Laws of Arithmetic

The big idea

Three laws, commutativity, associativity, and distributivity, govern how the four operations behave. These laws explain why calculation methods work, when calculations can be reorganised, and what algebra is built on. They are the grammar of the entire framework.

Why this matters for secondary maths

Pupils use these laws informally every time they calculate. When a pupil works out $18 \times 5$ as ( $20 - 2) \times 5 = 100 - 10 = 90$ , they are applying the distributive law. When they recognise that $7 \times 8 = 8 \times 7$ , they are using commutativity. When they regroup $25 \times 4 \times 7$ as $(25 \times 4) \times 7 = 100 \times 7$ , they are using associativity.

The problem is that these laws are typically left unnamed and unexamined. Pupils apply them but cannot articulate what they are doing or why it works. This matters because algebra demands the same reasoning in symbolic form: collecting like terms ( $3 x + 5 x = (3 + 5) x$ uses distributivity), expanding brackets ( $4 (x + 2) = 4 x + 8$ uses distributivity), and rearranging ( $a + b = b + a$ uses commutativity). Without explicit knowledge of the laws, algebra becomes a sequence of unexplained moves.

Making the laws explicit also helps pupils distinguish between legal and illegal moves. Knowing that subtraction is not commutative (7 − 3 ≠ 3 − 7) prevents errors that arise from careless rearrangement. Knowing that multiplication distributes over addition but addition does not distribute over multiplication prevents the common error of writing $(a + b)^{2} = a^{2} + b^{2}$ .

Key representations

The area model (for the distributive law)

A rectangle partitioned into sub-rectangles makes the distributive law visible: $a (b + c) = ab + a c$ is shown by a rectangle of height $a$ and width $(b + c)$ , split into two parts. This is the primary visual representation and is expanded in Concept 7, The Area Mode. When exploring the distributive law, the distributive property tool might be helpful.

Number line and bar models (for commutativity and associativity)

A number line can show that 3 + 5 and 5 + 3 both reach 8, regardless of the order of jumps. Bar models can show regrouping: three bars of length $a, b, c$ can be combined as $(a + b) + c$ or $a + (b + c)$ without changing the total.

Counterexamples (for non-commutative and non-associative operations)

Equally important are the representations that show where the laws do not hold. A number line showing 7 − 3 = 4 and 3 − 7 = −4 makes the non-commutativity of subtraction concrete. These counterexamples are essential teaching tools.

Worked examples

Example 1: Commutativity in practice

Calculate 4 + 37. Rewriting as 37 + 4 = 41 is easier to compute mentally. The commutative law guarantees this gives the same result.

Contrast: 10 − 3 = 7, but 3 − 10 = −7. Subtraction is not commutative. The operation and the order both matter.

Example 2: Associativity for efficiency

Calculate 25 × 7 × 4.

Regrouping: (25 × 4) × 7 = 100 × 7 = 700. The associative law allows this regrouping because the result of multiplication does not depend on how the factors are grouped.

Example 3: The distributive law for mental calculation

Calculate 7 × 98.

Rewrite: 7 × (100 − 2) = 700 − 14 = 686. The distributive law converts a harder product into an easier subtraction.

Example 4: The distributive law for collecting

Why is $3 \times 8 + 5 \times 8 = 8 \times 8$ ?

Because $3 \times 8 + 5 \times 8 = (3 + 5) \times 8 = 8 \times 8 = 64$ . The distributive law allows us to ‘factor out’ the common 8. This is collecting like terms in its arithmetic form.

The bridge to algebra

Every use of the laws in arithmetic has a direct algebraic parallel.

Commutativity: $a + b = b + a$ . When pupils rearrange terms in an expression (moving $3 x$ to the front, for example), they are using commutativity.

Associativity: $(a + b) + c = a + (b + c)$ . When pupils remove brackets in an addition, they are using associativity.

Distributivity - collecting like terms:

$3 x + 5 x = (3 + 5) x = 8 x$

This is the distributive law in reverse (‘factoring out’ $x$ ). It is the same reasoning as $3 \times 8 + 5 \times 8 = (3 + 5) \times 8$ .

Distributivity - expanding brackets:

$4 (x + 2) = 4 x + 8$

This is the distributive law applied directly. It is the same reasoning as $4 \times 12 = 4 \times 10 + 4 \times 2$ .

Knowing when a law does NOT apply:

The laws also tell us what we cannot do. Addition does not distribute over multiplication: $(a + b) \times c \neq = (a \times c) + (b \times c)$ is true, but $(a + b)^{2} \neq = a^{2} + b^{2}$ . Understanding which laws hold and which do not is what separates structural understanding from rote manipulation.

