Expressions, Equations, and Identities

The big idea

Algebra involves four distinct types of mathematical object: expressions, equations, identities, and formulae. Each one calls for a different kind of mathematical activity. Knowing which object you are working with determines what you can and should do with it: simplify an expression, solve an equation, prove an identity, or use a formula.

Why this matters for secondary maths

Pupils are often taught to ‘do algebra’ without being told clearly what kind of algebraic object they are working with. This leads to characteristic errors: trying to ‘solve’ an expression (there is no equation to solve), trying to ‘find $x$ ’ in an identity (it is true for all values of $x$ ), or treating a formula as an equation with one answer when it describes a relationship between varying quantities.

The four algebraic objects are:

An expression is a combination of numbers, variables, and operations with no equals sign. For example: $3 x + 5$ . An expression represents a quantity that varies depending on the value of the variable. You can simplify it, evaluate it by substitution, or factor it; but you cannot ‘solve’ it, because there is no equation.

An equation is a statement that two expressions are equal, and it is true only for specific values of the variable. For example: $3 x + 5 = 20$ . The task is to find those specific values. This connects directly to the ‘unknown’ role of a variable (Concept Section 15).

An identity is a statement that two expressions are equal for all values of the variable. For example: $2 (x + 3) \equiv 2 x + 6$ . It is not ‘solved’ but verified or proved. The triple-bar symbol ≡ is sometimes used to distinguish identities from equations, though in practice the equals sign is often used for both. This connects to the ‘generalised number’ role of a variable (Concept Section 15) and to the Laws of Arithmetic (Concept Section 2), since identities are typically consequences of these laws.

A formula is a rule connecting two or more variables, expressing how one quantity depends on others. For example: $A = π r^{2}$ . The task is not to ‘solve’ the formula but to use it: substitute known values to find the unknown one, or rearrange it to make a different variable the subject. This connects to the ‘varying quantity’ role of a variable.

These distinctions give teachers precise language for what they are asking pupils to do. ‘Simplify this expression’, ‘solve this equation’, ‘show that this identity is true’, and ‘use this formula to find the area’ are four different mathematical activities. Pupils who understand why they are different, and which tools and techniques are appropriate for each, have a structural understanding of algebra that goes far beyond procedural competence.

Key representations

The balance model (for equations)

An equation is like a balance: both sides must be equal. When solving, any operation performed on one side must be performed on the other to maintain the balance. This representation was introduced in Concept Section 3 (Equality, Equivalence, and the Equals Sign) and is most directly relevant to the ‘equation’ category.

Strengths: Makes the logic of equation solving visible. Reinforces the relational meaning of ‘=’.

Limitation: Less helpful for expressions (no balance to maintain) or identities (the balance never tips because both sides are always equal). Pupils may not have experience of balances in real life, especially in the context of scales.

Substitution testing (for distinguishing equations from identities)

Substituting several values of $x$ into both sides of a statement is a practical way for pupils to investigate whether a statement is always true (identity) or only sometimes true (equation). For example: Is $2 (x + 3) = 2 x + 6$ always true? Substitute $x = 1$ : left side = 8, right side = 8. Substitute x = 10: left side = 26, right side = 26. Substitute $x = - 5$ : left side = −4, right side = −4. It appears to be always true; this is an identity.

Contrast: Is $2 x + 3 = 11$ always true? Substitute $x = 1$ : left = 5, right = 11. Not equal. Substitute $x = 4$ : left = 11, right = 11. Equal. This is an equation, true only when $x = 4$ .

Strengths: Empirical and accessible. Helps pupils see the difference between ‘sometimes true’ and ‘always true’.

Limitation: Testing specific values can suggest but never prove an identity. The proof comes from algebraic manipulation (e.g. expanding using the distributive law). This distinction, between evidence and proof, is itself a valuable teaching point.

Function machines and tables (for formulae and expressions)

A function machine takes an input (the variable) and produces an output (the expression’s value). This representation connects to Concept Section 15 and helps pupils see that an expression defines a process, while a formula defines a relationship. A table of input–output pairs makes the variation visible.

Sorting activities

Presenting pupils with a set of algebraic statements and asking them to sort into ‘expression’, ‘equation’, ‘identity’, and ‘formula’ is an effective representation of the classification itself. The discussion that arises from disagreements is often more valuable than the final sorting.

Worked examples

Example 1: Identifying the four types

Classify each of the following:

(a) $3 x + 7$ (Expression: no equals sign)

(b) $3 x + 7 = 22$ (Equation: true when $x = 5$ )

(d) $P = 2 l + 2 w$ (Formula: connects perimeter to length and width)

The classification depends on the structure, not on the letters used or the difficulty of the algebra. Pupils should be able to justify each classification: (a) has no equals sign; (b) has an equals sign and is true only for $x = 5$ ; (c) has an equals sign and is true for every $x$ (by the distributive law); (d) has an equals sign connecting specific named variables in a rule.

