Ladybirds in the Garden

1816 /www/nrich/html/content/03/09/letme1/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> In Sam and Jill's garden there are two sorts of ladybirds. There are red Seven-Spot ladybirds with $7$ black spots and shiny black Four-Spot ladybirds with $4$ red spots. <mdo:image alt="a red ladybird with 7 black spots and a black ladybird with 4 red spots" height="82" src="LadyBs.gif" width="239"></mdo:image> Sam and Jill looked at a leaf with three ladybirds on it. <mdo:image alt="two black ladybirds, one red ladybird" height="202" src="LadyB2.gif" width="210"></mdo:image> "One Seven-Spot ladybird," said Sam, "and two Four-Spot ones." "That's $15$ spots altogether!" laughed Jill. "I wonder if we could find ladybirds whose spots add to other numbers. I know how to do $16$." "And $14$ is easy too," added Sam. How would you make $16$ and $14$ spots with the Seven-Spot and Four-Spot ladybirds? What other numbers can you make with adding $4$s and $7$s? Can you get lots of numbers from say $4$ to $35$? Are there some numbers you can't get? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> From Jacob, Luc and William at The Hall School in London we had the following, very thorough solution sent in, We looked at the number of spots that could be produced when you have a four spotted ladybird or a seven spotted ladybird or a combination of both. We were asked if it was possible to make $16$ and we found it is, by using four of the four spotted ladybirds. We discovered that the smallest number of spots we could produce was $4$. We were then asked what number of spots between $4$ and $35$ could be produced. We started off by writing a list of those numbers that did and didn't work. Then we made a table to show how we made the numbers of spots that were possible. We found seventeen different numbers could be made and three that could be made in two different ways. These were $28, 32$ and $35$. We have drawn a table to show the only number of spots that can be made between $4$ and $35$. For each number of spots that it is possible to make we have shown the number of four spotted and seven spotted ladybirds that make up the number of spots. <div style="text-align: center;"><mdo:image src="Ladybird%20table.png"></mdo:image> </div> We also had some ideas sent in from Christian at Heronsgate School, Olivia from Risley Lower Primary School both in England. From Australia we had a solution sent in from Maths Group $2$ at Brunswick South Primary School. Well done everyone! </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Ladybirds in the Garden</h2> In Sam and Jill's garden there are two sorts of ladybirds. There are red Seven-Spot ladybirds with $7$ black spots and shiny black Four-Spot ladybirds with $4$ red spots. <mdo:image alt="a red ladybird with 7 black spots and a black ladybird with 4 red spots" height="82" src="LadyBs.gif" width="239"></mdo:image> Sam and Jill looked at a leaf with three ladybirds on it. <mdo:image alt="two black ladybirds, one red ladybird" height="202" src="LadyB2.gif" width="210"></mdo:image> "One Seven-Spot ladybird," said Sam, "and two Four-Spot ones." "That's $15$ spots altogether!" laughed Jill. "I wonder if we could find ladybirds whose spots add to other numbers. I know how to do $16$." "And $14$ is easy too," added Sam. How would you make $16$ and $14$ spots with the Seven-Spot and Four-Spot ladybirds? What other numbers can you make with adding $4$s and $7$s? Can you get lots of numbers from say $4$ to $35$? Are there some numbers you can't get? </div> <h3>Why do this problem?</h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=1816&part=index">Ladybirds in the Garden</a> is a good problem for children to practise addition, subtraction and possibly multiplication. It is a useful context in which to encourage learners to "have a go" and "play" with numbers, but then you can focus on having a system to find all the possible totals and giving reasons why some can't be made.</div> <h3>Possible approach</h3> <div>It would be good to have the two pictures of the ladybirds in the problem displayed for the children to see, either printed off or on screen. You could get them started simply by asking how many spots there are altogether on the two, then how many on the picture of the three. You might say that you're picturing some ladybirds on a leaf whose spots add to $16$ and ask children in pairs to think about which ladybirds they are.