Give ten seconds of quiet think-time before any hands go up, then take three hands-up answers. Do not confirm or correct yet — leave the pairs on the board; the whole lesson is about how we settle which rule is right.

Expect a split: some pupils will spot +5 works for the first pair only, others may jump straight to a two-step rule. Hold both without judging — the next step shows how we decide.

Machine A (2→7, 5→16, 10→31): rule is × 3 + 1. Someone often guesses + 5 from 2→7 — test it on 5, where +5 gives 10 but the machine gives 16, so the jump grows (2→7 is a jump of 5, 5→16 is a jump of 11). Growing jump means multiplying. Three lots of input give 6, 15, 30, each output 1 more.

Machine B (2→8, 5→20): × 4, the simplest single-step one. Ask what 0 would give.

Machine C (0→7, 3→10): + 7. The 0→7 pair hands the add-number over instantly.

Machine D (4→5, 6→9): × 2 − 3, the two-step one. Double 4 is 8 take 3 is 5; double 6 is 12 take 3 is 9. Hold out for pupils to name both steps in order.

Point at each pair; let them try before you confirm the rule.

Talk this one through together — pupils take turns at the board and the class agrees or corrects out loud.

Feed 1 first: the class predicts, the machine gives 4. Then 0: predict, it gives 1 (this is the number that kills +5 stone dead — 0+5 would be 5). Then 3: predict, it gives 10. By now the class should be settling on 'times 3, then add 1'. Have a pupil build that rule on the buttons and press Check before you tap Show rule.

Keep the back rows in it: after each prediction, turn and ask a pupil to name what the machine did, rather than only the pupil at the board.

Listen for the reasoning, not just the answer — revoice a pupil who says "the jumps get bigger so it must be a times". That is the deduction move the lesson is teaching.

This is the practice round — pupils take turns at the board, check each answer, and the class confirms before moving on. Keep the board work brisk rather than over-explaining.

On each machine, ask "which number would settle it fastest?" before sending one through — 0 exposes the add-number, 1 halves the arithmetic. On the two-step machines (× 2 + 3 and × 4 + 2), hold out for both steps named in order; a pupil who says just "× 2" has half the rule.

The last machine is the one that lands the whole lesson and must not be skipped for time: it only gives itself away once pupils try a second number, because × 4 + 2 and + 5 agree on the first pair by accident. The description now shows this on the board — walk the class through the rival + 5 guess failing on the second pair.