# Interacting with the Geometry of the Circle

In July 2005 the NRICH website based a large number of its monthly problems around an interactive circular geoboard environment inspired by the ideas of Geoff Faux *(1).* That month's publication, based around an exploratory environment, reflects one of the ways in which the NRICH project is developing and changing its focus. In the past, the majority of problems were written so that they
could be used as a vehicle for consolidating and/or applying knowledge in a range of contexts. We wish our pupils to be proficient and confident in solving problems of this type and we will continue to offer such challenges. Over the last year or so we have also started to include problems that offer opportunities to learn new mathematics. Interactive environments, like Geoboards, can act as a
focus for such learning.

These types of interactivities have great potential as a resource in their own right, enabling problem posing by their users, as well as being used in conjunction with the problems we offer on the site. In this article we discuss what we hoped for when we published the environment and our vision for the future in terms of the creation and use of similar learning contexts *(2)* .

There are three major ways in which we imagine the environment and problems being used by teachers, and we list them here in order of the level of independence:

- firstly, using each problem independently as a focus for a lesson;
- secondly, selecting problems to tell a story and introduce some of the geometry of a circle over a period of time;
- thirdly, using the environment as a stimulus for posing your own (or your pupils') problems and supporting pupils' understanding of the properties of a circle.

### The environment and some of the decisions we made

### How could the environment support teaching and learning?

*(3),*Triangle Pin-Down

*(4),*Subtended Angles

*(5),*and Pegboard Quads

*(6).*

### How did the environment work in practice?

Figure 2

### And where to next?

### References

- Andrews, P. (2002) Angle measurement: an opportunity for equity
*, Mathematics in School,***31,**5 - DfEE (2001) Key Stage 3 National Strategy:
*Framework for teaching mathematics: years 7, 8 and 9* - DfEE and QCA (1999)
*Mathematics: The National Curriculum for England and Wales*

### Notes

- Real boards (18.5 cm in diameter, moulded in crystal clear ABS and suitable for using on an OHP) are available, together with a teacher's guide, from Geoff Faux at Education Initiatives .
- You can find most of the problems and the environment itself by searching for "geoboard".
- Framework reference: KS1, Y2:
*make and describe shapes; sort shapes and describe some of their features* - Framework reference: KS2, Y5:
*properties of numbers (factors and multiples); recognise properties of shapes* - Framework reference: KS3, Y7:
*Geometrical reasoning: lines, angles and shapes* - National curriculum reference:
*higher: properties of triangles and other rectilinear shapes; properties of circles* - For those of you who are interested, the mathematics behind the programming of the environment and some of the issues that needed to be addressed was also published on the NRICH website in July 2005.