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/www/nrich/html/content/00/07/six1/
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2011-02-01T00:00:01
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<p>An equilateral triangle can be dissected into four (equal)
smaller equilateral triangles.</p>
<p><mdo:image alt="" src="triangle.gif"></mdo:image></p>
<p>Can you dissect an equilateral triangle into six smaller ones,
not necessarily all the same size?</p>
<p>Is it possible to dissect a larger equilateral triangle into any
number of smaller equilateral triangles?</p>
<p>Are there any numbers that are impossible?</p>
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<p>The diagram shows an equilateral triangle dissected into 4 smaller equilateral triangles. . Which numbers of smaller triangles are possible and which are not, and why? This remains a Tough Nut as nobody has yet explained why it is impossible to dissect an equilateral triangle into certain numbers of smaller equilateral triangles.</p>
<p>Robert and Andrew, St James Middle School, Bury St Edmund's sent this solution with six equilateral triangles. They say they think this is the only solution with 6 smaller triangles other than rotating the shape.</p>
<mdo:image width="166" height="140" alt="" src="sol2.gif"></mdo:image>
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<p>Talei and Stephanie of Poltair Community School and Sports College, St Austell, Cornwall said they think that dissection into 2 or 3 smaller triangles is impossible and that 4 is the smallest number of triangles possible. They gave this as an example of dissection into 15 smaller triangles.</p>
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<p>Talei's also had the idea of allowing overlapping triangles and realised that this would make it possible to have any number of smaller triangles, even infinitely many, but that would be a different question.</p><br></br></mdoxml>
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<h2>Cut it Out</h2>
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<p>An equilateral triangle can be dissected into four (equal) smaller equilateral triangles.</p>
<p><mdo:image alt="" src="triangle.gif"></mdo:image></p>
<p>Can you dissect an equilateral triangle into six smaller ones, not necessarily all the same size?</p>
<p>Is it possible to dissect a larger equilateral triangle into any number of smaller equilateral triangles?</p>
<p>Are there any numbers that are impossible?</p>
</div>
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<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>At the most basic level, <a href="http://nrich.maths.org/public/viewer.php?obj_id=720&part=index">this problem</a> provides an opportunity to reinforce what an equilateral triangle is, and whilst that will be sufficient for some, this question requires much more sophisticated thinking than that. For the high flyers in your class, the fact that it is a 'tough nut' (ie no children's
solution has so far been submitted) might provide the extra incentive to persevere when the going gets tough.</div>
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<h3>Possible approach</h3>
<div>This problem requires some time to do properly. If you can introduce it whilst doing 2D shape as a main topic, you might want to offer it as a 'simmering' activity for a small able group who can return to it over several days, perhaps when they have completed their allotted tasks.</div>
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<div>The question is in three parts and these are best introduced one at a time. Give out <a href="/content/00/07/six1/CutItOutTrianglesnewicons.doc">sheets of equilateral triangles</a> and allow some time for the children to explore the first question either individually or in pairs. Some children might prefer to draw their triangles on <a href="http://www1.curriculum.edu.au/maths300/m300bits/000dotis.htm">isometric dotty paper</a> . Listen for statements which are accompanied by some sort of justification, for example "Four is the smallest because after one big triangle the next size up has two triangles along each side." Encourage the children to work systematically.</div>
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<div>If you think it appropriate, make available a section of display space where they can place their diagrams, cut out and sorted into 'families'. This way the individual findings can be a contribution to a greater whole.</div>
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<h3>Key questions</h3>
<div>What is the smallest number of triangles we can break one large one into?</div>
<div>How do you know?</div>
<div>Is there a largest number?</div>
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<h3>Possible extension</h3>
<div>Some children will want to find a general rule. This is a sophisticated piece of generalising.</div>
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<h3>Possible support</h3>
<div>You may want to give out triangular grid or <a href="http://www1.curriculum.edu.au/maths300/m300bits/000dotis.htm">dotty paper</a> and ask the children to draw equilateral triangles of different sizes and count how many smaller triangles are in each. This will help them to see that 1, 4, 9, 16 are useful numbers to use.</div>
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<mdoxml version="1.0">What is the smallest number of triangles you can break one triangle
in to?<br></br>
You could print out <a href="/content/00/07/six1/CutItOutTriangles.doc">this sheet of
triangles</a> or use <a href="http://www1.curriculum.edu.au/maths300/m300bits/000dotis.htm">isometric
dotty paper</a>.<br></br></mdoxml>
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Cut it Out
Can you dissect an equilateral triangle into 6 smaller ones? What
number of smaller equilateral triangles is it NOT possible to
dissect a larger equilateral triangle into?
Using, Applying and Reasoning about Mathematics
Generalising
2D Geometry, Shape and Space
Equilateral triangles
Admin
Upper primary mapping document