# Dominoes

### Task 33 ... Years 2 - 10

#### Summary

Dominoes are used to make addition sums on the assumption that the two sides of a domino are representing the tens and ones part of a number. For example the domino [2|5] would represent 25 and the domino [5|2] would represent 52. Trying to add these two dominoes would require the domino [7|7] which isn't in the set.

However, there are many problems which can be set where the answer domino is in the set. The task begins with sums which have no carrying and becomes more challenging as carrying is included.

#### Materials

• 1 set of Double Six dominoes

#### Content

• place value
• addition with and without carrying
• pattern
• problem solving strategies

#### Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

Clearly there is more than one answer to Questions 1 & 2. One might be tempted to ask:

• How many domino sums altogether and how do you know when you have found them all.
One strategy for approaching this is to break the problem into smaller parts by taking each domino in order and asking which sums can be made with it. For example:

[0|0]
None, because adding zero to a number leaves the number unchanged. Therefore a second copy of the other domino would be required, and there are no repeats in a Double Six set.

[0|1] ... [1|0]
[0|1] + ... [0|2], [0|3], [0|4], [0|5]
- NB: [0|0], [0|1] and [0|6] can't be used. Why?
[0|1] + ... [1|1], [1|2], [1|3], [1|4], [1|5]
- NB: Why not [1|0]?
[0|1] + ... [2|0], [2|1], [2|2], [2|3], [2|4], [2|5]
[0|1] + ... [3|0], [3|1], [3|2], [3|3], [3|4], [3|5]
[0|1] + ... [4|0], [4|1], [4|2], [4|3], [4|4], [4|5]
[0|1] + ... [5|0], [5|1], [5|2], [5|3], [5|4], [5|5]
[0|1] + ... [6|0], [6|1], [6|2], [6|3], [6|4], [6|5]

[1|0] + ... [0|2], [0|3], [0|4], [0|5], [0|6]
[1|0] + ... [1|1], [1|2], [1|3], [1|4], [1|5], [1|6]
[1|0] + ... [2|0], [2|1], [2|2], [2|3], [2|4], [2|5], [2|6]
[1|0] + ... [3|0], [3|1], [3|2], [3|3], [3|4], [3|5], [3|6]
[1|0] + ... [4|0], [4|1], [4|2], [4|3], [4|4], [4|5], [4|6]
[1|0] + ... [5|0], [5|1], [5|2], [5|3], [5|4], [5|5], [5|6]

[0|2] ... [2|0]
Continue the reasoning being careful not to include sums previously listed.

If the reasoning is followed through carefully all the possible sums would be found and that list could easily be split into those with and without carrying. But all these sums couldn't be made simultaneously. The nice twist in Question 3 is to use all the dominoes simultaneously in a suite of nine additions.

One solution without carrying is:
 00 22 06 30 31 21 41 10 20 23 44 50 16 24 15 04 52 33 11 66 56 46 55 36 45 62 53 34

One clue to finding a solution such as this is to realise that if there is no carrying, the problem really resolves to counting the number of dots. No new numbers are introduced by carrying. The total of the dots on the 28 dominoes is 168, so half of these must be in the question rows and half in the answer row, because that is how addition works. So, one way to approach the problem is to set up dominoes in an answer row so the total of dots is 84, then work backwards.

• Note:
If you check by counting dots across the answer lines above the total is 84. It does work, but remember you are just counting dots so the addition sequence (starting on the left) is: 6 + 6 + 5 + 6 + 4 +...

One solution with carrying is:
 00 15 16 10 41 35 11 30 23 12 25 26 56 05 20 33 06 22 31 40 42 66 46 55 44 36 45 43

Sorting out a solution like this depends on working through all the solutions as for Questions 1 & 2. Studying the carrying solutions shows that they all use an odd number of the 168 dots. For example:
15
25
40
uses 17 dots, there are 151 left to use in the remaining 8 sums. If there is no carrying in these, then the dots would be shared equally between the questions rows and the answer rows. Not possible with an odd number. Therefore there must be (at least) one other carrying sum, which in the solution above is:
16
26
42

Well, at least that's a starting point!

#### Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

Introduce the task to the class using a set of floor dominoes. Print each of the 28 pages of this PDF and store each in its own plastic envelope. Then your class set can be used over and over.

Once the concept of the problem is understood and the restrictions provided by the structure of a domino set have been identified, the work of finding all the solutions can be shared between groups in the class. This would give purpose to a whole class investigation and provide lots of arithmetic practice in context. Clearly it is best for these investigations if you have enough sets of dominoes for one between two. Perhaps, to start off a school collection, families could donate sets no longer used at home. Or you could design a table size print set based cutting, pasting and shrinking the drawings in the floor domino set. Print on card and laminate.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 66, Dominoes, which includes an Investigation Guide that concentrates on the additions that involve carrying. Its companion lesson Domino Trails (#95), supplies a printable set of table dominoes.

#### Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Dominoes task is an integral part of:

• MWA Number & Computation Years 5 & 6
• MWA Number & Computation Years 7 & 8

The Dominoes lesson is an integral part of:

• MWA Number & Computation Years 7 & 8
This task is also included in the Task Centre Kit for Aboriginal Students.