DominoesAuthor(s): Andy MartinSource: Mathematics in School, Vol. 32, No. 4 (Sep., 2003), pp. 41-43Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30212298 .

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Dominoes by Andy Martin

Introduction

The dominoes task is a mathematically rich activity that raises many "What if." and "How do you know?" type questions. It is accessible to pupils of all abilities but can also be used to challenge able pupils with higher level thinking skills. I have used this activity with parents as one problem that can be shared by the family for 'maths around the kitchen table!'.

The Task

DOMINOES MATERIALS

A full set of twenty-eight [28] dominoes

1. Can you place three [3] dominoes so that they make a correct addition?

2. Can you find a combination which involves carrying?

3. Can you arrange all twenty-eight [28] dominoes into nine sums which all show a correct addition?

This can be done without carrying, OR it can be done with some carrying.

Getting Started

This task is a wonderful resource to use with pupils at the concrete stage. The question "Can you find three pieces to make a correct addition?" has many potential solutions. In my experience all secondary age pupils can handle the task at this stage. The pupils should then be encouraged to work like a mathematician. At this point we look at the working mathematically process document from Maths300, which guides our problem solving activities. We discuss strategies that might work. For example, would it help to:

* Guess, check and improve. Put down any two dominoes and see if they work. Learn from this and modify appropriately.

* Work backwards. Start with a 'larger' domino that is likely to be the sum of others and search for those others.

* List all possibilities. There are 28 possible first tiles and 27 possible second tiles. So there are only 756 pairs to check. Could we list all the possibilities and then share the work of checking them?

Aiden worked on this task in year 7. He concluded that there were 384 different ways of making a domino sum. He listed in his book (see Figure 1) a summary of his discoveries for each possible total.

Dominoes in the Classroom

I have found this task works well on a whole-class basis. Pupils can work in pairs with a set of dominoes and the task card is used as a transparency on the

OH. I like to use

Aiden's summary as a challenge for the class. Do they agree with his results? Can a total be made in more ways than Aiden hypothesized? This can lead to very fruitful discussion between pupils, especially when they are unable to construct as many solutions as Aiden for a particular total.

The Nine-Sum Challenge

The extension on the task card has become an ongoing challenge for pupils. Can the 27 dominoes (the double blank is removed!) be used to make 9 domino sums simultaneously? My first pupil to find a solution was Sarah. We call this solution the Sarah Hutchinson Solution.

The Sarah Hutchinson Solution

41 44 22 31 42 52 60 15 23 12 10 33 30 20 11 04 50 43

53 54 55 61 62 63 64 65 66

This inspired other pupils to find and own a solution. Michael, Karen and Kelly duly became famous with their solutions.

The Michael Kirton Solution

10 06 52 33 50 30 51 41 35 34 40 02 22 11 32 12 24 31

44 46 54 55 61 62 63 65 66

Mathematics in School, September 2003 The MA web site www.m-a.org.uk 41

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7a~k lr P r~f ~t: 60 rS 61 S 16 6i~ y I rai 3 63 to

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Fig. 1 Aiden's summary of his discoveries of the number of ways to make each total

The Karen Shields Solution

04 20 26 33 22 03 13 60 25 12 15 10 11 23 43 42 05 41

16 35 36 44 45 46 55 65 66

The Kelly Plant Solution

15 21 33 06 31 44 52 42 43 01 05 02 30 14 11 04 22 23

16 26 35 36 45 55 56 64 66

Sarah revisited the problem two years later when she was in year 9. Working with her friend Clare she discovered another solution. We agreed to call this The Hutchinson-Wright Solution.

The Hutchinson-Wright Solution

40 42 33 12 41 52 60 51 23 05 11 22 44 20 10 03 13 43

43 53 55 56 61 62 63 64 66

I regularly displayed their work in my classroom as different classes worked on the task. New questions were posed:

* Can it be done if we carry digits from units to tens?

* Do we always have to use three doubles in one solution?

* Why do I always see consecutive numbers in the totals?

Scott Gillatt, then in year 8, cracked the second question. He produced a solution to show that the problem can be solved without using three doubles in one of the sums.

The Scott Gillatt Solution

11 02 12 06 44 33 51 43 50 14 24 23 30 01 13 04 22 16

25 26 35 36 45 46 55 65 66

So we had a classic case of disproof by counter-example and a proud pupil to have a solution named after him that was distinctly different from all the others that we had found!

