# Dice Differences

### Task 34 ... Years 2 - 10

#### Summary

The task is a bit like putting prisoners in cells and then releasing one at a time on the roll of two dice. The twist is that it is the difference between the two dice numbers which decides the cell from which a prisoner is allowed to be released. The challenge is to decide the best way to place the six prisoners in the cells (more than one in each cell is okay) so that all are removed with the fewest rolls of the dice.

#### Materials

• 2 playing boards (cell blocks)
• 2 dice
• 6 counters for each player

#### Content

• arithmetic - difference between
• probability - long run frequency and expectation
• notion of 'statistical confidence'
• probability distributions
• statistics - organising and displaying data

#### Iceberg

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

The major challenge is on the card and there is no definitive answer. However looking for the best solution leads to learning that some differences are more likely than others. Some students might guide their placement decisions with 100 trial rolls to indicate the likelihood of the differences 0, 1, 2, 3, 4, 5, 6. This is an empirical approach.

Others may realise here are 36 possible ways of pairing the dice and take a theoretical path using a table like:

 Dif. 1 2 3 4 5 6 1 0 1 2 3 4 5 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 5 4 3 2 1 0 1 6 5 4 3 2 1 0

The chart indicates that, for example, placing a counter in Cell 5 doesn't look too useful. It will, on average, in the long run, only be removed 2 in every 36 (1 in every 18) rolls.

Further investigations relate to the what happens if questions:

• What happens if we change the number of counters?
• What happens if we change the dice to ones with other faces, say 0-9 dice?

#### 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.

Involving the whole class in this investigation has been very fruitful for teachers from Year 2 to Year 9. Younger students benefit from the arithmetic challenges (it helps to have lots of Unifix or the like present to help some students work out the differences), the chance experiences and the display of data. Older students focus more on developing and comparing placement strategies, which involves deciding how much is enough data to decide that, on average, the number of days for releasing the counters using Strategy A is ...

At any level the task is a breeze to turn into a whole class lesson. To start you only need a dice and six counters for each person and scrap paper. If you have Poly Plug the plugs make excellent counters. Students sketch out the playing board quickly and you are away. Many teachers find it is great to introduce the investigation by drawing 6 large cell spaces on the white board and acting out the problem. The teacher becomes the governor of the prison and rolls the dice each (hypothetical) new morning to decide the cell from which one (and only one) prisoner is to be released.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 3, Dice Differences, which includes an Investigation Guide and companion software.

Visit Dice Differences in Menu Maths Pack C.

Visit A Good Task is the Tip of an Iceberg for one teacher's report of using this task with a Year 4/5 Aboriginal class.

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

• MWA Chance & Measurement Years 3 & 4
• MWA Chance & Measurement Years 9 & 10

The Dice Differences lesson is an integral part of:

• MWA Chance & Measurement Years 3 & 4
• MWA Chance & Measurement Years 9 & 10
This task is also included in the Task Centre Kit for Aboriginal Students and the Secondary Library Kit. Solutions for tasks in the latter kit can be found here.