Below you will find a comprehensive list of all the GCSE Maths foundation topics for Algebra you should revise if you are taking the foundation tier exams. Ensure that you revise all of the GCSE Maths foundation topics which are highlighted at the top of this page and practice using the Maths Made Easy GCSE Maths Revision page where you will get access to loads of exceptional resources.

Algebra Foundation Topics

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) 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 binomials
    (x –1)(x – 2) = x² – 3x + 2
    (a + 2b)(a – b) = a² + ab – 2b²
  • factorising quadratic expressions:
    x2 + 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
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

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;

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

12. Recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function y=1/x, with x ≠ 0;

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

13. 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.

14. Solve linear equations in one unknown algebraically (including those with the unknown on both sides of the equation);

Find approximate solutions using a graph

15. Solve quadratic equations like:
x2 + bx + c, (NOT including those that require rearrangement) algebraically by factorising;

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

16. Solve two simultaneous linear equations in two variables algebraically;

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

 

Find approximate solutions using a graph

17. 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

 

18. Solve linear inequalities in one variable;
e.g. Solve 3x – 1 = 5

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

 

 

19. 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

 

 

20. 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)

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

21. 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