# Nine-Pin Triangles

How many different triangles can you make on a circular pegboard that has nine pegs?

How many different triangles can you make on a circular pegboard that has nine pegs?

You may like to use the interactivity to try out your ideas. Click on two of the dots to create a line between them.

If you prefer to work on paper, you might find this sheet of nine-peg boards useful.

Once you've had a go at this, why not investigate the number of different triangles you can create on circular pegboards with more or fewer pegs?

You might also like to have a look at this task for some extension questions!

You may like to use the interactivity to try out your ideas. Click on two of the dots to create a line between them.

If you prefer to work on paper, you might find this sheet of nine-peg boards useful.

Once you've had a go at this, why not investigate the number of different triangles you can create on circular pegboards with more or fewer pegs?

You might also like to have a look at this task for some extension questions!

*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').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').

How will you record the triangles you've made? You might like to print off this sheet to use.

You could try drawing triangles which have a side made by joining two pegs next to each other. How many different triangles can you make in this way?

How will you know when you've got them all?

Cong who goes to St. Peter's RC Primary, Aberdeen, sent in a good solution to this problem. The key to answering it is to be sure you know what you mean by "different" triangles. Cong found 7 different triangles could be drawn on the nine-pin board which he drew:

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He also sent in a table which gave some more information about each triangle:

Number | Colour | Type |

1 | Green | Isosceles |

2 | Light blue | Scalene |

3 | Purple | Scalene |

4 | Orange | Isosceles |

5 | Pink | Scalene |

6 | Blue | Isosceles |

7 | Red | Equilateral |

Richard, a teacher from Cliff Lane Primary School, sent in the following:

There are 28 triangles to be made.

My class started by drawing the triangles from one peg. There were 7 possible triangles connecting peg 1 and 2, keeping that first short line constant.

They then kept peg 1 the same and connected it to peg 3, keeping this constant. They were able to make triangles using pegs 4, 5, 6, 7, 8 and 9. This results in 6 possible triangles.

They repeated for a constant line from peg 1 to peg 4 and found 5 triangles. They recognised the following pattern:

Constant line

Peg 1 - 2 7 triangles

Peg 1 - 3. 6 triangles

Peg 1 - 4. 5 triangles

Peg 1 - 5. 4 triangles

Peg 1 - 6. 3 triangles

Peg 1 - 7. 2 triangles

Peg 1 - 8. 1 triangle

Peg 1 - 9. 0 triangles

They added these up and found 28 triangles. This could be repeated for each new peg, but the triangles would be repeated, so would not count in the final total.

The next two solutions looked at a way of getting all the triangles by getting a series of numbers connected to the 9 points.

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This is a good idea, and, as so often happens in mathematics as well as helping, can produce a problem. So while 123,124,125,126,127,128,129, as a starting point is a good idea the problem arises when you go to ones starting with 234,235,236,237,238,239 it looks like we have made a further six but the size and shape of 123 is identical to 234. The same is true for the ones that would follow on. It's a matter of not losing sight of what the numbers are representing.

The next solutions did well to see that 123,321,213,312,231,132 are all the same.

Here is Tiffany's, then Claire's, work from Citipointe Christian College in Australia:

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Jiwoo and Sushi then Harry, Judy and Mai from The British International School Ho Chi Minh City in Vietnam sent these in:

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Thank you for these excellent thoughts and conclusions about this task, we hope to see you send in more solutions in the future.

### Why do this problem?

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.

### Possible approach

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.

### Key questions

How do you know your triangles are all different?

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

### Possible extension

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?

### Possible support

Children could begin by investigating the seven-point circle.