AS/A Level Mathematics a and Further Mathematics a Curriculum Planner Model 1

Curriculum Planning Document

Model 1 – Parallel over two years

Introduction

This GCE AS and A Level Curriculum Planning Document has been designed to help you develop schemes of work for delivering reformed Mathematics and Further Mathematics qualifications in parallel. This document details a possible programme for coteaching H230 OCR AS Level Mathematics A H240 OCR A Level Mathematics A H235 OCR AS Level Further Mathematics A H245 OCR A Level Further Mathematics A

The document should be used in conjunction with the full specification documents and the Teaching Order Framework. One key feature of the OCR A Level Mathematics and Further Mathematics specifications is their two-column structure, setting out the required content in a format to clearly show the progression through AS Level and A Level.

This document is fully customisable so it can be edited to suit your own particular cohort, however care should be taken to avoid introducing content without the required prerequisite knowledge. The route provided is a ‘best fit’, effective for whichever of the Further Mathematics options are taken (centres that do not offer all four Further Mathematics options will have less prior knowledge constraints that will need to be balanced and more flexibility for editing). The structure of the Curriculum Planning Document has been produced based on the assumption of A Level Maths and A Level Further Maths being taught as two fully timetabled qualifications, each covered by two teachers. In A Level Mathematics there is an assumption that Teacher A acts as a specialist in statistics and Teacher B acts as a specialist in mechanics (with both teaching aspects of the pure content). In A Level Further Maths there is an assumption that Teachers C and D each deliver one of the four options (with both teaching aspects of the core pure content).

Model 1 assumes that the optional content in Further Maths will be spread out over the course, rather than taught in a single block This is an early sight draft version to help with initial planning. Space has been provided for adding notes and links to resources. Notes on prior knowledge include reference to content learners would have seen during Key Stage 4, reference codes relate to OCR J560 GCSE (9-1) Mathematics. Download specifications, sample assessment materials, teaching and learning resources at ocr.org.uk/alevelmathematics ocr.org.uk/alevelfurthermaths

Version 2 1 © OCR 2017 Summary wk/ Teacher A Teacher B Teacher C & D Term Maths Maths Further Maths (Each teacher deliveries core pure + 1 applied component) (Each teacher delivers core pure + 1 option) [Students study all core pure + 2 options] [Students study all core pure + both applied components] Y532 Stats Y533 Mech Y534 Discrete Y535 Add Pure 1/T1 LDS and Sampling Algebra and Functions P & C Dim Analysis Graphs Number Theory 2.02 a 1.02 a, b, c, d, e 5.01 a 6.01 a, b, d 7.02 a, b, c, g 8.02 a, b, c, d

2/T1 Sampling Proof Applications of P Dim Analysis Graphs Number Theory & C 2.01 a, b, c, d 1.01 a, b, c 6.01 c, e 7.02 d, e, p, q, r 8.02 e, f, i, j, k 5.01 b

3/T1 Coordinate Geometry Binomial Expansion Proof by Induction The language of complex numbers 1.03 a 1.04 a 4.01 a 4.02 a, b, c

4/T1 Equations of lines Polynomials and Graphs The language of Matrices Basic Operations with complex numbers (Radians) 1.03 b, c 1.02 m, n, o 4.03 a, b, c 4.02 e, f

5/T1 Equations of circles Units and Kinematics Determinants and Inverses Solutions of equations 1.03 d, e, f 3.01 a, b 3.02 a, b 4.03 h, j, l, m, n, o, p 4.02 g, h, i, j

6/T1 Vectors Kinematic Graphs solutions of simultaneous equations Argand Diagrams and Loci 1.10 a, c, d 3.02 c 4.03 r 4.02 k, l, o, p

Version 2 2 © OCR 2017 wk/ Teacher A Teacher B Teacher C & D Term Maths Maths Further Maths (Each teacher deliveries core pure + 1 applied component) (Each teacher delivers core pure + 1 option) [Students study all core pure + 2 options] [Students study all core pure + both applied components] Y532 Stats Y533 Mech Y534 Discrete Y535 Add Pure 7/T1 Vectors Suvat Equations Chi Squared Energy Mathematical Groups Contingency Preliminaries 1.10 e, f, g 3.02 d 6.02 d, e 8.03 a, b Tables 7.01 a, b, c 5.06 a 8/T1 Probability Differentiation and Gradients Fitting Energy Mathematical Groups distributions Preliminaries 1.04 b, 2.03 a, b 1.07 a, b 6.02 i 8.03 c, d 5.06 b, d (ratio 7.01 d, e, f, g, I, k and proportion) 9/T1 Binomial Distribution Gradient Functions and 2 nd Probability Momentum Algorithms Groups derivatives Distributions 2.04 a, b, c 6.03 a, b 7.03 a, b, c 8.03 e, f 1.07 c, d, e 5.02 a, b, c

10/T Graphs and Transformations 1 st Principles of Differentiation Binomial, Uniform Restitution Algorithms Groups 1 and Geometric 1.02 p, q, r, w 1.07 g, i 6.03 i, j 7.03 j, l 8.03 g, h distributions 5.02 d, e, f, g, h 11/T Polynomial Equations Equations of tangents and Linear Transformations using matrices Vectors 1 normal 1.02 f, j 4.03 d, e, f 4.04 a, c, e, g 1.07 m

12/T Inequalities Stationary Points Invariance and scale factors Roots of equations 1 1.02 g, h, i 1.07 n, o 4.03 g, i, k, q 4.05 a, b

1/T2 Data Presentation Forces Fitting Resolving forces Graphs Properties of distributions (preliminary work) groups 2.02 b 3.03 a, f, g 7.02 j, k 5.06 b, d (Bin, U 8.03 i and Geo)

