Chocolate

34 /www/nrich/html/content/98/01/bbprob1/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> This challenge is about chocolate. You have to imagine (if necessary!) that everyone involved in this challenge enjoys chocolate and wants to have as much as possible. There's a room in your school that has three tables in it with plenty of space for chairs to go round. Table $1$ has one block of chocolate on it, table $2$ has two blocks of chocolate on it and, guess what, table $3$ has three blocks of chocolate on it. Now ... outside the room is a class of children. Thirty of them all lined up ready to go in and eat the chocolate. These children are allowed to come in one at a time and can enter when the person in front of them has sat down. When a child enters the room they ask themself this question: "If the chocolate on the table I sit at is to be shared out equally when I sit down, which would be the best table to sit at?" <table width="100%" border="0"> <tbody> <tr> <td width="33%"></td> <td width="33%"></td> <td width="33%"></td> </tr> </tbody> </table> <mdo:image width="563" height="141" src="Picture%203.jpg" alt="Pic3"></mdo:image> However, the chocolate is not shared out until all the children are in the room so as each one enters they have to ask themselves the same question. It is fairly easy for the first few children to decide where to sit, but the question gets harder to answer, e.g. It maybe that when child $9$ comes into the room they see: <ul> <li>$2$ people at table $1$</li> <li>$3$ people at table $2$</li> <li>$3$ people at table $3$</li> </ul> <div>So, child $9$ might think:</div> "If I go to: <ul> <li>table $1$ there will be $3$ people altogether, so one block of chocolate would be shared among three and I'll get one third.</li> </ul> <ul> <li>table $2$ there will be $4$ people altogether, so two blocks of chocolate would be shared among four and I'll get one half.</li> </ul> <ul> <li>table $3$, there will be $4$ people altogether, so three blocks of chocolate would be shared among four and I'll get three quarters.</li> </ul> Three quarters is the biggest share, so I'll go to table $3$." Go ahead and find out how much each child receives as they go to the "best table for them". As you write, draw and suggest ideas, try to keep a note of the different ideas, even if you get rid of some along the way. THEN when a number of you have done this, talk to each other about what you have done, for example: A. Compare different methods and say which you think was best. B. Explain why it was the best. C. If you were to do another similar challenge, how would you go about it? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">There were a few complete solutions sent in and many who showed us what the final result at the tables would be, i.e. $5$ at the $1$ table, $10$ at the $2$ table and $15$ at the $3$ table. The full solution showing fractions at each stage were received from Adriel, Emily and Aswaath from the Garden International School in Maylasia. Also Daniel at Staplehurst School and Megan at Twyford School. Here is Emily's contribution. Nrich Solution So after child $9$ has sat down, there are now: $2$ people at table $1$ $3$ people at table $2$ $4$ people at table $3$ This is a list of which table child $10-30$ would go to: The reason I listed down which table they would go to is to see whether there was a pattern Child $10$ would go to table $3$ and receive $¾$ of the chocolate Child $11$ would go to table $3$ and receive $½$ of the chocolate Child $12$ would go to table $2$ and receive $½$ of the chocolate Child $13$ would go to table $3$ and receive $3/7$ of the chocolate Child $14$ would go to table $2$ and receive $2/5$ of the chocolate Child $15$ would go to table $3$ and receive $3/8$ of the chocolate Child $16$ would go to table $1$ and receive $1/3$ of the chocolate Child $17$ would go to table $3$ and receive $1/3$ of the chocolate Child $18$ would go to table $2$ and receive $1/3$ of the chocolate Child $19$ would go to table $3$ and receive $3/10$ of the chocolate Child $20$ would go to table $2$ and receive $2/7$ of the chocolate Child $21$ would go to table $3$ and receive $3/11$ of the chocolate Child $22$ would go to table $1$ and receive $¼$ of the chocolate Child $23$ would go to table $2$ and receive $¼$ of the chocolate Child $24$ would go to table $3$ and receive $¼$ of the chocolate Child $25$ would go to table $3$ and receive $3/13$ of the chocolate Child $26$ would go to table $2$ and receive $2/9$ of the chocolate Child $27$ would go to table $3$ and receive $3/14$ of the chocolate Child $28$ would go to table $1$ and receive $1/5$ of the chocolate Child $29$ would go to table $2$ and receive $1/5$ of the chocolate Child $30$ would go to table $3$ and receive $1/5$ of the chocolate So now there are: $5$ people in table $1$ $10$ people in table $2$ $15$ people in table $3$ I found out a pattern from child 10 to child $30$. When it was child $16$’s turn to decide which was the best place to sit, child $16$ could just choose to randomly sit on any table because he’ll still get $1/3$ of the chocolate in any of the tables. Same with child $22$ and $28$. Child $22$ would either ways get $¼$ and child $28 1/5$. The numbers also increase by $6. 