions to equations. 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 of the family of solution curves. Complex numbers 1: De Moivre s Theorem Understand de Moivre s theorem and use it to find multiple angle formulae and sums of series. 4.1 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 4.5 Finding multiple angle identities using de Moivre's theorem To include using the results Z + 1 = 2 cos θ z z 1 = 2i sin θ z to find cos pθ, sin qθ and tan rθ in terms of powers of sin θ, cos θ, tan θ and powers of sin θ, cos θ, tan θ in terms of multiple angles. For sums of series, students should be able to show that, for example, 1 + z + z z n 1 Trig multiple angle formulae. = 1 + i cot ( π 2n ) where

7 z Complex Numbers 2: Complex Roots Complex Numbers 3: Exponential form Find the nth roots of a complex number for r 0 and know that they form the vertices of a regular n-gon in the Argand diagram. Use complex roots of unity to solve geometric problems. Know and use the definition e iθ = cos θ + i sin θ and the form z = re iθ Find the n distinct nth roots of re iθ for r Solving z n = The nth roots of unity 5.3 The nth roots of any complex number 5.4 Solving equations 6.1 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 = cos π n + i sin π n and n is a positive integer. cos θ = 1 2 (eiθ + e iθ ) sin θ = 1 2i (eiθ e iθ ) Second order equations 1: Homogeneous Solve equations of form y + ay + by = 0 where a and b are constants by using the auxiliary 1.1 Auxiliary equation method for 1st order DEs 1.2 Auxiliary equation method for 2nd order DEs

8 Second order equations 2: Nonhomogeneous Second order equations 3: equation. Understand and use the relationship between the cases when the discriminant of the auxiliary equation is positive, zero and negative and the form of solution of the equation. Solve equations of form y + a y + b y = f(x) where a and b are constants by solving the homogeneous case and adding a particular integral to he complementary function (in cases where f(x) is a polynomial, exponential or trigonometric function). Use equations in modelling in kinematics and in other contexts. 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 2.6 Particular integrals: special cases f(x) will have one of the forms: ke px A + Bx P + qx + cx 2 m cos ωx + n sin ωx

9 Modelling with 2nd order equations Solve the equation for simple harmonic motion x = ω 2 x and relate the solution to the motion. Model damped oscillations using second order Differential equations and interpret solutions. Analyse and interpret models of situations with one independent variable and two dependent variables as a pair of coupled first order Simultaneous equations and be able to solve them, for example predator-prey models. 3.1 Simple Harmonic Motion 3.2 Damped oscillations 3.3 Damped oscillations: plotting/interpreting solutions 3.4 Solving simultaneous equations 3.5 Finding particular solutions 3.6 Simultaneous equations: plotting/interpreting solutions Damped harmonic motion, with resistance varying as the derivative of the displacement, is expected. Problems may be set on forced vibration. Restricted to coupled first order linear equations of the form: x = ax + by + f(t) y = cx + dy + g(t)