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,

underlinedandboldtype 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**