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

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

Illustration for Watch and Notice A rule is what a machine does to a number: usually a multiply step, then an add or take-away step. Let's look at three machines together. Each one turns a number that goes IN into a different number that comes OUT, but the rule inside is hidden. Study the pairs and see what you notice.

Machine A: 1 to 4, 2 to 7, 3 to 10

Line up the IN numbers (1, 2, 3) against the OUT numbers (4, 7, 10). How many lots of the IN number can you see in each OUT number? And what is left over each time? Look first, then say what you think the machine does.

Machine B: 1 to 2, 2 to 4, 3 to 6, 4 to 8

What jumps from one OUT number to the next? Now find what the machine does to each IN number to make its OUT number.

Machine C: 1 to 5, 2 to 9, 3 to 13, 4 to 17

The OUT numbers climb in steps. Find the step first, then work back to what the machine does to each IN number.

Teacher Notes

This is the inquiry beat — do not announce any rule first. Let pupils notice, then put their own words on it.

Machine A — build this one fully as the worked case. Line positions 1, 2, 3 under outputs 4, 7, 10. Draw out from pupils that there are three lots of each position (3, 6, 9) with 1 left over each time, so the machine is ×3 then +1. Name the rule only once the class has said it.
Machine B — let pupils find the +2 step and that it starts at 2; let them say it is ×2. Do not pre-announce the doubling.
Machine C — step of +4 points to ×4; then the +1 lands it. Revoice: find the step first, then work back to the start.

The rule is the pay-off of this step, drawn from what pupils saw — not its opener.

👇 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

This round is the practice bank — 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

What we did today

We found the constant step that links each term to the next.
We cracked hidden rules from input-output pairs by finding the step first, then working back.
We used a rule to predict a term without listing every term before it.

Back to our first question

Worked example

The 100th term of 4, 7, 10, 13 is just the rule applied to position 100: multiply 100 by 3, then add 1, which gives 301. No need to list all 100 numbers.

Coming up

Next we will take the very rules we cracked today and set patterns out in a table of position against value, then use that table to write the rule down and leap straight to any term.

Teacher Notes

Keep this brief. Reinforce that a rule beats a list because it lets you jump to the 100th term directly — the question from Getting Started is now answered (301). The next lesson organises these rules in a table 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

Download Activity Book page (PDF)

End of lesson

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Teacher Class Feed

1 - Getting Started ~4 mins

2 - Watch and Notice ~9 mins

Machine A: 1 to 4, 2 to 7, 3 to 10

Machine B: 1 to 2, 2 to 4, 3 to 6, 4 to 8

Machine C: 1 to 5, 2 to 9, 3 to 13, 4 to 17

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

What we did today

Back to our first question

Coming up

Unlock the full learning experience

XP