## Lesson abstract

This inquiry explores the famous hypothesis of Diophantus, an ancient Greek mathematician, that any positive integer can be represented as the sum of four square numbers. Students explore the patterns that are generated by the sums of square numbers, as they work systematically to rediscover and test the hypothesis. There are patterns of differing complexity, so the investigation is accessible to all. Finally students use a number line to show between which two whole numbers a square root lies and consolidate their appreciation of the size of perfect squares.

## Mathematical purpose (for students)

Investigate how positive whole numbers can be written as the sum of square numbers.

## Mathematical purpose (for teachers)

Through this extended investigation, students glimpse something of the history of mathematics, and how it can take centuries for mathematical questions to be finally decided. They need to decide how they can record their work usefully (including their successful and unsuccessful approaches) and work systematically to find patterns. They will see the importance of collecting evidence, and organising it to show patterns, and also see the limitations of examples for proving that a mathematical result holds for all numbers. The lesson also builds fluency in identifying perfect squares and hence in approximating square roots.

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