Example 2: The same algebra, different objects

Consider the expression $2 x + 6$ . This is an expression, it has no equals sign and represents a varying quantity.

Now consider $2 x + 6 = 14$ . This is an equation, we can solve it to find $x = 4$ .

Now consider $2 (x + 3) = 2 x + 6$ . This is an identity, the left side expands to give the right side for any $x$ .

The same algebraic components appear in all three, but the mathematical objects are different because the structural context is different.

Example 3: Why you cannot ‘solve’ an expression

A pupil writes: ‘ $3 x + 5 = ?$ ’ and asks ‘What is the answer?’

There is no answer because there is no equation. $3 x + 5$ is an expression: its value depends on $x$ . If $x = 2$ , the expression equals 11. If $x = 10$ , it equals 35. Without an equation (with an equals sign and something on the other side), there is nothing to solve. The most we can do is simplify (if possible) or evaluate for a given $x$ .

Example 4: Proving an identity versus solving an equation

Solving an equation: $4 x - 2 = 10$ . Add 2: $4 x = 12$ . Divide by 4: $x = 3$ . There is one answer.

Proving an identity: Show that $4 (x - 2) + 8 \equiv 4 x$ . Expand: $4 x - 8 + 8 = 4 x$ . Simplify: $4 x = 4 x$ . This is true for every $x$ . We have not ‘found $x$ ’; we have shown that the two expressions are equivalent forms of each other, using the distributive law (Concept Section 2) and the additive identity (Concept Section 4: −8 + 8 = 0).

The bridge to algebra

This section is itself the bridge: it organises the algebraic landscape that pupils enter from their arithmetic foundations.

Expressions grow from arithmetic calculations.

In arithmetic, 3 × 7 + 5 = 26 is a completed calculation. In algebra, $3 x + 5$ is a calculation waiting for a value. The arithmetic provides the structure; the variable makes it general. Pupils who understand how arithmetic calculations are built from operations and numbers (as developed throughout Concept Sections 1–14) are ready to see algebraic expressions as the same structures with letters replacing specific numbers.

Equations grow from missing-number problems.

In arithmetic: ‘What number, tripled and increased by 5, gives 20?’ In algebra: $3 x + 5 = 20$ . The equation is the formalisation of the question. Solving it uses inverse operations (Concept Section 11) applied in a structured way.

Identities grow from the Laws of Arithmetic.

In arithmetic: 4 × (10 + 3) = 4 × 10 + 4 × 3. This is true regardless of the specific numbers, because the distributive law guarantees it. In algebra: $a (b + c) = ab + a c$ . The identity makes the law explicit. Every identity pupils meet in secondary mathematics is, at its root, a consequence of the Laws of Arithmetic (Concept Section 2).

Formulae grow from contextual relationships.

In arithmetic: ‘A rectangle has sides 5 and 8. Its area is 40.’ In algebra: $A = lw$ . The formula captures the general relationship that holds for every rectangle, not just one specific example. This connects to the varying-quantity role of the variable (Concept Section 15) and to proportional reasoning (Concept Section 6).

Key vocabulary

Term	Definition
Expression	A combination of numbers, variables, and operations with no equals sign. For example: $3 x + 5$ , $2 a^{2} - 7 b$ , $4 (y + 1)$ .
Equation	A statement that two expressions are equal, true for specific values of the variable(s). For example: $3 x + 5 = 20$ .
Identity	A statement that two expressions are equal for all values of the variable(s). For example: $2 (x + 3) \equiv 2 x + 6$ . Sometimes indicated by the triple-bar symbol ≡.
Formula	A rule connecting two or more variables, expressing how one quantity depends on others. For example: $A = π r^{2}$ , $v = d / t$ .
Simplify	To rewrite an expression in a more compact or efficient form without changing its value. For example: $3 x + 5 x$ simplifies to $8 x$ .
Solve	To find the value(s) of the variable that make an equation true.
Prove (an identity)	To show algebraically that two expressions are equivalent for all values of the variable, typically by expanding, simplifying, or factorising.
Subject (of a formula)	The variable written alone on one side of the formula. In $A = π r^{2}$ , the subject is $A$ . Rearranging can change the subject.
Equivalent expressions	Expressions that produce the same value for every possible input. If two expressions are equivalent, the statement equating them is an identity.

