Math // Multiplication & Division

Factor Pairs & Commutativity

Recognise and use factor pairs and commutativity in mental calculations

Ages
8 to 9
Type
Concept
Builds on
2 topics
Leads to
1 topic
Standards
1 code

Quick check

If the learner needs to multiply '4 × 15', can they spot that splitting it into '4 × 5 × 3 = 20 × 3 = 60' is a clever shortcut — using factor pairs to make it easier?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • List all factor pairs of a given number (e.g. 24: 1×24, 2×12, 3×8, 4×6)
  • Use commutativity to reorder a multiplication for easier mental calculation
  • Explain what a factor pair is and how commutativity helps

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

2 topics
  • All times tables to 12×12Required

    Multiplication & Division // Ages 8 to 9

    Finding factor pairs requires fluent table knowledge

  • Properties of OperationsHelpful

    Multiplication & Division // Ages 8 to 9

    Using commutativity in factor pairs draws on properties understanding

You are here

Factor Pairs & Commutativity

Ages 8 to 9 // Concept

Leads to

1 topic
  • Factors, multiples, and primesRequired

    Multiplication & Division // Ages 9 to 11

    Recognising factor pairs at Y4 is prerequisite to systematically finding all factor pairs

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

Teaching factor pairs and commutativity to 8–9 year-olds: a factor pair is two whole numbers that multiply to a target (24: 1×24, 2×12, 3×8, 4×6). Commutativity means 3×8 and 8×3 are the SAME unordered pair, so pairs are searched systematically from 1 upward and the search stops when the two factors 'meet in the middle' (near the square root). Each pair corresponds to a rectangle the number can form, which bridges to arrays/area and previews primes (a whole number greater than 1 with only the 1×n rectangle). Factor pairs also pay off as a mental-calculation tool via reordering/regrouping (e.g. 25×16 = 25×4×4 = 100×4 = 400). Manipulable rectangle-building is the core; static diagrams, checking tasks, and Socratic questioning serve distinct supporting roles.

Common misconceptions

  • Listing 3×8 and 8×3 as different pairs (not applying commutativity), inflating the count.
  • Missing factor pairs by searching unsystematically, especially the middle ones.
  • Confusing factors with multiples (factors go INTO the number; multiples are in its times table).
  • Forgetting 1 and the number itself are always a factor pair.
  • Not knowing when to stop — continuing past the square root and re-finding pairs in reverse.

Useful representations

  • Rectangle/array for each factor pair — factors as side lengths of all rectangles of that number
  • Systematic factor-pair list (1×24, 2×12, 3×8, 4×6) with the 'meet in the middle' stopping point
  • Factor 'rainbow' arcs pairing 1–24, 2–12, 3–8, 4–6 — shows commutativity and completeness
  • ×÷ fact family — factor-finding as division

Teaching practices

  • Search for factor pairs systematically — 1, 2, 3, … in order, pairing each with its partner — and use commutativity to know you've found them all when the pair 'meets in the middle'.
  • Represent each factor pair as a rectangle of the target number, so a factor is literally a side length and children see the finite set of rectangles a number allows.
  • Find factors through known table facts and the ×÷ inverse, reinforcing that factor-finding is division/known-fact work.
  • Use factor pairs and commutativity as mental-calculation tools (reorder/regroup for easy pairs like ×5×2=×10), not just as a listing exercise.
  • Seed prime/composite intuition: numbers with only the 1×n rectangle (one factor pair) hint at primes; square numbers have a repeated factor (the square).

Activities to try

  1. 1

    Rectangle hunt: build every rectangle of a number

    Why it works: Factors are the possible side-lengths of all rectangles a number can form; commutativity collapses 3×8 and 8×3 into one pair; a systematic search can stop where the factors meet in the middle because every larger factor has already appeared as the partner of a smaller one; and a whole number greater than 1 that makes only the 1×n rectangle is prime.

    1. Take 24 counters (or squared paper, tiles, or dots) and arrange them into complete, gap-free rectangles, recording the dimensions of each as a factor pair.
    2. Work systematically: ask 'does 1 fit? does 2 fit? does 3 fit?...' in order, logging each rectangle you can make (1×24, 2×12, 3×8, 4×6).
    3. After making 3×8, try to make 8×3 and observe it is the SAME rectangle turned a quarter-turn — record it as one pair already found, not a new one.
    4. Keep going (5 does not fit; 6 gives 6×4, the same rectangle as 4×6) and notice that no NEW rectangles appear once the two sides meet in the middle; explain why you can stop there.
    5. Repeat with 23 counters and discover only a single 1×23 line forms — a whole number greater than 1 that makes only a single 1×n line is prime.
  2. 2

    Factor pairs as a mental-calculation tool

    Why it works: Commutativity (changing the order) and regrouping (choosing which factors to multiply first, e.g. 25×(4×4) = (25×4)×4) let you use factor pairs to turn a calculation into easier steps, so factor-finding is a practical mental-maths tool, not just a listing exercise.

    1. Rewrite a multiplication by regrouping convenient factors: turn 25×16 into 25×4×4, then compute 100×4 = 400.
    2. Find a factor pair that makes a product easy (e.g. pairing ×5 with ×2 to make ×10), and explain why reordering is allowed.
    3. Find factors using known times-table facts and the multiplication/division inverse: 'what times what makes 24?'
  3. 3

    Prove the list is complete (systematic-search reasoning)

    Why it works: Completeness comes from a systematic sweep that pairs each small factor with its partner and stops where they meet, not from feeling finished; in a square number, one pair uses the same factor twice (e.g. 6×6 for 36), so that factor is counted once and the number of factors is odd.

    1. List all factor pairs of a number such as 36, then explain how you know none is missing.
    2. Respond to a challenge — 'Have we found them all? How do you know?' — by describing your sweep from 1 and your stopping point.
    3. Explain why 24 has an even count of factors (8) but a square number like 25 has an odd count (3: 1, 5, 25), because one pair uses the same factor twice (5×5, the square root), so that factor is counted only once.

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
The national curriculum in England: Key stages 1 and 2 framework document
Ma/KS2/Y4/MD/3

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