7301
/www/nrich/html/content/id/7301/
1
2011-02-01T00:00:01
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<div style="text-align: center;"><mdo:image width="371" height="245" src="Start%20shapes.jpg" alt="start"></mdo:image></div>
These pictures were made quite simply by starting with a square,
finding the half-way point on each side and joining those points
up. This creates a smaller shape (which also happens to be a
square) inside the original. The half-way points of this new
shape are then joined up to make a third shape. This way of making
new shapes is continued until it gets too small to do
properly. <br></br>
<br></br>
You can, of course, start with any straight-lined shape.<br></br>
<br></br>
Here's one where I've coloured each new halving line to help to see
what has happened more clearly.<br></br>
<br></br>
<div style="text-align: center;"> <mdo:image width="384" height="315" alt="odd" src="odd%20shape.jpg"></mdo:image></div>
<br></br>
So, it's your turn to have a go.<br></br>
<br></br>
It's probably good to start with a fairly large shape since it's
going to get smaller and smaller each time.<br></br>
<br></br>
<div style="text-align: left;">Here are some challenges for you to
pursue:</div>
<div style="text-align: left;"> </div>
<ul>
<li style="text-align: left;">Having made a design like one above,
cut out the triangles and the smallest inner shape and rearrange
the pieces to form a new shape/design.</li>
<li style="text-align: left;">Talk about and record the things you
notice as you have drawn more and more halving lines.</li>
<li style="text-align: left;">What is happening to the enclosed
area each time the sides are halved? (Try
investigating a regular shape first.)</li>
</ul>
<div style="font-style: italic;">This problem is based on an idea
suggested by Ian Short.</div>
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<p class="editorial">We had just three contributions sent in,
which is sometimes the case with these more visual challenges.
However it is good to receive anything you have to say about your
work having had a go at the challenge. Imogen-Rose from Sutton
Bonington wrote;</p>
<br></br>
You start with a square, then all you do is tilt the square into a
diamond and then back to a normal square and so on.<br></br>
<br></br>
<p class="editorial">That's interesting - I wonder what a 'normal'
square is? Does a square change shape if it is
tilted? </p>
<p class="editorial">A pupil from Pownhall Hall School
wrote;</p>
<br></br>
It got small really quickly at first and then it got smaller
slowly. The shapes got more regular as it went smaller. The
measurements got harder as the shape got smaller. <br></br>
<br></br>
<p class="editorial">Finally Chris from Northowram Primary School
wrote;</p>
<div> </div>
<div>To start with I started with a rectangle and marked halfway on
all four vertices with a dot then used a ruler to join the
dots up to make a tilted parallelogram. I did the same thing with
the parallelogram and then the shape was a rectangle. I did the
same again but with a trapezium, then an isosceles triangle, then a
hexagon, then finally an octagon.</div>
<div> </div>
<br></br>
<p><span class="editorial">Well Chris, it is good to see that you
went further in investigating this challenge. So, thank you for
these contributions and do feel encouraged to send in ideas
from any challenge even though you may not have 'completed'
it.</span></p>
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<h2>Shaping It</h2>
<br></br>
<div style="text-align: center;"><mdo:image alt="start" height="245" src="Start%20shapes.jpg" width="371"></mdo:image></div>
These pictures were made quite simply by starting with a square, finding the half-way point on each side and joining those points up. This creates a smaller shape (which also happens to be a square) inside the original. The half-way points of this new shape are then joined up to make a third shape. This way of making new shapes is continued until it gets too small to do properly. <br></br>
<br></br>
You can, of course, start with any straight-lined shape.<br></br>
<br></br>
Here's one where I've coloured each new halving line to help to see what has happened more clearly.<br></br>
<br></br>
<div style="text-align: center;"> <mdo:image alt="odd" height="315" src="odd%20shape.jpg" width="384"></mdo:image></div>
<br></br>
So, it's your turn to have a go.<br></br>
<br></br>
It's probably good to start with a fairly large shape since it's going to get smaller and smaller each time.<br></br>
<br></br>
<div style="text-align: left;">Here are some challenges for you to pursue:</div>
<div style="text-align: left;"> </div>
<ul>
<li style="text-align: left;">Having made a design like one above, cut out the triangles and the smallest inner shape and rearrange the pieces to form a new shape/design.</li>
<li style="text-align: left;">Talk about and record the things you notice as you have drawn more and more halving lines.</li>
<li style="text-align: left;">What is happening to the enclosed area each time the sides are halved? (Try investigating a regular shape first.)</li>
</ul>
<div style="font-style: italic;">This problem is based on an idea suggested by Ian Short.</div>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>This <a href="http://nrich.maths.org/7301&part=">activity</a> enables pupils with a wide range of attainment level to work on the same challenge to improve their concepts of shape and space. It also offers opportunities for further investigation and for pupils to create their own challenges.</div>
<br></br>
<h3>Possible approach</h3>
<div>This might need to vary according to your learners' experiences. For those who have good pencil, ruler and measuring skills, some of the examples here (<a href="/content/id/7301/ShapingIt.doc">Word document</a> or <a href="/content/id/7301/ShapingIt.pdf">pdf</a>) could be presented. You could ask children to describe what they see, with a partner first and then open it
out to the whole group. Discussion could follow that would allow pupils to decide what their own explorations would be. Some may want to create their own whereas those who struggle with motor control may wish to explore the given ones.</div>
<div> </div>
<div>It could be that you ask children to feed back about their discoveries orally, or you may wish them to create a poster of some kind. Encourage them to explain their observations. This activity could lend itself to being investigated over an extended period of time (a 'simmering activity') and it would be useful to dedicate a space on the wall for learners to contribute their work
during that time.</div>
<br></br>
<h3>Key questions</h3>
<div>What have you been exploring?</div>
<div>Tell me what you have found.</div>
<br></br>
<h3>Possible extension</h3>
Some pupils may be encouraged to use a spreadsheet to explore sizes of different lengths/areas. Others may be encouraged to compare results when the new lines are placed a quarter (or other fraction) along the previous lines instead of half way.<br></br>
<h3>Possible support</h3>
Some pupils may need to have help with the fine motor skills required whilst being interested in the overall effect of performing this kind of action on a shape.<br></br>
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It might help to have some large sheets of paper for this
activity.<br></br></mdoxml>
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Shaping It
These pictures were made by starting with a square, finding the
half-way point on each side and joining those points up. You could
investigate your own starting shape.
2D Geometry, Shape and Space
Regular polygons
2D Geometry, Shape and Space
Other polygons
Measures and Mensuration
Length/distance
Measures and Mensuration
Area
Using, Applying and Reasoning about Mathematics
Investigations
Admin
Lower primary mapping document
Admin
Upper primary mapping document