Notation, vocabulary and manipulation: 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

The idea

An equation such as 3x + 5 = 20 is a conditional statement: it is true only for particular values of x, here x = 5. An identity such as 3(x + 2) ≡ 3x + 6 is true for every value of x; the ≡ symbol signals this. To argue equivalence, expand brackets, collect like terms, or factorise until both expressions match exactly. For example, show 2(x + 3) + 4x and 6x + 6 are equivalent by expanding the first to 2x + 6 + 4x, then collecting to 6x + 6. You can also test several values, but a general algebraic argument is stronger. Use algebra to support arguments by forming expressions for unknown quantities, then manipulating them to justify a claim, such as proving a sum is always even.

What earns the marks

• Distinguishes an equation (true for specific values) from an identity (true for all values), using the ≡ symbol correctly.
• Expands brackets correctly, for example 3(x + 2) becomes 3x + 6.
• Collects like terms accurately to simplify expressions, for example 2x + 6 + 4x becomes 6x + 6.
• Constructs a general algebraic argument, such as representing consecutive integers as n and n + 1 and showing their sum is 2n + 1.
• Recognises that testing numerical values alone does not prove an identity for all values.

Common mistakes and how to put them right

• Treating an identity as an equation and solving it for a single value.

  Put it right: Recognise that an identity must be shown true for all values, often by algebraic simplification.

• Assuming two expressions are equivalent after checking only one or two values.

  Put it right: Provide a general algebraic argument that covers every possible value.

• Expanding a bracket incorrectly, for example writing 3(x + 2) as 3x + 2.

  Put it right: Multiply every term inside the bracket by the factor outside.

• Misreading the ≡ symbol as an ordinary equals sign and attempting to solve.

  Put it right: Interpret ≡ as indicating an identity that holds for all values.

Exam tips

• In multiple-choice questions, test each option by substituting a simple value such as x = 0 or x = 1, but remember that a single counterexample can disprove an identity while a proof needs algebra.
• When asked to show equivalence, start from one expression and manipulate it step by step to reach the other, showing each algebraic stage clearly.
• Use the correct symbol: = for equations that are conditional, ≡ for identities that are always true.
• If a question asks you to argue mathematically, write a short chain of reasoning using algebra rather than just numerical examples.