Nine-pin Triangles

This low threshold high ceiling problem will help learners extend their knowledge of properties of triangles. It requires visualisation, a systematic approach and is a good context for generalisation and symbolic representation of findings.

To start with, you could pose the problem orally, asking children to imagine a circle with nine equally spaced dots placed on its circumference. How many triangles do they think it might be possible to draw by joining three of the dots? Take a few suggestions and then ask how they think they could go about finding out.

Show the interactivity, or draw a nine-point circle on the board. Invite them each to imagine a triangle on this circle. How would they describe their triangle to someone else? Let the class offer some suggestions e.g. by numbering the dots and describing a triangle by the numbers at its vertices, and then return to the problem of the number of different triangles. Discuss ways in which they will be able to keep track of the triangles and how they will know they have them all. Some children may wish to draw triangles in a particular order, for example those with a side of 1 first (i.e. adjacent pegs joined), then 2 etc. Others may feel happy just to list the triangles as numbers. This sheet of blank nine-point circles may be useful. Encourage children to work in small groups to find the total number.

Bring them together to share findings and systems, using the interactivity to aid visualisation.

Quadrilaterals is a similar problem which pupils could try next.

How do you know your triangles are all different?

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

Triangles All Around is a good follow-up activity to this one.  You could challenge pupils to think about whether they could predict the number of different triangles which are possible for different point circles. How would they do about finding out? It may be useful to have sheets of other point circles available: four-point, five-point, six-point, eight-point. Are there any similarities between all the circles with an odd numbers of points? How about those with an even number?

Children could begin by investigating the seven-point circle.