Amy's dominoes
Problem
Amy's Dominoes printable sheet
You may like to have a look at Domino Sets before trying this problem.
Amy has a box containing ordinary domino pieces but she does not think it is a complete set.
She has 24 dominoes in her box and there are 125 spots on them altogether.
Which of her domino pieces are missing? How do you know?
If you do not have any dominoes, you might find our interactive Dominoes Environment useful.
Getting Started
How many spots are there altogether in a complete set of dominoes?
Student Solutions
We had quite a few solutions sent in for Amy and her dominoes. Don't forget that we are looking for solutions that include reasoning, not just the answer.
Ava from Clifton Hill Primary School in Australia sent in this picture:
Well done for counting these carefully, Ava!
Lyneham Primary Maths Challenge Group in Australia wrote:
Thanks for a great problem. Both groups worked on it and we made it a corridor display to get some more kids (and parents) thinking about it.
Their work can be seen here:
Lyneham-Amy dominoes-1.doc or Lyneham-Amy dominoes-1.pdf
Bjorn from Belfry Overstrand School in England wrote the following:
I wrote down all different dominoes in order (0:0, 0:1, 0:2...6:6), I counted them all and there were 28.
It told me that Amy had four dominoes missing.
I then added all the spots on the dominoes together and made a total of 168.
It told me that Amy was missing 43 spots.
I was looking for four dominoes that totalled 43 spots.
I began with the highest domino (6:6) because we had to get a high number (43) with a low number of dominoes.
I worked out that the missing were 6:6, 5:6, 5:5 and 4:6. These four dominoes totalled the missing amount of 43 spots!
Amy needs to look after her dominoes by putting them in a safer place!
That's very true Bjorn.
Matthew from the British School in Brussels listed all the dominoes in a very systematic way, which is a great idea if you don't have a domino set to hand:
Matthew went on to describe his method, which was very similar to Bjorn's above.
Alex from Forest of Galtres Primary School sent a very comprehensive solution:
She has 24 dominoes with 125 spots on.
I found our set of dominoes and I put them in order with 6s on the top, then 5s, 4s, 3s, 2s, 1s and blanks with doubles first.
I noticed that there are 8 of each number. So I did
6 x 8 = 48
5 x 8 = 40
4 x 8 = 32
3 x 8 = 24
2 x 8 = 16
1 x 8 = 8
0 x 8 = 0
and added them up to count the spots. There are 168 spots on a set of dominoes and there are 28 dominoes in a full set.
So I did 168 - 125 = 43.
Which meant there were 43 spots missing on four dominoes.
43 shared between the four dominoes meant there were about 10 spots on each domino. So I took out the two dominoes with 10 on which equalled 20. Then I took out the double six which took me to 32 so I needed an 11 and the only one was the 6 and 5.
I know this is the only answer because these are the four highest dominoes.
Thank you, Alex. Ella and Mia from Clatford Primary School also sent very clear solutions. Like Alex, Ella thought about what she could conclude from the fact that 43 spots were spread across only four dominoes. She said:
That is an average of over 10 spots per domino!
That's a helpful way of thinking about it, Ella. William from Churchdown Village Primary School also calculated an average number of dots per domino.
Suzanne journeyed through the challenge in a rather different way, working with a partner. I've summarised the method they used to work out how many dots there are in a set of dominoes.
They noticed that the number that you are looking at is always 2 more than that number you are considering. Two more because there's one of them with a zero and another that goes to make the double domino.
They called the number that you are considering n and the number that there are of them they called a and a is always 2 more than n.. They also realised that you had to add the numbers below n.
As an example when n = 5 the number of dots on all the fives is n x a, i.e. 5 x 7, plus 1 + 2 + 3 + 4. So there are 35 + 10, i.e. 45 dots on all the 5's.
We also received well explained solutions from: 23CG at Old Orchard Primary in Australia; Jake (who didn't give his school); Monty from the British School in Brussels; the Smarty Plants Bubble at Harlands Primary School; Bethany and Fatima from Arnhem Wharf Primary School; Clara from Meavy Church of England Primary School; Toby from St Johns
College School, Cambridge; Kieran from Aldro School; Jake and Johann from the British School Manila in the Philippines; Eleanor from St Nicholas C of E, East Challow and Y6 from Upton-upon-Severn C. of. E Primary School. Thank you to you all!
How about if there was one spot less missing - so that there were 126 spots. What might the possibilities be? Would there be more, fewer or could the problem not be solved?
Teachers' Resources
Amy's Dominoes
Amy's Dominoes printable sheet
You may like to have a look at Domino Sets before trying this problem.
Amy has a box containing ordinary domino pieces but she does not think it is a complete set.
She has 24 dominoes in her box and there are 125 spots on them altogether.
Which of her domino pieces are missing? How do you know?
If you do not have any dominoes, you might find our interactive Dominoes Environment useful.
Why do this problem?
This problem requires learners to understand the numbering system on dominoes and use this to solve a problem. Learners will need to use addition, subtraction and multiplication as well as logical reasoning.
Possible approach
If you have an interactive whiteboard, you may find our Dominoes Environment useful for this problem.
You could start by giving the whole group sets of dominoes to sort out in pairs or alternatively, if the children are already familiar with dominoes ask questions such as "How many domino pieces have four spots on them altogether?" and "How many domino pieces have five spots on them?"
Children could then work in pairs on the actual problem with a real set of dominoes or use the six spot dominoes from the printable resources page.
In a plenary, children could discuss not only the solution, but what information they needed to have to work it out and what calculations they had to do along the way. There will be several different approaches which will not only help other children but will also inform you about their thinking.
Key questions
How many dominoes are there in a complete set? So how many are missing in Amy's set?
How many lots of six spots are there in a set of dominoes? How many spots is that altogether?
How many spots are there altogether in a complete set of dominoes?
Possible support
Children could explore the structure of a set of dominoes first, which is encouraged in the task Domino Sets.
Possible extension
You could ask some follow-up questions, such as: If Amy had not 104, but 140 spots could you have found a solution? Is there just one possible answer or more? What was the fewest number of spots that could have been on the dominoes if four of them were missing? What could have been the greatest number of spots on a set that was missing four pieces?