16, 22, 28$. I noticed that sometimes there were tables that shares the same amount of chocolate so you could choose randomly between $2$ or all of the tables which was what I did when I tried finding out how many chocolates each child receives as they go to find the best table for them For example: Child $16$ gets to choose between any of the $3$ tables because he’ll still get $1/3$ of the chocolate in any of the tables Well done all of you for your work on quite a difficult challenge. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Chocolate</h2> This challenge is about chocolate. You have to imagine (if necessary!) that everyone involved in this challenge enjoys chocolate and wants to have as much as possible. There's a room in your school that has three tables in it with plenty of space for chairs to go round. Table $1$ has one block of chocolate on it, table $2$ has two blocks of chocolate on it and, guess what, table $3$ has three blocks of chocolate on it. Now ... outside the room is a class of children. Thirty of them all lined up ready to go in and eat the chocolate. These children are allowed to come in one at a time and can enter when the person in front of them has sat down. When a child enters the room they ask themself this question: "If the chocolate on the table I sit at is to be shared out equally when I sit down, which would be the best table to sit at?" <table style="" border=""> <tbody> <tr> <td width="33%" style=""> </td> <td width="33%" style=""> </td> <td width="33%" style=""> </td> </tr> </tbody> </table> <mdo:image alt="Pic3" height="141" src="Picture%203.jpg" width="563"></mdo:image> However, the chocolate is not shared out until all the children are in the room so as each one enters they have to ask themselves the same question. It is fairly easy for the first few children to decide where to sit, but the question gets harder to answer, e.g. It maybe that when child $9$ comes into the room they see: <ul> <li>$2$ people at table $1$</li> <li>$3$ people at table $2$</li> <li>$3$ people at table $3$</li> </ul> <div>So, child $9$ might think:</div> "If I go to: <ul> <li>table $1$ there will be $3$ people altogether, so one block of chocolate would be shared among three and I'll get one third.</li> </ul> <ul> <li>table $2$ there will be $4$ people altogether, so two blocks of chocolate would be shared among four and I'll get one half.</li> </ul> <ul> <li>table $3$, there will be $4$ people altogether, so three blocks of chocolate would be shared among four and I'll get three quarters.</li> </ul> Three quarters is the biggest share, so I'll go to table $3$." Go ahead and find out how much each child receives as they go to the "best table for them". As you write, draw and suggest ideas, try to keep a note of the different ideas, even if you get rid of some along the way. THEN when a number of you have done this, talk to each other about what you have done, for example: A. Compare different methods and say which you think was best. B. Explain why it was the best. C. If you were to do another similar challenge, how would you go about it?</div> <h3>Why do this problem?</h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=34&part=index">This</a> is an excellent problem for helping youngsters to develop their concepts of fractions. It's not so much to do with arithmetical manipulation of fractions but more to do with youngsters exploring and developing their ideas. By encouraging learners to share their methods, there is an opportunity to discuss which might be the 'best' (this might depend on the individual's preference too). </div> <h3>Possible approach</h3> <div>Children will need plenty of (the same sized) paper available for folding and tearing in order to explore sizes of fraction.</div> <div>To begin the activity, you could act out the problem using large sheets of paper to stand for the chocolate bars (or real bars!), placed on tables. The acting could go through the situation with the first six children coming up one by one to the tables. Encourage them to justify their decisions and make sure the whole group agrees with their choice.