Multiplication and Division

Status: Mandatory

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Concepts

This study delivers 5 primary concepts and 0 secondary concepts.

Primary concept: Prime numbers and composite numbers (MA-Y5-C003)

Type: Knowledge | Teaching weight: 3/6

A prime number has exactly two distinct factors: 1 and itself (2, 3, 5, 7, 11, 13, 17, 19...). A composite number has more than two factors. The number 1 is neither prime nor composite. Mastery means pupils can identify whether any number up to 100 is prime, recall primes up to 19, and explain why 1 is excluded from the definition of prime.

Teaching guidance: Use the Sieve of Eratosthenes: start with numbers 2-100, cross out all multiples of 2 (not 2 itself), then all multiples of 3, 5, 7 — the remaining numbers are prime. This systematic approach builds conceptual understanding. Connect to factors: a prime number has only one factor pair (1 and itself). The fact that there are infinitely many primes can be discussed as a fascinating mathematical result. Practise rapid identification of primeness for numbers under 20. Key vocabulary: prime number, composite number, factor, factor pair, divisible, Sieve of Eratosthenes, 1 is not prime Common misconceptions: Pupils frequently include 1 as a prime number (it has only one factor, not two). They may think all odd numbers are prime (9 = 3 × 3 is odd but not prime). The number 2 is the only even prime, which surprises pupils who think 'all primes are odd'. Large-number primeness testing requires systematic factor checking.

Differentiation

Model response (Entry): Factors of 12: 1, 2, 3, 4, 6, 12 — not prime (more than 2 factors). Factors of 13: 1, 13 — prime (exactly 2 factors).

Model response (Developing): Yes, 2 is prime — its only factors are 1 and 2. It is the only even prime because every other even number has 2 as a factor (in addition to 1 and itself), giving it more than 2 factors.

Model response (Expected): 51 is not prime: 51 = 3 × 17. I checked divisibility by 2 (no — it's odd), then 3 (5+1=6, divisible by 3). 1 is not prime because it has only one factor (1 itself), and primes must have exactly two factors.

Representation stages (CPA)

Primary concept: Square numbers and cube numbers (MA-Y5-C004)

Type: Knowledge | Teaching weight: 3/6

A square number is the product of an integer with itself: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100... written as n². A cube number is the product of an integer with itself three times: 1, 8, 27, 64, 125... written as n³. Mastery means pupils can list all square numbers up to 144 (12²) and the first five cube numbers, use the notation n² and n³, and connect these to area and volume.

Teaching guidance: Use physical square arrays (a 5 × 5 grid of squares = 25 = 5²) and cubic structures (a 3 × 3 × 3 arrangement of cubes = 27 = 3³) to make the concepts concrete. Square numbers connect directly to area of squares (area of a 4 cm square = 4² = 16 cm²); cube numbers connect to volume of cubes (volume of a 3 cm cube = 3³ = 27 cm³). Display a reference chart of squares and cubes. Note that 1 and 64 are both square and cube numbers. Key vocabulary: square number, cube number, squared, cubed, n², n³, power, notation, array Common misconceptions: Pupils confuse squaring with multiplying by 2 (thinking 5² = 10 rather than 25). Similarly, they may think 5³ = 15 (multiplying by 3) rather than 125 (multiplying 5 × 5 × 5). The notation n² is sometimes read as 'n two' rather than 'n squared'. Pupils may not connect square numbers to physical square arrays.

Differentiation

Model response (Entry): 16 counters. 4² = 16. It is called '4 squared' because it makes a square array.

Model response (Developing): 7² = 49 (7 × 7). 3³ = 27 (3 × 3 × 3).

Model response (Expected): Side length = 8 cm because 8² = 64. 64 is also a cube number: 4³ = 64.