Version 2 3 © OCR 2017 wk/ Teacher A Teacher B Teacher C & D Term Maths Maths Further Maths (Each teacher deliveries core pure + 1 applied component) (Each teacher delivers core pure + 1 option) [Students study all core pure + 2 options] [Students study all core pure + both applied components] Y532 Stats Y533 Mech Y534 Discrete Y535 Add Pure 2/T2 Bivariate Data Newton’s Laws Dependent and Impulse Network Properties of Independent Algorithms sequences 2.02 c, d, e 3.03 b, c, d, h 6.03 e, f Variables 7.04 a 8.01 a, b, h 5.09 a 3/T2 Average, Spread and Outliers Equilibrium Linear regression Restitution Network Properties of Algorithms sequences 2.02 f, g 3.03 I, j, r 5.09 b, c, d, e 6.03 i, j, k 7.04 b, f 8.01 c, d

4/T2 Working with LDS Connected Particles PMCC Work, Energy and Critical Path Fibonnaci and Power Analysis Solving relations 2.02 j 3.03 k, n 5.08 a, b, c 6.02 a, b, (i) 7.05 a 8.01 e, f

5/T2 Exponentials and Logarithms Trigonometry SRC Work, Energy and Critical Path Vector product Power Analysis and scalar triple 1.06 a, b, c 1.05 a, b, c 5.08 e, g product 6.02k, l, (i) 7.05 b, c 8.04 a, b, c, d

6/T2 Exponential Graphs Trigonometry Functions Recap of Proof and Matrices Recap of Complex numbers and Vectors 1.06 d, e, f 1.05 f, j, o

7/T2 Modelling with exponentials Fundamental Theorem of Poisson Uniform motion in Algorithms Surfaces Calculus a circle 1.06 g, h, i 5.02 i, j, k, l, 7.03 d, e, f, g 8.05 a 1.08 a, b 6.05 a

Version 2 4 © OCR 2017 wk/ Teacher A Teacher B Teacher C & D Term Maths Maths Further Maths (Each teacher deliveries core pure + 1 applied component) (Each teacher delivers core pure + 1 option) [Students study all core pure + 2 options] [Students study all core pure + both applied components] Y532 Stats Y533 Mech Y534 Discrete Y535 Add Pure 8/T2 Statistical Hypothesis Testing Definite Integrals Poisson Uniform motion in Graphical Linear Sections and 5.02 m , n + a circle Programming contours 2.05 a 1.08 d 5.06b, d 6.05 b 7.06 a, c 8.05 c

9/T2 Binomial Hypothesis Testing Area between curve and x- Hypothesis tests Uniform motion in Graphical Linear Partial Diff axis 5.08d a circle Programming 2.05 b 8.05 d 1.08 e 6.05 c 7.06 d

10/T Inference Variable Acceleration Hypothesis tests Motion in a Game Theory Stationary points 2 5.08f vertical circle 2.05 c 3.02 d, f 7.08 a, b, c, e 8.05 e 6.05 d

1/T3 Conditional Probability Radians and Trigonometry Non-parametric Hooke’s law Graphs and Finite (modular) Tests Networks arithmetic 2.03 c, d, e 1.05 d, e 6.02 g, h 5.07 a, b 7.02 f, h, i, 8.02 g

2/T3 Algebra and Functions Radians and Trigonometry Single Sample Linear momentum Graphs and Finite (modular) hypothesis tests in 2-D Networks arithmetic 1.02 u, v, x 1.05 g, h, i, k, o 5.07 c 6.03 c, d 7.02 l, m, n, o 8.02 h

3/T3 Series and Sequences Numerical Methods Paired-sample Oblique impact Network Fermat’s little and two sample Algorithms theorem and 1.04 c, d, e, f, g 1.09 a, b, c 6.03 g, h hypothesis test binomial theorem 7.04 c, d, e 5.07 d 8.02 l, o

Version 2 5 © OCR 2017 wk/ Teacher A Teacher B Teacher C & D Term Maths Maths Further Maths (Each teacher deliveries core pure + 1 applied component) (Each teacher delivers core pure + 1 option) [Students study all core pure + 2 options] [Students study all core pure + both applied components] Y532 Stats Y533 Mech Y534 Discrete Y535 Add Pure 4/T3 AP and GP Newton-Raphson Test for identity NEL Network Order Algorithms 1.04 h, i, j, k 1.09 d, e 5.07 e 6.03 l 8.02 m, n 7.04 c, d, e

5/T3 Parametric form Moments about a point Proof Exponential form of complex numbers and geometric effects 1.03 g 3.01 c, 3.04 a, b 4.01 b 4.02 d, m

6/T3 Modelling using parametric Modelling Static problems Solutions of equations and intersection Euler’s formula and de Moivre’s equations of planes theorem 3.04 c 1.03 h 4.03 s, t 4.02 n, q, r, s

7/T3 Normal Distribution Introduction to Differential Continous Centre of Mass of Arrangement and Groups Equations random variables symmetric lamina Selection 2.04 e, f 8.03 j, k, l problems 1.07 t 5.03 a 6.04 a, b 7.01 h, j 8/T3 Normal Distribution Analytical Solutions of Probability Composite Rigid Inclusion- Groups Differential Equations density functions bodies exclusion 2.04 d, g, h 8.03 m principle 1.08 k 5.03 b, c, d 6.04 c 7.01 l

Version 2 6 © OCR 2017 wk/ Teacher A Teacher B Teacher C & D Term Maths Maths Further Maths (Each teacher deliveries core pure + 1 applied component) (Each teacher delivers core pure + 1 option) [Students study all core pure + 2 options] [Students study all core pure + both applied components] Y532 Stats Y533 Mech Y534 Discrete Y535 Add Pure

1/T4 Partial Fractions Proof by Contradiction Summation of series Further Vectors 1.02 k, y 1.01 d 4.06 a 4.04 b, d, f

2/T4 Modulus Function 3D Vectors Method of differences Further Vectors 1.02 l, s, t 1.10 b 4.06 b 4.04 h, i, j

3/T4 Binomial Expansion Further Vectors and Cumulative Work in 2-D Efficiency and Scalar Vector Kinematics distribution complexity of 1.04 c 6.02 c 8.04 e functions Algorithms 1.10 h, 3.02 e 5.03 e, f, 7.03 h, i, 4/T4 Normal and Hypothesis Resolving Forces Cumulative Energy Strategies Surfaces distribution 2.05 d 3.03 e, l 6.02 f 7.03 k, m 8.05 b functions 5.03 g 5/T4 Normal and Hypothesis Forces in Equilibrium Random Conservation of Critical Path Stationary points Variables Energy Analysis 2.05 e 3.03 m, o 8.05 f 5.04 a 6.02 j 7.05 d, e

