# Domino sets

What if you were given lots of dominoes in a bag?

Before you started playing it might be a good idea to find out if you have a full set!

How would you go about it?

How could you be sure?

What if someone gave you some 0-9 dominoes?

How many do you think there would be in a full set?

If you do not have any dominoes, you might find our interactive Dominoes Environment useful.

*You may like to try Amy's Dominoes as a follow-up to this task.*

Are there the same number of dominoes with ones?

Could you organise the dominoes in some way to show you have/haven't got them all?

Well done to everybody who had a go at this activity. This task can be quite fiddly so we were sent several incorrect solutions, as well as a lot of correct ones! One of the solutions we received was more suitable for Amy's Dominoes, so we've uploaded that there instead.

The children from Linkwood Primary in Scotland sent in the following ideas:

We started with the doubles from 6 to 0. Then we thought about all the other dominoes that would have a 6. Then we did the same for 5, 4, 3, 2 and 1. When we had all of the dominoes we counted them all and found we had 28. We also noticed that for the dominoes with 6 there were 7, for 5 there would be 6 and so on.

Good ideas - this method was used by a lot of children!

We were sent in several similar solutions by the children at Brentwood Preparatory School in England. Bella said:

I used a set of 0-6 dominoes and discovered that the number of dominoes was always 1 more than the value of the face. e.g: there are 7 dominoes with a 6 on them, 6 with 5 dots, 5 with 4 dots .......

To check you have all the dominos, you could start with all the doubles and sort them by their faces, eg: 6-6 to 6-0.

In total there are 28 dominoes and I also found that, if you had 0-9, you would have 55 using the pattern that I found.

Chloe said:

You could arrange the dominoes in columns starting with a 0 on the left side of the domino and then 0-6 on the right side. In the next column you would have a 1 on the left and 0-6 on the right side. In every new column the left side would be increased by 1. The final column would have 6 on the left side. If all the dominoes are there you know that you have a full pack, but if there are some missing you know there is not a full pack.

Dhruv from The Glasgow Academy in the UK sent in the following picture, which can be clicked on to make it bigger. This the same arrangement that Chloe described in the previous solution.

Thank you all for sending in these very clear explanations, and well done for noticing that the arrangement will make a triangular shape.

Matilda and Eliza from Rainworth State School in Australia sent in this solution for the 0-9 dominoes:

Our solution is for the 0-9 dominoes. We lay the dominoes out in rows in order of the number on the top of the domino. When we finished laying them out, the pattern looked like a staircase because each one of the dominoes in the row below had been already used in the one above that. We counted the pattern - it went 1+2+3+4 and so on until +10. This equals 55 dominoes. We checked we were correct by counting the dominoes one by one. Our domino set was like the one described in the problem, the box was broken and not all the dominoes were there so we had to use dominoes from another set and we even had to draw one.

This is a very clear solution, and I like the way you've been creative with making a full set of dominoes when you couldn't find one. This is an excellent example of why we might need to be able to check how many dominoes we are expecting to find in a full set!

We received some excellent solutions from the following children at Ganit Kreeda, Vicharvatika, India: Ananya, Shivashree, Shreehari, Kanaa, Ruhi, Mrunmayee, Aarav, Shravani, Pushan, Ishan, Adithya, Aniket, Anirudh, Vikrant and Avyuktth. Take a look at Ganit Kreeda's full solutions to see their ideas.

Shreehari sent in the following video explaining some of these ideas:

Shaunak from Ganit Manthan, Vicharvatika, India also sent in a video to explain how they approached this task:

Thank you both for sending in these clear explanations of how you approached this task.

If you'd like to learn more about Shaunak's method, you might like to have a look at Shaunak's full solution to see an explanation of how the general version of this task can be solved using algebra.

Some children took a different approach and thought about how often each number would come up across a full set of dominoes. Ayden from St John's College in Cardiff, Wales said:

If you bought a pack of 0-6 dominoes, and they came in a bag, and if you wanted to check if you had a full set, you could count the number of dominoes with a particular number on one side, and then see if it adds up to 7, because there are 7 sets of each number. With this method, you can find out exactly what domino you are missing. For example, if you are counting for the number 3, and there are 6 dominoes, and 3 with 0, 1, 2, 3, 5, and 6, then you know that the 3/4 domino is missing. This is a more efficient, and quicker method than counting all the dominoes individually, but if you chose to do that, then you would know if there is a full set by having 28 dominoes in total.

In a set of 0-9 dominoes, you could use the same method mentioned above to find out if you have a full set. However, if you choose to count the dominoes individually, you would know if you are correct by having 55 dominoes in total.

We received a lot of similar solutions from the children at Park Hill Junior School in the UK. Isaac, Felix, Alba and Hugo had a different way of thinking about this task:

There should be 8 of each number so you count 8 sixes, 8 fives, 8 fours, 8 threes, 8 twos, 8 ones and 8 blanks.

The key to finding the missing ones is:

Line them up in rows or columns of its number.

OR:

There should be 28 dominoes in a pack so you count them.

4x7=28

This way of thinking about it is really interesting - I wonder why there are 8 of each number? And I wonder why that means there will be 28 dominoes in a pack?

Why do this problem?

When we use structured apparatus such as Cuisenaire rods, dominoes, Numicom etc. we often assume that children understand more about the structure than perhaps they do. This activity encourages children to explore the structure in their own way - and you may be surprised at how many different ways there are. You could use this activity in one of two ways, depending on your children. The most obvious way is to focus on the different patterns that can be built up, but you could also use the activity to focus on the explanation or justification of the children's description or argument. You might like to have a go yourself before offering this activity to the children...

Possible approach

You will need several sets of dominoes (you could make laminated card sets from this print out), one set between three or four children. If the dominoes are usually stored in a box, transfer them to a bag so that the children don't have a sense of a box being full.

Ask the children to tip the dominoes out onto the table and have a look at them, perhaps having an opportunity to play with them and arrange them into different patterns if this is the first time they have seen them.

Suggest that they may or may not be full sets and that before we can use them for a game we need to know if any are missing. Could they help you by finding out? Allow some time for this.

When a group gives you a definitive answer ask them how they know and challenge them to arrange the dominoes on the table in some way to show you that they have/have not got a full set. Again allow some time for this.

When, and if, appropriate you could suggest that the children move around the classroom to have a look at what other groups have done. You may then want to offer a little more time for those groups who haven't made much progress to have another go, having seen others' responses.

Groups will respond in different ways. In increasing order of sophistication, groups may:

- play a domino game where they match dominoes into a long train or loop, for example:

- make piles or groups of dominoes with numbers in common, for example:

- arrange the dominoes into a pattern or array, for example:

In each case, ask how the pattern/structure they have chosen helps. The most sophisticated explanations will use the idea of all possible pairings of the numbers 0-6.

The interactivity here may be helpful for whole class discussions of this activity and/or for learners to use as they work on the task, if real dominoes are not available.

Key questions

How many are there with blanks?

Are there the same number of dominoes with ones?

Tell me how you're arranging them. Why did you choose that way?

Could you organise them in a different way?

How does this show you that you have/haven't got them all?

Possible support

Reducing the number of dominoes can make a pattern easier to spot. Offer a subset, say 0-4, and allow time to play and look for patterns before asking if they have a full set.

Possible extension

What if you had a set of 0-9 dominoes (download a set here). How many do you think you would have? Do you need to make a set? How else could you record your ideas?

The task Amy's Dominoes is a natural extension to this activity and can be solved using the dominoes as a resource, or without them by more confident children.