6850 /www/nrich/html/content/id/6850/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <em>This activity has been particularly created for the excepionally mathematically able. (The pupils that you come across in many classrooms just once every few years.)<br></br> It can be used as a follow-on from  <a href="http://nrich.maths.org.uk/1169">Buckets of Thinking.</a>.</em><br></br> <br></br> <br></br> 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!<br></br> <div style="text-align: center;"><mdo:image alt="MultiB" height="184" src="multiBuckets.%20jpg" width="244"></mdo:image></div> <br></br> <div style="text-align: left;">Suppose we choose to use four buckets which each hold $5$ litres, like this:-</div> <br></br> <br></br> <div style="text-align: center;"><mdo:image alt="4,5 B" height="94" src="4%2C5%20Buckets.jpg" width="317"></mdo:image></div> <div style="text-align: left;">We&#39;re going to pour water into the buckets, sticking to these rules:</div> <br></br> <div style="text-align: left;">RULE $1$ :- All the buckets must have a different number of litres of water.</div> <div style="text-align: left;">RULE $2$ :- Every bucket must contain some water.</div> <div style="text-align: left;">RULE $3$ :- Only whole numbers of litres may be used (so no halves, thirds etc.).</div> <br></br> <div>So, I&#39;ll work this one with you:</div> <br></br> <div>In the four buckets you could have:</div> <br></br> <div>$1, 2, 3, 4$ litres</div> <div>or $2, 3, 4, 5$ litres</div> <div>or $1, 3, 4, 5$ litres</div> <div>or $1, 2, 3, 5$ litres</div> <div>or $1, 2, 4, 5$ litres</div> <br></br> <div style="text-align: left;">This looks like all the possibilities obeying the three rules above.</div> <div style="text-align: left;">You might like to check that you agree there aren&#39;t any other combinations.</div> <div style="text-align: left;">Can you explain how you know we&#39;ve got them all?</div> <br></br> <div style="text-align: left;">Your challenge is to</div> <div style="text-align: left;">A/ Choose a number of buckets</div> <div style="text-align: left;">B/ Decide on the size they will all be</div> <div style="text-align: left;">C/ Find all the different possibilities obeying the three rules above.</div> <br></br> <div style="text-align: left;">You might then like to try again with a different choice.</div> <div style="text-align: left;">When you&#39;ve done that you could compare the two sets of answers and maybe make some suggestions. Let us know what you come up with!</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p><span class="editorial">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.</span></p> <p><span class="editorial">First Andrew,</span></p> With $3$ buckets, each holding $4$ litres, there are only four solutions.<br></br> $123, 234, 134, 124$<br></br> <br></br> <p><span class="editorial">Second Matthew</span></p> 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.<br></br> <br></br> <p><span class="editorial">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):</span></p> <br></br> I took two $5$L pails, one $2$L pail, one $6$L pail and two $3$L pails. I put $5$L in one of the $5$L pails and $4$L in the other. I put $2$L in the $2$L one and $6$L in the $6$L one. And for the two $3$L pails, I put $1$L in one and put $3$L in the other.<br></br> <br></br> <p></p><br></br> <p class="editorial">Rajeev from Fair Field School sent in some very thorough thoughts and ideas as follows:</p> <br></br> <br></br> $6$ litres and $5$ buckets would have a combination of $6$<br></br> $6$ litres and $4$ buckets would have a combination of $15$<br></br> $7$ litres and $6$ buckets would have a combination of $7$<br></br> $7$ litres and $5$ buckets would have a combination of $21$<br></br> $7$ litres and $4$ buckets would have a combination of $35$<br></br> $7$ litres and $3$ buckets would have a combination of $35$<br></br> $7$ litres and $2$ buckets would have a combination of $21$<br></br> $7$ litres and $1$ buckets would have a combination of $7$<br></br> <br></br> and with $12$ litres and $6$ buckets it would be $924$<br></br> and with $13$ litres and $6$ buckets it would be $1716$<br></br> <br></br> <p class="editorial">He also showed how these can all be found by exploring Pascal's Triangle. Well Done.</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div><a href="http://nrich.maths.org/6850&amp;part=">This activity</a> 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 leaves scope for children to tackle it 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.</div> <br></br> <h3>Possible approach</h3> <div>It is not essential for children to have tackled <a href="http://nrich.maths.org/1169&amp;part=">Buckets of Thinking</a> but it may be helpful to have done so.</div> <br></br> <div>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.</div> <br></br> <div>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. </div> <br></br> <h3>Key questions</h3> <div>Tell me about the answers you've got.</div> <div>How do you know you have got all the possibilities?</div> <div>What do you now think is a good way of doing this kind of challenge?</div> <div>Can you predict the number of different possibilities you might get before you work them out?</div> <div>How did you make your prediction?</div> <br></br> <h3>Possible extension</h3> 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.<br></br> <h3>Possible support</h3> 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.<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">How do you know you have got all the possibilities? <br></br> 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. <br></br> How will you keep track of the possibilities you have found? <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Here is a table of the possibilities.<br></br> <div style="text-align: center;"><mdo:image width="409" height="386" src="Bucket%20Spread.jpg" alt="Spread"></mdo:image></div> <br></br> which is likely to be recognised as Pascal on its side.<br></br></mdoxml> 2 3 0 1 0 0 0 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? Decision Mathematics and Combinatorics Combinations Using, Applying and Reasoning about Mathematics Working systematically Using, Applying and Reasoning about Mathematics Questioning