More and more buckets
In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?
Age
7 to 11
Challenge level
Problem
It can be used as a follow-on from Buckets of Thinking.
In this challenge, buckets come in five different sizes: the capacity of a bucket is $2$ litres, $3$ litres, $4$ litres, $5$ litres or $6$ litres. You can choose any number of buckets from two to six, including two and including six. Here are some pictures of buckets. The colours do not matter - they are just to make them look nice!
Image
Suppose we choose to use four buckets which each hold $5$ litres, like this:-
Image
We're going to pour water into the buckets, sticking to these rules:
RULE $1$ :- All the buckets must have a different number of litres of water.
RULE $2$ :- Every bucket must contain some water.
RULE $3$ :- Only whole numbers of litres may be used (so no halves, thirds etc.).
So, I'll work this one with you:
In the four buckets you could have:
$1, 2, 3, 4$ litres
or $2, 3, 4, 5$ litres
or $1, 3, 4, 5$ litres
or $1, 2, 3, 5$ litres
or $1, 2, 4, 5$ litres
This looks like all the possibilities obeying the three rules above.
You might like to check that you agree there aren't any other combinations.
Can you explain how you know we've got them all?
Your challenge is to
A/ Choose a number of buckets
B/ Decide on the size they will all be
C/ Find all the different possibilities obeying the three rules above.
You might then like to try again with a different choice.
When you've done that you could compare the two sets of answers and maybe make some suggestions. Let us know what you come up with!
Getting Started
How do you know you have got all the possibilities?
It might help to draw large outlines of your four buckets on paper or card then to use digit cards to show the amount of water each one contains.
How will you keep track of the possibilities you have found?
It might help to draw large outlines of your four buckets on paper or card then to use digit cards to show the amount of water each one contains.
How will you keep track of the possibilities you have found?
Student Solutions
Well done to everyone who engaged themselves in this challenge. Here are just some of the solutions we had emailed to us. Eskdale School sent in two solutions that I'm showing here.
First Andrew,
With 3 buckets, each holding 4 litres, there are only four solutions.123, 234, 134, 124
Second Matthew
If we are only allowed to use four numbers out of five, then we must leave out one number. Once we have used our numbers there is always one left. There are five numbers to pick out of. So there are five possibilities.James, from St. John French Immersion School in Ontario sent in this good one (and like Jack and Gill that went up the hill he used pails instead of buckets):
I took two 5L pails, one 2L pail, one 6L pail and two 3L pails. I put 5L in one of the 5L pails and 4L in the other. I put 2L in the 2L one and 6L in the 6L one. And for the two 3L pails, I put 1L in one and put 3L in the other.
Rajeev from Fair Field School sent in some very thorough thoughts and ideas as follows:
6 litres and 5 buckets would have a combination of 6
6 litres and 4 buckets would have a combination of 15
7 litres and 6 buckets would have a combination of 7
7 litres and 5 buckets would have a combination of 21
7 litres and 4 buckets would have a combination of 35
7 litres and 3 buckets would have a combination of 35
7 litres and 2 buckets would have a combination of 21
7 litres and 1 buckets would have a combination of 7
and with 12 litres and 6 buckets it would be 924
and with 13 litres and 6 buckets it would be 1716
He also showed here.doc or here.pdf how these can all be found by exploring Pascal's Triangle. Well Done.
Teachers' Resources
More and More Buckets
It can be used as a follow-on from Buckets of Thinking.
In this challenge, buckets come in five different sizes: the capacity of a bucket is $2$ litres, $3$ litres, $4$ litres, $5$ litres or $6$ litres. You can choose any number of buckets from two to six, including two and including six. Here are some pictures of buckets. The colours do not matter - they are just to make them look nice!
Image
Suppose we choose to use four buckets which each hold $5$ litres, like this:-
Image
We're going to pour water into the buckets, sticking to these rules:
RULE $1$ :- All the buckets must have a different number of litres of water.
RULE $2$ :- Every bucket must contain some water.
RULE $3$ :- Only whole numbers of litres may be used (so no halves, thirds etc.).
So, I'll work this one with you:
In the four buckets you could have:
$1, 2, 3, 4$ litres
or $2, 3, 4, 5$ litres
or $1, 3, 4, 5$ litres
or $1, 2, 3, 5$ litres
or $1, 2, 4, 5$ litres
This looks like all the possibilities obeying the three rules above.
You might like to check that you agree there aren't any other combinations.
Can you explain how you know we've got them all?
Your challenge is to
A/ Choose a number of buckets
B/ Decide on the size they will all be
C/ Find all the different possibilities obeying the three rules above.
You might then like to try again with a different choice.
When you've done that you could compare the two sets of answers and maybe make some suggestions. Let us know what you come up with!
Why do this problem?
This activity is a good one to try when pupils are used to doing some investigations with just a little prompting from the teacher. It is not obvious how to go about working on solutions, and so this allows children to tackle the problem in many different ways. No particular skills of the four rules of number are required, so it is
appropriate for a very wide attainment range. It can be a catalyst to encourage pupils to work in a systematic way.
Possible approach
It is not essential for children to have tackled Buckets of Thinking but it may be helpful to have done so.
You could create some images of buckets on an interactive whiteboard, or on card to fix to an ordinary board. This may help as you can drag buckets around the board so that the rules become clear. You could introduce the example and ask everyone to write down on a strip of paper one way of filling the buckets with water. Ask them to compare what they have written with others sitting near
them and encourage them to find other solutions. You could invite learners to pin up their strips on the board so that you can order them in some way. This will help the whole group decide whether any have been missed out. Of course there are many different ways to order the possibilities so encourage and discuss different approaches.
Leave pupils to work on some examples of their own and then to share anything they notice with a partner or in a small group. This investigation would lend itself to being worked on over an extended period of time. You could devote an area of the classroom wall to it and ask learners to contribute findings, comments and questions to this wall space over the next few days. It may be that they
will be able to predict the number of possibilities as they identify and explain patterns.
Key questions
Tell me about the answers you've got.
How do you know you have got all the possibilities?
What do you now think is a good way of doing this kind of challenge?
Can you predict the number of different possibilities you might get before you work them out?
How did you make your prediction?
Possible extension
Some children can be challenged to produce a table or spreadsheet to show what you get with all the possible choices of groups of buckets that can be made.
Possible support
Having pictures of buckets of different sizes cut out of card and laminated will help some children with this problem. They can be encouraged to write (using a 'wipeable' marker) on each bucket the amount of water it contains or they could use digit cards to place on the buckets.