Number - Multiplication and Division
KS2MA-Y5-D003
Identifying multiples, factors, common factors, prime numbers, square numbers and cube numbers; long multiplication (4-digit × 2-digit); short and long division; problem solving with all four operations.
National Curriculum context
Year 5 introduces the major formal written methods of long multiplication and long division, extending the short multiplication and division of Year 4. The non-statutory guidance specifies that pupils should practise long multiplication for up to a four-digit number multiplied by a two-digit number, and short and long division with remainders. Number theory is introduced formally: pupils learn to identify factors, common factors and highest common factor; multiples and lowest common multiple; prime numbers, prime factor decomposition (informally); and square and cube numbers. These number theory concepts underpin the fraction arithmetic of this year (finding common denominators uses LCM; simplifying fractions uses HCF) and the algebra and ratio of Year 6.
5
Concepts
2
Clusters
5
Prerequisites
5
With difficulty levels
Lesson Clusters
Understand factors, multiples, primes, squares and cubes
introduction CuratedPrime/composite numbers and factors/multiples are mutually co-taught (C003 and C005 co-teach). Square and cube numbers naturally accompany this cluster as related number theory. Together they build multiplicative number sense before formal algorithms.
Multiply and divide using formal written methods
practice CuratedLong multiplication (4-digit × 2-digit) and short division with remainders are the two formal written method targets for Year 5. Both are high teaching-weight and procedurally distinct from the number theory cluster.
Teaching Suggestions (1)
Study units and activities that deliver concepts in this domain.
Multiplication and Division
Mathematics Pattern SeekingPrerequisites
Concepts from other domains that pupils should know before this domain.
Domain Vocabulary
42 terms across 5 concepts (42 domain-specific)(3 shared)
1 is not prime(phrase)
A key mathematical fact: the number 1 has only one factor (itself), so it does not meet the definition of a prime number.
array(noun)
Objects arranged in equal rows and columns, used to show multiplication and division.
bus stop method(noun)
An informal name for the short division algorithm, where the dividend sits inside the division bracket.
carry(verb)
To transfer a value from one place-value column to the next when a column total exceeds 9.
common factor(noun)
A number that divides exactly into two or more other numbers with no remainder.
common multiple(noun)
A number that appears in the times tables of two or more numbers.
composite number(noun)
A whole number greater than 1 that has more than two factors; the opposite of a prime number.
context(noun)
The real-life situation or scenario in which a mathematical problem is set, helping children see the purpose of the calculation.
cube number(noun)
The result of multiplying a whole number by itself three times (e.g. 2 × 2 × 2 = 8).
cubed(adjective)
Raised to the power of three; multiplied by itself twice (n × n × n), written as n³.
dividend(noun)
The number being divided in a division calculation.
divisible(adjective)
A number that can be divided by another number with no remainder.
Shared by 2 concepts
divisor(noun)
The number you divide by in a division calculation.
factor(noun)
A whole number that divides exactly into another number with no remainder.
Shared by 2 concepts
factor pair(noun)
Two whole numbers that multiply together to give a particular product.
Shared by 2 concepts
formal method(noun)
A standard written algorithm for calculation (e.g. column addition, short multiplication) as distinct from mental or informal strategies.
fraction(noun)
A number that represents part of a whole or part of a group, written with a numerator over a denominator.
hcf(noun)
An abbreviation for Highest Common Factor — the largest number that divides exactly into two or more given numbers.
highest common factor(noun)
The largest whole number that divides exactly into two or more given numbers with no remainder.
indent(noun)
A notch or small cut into the edge of a shape, creating an interior angle greater than 180° (a reflex angle).
interpret(verb)
To read and make sense of information presented in graphs, charts, tables, or diagrams.
lcm(noun)
An abbreviation for Lowest Common Multiple — the smallest number that appears in the times tables of two or more given numbers.
long multiplication(noun)
A written method for multiplying by a two-digit or larger number, using partial products that are then added together.
lowest common multiple(noun)
The smallest whole number that is a multiple of two or more given numbers.
multiple(noun)
A number that can be divided by another number with no remainder; a result of a times table.
n²(noun)
The notation for a number squared — multiplied by itself once — written with a small 2 above the number.
notation(noun)
A system of symbols used to write numbers, operations, or mathematical ideas.
ones(noun)
The place-value column for single units (0-9); also called units.
partial product(noun)
An intermediate result in a multiplication, found by multiplying part of one number by part of another.
placeholder(noun)
A zero used to hold a place-value position, ensuring digits are in the correct column.
power(noun)
A way of expressing repeated multiplication; the exponent tells you how many times to multiply the base by itself.
prime number(noun)
A whole number greater than 1 that has exactly two factors: 1 and itself.
