Generalising Number Patterns and Sequences

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1 Getting Started
2 Watch and Notice
3 Try It Together
4 Write the Rules in Your Copy
5 Class Challenge
6 What Did We Notice?
7 What's Next
Pupil practice

1 - Getting Started ~4 mins

Here is a pattern: 4, 7, 10, 13, and it keeps going. Each number in the sequence is called a term, and we count them in order — 4 is the 1st term, 7 is the 2nd term, and so on. What would the 100th term be?

Have a think

You could write out all hundred terms, one after another, and find it that way. But that would take a long time and it is easy to slip up. Is there a faster way to leap straight to the 100th term without listing the ninety-nine before it? Have a think.

Teacher Notes

This is an inquiry opener: pose the problem and let pupils genuinely sit with it. Take three or four hands-up ideas, do not reveal a method yet — the whole point is for the class to feel why listing is slow before we hunt for a rule. Listen for anyone who notices the +3 step; revoice it as 'so it grows by the same amount each time' but leave the leap-to-100 question hanging for the next step.

👇 Mark your steps as done as you complete them.

+5 XP

2 - Watch and Notice ~10 mins

1, 2, 3 → 4, 7, 10

Let's look at this machine together. We feed in 1 and out comes 4. We feed in 2 and out comes 7. We feed in 3 and out comes 10. The machine's rule is hidden, shown as a question mark. A rule lets you work out any term straight from its position, without listing them all — so our job is to find this machine's hidden rule.

First, the step: the outputs go up by 3 each time. Notice why that matters. Every time the position goes up by one, the output goes up by the step — and that is exactly what multiplying the position by the step does. So the step of 3 tells us to multiply the position by 3. But check: position 1 × 3 = 3, yet the output is 4. We are one short, so we add 1. The hidden rule is ×3 then +1.

1, 2, 3, 4 → 2, 4, 6, 8

Here is a second machine, rule still hidden. In go 1, 2, 3, 4 and out come 2, 4, 6, 8. What is the step each time? Use the same idea: the step tells us what to multiply the position by, then we check whether we need to adjust to land on the first number.

1, 2, 3, 4 → 5, 9, 13, 17

A third hidden machine: 1 makes 5, 2 makes 9, 3 makes 13, 4 makes 17. The outputs jump by the same amount each time. Spot the step, multiply the position by it, then adjust to fit the first number — just like we did on the first machine.

Teacher Notes

Keep the rules hidden — pupils must deduce them from the numbers, not be told. This is the inquiry beat: the class gathers the evidence, then we name the rule, drawn from what they saw.

4, 7, 10 — fully model the bridge on screen: find the +3 step, link it to ×position, then show position 1 × 3 = 3 is one short of 4, so we add 1, giving ×3 then +1. This is the machine where the work-back is demonstrated.
2, 4, 6, 8 — the cleaner case: step is +2 and the machine simply doubles, no adjustment needed. Use this to show step and rule are not always the same wording.
5, 9, 13, 17 — now pupils apply the modelled work-back: step is +4, position 1 × 4 = 4 is one short of 5, so add 1, rule is ×4 then +1.

Close by naming the big idea: 'the step tells us what to multiply by, then we adjust to fit the first number.'

👇 Mark your steps as done as you complete them.

+10 XP

3 - Try It Together ~8 mins

Today we're going to explore a hidden-rule machine together. We'll feed in numbers, read what comes out, and work out the secret rule. Once we think we know it, we'll test it on a fresh number to check we are right.

Crack the hidden rule

Teacher Notes

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

Start with the three preset pairs (1→5, 2→7, 3→9). Ask for the step first (+2), then ask what to do to the input to land on the output: position 1 × 2 = 2, but the output is 5, so we add 3 — the rule is ×2 then +3. Let a pupil feed in a fresh number (say 7) and have the class predict the output before the machine reveals it — that prediction is the test of their rule. Revoice a strong contribution: 'so the step of two told us to multiply by two, then we adjusted by three to fit.'

+15 XP

4 - Write the Rules in Your Copy ~3 mins

COPYBOOK MOMENT

In your maths copy, write out these three sequences, one under the other. Beside each one, write its constant step and its rule in words. Circle the step on each.

4, 7, 10, 13
2, 4, 6, 8
5, 9, 13, 17

Teacher Notes

Walk the room glancing at whether pupils have circled the step and written a rule in words (not just a number) — this is whole-class copybook practice, not marking. Prompt anyone stuck with 'how much does it grow each time?'

+10 XP

5 - Class Challenge ~13 mins

Now we crack some hidden-rule machines together. We'll warm up on a couple of one-step machines, then work out our headline rule for 6, 11, 16, 21. For each one, find the step, then the secret rule, then test it on a fresh number. If there's time, two trickier two-step machines are waiting as a stretch.

Tip

To finish, here is one to puzzle over out loud: 3, 8, 15, 24. Does it grow by the same amount each time? See if our usual trick still works — you may be surprised.

Deduce the hidden rule

Teacher Notes

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.

Work three machines properly with the class: the first two single-step ones (add 4, ×5) to warm up step-spotting, then the headline two-step 6, 11, 16, 21 (step +5; position 1 × 5 = 5 is one short of 6, so add 1 → ×5 then +1). The last two machines (1,4,7,10 and 8,14,20,26) are stretch only — reach them if time allows. For each, get the step first, then the rule, then a prediction on a fresh input before Check.

To close, pose 3, 8, 15, 24 verbally only — it is not in the machine bank. It grows by an increasing step (+5, +7, +9), so it has no single constant-step rule. Ask 'why won't our usual trick work here?' and let pupils notice the step itself is changing.

+15 XP

Pupil practice

Module 10 · Algebra: Patterns, Expressions and Equations Mixed

Lesson 104 · Generalising Number Patterns and Sequences

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Teacher Class Feed

1 - Getting Started ~4 mins

2 - Watch and Notice ~10 mins

1, 2, 3 → 4, 7, 10

1, 2, 3, 4 → 2, 4, 6, 8

1, 2, 3, 4 → 5, 9, 13, 17

3 - Try It Together ~8 mins

Crack the hidden rule

4 - Write the Rules in Your Copy ~3 mins

5 - Class Challenge ~13 mins

Deduce the hidden rule

Unlock the full learning experience

XP