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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
- 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 idea of teachers and pupils posing their own questions is
the aspect of environments which we find most exciting, and it is
interesting to note that when one of us recently used the material
in a classroom this is exactly what happened. In this article we
will describe the properties of the environment and discuss some of
the problems in a little more detail. Finally, we will talk about
our experience of using the environment in a classroom.
The environment and some of the decisions we made
In creating the environment we made a number of decisions;
most important was that the environment should be flexible enough
to enable pupils and teachers to explore it and pose their own
questions, not just use it as a means of tackling the problems we
set on the site that month. That is, the environment should be a
useful tool for teachers to use in the classroom on its own.
We wanted a means of recording what users had done, so in all
cases, the environment has different coloured "rubber bands", and
stores the shapes created in a simple table. We wanted to offer the
flexibility to change the number of points around the circle so
that it was possible to make conjectures based on a number of cases
and then test generalisations. So in the general environment users
can change the number of pegs on the circumference of the circle
and decide whether to include a peg in the centre (see figure
For each of the problems on the site the environment was set
up with a particular starting point in mind (eg a circular geoboard
with ten pegs: nine around the edge and one in the middle).
Downloadable sheets with different numbers of pegs are also
available on the site. Of course, a set of nine-point plastic
geoboards for hands-on experimentation would also be a great
How could the environment support teaching and learning?
So, once we had the environment what did we do with it? We
wanted to pose questions that might lead the user to learning some
new mathematics. What mathematics could potentially emerge? Having
considered some problems as useful starting points, we looked at
the numeracy framework (DfEE, 2001) and the mathematics national
curriculum (DfEE and QCA, 1999) and were able to identify
mathematical content from each of years 2 to 12. Examples are
Nine-pin Triangles (3),
Triangle Pin-Down (4),
Subtended Angles (5) a
Pegboard Quads (6).
What this demonstrates is the flexibility of this environment,
which can be used at a very wide variety of levels and with a
variety of outcomes. Although we have given references to national
curriculum and framework documents, it is perfectly possible that
pupils will be working at all these levels in a mixed ability
class. The environment also offers the potential to be a far more
stimulating and engaging way in which to learn circle theorems than
the usual ones (see also Andrews, 2002).
Using the environment in the ways described above opens the
possibility of learning mathematics through problem solving. Such
environments also allow pupils to start from levels of
understanding with which they feel comfortable.
How did the environment work in practice?
We recently led a two-hour session with pupils who had access
to sheets with nine-point circles and plain circles (to which any
number of points could be added) and a computer and data projector
for using the interactivity. The session started with pupils being
asked to consider this task:
How many different isosceles triangles can you draw on
a nine-point circle containing one vertex at the
Calculate the size of their angles.
Pupils were asked to work on their own for several minutes
before sharing their ideas with each other and/or the rest of their
group, with some time given to questions that sought to clarify the
problem, such as "What do you mean by different?"
The groups worked at quite different paces and took different
routes, with one group extending their work to find the number of
different isosceles triangles in a circle with any number of points
and the sizes of the angles in those triangles, developing general
rules in each case. This came as quite a surprise, as the question
had seemed a fairly closed one and represented a starter to lead
into the rest of the session but also gave us lots of feedback on
the "comfort zones" of pupils in the group. Some needed to spend
quite some time convincing themselves that all the triangles were
isosceles and that there were a very limited number of different
ones; others quickly generated a general rule.
It was intended that the work on isosceles triangles with a
vertex at the centre would lead naturally into considering the
number of different triangles with all three vertices on the
circumference of the circle and to calculating the sizes of their
angles (without recourse to circle theorems, which they did not
know). For a few, the knowledge gained concerning isosceles
triangles was brought to bear on the new problem without prompting;
for some a diagram of a triangle with an isosceles triangle imposed
on it (figure 2) was enough to make the connection; and for others
a gentle nudge about using what they had already found was helpful.
There were also two contrasting approaches to the new challenge.
Some pupils immediately drew a triangle and, after some discussion,
constructed isosceles triangles and used them to calculate angles.
Other groups became engaged in ensuring that they could find all
the triangles first, with some of them wanting to generalise the
rule for the number of triangles for any point-circle and comparing
their findings with those for the isosceles triangles.
The original intention had been to move the pupils on to
finding the angles of the circumscribed triangles, to look for
relationships and to try to explain them. The two relationships
that we had anticipated might emerge were that angles subtended by
the same arc are equal and that the angle subtended by an arc at
the centre is twice the angle subtended by the same arc at the
circumference. This would require a systematic search for patterns
and relationships. For example, to ensure that they had found all
the triangles they might need to draw them in an ordered way,
starting with one side formed by joining two adjacent points, then
two points one point apart and so on. Again, pupils worked at
different paces and developed slightly different lines of inquiry,
all of which appeared to be helping pupils to establish the
groundwork for the identification of a general rule and then an
explanation of why it might be the case, possibly by first looking
at circles with different numbers of points. The impressive thing
was the amount of problem posing that was going on and the evidence
that pupils were working from their own understandings and building
on them with confidence. The session ended with a discussion of
strategies that might lead to identifying and describing
The next time we meet, we will continue the journey, probably
moving on to identifying, generalising and justifying the angle
properties of triangles in a circle and looking at the special case
of the angle in a semicircle and angles of a cyclic quadrilateral.
Now the latter does not feel like an extension of the work, just a
What was clear was that the environment and findings were
making sense to many of the pupils who had the freedom to start off
from a point where they felt comfortable.
And where to next?
We will continue to develop environments and offer some ideas
for problems at a range of levels, but one of our main hopes is
that teachers will make them their own, encouraging pupils to pose
their own questions alongside the ones suggested. A completely open
starter might be "Investigate this environment and describe and
justify what you discover."
We would welcome ideas for more environments.
- 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
- 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 "
July 2005" in the monthly search on the site or entering
" into the word search.
- 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
- 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.