Math // Multiplication & Division

Using inverse operations

Recognise and use relationships between operations including inverse operations; use these relationships to check answers and simplify calculations

Ages
11 to 12
Type
Concept
Builds on
2 topics
Leads to
2 topics
Standards
1 code

Quick check

If the learner works out '252 ÷ 7 = 36', can they immediately use the inverse to check — '36 × 7 = 252' — without having to redo the division?

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.
  • Use addition and subtraction as inverse operations to check and solve problems
  • Use multiplication and division as inverse operations to check and solve problems
  • Recognise that squaring and square-rooting are inverse operations

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
  • Order of operationsRequired

    Multiplication & Division // Ages 10 to 11

    Understanding order of operations (BODMAS) is required before studying how operations undo each other via inverse relationships

  • Multi-step problems: choosing operationsRequired

    Multiplication & Division // Ages 10 to 11

    Fluency with all four operations in context is required before formalising inverse relationships between them

You are here

Using inverse operations

Ages 11 to 12 // Concept

Leads to

2 topics
  • Solving Linear EquationsRequired

    Algebra // Ages 11 to 14

    Solving linear equations by rearrangement requires applying inverse operations to isolate an unknown

  • Simple formulaeRequired

    Algebra // Ages 12 to 14

    Rearranging formulae to change the subject is an application of inverse operations to algebraic expressions

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 11–12, teach inverse operations (add↔subtract, multiply↔divide by a non-zero number, square↔square-root for non-negative numbers) as a way to UNDO calculations — both to check results and to solve for an unknown by isolating it. Core principles: undoing a multi-step process reverses both the operations AND their order (last done, first undone); and 'do the same to both sides' keeps an equation balanced. This is the concrete foundation for solving and rearranging equations. Build the do/undo intuition through function-machine and balance-scale work, then consolidate and probe reasoning.

Common misconceptions

  • Undoing operations in the WRONG order (undoing the × before the + in a two-step machine) — not reversing the order of operations.
  • Applying the inverse to only one side of an equation, breaking the balance.
  • Confusing inverse (undo) with commutative/associative relationships, or thinking subtraction 'undoes' multiplication.
  • Believing squaring and square-rooting are perfect inverses without the ± / non-negative subtlety.
  • Treating 'inverse' as merely reversing the numbers (12÷156 to check 156÷12) rather than applying the inverse operation.

Useful representations

  • Function machine run forwards and BACKWARDS, each step replaced by its inverse in reverse order.
  • Balance-scale / equation model showing the same inverse applied to both sides to isolate the unknown.
  • Fact-family triangle (a, b, a×b) linking a multiplication to its two divisions — inverse made concrete.
  • Inverse-pair chart: + ↔ −, × ↔ ÷, square ↔ √.

Teaching practices

  • Make CHECKING routine: after any calculation, undo it with the inverse (13×12 to check 156÷12) — embedding inverse relationships as a self-verification habit, connecting to estimation-checking.
  • Use the BALANCE / equation model — whatever you do to one side you do to the other — so applying an inverse is seen as keeping equality, not a trick; build from fact families and bar models already secured.
  • Teach inverse operations in ORDER-OF-OPERATIONS reverse when unpicking multi-step expressions (undo +/− outer layer, then ×÷), foreshadowing equation rearrangement — the prerequisite BODMAS understanding (mt_jHgRQ4hR0g) is exactly why the undo order matters.
  • Include squaring↔square-rooting as an inverse pair, noting the ± subtlety (√ of a square recovers the magnitude), to broaden 'inverse' beyond the four basic operations.
  • Use empty-box and function-machine tasks (input→×3→+2→output; run backwards to find the input) so inverse becomes 'run the machine in reverse'.

Activities to try

  1. 1

    Function machine: run it forwards, then run it backwards

    Why it works: Each operation is undone by its inverse, and to undo a sequence you must reverse BOTH the operations and their order.

    1. Build a two-step machine, e.g. '×3 then +2'. Drop in an input (say 5) and record the output (17).
    2. Now recover the original input WITHOUT being told it: work out which operation to undo first and which second. (Undo +2 first → 15, then ÷3 → 5.)
    3. Deliberately try undoing in the forward order instead (÷3 first, then −2) and check whether you get back the start number. Notice it does NOT return 5 — this shows why the order must reverse.
    4. State the rule in your own words: to undo a chain, reverse both the operations and their order (last done is first undone).
    5. If a live model is available, flip the machine into REVERSE mode so each step is automatically swapped for its inverse and reordered, and compare it with a manual attempt that undoes the steps in the forward order; that wrong-order attempt visibly fails to return the starting value.
  2. 2

    Balance scale: solving one-step unknowns by keeping both sides equal

    Why it works: Applying an inverse isolates the unknown, and doing it to both sides preserves equality — it is not a trick but a consequence of keeping the balance level.

    1. Look at a balance representing □ + 7 = 20. Decide which inverse isolates the unknown (subtract 7).
    2. Apply that inverse to BOTH sides and read off □ = 20 − 7 = 13.
    3. Try applying the inverse to only one side and predict what happens to the balance — it tilts, because the two sides are no longer equal.
    4. Repeat with a multiplication case, e.g. 3·□ = 21 (divide both sides by 3 → □ = 7).
    5. Explain why 'do the same to both sides' keeps the statement true.
  3. 3

    Checking a calculation by undoing it

    Why it works: Inverse operations provide an independent verification of a result; redoing the same operation does not.

    1. Compute a division such as 156 ÷ 12 = 13.
    2. Check it by applying the inverse: 13 × 12. Confirm it returns exactly 156.
    3. Discuss why re-dividing (156 ÷ 12 again) is a weaker check: repeating the same operation tends to repeat the same mistake (especially a mistake in method), whereas the inverse is a more independent check.
    4. Try a square/root pair: check √49 = 7 by squaring (7² = 49). Note the subtlety that √ of a square recovers the non-negative magnitude.
    5. Make checking a habit: after any calculation, undo it with its inverse.

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
KS3.Maths.Num.6

Math // Multiplication & Division

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