---
title: "Generalising from repeated reasoning"
description: "Recognise and use repeated reasoning to generalise: extend patterns in times tables and equivalent fractions, derive unknown facts from known facts efficiently, describe general rules"
canonical: https://lightmysky.com/learn/mathematics/generalising-from-repeated-reasoning-mt_aivrWs6jrS
source: https://lightmysky.com/learn/mathematics/generalising-from-repeated-reasoning-mt_aivrWs6jrS.md
retrieved: 2026-09-02
---

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# Generalising from repeated reasoning

Recognise and use repeated reasoning to generalise: extend patterns in times tables and equivalent fractions, derive unknown facts from known facts efficiently, describe general rules

Subject: Mathematics · Area: Mathematical Thinking · Ages 8 to 9
Page: https://lightmysky.com/learn/mathematics/generalising-from-repeated-reasoning-mt_aivrWs6jrS

## Ready when they can

- Notice that all fractions equivalent to 1/2 have a numerator that is half the denominator
- Use the pattern 3×4=12, 3×40=120, 3×400=1200 and explain the generalisation
- Derive 8×7 from 8×5=40 plus 8×2=16 and describe the strategy as a general approach

## Lesson: Finding the Rule Behind the Facts

You already know lots of times tables facts, and a pattern hides inside them. Look at 3×4=12, 3×40=120, 3×400=1200. Every time one number gets 10 times bigger, the answer gets 10 times bigger too. One extra zero going in means one extra zero coming out. That is a rule you can say out loud and use on facts you never learned by heart.

*(drawing: Every time one number gets 10 times bigger, the answer gets 10 times bigger too. One extra zero in, one extra zero out.)*

**Example.** Mo needs 8×7 but only remembers 8×5 and 8×2. He splits it: 8×5=40 and 8×2=16. Then he adds them: 40+16=56. So 8×7=56, and Mo used the same splitting trick to solve a fact he had not memorized.

*(drawing: 3 out of 6 shaded is the same amount as 1 out of 2. The top number is always half the bottom number.)*

Equivalent fractions follow a rule too. 3 out of 6 shaded is the same amount as 1 out of 2. Look at the numbers: 3 is half of 6. Any fraction equal to 1/2 has a top number that is exactly half the bottom number, so 6/12 works and 5/11 does not.

**Recap.** Spot the pattern, say the rule in your own words, then use it to work out any new fact.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Extending Table Patterns](https://lightmysky.com/learn/mathematics/extending-table-patterns-mt_2jbUekyTu4)
- [Equivalent fractions on a number line](https://lightmysky.com/learn/mathematics/equivalent-fractions-on-a-number-line-mt_Ep7TDFuYUa)
- [Rectangle area by multiplying](https://lightmysky.com/learn/mathematics/rectangle-area-by-multiplying-mt_GzcJEVkNRn)
- [Describing Rules & Patterns](https://lightmysky.com/learn/learning-to-learn/describing-rules-and-patterns-mt_hbe_kdE_7C)

## Opens up

- [Reasoning with Equivalences](https://lightmysky.com/learn/mathematics/reasoning-with-equivalences-mt_jCy07DyBNU)