</div> <div>Set the class off on the task of finding other possible totals, saying very little more. Wander round the group as they work, looking for different ways of recording and different ways of working. After 5/10 minutes, draw attention to some of these. Some might be writing number sentences, others might be drawing pictures. Some might be starting with a total, others might be starting from a combination of ladybirds. It is important to stress that all of these are valid and it depends what each pair finds most helpful. At this stage, remind the children that you'd like them to be sure they have found all the possible totals. How will they know? Invite them to share their ideas which could lead into a discussion of having a system, for example working in numerical order through the totals, or by using only four-spot ladybirds first, then only seven-spot, then both.</div> <div>You could bring their ideas together by asking them to write each different total, and how they made it, on a strip of paper. Gather these strips on the board or a wall and ask the children to arrange them in numerical order. Draw out a list of numbers which haven't been made and ask all pairs to check that they're sure they are impossible. The important point here is for pupils to try to explain why these totals can't be made. Ask pairs to convince each other and then invite some explanations for all to hear.</div> <h3>Key questions</h3> <div>Which totals have you found so far? Which ladybirds made these totals?</div> <div>What totals could we make if there are no four-spot ladybirds?</div> <div>What totals could we make if there are no seven-spot ladybirds?</div> <div>What totals could we made if there are both four-spot and seven-spot ladybirds?</div> <div>How do you know you've made all the totals that are possible?</div> <div>Why can't we make these totals?</div> <h3>Possible extension</h3> <div>The article, <a href="http://nrich.maths.org/public/viewer.php?obj_id=5657&part=index">Opening Out</a> , suggests possible routes for further investigation in general, but this problem is also mentioned specifically. Children could be asked what combination of spots would be needed to be able to generate all the numbers under $30$. Would it be possible to do this without a one-spot ladybird? If we couldn't use a one-spot ladybird, what totals would be possible?</div> <h3>Possible support</h3> <div>Some children might find it useful to have a sheet with consecutive numbers from $1$ to $29$ written on so that they can fill in ways of making each total as they go along. Calculators might be available, or other apparatus to support the arithmetic, for example a hundred square, number line, Cuisenaire rods or counters/cubes. <a href="/content/03/09/letme1/Ladybirds%20in%20the%20Garden.pdf">This document</a> contains multiple copies of each ladybird which could be cut into cards. To make the task even more accessible, you could choose and create ladybirds with $2$ or $3$ spots instead.</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">For ideas about how to open up this activity, you may like to read Bernard Bagnall's article Opening Out . We had lots of responses to this problem - thank you to you all. Lachlan from St Faith's School sent a solution to the first part of the question: two sevens =14 four fours=16 Ann and Daniel who are both 10, wrote this a slightly different way: 7+7=14 4+4+4+4=16 Ann and Daniel then continued with the rest of the question, writing the sums in order of size. Julia from St Anne's C of E Primary School and Michael from St Alfred's College also tackled the problem in this way. Here is Michael's solution: 4=4 7=7 4+4=8 4+7=11 4+4+4=12 7+7=14 4+7+4=15 4+4+4+4=16 4+7+7=18 4+4+4+7=19 4+4+4+4+4=20 7+7+7=21 4+7+4+7=22 4+4+4+4+7=23 4+4+4+4+4+4=24 4+7+7+7=25 4+7+4+7+4=26 4+4+4+4+4+7=27 7+7+7+7=28 4+7+4+7+7=29 Dylan, Robert, Shani and Amy who are all 7 years old, approached it a slightly different way (and have come up with some different solutions) which is just as good, although they have missed out a few: 4=4 4+4=8 4+4+4=12 4+4+4+4=16 4+4+4+4+4=20 4+4+4+4+4+4=24 4+4+4+4+4+4+4=28 7=7 7+7=14 7+7+7=21 7+7+7+7=28 7+4=11 7+4+4= 15 7+4+4+4=19 7+7+4=18 7+7+4+4+4=26 7+7+4+4+4+4= 30 Can you see where the missing sums would fit into the pattern they've created? In conclusion, Michael says that the impossible numbers are: 1, 2, 3, 5, 6, 9, 10, 13, 17.</mdoxml> 2 4 1 0 0 0 0 Ladybirds in the Garden In Sam and Jill's garden there are two sorts of ladybirds with 7 spots or 4 spots. What numbers of total spots can you make? Using, Applying and Reasoning about Mathematics Working systematically Calculations and Numerical Methods Addition & subtraction Decision Mathematics and Combinatorics Combinations Admin Lower primary mapping document