AS Mathematics pure content Pure Mathematics Unit Title Estimated hours Algebra and functions a Algebraic expressions – basic algebraic manipulation, indices and surds 4 b Quadratic functions – factorising, solving, graphs and the discriminants 4 c Equations – quadratic/linear simultaneous 4 d Inequalities – linear and quadratic (including graphical solutions) 5 e Graphs – cubic, quartic and reciprocal 5 f Transformations – transforming graphs – f(x) notation 5 2 Coordinate geometry in the (x, y) plane a Straight-line graphs, parallel/perpendicular, length and area problems 6 b Circles – equation of a circle, geometric problems on a grid 7 3 Further algebra a Algebraic division, factor theorem and proof 8 b The binomial expansion 7 4 Trigonometry a Trigonometric ratios and graphs 6 b Trigonometric identities and equations 10 5 Vectors (2D) a Definitions, magnitude/direction, addition and scalar multiplication 7 b Position vectors, distance between two points, geometric problems 7 6 Differentiation a Definition, differentiating polynomials, second derivatives 6 b Gradients, tangents, normals, maxima and minima 6 7 Integration a Definition as opposite of differentiation, indefinite integrals of xn 6 b Definite integrals and under curves 5 8 Exponentials and logarithms: Exponential functions and natural logarithms 12 120 hours AS Mathematics applied content
AS Mathematics applied content Section B – Mechanics Unit Estimated hours 6 Quantities and units in mechanics a Introduction to mathematical modelling and standard S.I. units of length, time and mass 1 b Definitions of velocity, speed, acceleration and weight and displacement; Vector and scalar quantities 2 7 Kinematics 1 (constant acceleration) a Graphical representation of velocity, acceleration and displacement 4 b Motion in a straight line under constant acceleration; suvat formulae for constant acceleration; Vertical motion under gravity 6 8 Forces & Newton’s laws a Newton’s first law, force diagrams, equilibrium, introduction to i, j system 4 b Newton’s second law, ‘F = ma’, connected particles (no resolving forces or use of F = μR); Newton’s third law: equilibrium, problems involving smooth pulleys 6 9 Kinematics 2 (variable acceleration) a Variable force; Calculus to determine rates of change for kinematics 4 b Use of integration for kinematics problems i.e. 3 30 hours UNIT 6: Quantities and units in mechanics SPECIFICATION REFERENCES 6.1 Understand and use fundamental quantities and units in the S.I. system: length, time, mass. Understand and use derived quantities and units: velocity, acceleration, force, weight PRIOR KNOWLEDGE GCSE (9-1) in Mathematics at Higher Tier
UNIT 5: The binomial theorem SPECIFICATION REFERENCES 4.1 Understand and use the binomial expansion of (a + bx)n for rational n, including its use for approximation; be aware that the expansion is valid for < 1 (proof not required) KNOWLEDGE Covered so far Series and sequences (See Unit 4 of the SoW) GCSE (9-1) in Mathematics at Higher Tier A4 Algebraic fractions AS Mathematics – Pure Mathematics content 2.6 Algebraic division, factor theorem (See Unit 3a of the SoW) 4.1 Binomial expansion of the form (a + bx)n , where n is a positive integer (See Unit 3b of the SoW) KEYWORDS Binomial, expansion, theorem, integer, rational, power, index, coefficient, validity, modulus, factorial, nCr, combinations, Pascal’s triangle, partial fractions, approximation, converges, diverges, root. The formula book includes formulae for the binomial expansion: (a + b)n = an + an – 1b + an – 2b2 + … + an – rbr + … + bn where nCr = (1 + x)n = 1 + nx + (|x| < 1, n ) This unit links with the binomial distribution in the statistics section of the remaining A level Mathematics content of statistics. 5a. Expanding (a + bx)n for rational n; knowledge of range of validity (4.1) Teaching time 4 hours
If we now go up from the point b, hit the curve and then construct another tangent (as in the diagram below) then, a similar argument, gives a better approximate root at c (nearer than b). Therefore we would get c = b – . So if we continued this process we would get d = c – and generally xn + 1 = xn – . Sometimes the process fails for some curves or starting points. What happens to the tangent if we try to the process here? An example of the type question which may be seen: f (x) = x3 + 8x – 19. Obtain an approximation to the real root of f(x) = 0 by performing two applications of the procedure to f(x), using x = 2 as the first approximation. Give your answer to 3 decimal places. OPPORTUNITIES FOR REASONING/PROBLEM SOLVING Try different methods to find the roots of the same function. Which is the most efficient method or leads to the more accurate approximation? Consider, for example, iteration vs Newton–Raphson. COMMON MISCONCEPTIONS/EXAMINER REPORT QUOTES Marks are often lost for sign errors and other numerical slips. Students must show full working leading to the correct answer for full marks. Giving a correct answer either without working or following wrong working will result in zero marks. NOTES Graph drawing packages are an essential way to ‘look’ at the curve and the potential position of the roots depending on the first approximation of the root. There will be a rich source of questions from the legacy FP1 papers as this topic was part of that specification. Functions used will be consistent with the differentiation unit, e.g. e2x, etc. 9d. Problem solving (9.4) Teaching time 2 hours OBJECTIVES By the end of the sub-unit, students should: be able to use numerical methods to solve problems in context. TEACHING POINTS
TEACHING POINTS Begin by showing some 3D coordinates on x, y, z axes. (Graph drawing packages are very useful here, especially if you can turn the grid to view from different positions.) Consider a cuboid (2 by 3 by 4), with one corner at the origin. Ask the class to write down the coordinates of all the vertices. Remind students of 2D work and extend to 3D column vectors, orthogonal unit vectors i, j, k and position vectors. Write all the vectors from the ‘origin’ corner of the cuboid as position vectors (e.g. OA = a, etc.) Calculate the magnitude of these vectors as for example. Extend this idea to calculating the distance d between two points (x1, y1, z1) and (x2, y2, z2) using d 2 = (x1 – x2)2 + (y1 – y2)2 + (z1 – z2)2 Extend the ideas of vector addition and subtraction to 3D: = b – a. Cover the triangle and parallelogram laws of addition, as well as demonstrating parallel vectors. Show how to find a unit vector in the direction of a, and make sure students are familiar with the notationa (extended to 3D). Use vectors to solve problems in pure mathematics and discuss the 3D geometrical interpretations of solutions. OPPORTUNITIES FOR REASONING/PROBLEM SOLVING Link examples to mechanics (kinematics and forces). For example, consider questions such as: The velocity of an object is given by vector v = 3ti + t2j + 4k What is its speed after 5 seconds? COMMON MISCONCEPTIONS/EXAMINER REPORT QUOTES Encourage students to draw diagrams to help geometrical thinking when answering vector questions. Stress the importance of reading the question carefully and giving answers in the correct way, for example coordinates or column vectors may be requested. Emphasise the importance of good notation. Students do not always understand that represents the square of the length AP. NOTES Please note that vector equations of straight lines and the scalar product are not on this specifications.