Ice Cream

77 /www/nrich/html/content/00/04/bbprob1/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Let's all go down to the favourite cafe that sells ice cream which you choose from different tubs. <mdo:image alt="Three different flavours of ice cream." src="icecream.jpg"></mdo:image> Suppose that there's Apricot, Banana and Citrus. There is just one rule about what you can choose, and here it is: YOU CANNOT CHOOSE A SELECTION OF ICE CREAM FLAVOURS THAT INCLUDES TOTALLY WHAT SOMEONE HAS ALREADY CHOSEN! This means that if someone has chosen Banana and Citrus I cannot then go and choose all three but I could choose to have Apricot on its own. So perhaps something like this happens:- <div style="text-align: center;"><mdo:image alt="Sarah, thinking of apricot and citrus!" src="sarah.jpg"></mdo:image></div> Sarah, the first child, chooses Apricot and Citrus. Tim, the second child chooses Banana and Citrus [this obeys the rule because Sarah's choice was not Banana on its own nor was it Citrus on its own]. Raj, the third child, chooses Citrus. Zoe, the fourth child, chooses Banana. Matt, has to be the last child because he can only choose Apricot and then there are no other choices left. In this example, with these children making these choices, only five children can have ice cream [using our rule]. But suppose more children wanted ice cream and so they got together to work out how this could be done. They might come up with an idea like this:- [I'm using the short-hand this time of A B C where A is Apricot, B is Banana and C is Citrus.] 1st. choice > A B C 2nd. choice > A B 3rd. choice > A C 4th. choice > B C 5th. choice > A 6th. choice > B 7th. choice > C So seven children altogether. I think that the children can have different sized scoops so that even if they only have one flavour they have as much ice cream as someone choosing three flavours! If these children weren't very good at working things out they could come up with the worst way; that would be like this:- 1st. choice > A 2nd. choice > B 3rd. choice > C AND THAT'S ALL! Well that's what it's like when there are three flavours. At the most, seven children can go in that order and get those choices of ice cream. At worst, only three children can go and get ice cream. Mind you I think that there are other ways of getting seven. Have a go and find all the different ways of having seven children getting ice cream. Remember that when someone goes up and makes their choice they have to obey that rule: YOU CANNOT CHOOSE A SELECTION OF ICE CREAM FLAVOURS THAT INCLUDES TOTALLY WHAT SOMEONE HAS ALREADY CHOSEN! <comment> #set var="roll-text" value="Be a detective!" </comment> <comment> #set var="roll-text" value="" </comment> So seven is the most, three is the least for three flavours. What about four flavours now that Damson is available? What about five ...? <comment> #set var="roll-text" value="Be a detective!" </comment> <comment> #set var="roll-text" value="" </comment> Are you able to go on and on? Do you know how many different ways of getting the most children to have ice cream for four flavours? For five flavours? <comment> #set var="roll-text" value="Be a detective!" </comment> <comment> #set var="roll-text" value="" </comment> So you can work all these different things out. Here's the next bit of the challenge - can you find out how many there are for seven flavours, without working out the answers to six flavours first? <comment> #set var="roll-text" value="Be a detective!" </comment> <comment> #set var="roll-text" value="" </comment> And FINALLY ... I wonder what would happen if ...? Please send in any results that you get along the way. I have to go - my mouth is watering for some ice cream! </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> We had a few solutions sent in including one that got to the heart of the matter and produced a really good account. This is it from Jacob in the United Kingdom . We started with the maximum number of combinations of ice cream that you can have with four flavours of ice cream. We counted $15$ combinations: The first person would have all four flavours. 4 more people could then have three of the flavours. 6 more people could have two flavours. 4 more people could have one flavour. $1 + 4 + 6 + 4 = 15$ We added in another flavour of ice cream, which gave us five flavours. We worked out that there were $31$ combinations for five flavours: The first person would have all five flavours. Five more people could then have four of the flavours. Ten more people could have three flavours. Ten more people could have two flavours. Five more people could have one flavour. $1 + 5 + 10 + 10 + 5 = 31$ We noticed a pattern that we had seen before! This pattern is in a number triangle where you add up the two numbers in the row above. When we added up the total of each row in the number triangle, they added up to one more than the number of ice cream combinations! We then used the number triangle to work out how many ice cream combinations there would be for six flavours, seven flavours, eight flavours, nine flavours and ten flavours. When we had worked out these combinations, we also realised that each total in the number triangle was the previous total multiplied by two. This made it very quick to work out how many ice cream combinations there would be for any number of flavours of ice cream! My Dad helped me with building the number table on Excel and attaching it to this solution. <div style="text-align: center;"><mdo:image height="293" width="542" src="Jacob%27s%20Ice%20Creams.jpg" alt="jacobs"></mdo:image></div> I really enjoyed finding the patterns in this problem. Thank you for sending in your solution to us, Jacob. It looks like you worked hard on this investigation. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Ice Cream</h2> Let's all go down to the favourite cafe that sells ice cream which you choose from different tubs. <mdo:image alt="Three different flavours of ice cream." src="icecream.jpg"></mdo:image> Suppose that there's Apricot, Banana and Citrus. There is just one rule about what you can choose, and here it is: YOU CANNOT CHOOSE A SELECTION OF ICE CREAM FLAVOURS THAT INCLUDES TOTALLY WHAT SOMEONE HAS ALREADY CHOSEN! This means that if someone has chosen Banana and Citrus I cannot then go and choose all three but I could choose to have Apricot on its own. So perhaps something like this happens:- <div style="text-align: center;"><mdo:image alt="Sarah, thinking of apricot and citrus!" src="sarah.jpg"></mdo:image></div> Sarah, the first child, chooses Apricot and Citrus. Tim, the second child chooses Banana and Citrus [this obeys the rule because Sarah's choice was not Banana on its own nor was it Citrus on its own]. Raj, the third child, chooses Citrus. Zoe, the fourth child, chooses Banana. Matt, has to be the last child because he can only choose Apricot and then there are no other choices left. In this example, with these children making these choices, only five children can have ice cream [using our rule]. But suppose more children wanted ice cream and so they got together to work out how this could be done. They might come up with an idea like this:- [I'm using the short-hand this time of A B C where A is Apricot, B is Banana and C is Citrus.] 1st. choice &gt; A B C 2nd. choice &gt; A B 3rd. choice &gt; A C 4th. choice &gt; B C 5th. choice &gt; A 6th. choice &gt; B 7th. choice &gt; C So seven children altogether. I think that the children can have different sized scoops so that even if they only have one flavour they have as much ice cream as someone choosing three flavours! If these children weren't very good at working things out they could come up with the worst way; that would be like this:- 1st. choice &gt; A 2nd. choice &gt; B 3rd. choice &gt; C AND THAT'S ALL! Well that's what it's like when there are three flavours. At the most, seven children can go in that order and get those choices of ice cream. At worst, only three children can go and get ice cream. Mind you I think that there are other ways of getting seven. Have a go and find all the different ways of having seven children getting ice cream. Remember that when someone goes up and makes their choice they have to obey that rule: YOU CANNOT CHOOSE A SELECTION OF ICE CREAM FLAVOURS THAT INCLUDES TOTALLY WHAT SOMEONE HAS ALREADY CHOSEN! <comment> #set var="roll-text" value="Be a detective!" --&gt;&lt;!-- #set var="roll-text" value="" --&gt;&lt;/p&gt; &lt;p&gt;So seven is the most, three is the least for three flavours.&lt;/p&gt; &lt;p&gt;What about four flavours now that Damson is available?&lt;/p&gt; &lt;p&gt;What about five ...?&lt;/p&gt; &lt;p&gt;&lt;!-- #set var="roll-text" value="Be a detective!" --&gt;&lt;!-- #set var="roll-text" value="" --&gt;&lt;/p&gt; &lt;p&gt;Are you able to go on and on?&lt;/p&gt; &lt;p&gt;Do you know how many different ways of getting the most children to have ice cream for four flavours? For five flavours?&lt;/p&gt; &lt;p&gt;&lt;!-- #set var="roll-text" value="Be a detective!" --&gt;&lt;!-- #set var="roll-text" value="" --&gt;&lt;/p&gt; &lt;p&gt;So you can work all these different things out.&lt;/p&gt; &lt;p&gt;Here's the next bit of the challenge - can you find out how many there are for seven flavours, without working out the answers to six flavours first?&lt;/p&gt; &lt;p&gt;&lt;!-- #set var="roll-text" value="Be a detective!" --&gt;&lt;!-- #set var="roll-text" value="" </comment> And FINALLY ... I wonder what would happen if ...? Please send in any results that you get along the way. I have to go - my mouth is watering for some ice cream! <h3>Why do this problem?</h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=77&part=">This activity</a> , when done practically, usually gives great excitement and leads to an extensive discussion as learners try to explain their reasoning to one another.</div> </div> <h3>Why do this problem?</h3> <a href="/77">This investigation</a> offers an opportunity for children to work creatively as there is no obvious place to start. However, it may well give you chance to discuss systematic approaches once they have been working on it for a while. <h3> Possible approach</h3> <div>You'll probably get the best discussion and thoughts from the pupils if you 'act' out the situation. You are the ice cream vendor with a selection of cards, counters, cubes etc. to represent the ice cream. Make sure that the class can see the choices that the previous pupil(s) made and encourage them to check if they've kept to the rule, helping as little as possible.</div> <div>Once they have got the idea, invite them to work on the problem, perhaps in pairs. Allow them to make their own decisions as to the equipment they use and the way they record - sharing these can be just as valuable as sharing solutions.</div> <div>This activity would lend itself to a display and so you may want children to work on large sheets of paper which could be the stimulus for a plenary and could then be put straight up on the classroom walls. You could invite some children to explain how they have found different ways of seven children having ice cream. Draw attention to those who have found a systematic way of finding solutions.</div> <h3>Key questions</h3> <div>Why did you choose that flavour?</div> <div>Is there a way of choosing to let more pupils have a choice?</div> <h3>Possible extension</h3> <div>As the problem suggests, the thinking can be extended by looking at larger numbers of flavours.</div> <div> </div> <h3>For the exceptionally mathematically able</h3> <div>From the "Possible Extension" the challenge would be to find a general formulae that would take into account the number of flavours and how many scoops were alloed each time to produce the largest numbers of hungry ice cream addicts who could have their fill. Then there would be the need to prrove it to be correct.</div> <h3>Possible support</h3> <div>This problem is a good context in which to try to stand back and encourage pupils to support each other and explain ideas themselves. Working in pairs might be an appropriate way to support this.</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> This investigation is a little different from many because the order is so very important - have you checked you have stuck to the rule? What could you choose next? Is there a way you can select flavours so that more people will have a choice after you? It might help to record your working. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><mdo:image height="473" width="151" src="Ice%20Cream%2077%20poss.jpg" alt=""></mdo:image> </mdoxml> 2 5 0 1 0 0 0 Ice Cream You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream. Using, Applying and Reasoning about Mathematics Investigations Decision Mathematics and Combinatorics Combinations Using, Applying and Reasoning about Mathematics Working systematically