Lesson 16 Three dice

# Episode 1

Reasoning Resources: Blank ‘tickets’ (small pieces of paper), Worksheet 1, dice in three different colours
Whole class preparation
Accessible start for all, with little mathematical sense. Personal preferences dominate e.g. ‘birthday dates’ and ‘numbers I like.'

Gradual understanding of context, mathematically.

Range of possible outcomes.

Minimum and maximum.

Teacher demonstration of what pupils should do on their own.
Start with a story on the National Lottery or something similar, and how people choose the numbers. Then ask pupils to think about a game using three dice, where the winning score is the sum of the three numbers. What score do you think is most likely to come up?

Hand out blank ‘tickets’ and ask each pupil to write on it a number they like. Collect the slips of paper and draw six ‘from a hat’ listing the numbers on the board as the first ‘class ticket’. Include numbers less than 3 and over 18, hoping that some pupils will object.

At this point clarify that there are minimum and maximum possible outcomes, or a range of possible outcomes. Replace the impossible numbers with possible ones. Accept repeats — they have to be ‘hit’ separately.
Independent and collaborative pupils’ work often requires coaching on taking turns. But it should not be lengthy ‘busy work'. Ask a group, or different pupils one at a time, to roll three dice 25 times, while you tally the totals in a table on the board. Tick off the class ticket numbers if they are ‘hit!

Ask the class to describe the results. Typical answers: ‘There are more hits for the middle numbers, ‘easier to score 10 than 3 or 18,’ there are more ways of getting 8 than 3'. Leave the tally table of scores on the board as a model for the pair work.
Pair and small group work

Beginning of recognition that middle numbers are more likely to come up.

Pupils change selection of numbers based on own experience only, or on intuitively recognising probability patterns.

The move to collecting the class results allows the effect of sample size on sample variation to be evident.
Pupils need to work in pairs or small groups for the data gathering. Give each pair/group three dice of different colours and Worksheet 1. They are to choose their six numbers, roll the dice and record the results, and prepare answers for group sharing. Suggest that they take turns, or divide labour to roll the dice, add up the score, and record the results on the tally chart, ensuring they make 25 throws, even if they have hit all their ticket numbers.

Encourage pairs to review their results, perhaps asking; What do you notice? and What numbers would be best to choose for a ticket? Then ask pupils to choose another six numbers and repeat for a further 25 dice rolls.
Distribution of scores is now evident. Variations between groups are recognised as minor as well as random and cancel each other. Whole class sharing and discussion

First find out which groups hit all the numbers in their first ticket? How early or late in the 25 throws? Which did not hit all the numbers? Which scores seemed to come up least? Discuss the changes they made to their numbers for their second ticket and why. Some pupils would have just used the numbers that came up most and avoided the ones that didn't, without going further.

Some pupils would intuitively approach difference within the range, e.g.‘end numbers are no good’ and ‘middle numbers come up more’ At this point you can move to collect all the data from all the groups. List the number of hits for each score for each group and calculate the frequency. This allows comparisons of groups and recognition that there is a ‘bulge’ in the middle of the range.
Whole class reflection

How is choosing numbers for the three dice game different from choosing numbers for the National Lottery? Pupils should talk about the ‘hits’ being not actually random, some scores being more likely than others, and scores can be repeated.

## License

Thinking Mathematics Lessons Copyright © by Michael Shayer and Mundher Adhami. All Rights Reserved.