Order of operations

For ages 10–11, learners understand and apply the conventional order of operations (BODMAS/PEMDAS) across the four operations so that a multi-operation string has one agreed value. The core insight is structural: brackets sit at the top, then orders/exponents, then multiplication and division as one equal-rank tier swept left→right, then addition and subtraction as one equal-rank tier swept left→right. It is a social convention for removing ambiguity, not a law of nature, and brackets exist to override it. Activities build and check the sequencing skill, pre-empt the dominant 'do all × before any ÷' error, and surface why the convention is needed.

• Strict left-to-right computation ignoring precedence: reading 8 + 2 × 5 as 50 because they add first.
• Treating the acronym literally: doing Division before Multiplication (or Addition before Subtraction) because D/A come first in BODMAS, e.g. 12 ÷ 2 × 3 wrongly done as 12 ÷ 6 = 2.
• Believing multiplication ALWAYS beats division rather than sharing a tier resolved left→right.
• Ignoring or mis-scoping brackets — evaluating outside a bracket first, or dropping the bracket once opened.
• Thinking the convention is a deep mathematical truth rather than an agreed notation choice, which blocks understanding of why brackets can override it.

• A tiered 'staircase' or ladder diagram: Brackets on the top step, Orders next, ×÷ on one shared step, +− on the bottom shared step — visually encoding that same-step operations are equal and read left→right.
• Colour-coded expression annotation (circle brackets, underline ×÷, box +−) applied to the live expression before evaluation.
• Step-by-step rewrite chains where the expression shrinks one line at a time (12 ÷ 3 + 4 × 2 → 4 + 8 → 12), each line showing exactly one tier resolved.
• Number line / equal-groups callback for WHY 8 + 2 × 5 = 18: the 2 × 5 is one chunk of ten added to 8, not (8+2) copied five times.

• Motivate the NEED first: give the class 8 + 2 × 5 and collect the different answers (18 and 50) before naming the rule — the convention exists precisely to end that disagreement, so let them feel the ambiguity.
• Teach ×÷ as one equal-rank tier and +− as one equal-rank tier, always swept left→right, to pre-empt the dominant 'do all × before any ÷' error; use worked pairs like 12 ÷ 2 × 3 (=18, not 2) that break if you reorder within the tier.
• Underline operations tier-by-tier in colour before computing — first circle brackets, then underline the ×÷ tier, then the +− tier — so the child sees the STRUCTURE of the expression before touching a single number.
• Connect to already-secured brackets work (mt_8jFSnXxqQD): brackets are the top of the hierarchy and are the tool for overriding default precedence, so writing 8 + 2 then ×5 as (8 + 2) × 5 makes the intended order explicit.
• Avoid over-reliance on the mnemonic: BODMAS/PEMDAS wrongly suggests D before M and A before S. Reinforce the two equal-rank tiers explicitly and rehearse counterexamples so the acronym doesn't cement the misconception.