Generalising Number Patterns and Sequences

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

1 - Getting Started ~4 mins

Illustration for Getting Started Here is a pattern that keeps going: 4, 7, 10, 13, and on and on. Roughly how big do you think the 100th number would be?

Tip

Have a guess first. Then think: could we find it exactly without writing out all 100 numbers, one after another?

Teacher Notes

Take three quick estimates, not open call-outs. Don't reveal a method yet — the point is for pupils to feel how slow listing 100 terms would be. Could there be a shortcut? is the hook to leave hanging.

👇 Mark your steps as done as you complete them.

+5 XP

2 - Watch and Notice ~9 mins

A rule is what a machine does to a number: usually a multiply step, then an add or take-away step. Each machine turns an IN number into a different OUT number, but the rule inside is hidden. Study the pairs and see if you can work out what each machine is doing.

Teacher Notes

Inquiry beat: no rule announced first. Pupils notice, then put it in their own words; name each rule only once the class has said it.

Machine A (1 to 4, 2 to 7, 3 to 10) is the worked case, build it fully. Line 1, 2, 3 under 4, 7, 10. Ask how many lots of the IN number in each OUT. Draw out three lots (3, 6, 9) with 1 left over each time, so rule is x3 then +1.

Machine B (1 to 2, 2 to 4, 3 to 6, 4 to 8): let them find the +2 step and that it starts at 2, then say it is x2. Do not pre-announce the doubling.

Machine C (1 to 5, 2 to 9, 3 to 13, 4 to 17): step of +4 points to x4, then +1 lands it. Find the step first, then work back to the start.

👇 Mark your steps as done as you complete them.

+10 XP

3 - Try It Together ~9 mins

One pupil works at the board while the rest of us say the step aloud together. The rule inside this machine is hidden. We feed a number in, read what comes out, and use the pairs to work out the secret rule. Find the step between the outputs first; then work back to what the machine does to each number. Once we think we have the rule, we predict the next output before we check.

Crack the hidden rule

Teacher Notes

This round is for talking it through together — one pupil at the board, the watching class says the step aloud and agrees or corrects.

Hide the rule, then let individual pupils feed inputs and read outputs to reverse-engineer it. Insist the class states the step first (up in 5s, so there's a ×5) before building the full rule. Rotate four pupils. Revoice a strong deduction: so you used two pairs to be sure, not just one.

+15 XP

4 - Note the Step and the Rule in Your Copy ~3 mins

COPYBOOK MOMENT

In your maths copy, write out these three sequences. Under each term write its position number (1, 2, 3, 4) — that position is the number going IN. Then circle the step between the terms, and write the rule as 'multiply the position by __, then __'.

3, 8, 13, 18
6, 11, 16, 21
4, 10, 16, 22

Teacher Notes

Walk the room and glance for three things: the position numbers written under each term, the circled step, and a rule written in words, not just numbers. This is whole-class copybook practice, not marking. Prompt anyone stuck with what number is going in to make this term come out?

+10 XP

5 - Class Challenge ~8 mins

Today we work through five hidden-rule machines together, each a step trickier than the last: first 6, 11, 16, 21; then 4, 8, 12, 16; then 5, 8, 11, 14; then 7, 17, 27, 37; and finally a stretch with 7, 13, 19, 25. Every one has a constant step, so a multiply-and-add (or take-away) rule will crack it. Feed numbers in, read the outputs, find the step first, then crack the rule before we check.

Crack the rule

Teacher Notes

These are the practice questions — one pupil at a time works at the board, checks each answer, and the class confirms before moving on. Keep the board work brisk rather than over-explaining.

All five machines have a constant step, so each has a clean multiply-then-add-or-subtract rule. Note that the 7, 17, 27, 37 machine takes away (×10 then −3), so watch for pupils trying to add. The 7, 13, 19, 25 stretch is ×6 then +1. Fast finishers wait and mouth the next output rather than working ahead. Save the changing-step contrast for the maths-talk that follows.

+15 XP

6 - What Did We Notice? ~3 mins

MATHS TALK

Every machine we cracked today had a constant step, like 6, 11, 16, 21 going up in 5s, so one multiply-and-add rule worked for every term. Now look at a new one on the board: 3, 8, 15, 24. The jumps are +5, then +7, then +9. What is different about this pattern? Could a single multiply-and-add rule still crack it?

Teacher Notes

Write 3, 8, 15, 24 on the board and mark the jumps +5, +7, +9 so pupils see the step itself growing — this is a fresh example, not one they cracked. Listen for pupils naming the constant step as the thing that makes a simple multiply-and-add rule possible. Revoice: when the step holds still, one rule works for every term; when the step itself changes, we need to think differently. This lesson owns cracking hidden rules; next lesson takes those same rules and organises them in a position-against-value table, so this is a forward bridge.

+10 XP

7 - What's Next ~4 mins

Today we found the constant step in a pattern, cracked hidden rules from pairs, and used a rule to jump straight to any term. We will now go back to our very first question and answer it in one move — watch how the rule reaches the 100th term without listing all the ones before it.

Teacher Notes

Recap the three moves briefly: find the step, work back to the rule, use the rule to predict.

Back to the Getting Started question — 100th term of 4, 7, 10, 13. Step is 3, rule is multiply position by 3 then add 1.

Apply to position 100: 100 x 3 = 300, + 1 = 301. Stress a rule beats a list — no need to write out all 100 terms.

Coming up: next lesson sets these rules in a position-against-value table and reads the rule off it, rather than re-teaching how to crack them.

+5 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 ~9 mins

3 - Try It Together ~9 mins

Crack the hidden rule

4 - Note the Step and the Rule in Your Copy ~3 mins

5 - Class Challenge ~8 mins

Crack the rule

6 - What Did We Notice? ~3 mins

7 - What's Next ~4 mins

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