What we don’t say

Avoid	Why	Say instead
‘Solve the expression’	Expressions cannot be solved because there is no equation. You can simplify, evaluate, or factorise an expression, but solving requires an equals sign and a statement to satisfy.	‘Simplify the expression’ or ‘Evaluate the expression when $x = 3$ .’
‘Find $x$ ’ (for identities)	In an identity, $x$ does not have a single value to find. The statement is true for all values. Asking pupils to ‘find $x$ ’ in $2 (x + 3) = 2 x + 6$ creates confusion.	‘Show that this identity is true for all values of $x$ ’ or ‘Prove that the left side is equivalent to the right side.’
‘Equation’ used loosely for any algebraic statement	Using ‘equation’ to mean any algebraic writing prevents pupils from distinguishing the four types. An expression is not an equation; an identity is not the same as an equation.	Use the precise term: ‘expression’, ‘equation’, ‘identity’, or ‘formula’, and explain why each is different.
‘Just expand and simplify’ (without stating the goal)	Expanding and simplifying are techniques, not goals. The goal depends on the object: proving an identity, simplifying an expression, or transforming an equation to solve it. Pupils need to know why they are expanding.	‘We are expanding to show that these two expressions are equivalent’ or ‘We are simplifying this expression so it is easier to work with.’

Common misconceptions and how to surface them

Misconception 1: Treating every algebraic statement as an equation to solve.

Pupils who see an equals sign and immediately try to ‘find $x$ ’ have not distinguished between equations and identities. Present $5 (x + 1) = 5 x + 5$ and ask: ‘Can you find the value of $x$ ?’ A pupil who says $x = 0$ , or tries several values and gets confused, has not recognised the identity. Follow up: ‘Try $x = 0$ . Now try $x = 3$ . Now try $x = 100$ . What do you notice?’

Misconception 2: Thinking an expression is ‘unfinished’ because it has no equals sign.

Pupils sometimes feel that $3 x + 5$ is incomplete, that there should be an answer. They may write $3 x + 5 = 3 x + 5$ or try to ‘simplify’ it to a single number. Ask: ‘What is $3 x + 5$ when $x = 2$ ? When $x = 10$ ?’ Help them see that an expression is a complete mathematical object whose value depends on the variable.

Misconception 3: Confusing a formula with an equation.

In A = πr², a pupil might ask ‘What is A?’ as though there is a single answer. But A depends on r: different values of r give different values of A. The formula is a relationship, not a single-answer equation. Ask: ‘If r = 3, what is A? If r = 5, what is A? Does A always have the same value?’

Misconception 4: Believing that an identity can be ‘proved’ by verifying specific examples.

If a pupil finds that both sides of $2 (x + 3) = 2 x + 6$ give the same result for $x = 4$ , they may say ‘it’s true because it works for 4.’ If they then try $x = - 1$ and it still works, they may say ‘it works for everything I’ve tried, so it must be an identity.’ The teaching point: examples can support but not prove an identity; disproof requires only one counter-example. For identities, proof comes from algebraic manipulation (using the distributive law, Concept Section 2).

Diagnostic questions

Question 1: Sort the following into expressions, equations, identities, and formulae.

(a) $5 x - 3$

(b) $5 x - 3 = 17$

(d) $d = s t$

What this reveals: (a) Expression: no equals sign. (b) Equation: true only when $x = 4$ . (c) Identity: expanding gives $5 x - 5 + 5 = 5 x$ , which is $5 x = 5 x$ , true for all $x$ . (d) Formula: a rule connecting distance, speed, and time.

A pupil who can classify all four and justify each classification demonstrates structural understanding. A pupil who classifies (c) as an equation ‘because it has an equals sign’ has not grasped the equation–identity distinction.

Question 2: A pupil says: ‘I tried $x = 2$ and both sides of $3 (x + 4) = 3 x + 12$ gave 18, so $x = 2$ .’ What would you say to this pupil?

What this reveals: The pupil has verified one case of an identity and reported it as if it were a solution to an equation. A structurally fluent response would be: ‘It’s true for $x = 2$ , but it’s also true for every other value of $x$ . This is an identity, not an equation. We can prove it by expanding the left side using the distributive law: $3 (x + 4) = 3 x + 12$ .’ This question reveals whether pupils understand the difference between finding a solution and recognising a general truth.

Progression spine

Stage	Key ideas	Notes
Primary (Y5–Y6)	Simple equations presented as missing-number problems. Expressing relationships in words (e.g. ‘the perimeter is twice the length plus twice the width’). Substituting into simple formulae given in words or simple notation.	Pupils work with equations and formulae informally. The distinction between expressions and equations, or between equations and identities, is not yet made explicit.
Year 7	Explicit classification of expressions, equations, identities, and formulae. Writing and simplifying expressions. Setting up and solving simple equations. Verifying identities by substitution and by algebraic expansion. Using formulae from context.	The four-way classification should be introduced early and revisited whenever pupils meet a new algebraic statement. Sorting activities are particularly effective.
Years 8–11	Solving increasingly complex equations (linear, quadratic, simultaneous). Proving identities algebraically. Rearranging formulae to change the subject. Working with expressions in factorised and expanded forms. Distinguishing identities from equations in exam contexts. Algebraic proof.	The classification becomes a tool for strategic thinking: ‘Is this an equation or an identity? That determines what I should do with it.’ By KS4, pupils should be able to classify any algebraic statement and explain their reasoning.