Task taken from

The following task is taken from NRICH https://nrich.maths.org/problems/non-transitive-dice:

The Task: Non-transitive dice

Here are three dice that are used to play a game for two players:

There are three dice each showing three faces. On the first die the faces show two dots, four dots and nine dots. On the second die the faces show one dot, eight dots and six dots. On third die the faces show three dots, five dots and seven dots.

The red die has the numbers {1, 1, 6, 6, 8, 8}
The green die has the numbers {2, 2, 4, 4, 9, 9}
The blue die has the numbers {3, 3, 5, 5, 7, 7}

Each player chooses a different die.
They roll their dice.
The winner is the person whose die shows the bigger number.

Alison and Charlie are playing the game. Charlie wants to go first so Alison lets him.

Was that such a good idea?

Can you advise Alison on which die to choose once she knows which die Charlie has selected?

The Teachers' Resources for https://nrich.maths.org/problems/non-transitive-dice?tab=teacher include guidance on how to use this task. The following notes build on this guidance from NRICH and add specific guidance about supporting classroom communication and mathematics together.

One of the NRICH suggestions for how to work on this task with students is the idea of playing the game, with either the teacher playing against a student or students playing in pairs. In this way, students can become familiar with the game and start thinking and talking about a strategy to win. Working in pairs or groups of three as students think through the game and seek a strategy to win gives them opportunities to explain their reasoning and to convince others.

In individual discussions or in whole class discussion, some students might need language support for expressing their ideas. This might include support for talking about how they can work systematically to consider every possibility, or why they did not have to do that within their strategy. Important phrases and technical vocabulary might be “"outcome”, and phrases for comparing probabilities, e.g. “more/less likely than”.

As students get started on the task, you may hear or see attempts to articulate or represent the problem. As there are three possible pairs of dice, students may need support to break the problem down so that they can start making specific comparisons to support their reasoning. As the NRICH team suggest, the following questions could help groups to get started. “What are the possible outcomes when red plays green? What about when red plays blue? What about when green plays blue?”, “How could you work out the probability of each outcome?”

Using and connecting representations

To support thinking and communication in small groups and to encourage students to work systematically and make sense of what this might look like in this task context, we recommend inviting students to link different representations to find a strategy to win. For example, besides talking about their ideas, it is probably very helpful to use sample space diagrams, tree diagrams or a structured list of possible outcomes. However, formulating representations and making links between different representations involves linguistic demands such as naming elements of the representation and how these relate to the particular case of the problem.

The Student Solutions from https://nrich.maths.org/problems/non-transitive-dice?tab=solutions exemplify the variety of representations students have used in this task and how they used these representations in their reasoning. These include text with symbolic notation and technical terms from probability such as “chance” or listing results of which die to pick. Some students used tables for pairs of dice to support counting and comparing the ways each die in the pair could win, while others used tree diagrams. These are each illustrated below for the case that Charlie chooses the red die and Alison chooses the green die.

	1	1	6	6	8	8
2	G	G	R	R	R	R
2	G	G	R	R	R	R
4	G	G	R	R	R	R
4	G	G	R	R	R	R
9	G	G	G	G	G	G
9	G	G	G	G	G	G

Red die is along the top, green die is down the side. Probability of red win =16/36, Probability of green win = 20/36, so green is more likely to win than red.

Tree diagram with two levels of branches. The first set of outcomes are 1, 6, 8 from the red die, each with probability one third. The second set of outcomes from each of these is 2, 4, 9 from the green die, each with probability one third again. The combined outcomes where the person with the green die wins are 1,2 1,4 1,9 6,9 8,9 These each have one ninth written next to them.

Whatever representations students use, it is important for them to have an opportunity to use these to develop their argument.

Discussing approaches

Rather than you giving immediate feedback to each group, you can use whole-class discussion between phases of small group work to make ideas from different groups available and understandable to the whole classroom. Using discussion to support progress can be particularly challenging. By listening to different approaches during small group work, you might be able to plan this discussion by selecting groups with different approaches to present their ideas.

Students may find it challenging to present their arguments, particularly if they are only part way through working on the task, but this can be particularly useful for others. One way of handling this is to give groups warning that you will be inviting them to come to the front of the classroom and that they should focus on sharing their thinking and the representations they used are using. Giving students an opportunity to prepare what they will say and how they will make their representations and notes visible to the whole classroom makes it easier to present as well as easier to follow. From a language perspective, students often find it easier to communicate their thinking if they can refer to a shared picture and point to things.

Another way of sharing representations is to collect ideas of different representations while students are working in small groups or pairs and write some of these up on the board, so they are ready to use when you bring the class back together. When students remain in their seats to explain their thinking, pointing becomes less useful and students need to be more precise in how they are using the representation you have shared.

Some students may not feel ready to engage with alternative approaches and representations, particularly if they feel they are making progress with their own approach. For these students, it could be helpful to ask them to focus on looking for connections between their own work and the representations and the explanations other students are sharing. You could ask the students about similarities and differences between the approaches, or if they could use a different representation to support their argument.

Students can find it hard to understand others’ ideas without additional support, particularly when it is very different from their own. Following an explanation, you could invite the rest of the class to ask questions to the presenting student(s). You could ask other groups with similar approaches to build on ideas that have just been presented. Another option would be to ask another student to rephrase what has just been presented in a slightly different way.

Can you advise Alison on which die to choose once she knows which die Charlie has selected?

Students may have already answered this question orally as part of their work, but this question could be treated as an opportunity for a written task, for example as homework. In this way, students need to re-think their solution while writing a text and they may need to develop their argument in more detail than when they presented their work orally.

Feedback on this writing could be given by the teacher or by other newly formed groups of students. In this case, students also need to think through another’s idea and how it is communicated, which opens up important mathematical learning opportunities about how to communicate advice and the mathematical argument that it is based on.

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