Twenty Divided into Six

1047 /www/nrich/html/content/01/03/penta4/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Katie had a pack of twenty cards numbered from $1$ to $20$. <mdo:image alt="20 cards numbered from 1 to 20." src="cards.gif"></mdo:image> She arranged the cards into six unequal piles. The numbers on the cards in each pile added to the same total. What was the total and how could this be done? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Every month, one problem will attract more problem solvers than any other. This month it was <a href="http://nrich.maths.org/public/viewer.php?obj_id=1047&part=index"> Twenty Divided into Six</a> . What an enormous number of replies we had to this. Many of the replies came from whole classes and different years in schools from many different places. A special welcome to the some newcomers, pupils at St Aldhelm's , C.E.V.A. Combined School in Poole in Dorset. We hope you continue to use the site and use all the great skills you showed this month to solve the NRICH problems. Thank you to special friends of NRICH - Moorfield Junior School , Tattingstone Primary School and Burgoyne Maths Club as well as Hutt Intermediate School in Wellington, New Zealand, Nan Hua Primary School , Singapore, Higher Bebington Junior School , Rosebank Primary School Leeds, and Lazonby C of E School. Not only were there many responses there were also many solutions to this problem. Below, are a selection of the possibilities. I wonder if you could add to the list. In fact, I wonder if you could find a way of discovering how many possible answers there are to this question. Now there's a challenge!! The problem took more planning and moving of numbers than people realised at first. One strategy was explained by pupils Emma and Hollie of Moorfields Junior School: We added up all the numbers up to 20 and divided the total by 6. Next step was described by their classmates Steve , Matthew and Luke: ... we paired up all of the numbers and then we started to move the numbers into groups. Burgoyne Maths Club wrote: After various attempts to make six equal piles with twenty cards, we abandoned this because we couldn't make twenty (the number of cards) with 6 different numbers. Georgereported: I began with 20+15 and....the rest was just matching up. Have a look at these solutions to see if yours is amongst them or if you have the same solution. Perhaps you had some solutions from one set and some from another. Would it be possible to have one answer from many different sets of possibilities? What reason do you have for your answer to that question? <table cellspacing="0" cellpadding="5" border="1"> <tbody> <tr> <td>20, 15 1, 2, 3, 4, 5, 9, 11 18, 17 10, 13, 12 19, 16 6, 7, 8, 14 </td> <td>20,15 1, 14, 11, 9 10, 19, 6 3, 4, 12, 16 17, 13, 5 2, 7, 8, 18 </td> <td>20, 10, 5 15, 1, 19 6, 13, 16 3, 12, 2, 18 8, 7, 9, 11 17, 4, 14 </td> </tr> <tr> <td>20, 15 19, 9, 7 18, 17 16, 13, 5, 1 14, 12, 6, 3 11, 10, 8, 4, 2 </td> <td>17, 18 19, 16 20, 10, 5 4, 7, 15, 6, 3 12, 9, 14 13, 11, 2, 1, 8 </td> <td>17, 13, 5 9, 15, 11 1, 4, 12, 18 6, 8, 7, 14 19, 16 2, 10, 20, 3 </td> </tr> </tbody> </table> These solutions were a selection representing the hundreds that were sent in. The senders of these were George and Matthew from Rosebank Primary in Leeds, Ong of Nan Hua Primary in Singapore, Martin from Robert Kett Junior School in Wymondham, Suzanne and Lucy from Lazonby C of E School, Joshua from Higher Bebington Juniors, Steve , Matthew and Luke of Moorfiled Junior School, and members of the Burgoyne Maths Club. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Twenty Divided Into Six</h2> Katie had a pack of twenty cards numbered from $1$ to $20$. <mdo:image alt="20 cards numbered from 1 to 20." src="cards.gif"></mdo:image> She arranged the cards into six unequal piles. The numbers on the cards in each pile added to the same total. What was the total and how could this be done? </div> <h3>Why do this problem?</h3> <a href="http://nrich.maths.org/public/viewer.php?obj_id=1047&part=index">This problem</a> is one that can be accessed easily - everyone can make a start. It offers opportunities for learners to practise addition and subtraction, along with some multiplication and division, and requires a systematic approach. <h3>Possible approach</h3> <div>You could start by asking the group to work on the problem in pairs with digit cards numbered from $1$ to $20$ without saying very much else at this stage. (You could print and cut out cards from <a href="/content/01/03/penta4/Cards1-20.pdf">this sheet</a> if you do not have enough to go round.) Learners might find it useful to make jottings on mini-whiteboards or paper as they explore the problem.</div> <div>After some time, draw the class together to find out how they are getting on. Invite some pairs to share their approach so far with the whole group. Some children might be using trial and improvement, some may have worked out what the total of each pile needs to be and then used trial and improvement. You may need to talk about how they work out the total of each pile if this does not come up naturally. Can they think of a quick way of doing it without a calculator?</div> <div>They could then continue to work in pairs on the problem. After the initial calculations the problem is a fairly simple one of adding and building the piles but there are many ways of doing it. It would be interesting and instructive to listen to the way that the various pairs are working on the problem, and you may like to gather solutions as a whole class on the board.</div> <div>In a plenary at the end of the lesson, you could talk about how they have found the different solutions and you may want to ask whether they think they have got them all. This might be a good opportunity to share ways of working systematically so that they could convince you they would be able to find every solution.</div> <h3>Key questions</h3> <div>What number must all the piles add to?</div> <div>What is the total of each pile?</div> <div>How do you know you have all the solutions?</div> <h3>Possible extension</h3> <div>Learners could find as many completely different solutions to this problem as possible and some children will be able to suggest a way to find them all.</div> <h3>Possible support</h3> If you want to focus on finding all possibilities, some learners might benefit from using a calculator so they are not held up by the mental arithmetic. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> What number do all the cards add to? What will the total of each pile be? How will you keep track of the piles you have made? You might find digit cards useful. </mdoxml> 2 4 0 1 0 0 0 Twenty Divided into Six Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done? Calculations and Numerical Methods Multiplication & division Using, Applying and Reasoning about Mathematics Trial and improvement Mathematics Tools Digit cards Calculations and Numerical Methods Addition & subtraction Using, Applying and Reasoning about Mathematics Working systematically Admin Upper primary mapping document