Version 2 7 © OCR 2017 wk/ Teacher A Teacher B Teacher C & D Term Maths Maths Further Maths (Each teacher deliveries core pure + 1 applied component) (Each teacher delivers core pure + 1 option) [Students study all core pure + 2 options] [Students study all core pure + both applied components] Y532 Stats Y533 Mech Y534 Discrete Y535 Add Pure 6/T4 Points of Inflection Resultant Forces in motion Normal Random Power Critical Path Tangent planes Variables Analysis 1.07 f, p 3.03 p, q, s, t, u 6.02 m 8.05 g 5.04 b 7.05 d, e

7/T4 Product and Quotient Rule Area between two curves Polar Coordinates Hyperbolic functions 1.07 q 1.08 f 4.09 a, b 4.07 a, b

8/T4 Chain Rule Integration as limit of a sum Area enclosed by polar curve Hyperbolic functions 1.07 r 1.08 g 1.09 f 4.09 c 4.07 c, d

9/T4 Trigonometry Identities Integration by substitution 5.06 Chi Squared Statics of solids Game Theory Second Order Tests Recurrence 1.05 k. l, m 1.08 h 6.04 d, e 7.08 d, f Relation 5.06 c 8.01 g, i 10/T Trigonometry Identities Motion in two dimensions Differential Equations Volumes of solids of revolution 4 1.05 n, o, p, q 3.02 e, h, i 4.10 a, c 4.08 d

11/T Further Numerical Methods Projectile model 2nd order homogeneous differential Volumes of solids of revolution defined 4 equations parametrically 1.09 g 3.02 i 4.10 d (real) 4.08 d

Version 2 8 © OCR 2017 wk/ Teacher A Teacher B Teacher C & D Term Maths Maths Further Maths (Each teacher deliveries core pure + 1 applied component) (Each teacher delivers core pure + 1 option) [Students study all core pure + 2 options] [Students study all core pure + both applied components] Y532 Stats Y533 Mech Y534 Discrete Y535 Add Pure 1/T5 Pearson’s product moment Exponential calculus Central Limit Motion in a Circle Graphical Linear Further Calculus correlation Theorem review Programming 1.07 j, l, 1.08 c 8.06 a 2.05 f 5.05 a 7.06 b, e, f

2/T5 Hypothesis testing and Trigonometrical calculus Population Mean Radial and The Simplex Further Calculus correlation and Variance tangential Algorithm 1.07 h, k, q, r 8.06 a components 2.05 g 5.05 b 7.07 a, b, c, 6.05 e 3/T5 Functions and Modelling Trigonometrical calculus Hypothesis Tests Free motion The Simplex Further Calculus Algorithm 1.02 z 1.08 c 5.05 c, 5.07 f 6.05 f 8.06 b 7.07 d

4/T5 Integration of parametric Further Kinematics Confidence Linear Motion The Simplex Further Calculus functions Intervals under variable Algorithm 3.03 v 8.06 b force 1.08 f 5.05 d 7.07 e, f 6.06 a 5/T5 Integration by parts Further Kinematics Differential Equations Inverse Hyperbolic functions 1.08 i, j 3.02 g 4.10 b 4.07 e, f

6/T5 Further Differential equations Further Parametrics 2nd order homogeneous differential Further Integration equations 1.07 t, 1.08 k, l 1.03 g, h, 1.07 s, 1.08 f 4.08 g 4.10 d (real and complex)

Version 2 9 © OCR 2017 wk/ Teacher A Teacher B Teacher C & D Term Maths Maths Further Maths (Each teacher deliveries core pure + 1 applied component) (Each teacher delivers core pure + 1 option) [Students study all core pure + 2 options] [Students study all core pure + both applied components] Y532 Stats Y533 Mech Y534 Discrete Y535 Add Pure 7/T5 Pure Revision Pure Revision 2nd order non-homogeneous Further Integration differential equations 4.05 c 4.08 h 4.10 e

8/T5 Statistics Revision Mechanics Revision Simple Harmonic Motion Maclaurin series 4.10 f 4.08 a, b

9/T5 Statistics Revision Mechanics Revision Damped Oscillations Improper integrals 4.10 g 4.08 c

10/T Pure Revision Pure Revision Linear Systems Further Calculus 5 4.10 h 4.08 e, f

1/T6 Pure Revision Pure Revision Core Revision Core Revision

2/T6 Statistics Revision Mechanics Revision Option Revision Option Revision Option Revision Option Revision

Version 2 10 © OCR 2017 wk/ Teacher A Teacher B Teacher C & D Term Maths Maths Further Maths (Each teacher deliveries core pure + 1 applied component) (Each teacher delivers core pure + 1 option) [Students study all core pure + 2 options] [Students study all core pure + both applied components] Y532 Stats Y533 Mech Y534 Discrete Y535 Add Pure 3/T6 Statistics Revision Mechanics Revision Core Revision Core Revision

4/T6 Pure Revision Pure Revision Option Revision Option Revision Option Revision Option Revision

5/T6 Pure Revision Pure Revision Core Revision Core Revision

Term 1 (AS and A Level) Week Teacher A Teacher B Prior knowledge Prior knowledge 1/T1 GCSE review 12.03a GCSE review 3.01c, 3.03b, 6.01f, 6.03b, 6.03c, 6.03d Large data Set and Sampling Algebra and Functions Be able to interpret tables and diagrams for single-variable Understand and be able to use the laws of indices for all data. rational exponents.

Version 2 11 © OCR 2017 Week Teacher A Teacher B Be able to use and manipulate surds, including rationalising the denominator.

Be able to solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation.

Be able to work with quadratic functions and their graphs, and the discriminant (D or) of a quadratic function, including the conditions for real and repeated roots.

Be able to complete the square of the quadratic polynomial .

