5579
/www/nrich/html/content/id/5579/
1
2011-02-01T00:00:01
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Ben is on the allotment with his Mum. They would like to grow some garlic and are deciding how to plant the garlic cloves.<br></br>
<br></br>
Ben arranges the cloves into three rows and finds that he has one spare clove. How many cloves might he have had to start with?<br></br>
<br></br>
Ben plants cloves of garlic in two rows and has one clove left over. So he tries again.<br></br>
He plants cloves in three rows and has one left over. So he tries again.<br></br>
He plants cloves in four rows and has one left over. So he tries again.<br></br>
He plants cloves in five rows and has one left over. So he tries again.<br></br>
He plants cloves in six rows and has one left over.<br></br>
We know that he has less than $100$ garlic cloves. How many did he have?<br></br>
You could think about how many cloves he might have had if there were more than $100$.<br></br>
<br></br>
You might like to use this interactivity to investigate the problem. You can alter the number of bulbs using the up and down arrows, and you can drag them into different arrangements using the little white square.
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<div style="clear: both;"><a href="/content/id/5579/Garlic.swf">Full Screen Version</a></div>
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<p class="editorial">We had lots of correct solutions to this
problem and many of you explained your thinking very clearly. Well
done!</p>
<p class="editorial">Musab from Orchard School answered the first
part of the problem:</p>
Ben might have started with a number 1 more than a multiple of 3
because it says when Ben put the garlic into groups of 3 there was
1 garlic plant left. For example Ben could have started with 7.
<p class="editorial">You're right, Musab. What other possibilities
could there have been then? Pupils at St Mary's in Tetbury
approached the second part of the problem in a very logical (or
systematic) way. They said:</p>
<div>We decided to work out all the numbers it could be.</div>
<br></br>
We started with 6 rows with one left over, and wrote down all the
multiples of 6, add 1, up to 100.<br></br>
Then we looked at 5 rows with one left over, and wrote down all the
multiples of 5, add 1, up to 100. <br></br>
We rubbed out all the numbers that weren't in both. <br></br>
<br></br>
Then we looked at 4 rows with one left over, and wrote down any
multiples of 4, add 1, that appeared in our lists for 5 rows and 6
rows. <br></br>
Then we looked at 3 rows with one left over, and there was only one
number that worked for all of them. We think the solution is
61.<br></br>
<br></br>
<p class="editorial">PJ, Josh and Michael tried a few things out
and gradually got close to the answer. This can be a very useful
way of solving a problem (we call it trial and improvement):</p>
<div>We started by just trying numbers. Then Josh started doing
multiples, and we found out some numbers that have one <span class="editorial">(in the units column?)</span> work so we tried 41 then
51 then we found that 61 worked, and that's how we got our answer:
61.</div>
<br></br>
<p class="editorial">Qiuying of Wimbledon High School explained the
solution in slightly higher level maths. (Don't worry if you
haven't come across these ideas yet.)</p>
<div>Less than 100:</div>
<div>The Number of garlic cloves equals the Lowest Common Multiple
of 2, 3, 4, 5, 6 and then plus 1.</div>
<div>LCM = 60 So the Number is 60 + 1 = 61</div>
<div>The first one more than 100:</div>
<div>The numbers of garlic cloves equals 60 x 2 + 1 = 121</div>
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<h2>Growing Garlic</h2>
<br></br>
Ben is on the allotment with his Mum. They would like to grow some garlic and are deciding how to plant the garlic cloves.<br></br>
<br></br>
Ben arranges the cloves into three rows and finds that he has one spare clove. How many cloves might he have had to start with?<br></br>
<br></br>
Ben plants cloves of garlic in two rows and has one clove left over. So he tries again.<br></br>
He plants cloves in three rows and has one left over. So he tries again.<br></br>
He plants cloves in four rows and has one left over. So he tries again.<br></br>
He plants cloves in five rows and has one left over. So he tries again.<br></br>
He plants cloves in six rows and has one left over.<br></br>
We know that he has less than $100$ garlic cloves. How many did he have?<br></br>
You could think about how many cloves he might have had if there were more than $100$.<br></br>
<br></br>
You might like to use this interactivity to investigate the problem. You can alter the number of cloves using the up and down arrows, and you can drag them into different arrangements using the little white square.
