6227
/www/nrich/html/content/id/6227/
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2011-02-01T00:00:01
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Place the numbers $1$ to $6$ in the circles so that each number is the difference between the two numbers just below it.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="6 circles" height="151" src="FindDa.gif" width="170"></mdo:image></div>
Example: <mdo:image alt="" height="77" src="FindD.gif" width="86"></mdo:image>$5 - 2 = 3$<br></br>
<br></br>
You could try it in this interactivity.<br></br>
<a href="/content/id/6227/FindDiff.swf">Full screen version</a>
<div style="text-align: center;"><mdo:flash height="400" id="/content/id/6227/FindDiff.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param>
<param name="movie" value="/content/id/6227/FindDiff.swf"></param>
<param name="flashplayerversion" value="8"></param>
</mdo:flash></div>
When you have tried this and got more than one answer you could try other things like having $4$ rows of discs and use the number $1$ to $10$<br></br>
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<p><span class="editorial">Many of you sent us answers for this problem but very few gave us details about how you solved it. Please do try to explain how you reach your solutions.</span></p>
<p><span class="editorial">Stephen, who is home-educated, told us:</span></p>
I thought about all the possible subtractions and eliminated all the subtractions that have the same number repeated in them. (e.g. 6 - 3 = 3). That's a good way to start, Stephen. It's a shame that you didn't say what you did next.<br></br>
<br></br>
<p><span class="editorial">However, the Junior Class at Ysgol Bryncrug in Gwynedd, Wales sent an extremely well-explained solution:</span></p>
We thought which number to put on the top. We started with 1. That meant 2, 3 or 3, 4 or 4, 5 or 5, 6 could go under it.<br></br>
We tried 2, 3 that left us with 4, 5, 6 for the bottom line. None of these gave us a difference of 3.<br></br>
Then we tried 3, 4 and this left us with 2, 5, 6 which worked.<br></br>
Then we tried 4, 5 and this left us with 2, 3, 6 for the bottom. None of these gave a difference of 5.<br></br>
Then we tried 5, 6 on the second line and it is impossible to make a difference of 6 because it is the highest number.<br></br>
<br></br>
So only 3, 4 works under 1:<br></br>
<br></br>
<mdo:image alt="first solution" height="197" src="sol1.gif" width="235"></mdo:image><br></br>
<br></br>
We then saw that we could swap 3 and 4 around if we also swapped 5 and 6:<br></br>
<br></br>
<mdo:image alt="second solution" height="205" src="sol2.gif" width="248"></mdo:image><br></br>
Then we tried 2 at the top. Working the same way we found out that only two ways could work:<br></br>
<br></br>
<mdo:image alt="third and fourth solutions" height="188" src="sol3_4.gif" width="400"></mdo:image><br></br>
<br></br>
Then we tried 3 at the top. Working the same way we found out that four ways could work:<br></br>
<br></br>
<mdo:image alt="fifth, sixth, seventh and eighth solutions" height="400" src="sols5_8.gif" width="400"></mdo:image><br></br>
<br></br>
Then we tried 4 at the top. On the second line we would have to use 1, 5 or 2, 6.<br></br>
1, 5 didn't work out because we can't make a difference of 5 on the bottom line (6+1) because we have used 1 in the middle line.<br></br>
2, 6 didn't work because we can't make a difference of 6 because it is the biggest number and it must go on the bottom line.<br></br>
<br></br>
When we tried 5 on the top we have to use 1, 6 in the middle and we can't make a difference of 6 on the bottom.<br></br>
<br></br>
There wouldn't be a point for us to try 6 at the top because no two numbers can give us a difference of 6 because it is the highest number.<br></br>
<br></br>
So we think there are 8 solutions.<br></br>
<br></br>
<p class="editorial">What fantastic logical reasoning, well done. I like the way you have gone through each possible top number in order. Having a system like that ensures that you find all the solutions.</p>
<br></br>
<br></br>
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<mdoxml version="1.0"><div class="embed">
<h2>Find the Difference</h2>
<br></br>
Place the numbers $1$ to $6$ in the circles so that each number is the difference between the two numbers just below it.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="6 circles" height="151" src="FindDa.gif" width="170"></mdo:image></div>
Example: <mdo:image alt="" height="77" src="FindD.gif" width="86"></mdo:image>$5 - 2 = 3$<br></br>
<br></br>
