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### Number and algebra

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### Working mathematically

### For younger learners

### Advanced mathematics

# Dicey Array

## Dicey Array

### Why play this game?

### Possible approach

### Key questions

### Possible support

### Possible extension

## You may also like

### Prompt Cards

### Round and Round the Circle

### What's Left?

Links to the University of Cambridge website
Links to the NRICH website Home page

Nurturing young mathematicians: teacher webinars

30 April (Primary), 1 May (Secondary)

30 April (Primary), 1 May (Secondary)

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Age 5 to 11

Challenge Level

- Game
- Getting Started
- Student Solutions
- Teachers' Resources

This is a game for two players.

You will need two 0-9 dice (or you could use our interactive dice), some counters, pen and paper.

Watch the video below (which has not got any sound). It shows two people playing the game.

**Can you work out how to play?
What do you think the rules might be? **

If you are unable to view the video, the rules of the game are hidden below.

**The aim of the game:**

To be the first player to reach the target score.

**How to play:**

1. Players agree a target score to reach.

2. Player 1 rolls the two dice and finds the total of the two numbers.

3. Player 1 counts out that number of counters and creates as many rectangular arrays as they can using that number of counters. They score a point for each correct array. (Each array must use all the counted-out counters.)

4. Player 2 rolls the dice in the same way and uses that number of counters to create as many rectangular arrays as possible. Again, they score a point for each correct array.

5. Play continues like this with players taking it in turns to roll the two dice. The points scored in each player's second turn are added to the number of points they scored in their first turn to make a running total. Points scored in their third turn are added on again, and so on.

6. The winner is the player who reaches the target number of points first.

Have a go at playing the game several times.

**Which dice totals are good to get? Why?**

**Which dice totals are not so good to get? Why?**

**Does it matter if you go first or second? **

**What happens if you can choose to add or subtract the two dice numbers? How does this change the game?**

This low threshold high ceiling game offers a meaningful and motivating context in which learners can deepen their understanding of factors, multiples and primes, and develop their fluency with finding factor pairs.

*This problem featured in an NRICH Primary webinar in April 2021.*

Play the video to the group, simply saying that you'd like them to watch carefully, in silence, to see whether they can work out the rules of the game. The video has no sound and includes three turns altogether, so you might choose to play the whole clip straight away, or you could pause it after player 1's first turn (about 1:20 in).

Invite learners to talk in pairs about the possible rules and how the game is won. Emphasise that they may not be completely sure and that is alright. They may even have some questions to seek clarity. After a suitable length of time, continue to play the video or show it again so that learners can check their initial thoughts.

Bring everyone together again to facilitate a whole group discussion in order to agree on the rules. You could reveal them on screen by clicking the 'Show' button on the problem page. Once everyone is clear, allow them time to play the game several times in pairs without saying much more yourself. It is important that learners are able to 'get into' the game before being expected to
analyse it in detail. You may decide that you give children the freedom to develop their own ways of recording their arrays.

You could then invite the group to begin to think about which dice totals are good to get and why (if they haven't done so already). At this point, you could put them in groups of four so that they play two against two. This gives them the opportunity to discuss their thinking with their partner. Watch out for those pairs who have developed a systematic way of working so they are
ensuring they find all possible arrays. You may wish to lead a mini plenary to focus on this.

In the final plenary, you could share thoughts about 'good' and 'not so good' dice totals. Encourage learners to articulate their reasoning clearly and suggest that anyone in the class can ask questions to seek clarification. You can ask another member of the group to re-phrase the reason in their own words, which not only helps that individual but also may help other learners who had not followed the reasoning the first time round. Where appropriate, support learners to use correct mathematical vocabulary, for example 'factor', 'multiple', 'prime', or use this as an opportunity to introduce that vocabulary if your class has not met it before.

How do you know you have created all the arrays that can be made using that number of counters?

Which dice totals have you found make lots of arrays?

Which dice totals don't make many arrays?

Can you explain why?

You may prefer to have pupils play in groups of four from the start, with one pair playing against another pair. That way, they can collaborate as they create their arrays.

Learners could tweak the rules of the game. What would happen if they choose a different target score? What would happen if they use different dice? Encourage learners to ask their own 'what if...?' questions too.

These two group activities use mathematical reasoning - one is numerical, one geometric.

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Use this grid to shade the numbers in the way described. Which numbers do you have left? Do you know what they are called?