Key vocabulary

Term	Definition
Commutative law	For addition and multiplication, the order of the numbers does not change the result: $a + b = b + a$ and $ab = ba$ .
Associative law	For addition and multiplication, the grouping does not change the result: $(a + b) + c = a + (b + c)$ and $(ab) c = a (b c)$ .
Distributive law	Multiplication distributes over addition and subtraction: $a (b + c) = ab + a c$ and $a (b - c) = ab - a c$ .
Priority of operations	The convention that multiplication and division are performed before addition and subtraction, and that brackets override this. Arises from mathematical structure, not arbitrary rules.
Identity	The element that leaves a number unchanged under an operation: 0 for addition, 1 for multiplication.

What we don’t say

Avoid	Why	Say instead
‘BIDMAS says we do multiplication first’	BIDMAS is a mnemonic, not a mathematical reason. It implies division always precedes multiplication, and addition always precedes subtraction, both false. Multiplication and division have equal priority (left to right), as do addition and subtraction.	‘Multiplication is performed before addition because a product is a single quantity. We can use brackets to change the order if we need to.’
‘You can always swap the numbers around’	Only true for addition and multiplication (commutativity). Pupils who believe this applies to subtraction and division will make errors: 5 − 3 ≠ 3 − 5.	‘You can swap the order in addition and multiplication because they are commutative. Subtraction and division are not commutative, so order matters.’
‘Brackets first, then everything else’	Brackets are not an operation; they are a grouping device. Saying ‘brackets first’ treats them as something to ‘do’ rather than as a structural indicator.	‘Work out what is inside the brackets first, because the brackets tell us that this quantity is treated as a single unit.’
‘Multiply everything out’ (without naming the law)	Expansion works because of the distributive law. If pupils do not know this, they cannot distinguish legal from illegal expansions.	‘We use the distributive law to expand: every term inside the bracket is multiplied by the term outside.’

Common misconceptions and how to surface them

Misconception 1: All operations are commutative.

Pupils often assume that because 3 + 5 = 5 + 3, it must also be true that 5 − 3 = 3 − 5. Present both calculations and discuss the results. Then ask: ‘Is 12 ÷ 4 the same as 4 ÷ 12?’ Building a clear mental map of which operations are commutative and which are not is essential.

Misconception 2: BIDMAS as a rigid linear sequence.

Pupils often evaluate 8 − 3 + 2 as 8 − 5 = 3, because they think addition comes ‘before’ subtraction in BIDMAS. The correct answer is 8 − 3 + 2 = 7 (left to right, because addition and subtraction have equal priority). Ask: ‘Does it matter which order we do 8 − 3 + 2? Why?’

Misconception 3: Distributing incorrectly with subtraction.

The expression $5 (x - 3)$ is often expanded as $5 x - 3$ rather than $5 x - 15$ . The error arises because the pupil does not distribute the 5 to both terms. Use the area model: a rectangle with height 5 and width $(x - 3)$ has two regions, $5 x$ and $- 5 \times 3 = - 15$ .

Misconception 4: Treating brackets as decoration.

In $2 (x + 3)$ , some pupils ignore the brackets and write $2 x + 3$ . They do not see the brackets as indicating that $(x + 3)$ is a single quantity being multiplied by 2. The area model helps: the bracket defines one complete side of the rectangle.

Diagnostic questions

Question 1: Without calculating, explain why $6 \times 17 + 4 \times 17 = 10 \times 17$ .

What this reveals: A pupil who can explain this using the distributive law (‘six 17s plus four 17s makes ten 17s’, or ‘ $(6 + 4) \times 17 = 10 \times 17$ ’) demonstrates structural understanding. A pupil who calculates both sides separately and checks they match has procedural verification but not structural reasoning.

Question 2: Is this true or false: $8 + 5 \times 2 = 26$ ? Explain.

What this reveals: The answer is false ( $8 + 5 \times 2 = 8 + 10 = 18$ , not $(8 + 5) \times 2 = 26$ ). A pupil who gets 26 is applying operations left to right without regard to priority. A pupil who can explain why multiplication takes priority, because 5 × 2 forms a single product before it is combined with 8, understands the structural reason, not just the rule.

Progression spine

Stage	Key ideas	Notes
Primary (Y1–Y6)	Informal use of commutativity and the distributive law in mental calculation. Early Priority of Operations (often framed as BIDMAS). Some grouping and partitioning strategies.	Pupils use the laws without naming them. Making them explicit in Year 7 gives pupils the language to articulate what they already do.
Year 7	Explicit naming and exploration of the three laws. Counterexamples for non-commutative/non-associative operations. Priority of operations as structural, not mnemonic. Connection to area model.	The laws should be presented as the ‘grammar’ that underpins everything else in the framework.
Years 8–11	Collecting like terms (distributive). Expanding brackets (distributive). Factorising (distributive in reverse). Index laws (extensions of associativity). Algebraic proof (justifying steps by naming the law).	By KS4, pupils should be able to justify algebraic steps by reference to the laws. This is the foundation for algebraic proof.