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2011-02-01T00:00:01
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<p>I don't know about you but when I'm walking or cycling I like to cut a corner if it's possible and safe. I do this most when I'm walking along a path and the next path is a long way ahead and going off to the left at right angles. Looking from above this might look like this :-</p>
<p> </p>
<p style="text-align: center;"><mdo:image alt="" height="150" src="Path1.gif" width="150"></mdo:image></p>
<br></br>
<p>and so I decide to "cut the corner" as it saves time, it's a shorter distance to go, and you can overtake someone who is walking around the path. So I go along the dotted route.</p>
<div style="text-align: center;"><mdo:image alt="" height="150" src="Path2.gif" width="150"></mdo:image></div>
<br></br>
<p>Hey this makes a rather good triangle, also, I guess, rather special. The two shorter sides are the same length and are at right angles to each other.</p>
<p>You may, at school, have a collection of these kinds of triangles and you might be able put them together in different kinds of designs and patterns.</p>
<div style="text-align: center;"><mdo:image alt="" src="triangles.gif"></mdo:image></div>
<br></br>
<p>I thought it would be good to have a very "OPEN" challenge this time since this triangle came from the open-air! Here it is drawn on its own.</p>
<br></br>
<p style="text-align: center;"><mdo:image alt="big Tri" height="452" src="CornerTri.jpg" width="448"></mdo:image></p>
<p>I've taken a big one of these triangles and cut it several times in a special way. The first cut was from the bottom right hand corner and cut the big triangle in half. I've then gone on to cut in a "Zig-Zag" fashion each cut halves the previous triangle, each time halving the remaining triangle.</p>
<br></br>
<div style="text-align: center;"><mdo:image alt="corner Tris" height="447" src="CornerTRis.jpg" width="447"></mdo:image></div>
<br></br>
These cuts give us lots of triangles and my challenge to you, by using many of these triangles is to come up with the MOST,<br></br>
the MOST Extraordinary,<br></br>
the MOST AMAZING,<br></br>
the MOST UNUSUAL,<br></br>
the MOST . . . . . . !? PATTERNS/DESIGNS<br></br>
<br></br>
<p>You might do it on paper or card. But you might be able to use a draw program on your computer to make these Triangles [if you do that make sure that they are half of the size of the last one].</p>
<br></br>
<p>Finally - - as always -- "I wonder what would happen if we were allowed to . . . . ?"<br></br>
</p>
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There are lots of answers to this problem, depending on what
questions you choose to ask.<br></br>
<br></br>
Have a go yourself, and if you discover anything interesting,
<a href="mailto:%20nrich@damtp.cam.ac.uk">e-mail</a> us to tell us
what you've done! Please don't worry that your solution is not
"complete" - we'd like to hear about anything you have tried.
Teachers - you might like to send in a summary of your
children's work. <br></br></mdoxml>
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<h2>Cutting Corners</h2>
<br></br>
<p>I don't know about you but when I'm walking or cycling I like to cut a corner if it's possible and safe. I do this most when I'm walking along a path and the next path is a long way ahead and going off to the left at right angles. Looking from above this might look like this :-</p>
<p> </p>
<p style="text-align: center;"><mdo:image alt="" height="150" src="Path1.gif" width="150"></mdo:image></p>
<br></br>
<p>and so I decide to "cut the corner" as it saves time, it's a shorter distance to go, and you can overtake someone who is walking around the path. So I go along the dotted route.</p>
<div style="text-align: center;"><mdo:image alt="" height="150" src="Path2.gif" width="150"></mdo:image></div>
<br></br>
<p>Hey this makes a rather good triangle, also, I guess, rather special. The two shorter sides are the same length and are at right angles to each other.</p>
<p>You may, at school, have a collection of these kinds of triangles and you might be able put them together in different kinds of designs and patterns.</p>
<div style="text-align: center;"><mdo:image alt="" src="triangles.gif"></mdo:image></div>
<br></br>
<p>I thought it would be good to have a very "OPEN" challenge this time since this triangle came from the open-air! Here it is drawn on its own.</p>
<br></br>
<p style="text-align: center;"><mdo:image alt="big Tri" height="452" src="CornerTri.jpg" width="448"></mdo:image></p>