Resource links Resource links OCR LDS 1.02 Delivery Guide 2.02 Delivery Guide Bridging the gap between GCSE and AS/A Level – A Student Guide

Prior knowledge Prior knowledge 2/T1 GCSE review 12.01a GCSE review 6.01a, 6.01b Statistical Sampling Proof Understand and be able to use the terms ‘population’ and Understand and be able to use the structure of mathematical ‘sample’. proof, proceeding from given assumptions through a series of logical steps to a conclusion. Be able to use samples to make informal inferences about the population. Understand and be able to use the logical connectives . Understand and be able to use sampling techniques, including simple random sampling and opportunity sampling. Be able to show disproof by counter example. Be able to select or critique sampling techniques in the context of solving a statistical problem, including

Version 2 12 © OCR 2017 Week Teacher A Teacher B understanding that different samples can lead to different conclusions about the population.

Resource links Resource links 2.01 Delivery Guide 1.01 Delivery Guide

Prior knowledge Prior knowledge 3/T1 GCSE review 7.02a GCSE review 11.02b Coordinate Geometry Binomial Expansion Understand and be able to use the equation of a straight Understand and be able to use the binomial expansion of line, including the forms , and . for positive integer and the notations and , or , with .

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Resource links Resource links 1.03 Delivery Guide 1.04 Delivery Guide

Prior knowledge Prior knowledge 4/T1 GCSE review 7.02b GCSE review 7.01b, 7.01c Equations of Lines Polynomials and Graphs Be able to use the gradient conditions for two straight lines Understand and be able to use graphs of functions. to be parallel or perpendicular. Be able to sketch curves defined by simple equations Be able to use straight line models in a variety of contexts. including polynomials. Be able to sketch curves

Version 2 14 © OCR 2017 Week Teacher A Teacher B Resource links Resource links 1.03 Delivery Guide 1.02 Delivery Guide

Prior knowledge Prior knowledge 5/T1 GCSE review 7.01f, 8.05c, 8.05e, 8.05f, GCSE 10.01a, 10.01b, 6.02e, 7.04a 10.05a Equations of Circles Units and Kinematics Understand and be able to use the coordinate geometry of a Understand and be able to use the fundamental quantities circle including using the equation of a circle in the form . and units in the S.I. system: length (in metres), time (in seconds), mass (in kilograms). Be able to complete the square to find the centre and radius of a circle. Be able to use the following circle properties in the context of Understand and be able to use derived quantities and units: problems in coordinate geometry: velocity (m/s or ms-1), acceleration (m/s2 or ms-2), force (N), 1. the angle in a semicircle is a right angle, weight (N). 2. the perpendicular from the centre of a circle to a chord bisects the chord, 3. the radius of a circle at a given point on its circumference Understand and be able to use the language of kinematics: is perpendicular to the tangent to the circle at that point. position, displacement, distance, distance travelled, velocity, speed, acceleration, equation of motion.

Understand, use and interpret graphs in kinematics for motion in a straight line.

Resource links Resource links 1.03 Delivery Guide 3.01 Delivery Guide 3.02 Delivery Guide

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Prior knowledge Prior knowledge GCSE review 9.03a, 9.03b GCSE review 7.04b, 7.04c Vectors Kinematic Graphs Be able to use vectors in two dimensions. Be able to interpret displacement-time and velocity-time Be able to calculate the magnitude and direction of a vector graphs, and in particular understand and be able to use the and convert between component form and facts that the gradient of a displacement-time graph magnitude/direction form. represents the velocity, the gradient of a velocity-time graph represents the acceleration, and the area between the graph Be able to add vectors diagrammatically and perform the and the time axis for a velocity-time graph represents the algebraic operations of vector addition and multiplication by displacement. scalars, and understand their geometrical interpretations. 6/T1

Resource links Resource links 1.10 Delivery Guide 3.02 Delivery Guide

Prior knowledge Prior knowledge 7/T1 Vectors 1.10 a, c, d GCSE review 6.02e: Kinematic Graphs 3.02 b, c

Version 2 16 © OCR 2017 Week Teacher A Teacher B Vectors 2 Suvat Understand and be able to use position vectors. Understand, use and derive the formulae for constant acceleration for motion in a straight line: Be able to calculate the distance between two points represented by position vectors. Be able to use vectors to solve problems in pure mathematics and in context, including forces.

Resource links Resource links 1.10 Delivery Guide 3.02 Delivery Guide

Prior knowledge Prior knowledge 8/T1 GCSE review 11.01d, 11.02a, 11.02c, 11.02d, GCSE review 7.02a, 7.02b, 7.04b 11.02e Polynomials and Graphs 1.02 j, m, n Binomial Expansion 1.04 a

Version 2 17 © OCR 2017 Week Teacher A Teacher B Probability Differentiation Understand and know the link to binomial probabilities. Understand and be able to use the derivative of as the gradient of the tangent to the graph of at a general point . Understand and be able to use mutually exclusive and independent events when calculating probabilities. Understand and be able to use the gradient of the tangent at a point where as: Be able to use appropriate diagrams to assist in the 1. the limit of the gradient of a chord as tends to calculation of probabilities. 2. a rate of change of with respect to .

Resource links Resource links 1.04 Delivery Guide 1.07 Delivery Guide 2.03 Delivery Guide

Prior knowledge Prior knowledge 9/T1 Binomial Expansion 1.04 a Differentiation 1.07 a, b Binomial Distributions Differentiation Understand and be able to use simple, finite, discrete Understand and be able to sketch the gradient function for a probability distributions, defined in the form of a table or a given curve. formula Understand and be able to find second derivatives. Understand and be able to use the binomial distribution as a Version 2 18 © OCR 2017 Week Teacher A Teacher B model. Learners should be able to use the notations and and recognise their equivalence.

Be able to calculate probabilities using the binomial Understand and be able to use the second derivative as the distribution, using appropriate calculator functions. rate of change of gradient.

