Math // Multiplication & Division

Priority of operations

Use conventional notation for the priority of operations including brackets, powers, roots, and reciprocals; apply BIDMAS/BODMAS consistently to evaluate complex numerical expressions

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

Quick check

Can the learner evaluate a complex expression like '5 + 2³ ÷ 4' correctly — applying the right order of operations, handling the power before the division and division before the addition?

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.
  • Evaluate expressions involving brackets, indices, and all four operations in the correct order
  • Explain why the order of operations is necessary to avoid ambiguity
  • Use the reciprocal of a number and understand that a number multiplied by its reciprocal gives 1

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

1 topic
  • Order of operationsRequired

    Multiplication & Division // Ages 10 to 11

    Order of operations with powers/roots extends Y6 BODMAS understanding

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Priority of operations

Ages 11 to 12 // Concept

Leads to

1 topic
  • Square and cube numbersRequired

    Number Representation & Place Value // Ages 11 to 14

    Powers and roots require understanding of order of operations notation

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: extend the order of operations to include an Indices tier (powers and roots) sitting between brackets and ×÷, and introduce the reciprocal as the multiplicative inverse (n × 1/n = 1, with ÷n = ×1/n). Learners evaluate complex expressions tier by tier, confront power-specific traps (2³ ≠ 2×3; −3² vs (−3)²; a root sign implies grouping), and grasp that the whole convention is an agreed anti-ambiguity notation, not an arbitrary rule. Activities range from a live expression-resolving tool (primary proposal) through anchor charts, consolidation questions, and Socratic justification.

Common misconceptions

  • Evaluating powers as multiplication: 2³ = 6 instead of 8.
  • Applying the acronym literally — doing division before multiplication or addition before subtraction because I/D/A appear first.
  • Mishandling sign vs power: −3² taken as (−3)² = 9 rather than −(3²) = −9.
  • Thinking the reciprocal of n is −n (confusing multiplicative inverse with additive inverse) rather than 1/n.
  • Treating roots as outside the hierarchy, or evaluating under a root before resolving the bracket-like grouping the root implies (√(9+16) ≠ √9 + √16).

Useful representations

  • Extended hierarchy ladder with the new Indices step inserted between Brackets and ×÷.
  • Colour-tiered annotation of an expression (brackets, then powers/roots, then ×÷, then +−) before evaluation.
  • Reciprocal pairing: n and 1/n shown multiplying to 1 (e.g. on a number line or as a product returning to unity).
  • Step-by-step rewrite chain collapsing one tier per line (3 + 2³×4 → 3 + 8×4 → 3 + 32 → 35).

Teaching practices

  • Slot the new INDICES tier explicitly into the already-secured Y6 hierarchy (prerequisite mt_jHgRQ4hR0g): powers and roots sit just below brackets and above ×÷ — present it as an extension of a known ladder, not a fresh acronym.
  • Keep reinforcing that ×÷ share a tier and +− share a tier (swept L→R); the BIDMAS acronym still misleads on D-before-M and A-before-S, so rehearse counterexamples with powers added.
  • Teach the reciprocal via the defining property n × 1/n = 1 and connect ÷n = ×1/n — seeding fraction division and later algebra; use it to show why reciprocals are 'undo by multiply'.
  • Confront power-specific traps directly: 2³ means 2×2×2 = 8 (not 2×3 = 6), and −3² vs (−3)² differ because the power binds tighter than the negative sign unless bracketed.
  • Motivate with genuine ambiguity: show a calculator or two people disagreeing on 3 + 2³ × 4, then resolve with the extended hierarchy, so the convention is felt as necessary.

Activities to try

  1. 1

    Resolve the expression one tier at a time

    Why it works: Indices (powers and roots) rank just below brackets and above ×÷; evaluating in the agreed order produces one unambiguous value, and a root implies grouping of what sits under it.

    1. Work with a live expression such as 3 + 2³ × 4 and decide which operation to resolve next, following the hierarchy Brackets → Indices (powers & roots) → ×÷ (left to right) → +− (left to right).
    2. Resolve the power first: 2³ = 8 (write out 2×2×2 to confirm it is not 2×3 = 6), then ×4 = 32, then +3 = 35.
    3. Deliberately try the wrong order — doing the addition before the multiplication: 3 + 2³ = 11, then × 4 = 44 — and compare 44 with the correct 35 so the two answers are seen to diverge.
    4. Annotate an expression by shading each tier a different colour (brackets, then powers/roots, then ×÷, then +−) before collapsing it one tier per line: 3 + 2³×4 → 3 + 8×4 → 3 + 32 → 35.
    5. Evaluate √(9+16): resolve the grouping under the root first (9+16 = 25) then take √25 = 5, and check that this is not √9 + √16 = 3+4 = 7.
  2. 2

    The sign-versus-power boundary: −3² vs (−3)²

    Why it works: Without brackets, a power applies only to the number immediately beneath it, not to a preceding minus sign; bracketing changes what the exponent acts on.

    1. Evaluate −3² by noting the power binds tighter than the negative sign: −3² = −(3²) = −9.
    2. Evaluate (−3)² where the bracket groups the negative with the base: (−3)² = (−3)×(−3) = 9.
    3. Add or remove a bracket around a negative number and predict how the value flips between −9 and 9 before confirming.
  3. 3

    Reciprocals as multiplicative inverse

    Why it works: Any non-zero number times its reciprocal is 1, so the reciprocal is what you multiply by to 'undo' multiplication, and dividing by a non-zero number is the same as multiplying by its reciprocal (0 has no reciprocal).

    1. State the reciprocal of a non-zero number n as 1/n and multiply n × 1/n, confirming the product is exactly 1 (e.g. 5 × 1/5 = 1); note that 0 has no reciprocal.
    2. Compare dividing by n with multiplying by 1/n on the same number, checking the two operations give the same result (÷n = ×1/n).
    3. Distinguish the reciprocal (multiplicative inverse, 1/n) from the negative (additive inverse, −n): the reciprocal of 5 is 1/5, not −5.
    4. Match each number to its reciprocal (e.g. 4 ↔ 1/4, 2/3 ↔ 3/2).

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.5

Math // Multiplication & Division

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