Involving Parents

I invited parents to a promotion evening this year where I explained that a group of pupils would be using tasks like this for Home-Lending. Parents were invited into school to support their child and participated in a Parents workshop. During the evening the following solutions emerged:

The Green Family Solution

14 04 03 20 24 44 51 05 23 12 31 33 25 22 11 10 60 43

26 35 36 45 46 55 61 65 66

The George Pugh and Lynne Hibbard Solution

14 05 32 25 63 15 30 04 44 10 26 11 21 02 45 33 61 22

24 31 43 46 55 60 63 65 66

I am indebted to George Pugh and Lynne Hibbard for their work, which has demonstrated that the problem could be solved using carrying. We discussed this solution at the year 7 parents evening, a refreshing change - to discuss original work from parents that had been done with their children.

The fact that a solution was found with carrying created more interest in the task. Two pupils in year 10 offered other solutions.

The Tony Andrew Solution

06 14 12 16 11 51 32 42 35 04 20 33 36 44 05 30 22 31

10 34 45 52 55 56 62 64 66

The Peter Noble Solution

06 40 11 35 52 43 62 23 54 16 02 33 15 03 13 01 41 12

22 42 44 50 55 56 63 64 66

The naming of solutions caused immense interest with my current year 7 class. Hannah and Harriet noticed that they could adapt Scott's solution by reversing some of the dominoes. Since the totals were now different it had to be a

42 Mathematics in School, September 2003 The MA web site www.m-a.org.uk

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different solution! They argued that this could be done with more than one of his sums so that they could claim a lot of answers on the basis of his one solution. A wonderful opportunity for year 7 to begin to consider permutations and the concept of a 'different solution'.

Involving Staff

We used this task during the school training day for 'Numeracy Across the Curriculum'. We had the opportunity for the whole staff to work with dominoes and a range of domino problems. Whilst nobody generated a new solution to the 9-sum problem during the day, the husband of one of the staff did send me a new solution which he did in minutes!

The lan Scott Solution

23 50 11 10 60 21 33 22 41 30 04 44 51 02 42 31 43 25

53 54 55 61 62 63 64 65 66

I had also visited the Maths300 web site to look at the work being done using this task by Sally Collins in Denver, USA. Sally had been working on the problem "What is the greatest number of domino pieces you can use in one sum?". Sally and her husband had found a solution with 9 domino pieces. A colleague of mine produced a solution using 10 domino pieces. We call this The Warren Beaumont Solution to 'the maximum number of domino pieces in one sum' challenge.

The Warren Beaumont Solution

01 02 03 04 05 06 11 22 12 66

Warren's solution is now displayed on the Maths300 web site beside that of Sally and Ed. Who will be the next contributor?

I held a second meeting with parents of year 7 pupils who had not explored the task. After some frantic working with their children we had another solution offered by Paul.

The Paul Booth Solution

06 04 03 22 26 45 50 44 32 14 31 33 24 25 10 11 21 34

20 35 36 46 51 55 61 65 66

The following day I was presented with a solution from his parents. They had worked on the task after the meeting and tried to find their own solution.

Paul's Mum and Dad's Solution!

02 22 12 14 45 16 40 06 53 23 11 24 30 01 34 15 56 13

25 33 36 44 46 50 55 62 66

In the same lesson I was presented with another different pupil solution, Wendy had also worked overnight on the challenge.

The Wendy Reynolds Solution

22 25 03 21 31 60 44 51 43 04 11 50 33 24 01 20 14 23

26 36 53 54 55 61 64 65 66

So now my classroom had 13 different solutions to the challenge on display.

Conclusion

There is nothing new for many teachers in using dominoes for mathematics. The use of dominoes for number sums has been very fruitful for me and created a lot of enthusiastic problem solving. I still don't know how many different solutions exist for the 9-sum challenge or if a single sum can be created from 10, or more, dominoes in other ways. However, I am sure that I will work with pupils who will want to explore this further! As for my able pupils, I hope they will try to solve the third problem posed a few years ago. Now, why do we always seem to get consecutive totals?

Keywords: Dominoes.

Author Andy Martin, Thorne Grammar School, St. Nicholas Road, Thorne, Doncaster DN8 5BQ. e-mail: andy@lmartinl.freeserve.co.uk

Crossnumber Puzzles

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Mathematics in School, September 2003 The MA web site www.m-a.org.uk 43

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