Representation stages (CPA)

Primary concept: Factors, common factors and multiples (MA-Y5-C005)

Type: Knowledge | Teaching weight: 3/6

Factors of a number divide it exactly; multiples of a number are its products with positive integers. Common factors of two numbers are factors shared by both; the highest common factor (HCF) is the largest. Common multiples are multiples shared by two numbers; the lowest common multiple (LCM) is the smallest. Mastery means pupils can find all factor pairs of any number to 100, identify common factors and HCF of two numbers, identify the LCM of two single-digit numbers, and use these in simplifying fractions and finding common denominators.

Teaching guidance: Systematic factor pair listing: start from 1 × n and work upward until the factors meet in the middle (e.g. for 24: 1×24, 2×12, 3×8, 4×6 — stop at 4 because 5 does not divide 24 exactly and 5 × 5 = 25 > 24). Venn diagrams showing factors of two numbers overlap at their common factors. Connect directly to fractions: HCF is used to simplify (24/36: HCF is 12, so 24/36 = 2/3); LCM is used to find common denominators (adding 1/4 + 1/6: LCM of 4 and 6 is 12). Key vocabulary: factor, multiple, factor pair, common factor, highest common factor, HCF, common multiple, lowest common multiple, LCM, divisible Common misconceptions: Factors and multiples are persistently confused: factors divide a number (factors of 12 are 1, 2, 3, 4, 6, 12); multiples are the products of multiplying a number by positive integers (multiples of 12 are 12, 24, 36...). When finding common factors, pupils often stop after finding one rather than listing all. HCF is confused with product of factors.

Differentiation

Model response (Entry): 1 × 24, 2 × 12, 3 × 8, 4 × 6. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.

Model response (Developing): Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Common factors: 1, 2, 3, 6. HCF = 6.

Model response (Expected): HCF of 18 and 24 is 6. 18/24 = 3/4. LCM of 4 and 6: multiples of 4 are 4, 8, 12, 16...; multiples of 6 are 6, 12, 18... LCM = 12.

Representation stages (CPA)

Primary concept: Long multiplication (4-digit × 2-digit) (MA-Y5-C006)

Type: Skill | Teaching weight: 4/6

Long multiplication extends short multiplication to a two-digit multiplier. The product is computed as the sum of two partial products: one where the multiplier is the ones digit (with the result placed on the first line) and one where the multiplier is the tens digit (with the result indented by one place, equivalent to multiplying by ten). Mastery means pupils can reliably compute any up-to four-digit number multiplied by any two-digit number using the formal long multiplication method.

Teaching guidance: Build from short multiplication. Show that 47 × 23 = 47 × 20 + 47 × 3 = 940 + 141 = 1081 using grid method first. Then show how this is compressed into the long multiplication layout: first row: 47 × 3 = 141; second row: 47 × 20 = 940 (written with 0 in the ones column as a placeholder, then multiplied by 2 for the tens); add the two rows: 141 + 940 = 1081. Estimation before multiplying: 47 × 23 ≈ 50 × 20 = 1000, so 1081 is plausible. Key vocabulary: long multiplication, partial product, placeholder, indent, two-digit multiplier, ones, tens, formal method, carry Common misconceptions: The most common error is forgetting to write the zero placeholder in the second row (so the tens product is not shifted left correctly). Pupils also forget to add carries from the first row before starting the second. Some pupils treat long multiplication as two separate short multiplications and forget to add the partial products at the end.

Differentiation

Model response (Entry): Grid: 30×20=600, 30×3=90, 4×20=80, 4×3=12. Total: 600+90+80+12 = 782.

Model response (Developing): 156 × 7 = 1,092 (first row). 156 × 20 = 3,120 (second row, with 0 placeholder). 1,092 + 3,120 = 4,212.

Model response (Expected): Estimate: 2,000 × 50 = 100,000. Calculation: 2,345 × 6 = 14,070; 2,345 × 40 = 93,800. Total: 14,070 + 93,800 = 107,870.

