Math // Multiplication & Division

Commutative Multiplication

Understand and apply the commutative property of multiplication and recognise that division is not commutative

Ages
6 to 7
Type
Concept
Builds on
2 topics
Leads to
1 topic
Standards
1 code

Quick check

Does the learner know that 4 × 7 gives the same answer as 7 × 4 — so they can choose whichever order is easier to remember from their times tables?

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.
  • Explain that 3 × 5 = 5 × 3 and show this with an array rotated
  • Use commutativity to choose the easier calculation
  • Demonstrate that 12 ÷ 3 ≠ 3 ÷ 12

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

You are here

Commutative Multiplication

Ages 6 to 7 // Concept

Leads to

1 topic
  • Properties of OperationsRequired

    Multiplication & Division // Ages 8 to 9

    Applying all three properties extends Y2 commutativity understanding

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

For ages 6–7 (UK Year 2; in the US this content is formally taught in Grade 3, usually ages 8–9): teach that multiplication is commutative (a×b = b×a) by rotating an array so the same objects give the same total read two ways, use this to nearly halve fact-learning, and firmly teach the boundary that division is NOT commutative (12÷3 ≠ 3÷12) to prevent over-generalising. Learners should predict then prove commutativity by action, use it as a strategy to derive and check facts, and refute the claim that order never matters.

Common misconceptions

  • Over-generalising commutativity to division and subtraction (thinking 12÷3 = 3÷12, or 8−3 = 3−8).
  • Believing 3×4 and 4×3 are different facts needing separate memorisation.
  • Accepting commutativity as a stated rule without being able to justify it (no array reasoning).
  • Confusing 'same product' with 'same situation' — not seeing that different stories can share an answer.
  • Applying commutativity correctly to × but then assuming order never matters in any calculation.

Useful representations

  • The array rotated 90° (the proof of commutativity)
  • The multiplication square's diagonal symmetry
  • Counters showing 12÷3 ≠ 3÷12 (the non-commutative counterexample)
  • Paired context pictures (3 groups of 4 vs 4 groups of 3)

Teaching practices

  • Prove commutativity concretely by rotating an array a quarter turn — same objects, same total, different reading — rather than stating it as a rule; the array makes it self-evident.
  • Explicitly use commutativity to lighten fact learning: knowing 3×8 gives 8×3 for free, so the multiplication square is nearly halved by its diagonal symmetry.
  • Teach the boundary firmly: division is NOT commutative — have children test 12÷3 vs 3÷12 with counters and see they differ, so the property isn't wrongly generalised (a very common error, also later applied wrongly to subtraction).
  • Connect to context: 3 bags of 4 and 4 bags of 3 are different situations but the SAME product — distinguishing 'same answer' from 'same story'.
  • Use commutativity as a checking strategy (compute both ways, compare) to build self-correction.

Activities to try

  1. 1

    Predict then rotate the array to prove 3×4 = 4×3

    Why it works: The same set of objects, read in two directions, gives the same product, so multiplication can be done in either order — this is predicted and then seen by rotating, not merely stated as a rule.

    1. Build an array of 3 rows of 4 counters (or dots) and count the total: 12.
    2. Before you turn the array, predict whether the total will change if you turn it, and give your reason.
    3. Rotate the whole array a quarter turn so it becomes 4 rows of 3 — do not add or remove any counters.
    4. Count again and confirm the total is still 12; record 3×4 = 4×3.
    5. Repeat with another pair you choose (e.g. 2×5 and 5×2) and state the matching fact.
  2. 2

    Use commutativity to get facts for free

    Why it works: Because order doesn't matter for multiplication, each known fact gives its partner free, nearly halving the facts to learn, and you can pick the easier order or check your work.

    1. Given a known fact (e.g. 3×8 = 24), state its commuted partner (8×3 = 24) without recomputing.
    2. Look at a multiplication square and find the diagonal line of symmetry; notice that facts on either side of it mirror each other.
    3. Decide which of a pair is easier to work out (e.g. 2×9 vs 9×2) and explain why using the easier order helps.
    4. Use 'compute both ways and compare' to check one of your own answers.
  3. 3

    Test the boundary: division does not swap

    Why it works: Multiplication's array can be turned freely without changing the total, but swapping the two numbers in a division (or subtraction) usually changes the answer (it only stays the same when the two numbers are equal) — so the swapping shortcut has a precise stopping point.

    1. Act out 12 ÷ 3 with counters: share 12 counters into 3 equal groups and count one group (4).
    2. Now act out 3 ÷ 12: try to share 3 counters among 12 — you cannot give everyone a whole counter, so each would get less than one.
    3. Compare the two results and state clearly that 12÷3 ≠ 3÷12 (4 is not the same as less than one).
    4. Try to 'rotate' the sharing the way you rotated a multiplication array, and describe why it does not work.
    5. Decide for each pair whether order can swap: sort statements (mixing × and ÷) into 'order can swap' and 'order matters'.

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
Maths/Y2/MD/3

Math // Multiplication & Division

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