## Who is this Sequence for?

These lessons assume only elementary knowledge of circles – that all the points on the circumference are the same distance from the centre n- and an intuitive understanding of ratio. Circle calculations require adequate numerical skills.

## Summary of learning goals

This sequence introduces the properties of circles focussing on the circumference and the relationship with its diameter (and radius). Common objects, such as balls to represent spheres, are used to demonstrate the constant ratio between circumference and diameter for every circle. Further tasks offer different approaches to identify a more accurate value for the ratio. Challenges are provided allowing for reasoned estimation, generalisation and multi-step problem solving.

## Rationale for this sequence

This sequence provides a very broad treatment of circles displaying the many ways in which they are evident in the world around us – in plates and balls, and rolling wheels, and wrapping around and unwrapping, and as a limit of polygons, as a locus of points at the end of a rope and in two and three dimensional objects. This breadth is demonstrated in both the mathematical investigations and the real world applications that are shown.

The consolidation activities demonstrate the range of problem types to which students should be exposed: *C* from *r* and *d* and vice versa, single step and multistep problems, with numbers of different types, and demonstrating many different real world occurrences. By thinking about the circle formulas in such a broad way, with variation of tasks based on the variation of concepts, background and complexity of situations, and avoiding mechanical repetition in the design of exercises for practice, strong flexible knowledge is constructed. This is called ‘the principle of variation’.

**reSolve Mathematics is Purposeful**

Fluency – Students increase their mathematical fluency through a variety of tasks which are encountered so that practice is not mechanical.

Problem solving – Many photos are used as problem prompts, giving a strong connection to the real world. Estimation and an appreciation of sensible levels of accuracy is a strong feature across the lessons. There is opportunity for investigating a famous puzzle and for creative problem solving.

Reasoning – Students will use the circumference formula in conjunction with common sense knowledge (e.g. to find the arc length of a sixth part of a circle not by learning a special formula). Some of the approaches to finding a numerical value of *π* are based on historically significant mathematical methods (such as approximating a circle by inscribed polygons).

**reSolve Tasks are Challenging Yet Accessible**

The initial activity provides a common experience that allows all students to access subsequent tasks. Other lessons also proceed from physical experience of circles in many guises providing a strong experiential base for subsequent reflection and analysis.

A range of activities are made available, which increases the likelihood that students will find at least one approach helpful and indeed memorable.

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**

The practical tasks are carried out in groups so that students can work together to get accurate measurements, clarify the tasks, check calculations, and evaluate results. Some of the activities use classroom response cards, so that teachers can monitor students’ progress in a time-efficient way.