<div style="clear: both;">
<div style="clear: both;">
<div style="clear: both;">
<div style="clear: both;">
<div style="clear: both;"> </div>
<div style="clear: both;"><a href="/content/id/5579/Garlic.swf">Full Screen Version</a></div>
<div style="clear: both;"> </div>
</div>
</div>
</div>
</div>
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<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>The idea of <a href="http://nrich.maths.org/public/viewer.php?obj_id=5579&part=index">this problem</a> is to encourage children to spot and describe a pattern, to extend this into a general rule and, depending on their experience, to use relevant vocabulary associated with factors and multiples.</div>
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<h3>Possible approach</h3>
<div>You could use this problem with the whole class as it is an example of a low threshold, high ceiling activity - accessible to all but challenging for the most confident.</div>
<br></br>
<div>You may want to start by asking some simpler questions and model them using the interactivity, for example:</div>
<ul>
<li>Ben has $15$ bulbs of garlic. (Scroll down to $15$)</li>
<li>Can he plant them in rows of two? Why? Check using the interactivity.</li>
<li>Can he plant them in rows of three? Why? Check using the interactivity.</li>
<li>Can he plant them in rows of four? Why? Check using the interactivity.</li>
<li>What else could you tell me?</li>
</ul>
<br></br>
<div>Then, without using the interactivity, ask the children to find how many bulbs Ben could have if he plants them in rows of, say, four, but has one left over. Have cubes or counters available for them to use. You might also want to have squared paper available. Take answers and list them, using the interactivity to check them. Put them in order of size to help pattern spotting. Encourage
comments which indicate the children have spotted the pattern (it's one more than the four times table) rather than specific (it could be $41, 5, 25 \ldots$)</div>
<br></br>
<div>Pose the first part of the written problem and invite pupils to 'think, pair, share' - think on their own, then talk to a partner about what they think the solution might be, and/or how they might work it out. Take suggestions and if appropriate let one or two children come to the IWB to model their ideas.</div>
<br></br>
<div>Pose the rest of the problem and leave the children to pursue their ideas, working in pairs. Emphasise that you are interested in the answer, but also interested in the way that the children found it. If you are working in a room where there are lots of computers the children can use the interactivity to check their solutions. Otherwise pegboards, cubes, or any other units such as bottle
tops can be helpful.</div>
<br></br>
<div>Bring the children together and share their findings, again encouraging working that is systematic and helping them to differentiate between specific answers and general ones ('if there is one left over with two rows it must be an odd number', rather than 'it must be $21$ or $15$ or $21$').</div>
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<h3>Key questions</h3>
<div>What is the same about these answers?</div>
<div>How could we record our results to help us to spot a pattern?</div>
<br></br>
<h3>Possible extension</h3>
<div>Children who find this easy could make up their own questions for a partner. Devising a question which has a unique answer is quite difficult.</div>
<br></br>
<h3>Possible support</h3>
<div>Physical apparatus helps to consolidate the idea of 'one more than a multiple'. Listing possible answers in order of size can also help the children to spot patterns - encourage them to work systematically.</div>
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<mdoxml version="1.0">Have you tried out your ideas using the interactivity?
Or you could use some counters or cubes to stand for the garlic
cloves and arrange them in rows.<br></br>
If Ben had only planted one clove in each row, how many cloves
would he have had?<br></br>
What other numbers of cloves could he have planted in each
row? So how many cloves would he have had altogether
then?<br></br></mdoxml>
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one more than a multiple of 3 eg 4, 7, 10, 13, 16 etc...<br></br>
61<br></br></mdoxml>
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Growing Garlic
Ben and his mum are planting garlic. Use the interactivity to help
you find out how many cloves of garlic they might have had.
Information and Communications Technology
Interactivities
Using, Applying and Reasoning about Mathematics
Working systematically
Using, Applying and Reasoning about Mathematics
Generalising
Numbers and the Number System
Factors and multiples
Admin
Upper primary mapping document