Resource links Resource links 2.04 Delivery Guide 1.07 Delivery Guide

Prior knowledge Prior knowledge 10/T1 GCSE review 6.03d, 7.03a Differentiation 1.07 a, b, c, d, e Graphs and Transformations Differentiation Be able to interpret the algebraic solution of equations Be able to show differentiation from first principles for small graphically. positive integer powers of .

In particular, learners should be able to use the definition Be able to use intersection points of graphs to solve including the notation. equations. [Integer powers greater than 4 are excluded.]

Be able to differentiate , for rational values of n, and related Understand and be able to use proportional relationships constant multiples, sums and differences.

Version 2 19 © OCR 2017 Week Teacher A Teacher B and their graphs.

Understand the effect of simple transformations on the graph of including sketching associated graphs, describing transformations and finding relevant equations: ,, and , for any real a.

Resource links Resource links 1.02 Delivery Guide 1.07 Delivery Guide

Prior knowledge Prior knowledge 11/T1 Algebra and Functions 1.02 b, c Equations of Lines 1.03 b, Differentiation 1.07 i Polynomial Equations Equations of tangents and normals Be able to solve quadratic equations including quadratic Be able to apply differentiation to find the gradient at a point equations in a function of the unknown. on a curve and the equations of tangents and normals to a curve. Be able to manipulate polynomials algebraically, including the factor theorem

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Resource links Resource links 1.02 Delivery Guide 1.07 Delivery Guide

Prior knowledge Prior knowledge 12/T1 GCSE review 6.04b: Polynomial Equations Differentiation 1.07 i 1.02f Inequalities Stationary Points

Be able to solve linear and quadratic inequalities in a single Be able to apply differentiation to find and classify stationary variable and interpret such inequalities graphically, including points on a curve as either maxima or minima. inequalities with brackets and fractions. Be able to identify where functions are increasing or Be able to express solutions through correct use of ‘and’ and decreasing. ‘or’, or through set notation.

Be able to represent linear and quadratic inequalities such as and graphically.

Version 2 21 © OCR 2017 Week Teacher A Teacher B Resource links Resource links 1.02 Delivery Guide 1.07 Delivery Guide

Version 2 22 © OCR 2017 Term 2 (AS and A level) Week Ref Teacher A Ref Teacher B Prior knowledge Prior knowledge GCSE review 12.02b GCSE review 10.01b Vectors 1.10 a, c, d, g Data Forces Presentation 2.02 b 3.03 a Understand the concept and vector nature of a force. Understand that area in a histogram Understand and be able to use the weight represents 3.03 f () of a body to model the motion in a straight line under frequency. gravity.

Understand the gravitational acceleration, g, and its value in 1/T2 3.03 g S.I. units to varying degrees of accuracy.

Resource links Resource links 2.02 Delivery Guide 3.03 Delivery Guide

Prior knowledge Prior knowledge 2/T2 GCSE review 12.03c Suvat 3.02 d

Version 2 23 © OCR 2017 Week Ref Teacher A Ref Teacher B Statistical Sampling 2.01 a Bivariate Newton’s Laws Data 2.02 c 3.03 b Understand and be able to use Newton’s first law. Be able to 3.03 c Understand and be able to use Newton’s second law interpret () for motion in a straight line for bodies of constant mass scatter moving under the action of constant forces. diagrams and regression 2.02 d lines for 3.03 d Understand and be able to use Newton’s second law bivariate data, () in simple cases of forces given as two dimensional including vectors. recognition of 2.02 e scatter 3.03 h Understand and be able to use Newton’s third law. diagrams which include distinct sections of the population.

Be able to understand informal interpretation of correlation.

Be able to understand that correlation does not imply causation.

Version 2 24 © OCR 2017 Week Ref Teacher A Ref Teacher B

Resource links Resource links 2.02 Delivery Guide 3.03 Delivery Guide

Prior knowledge Prior knowledge 3/T2 GCSE review 12.03a, b, d Forces 3.03 f, g

Average, Equilibrium spread and 2.02 f 3.03 i Understand and be able to use the concept of a normal outliers reaction force. Be able to calculate and 3.03 j Be able to use the model of a ‘smooth’ contact and 2.02 g interpret understand the limitations of the model. measures of central 3.03 r Understand the concept of a frictional force and be able to tendency and apply it in contexts where the force is given in vector or variation, component form, or the magnitude and direction of the force including are given 2.02 h mean, median, mode, 2.02 i percentile, quartile, interquartile range, standard deviation and

Version 2 25 © OCR 2017 Week Ref Teacher A Ref Teacher B variance Be able to calculate mean and standard deviation from a list of data, from summary statistics or from a frequency distribution, using calculator statistical functions.

Recognise and be able to interpret possible outliers in data sets and statistical diagrams.

Be able to select or critique data presentation techniques in the context of a statistical problem.

Version 2 26 © OCR 2017 Week Ref Teacher A Ref Teacher B

Resource links Resource links 2.02 Delivery Guide 3.03 Delivery Guide

Prior knowledge Prior knowledge 4/T2 Data Presentation 2.02 a, Forces 3.03 a, b, c, d, f, g, h, I, j b, c, f, g, h, i Suvat 3.02 d

Working with Connected Particles Large Data 2.02 j 3.03 k Be able to use the concept of equilibrium together with one Set dimensional motion in a straight line to solve problems that Be able to involve connected particles and smooth pulleys. clean data, including 3.03 n Be able to solve problems involving simple cases of dealing with equilibrium of forces on a particle in two dimensions using missing data, vectors, including connected particles and smooth pulleys. errors and outliers.

Resource links Resource links

Version 2 27 © OCR 2017 Week Ref Teacher A Ref Teacher B 2.02 Delivery Guide 3.03 Delivery Guide

Prior knowledge Prior knowledge 5/T2 GCSE review 7.01d GCSE review 6.02d, 10.05b, 10.05d, 10.05e, 10.03a Indices 1.02 a Surds 1.02 b Exponentials Trigonometry (in degrees) and 1.06 a 1.05 a Understand and be able to use the definitions of sine, cosine Logarithms and tangent for all arguments. Know and use the function ax Understand and be able to use the sine and cosine rules. and its graph, 1.05 b where a is 1.06 b positive. Understand and be able to use the area of a triangle in the 1.05 c Know and use form . the function ex 1.06 c and its graph.

