Lesson 16 Three dice

# Episode 2

Reasoning Resources: Whole class results from Episode 1, Worksheet 2 on OHT (optional)
Whole class preparation
It is probably straightforward for pupils to see that a score of 7 arises from summing (1,3, 3) or (1, 2,4) or (1, 7, 5) or (2, 2,3). It is more difficult for them to see that the permutations of these number combinations have also to be taken into account when working out the total number of ways to score 7. It is probably easier to talk about as the number of possible ways we can arrange the three numbers. Show the whole class results, which should show that scores of, say, 9-13 are much more frequent than scores of 3-5 or 16-18. How can we explain what is happening? If necessary focus attention on scores of 3 and 18, which are produced by (1, 1, 1) and (6, 6, 6) respectively. Then consider a score such as 6. Illustrate firstly that this can be obtained by combining (1, 2, 3) or (2, 2, 2) and then that these scores can turn up on different dice, i.e.in different arrangements (permutations). Using coloured dice and showing how the numbers for the (1, 2, 3), (1, 2, 2) and (2,2, 2) combinations could appear for each colour can help them to get some idea of how to work out the permutations.
Understanding the difference between combinations and permutations will prove difficult for many pupils.
Teachers may need to limit the task to finding combinations and then telling them how to calculate the permutations.
Since we are not teaching permutations and combinations it is sufficient for many pupils to give the facts that: 3 dice the same 1 arrangement, 2 dice the same 3 different arrangements, 3 dice different 6 different arrangements.
Pair and small group work
The ability to develop the hypothetical model allows the next step in probabilistic thinking — how well does the model fit the experimental data?

The deviation of the experimental data from the theoretical model challenges the pupils to account for the
differences (sampling variation).
Pairs are given two scores. For each they work out the total number of ways it can be produced with three dice.

Allocate each pair, one score from the middle of the range and one from either end, e.g. 4 and 10, or 10 and 17.( A difference of 6/7 between the two scores should ensure sufficient variety.) Give the same scores to more than one pair of pupils to cross-check that they have the correct totals for arrangements.

Pairs should first find the combinations that make up the scores. Then for each combination generate the arrangements systematically and check with the rules above, or just use the rules.
Whole class sharing/discussion
On the table of class results from Episode 1 (experimental results) create a new heading for theoretical results, Call this something like‘ Arrangement results: (Worksheet 2 shows a fictional set of class results and the theoretical results, which can be shown.)

Discuss the total number of all the possible arrangements. Some pupils would follow the logic: first die has 6 possibilities, for each of which die 2 has also six possibilities. So with two dice there are 36 possibilities. For each of these the 3rd die has 6 possibilities. That makes 216 possibilities altogether.

Is the pattern in the actual results similar or different from the pattern of the arrangement results? Some pupils should recognise that the totals are not the same, and a discussion can follow on how to make a better comparison. Some pupils may recognise they can use percentages for each column separately then compare the percentages. A rich discussion can then be had on how similar or different the two are, whether the differences are important, and whether larger samples will make a difference.
Each pupil will go as far as they are able in integrating combinatorial thinking with hypothetical thinking. However the struggle to make sense of the problem activates the pupil's mind to the fact that there is a higher level of understanding to be developed and success to date, and other pupils’ progress, encourages the quest to move to higher thinking.
End of Lesson Reflection
Give opportunity for the pupils to think back over the lesson and make brief comments about some feature of it.