Lots of Lollies

2360 /www/nrich/html/content/id/2360/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> If you are a teacher, click <a href="/2360&part=note">here</a> for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on ... Frances and Rishi were given a bag of lollies. <mdo:image align="top" alt="lollypops" bgcolor="" src="lollies.jpg" style="width: 255px; height: 236px;"></mdo:image> They shared them out evenly and had one left over. Just as they had finished sharing them their friends Kishan, Hayley and Paul came along. They wanted some lollies too so the children shared them out again between all of them. This time they had two lollies left over. How many lollies could there have been in the bag? Once you've had a chance to think about it, click below to see how three different groups of pupils began working on the task. Sarah, Danielle and Sally said: <div class="toggle" style="margin-left: 36pt;">"We noticed that $17$ works as when there are only two of them they get $8$ each, with one left over. But when their friends come along they get three each with $2$ left over. We also notice that $7$ works and $27$ works, as well as $107$." </div> Poppy began like this: <div class="toggle" style="margin-left: 36pt;"> If the two children end up with one lolly it must be an odd number of lollies. Then three more children come making the total number of children $5$. Say they had $1$ lolly each when they shared them, the number of lollies would be $7$ because $1$ times $5$ is $5$ add on $2$ for the left over ones and it makes seven. If we carry this on to $10$ lollies each it shows: $1$ lolly each - $7$ lollies $2$ lollies each - $12$ lollies $3$ lollies each - $17$ lollies $4$ lollies each - $22$ lollies $5$ lollies each - $27$ lollies $6$ lollies each - $32$ lollies $7$ lollies each - $37$ lollies $8$ lollies each - $42$ lollies $9$ lollies each - $47$ lollies $10$ lollies each - $52$ lollies</div> Here is the start of Phoebe and Alice's work: <div class="toggle"><mdo:image src="phoebe.gif"></mdo:image></div> Can you take each of these starting ideas and develop it into a solution? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> We had a good number of responses to this challenge. The first ones are from Culgaith School in the UK who sent in the following; Nathan said that the answer has to be an odd number as it is shared between two children with $1$ left over. We found the first solution was $7$ - Emma said this. We then tried to find out if there were any other answers. We tried $8$ - but then realised that this was an even number so we looked at $9$ - too many left over. We looked at $11$ - this time too few left over. Alex suggested $17$ - why did he choose this? He said the numbers were "basically the same". Mr Dodd said this wasn't quite true as he would rather have $17$ chocolate bars than $7$! Millie said that the $10$ lollipops different between $7$ and $17$ would give each of the two children two lollipops each and so $17$ would work. We checked this out and it did! We then used the number square to show us all the solutions up to $100$ - they all end in "$7$". We worked out that this was because the tens digit would alway share out evenly between $5$ children and the $7$ would give each child $1$ lollipop and have two left over. We were really pleased with our thinking and problem solving. Looking forward to the next problem! From Deidre at Calgary Science School in Canada we had a really lovely suggestion. Grade 4 class, she says approached a similar problem they called the Candy problem. They posted a Video of how the students arrived at solutions at - <a href="http://savouringtheish.blogspot.com">http://savouringtheish.blogspot.com</a>. Thank you so much for that it's certainly worth watching! Stella from Chigwell School in England wrote a very thorough account of the challenge; Any number ending in $7$ works as if you divide it by $2$ you get $1$ left over and if you divide it by $5$ you get $2$ left over which fits in with the problem:- They shared them out evenly and had one left over. Just as they had finished sharing them their friends Kishan, Hayley and Paul came along. They wanted some lollies too so the children shared them out again between all of them. This time they had two lollies left over. How many lollies could there have been in the bag? Because they start off with $2$ children and they share them out equally you have to divide by 2 but so that it does't go in exactly you have $1$ left over which means that it has to be an odd number. but when they have just finished sharing them out properly $3$ more children come along making the total of children go up to $5$, so now you have to divide by $5$ so it can't be any thing ending in $9 , 3 , 5$ or $1$ so that only leaves $7$. Luke from Locks Heath Junior School in England presented us with the following good account. There were two people at the start and there was remainder $1$, so it had to be odd then , $3$ more people came over and there was remainder $2$, so I thought of every odd number up to $9$ so there is $1 3 5 7 9$ I chose every odd . On number $1$ only $1$ person could get a lolly then I thought of $3$ then, only $3$ people could get a lolly, then I tried $5$ then only everybody could get $1$ each then there was no reminder. Then I skipped $7$ for no reason then I went on to $9$ there would of been to many left. So the answer is $7$ . Well done all of you. Thank you so much for letting us into your thoughts and ideas while sorting out solutions. We had a bit of blogging going on in our <a class="blogbutton" href="http://nrich.maths.org/z/infinities">blog</a>. We hope to hear from you all again soon. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Lots of Lollies</h2> If you are a teacher, click <a href="/2360&part=note">here</a> for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on ... Frances and Rishi were given a bag of lollies. <mdo:image align="top" alt="lollypops" bgcolor="" src="lollies.jpg" style="width: 255px; height: 236px;"></mdo:image> They shared them out evenly and had one left over. Just as they had finished sharing them their friends Kishan, Hayley and Paul came along. They wanted some lollies too so the children shared them out again between all of them. This time they had two lollies left over. How many lollies could there have been in the bag? Once you've had a chance to think about it, click below to see how three different groups of pupils began working on the task. Sarah, Danielle and Sally said: <div class="toggle" style="margin-left: 36pt;">"We noticed that $17$ works as when there are only two of them they get $8$ each, with one left over. But when their friends come along they get three each with $2$ left over. We also notice that $7$ works and $27$ works, as well as $107$." </div> Poppy began like this: <div class="toggle" style="margin-left: 36pt;"> If the two children end up with one lolly it must be an odd number of lollies. Then three more children come making the total number of children $5$. Say they had $1$ lolly each when they shared them, the number of lollies would be $7$ because $1$ times $5$ is $5$ add on $2$ for the left over ones and it makes seven. If we carry this on to $10$ lollies each it shows: $1$ lolly each - $7$ lollies $2$ lollies each - $12$ lollies $3$ lollies each - $17$ lollies $4$ lollies each - $22$ lollies $5$ lollies each - $27$ lollies $6$ lollies each - $32$ lollies $7$ lollies each - $37$ lollies $8$ lollies each - $42$ lollies $9$ lollies each - $47$ lollies $10$ lollies each - $52$ lollies</div> Here is the start of Phoebe and Alice's work: <div class="toggle"><mdo:image src="phoebe.gif"></mdo:image></div> Can you take each of these starting ideas and develop it into a solution? </div> <h3>Why do this problem?</h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=2360&part=index">This problem</a> requires children to apply their knowledge of factors and multiples, and is a good way of making the link between sharing, division and multiples/factors. It may also be used to introduce learners to the fact that a problem can have more than one solution and that the solutions can be generalised. It can be approached in many different ways so can be a useful context in which to talk about different ways of recording and different methods of solving problems.</div> <h3>Possible approach</h3> <div>You could introduce this challenge by acting it out. Invite two children to the front of the class and ask everyone to imagine that you have a bag of lollies. Explain that you give them out to the two volunteers so that they have the same number each, but there is one left in the bag. You could mime giving out some lollies so that everyone gets the idea. Then invite three more children to the front. Mime gathering the lollies back into the bag and then distributing them equally again, this time explaining that you have two left over. Pose the question "I wonder how many lollies could have been in the bag?". Give the class a few minutes to consider, individually, how they might go about tackling the problem, then pair them up and suggest that they talk to their partner about their ideas so far. Try to stand back and observe, and resist the temptation to make helpful suggestions! Allow pairs to work on the task so that you feel they have made some progress, but do not worry if they have not completed it or if they report being stuck. The aim at this stage is for everyone to 'get into' the problem and work hard on trying to solve it, but not necessarily to achieve a final solution. At a suitable time, hand out this (<a class="doclink" href="/content/id/2360/BB%20lots%20of%20lollies%20word.doc">doc</a> <a class="pdflink" href="/content/id/2360/BB%20lots%20of%20lollies.pdf">pdf</a> ) to pairs. Suggest to the class that when they've finished or can't make any further progress, they should look at the sheet showing three approaches used by children working on this task. Pose the question, "What might each do next? Can you take each of their starting ideas and develop them into a solution?". You may like pairs to record their work on large sheets of paper, which might be more easily shared with the rest of the class in the plenary. Allow at least fifteen minutes for a final discussion. Invite some pairs to explain how the three different methods might be continued. You may find that some members of the class used completely different approaches when they worked on the task to begin with, so ask them to share their methods too. You can then facilitate a discussion about the advantages and disadvantages of each. Which way would they choose to use if they were presented with a similar task in the future? Why? (You might find it helpful to adapt this <a href="/content/id/2360/Sharing_lollies.notebook">Smart Notebook file</a> for use on the interactive whiteboard. Thank you to Gemma for giving us permission to include it here.) </div> <h3>Key questions</h3> <div>How many children are there altogether when they share the lollies the second time? What is the smallest number of lollies there could be? Is this the only number of lollies there could be? Tell me about this approach. What do you think she/they were doing? How do you think this will help to solve the problem? What do you think she/they would have done next?</div> <div> </div> <h3>Possible extension</h3> <div>At the highest level, encourage learners to articulate a full generalisation about what kind of numbers of lollies there could be in the bag. This might be worded in terms of multiples, or in terms of the units digit of possible solutions. <a href="http://nrich.maths.org/public/viewer.php?obj_id=5579&part=index">Growing Garlic</a> is a similar problem which pupils could have a go at as a follow-up to this one.</div> <h3>Possible support</h3> <div>Learners might request a range of different resources to help them tackle this challenge, for example a hundred square, or some counters/cubes. Try not to pre-empt their requests by placing equipment out on tables at the start, but do make sure these kind of resources are easily accessible to the children, should they want to use them and do your best to accommodate any requests which you hadn't anticipated!</div></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> How many children are there altogether when they share the lollies the second time? Try using counters or blocks instead of lollies to help. What is the smallest number of lollies there could be? Is this the only number of lollies there could be? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Number of lollies is one bigger than a multiple of 2 and two bigger than a multiple of 5. Number of lollies could be 7, 17, 27 etc. </mdoxml> 5 5 1 0 0 0 0 Lots of Lollies Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag? Using, Applying and Reasoning about Mathematics Generalising Numbers and the Number System Factors and multiples Using, Applying and Reasoning about Mathematics Working systematically Admin Lower primary mapping document