Representation stages (CPA)

Primary concept: Short division with remainders (MA-Y5-C007)

Type: Skill | Teaching weight: 4/6

Short division (the 'bus stop' method) divides a multi-digit number by a single-digit number, recording the working compactly above the number. In Year 5, this extends to four-digit dividends, and remainders must be interpreted appropriately — as a whole number remainder, as a fraction, or by rounding up or down depending on context. Mastery means pupils can accurately complete any four-digit ÷ one-digit calculation using short division and interpret the remainder correctly for the problem context.

Teaching guidance: Establish the procedure with two-digit ÷ one-digit first to consolidate the method. Extend to three then four digits. The critical skill is interpreting remainders: dividing 43 children into groups of 5: 43 ÷ 5 = 8 remainder 3 → 9 groups needed (round up for groups of people). Dividing £43 equally among 5 people: 43 ÷ 5 = £8.60 (express as a decimal). Share 43 biscuits equally among 5: 43 ÷ 5 = 8 with 3 left over (whole number remainder). Context determines the form of the answer. Key vocabulary: short division, bus stop method, dividend, divisor, quotient, remainder, interpret, round up, round down, fraction, context Common misconceptions: Pupils forget to bring down a digit when there is a remainder from one column to the next, particularly when a zero appears in the quotient (e.g. 3024 ÷ 4 = 756, where 0 ÷ 4 = 0 must still be written). Interpreting remainders contextually is frequently missed — pupils give a remainder answer when the context requires rounding.

Differentiation

Model response (Entry): 8 ÷ 4 = 2 (write 2 above the 8). 4 ÷ 4 = 1 (write 1 above the 4). Answer: 21.

Model response (Developing): 4 ÷ 6 = 0 remainder 4 (carry 4 to make 42). 42 ÷ 6 = 7. 1 ÷ 6 = 0 remainder 1 (carry 1 to make 18). 18 ÷ 6 = 3. Answer: 703.

Model response (Expected): 435 ÷ 8 = 54 remainder 3. So 54 full teams with 3 children left over. As a decimal: 54.375 (continue dividing: 30 ÷ 8 = 3 r6, 60 ÷ 8 = 7 r4, 40 ÷ 8 = 5).

Representation stages (CPA)

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Thinking lens: Patterns (primary)

Key question: What patterns can I notice here, and what do they allow me to predict? Why this lens fits: Long multiplication and short division apply systematic patterns — each step in long multiplication follows the same partial-product logic, and short division applies the same remainder-carry pattern at every column. Question stems for KS2:

What pattern can you see?

Does this always happen, or can you find an exception?

What rule connects these examples?

What would you predict for the next one? Why?

Secondary lens: Scale, Proportion and Quantity — Multiplying a 4-digit number by a 2-digit number and interpreting remainders in division both require pupils to reason about the proportional scale of the result relative to the inputs.

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Session structure: Pattern Seeking + Worked Example Set

This study uses 2 vehicle templates:

Pattern Seeking (main structure)

Enquiry focused on identifying relationships and regularities in data. Pupils pose questions about possible correlations, gather data through observation or measurement, organise and represent data graphically, identify patterns, and attempt to explain the underlying relationship.

question → data_gathering → graphing → pattern_identification → explanation Assessment: Data presentation with appropriate graph or chart, written description of the pattern found, and explanation of the possible reasons for the pattern, including evaluation of the strength of evidence. Teacher note: Use the PATTERN SEEKING template: pose a question that pupils investigate by collecting data and looking for relationships. Guide them to gather data systematically, present it in tables or graphs, and describe any patterns they find. Encourage them to suggest explanations for the patterns and consider whether the pattern always holds true. KS2 question stems:

What data do we need to collect to answer this question?

What does the graph or table show? Can you describe the pattern?

Does this pattern always happen, or are there exceptions?

What might explain the pattern you have found?