<p>I've taken a big one of these triangles and cut it several times in a special way. The first cut was from the bottom right hand corner and cut the big triangle in half. I've then gone on to cut in a "Zig-Zag" fashion each cut halves the previous triangle, each time halving the remaining triangle.</p>
<br></br>
<div style="text-align: center;"><mdo:image alt="corner Tris" height="447" src="CornerTRis.jpg" width="447"></mdo:image></div>
<br></br>
These cuts give us lots of triangles and my challenge to you, by using many of these triangles is to come up with the MOST,<br></br>
the MOST Extraordinary,<br></br>
the MOST AMAZING,<br></br>
the MOST UNUSUAL,<br></br>
the MOST . . . . . . !? PATTERNS/DESIGNS<br></br>
<br></br>
<p>You might do it on paper or card. But you might be able to use a draw program on your computer to make these Triangles [if you do that make sure that they are half of the size of the last one].</p>
<br></br>
<p>Finally - - as always -- "I wonder what would happen if we were allowed to . . . . ?"<br></br>
</p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>This is such an open investigation - I really feel that it is good, every now and again, to include such an <a href="http://nrich.maths.org/public/viewer.php?obj_id=62&part=">activity</a> as this in the children's mathematical experience. The pupils' understanding of the properties of these isosceles triangles will be enhanced. The concept of area is often challenged in this
investigation as our eyes seem to play tricks on us and we have to verify things in a more certain way. With some pupils you may even start thinking about Pythagoras and square numbers and square roots.</div>
<br></br>
<h3>Possible approach</h3>
<div>You'll need to be enthusiastic in a way that will capture the pupils' interest and imagination. You could begin by working on the board and demonstrating how to cut the triangle in half (turning it on its side helps to see the symmetry for some children). Then offer the children a large ready-made triangle each for them to halve and halve again. If you use different coloured paper for
each child then you might encourage them to swap a piece of the same size and shape with a another child, so making their pattern more interesting.</div>
<br></br>
<h3>Key questions</h3>
<div>Do you notice anything about the pattern/shape you've made?</div>
<div>Can you make other shapes with the triangles you've chosen to use?</div>
<br></br>
<h3>Possible extension</h3>
<div>Is it possible to halve all triangles like this? Why or why not?</div>
<br></br>
<h3>Possible support</h3>
<div>It may be necessary to help those who lack confidence in manipulation to draw lines with a ruler and cut the triangles. You might prefer to draw lines ready to cut, and focus the activity instead on the language of shape and size.</div>
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I really feel that it is good to have such an open investigation
every now and again. From my experience it will need to be
introduced in some enthusiastic way that will capture the pupils'
interest and imagination. <br></br>
<br></br>
The pupils' understanding of the properties of these isosceles
triangles will no doubt be enhanced. With some pupils you may get
into the whole idea of pythagoras and square numbers and square
roots. The concept of area is often challenged in this
investigation as our eyes seem to play trick on us and we have to
verify things in a more certain way. <br></br>
<br></br>
Take for example this, where the half triangle has been placed in
the bottom right hand corner. <br></br>
<br></br>
Triangles small-drawfile3<br></br>
<mdo:image width="284" height="284" src="small_DrawFile3.gif" alt="drawfile3"></mdo:image><br></br>
<br></br>
It looks to many people that it does not show a half, and what
about putting the smaller triangle in the top corner so that it
fits snugly. <br></br>
<br></br>
There is also a good opportunity in this investigation to consider
Infinity, and particularly in respect of fractions. <br></br>
<br></br>
It's not possible to predict what will happen but I have found
these activity to be very rich in what the pupils learn, develop,
image and communicate. There's not much more that we would
want!<br></br></mdoxml>
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Cutting Corners
Can you make the most extraordinary, the most amazing, the most
unusual patterns/designs from these triangles which are made in a
special way?
Transformations and their Properties
Compound transformations
2D Geometry, Shape and Space
Right angled triangles
Using, Applying and Reasoning about Mathematics
Investigations
Using, Applying and Reasoning about Mathematics
Practical Activity