quotient(noun)
The result of a division calculation.
remainder(noun)
The amount left over when a number cannot be divided exactly into equal groups.
round down(verb)
To reduce a number to a lower value when rounding, because the deciding digit is less than 5.
round up(verb)
To increase a number to a higher value when rounding, because the deciding digit is 5 or more.
short division(noun)
A compact written method for dividing a multi-digit number by a single-digit divisor, carrying remainders mentally.
sieve of eratosthenes(noun)
A systematic method for finding all prime numbers up to a given value by repeatedly crossing out multiples.
square number(noun)
The result of multiplying a whole number by itself (e.g. 4 × 4 = 16); can be represented as a square array of dots.
squared(adjective)
Multiplied by itself once; raised to the power of 2, written as n².
tens(noun)
The place-value column for groups of ten; the second digit from the right.
two-digit multiplier(noun)
A number between 10 and 99 used as the multiplier in a multiplication calculation, requiring the long multiplication method.
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Concepts (5)
Prime numbers and composite numbers
knowledge AI DirectMA-Y5-C003
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.
Vocabulary (7 terms)
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.
Difficulty levels
Identifying whether numbers up to 20 are prime or not by listing their factors.
Example task
List the factors of 12. Is 12 prime? List the factors of 13. Is 13 prime?
Model response: Factors of 12: 1, 2, 3, 4, 6, 12 — not prime (more than 2 factors). Factors of 13: 1, 13 — prime (exactly 2 factors).
Using the Sieve of Eratosthenes to identify prime numbers up to 50, and knowing that 2 is the only even prime.
Example task
Is 2 prime? Why is it the only even prime number?
Model response: 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.
Identifying whether any number up to 100 is prime, recalling primes up to 19, and explaining why 1 is not prime.
Example task
Is 51 prime? Explain your method. Why is 1 not a prime number?
Model response: 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.
CPA Stages
concrete
Using counters to build arrays for each number 2-30, identifying which numbers can only make a 1-by-n array (primes) and which can form multiple arrays (composites), following the Sieve of Eratosthenes with a hundred square
Transition: Child explains that a prime number has exactly two factors (1 and itself) and can identify primes up to 20 without the sieve
pictorial
Recording factor pairs to test for primeness, drawing factor trees, and using the completed Sieve of Eratosthenes as a reference to identify primes on paper
Transition: Child tests any number up to 100 for primeness using systematic factor checking and identifies prime factors without the sieve
abstract
Identifying prime and composite numbers mentally, recalling primes up to at least 19, and explaining why 1 is not prime and why 2 is the only even prime
Transition: Child identifies primes and composites up to 100 confidently, explains the definition, and uses prime factorisation to decompose numbers
Delivery rationale
Upper primary maths (Y5) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.
Square numbers and cube numbers
knowledge AI DirectMA-Y5-C004
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.
Vocabulary (8 terms)
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.
Difficulty levels
Building square numbers using arrays of counters and recognising the pattern: 1, 4, 9, 16, 25.
Example task
Make a 4 × 4 square from counters. How many counters? Write this using the ² notation.
Model response: 16 counters. 4² = 16. It is called '4 squared' because it makes a square array.
Recalling square numbers up to 12² = 144 and the first five cube numbers, using the notation n² and n³.
Example task
What is 7²? What is 3³?
Model response: 7² = 49 (7 × 7). 3³ = 27 (3 × 3 × 3).
Using square and cube numbers in context, connecting to area and volume, and identifying numbers as square or cube.
Example task
A square has area 64 cm². What is its side length? Is 64 also a cube number?
Model response: Side length = 8 cm because 8² = 64. 64 is also a cube number: 4³ = 64.
CPA Stages
concrete
Building square arrays from tiles (3×3=9, 4×4=16, 5×5=25...) and cubic structures from linking cubes (2×2×2=8, 3×3×3=27) to see square and cube numbers as physical shapes
Transition: Child states all square numbers up to 144 and the first five cube numbers from memory, connecting them to area and volume
pictorial
Drawing square arrays on squared paper, recording the square and cube number sequences, and using the n² and n³ notation
Transition: Child uses the n² and n³ notation correctly, identifies whether a given number is square or cube, and generates these sequences without drawing
abstract
Recalling square and cube numbers instantly, using the notation, and applying them in calculations and reasoning
Transition: Child answers any square/cube number question within 3 seconds and connects them to area and volume problems
Delivery rationale
Upper primary maths (Y5) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.
Factors, common factors and multiples
knowledge AI DirectMA-Y5-C005
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).
Vocabulary (10 terms)
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.
Difficulty levels
Finding all factor pairs of numbers up to 30 by systematic trial.
Example task
Find all the factor pairs of 24.
