# Duelling Dice

### Task 46 ... Years 4 - 10

#### Summary

On the surface there is nothing much to this task. It is a game between two players, each of whom chooses one dice and rolls it. Higher number wins. But the dice aren't 'normal', so which is the better one to choose - red, blue, green or black? Presumably the one likely to win more often. How do you decide?

#### Materials

• 4 coloured dice with numbered faces as listed below

#### Content

• long run frequency
• sample space
• collection and organisation of data
• statistical inference

#### Iceberg

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

• The game is for two dice at a time. List all the possible battles between dice eg: Red vs Blue.
• Play each battle a number of times and record the win/loss scores, eg: (7, 5) or (2, 7). A game ends when one player scores 7 points. How many times is enough to make a decision about the best dice? Perhaps totalling the columns will help decide?
• Is there a place for a poster in the Maths Corner for collecting on-going records of win/loss scores for each possible battle?
• One of a mathematician's strategies is to draw a diagram. Could you draw a picture to show which dice is better than which?
• Without even rolling a dice could you decide which is the better dice in each battle? For example, in a Red vs Blue battle, what happens if ... Red rolls 0? ... Red rolls 1? ... Red rolls 7? etc.
• Would it help to make a table to record each of these theoretical battles?
Teachers might also make use of a mathematician's exploration of a similar problem recorded here. It raises the questions what happens if we change the numbers on the dice or change the number of dice used. And we could also ask, what happens if we use a dice with a number of faces other than 6?

#### 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.

This could become a class investigation over time using the one task and the on-going class record sheet as suggested above. Alternatively, you will need sets of wooden cubes in four colours which will have to be prepared with the necessary faces. The investigation will be guided by the questions above. The Maths300 lesson has companion software which allows thousands of battles to be decided in real time.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 59, Duelling Dice, which also an Investigation Guide.

#### 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 Duelling Dice task is an integral part of:

• MWA Chance & Measurement Years 3 & 4
• MWA Chance & Measurement Years 7 & 8

The Duelling Dice lesson is an integral part of:

• MWA Chance & Measurement Years 5 & 6
• MWA Chance & Measurement Years 7 & 8