Know that the gradient of is equal to and hence understand why the exponential model is suitable in many applications.

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Know and use the definition of (for ) as the inverse of (for all ), where is positive.

Resource links Resource links 1.06 Delivery Guide 1.05 Delivery Guide

Prior knowledge Prior knowledge 6/T2 Exponentials and GCSE review 7.01e logarithms 1.06 a, b, c Trigonometry 1.05 a Exponential Trigonometry (in degrees) Graphs 1.06 d 1.05 f Understand and be able to use the sine, cosine and tangent Know and use functions, their graphs, symmetries and periodicities. 1.06 e the function Understand and be able to use and . and its graph. 1.05 j 1.06 f Know and use Be able to solve simple trigonometric equations in a given as the inverse interval, including quadratic equations in , and and function of . equations involving multiples of the unknown angle. 1.05 o Understand and be able to use the laws of logarithms

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Resource links Resource links 1.06 Delivery Guide 1.05 Delivery Guide

Prior knowledge Prior knowledge 7/T2 GCSE review 5.03a GCSE review 7.04c Exponential Graphs 1.06 d Differentiation 1.07 e Modelling Fundamental Theorem of Calculus with Exponentials Know and be able to use the fundamental theorem of 1.06 g 1.08 a calculus. Be able to Be able to integrate where and related sums, differences solve 1.06 h 1.08 b and constant multiples. equations of the form for .

Be able to use logarithmic graphs to 1.06 i estimate parameters in relationships of the form and , given data for and .

Version 2 30 © OCR 2017 Week Ref Teacher A Ref Teacher B

Understand and be able to use exponential growth and decay and use the exponential function in modelling.

Resource links Resource links 1.06 Delivery Guide 1.08 Delivery Guide

Prior knowledge Prior knowledge 8/T2 Binomial Distributions 2.04 Fundamental Theorem of Calculus 1.08 a, b b, c Statistical Definite Integrals Hypothesis 2.05 a 1.08 d Be able to evaluate definite integrals. Testing Understand and be able to use the language of statistical

Version 2 31 © OCR 2017 Week Ref Teacher A Ref Teacher B hypothesis testing, developed through a binomial model: null hypothesis, alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p- value.

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Prior knowledge Prior knowledge 9/T2 Statistical Hypothesis Integration 1.08 b, d Testing 2.05 a Binomial Area between curve and x-axis Hypothesis Testing 2.05 b 1.08 e Be able to use a definite integral to find the area between a curve and the -axis. Be able to x conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context.

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Prior knowledge Prior knowledge 10/T2 Binomial Hypothesis Constant Acceleration 3.02 d Testing 2.05 b, c Inference Variable Acceleration 2.05 c Understand 3.02 d derive the constant acceleration formulae using a variety of that a sample calculus techniques: is being used by integration, e.g. . to make an inference 3.02 f Be able to use differentiation and integration with respect to about the time in one dimension to solve simple problems concerning population the displacement, velocity and acceleration of a particle and appreciate that the significance level is the probability of incorrectly rejecting the null hypothesis.

Version 2 34 © OCR 2017 Week Ref Teacher A Ref Teacher B

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Version 2 35 © OCR 2017 Term 3 (AS revision and A Level) Week Teacher A Teacher B Prior knowledge Prior knowledge Probability 2.03 a, b Trigonometry 1.05 a, b, c Conditional Probability Radians and Trigonometry Understand and be able to use conditional probability, Be able to work with radian measure, including use for arc including the use of tree diagrams, Venn diagrams and two- length and area of sector. way tables. Learners should know the formulae and . Understand the concept of conditional probability, and calculate it from first principles in given contexts. Learners should be able to use the relationship between degrees and radians. Be able to model with probability, including critiquing Understand and be able to use the standard small angle assumptions made and the likely effect of more realistic 1/T3 approximations of sine, cosine and tangent: assumptions. 1., 2., 3., where is in radians.

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Prior knowledge Prior knowledge 2/T3 Algebra and Functions 1.02 j, w Trigonometry 1.05 f, j Differentiation 1.07 a, d Integration 1.08 a Algebra and functions Radians and Trigonometry Understand and be able to use the definition of a function. Know and be able to use exact values of and for and multiples thereof, and exact values of for and multiples Understand and be able to use inverse functions and their

Version 2 36 © OCR 2017 Week Teacher A Teacher B graphs, and composite functions. Know the condition for the thereof. inverse function to exist and be able to find the inverse of a Understand and be able to use the definitions of secant function either graphically, by reflection in the line , or (), cosecant () and cotangent algebraically. () and of , and and their relationships to , and respectively. Understand the effect of combinations of transformations on Understand the graphs of the functions given in 1.05h, their the graph of including sketching associated graphs, ranges and domains. describing transformations and finding relevant equations. Understand and be able to use and . Be able to solve simple trigonometric equations in a given interval, including quadratic equations in , and and equations involving multiples of the unknown angle. Resource links Resource links 1.03 Delivery Guide 1.05 Delivery Guide

Prior knowledge Prior knowledge 3/T3 Binomial Expansion 1.04 a GCSE review 6.03e Polynomials 1.02 j, q Series and Sequences Numerical Methods Be able to extend the binomial expansion of to any rational , Be able to locate roots of by considering changes of sign of including its use for approximation. in an interval of on which is sufficiently well-behaved. Know that the expansion is valid for . Be able to work with sequences including those given by a Understand how change of sign methods can fail. formula for the term and those generated by a simple relation of the form . Be able to solve equations approximately using simple Understand the meaning of and work with increasing iterative methods and be able to draw associated cobweb sequences, decreasing sequences and periodic sequences. and staircase diagrams. Understand and be able to use sigma notation for sums of series.

