## Who is this Sequence for?

This sequence is for students who know basic properties of circles, and the formula for the circumference of a circle. They know π as the ratio of circumference to diameter of a circle. There are opportunities for algebra, but this is not essential. The first lesson is to prove the formula for the area of a circle. This formula provides one of the best opportunities for students at this level to see an age-appropriate proof of a really famous mathematical principle. For some students, teachers might replace the proofs by numerical investigation (e.g. counting squares covered by circles on grids).

## Summary of learning goals

This sequence looks at the famous formula for the area of a circle from many points of view. It considers estimations and approximations and exact values. Students meet the idea of proof, engage with several different proofs and develop deeper understanding and skills of communication through explaining to others. They apply their knowledge to real world problems, including where formulating an everyday statement mathematically is a special challenge. There are opportunities for creativity.

## Rationale for this sequence

This sequence provides a broad treatment of the formula for the area of a circle, emphasising the place of this formula for practical measurement on the one hand and as a part of pure mathematics on the other. There are real world examples, for calculation and approximation. There is also a strong emphasis in one lesson on mathematical proof, in a way which will probably be new to students. Students learn that results in mathematics are proved, rather than only tested from numerical evidence, and that there are frequently many different ways to prove a result. Other parts of the lessons emphasise approximations in real world settings.

### reSolve Mathematics is Purposeful

Problem solving – In lesson 3, the square peg and round hole investigation requires students to solve a general problem using strategies of their own selection.

Reasoning – Tasks in lesson 1 emphasis reasoning and communication of a variety of proofs.

Fluency – Lesson 2 provides many opportunities.

### reSolve Tasks are Challenging Yet Accessible

In lesson 1, teachers can match the mathematical demands of the different proofs to the mathematical capacities of groups of students. In lesson 3, the square peg and round hole investigation can be tackled numerically or algebraically by students. Teachers may choose to offer the structured approach to the investigation or might present it in a very open fashion for more confident problem solvers.

Teachers’ attention is drawn to practical issues such as ensuring that students can use their calculators to work with fractions and decimals together.

### reSolve Classrooms Have a Knowledge Building Culture

Students work together in groups, and explain to each other, in order to increase their own understanding through the comments of others, and in order to improve communication skills.