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/www/nrich/html/content/02/09/penta2/
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2011-02-01T00:00:01
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Here is a grid of four "boxes":<br></br>
<p><mdo:image alt="Square divided into four." src="square1.gif"></mdo:image></p>
<p>You must choose four different digits from $1 - 9$ and put one
in each box. For example:</p>
<p><mdo:image alt="Square divide into four, with 5,2,1 & 9 in the sections." src="square2.gif"></mdo:image></p>
<p>This gives four two-digit numbers:</p>
<table cellpadding="5" border="0">
<tbody>
<tr>
<td>$52$</td>
<td>(reading along the $1$st row)</td>
</tr>
<tr>
<td>$19$</td>
<td>(reading along the $2$nd row)</td>
</tr>
<tr>
<td>$51$</td>
<td>(reading down the left hand column)</td>
</tr>
<tr>
<td>$29$</td>
<td>(reading down the right hand column)</td>
</tr>
</tbody>
</table>
<p>In this case their sum is $151$.</p>
<p>Your challenge is to find four different digits that give four
two-digit numbers which add to a total of $100$.</p>
<p>How many ways can you find of doing it?</p>
<p><span style="font-style: italic;">This problem is adapted from
Make 200 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;">here</a> <span style="font-style: italic;">.</span></p>
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<table width="100%" cellspacing="0" cellpadding="10" border="0" id="tableedit1">
<tbody>
<tr>
<td colspan="4" class="editorial">Shannon, Ellie, Stephen, Tom and
Zachary all at Trinity Middle School, Newport, Isle of Wight found
solutions to "Reach 100". Shannon and Ellie realised that the
numbers 12, 14, 27 and 47 total 100. Tom and Stephen each suggested
a different way the digits could be arranged in the grid:</td>
</tr>
<tr>
<td width="25%"></td>
<td width="25%">
<table width="100" height="100" cellspacing="0" cellpadding="0" bordercolor="#000000" border="1" align="center" class="squarelarge" summary="top left cell contains 1 top right cell contains 2 bottom left cell contains 4 bottom right cell contains 7">
<tbody>
<tr>
<td align="center">1</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center">4</td>
<td align="center">7</td>
</tr>
</tbody>
</table>
</td>
<td width="25%">
<table width="100" height="100" cellspacing="0" cellpadding="0" bordercolor="#000000" border="1" align="center" class="squarelarge" summary="top left cell contains 1 top right cell contains 4 bottom left cell contains 2 bottom right cell contains 7">
<tbody>
<tr>
<td align="center">1</td>
<td align="center">4</td>
</tr>
<tr>
<td align="center">2</td>
<td align="center">
<p>7</p>
</td>
</tr>
</tbody>
</table>
</td>
<td width="25%"></td>
</tr>
<tr>
<td colspan="4" class="editorial">Zachary spotted both these - very
well done - and also sent this solution:</td>
</tr>
<tr>
<td colspan="4">
<table width="100" height="100" cellspacing="0" cellpadding="0" bordercolor="#000000" border="1" align="center" class="squarelarge" summary="top left cell contains 2 top right cell contains 3 bottom left cell contains 1 bottom right cell contains 8">
<tbody>
<tr>
<td align="center">2</td>
<td align="center">3</td>
</tr>
<tr>
<td align="center">1</td>
<td align="center">8</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td colspan="4" class="editorial">Kirsty, from St Aldhelms School
in Poole, went one stage further. She rearranged these digits to
make the final solution:</td>
</tr>
<tr>
<td colspan="4">
<table width="100" height="100" cellspacing="0" cellpadding="0" bordercolor="#000000" border="1" align="center" class="squarelarge" summary="top left cell contains 2 top right cell contains 3 bottom left cell contains 1 bottom right cell contains 8">
<tbody>
<tr>
<td align="center">2</td>
<td align="center">1</td>
</tr>
<tr>
<td align="center">3</td>
<td align="center">8</td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
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<h2>Reach 100</h2>
<br></br>
Here is a grid of four "boxes":<br></br>
<p><mdo:image alt="Square divided into four." src="square1.gif"></mdo:image></p>
<p>You must choose four different digits from $1 - 9$ and put one in each box. For example:</p>
<p><mdo:image alt="Square divide into four, with 5,2,1 & 9 in the sections." src="square2.gif"></mdo:image></p>
<p>This gives four two-digit numbers:</p>
<table style="" border="">
<tbody>
<tr>
<td>$52$</td>
<td>(reading along the $1$st row)</td>
</tr>
<tr>
<td>$19$</td>
<td>(reading along the $2$nd row)</td>
</tr>
<tr>
<td>$51$</td>
<td>(reading down the left hand column)</td>
</tr>
<tr>
<td>$29$</td>
<td>(reading down the right hand column)</td>
</tr>
</tbody>
</table>
<p>In this case their sum is $151$.</p>
<p>Your challenge is to find four different digits that give four two-digit numbers which add to a total of $100$.</p>
<p>How many ways can you find of doing it?</p>
<p><span style="font-style: italic;">This problem is adapted from Make 200 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;">here</a>
<span style="font-style: italic;">.</span></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=1130&part=index">This problem</a> challenges pupils' understanding of place value and is a good way to practise a particular method of written addition.</div>
<br></br>
<h3>Possible approach</h3>
<div>It would be good to start with the grid drawn on the board and for you to explain the challenge orally to the group. You could use the example in the problem itself so that the task is clear.</div>
<br></br>
<div>Ask for suggestions as to how they might start and give learners a few minutes to think on their own, then share their ideas with a partner. Open this out to the whole class so that a few pairs share their thoughts with everyone. The suggestions are likely to be quite general at this stage.</div>
<br></br>
<div>Allow children to work together on the problem in their pairs. They will find mini-whiteboards or paper useful for keeping track of their calculations. After some time, bring the group together again and discuss any insights they have gained. Some pairs may have thought about the cells in the grid which make up the units digits of the four numbers, others may have concentrated on the cells
which contribute to the tens digits. In either case, encourage them to explain the restrictions they have noticed, and look out for those pairs who are working systematically through the options.</div>
<br></br>
<div>You may like to leave time (perhaps in a subsequent lesson) for each pair to produce a poster describing how they arrived at the solution(s).</div>
<br></br>
<h3>Key questions</h3>
<div>Where could you start?</div>
<div>How do the four digits you choose contribute to the zero in the units column of $100$?</div>
<div>What can you say about the size of the digit in the top left cell?</div>
<br></br>
<br></br>
<h3>Possible extension</h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=781&part=index">Two and Two</a> requires similar systematic thinking and could be a good problem for some children to try next.</div>
<br></br>
<h3>Possible support</h3>
<div>Having digit cards available for learners to physically manipulate will help those who are reluctant to commit ideas to paper.</div>
<br></br>
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<p>Simply writing out the numbers $1 - 9$ along the top of your
paper can be a great help.</p>
<p>Imagine arranging your four two-digit numbers in columns to do
the addition sum, like this:</p>
<p>$5 \: 2$<br></br>
$1 \; 9$<br></br>
$5 \; 1$<br></br>
$2 \; 9$</p>
<p>If the total has to be $100$, think about how the four digits
you choose could add to make the $0$ in the units column, then look
at the tens column.</p>
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Reach 100
Choose four different digits from 1-9 and put one in each box so
that the resulting four two-digit numbers add to a total of 100.
Using, Applying and Reasoning about Mathematics
Working systematically
Numbers and the Number System
Place value
Calculations and Numerical Methods
Addition & subtraction
Mathematics Tools
Digit cards
Admin
Upper primary mapping document