Version 2 37 © OCR 2017 Week Teacher A Teacher B

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Prior knowledge Prior knowledge Series and Sequences 1.04 e, f, g Polynomials 1.02 q Series and Sequences 1.04 e Differentiation 1.07 i AP and GP Numerical Methods Understand and be able to work with arithmetic sequences Be able to solve equations using the Newton-Raphson and series, including the formulae for the term and the sum method and other recurrence relations of the form . to terms. Understand and be able to show how such methods can fail. Understand and be able to work with geometric sequences and series including the formulae for the term and the sum 4/T3 of a finite geometric series. Understand and be able to work with the sum to infinity of a convergent geometric series, including the use of and the use of modulus notation in the condition for convergence. Be able to use sequences and series in modelling.

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Version 2 38 © OCR 2017 Week Teacher A Teacher B Prior knowledge Prior knowledge Polynomials 1.02 j Simultaneous Equations 1.02 c Forces 3.03 b, 3.03 h Parametric form Moments Understand and be able to use the parametric equations of Understand and be able to use the unit for moment (N m). curves and be able to convert between cartesian and parametric forms. Be able to calculate the moment of a force about an axis through a point in the plane of the body. Understand that when a rigid body is in equilibrium the resultant moment is zero and the resultant force is zero.

5/T3

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Prior knowledge Prior knowledge 6/T3 Parametric form 1.03 g Moments 3.04 a, b

Version 2 39 © OCR 2017 Week Teacher A Teacher B Modelling using parametric equations Modelling using moments Be able to use parametric equations in modelling in a variety Be able to use moments in simple static contexts. of contexts.

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Prior knowledge Prior knowledge 7/T3 LDS and Sampling 2.02 a Differentiation 1.07 i Integration 1.08 d Normal Distribution Model Introduction to Differential Equations

Version 2 40 © OCR 2017 Week Teacher A Teacher B Understand and be able to use the normal distribution as a Be able to construct simple differential equations in pure model. mathematics and in context (contexts may include kinematics, population growth and modelling the relationship between price and demand). Be able to find probabilities using the normal distribution, using appropriate calculator functions.

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Prior knowledge Prior knowledge 8/T3 Normal Distribution Model 2.04 e, f Introduction to Differential Equations 1.07 t Normal Distribution Analytical Solutions of Differential Equations Be able to evaluate the analytical solution of simple first Know and be able to use the formulae and when choosing order differential equations with separable variables, a particular normal model to use as an approximation to a including finding particular solutions. binomial model.

Understand links to histograms, mean and standard

Be able to select an appropriate probability distribution for a context, with appropriate reasoning, including recognising when the binomial or normal model may not be appropriate.

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Term 4 (A Level) Week Teacher A Teacher B Prior knowledge Prior knowledge 1/T4 Polynomials 1.02 j Proof 1.01 a Partial Fractions Proof by contradiction

Understand and be able to use proof by contradiction. Be able to simplify rational expressions.

Be able to decompose rational functions into partial fractions (denominators not more complicated than squared linear terms and with no more than 3 terms, numerators constant or linear).

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Prior knowledge Prior knowledge Proof 1.01 b Vectors 1.10 a Indices 1.02 a Straight Lines 1.03 a Modulus Function 3D Vectors

Understand and be able to use the modulus function Be able to use vectors in three dimensions. Be able to sketch the graph of the modulus of a linear i.e. Learners should be able to use vectors expressed as or function involving a single modulus sign. as a column vector . Be able to solve graphically simple equations and Includes extending 1.10c to 1.10g to include vectors in three inequalities involving the modulus function. dimensions, excluding the direction of a vector in three 2/T4 dimensions.

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Version 2 43 © OCR 2017 Week Teacher A Teacher B Prior knowledge Prior knowledge Binomial Expansion 1.04 a, b Vectors 1.10 b Constant Acceleration 3.02 d Binomial Expansion Further Vectors and Kinematics Be able to extend the binomial expansion of to any rational Be able to use vectors to solve problems in kinematics. n, including its use for approximation

Be able to extend the constant acceleration formulae to motion in two dimensions using vectors.

3/T4

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Prior knowledge Prior knowledge 4/T4 Measures of average and spread 2.02 f Newton’s second law 3.03 c, d Normal distribution 2.04 e, f, g Newton’s third law 3.03 h, I, j, k Language of hypothesis testing 2.05 a

Version 2 44 © OCR 2017 Week Teacher A Teacher B Hypothesis Testing using Normal Distribution Resolving Forces Recognise that a sample mean,, can be regarded as a Be able to extend use of Newton’s second law to situations random variable. where forces need to be resolved (restricted to two dimensions).

Be able to extend use of Newton’s third law to situations where forces need to be resolved (restricted to two dimensions).

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Prior knowledge Prior knowledge 5/T4 Normal Distribution 2.05 d Forces 3.03 e, l

Hypothesis Testing using Normal Distribution Forces in Equilibrium Be able to use the principle that a particle is in equilibrium if Be able to conduct a statistical hypothesis test for the mean and only if the sum of the resolved parts in a given direction of a normal distribution with known, given or assumed is zero. variance and interpret the results in context. Be able to resolve forces for more advanced problems

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Prior knowledge Prior knowledge 6/T4 Gradients 1.07 d Vectors 1.10 a, b, c, d Tangents 1.07 m Kinematics 3.02 e Forces 3.03 e, l Points of Inflection Resultant Forces in motion Understand and be able to use the second derivative in Understand the term ‘resultant’ as applied to two or more connection to convex and concave sections of curves and forces acting at a point and use vector addition in solving points of inflection. problems involving resultants and components of forces.

In particular, learners should know that: 1. if on an interval, the function is convex in that interval; Be able to solve problems involving the dynamics of motion 2. if on an interval the function is concave in that interval; for a particle moving in a plane under the action of a force or 3. if and the curve changes from concave to convex or vice forces. versa there is a point of inflection. Be able to represent the contact force between two rough surfaces by two components (the ‘normal’ contact force and Version 2 46 © OCR 2017 Week Teacher A Teacher B Be able to apply differentiation to find points of inflection on the ‘frictional’ contact force). a curve. Understand and be able to use the coefficient of friction and the model of friction in one and two dimensions, including the concept of limiting friction.

