## Who is this Sequence for?

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. However, the 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.

## Summary of learning goals

The 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.

## Rationale for this sequence

This lesson is designed as a carefully structured investigation into pure mathematics. The structure supports students through the phases of mathematical problem solving, as they become acquainted with the mathematical ideas involved, then immerse themselves into deep mathematical thinking, and finally reflect on what has been found. The lesson is staged by first looking at examples and counter-examples, then assembling evidence systematically, all the while looking for patterns and conjectures and seeking reasons. Teachers may also choose to have students write a record of their work, to develop their communication capability.

Being able to conduct a substantial mathematical investigation is an important goal of the Australian Curriculum: Mathematics. This lesson gives students an experience of this with strong classroom support. Teachers can plan that over time, students will undertake investigations with progressively less support and with more responsibility for deciding on the best paths to take.

**reSolve Mathematics is Purposeful**

By examining a mathematics problem of historical interest students see mathematics as a living, breathing part of human society. They experience a substantial mathematical investigation, with many avenues to explore.

Recognising perfect squares that can be used to sum to other numbers builds fluency and estimation skills.

**reSolve Tasks are Challenging Yet Accessible**

The introductory activity is both accessible and intriguing. The start of the lesson develops a sense of curiosity where students wonder what is happening and what the lesson is about. The obvious question that arises through the activity is why some numbers require three or four squares, yet others only need two. Explaining this leads to some interesting and challenging mathematics related to modulo arithmetic. Students can participate in mathematical activity at an appropriate depth for them, from conducting arithmetic trials systematically, on to pattern spotting and testing, whilst some will move towards mathematical proofs that the observations they make are always true.

**reSolve Classrooms Have a Knowledge Building Culture**

The lesson is carefully designed to encourage students to display their results so that everyone in the class can see them. The use of post-it notes enables self-correction and provides the opportunity for students to improve on the results of others.