This lesson is an extended investigation into a famous problem in mathematics. 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 evidence for proving a mathematical result holds for all numbers. The lesson also builds fluency in identifying perfect squares and hence in approximating square roots.

Students need very little mathematical content knowledge to undertake this lesson. They need to be able to square whole numbers and add and subtract them. Rhe lesson calls upon, and will further develop, students’ strategic skills for conducting an investigation, and the capacity to look for patterns and regularities and make conjectures. The lesson is structured to help students do these things.

### Lesson 1: Diophantus and Lagrange

This inquiry explores the 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 to find, 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.

Last updated June 20 2020.