Worked Example Set

A mastery-oriented mathematics sequence moving through the concrete-pictorial-abstract progression with activation and reasoning extension phases. Begins by activating prior knowledge, introduces new concepts with physical manipulatives, transitions to pictorial representations, develops abstract fluency, applies in context, and extends through reasoning challenges.

activation → concrete → pictorial → abstract → application → reasoning_extension Assessment: Graduated practice set moving from guided examples to independent application, with reasoning task requiring explanation of method and justification of answers. Teacher note: Use the WORKED EXAMPLE SET template: activate prior knowledge and address common misconceptions. Guide pupils through the concrete-pictorial-abstract progression, modelling each step with clear mathematical language. Provide varied practice that builds fluency, then extend with reasoning problems that require pupils to explain, justify, or spot errors. Use bar models and diagrams to build conceptual understanding. KS2 question stems:

What do you already know that could help you here?

Can you draw a bar model or diagram to represent this problem?

Where has this gone wrong, and how would you correct it?

Can you explain why this method works, not just how?

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Mathematical reasoning skills (KS2)

These disciplinary skills should be woven through teaching, not taught in isolation:

Algebraic reasoning and generalisation — Express generalisations symbolically using algebraic notation, reason about the properties of unknown quantities, and use algebra to prove or disprove conjectures about numbers and geometric relationships.

Deductive reasoning and logical argument — Construct and present logical chains of deductive reasoning, recognising what has been assumed and what must be proved, moving towards formal mathematical argument and beginning to distinguish between a demonstration and a proof.

Algebraic and procedural fluency — Manipulate algebraic expressions, formulae and equations accurately and efficiently, applying learned procedures to a wide range of numerical and symbolic contexts, including working with negative numbers, surds, indices and standard form.

Generalisation from patterns and relationships — Identify, describe and represent patterns in numbers, sequences and shapes, formulating a general rule in words and testing it against further examples, progressing towards expressing generality using symbolic or algebraic notation.

Solving problems in familiar contexts — Apply known mathematical procedures to solve simple one- and two-step problems set in practical, concrete contexts, selecting the appropriate operation and checking that the answer makes sense.

Checking and verifying results — Use inverse operations, estimation or an alternative method to check whether a result is reasonable, and adjust working when an answer does not make sense in context.

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Vocabulary word mat

Prior knowledge (retrieval plan)

Pupils should already know the following from earlier units:

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Assessment alignment (KS2)

KS2 test framework content domain codes assessed by this study:

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Scaffolding and inclusion (Y5)

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Knowledge organiser

Core facts (expected standard):

Prime numbers and composite numbers: Identifying whether any number up to 100 is prime, recalling primes up to 19, and explaining why 1 is not prime.

Square numbers and cube numbers: Using square and cube numbers in context, connecting to area and volume, and identifying numbers as square or cube.

Factors, common factors and multiples: Finding HCF and LCM, and applying them to simplify fractions and find common denominators.

Long multiplication (4-digit × 2-digit): Reliably computing any up-to 4-digit × 2-digit multiplication using long multiplication, with estimation to check.

Short division with remainders: Completing short division for any four-digit ÷ one-digit, interpreting remainders as whole numbers, fractions or decimals depending on context.

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Graph context

Node type: MathsTopicSuggestion | Study ID: MTS-Y5-003 Concept IDs:

MA-Y5-C003: Prime numbers and composite numbers (primary)

MA-Y5-C004: Square numbers and cube numbers (primary)

MA-Y5-C005: Factors, common factors and multiples (primary)

MA-Y5-C006: Long multiplication (4-digit × 2-digit) (primary)

MA-Y5-C007: Short division with remainders (primary)

Cypher query:

``cypher

MATCH (ts:MathsTopicSuggestion {suggestion_id: 'MTS-Y5-003'})

-[:DELIVERS_VIA]->(c:Concept)

-[:HAS_DIFFICULTY_LEVEL]->(dl)

RETURN c.name, dl.label, dl.description

``

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Generated from the UK Curriculum Knowledge Graph — zero LLM generation.