Take three hands-up answers, not open call-outs. Some pupils will keep counting up the pattern (9, 11, 13...) — that's a fine starting point. Hold the 'without drawing all ten' part as the hook tension; don't resolve it yet.

Walk each example aloud, one at a time, and pause between them so each rule lands before the next starts.

• Example 1 — write the pairs into a two-row table on the board beside the machine; ask 'what jumps from one value to the next?' (it goes up by 2). Take two hands-up answers, then make the link on the board: the +2 jump is the ×2 in the rule, and the +1 lands you on the first value, 3.
• Example 2 — pause first and ask the class to name the jump (3) before you reveal anything. The cleanest case: the value is just term-number × 3, with nothing to add. Use it to show the constant step (3) sits in the rule.
• Example 3 — the take away one at the end catches people out. Have pupils name the step of 4 first, then ask what we'd need to do to land on 3 rather than 4. Reveal the rule only after they have spotted the step.

The key move: the jump between values tells you what to multiply by; then you adjust with a + or − to land on the first value. This bridge is now on the board for pupils to read, not only spoken.

This round is for talking it through together — pupils take turns at the board and the class agrees or corrects out loud.

Set the machine to the ×2 then +1 rule in explore mode (the rule is visible here, unlike the earlier hidden-rule work). Call a term-number, ask the class to predict the value before an individual pupil sends it through, then the class reads the value and you add the pair to the term-number/value table on the board. Rotate four pupils through terms 4, 5, 6, then jump to term 10.

Watch for: pupils who add 1 first then multiply (gives the wrong value). Revoice: 'multiply by two FIRST, then add the one.' When you reach term 10, ask the class to predict before sending it through.

This round is the practice bank — 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.

The interactive holds four hidden-rule sub-challenges. In your 8 minutes, run the warm-up (×3) and the matchstick squares (×3 then +1) as the core work; treat the harder (×4 +1) and stretch (×5 −2, then leap to term 50) as stretch if time allows.

• Send 1, 2, 3 through and tabulate what comes out (for the matchstick squares: 1 → 4, 2 → 7, 3 → 10).
• Start with the visible step (going up by 3) before naming the ×3.
• The +1 is the tricky adjustment — link it to the first square needing four matchsticks, not three.
• For the 50th term, push pupils to use the rule (50 × 3 + 1 = 151), not to count up.

Let pupils probe a few inputs in the hidden-rule challenge, then build the rule from the picker and Check. Fast finishers wait quietly, mouth the next term, or watch the board.