</div> <div>Then learners could work in pairs on what happens when further children come to the tables. By listening to their conversations you can get a good insight into the ways that those youngsters think about and visualise fractions. Some surprises are very likely!</div> <div> </div> <div>After some time, bring everyone together again to talk about their ways of working. Invite comments about each method and then once all the different ways have been explained, ask pairs to discuss which method they would use now they have seen so many. You can then suggest they continue working on the problem, choosing any approach (or see the extension below). It would be interesting to talk to those pairs who have changed the way they tackle the task to find out what it is about their new method that they preferred to the original one. Some of their reflections could be recorded for display.</div> <h3>Key questions</h3> <div>Tell me about this. (Probably in reference to a torn-up piece of paper.)</div> <div>What size do you think this is ...?</div> <div>Why? (In response to the answer to the above.)</div> <h3>Possible extension</h3> <div>Challenge children to come up with a system or pattern that would help them to solve similar challenges.</div> <h3>For the exceptionally mathematically able</h3> <h4 style="font-weight: 400;">These pupils can then move onto the situation of four tables set out with $1, 2, 3, 4$ chocolate bars on them. The two different activities can then be compared, looking at similarities and differences, and giving proofs where appropriate.</h4> <h3>Possible support</h3> <div>You could start off with just two tables and a total of three bars of chocolate.</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> How much chocolate is on the first table? How many people would it have to be shared between? So, how much chocolate would each person get if you went to the first table? How about on the second table? ... third table? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> The sharing of the three tables with chocolate bars on. Well this really proved to be quite difficult! I have three pieces of work that have resulted from trying this problem before it went on the Internet. Robert's solution Number A,B or C How much 1 A 3 2 B 2 3 A 1 1/2 4 C 1 5 B 1 6 A 1 7 A 3/4 8 B 2/3 9 A 6/10 10 C 1/2 11 B 1/2 12 A 1/2 13 A 15/35 14 B 4/10 15 A 15/40 16 C 1/3 17 A 1/3 18 B 1/3 19 A 3/10 20 B 2/7 21 A 3/11 22 C 1/4 23 B 1/4 24 A 1/4 25 A 3/13 26 B 2/9 27 A 3/14 28 C 1/5 29 B 1/5 30 A 1/5 You get A B A everytime in the fractions between 1, 1/2, 1/3, 1/4, 1/5: Fraction A Fraction B Fraction A 1 1 1 Fraction A Fraction B Fraction A 1/2 1/2 1/2 Fraction A Fraction B Fraction A etc. This was rather nice and he noticed a good pattern that had emerged in the choice of table visited and the fraction received. Table A, B, C have 3, 2 & 1 piece of chocolate. So he noticed that the people come in and go to table A then B then A getting a different amount each and then the next three people get the same amount; this pattern, as he shows repeats itself. It's also rather nice that the three equal fractions that occur after the A B A gradually go down in order :- 1, 1/2, 1/3, 1/4, 1/5. This helps you to see very easily how much children would get when there were so many at any table. However, as the note says, she needed to get pieces of paper and do some folding and tearing in order to compare things like 2/3 and 3/4 to find out which was bigger. The last example was done jointly between three 11 year old boys. Looking down the column of 3 bars, 2 bars and 1 bar you see the number of the person entering the room and the amount that they would have received if the sharing had taken place. Doing it this way shows the interesting pattern that comes from the number of those people sitting around a table. So for example, on the 2-bar table at the end were people who had come in as number, 2, 5, 8, 1, etc. all with a rise of 3 each time. It was good to spot that and to notice that the three patterns could be arranged to give a total of 6. These three patterns all add up to 6: 2 + 1 + 3 = 6 3 + 3 = 6 6 = 6 It leads me to want to find out whether if there had been 4 tables and so 10 blocks of chocolate altogether, the patterns would all have added up to ten!These three patterns all add up to 6: 2 + 1 + 3 = 6 3 + 3 = 6 6 = 6 It leads me to want to find out whether if there had been 4 tables and so 10 blocks of chocolate altogether, the patterns would all have added up to ten! </mdoxml> 5 4 0 1 1 0 0 Chocolate There are three tables in a room with blocks of chocolate on each. Where would be the best place for each child in the class to sit if they came in one at a time? 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