# GCSE: Higher level content

Reformed GCSE subject content includes three types of content: standard, underlined and bold. In the words of he document,

The expectation is that:

• All students will develop confidence and competence with the content identified by standard type
• All students will be assessed on the content identified by the standard and the underlined [here, for technical reasons, emphasised — AB] type; more highly attaining students will develop confidence and competence with all of this content
• Only the more highly attaining students will be assessed on the content identified by bold type. The highest attaining students will develop confidence and competence with the bold content.

The distinction between standard, underlined and bold type applies to the content statements only, not to the assessment objectives or to the mathematical formulae in the appendix.

What follows is the list of items in the  Mathematics GCSE subject content and assessment objectives which contain bold type, higher content.I think this short lists clearly marks the boundaries of GCSE — AB

Number

5. apply systematic listing strategies including use of the product rule for counting
6. use positive integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5; estimate powers and roots of any given positive number
7. calculate with roots, and with integer and fractional indices
8. calculate exactly with fractions, surds and multiples of π; simplify surd expressions involving squares […] and rationalise denominators

10. work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.375 or 3/8 ); change recurring decimals into their corresponding fractions and vice versa

16. apply and interpret limits of accuracy, including upper and lower bounds

Algebra

4. simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by: […]

expanding products of two or more binomials
factorising quadratic expressions of the form x^2 + bx + c, including the difference of two squares; factorising quadratic expressions of the form ax^2 + bx+ c

6. know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs

7. where appropriate, interpret simple expressions as functions with inputs and outputs; interpret the reverse process as the ‘inverse function’; interpret the succession of two functions as a ‘composite function’.

9. plot graphs of equations that correspond to straight-line graphs in the coordinate plane; use the form y = mx + c to identify parallel and perpendicular lines; find the equation of the line through two given points, or through one point with a given gradient

11. identify and interpret roots, intercepts, turning points of quadratic functions graphically; deduce roots algebraically and turning points by completing the square
12. recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function y = 1/x with x ≠ 0, exponential functions y = k^x for positive values of k, and the trigonometric functions (with arguments in degrees) y = sin x , y = cos x and y = tan x for angles of any size
13. sketch translations and reflections of a given function
14. plot and interpret graphs (including reciprocal graphs and exponential graphs) and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration

15. calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts
16. recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point.

18. solve quadratic equations (including those that require rearrangement) algebraically by factorising, by completing the square and by using the quadratic formula; find approximate solutions using a graph
19. solve two simultaneous equations in two variables (linear/linear or linear/quadratic) algebraically; find approximate solutions using a graph
20. find approximate solutions to equations numerically using iteration

22. solve linear inequalities in one or two variable(s), and quadratic inequalities in one variable; represent the solution set on a number line, using set notation and on a graph

24. recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions ( r n where n is an integer, and r is a rational number > 0 or a surd) and other sequences
25. deduce expressions to calculate the nth term of linear and quadratic sequences

Ratio, proportion and rates of change

13. understand that X is inversely proportional to Y is equivalent to X is proportional to 1/Y; construct and interpret equations that describe direct and inverse proportion
14. interpret the gradient of a straight line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion
15. interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of average and instantaneous rate of change (gradients of chords and tangents) in numerical, algebraic and graphical contexts
16. set up, solve and interpret the answers in growth and decay problems, including compound interest and work with general iterative processes.

Geometry and measures

7. identify, describe and construct congruent and similar shapes, including on coordinate axes, by considering rotation, reflection, translation and enlargement (including fractional and negative scale factors)

8. describe the changes and invariance achieved by combinations of rotations, reflections and translations

10. apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results

22. know and apply the sine rule, and cosine rule,  to find unknown lengths and angles
23. know and apply Area = 1/2 ab sin C to calculate the area, sides or angles of any triangle.

25. apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors; use vectors to construct geometric arguments and proofs

Probability

9. calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams

Statistics

3. construct and interpret diagrams for grouped discrete data and continuous data, i.e. histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use