1129
/www/nrich/html/content/02/09/penta1/
1
2011-02-01T00:00:01
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<p>This represents the multiplication of a $4$-figure number by
$3$.</p>
<div style="text-align: center;"><mdo:image height="134" width="226" alt="a multiplication of a 4-figure number by 3" src="multiplication.gif"></mdo:image></div>
<p>The whole calculation uses each of the digits $0 - 9$ once and
once only.</p>
<p>The $4$-figure number contains three consecutive numbers, which
are not in order. The third digit is the sum of two of the
consecutive numbers.</p>
<p>The first, third and fifth figures of the five-digit product are
three consecutive numbers, again not in order. The second and
fourth digits are also consecutive numbers.</p>
<p>Can you replace the stars in the calculation with figures?</p>
<br></br>
<div><span style="font-style: italic;">A practical version of this
activity is included in one of the Brain Buster Maths Boxes which
contains hands-on challenges developed by members of NRICH and
produced by BEAM. For more details and ordering information, please
scroll down</span> <a style="font-style: italic;" href="http://nrich.maths.org/public/viewer.php?obj_id=4833&part=index">
this page</a> .</div>
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<p class="editorial">Zachary, Stephen and Natasha who are all
pupils at Trinity Middle School in Newport, Isle of Wight sent us
correct answers to this problem.Stephen said:</p>
<p>I decided that if you don't use a 9 last in the 4 numbers at the
top it would work out.</p>
<p class="editorial">Yes - we agree Stephen, it is the third digit
in the 4-figure number which is the sum of consecutive numbers.
Well spotted!</p>
<p class="editorial">Natasha looked up the clues. She wrote the
numbers on pieces of card and then jiggled them about.</p>
<p class="editorial">Kirsty from St Aldhelms School in Poole agreed
that it must be the third digit of the 4-digit number which is the
sum of consecutive numbers. She explained very clearly how she
arrived at her answer:</p>
<p>The only three consecutive numbers that can go in the 4-figure
number are 4, 5 and 6. 7, 8 and 9 are too big. The sum of any two
of these is greater than 9. For example:<br></br>
7 + 8 = 15<br></br>
8 + 9 =17<br></br>
9 + 7 = 16<br></br>
0, 1 and 2 cannot go on the first line because:<br></br>
0 x 3 = 0 (same number twice)<br></br>
1 x 3 = 3 (same number twice)<br></br>
Therefore the third number must be 9 (5 + 4) beause 6 + 5 and 6 + 4
are both too big.<br></br>
The fourth number in the 4-figure number cannot be 5 as 5 x 3 = 15
(repeat digit 5).<br></br>
The fourth number also cannot be 6 as then we would get 8 twice, so
it must be 4.<br></br>
So, the last two digits must be 5 then 6 so they're not in
order.</p>
<p>This is the answer all four agreed on:</p>
<table width="124" cellspacing="0" cellpadding="0" border="0" summary="5694 x 3 = 17082">
<tbody>
<tr>
<td></td>
<td align="center">5</td>
<td align="center">6</td>
<td align="center">9</td>
<td align="center">4</td>
</tr>
<tr>
<td align="center">x</td>
<td></td>
<td></td>
<td></td>
<td align="center">3</td>
</tr>
<tr>
<td align="center" colspan="5">
<hr></hr></td>
</tr>
<tr>
<td align="center">1</td>
<td align="center">7</td>
<td align="center">0</td>
<td align="center">8</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center" colspan="5">
<hr></hr></td>
</tr>
</tbody>
</table>
<p class="editorial">Well done, particularly to Kirsty for her
excellent reasoning!</p>
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<h2>All the Digits</h2>
<br></br>
<p>This represents the multiplication of a $4$-figure number by $3$.</p>
<div style="text-align: center;"><mdo:image alt="a multiplication of a 4-figure number by 3" height="134" src="multiplication.gif" width="226"></mdo:image></div>
<p>The whole calculation uses each of the digits $0 - 9$ once and once only.</p>
<p>The $4$-figure number contains three consecutive numbers, which are not in order. The third digit is the sum of two of the consecutive numbers.</p>
<p>The first, third and fifth figures of the five-digit product are three consecutive numbers, again not in order. The second and fourth digits are also consecutive numbers.</p>
<p>Can you replace the stars in the calculation with figures?</p>
<br></br>
<div><span style="font-style: italic;">A practical version of this activity is included in one of the Brain Buster Maths Boxes which contains hands-on challenges developed by members of NRICH and produced by BEAM. For more details and ordering information, please scroll down</span> <a href="http://nrich.maths.org/public/viewer.php?obj_id=4833&part=index" style="font-style: italic;">this
page</a> .</div>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=1129&part=index">This problem</a> requires learners to think about place value and the way that standard column multiplication works. Although the problem can be done by trial and improvement, it is solved more efficiently if worked through systematically.<br></br>
<h3>Possible approach</h3>
<div>You could start by showing the problem to the whole group and discussing what is required to do it. Do they understand what consecutive numbers are? Are they confident about the meaning of 'sum' and 'product'?</div>
<br></br>
<div>After this introduction the group could work in pairs on the problem so that they are able to talk through their ideas with a partner. <a href="/content/02/09/penta1/1129.pdf">This sheet</a> is intended for rough working and the solution, and <a href="/content/02/09/penta1/1129B.pdf">this sheet</a> gives the blank calculation and digit cards to cut out. Give the children time to make a start
and then after a suitable length of time, bring the group back together to talk about how they are getting on so far. This is a good opportunity to share some initial insights. For example, some pairs may have worked out which digits must be in the four-digit number, even if they don't know the order yet. Some may have started in a different way, for example by looking for the digit which could
go in the units column of the four-digit number. Draw attention to those pairs that have adopted a system in their working which means they are trying numbers in an ordered way. This means that they are guaranteed not to leave out any possibilities. You could then leave learners to continue with the problem.</div>
<br></br>
<div>At the end, the whole class could discuss the steps in their reasoning and how they reached a full solution. Did they use all the information in the question right from the start? Which parts were most helpful and why?</div>
<br></br>
<h3>Key questions</h3>
<div>What could the first figure of the product be if the multiplication is by $3$?</div>
<div>Which consecutive numbers could be in the four-digit number?</div>
<div>Which other digit could appear in the four-digit number?</div>
<br></br>
<h3>Possible extension</h3>
Challenge those pupils who finish quickly to prove to you that there is only one solution. How many solutions would there be if the clues about consecutive numbers did not hold?<br></br>
<h3>Possible support</h3>
Suggest working with digit cards and possibly a mini-whiteboard. <a href="/content/02/09/penta1/1129B.pdf">This sheet</a> gives the blank calculation and digit cards to cut out.<br></br>
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<p>Use counters or scraps of paper with the digits $0 - 9$ written
on them.</p>
<p>Make a list of $3$ consecutive numbers $0 - 9$ remembering that
$3$ has already been accounted for.</p>
<p>What could the first figure of the product be if the
multiplication is by $3$?</p>
<p>Which consecutive numbers could be in the four-digit number?</p>
<p>Which other digit could appear in the four-digit number?</p>
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All the Digits
This multiplication sum uses each of the digits 0 - 9 once and once
only. Using the information given, can you replace the stars in the
calculation with figures?
Numbers and the Number System
Place value
Using, Applying and Reasoning about Mathematics
Working systematically
Calculations and Numerical Methods
Multiplication & division
Mathematics Tools
Digit cards
Admin
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