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The ancient Egyptians were said to make right-angled triangles
using a rope which was knotted to make $12$ equal sections.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image height="226" width="269" alt="" src="knots.jpg"></mdo:image> <mdo:image height="226" width="280" alt="" src="tri.jpg"></mdo:image></div>
<br></br>
<br></br>
If you have a rope knotted like this, what other triangles can you
make? (You must have a knot at each corner.)
<p style="text-align: left;">What regular shapes can you make -
that is, shapes with equal sides and equal angles?</p>
<br></br>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=6832&part=">
Click here for a poster of this
problem</a>. </div>
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<p class="editorial">Randley School sent in four good
solutions.</p>
<br></br>
<p class="editorial">Naziya said:</p>
I found out that I could make a square with three sections at each
side. I also found out that I could make three different triangles:
an isoscelos triangle $2$ along the bottom and $5$ up each side, a
right angled which had $3$ along the bottom, $4$ up one side and
$5$ on the other side and an equilateral with all three sides the
same. I could also make two different rectangles, like this: $5, 1,
5, 1$ or $4, 2, 4, 2$.<br></br>
<br></br>
<p><span class="editorial">Matthew and Jordan said:</span></p>
I made three lots of triangles and I made an isosceles triangle
with $2, 5$ and $5$. Then I made an equilateral triangle with three
lots of $4$. The last triangle was a right angled triangle with $5,
4$ and $3$. I made two rectangles one had $4, 2, 4, 2$. The last
rectangle had $5, 1, 5, 1$. The square had $3, 3, 3, 3$. Then we
made a hexagon and it had $2, 2, 2, 2, 2, 2$.<br></br>
<br></br>
<p><span class="editorial">Ffion said:</span></p>
I made a square with three sections at each side. I made two
rectangles one of them was $5$ across and $2$ down, the second one
was $4$ across and $2$ down. I made three triangles: a right angled
triangle which has $3$ across, $4$ up one side and $5$ on the other
side, the isosceles triangle which has $2$ across and $5$ up, the
other triangle was the equilateral triangle with all the sides had
$4$. I made a hexagon which has $12$ sides.<br></br>
<br></br>
<p class="editorial">I'm sure there'll be some rethinking of that
last bit - do you know the name of a $12$-sided shape?</p>
<p class="editorial">Well done everyone.</p>
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<h2>Egyptian Rope</h2>
<br></br>
The ancient Egyptians were said to make right-angled triangles using a rope which was knotted to make $12$ equal sections.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="" height="226" src="knots.jpg" width="269"></mdo:image> <mdo:image alt="" height="226" src="tri.jpg" width="280"></mdo:image></div>
<br></br>
<br></br>
If you have a rope knotted like this, what other triangles can you make? (You must have a knot at each corner.)
<p style="text-align: left;">What regular shapes can you make - that is, shapes with equal sides and equal angles?</p>
<br></br>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=6832&part=">Click here for a poster of this problem</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=982&part=index">This problem</a> is one that combines knowledge of properties of shapes while using the operations of addition, subtraction, multiplication and division with small numbers. It also provides an opportunity for learners to consider the effectiveness of alternative strategies.<br></br>
<br></br>
<h3>Possible approach</h3>
<div>You could use this problem during work on either number or shape. It could be introduced by looking at the picture of the triangle made from rope and asking children what they see. If it does not come up naturally, draw their attention to the fact there are twelve sections in the rope and ask learners to investigate other possible triangles, using headless matches (or something similar such
as lolly sticks or cut-up drinking straws). It would be a good idea to work in pairs so that they are able to talk through their ideas with a partner.</div>
<br></br>
<div>They could then go on to the second part of the problem to find regular shapes that can be made using all twelve sticks.</div>
<br></br>
<div>At the end of the lesson it would be useful to discuss why no other triangles are possible with the twelve sticks. Some children may well have come up with 'rules' for the possible triangles which would be worth talking about together.<br></br>
</div>
<h3>Key questions</h3>
<div>Why do you think these two sides will not make a triangle with the other sticks?</div>
How do you know you have found them all? Can you tell me why no other ones are possible?<br></br>
<div>What numbers are factors of $12$? Can this help you to make some regular shapes?</div>
<br></br>
<h3>Possible extension</h3>
Learners could investigate the possible triangles made with different numbers of sticks as in the problem <a href="http://nrich.maths.org/public/viewer.php?obj_id=5045&part=index">Sticks and Triangles</a>.<br></br>
<br></br>
<h3>Possible support</h3>
Having twelve sticks of equal length (such as headless matches, or even pencils) to build the shapes makes this problem accessible to all children.<br></br>
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You could try using twelve sticks of equal length such as headless
matches.<br></br></mdoxml>
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<strong class="editorial">Daniel</strong> <span class="editorial">and</span> <strong class="editorial">Jaimee</strong>
<span class="editorial">(Tattingstone School) both sent in
carefully drawn solutions for this problem. Each found that three
regular shapes could be made with the rope. Each explained how they
used multiplication or division to work out which shapes could be
made.</span> <br></br>
<br></br>
Jaimee worked out "which numbers can be multiplied to make 12",
which showed how many sides the shapes would have. <br></br>
<br></br>
<div style="text-align: center;"><mdo:image width="332" height="295" alt="Regular polygons" src="regular%20shapes.gif"></mdo:image></div>
<br></br>
<br></br>
Thinking about it slightly differently, Daniel explained that the
number of knots divided by the number of sides gives how many
'sections' (between knots) there are on each side. For example:
12/4 = 3 means that a shape with four sides (square) has three
knot-sections along each side.
<p><br></br>
<span class="editorial">Christina (Marlborough Primary School,
London) found two other types of triangles, in addition to the
equilateral triangle above. She says:</span></p>
<br></br>
<div style="text-align: center;"><mdo:image width="268" height="174" src="fig2-2.gif" alt="two triangles"></mdo:image></div>
<br></br>
<br></br>
"I know that the two shortest sides in a triangle must add up to
more than the length of the third side, so the longest side of the
triangle can be at most five. The only possibilities are then<br></br>
<br></br>
<ul>
<li>5, 5, 2</li>
<li>5, 4, 3</li>
<li>4, 4, 4</li>
</ul>
and these are exactly the triangles illustrated."<br></br>
<br></br>
<span class="editorial">Well done everyone!</span> <br></br>
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Egyptian Rope
The ancient Egyptians were said to make right-angled triangles
using a rope with twelve equal sections divided by knots. What
other triangles could you make if you had a rope like this?
Using, Applying and Reasoning about Mathematics
Investigations
2D Geometry, Shape and Space
Regular polygons
2D Geometry, Shape and Space
Mixed triangles
Using, Applying and Reasoning about Mathematics
Working systematically
Using, Applying and Reasoning about Mathematics
Practical Activity
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