Multiplying fractions (age 11+)

Activities to try

1. 1

   Operator machine: set an amount, set an operator, watch it partition

   Why it works: 'Of' is one action — partition the whole into equal parts, then take a copies — whether it wears a fraction hat or a percent hat. Whether the result is bigger or smaller than you started is decided by the operator's size, not by multiplication 'always enlarging'.

   1. Enter an amount shown as a whole bar or hundred-grid (e.g. 200).
   2. Work one example at a time, each with its own entered amount: with 200 in the bar apply the fraction 3/4; then re-enter 60 and apply the percentage 20% — watch the same machine partition into equal parts and shade the taken portion each time.
   3. Read the equation printed beside each partition: 3/4 × 200 = 150; and separately 20% = 20/100 = 0.2, so 0.2 × 60 = 12.
   4. Move the operator above 1 (e.g. 150% of 200 = 300, or 5/4 of 200 = 250) and below 1, and describe in words whether the result grew or shrank from the starting amount and why.
   5. Second stage — a shop: start from a £45 price, apply '20% off' so 20% of £45 = £9 is shaded (a double number line pairing £0–£45 with 0–100% can show where 20% sits), then subtract to reach the £36 paid; state the two separate moves — find 20% OF the price, THEN take it away.
2. 2

   Worked bar in stages: make the two-step structure inspectable

   Why it works: Finding a non-unit fraction of an amount is two linked steps — divide to find the unit part, then multiply to take a copies — and percentage-of is the same partition read out of a hundred.

   1. On a whole bar for 3/4 of 200, split it into four equal parts and name one part (1/4 = 50).
   2. Take (shade) three of those parts and record the second move explicitly (×3 → 150).
   3. Repeat the same staged partition on a hundred-grid for 20% of 60, reading '20 of the hundred parts' and reaching 12; a double number line pairing 60 with 0–100% can sit alongside to show 20% as a position.
   4. Answer the prompt 'why is the answer smaller than 200?' in your own words.
3. 3

   In-role market trader: reasoning about more-or-less

   Why it works: The operator is applied to a specific quantity, so the amount taken off scales with that quantity; judging reasonableness ('more or less than I started') is part of operating correctly.

   1. Take on a market-trader/shopkeeper exchange where you compute successive small-number discounts and portions mentally.
   2. Before each calculation, answer the trader's probe: 'is that more or less than you started with?'
   3. Explain to the trader what the money taken off actually is — e.g. '£9 is 20% OF the £45 price'.
   4. Reason about a what-if: if the price doubled from £45 to £90, would the £-off double? Justify (20% of £90 = £18, double the £9).