Understand and be able to solve problems regarding the static equilibrium of a body on a rough surface and solve problems regarding limiting equilibrium. Resource links Resource links 1.07 Delivery Guide 3.03 Delivery Guide

Prior knowledge Prior knowledge 7/T4 Polynomials 1.02 j, k Integration 1.08 e Differentiation 1.07 i, j, k, l Product and Quotient Rules Area between two curves Be able to differentiate using the product rule and the Be able to use a definite integral to find the area between quotient rule. two curves.

Learners should understand the relationship between this method and the chain rule

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Prior knowledge Prior knowledge 8/T4 Polynomials 1.02 j, k GCSE 7.04c Differentiation 1.07 i, j, k, l, q Definite Integrals 1.08 d Chain Rule Integration as a limit of a sum Be able to differentiate using the chain rule, including Understand and be able to use integration as the limit of a problems involving connected rates of change and inverse sum. functions.

Understand and be able to use numerical integration of functions, including the use of the trapezium rule, and estimating the approximate area under a curve and the limits that it must lie between.

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Prior knowledge Prior knowledge Trigonometry Identities 1.05 j Chain Rule 1.07 r

Trigonometry Identities Integration by substitution Understand and be able to use and . Be able to carry out simple cases of integration by substitution for polynomials

Understand and be able to use double angle formulae and the formulae for , and .

9/T4 Understand the geometrical proofs of these formulae.

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Prior knowledge Prior knowledge 10/T4 Trigonometry Identities 1.05 f, g, j, k Constant acceleration 3.02 d

Version 2 49 © OCR 2017 Week Teacher A Teacher B Vectors 1.10 c Trigonometry Identities Motion in two dimensions Understand and be able to use expressions for in the Be able to extend the constant acceleration formulae to equivalent forms of or . motion in two dimensions using vectors.

Extend their knowledge of trigonometric equations to include radians and the trigonometric identities in Stage 2. Be able to model motion under gravity in a vertical plane using vectors where or . Be able to construct proofs involving trigonometric functions and identities. Be able to model the motion of a projectile as a particle moving with constant acceleration and understand the Be able to use trigonometric functions to solve problems in limitation of this model. context, including problems involving vectors, kinematics and forces.

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Prior knowledge Prior knowledge 11/T4 Numerical Methods 1.09 a, b, c, d, e, f Constant acceleration 3.02 d Vectors 1.10 c Further Numerical Methods Projectile model Be able to model the motion of a projectile as a particle Be able to use numerical methods to solve problems in moving with constant acceleration and understand the context. limitation of this model.

Includes being able to: 1.Use horizontal and vertical equations of motion to solve problems on the motion of projectiles.

Version 2 50 © OCR 2017 Week Teacher A Teacher B 2. Find the magnitude and direction of the velocity at a given time or position. 3. Find the range on a horizontal plane and the greatest height achieved. 4. Derive and use the Cartesian equation of the trajectory of a projectile.

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Version 2 51 © OCR 2017 Term 5 (A Level) Week Teacher A Teacher B Prior knowledge Prior knowledge Bivariate Data 2.02 c, d, e Indices 1.02 a Exponentials 1.06 a, b, i Differentiation 1.07 q, r Integration 1.08 a Pearson’s product moment correlation Exponential Calculus Understand Pearson's product-moment correlation Be able to differentiate and , and related sums, differences coefficient as a measure of how close data points lie to a and constant multiples. straight line. 1/T5 Understand and be able to use the derivative of . Be able to integrate , and related sums, differences and constant multiples.

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Prior knowledge Prior knowledge 2/T5 Language of Hypothesis Testing 2.05 a Graphs of Trigonometric functions 1.05 f, g Pearson Product Moment Correlation Differentiation from first principles 1.07 g 2.05 f

Hypothesis Testing and Correlation Trigonometrical Calculus Be able to show differentiation from first principles for and . Use and be able to interpret Pearson's product-moment Be able to differentiate , , and related sums, differences and Version 2 52 © OCR 2017 Week Teacher A Teacher B correlation coefficient in hypothesis tests, using either a constant multiples. given critical value or a p-value and a table of critical values.

When using Pearson's coefficient in an hypothesis test, the Differentiate trigonometric functions using the product rule data may be assumed to come from a bivariate normal and the quotient rule. distribution.

Differentiate trigonometric functions using the chain rule.

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Prior knowledge Prior knowledge 3/T5 Algebra and Functions 1.02 k, l, u, v, y Graphs of Trigonometric functions 1.05 f, g Differentiation of Trigonometric functions 1.07 k Functions and Modelling Trigonometrical Calculus Be able to integrate , and related sums, differences and constant multiples. Be able to use functions in modelling.

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Prior knowledge Prior knowledge 4/T5 Parametric equations 1.03 g Frictional forces 3.03 r, s, t, u, v Integration of parametric functions Further Kinematics Be able to use a definite integral to find the area between Understand and be able to solve problems regarding the two curves defined parametrically motion of a body on a rough surface.

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Prior knowledge Prior knowledge 5/T5 Product Rule 1.07 q Differentiation 1.07i, t Integration 1.08 d, e, f Non-uniform acceleration 3.02 f Vectors 1.10 h Integration by parts Further Kinematics Be able to carry out simple cases of integration by parts. Be able to extend the application of differentiation and Be able to integrate functions using partial fractions that integration to two dimensions using vectors. have linear terms in the denominator.

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Prior knowledge Prior knowledge Differentiation 1.07 i Polynomials 1.02 j Integration 1.08 d Further Differential equations Further Parametrics

Be able to construct simple differential equations in pure Understand and be able to use the parametric equations of mathematics and in context (contexts may include curves and be able to convert between cartesian and kinematics, population growth and modelling the relationship parametric forms. between price and demand)

Be able to evaluate the analytical solution of simple first order differential equations with separable variables, including finding Be able to use parametric equations in modelling in a variety particular solutions. of contexts.

6/T5 Be able to interpret the solution of a differential equation in Be able to differentiate simple functions and relations the context of solving a problem, including identifying defined implicitly or parametrically for the first derivative limitations of the solution. only.

Be able to use a definite integral to find the area between two curves.

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