Number - Addition, Subtraction, Multiplication and Division

Ensure pupils have thoroughly understood short multiplication (Year 5) before introducing the two-row long multiplication format. Use place value understanding to explain why the second partial product row is shifted one place to the left (multiplying by tens). Grid multiplication provides an effective bridge between informal methods and the formal algorithm: show how the four cells of the grid correspond to the four partial products in long multiplication. Estimation should always precede formal calculation. Regular practice with varied digit configurations (including zeros within numbers) builds fluency.

The most common error in long multiplication is forgetting to shift the second row of partial products one place to the left when multiplying by the tens digit. Pupils also make errors with carrying, particularly when sums exceed 10 in multiple columns simultaneously. Some pupils misalign digits in the answer, especially when the number contains zeros. Estimation before calculating helps pupils identify unreasonable answers.

Upper primary maths (Y6) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.

Long division is one of the most cognitively demanding written methods in the primary curriculum and should be introduced only after pupils are completely secure with short division. Begin with divisors from the times tables (e.g., 13, 15, 25) before progressing to less familiar divisors. The 'chunk' method (subtracting known multiples) can serve as a bridge to the formal algorithm for pupils who struggle. Explicitly teach the four-step cycle: estimate how many times the divisor goes into the partial dividend, write the quotient digit, multiply and write the product below, subtract. Practise identifying from the context whether to express the remainder as r___, as a fraction, or as a decimal.

Pupils frequently lose track of place value during long division, producing quotient digits in the wrong columns. A common error is not bringing down the next digit when the partial dividend is smaller than the divisor (the quotient digit at that step should be zero, which is often omitted). Pupils also confuse when to express the remainder as a fraction (½ of a pizza) versus rounding up (number of buses needed) — always connect to the problem context.

Upper primary maths (Y6) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.

Introduce through concrete examples that demonstrate the need for an agreed convention: show that 3 + 4 × 5 can be evaluated as either 35 or 23 without a rule, then establish that mathematicians have agreed on the order of operations to avoid ambiguity. Use the acronym BODMAS (Brackets, Other, Division, Multiplication, Addition, Subtraction) or BIDMAS as a memory aid but emphasise understanding over rote application. Provide exercises where pupils must insert brackets to produce a given answer (e.g., make 3 + 4 × 5 equal 35 by inserting brackets). Connect to calculator use: demonstrate that most calculators apply the order of operations automatically.

The most prevalent misconception is evaluating left-to-right without applying the order of operations (treating all operations as having equal priority). Pupils also commonly neglect to evaluate the contents of brackets first when brackets are nested or when brackets appear later in the expression. Some pupils misremember BODMAS as meaning division always before multiplication, when in fact they have equal priority and should be evaluated left to right.

Upper primary maths (Y6) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.

Build on Year 5 work on factors, multiples and primes by introducing the terms 'common factor', 'common multiple', 'highest common factor' and 'lowest common multiple'. Use Venn diagrams to organise the factors or multiples of two numbers visually, identifying the overlapping region as the common factors/multiples. Connect explicitly to fraction simplification (divide numerator and denominator by their highest common factor) and finding common denominators (use the lowest common multiple of the two denominators). Divisibility rules (by 2, 3, 5, 7, 9, 10, 11) help pupils work efficiently without listing all factors.

Pupils often confuse factors and multiples, and further confuse highest common factor (a factor, smaller than or equal to the number) with lowest common multiple (a multiple, larger than or equal to the number). Some pupils include 1 as a prime number (it is not, as it has only one factor). When finding common factors, pupils often stop at the first common factor they find rather than the highest. Systematic listing of all factors of both numbers into a Venn diagram prevents most of these errors.

Upper primary maths (Y6) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.

Mathematical fluency is developed through regular, varied practice rather than single-topic drills. Use mixed-fact retrieval exercises, timed challenges (with the emphasis on accuracy as well as speed), and application of facts in problem-solving contexts where fluent recall enables higher-level thinking. Regularly revisit all four operations and their inverses, including in contexts involving large numbers, negative numbers and non-integer values. Distinguish fluency (doing the right thing quickly and accurately) from speed alone: a pupil who makes errors quickly is not fluent. Connect fluency to estimation: a fluent pupil can also rapidly assess the approximate size of an answer.

Pupils and teachers sometimes confuse fluency with speed, neglecting accuracy. Some pupils develop apparent fluency through pattern-matching without conceptual understanding, leading to errors when the surface features of a problem change. The most damaging gap is in multiplication fact recall: without secure times-table knowledge, all four operations (including division, fractions and percentages) are impeded at every stage.

Upper primary maths (Y6) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.

Teach problem-solving strategies explicitly and by name so that pupils can consciously select and apply them. Use rich problems from a variety of contexts, including problems with more than one valid approach. Encourage pupils to articulate their strategy before beginning and to evaluate their strategy after completing the solution. Collaborative problem-solving, in which pupils explain their approaches to each other, develops both mathematical communication and metacognitive awareness. Include problems that have no solution, problems with multiple solutions, and problems where the initial approach does not work and pupils must try a different strategy — this builds the resilience and flexibility that distinguish strong mathematical problem-solvers.

Pupils often attempt to apply a recently learned procedure to a new problem without first reading to understand what the problem is asking. A very common error is computing an answer without checking whether it is reasonable in context (e.g., a negative number of people, or a fraction of a journey taking more time than the whole journey). Some pupils give up quickly on problems that do not yield to their first approach, rather than trying a different strategy. Regular problem-solving with deliberate strategy discussion and reflection builds the persistence required.

Upper primary maths (Y6) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.