GCSE Maths Revision List – Algebra Higher

Below you will find a comprehensive list of all the GCSE Maths Higher Tier Algebra topics you will need to revise for AQA, Edexcel and OCR exam boards. Ensure that you revise all of the GCSE Maths Algebra topics and practice using the Maths Made Easy GCSE Maths Revision page where you can fully revise the 9-1 GCSE Maths course.

ALGEBRA:H

1. Use and interpret algebraic notation, including:

  • ab in place of a × b
  • 3y in place of y + y + y and 3 × y
  • a2 in place of a × a
  • a3 in place of a × a × a; a2b in place of a × a × b
  • fraction-r in place of a ÷ b
  • coefficients written as fractions rather than as decimals
  • brackets

2. Substitute numerical values into formulae and expressions, including scientific formulae

e.g. Given that v = u + at,
find v when t = 1, a = 2 and u = 7

v = √(u² + 2as)

with u = 2.1, s = 0.18, a = -9.8.

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:

  • collecting like terms
    2a + 3a = 5a
  • multiplying a single term over a bracket
    2(a + 3b) = 2a + 6b
    2(a + 3b) + 3(a – 2b) = 5a
  • taking out common factors(factorise)
    3a – 9b = 3(a – 3b)
    2x + 3x² = x(2 + 3x)
  • expanding products of two or more binomials
    (x –1)(x – 2) = x² – 3x + 2
    (a + 2b)(a – b) = a² + ab – 2b²
  • factorising quadratic expressions:
    ax2 + bx + c, and the difference of two squares
    x² – x – 6 = (x – 3)(x + 2)
    x² – 16 = (x – 4)(x + 4)
    x² – 3 = (x – √3)(x + √3)
  • simplifying expressions involving sums, products and powers, including the laws of indices
    a × a × a = a³ ; 2a × 2b = 4ab
    a³ × a² = a5 ; 3a3 ÷ a = 3a³
    y2 × y5 = y7 ; y8 ÷ y3 = y5
    (y2)4 = y×y × y×y × y×y × y×y = y8

5. Understand and use standard mathematical formulae;

Circumference circle: 2πr = πd
Area circle: πr2
Pythagoras’ theorem: a2 = b2 + c2

Trigonometry formulae:

Rearrange formulae to change the subject

1. where the subject appears once only.
Make d the subject of the formula
c = πd.
Make x the subject of the formula
y = 3x – 2.

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)
e.g. 3a + 2a ≡ 5a; x2 + x2 ≡ 2x2

Argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs
e.g. show that (x + 1)² + 2 = x² + 2x + 3

7. Where appropriate, interpret simple expressions as functions with inputs and outputs.

e.g. y = 2x + 3 as

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 straight-line 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:
e.g. y = 2x + 3; y = 2x2 + 1

y = x3 – 2x ; y=1/x + x

10. Identify and interpret gradients and intercepts of linear functions graphically and algebraically

y = 2x + 1 has gradient = 2, intercept = 1
y = 3 – x has gradient = –1, intercept = 3

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;

e.g. y = 2 ; x = 1 ; y = 2x ; y=x2
y = x3 – 2x ;

exponential functions y=kx 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 non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration.

i.e. distance-time, money conversion, temperature conversion

Straight line gradients = rates of change.
Gradient distance-time graph = velocity.

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

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:
x2 + bx + c, (including those that require rearrangement) algebraically by factorising; by completing the square and by using the quadratic formula;

Find approximate solutions using a graph

Solve x2 – 5x + 6 = 0,

Find x for an x cm by (x + 1)cm rectangle of area 42cm2

19. Solve two simultaneous linear equations in two variables linear/linear or linear/quadratic algebraically;

e.g. Solve simultaneously
2x + 3y = 18 and
y = 3x – 5

 

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;

e.g. Cost of car hire at £50 per day plus 10p per mile. The perimeter of a rectangle when the length is 2 cm more than the width.

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.
e.g. Solve 3x – 1 = 5

Represent the solution set on a number line, using set notation and on a graph.
e.g 2x + 1 ≥ 7 and 1 < 3x – 5 ≤ 10

23. Generate terms of a sequence from either a term-to-term or a position-to-term rule

Continue the sequences
1, 4, 7, 10, …
1, 4, 9, 16, …
3, 4, 5, … n + 2

24. Recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions ( rn where n is an integer, and r is a rational number > 0 or a surd) and other sequences

Triangular numbers: 3, 6, 10, 15,
Square numbers: 1, 4, 9, 16, 25,
Cube numbers: 1, 27, 81, 256,

Fibonacci: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34
The next number is found by adding up the two numbers before it.
The Rule is xn = xn –1 + xn–2

25. Deduce expressions to calculate the nth term of linear sequence.

e.g. nth term = n2 + 2n gives 3, 8, 15, …

 

Find a formula for the nth term of an arithmetic sequence.

3, 7, 11, 14… 4n – 1
40, 37, 34, 31… 43 – 3n