## Triangles All Around

You might like to have a look at

Nine-Pin Triangles before trying this problem.

How many different triangles can you draw on a circular pegboard which has four equally spaced pegs?

What are the angles of each triangle?

If you have a six-peg circular pegboard, how many different triangles are possible now?

What are their angles?

How many different triangles could you draw on an eight-peg board?

Can you find the angles of each?

You may like to use the interactivity to try out your ideas. When you have selected the number of dots you need, select the line drawing tool, then click on two dots to draw a line between them.

*Many thanks to Geoff Faux who introduced us to the merits of the nine-pin circular geoboard.*

*For further ideas about using geoboards in the classroom, please see Geoff's publications available through the Association of Teachers of Mathematics (search for 'geoboards').*

Why do this problem?

This problem is designed to provoke learners' curiosity. Some may be intrigued by what constitutes a 'different' triangle; others might be keen to know they have found all the possibilities for a given pegboard. The interactivity will help learners try out their ideas and deepen their understanding of the concept of a triangle. The task offers an opportunity for pupils to work in a
systematic way, using their knowledge of the properties of triangles, angles in circles and angles in triangles. (A knowledge of circle theorems is not required.) The problem encourages learners to be clear about what they do know and what they can work out from it.

#### Possible approach

It would be a good idea to try

Nine-pin Triangles before tackling this task. You may like to read the

teachers' notes of that task and follow a similar approach.

After learners have been working on the four-peg board for a short while, it would be helpful to facilitate a discussion about which triangles are the same and which are different, and why. Once the possibilities have been shared, it might be useful to suggest that if two triangles could be picked up and placed exactly on top of each other, then they are the same (i.e. that orientation on
the pegboard does not matter). This makes the total number to be found more manageable in each case.

If working on paper rather than using the interactivity, pupils may find it helpful to print these sheets off:

Sheet of

four-peg boards

Sheet of

six-peg boards

Sheet of

eight-peg boards

In addition to being a useful tool for working on the problem, the interactivity will allow you to share learners' findings during a final plenary. You could warn a few pairs that you'd like them to draw their triangle/s on the board using the interactivity and to explain how they calculated the angles. It would be interesting to see whether other pairs had different methods for working out
the angles. Another pair could be chosen to justify the total number of triangles they had found for a particular size pegboard.

#### Key questions

How do you know your triangles are all different?

How do you know you have got all the different triangles?

What do you know about the angles in a triangle?

If you mark the centre of your circle and draw in some radii, how might that help?

How many degrees are there in a full turn?

#### Possible support

Working in pairs will help learners access this task. After some time, you could encourage two pairs to join forces and compare their ways of working.