Here is a suggested scheme of learning for Further Mathematics GCSE. This scheme of learning is not exhaustive but covers concepts that are not covered by the GCSE syllabus and enables classes to learn the concepts required in approximately an hour a week throughout Year 11.



  • Multiply matrices up to and including 2 x 2 by 2 x 2.
  • Understand that, in general, matrix multiplication is not commutative
  • Understand that matrix multiplication is associative
  • Understand that AI = IA = A
  • Work out the image of any vertex of the unit square given the matrix operator
  • Work out or recall the matrix operator for a given transformation
  • Understand that the matrix product PQ represents the transformation with matrix Q followed by the transformation with matrix P
  • Work out the matrix which represents a combined transformation



  • Understand the concepts of gradient of a curve and rates of change
  • Work out the gradients of curves and rates of change
  • State the gradient of a curve at a point given the gradient or equation of the tangent at that point
  • Find the derivative of polynomials
  • Simplify expressions before differentiating if necessary
  • Understand the meaning of increasing and decreasing functions
  • Understand the meaning of maximum points, minimum points and points of inflection
  • Prove whether a stationary point is a maximum, minimum or point of inflection



  • Sketch and use sin, cos and tan graphs to solve problems
  • Use the identities to simplify expressions
  • Use the identities to prove other identities
  • Use the identities in solution of equations
  • Work out all solutions in a given interval

Factor Theorem

  • Understand and use the factor theorem to factorise polynomials up to and including cubics
  • Find integer roots of polynomial equations up to and including cubics
  • Show that x – a is a factor of a function f(x) by checking f(a) = 0
  • Solve cubic equations where at least one of the roots is an integer



  • Work out the limiting value for a given sequence or for a given nth term as n approaches infinity

Equations of Circles

  • Recognise the equation of a circle, centre (a, b), radius r
  • Write down the equation of a circle given centre (a, b) and radius
  • Work out coordinates of points of intersection of a given circle and a given straight line