You could try it in this interactivity.<br></br>
<a href="/content/id/6227/FindDiff.swf">Full screen version</a>
<div style="text-align: center;"><mdo:flash height="400" id="/content/id/6227/FindDiff.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param>
<param name="movie" value="/content/id/6227/FindDiff.swf"></param>
<param name="flashplayerversion" value="8"></param>
</mdo:flash></div>
When you have tried this and got more than one answer you could try other things like having $4$ rows of discs and use the number $1$ to $10$<br></br>
</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=6227&part=index">This problem</a> is a challenging way of practising subtraction at the same time as being logical about arranging the numbers. The idea of 'difference' can be hard for children to grasp and this problem is an ideal way of coming to terms with it. You could also use this problem to focus on how children record their
workings.<br></br>
<h3>Possible approach</h3>
<div>You could start by putting the numbers in any places in the pyramid and asking children to describe what they see. They may notice some accidental number patterns, but also which numbers are used. Then put two numbers into the pyramid, for example $4$ and $5$ in the bottom row, next to each other. Introduce the idea of the problem and invite pupils to suggest what number should go above the
$4$ and $5$. You could repeat this a few times with different pairs of numbers so that children are happy with 'difference' and with the way the pyramid will be structured. You may need to emphasise that the smaller number will be taken away from the larger number each time.</div>
<br></br>
Introduce the task as stated in the problem and encourage learners to work in pairs. They may want to use the <a href="/content/id/6227/FindDiff.swf">interactivity</a> on computers and/or they could work on paper, using <a href="/content/id/6227/Find%20the%20Difference%20Circles.pdf">this sheet</a> of blank pyramids for rough working and recording. If you want a whole class to work on the problem
at the same time, <a href="/content/id/6227/Find%20the%20Difference%20Problem.pdf">this sheet</a> gives the pyramid of circles, the numbers from 1 to 6 to cut out and leaves space for adding another row of circles should this be required. After a short time, it would be useful to encourage learners to share the ways they are working and what they are recording. Some might be trying numbers then
adjusting them, others might have thought about where, for example, the largest must go, others might be taking one number at a time and looking at the different places it could go in turn. The key point to make is that their recording system should somehow enable them to keep track of what arrangements they have tried so far so they don't repeat themselves. It is an advantage, therefore, not to
use an eraser!<br></br>
<br></br>
<div>The plenary can be used to compare different solutions, using the interactivity, and to discuss the advantages of the different recording methods.</div>
<h3>Key questions</h3>
<div>Where could the largest number go? Why?</div>
What do we need to do to find the difference between two numbers?
<h3>Possible extension</h3>
<div>Some children could be challenged to find as many different arrangements as possible.<br></br>
They might also like to try this very challenging <a href="http://nrich.maths.org/public/viewer.php?obj_id=927&part=index">problem</a> which uses the numbers from 1 - 10.</div>
<br></br>
<h3>Possible support</h3>
Some children would benefit from using numbered counters that can be moved about if they don't have access to the interactivity. Some children may need to use manipulatives to support them in finding the differences.<br></br>
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<mdoxml version="1.0"><br></br>
You could use numbered counters that can be moved about if you're
not using the interactivity.<br></br>
Where might the largest number go?<br></br>
You might find <a href="/content/id/6227/FindDifa.pdf">this sheet
of blank pyramids</a> useful.<br></br></mdoxml>
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Find the Difference
Place the numbers 1 to 6 in the circles so that each number is the
difference between the two numbers just below it.
Using, Applying and Reasoning about Mathematics
Trial and improvement
Calculations and Numerical Methods
Addition & subtraction
Using, Applying and Reasoning about Mathematics
Working systematically
Information and Communications Technology
Interactivities
Admin
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