Fractions of amounts (harder)

Finding a non-unit fraction of a quantity (e.g. 3/8 of 24) is a two-step operator procedure for 8-9 year olds: divide the whole by the denominator to find one equal part, then multiply that part by the numerator. Children already know unit fractions of sets, so the new work is coordinating the divide-then-multiply sequence, grounding it in the meaning of numerator and denominator, and translating worded problems into that structure. Make the two steps explicit and visible before compressing them.

• Divides by the numerator instead of the denominator (3/8 of 24 read as 24 ÷ 3)
• Stops after one step — finds the unit fraction (24 ÷ 8 = 3) and gives that as the answer without multiplying by the numerator
• Multiplies the whole by the numerator (24 × 3) giving an answer larger than the whole
• Multiplies numerator and denominator into the whole in the wrong order or muddles which is which under worded-problem load

• Bar model partitioned into denominator equal parts with numerator shaded (whole labelled, one part labelled)
• Array / equal groups (24 counters in 8 groups of 3; take 3 groups)
• Double number line: 0 ... whole on top, 0 ... fraction on bottom, marking one part then stepping along
• Written two-step record: 24 ÷ 8 = 3, then 3 × 3 = 9

• Make the two steps explicit: ring-fence 'divide by the denominator to get one part' as step 1, then 'multiply by the numerator' as step 2, before letting children compress it — many errors are a collapsed or reversed sequence, not arithmetic slips.
• Anchor the procedure in the fraction-as-operator meaning: the denominator says how many equal groups to split into, the numerator says how many of those groups to take. This lets children self-check which number divides and which multiplies.
• Use bar models / arrays: draw the whole as a bar, partition into 'denominator' equal parts, shade 'numerator' of them, and read off the total — the model makes the divide-then-multiply visible and catches the 'divide by numerator' error.
• Sequence from unit to non-unit: confirm 1/8 of 24 = 3 first, then 3/8 = 3 x 3 = 9, so the new step (multiply by numerator) is isolated against secure ground.
• Include a plausibility check: a proper fraction of a quantity must be smaller than the whole (9 red sweets out of 24), which catches multiply-the-whole errors.
• Vary the worded contexts (sweets, money, measures, sets of people) so children practise translating language into the divide-then-multiply structure rather than pattern-matching one phrasing.