2004
/www/nrich/html/content/03/11/penta1/
1
2011-02-01T00:00:01
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<div align="center"><mdo:image src="sum1.gif" alt="sum: 1abcde multiplied by 3 equals abcde1." width="200" height="86"></mdo:image></div>
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<div align="center"><mdo:image src="sum2.gif" alt="sum: 2fghij multiplied by 3 equals fghij2" width="200" height="86"></mdo:image></div>
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<p>Can you replace the letters with numbers?<br></br>
Is there only one solution in each case?</p>
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<p class="editorial">This problem looked more difficult than I
think it really was, although it did require a great deal of
logical thinking. Abdullah from Higher Bebington Junior School
emailed us with his solution. He says:</p>
<p>For each problem I first looked to find a number that would make
the units column accurate, then I substituted the number for the
answer in the tens column and then continued the process until the
calculation was complete.</p>
<p class="editorial">This is a very good way of going about solving
the problem, well done Abdullah. Here are Abdullah's final
answers:</p>
<p>1 4 2 8 5 7</p>
<p><span style="text-decoration: underline;">x 3</span></p>
<p><span style="text-decoration: underline;">4 2 8 5 7 1</span></p>
<p>and</p>
<p>2 8 5 7 1 4</p>
<p><span style="text-decoration: underline;">x 3</span></p>
<p><span style="text-decoration: underline;">8 5 7 1 4 2</span></p>
<p class="editorial">Joshua from Tattingstone School tells us how
he approached the problem in more detail:</p>
<p>I wrote out single digit multiples of three up to 9 because each
letter was one digit. I noticed that the numbers 1 to 9 only
appeared once in the units column of the answers. I looked at the
question and realised that 3 x e had to be 21 because it was the
only answer ending in 1. This meant that e had to be 7. I carried
the 2 and took it from 7 (the other e) and got 5. So d x 3 had to
end in 5 which meant d had to be 5 because 5 x 3 = 15. I then
repeated the process for the rest of questions 1 and 2.</p>
<p class="editorial">This is very clearly explained - thank you
Joshua.</p>
<p class="editorial">Goh Wei Hao from Corporation Primary School in
Singapore thinks that these are the only solutions. He says:</p>
<p>There is no multiple of three that has 1 in the ones place
except for 7. (In the first sum)</p>
<p>There is no multiple of three that has 2 in the ones place
except for 4. (In the second sum)</p>
<p class="editorial">This is certainly true (and Joshua mentions
this too), although we would need to take the explanation a bit
further to justify that there aren't any other solutions.</p>
<p class="editorial">Well done also to the following who sent
correct solutions:</p>
<p class="editorial">Ang Boon, Ang Li, Bay Jia, Goh Bee, Khoo Kian,
Tan Li, Tan Yang and Teoh Wei all from Corporation Primary School
in Singapore</p>
<p class="editorial">Harry from St Mary's Stansted</p>
<p class="editorial">Katrina and Helen from British School
Manila</p>
<p class="editorial">Tutku and Adil from Istanbul</p>
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<h2>Trebling</h2>
<table style="" border="">
<tbody>
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<div align="center"><mdo:image alt="sum: 1abcde multiplied by 3 equals abcde1." height="86" src="sum1.gif" width="200"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image alt="sum: 2fghij multiplied by 3 equals fghij2" height="86" src="sum2.gif" width="200"></mdo:image></div>
</td>
</tr>
</tbody>
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<p>Can you replace the letters with numbers?<br></br>
Is there only one solution in each case?</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=2004&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.</div>
<br></br>
<h3>Possible approach</h3>
<div>You could start by showing the first sum to the whole group and discussing what is required to do it. It would be good for learners to work in pairs, perhaps using <a href="/content/03/11/penta1/TreblingTemplates.doc">this sheet</a> . (The first page contains the first sum with several lighter versions for children to write on. The second page is the second sum done in the same way.)
Alternatively, pairs could use digit cards to move around on a sheet of paper, or mini-whiteboard.</div>
<br></br>
<div>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 digit 'e' must be by looking at the units digit of the answer. Some may have started in a different way, for example by trying
digits at random to see what happens when the multiplication takes place. Draw attention to those pairs that have adopted a system in their working and highlight that this means the solution will be found more efficiently. You may need to clarify that all the letters appear in the answer too, in other words that 'e' stands for the same number wherever it appears (the beginnings of algebraic
notation). You could then leave learners to continue with the problem.</div>
<br></br>
<div>A discussion of all the steps in the reasoning could take place in the plenary.</div>
<br></br>
<h3>Key questions</h3>
<div>How does the $1$ (or $2$) in the units column of the answer help?</div>
<div>What are the units digits of the multiples of $3$?</div>
<br></br>
<h3>Possible extension</h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=1129&part=index">All the Digits</a> makes a good extension activity as the mathematics is similar, but slightly more sophisticated reasoning is required.</div>
<br></br>
<h3>Possible support</h3>
<div>It might help some children to write out their three times table to begin with, or to be able to refer to a multiplication square as they tackle this problem.</div>
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Would it help to start by looking at the units column when
multiplied by $3$? <br></br>
Don't forget that the digits appear in the answer too!<br></br></mdoxml>
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Trebling
Can you replace the letters with numbers? Is there only one
solution in each case?
Calculations and Numerical Methods
Multiplication & division
Using, Applying and Reasoning about Mathematics
Working systematically
Numbers and the Number System
Place value
Algebra
Introducing algebra
Admin
Upper primary mapping document