GCSE (91) Higher Mathematics is broken down into six main areas:
ALGEBRA:H 

1. Use and interpret algebraic notation, including:

2. Substitute numerical values into formulae and expressions, including scientific formulae 
3. Understand and use the concepts and vocabulary of expressions, equations, inequalities, terms and factors 
4. Simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by:

5. Understand and use standard mathematical formulae; Trigonometry formulae Rearrange formulae to change the subject 
ALGEBRA:H 

6. Know the difference between an equation and an identity; An Identity is an equation that is true for all values (use the ≡ identical symbol) 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' 
8. Work with coordinates in all four quadrants. 
9. Plot graphs of equations that correspond to straightline graphs in the coordinate plane; Use the form y = mx + c to identify parallel lines and perpendicular lines Find the equation of the line through two given points, or through one point with a given gradient Use a table of values to plot graphs: 
10. Identify and interpret gradients and intercepts of linear functions graphically and algebraically 
11. Identify and interpret roots, intercepts, turning points (stationary 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 , 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 graphs of nonstandard 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 nonlinear graphs), and interpret results in cases such as distancetime graphs, velocitytime 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 
17. Solve linear equations in one unknown algebraically (including those with the unknown on both sides of the equation); Find approximate solutions using a graph 
18. Solve quadratic equations like: Find approximate solutions using a graph Solve x^{2} – 5x + 6 = 0, Find x for an x cm by (x + 1)cm rectangle of area 42cm^{2} 
19. Solve two simultaneous linear 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 
21. Translate simple situations or procedures into algebraic expressions or formulae; Derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution 
22. Solve linear inequalities in one or two variable; and quadratic inequalities in one variable. Represent the solution set on a number line, using set notation and on a graph. 
23. Generate terms of a sequence from either a termtoterm or a positiontoterm rule 
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 sequence. Find a formula for the nth term of an arithmetic sequence. 