 # IGCSEFM

This page is no longer updated. IF YOU ARE A TIFFIN STUDENT, use the existing Year 10-11 scheme of work that combines GCSE and IGCSE Further Maths.

# Summer Term

## Summer 2

#TopicResources
1 Coordinate Geometry (IGCSEFM/C1)
2 Equations of Circles
(a) Determine the equation of a circle given its radius and centre, or give an equation in another form.
(b) Use coordinate geometry/Pythagoras to establish the centre/radius of a circle.
(c) Find the points of intersection of a circle with the coordinate axes or another line.
(d) Determine the equation of the tangent of a circle.
(e) (C1 only) Use the discriminant to prove that a line does not intersect a circle.
(f) Determine the points where a straight line and circle intersect by solving simultaneous equations.

## Autumn 1

#TopicResources
3 Indices (IGCSEFM/C1)
(a) (GCSE) Know your laws of indices, including negative and fractional indices.
(b) Be able to use laws of indices backwards, e.g. solve x(3/2) = 1/27.
4 Expanding Brackets and Collecting Terms (IGCSEFM/C1)
(a) Be able to expand two brackets which have more than two terms in each, e.g. (x + y + 1)(y - 1)
(b) Be able to expand three (or more) brackets, e.g. (x + 1)(x + 2)(x - 3) or (x + 2)3.
5 Surds (IGCSEFM/C1)
(a) Be able to simplify surds as per GCSE, such as multiplying surds, expanding brackets with surds, and simplifying surds.
(b) Rationalise denominators of fractions with surds, e.g. (3?2 + 4)/(5?2 - 7)
6 Factorising (IGCSEFM/C1)
(a) Be able to re-factorise expressions where parts have been factorised, e.g. for (2x+3)2 - (2x-5)2 using difference of two squares, or (x+1)2 + (x+1) = (x+1)(x+1+1)
(b) Be able to factorise more complex expressions by 'intelligent guessing', e.g. 15x2 - 34xy - 16y2.
(c) Factorise expressions involving multiple factorisation steps, e.g. x4 - 25x2
7 Completing the Square (IGCSEFM/C1)
(a) As per GCSE, be able to put quadratics of form x2 + bx + c into the form (x+_)2 + _ (b) Also be able to complete the square when the coefficient of x2 is not 1, i.e. put in the form a(x+b)2 + c or c - a(x+b)2
8 Solving Equations (IGCSEFM/C1)
(a) Solve linear equations, possibly involving fractions.
(b) Be able to use the quadratic formula.
(c) Solve equations involving algebraic fractions.
9 The Discriminant (C1)
(a) Understand that the value of discriminant b2 - 4ac determines whether a quadratic has two distinct roots (if it is positive), equal roots (if it is 0) or no real real (if negative).
(b) Understand how the discriminant relates to the sketch of a quadratic, e.g. the graph will 'touch' the x-axis if b2 - 4ac = 0.
(c) Solve discriminant problems involving unknown coefficients of a quadratic.
10 Sketching Graphs (IGCSEFM/C1)
(a) Recognise the shapes of different types of graphs (quadratic, cubic, reciprocal).
(b) Understand the features that make up a graph, i.e. roots, y-intercept, max/min points and asymptotes. Understand that an asymptote is a line that a graph approaches towards infinity.
(c) Be able to sketch quadratic graphs, including using the completed square to identify the min/max point.
(d) Be able to sketch cubics, recognising when the graph crosses at a root (non-repeated factor), touches (squared factor) or has a point of intersection (cubed factor).
(e) [IGCSEFM only] Be able to sketch and reason about functions defined in 'pieces', including finding possible inputs for a given output.
(f) [C1 only] Sketch and reason about graph transformations (as per GCSE).
11 Domain and Range of Functions (IGCSEFM)
(a) Understand how functions work, e.g. find f(x2) if f(x) = 3x - 5.
(b) Understand that the domain is the set of possible inputs of a function and the range the possible outputs.
(c) Find the domain and range of common functions, particular quadratic and trigonometric.
(d) Reason about the range of functions defined in pieces (see Topic 10).
(e) Construct a function for a straight line based on a given domain/range.

## Autumn 2

#TopicResources
12 Inequalities (IGCSEFM/C1)
(a) Solve linear inequalities (as per GCSE).
(b) Reason about a modified inequality, e.g. find the smallest and largest value of x2 if -3 <= x <= 2.
(c) Solve quadratic inequalities, e.g. solve x2 - 4x - 5 < 0.
(d) [C1 only] Solve quadratic inequalities in the context of discriminants.
13 Differentiation (IGCSEFM/C1)
(a) Understand that dy/dx is the gradient function, and allows us to find the gradient of a curve for any value of x. Understand that dy/dx means 'the rate of change of y with respect to x'.
(b) Differentiate kxn where n is a positive integer or 0, and the sum of such functions.
(c) Find the equation of a tangent or normal of at any point on a curve.
(d) State the equations of tangents and normals to a curve at stationary points.
(e) Use differentiation to find stationary points on a curve: maxima, minima and points of inflection.
(f) Find the point(s) on a curve where the curve has a given gradient.
(g) Sketch a curve (e.g. a cubic) with known stationary points.

# Spring

## Autumn 1

#TopicResources
14 Simultaneous Equations (IGCSEFM/C1)
• Solve linear equations which may not be in the conventional ax + by = c form.
• Solve simultaneous equations where one is linear and one is quadratic.
15 Sequences (IGCSEFM)
• Find the nth term of both linear sequences (i.e. 'first difference' is constant) and quadratic sequences (i.e. 'second difference' is constant).
• Find the 'limiting value' of a sequence as n tends towards infinity.
16 Matrix Transformations (IGCSEFM)
• Be able to multiply matrices (only up to 2×2 in size).
• Know the (2×2) identity matrix I, where AI = IA = A.
• Determine the 2×2 matrix that represents 90/180 rotations, reflections in y = x, y = -x, y = k, x = k and enlargements.
• Be able to combine transformations, understanding that the matrix BA represents a transformation of A followed by a transformation of B.

## Autumn 2/Summer 1

#TopicResources
17 Proof
• Algebraic proofs, e.g. that some expression is divisible by a number.
• Understand that generic odd and even numbers can be represented respectively by 2n + 1 and 2n.
• Can apply and quote GCSE circle theorems.
• Can form angle proofs, i.e. a sequence of angle justifications with appropriate referencing of angles.
18 Algebraic Manipulations
Slides
19 Factor Theorem
• Understand that if we divide a polynomial f(x) by (x - a), the remainder is f(a).
• Therefore understand that if the remainder is 0, then (x-a) must be a factor of f(x).
• Be able to fully factorise a cubic.
20 Trigonometric Angles
• Sketch and use graphs of y = sin(x), y = cos(x), y = tan(x).
• Solve trigonometric equations, e.g. sin(x) = 2cos(x), finding all solutions within some range.
• Know how to find sin, cos and tan of 30, 45, 60, 90.
• Know the identities sin(x)/cos(x) = tan(x) and sin2(x) + cos2(x) = 1.
• Proof other identities using the above identities.
21 3D Pythagoras, Trigonometry, Sine/Cosine Rules
• Find the angle between a line and a plane.
• Find the angle between two planes.
• Find lengths within 3D shapes, e.g. height of a pyramid or internal diagonal of a cuboid.
• Use sine/cosine rules, including with sides that are algebraic or surds.
• Understand that bearings are measured clockwise from North.
• Know the area of a non-angled triangle is 1/2 ab sin C.