Lesson 16 Three dice

# Three dice

Overview Resources
Pupils roll a set of three dice and work out the frequency of the total scores. Analysis of the patterns of scores and sample size lead to consideration of the theoretical probability and ratio for each score. The activity allows an intuitive approach to combinations and permutations. This activity is the first of five activities which explore probability. Blank 'tickets', e.g. A4 sheet cut into 30. Worksheets 1 and 2. Dice in three different colours (or marked with coloured pen), one set of three per pair (or group) of pupils. Paper plates, cardboard-box lids, or paper sheets with edges folded, to deaden the noise of dice-rolling (optional).
• Combinatorial thinking
• Sample variation
• Experimental and theoretical probability.
EPISODE 1
What's the score in different size samples?
First, pupils make a mostly random choice of numbers for a‘winning ticket! where the winning scores are generated by adding the numbers on three dice. Pupils then look at the patterns in their own results and suggest another set of six numbers. The whole class results are discussed, which should lead to the recognition of scores which occur more or less frequently than others. Also the effect of increased sample size in giving more confidence in the pattern of results should arise as variations in different groups’ results ‘cancel each other out!
EPISODE 2
How do we explain the pattern of scores?
Pupils explore how individual scores are obtained. First they consider the three numbers to give the total 3 or 18. Then they consider different score combinations that give 6, and the different ways these can occur on three different coloured dice. Both the combinations and permutations are needed to find the number of all the possible ways. Group sharing of the results enables a theoretical model to be developed to account for the distribution of the 216 possible outcomes. A final discussion compares the actual (experimental) class results to the theoretical model, and considers the effect of increasing sample size.
Before you teach the lesson
Decide on how you will manage the dice-rolling part of the activity. Pupils should generate the data themselves, but it is not an end in itself, and it should be done efficiently and quickly. Different groups and pairs may give more or less, but a total of 200-600 scores for the class will give a good enough match to the theoretical distribution of scores.

The first episode should be accessible to all year 8 pupils. The second episode touches on the difference between combinations and permutations, which many pupils find difficult. The teacher mediator role is to act in a‘seed-sowing’ mode rather than pressing for full understanding. Pupils are implicitly challenged to begin to integrate combinatorial thinking with hypothetical thinking. Each pupil will only go as far towards this aim as is possible for them to grasp. There is no way that there could be just one ‘learning objective’ for the whole class.