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/www/nrich/html/content/02/06/penta3/
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2011-02-01T00:00:01
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<p>Ram divided $15$ pennies among four small bags.</p>
<p><mdo:image alt="bags" src="bags.gif"></mdo:image></p>
<p>He labelled each bag with the number of pennies inside it.</p>
<p>He could then pay any sum of money from $1$p to $15$p without opening any bag.</p>
<p>How many pennies did Ram put in each bag?</p>
<p><span style="font-style: italic;">This problem is based on</span> <em>Money Bags</em> <span style="font-style: italic;">from 'Mathematical Challenges for Able Pupils Key Stages 1 and 2', published by DfES. You can find out more about this book, including how to order it, on the Standards website</span> <a href="http://www.standards.dfes.gov.uk/primary/publications/mathematics/able_pupils_challenges/" style="font-style: italic;"></a><a href="http://webarchive.nationalarchives.gov.uk/20110202093118/http:/nationalstrategies.standards.dcsf.gov.uk/node/85260"><span style="font-style: italic;">here</span> <span style="font-style: italic;">.</span></a></p>
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<p>Matthew, James and Yuji from Moorfield Junior School, and Sarah
L and Joe from Tattingstone School agreed that the money should be
split in this way:</p>
<blockquote>
<table cellpadding="10" summary="">
<tbody>
<tr>
<td>1p</td>
<td>2p</td>
<td>4p</td>
<td>8p</td>
</tr>
</tbody>
</table>
<p>They all then went on to show how each sum of money from 1p to
15p can be made with these four bags:</p>
<p>1p<br></br>
2p<br></br>
1p+2p=3p<br></br>
4p<br></br>
4p+1p=5p<br></br>
4p+2p=6p<br></br>
4p+1p+2p=7p<br></br>
8p<br></br>
8p+1p=9p<br></br>
8p+2p=10p<br></br>
8p+2p+1p=11p<br></br>
8p+4p=12p<br></br>
8p+1p+4p=13p<br></br>
8p+4p+2p=14p<br></br>
8p+4p+2p+1p=15p</p>
</blockquote>
<div>Aditya from Bristol Grammar has been very busy this month and
also gave us the reasoning behind this answer:</div>
<br></br>
<div>If we call the numbers of pennies in each bag a , b , c and d
, we know that a+b+c+d=15, otherwise the problem would have stated
that Ram can pay any sum of money from 1 to more than 15.<br></br>
<div style="clear: both;">Just for convenience, let a < b < c
< d.<br></br>
<div style="clear: both;">The smallest value, a, must equal
1.<br></br>
<div style="clear: both;">As there is no use in having two bags
with one penny in them, b must equal 2.<br></br>
<div style="clear: both;">If Ram was to pay 3p, he would hand over
bag a and b.<br></br>
<div style="clear: both;">But if he were to pay 4p (this is the
same as if he had to pay 2p) it would be pointless repeating
another bag of 2p.<br></br>
<div style="clear: both;">So bag c has 4p in it.<br></br>
<div style="clear: both;">As a+b+c+d = 15, and a+b+c = 7, d
therefore is 8.</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
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<h2>Money Bags</h2>
<br></br>
<p>Ram divided $15$ pennies among four small bags.</p>
<p><mdo:image alt="bags" src="bags.gif"></mdo:image></p>
<p>He labelled each bag with the number of pennies inside it.</p>
<p>He could then pay any sum of money from $1$p to $15$p without opening any bag.</p>
<p>How many pennies did Ram put in each bag?</p>
<p><span style="font-style: italic;">This problem is based on</span> <em>Money Bags</em> <span style="font-style: italic;">from 'Mathematical Challenges for Able Pupils Key Stages 1 and 2', published by DfES. You can find out more about this book, including how to order it, on the Standards website</span> <a href="http://webarchive.nationalarchives.gov.uk/20110202093118/http:/nationalstrategies.standards.dcsf.gov.uk/node/85260"><span style="font-style: italic;">here</span> <span style="font-style: italic;">.</span></a></p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=1116&part=index">This problem</a> is a good example of a challenge which does not require high-level mathematics, but does need a systematic approach.</div>
<br></br>
<h3>Possible approach</h3>
<div>In order that children understand the requirements of this problem, it would be a good idea to begin by reversing the situation. Draw two bags on the board and label them $1$p and $3$p. Invite children to suggest the amounts you could pay using these bags if you are not allowed to open either of them [$1$p, $3$p and $4$p]. Introduce another bag to go with the first two which contains $5$p.
This time, ask the children which amounts of money they <span style="font-style: italic;">wouldn't</span> be able to make using these bags [$2$p, $7$p, anything of $10$p and above]. You could give them time in pairs to work on this, perhaps using a mini-whiteboard to make jottings.</div>
<br></br>
<div>Invite pairs to share their solutions and highlight those that have used a systematic approach, for example by starting with the smallest amount of money and working up in steps of a penny at a time. You can then introduce the problem as it stands. Give pairs chance to work together before once again sharing their ideas. Before they give the answer, ask children to explain how they went
about finding the solution.</div>
<br></br>
<div>The whole class could then check the solution offered by trying to make all the different amounts from $1$p to $15$p.</div>
<br></br>
<h3>Key questions</h3>
<div>How will you make $1$p?</div>
<div>How will you make $2$p? ...</div>
<div>How will you make sure you can pay for all the amounts from $1$p to $15$p?</div>
<br></br>
<h3>Possible extension</h3>
<div>You could challenge children to extend this problem by asking what amount they would need in a fifth bag to be able to make as many amounts as possible over $15$p.</div>
<br></br>
<h3>Possible support</h3>
<div>Having real or fake coins available, along with some small bags, might help some children grasp this problem. Encouraging them to try ideas out is vital.</div>
<br></br>
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<p>How would you make $1$p?</p>
<p>What do you need to do now to be able to make $2$p?</p>
<p>Does this help to make $3$p?</p>
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Money Bags
Ram divided 15 pennies among four small bags. He could then pay any
sum of money from 1p to 15p without opening any bag. How many
pennies did Ram put in each bag?
Measures and Mensuration
Money
Calculations and Numerical Methods
Addition & subtraction
Using, Applying and Reasoning about Mathematics
Working systematically
Mathematics Tools
Coins
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