---
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: 2026-11-12
---

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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-table facts. Patterns hide inside them. Look at 3 × 4 = 12, 3 × 40 = 120, and 3 × 400 = 1200. When one factor gets 10 times bigger, the answer gets 10 times bigger too. One extra zero in gives one extra zero out. Say this rule aloud and use it for facts you have not learned by heart.

*(drawing: When 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

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

## Needs first

- [Extending Table Patterns](https://lightmysky.com/learn/mathematics/extending-table-patterns-mt_2jbUekyTu4)

## Optional connections

- [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)