Model response: 1 × 24, 2 × 12, 3 × 8, 4 × 6. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
Finding common factors and common multiples of two numbers using Venn diagrams.
Example task
Find the common factors of 18 and 24. What is the highest common factor?
Model response: 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.
Finding HCF and LCM, and applying them to simplify fractions and find common denominators.
Example task
Simplify 18/24 using the HCF. Find the LCM of 4 and 6.
Model response: 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.
CPA Stages
concrete
Building arrays to find all factor pairs, using counters on Venn diagram sorting hoops to identify common factors, and laying out multiples with Cuisenaire rods to find common multiples
Transition: Child lists all factor pairs systematically and identifies common factors, HCF, and LCM without arrays or hoops
pictorial
Drawing Venn diagrams of factor sets, recording factor trees, listing multiples to find LCM, and connecting HCF and LCM to fraction operations
Transition: Child finds HCF and LCM on paper without arrays and uses them to simplify fractions and find common denominators
abstract
Finding factors, common factors, HCF, multiples and LCM mentally, and applying them fluently to fraction operations
Transition: Child identifies HCF and LCM of any pair of numbers up to 100 within 5 seconds and applies them to fraction problems
Delivery rationale
Upper primary maths (Y5) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.
Long multiplication (4-digit × 2-digit)
skill AI DirectMA-Y5-C006
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.
Vocabulary (9 terms)
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.
Difficulty levels
Multiplying a two-digit number by a two-digit number using the grid method as a bridge to the formal layout.
Example task
Use the grid method to work out 34 × 23.
Model response: Grid: 30×20=600, 30×3=90, 4×20=80, 4×3=12. Total: 600+90+80+12 = 782.
Using the formal long multiplication layout for up to 3-digit × 2-digit, with the zero placeholder in the second row.
Example task
Use long multiplication: 156 × 27.
Model response: 156 × 7 = 1,092 (first row). 156 × 20 = 3,120 (second row, with 0 placeholder). 1,092 + 3,120 = 4,212.
Reliably computing any up-to 4-digit × 2-digit multiplication using long multiplication, with estimation to check.
Example task
Estimate, then calculate: 2,345 × 46.
Model response: Estimate: 2,000 × 50 = 100,000. Calculation: 2,345 × 6 = 14,070; 2,345 × 40 = 93,800. Total: 14,070 + 93,800 = 107,870.
CPA Stages
concrete
Using Dienes blocks to model the two partial products in long multiplication: partitioning the two-digit multiplier into tens and ones and building each product physically
Transition: Child explains why long multiplication has two rows (ones product and tens product) and transitions to the written method without blocks
pictorial
Using the grid method side by side with the formal long multiplication layout, showing how the partial products correspond, then practising the compact method
Transition: Child completes long multiplication using the compact method independently, correctly placing the zero placeholder and managing carries across both rows
abstract
Performing long multiplication of up to 4-digit × 2-digit numbers fluently using the compact method, with estimation to check
Transition: Child completes any long multiplication with correct carrying and zero handling, routinely estimating before calculating
Delivery rationale
Upper primary maths (Y5) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.
Short division with remainders
skill AI DirectMA-Y5-C007
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.
Vocabulary (11 terms)
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.
Difficulty levels
Dividing a two-digit number by a one-digit number using the short division ('bus stop') layout with no remainders.
Example task
Use short division: 84 ÷ 4.
Model response: 8 ÷ 4 = 2 (write 2 above the 8). 4 ÷ 4 = 1 (write 1 above the 4). Answer: 21.
Dividing three- and four-digit numbers by a one-digit divisor, including cases with remainders and zeros in the quotient.
Example task
Use short division: 4,218 ÷ 6.
Model response: 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.
Completing short division for any four-digit ÷ one-digit, interpreting remainders as whole numbers, fractions or decimals depending on context.
Example task
435 children are put into teams of 8. How many full teams? How many children left over? Express the answer as a decimal.
Model response: 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).
CPA Stages
concrete
Using Dienes blocks and place value counters to model sharing (division) physically, demonstrating the bus stop method with concrete regrouping when a column does not divide evenly
Transition: Child models division with regrouping correctly with blocks and explains what happens when there is a remainder
pictorial
Recording short division using the bus stop layout on paper, showing the carried digits, and practising remainder interpretation with word problems
Transition: Child completes short division on paper with correct carrying (including zero quotients) and interprets remainders appropriately for the context
abstract
Performing short division of up to 4-digit numbers by 1-digit divisors fluently, expressing remainders as whole numbers, fractions or decimals as context requires
Transition: Child completes any short division fluently and selects the correct remainder form for the problem context without prompting
Delivery rationale
Upper primary maths (Y5) — most pupils at pictorial/abstract stage. AI can deliver with virtual representations.