# McMahon's Triangles 1

### Task 148 ... Years 4 - 10

#### Summary

In essence this is a colouring problem. Four colours and three spaces - find all the unique ways to colour in the spaces, given that if a colouring can be rotated to appear the same as another then it is not different. The clever thing about the presentation of this task is that it doesn't waste time colouring. Counters are placed to represent the colouring. Perhaps the other clever thing is that the materials don't imply the answer. The challenge is to find all the possibilities and know that you have found them all.

#### Materials

• About 80 counters in four colours
• About 30 triangle cutouts

#### Content

• sorting & ordering
• spatial perception - especially rotation
• problem solving strategies
• combination theory

#### Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

The first challenge (to colour 10 triangles) is designed to put students in the position of judging whether solutions are unique. Somewhere in that collection they will have to ask: Have we made that one yet?. The 24 ways to do the colouring are shown below and the 'properly coloured' version of them creates the pieces for McMahon's Triangles 2.

#### Extensions

• Suppose there are only three colours. How many ways could the triangles be coloured?
• Suppose four squares are arranged as a square. How many ways can the bigger square be coloured using four colours?
• The solution of this problem relates to one way to tackle Task 112, Coloured Squares.

#### Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

The problem with developing the whole class lesson life for this task is the number of counters required. Eighty for each pair if you have even 24 students in a class is a lot. However, this can be managed by using groups of 4 and providing each group with four small sandwich bags, each with 20 counters of one colour. Each person is in charge of using one colour (and packing it up). You will also need to provide triangles for each group. Give two copies of this 16 Triangle Master to each group and provide scissors. One student cuts the sheet into four strips, then each student cuts one of the strips into four triangles. There is no need to cut the second sheet until the students realise that they need it.

The challenge in the problem is for the class to find all the ways to colour the triangles. It is not about being the first group to 'get them' all. This is an opportunity for a community of mathematicians to work on a problem. It may also be an opportunity for you to use your electronic whiteboard tools to record ways of colouring as they are found.

At this stage, McMahon's Triangles 1 does not have a matching lesson on Maths300.

#### Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The McMahon's Triangles 1 task is an integral part of:

• MWA Space & Logic Years 3 & 4
• MWA Space & Logic Years 9 & 10