1840
/www/nrich/html/content/03/07/six1/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p>In how many ways can you fit all three pieces together to make shapes with line symmetry?</p>
<p><font color="#FF0000"><mdo:image alt="Printable Image of the problem" height="281" src="rotate_image.gif" width="481"></mdo:image></font></p>
<p>There are more than five solutions to this problem - how many can you find?</p>
<p>Can you find them all?</p>
<p><a href="/content/03/07/six1/Printable_Rotate_Images.gif">Additional printable copies of the shapes can be found here.</a></p>
<p>Try changing the question a bit:</p>
<p>For example, design your own set of three shapes (drawn on a square grid, as above), keep the total area 10. How many ways can they be arranged to make symmetrical shapes?<br></br>
Can you find a set of three such shapes, which have <span style="text-decoration: underline;">more</span> valid arrangements than the shapes from the original problem?<br></br>
Can you find three such shapes which can never be arranged to make a symmetrical shape?</p>
<h5>This problem is based on one found in the Dime "Line Symmetry A" pack, produced by <a href="http://www.tarquin-books.demon.co.uk">Tarquin Publications</a></h5>
<br></br>
<a href="http://nrich.maths.org/7198">Click here for a poster of this problem</a>. <br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p class="editorial">An excellent solution from Andrei of School
205 Bucharest. I like this very much because of the effective and
systematic approach he took to investigating the problem.</p>
<p class="editorial">Here is Andrei's solution:</p>
<p>I worked systematically. I took the big (yellow) piece and I
placed the rectangle in all the possible locations around it (both
vertically and horizontally). Then I looked if the figure admitted
a symmetry line (vertical, horizontal, or at a 45$^{\circ}$ angle)
if I added a small square (blue). Here is all my work and the
solutions I found, with the corresponding symmetry line:</p>
<table width="95%" cellspacing="0" cellpadding="0" border="0" align="center">
<tbody>
<tr>
<td width="34%">
<div align="center"><mdo:image width="165" height="100" src="square1.gif"></mdo:image></div>
</td>
<td width="34%">
<div align="center"><mdo:image width="198" height="100" src="square2.gif"></mdo:image></div>
</td>
<td width="34%">
<div align="center"><mdo:image width="132" height="99" src="square3.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image width="165" height="100" src="square4.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="165" height="100" src="Square5.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="133" height="133" src="square6.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image width="174" height="133" src="square7.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="133" height="166" src="square8.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="133" height="132" src="square9.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image width="133" height="166" src="square10.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="133" height="132" src="square11.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="133" height="166" src="square12.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image width="166" height="133" src="square13.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="166" height="133" src="square14.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="198" height="100" src="square15.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image width="165" height="100" src="sq16.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="198" height="100" src="sq17.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="165" height="100" src="sq18.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image width="165" height="160" src="sq19.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="165" height="131" src="sq20.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="165" height="133" src="sq21.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image width="147" height="147" src="sq22.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="157" height="155" src="sq23.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="184" height="148" src="sq24.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image width="140" height="140" src="sq25.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="168" height="100" src="sq26.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="133" height="100" src="sq27.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image width="133" height="133" src="sq28.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="133" height="133" src="sq29.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="133" height="166" src="sq30.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image width="133" height="133" src="sq31.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="132" height="165" src="sq32.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image width="165" height="133" src="sq33.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image width="166" height="133" src="sq34.gif"></mdo:image></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
</tbody>
</table>
<br></br>
<div>Sara spotted that Andrei had missed a solution. <a href="/content/03/07/six1/Reflecting%20Squarely%20Solutions.xls">Here</a>
is Sara's complete set - her last solution is the one missing in
the list above.</div>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<h3>Why do this problem?</h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=1840">This
problem</a> makes students think about the symmetries of many
different shapes. There will be opportunities for instant feedback,
and a challenge that will engage the whole class in an activity
that offers choice and room for creativity. The students have to
take responsibility for checking each other's work, since the
teacher clearly can't be expected to anticipate all the possible
combinations that might be generated.<br></br>
<h3>Possible approach</h3>
This problem could easily be used as it stands, as one of many
activities on reflection symmetry. It can also be expanded, leading
into a richer task:<br></br>
<br></br>
As students enter, have copies of the three pieces available (on an
OHP/blu-tacked to the board/on desks - card shapes or multilink
(interlocking cubes)). Ask the class to find a few arrangements
that are symmetrical. Keep a record of correct solutions as they
are suggested.<br></br>
<br></br>
Give students time to work individually/in pairs trying to find all
the others. After a while students could be invited to draw new
solutions on the board. Say that students have found only six
arrangements, suggest that the set of three shapes is called a
'6-ways-set'.<br></br>
<br></br>
"Now it is time to 'beat the problem'. You can design your own
three shapes, like the original, all made from squares on a square
grid, with a total area of 10. Do you think we can find 3 shapes
which can be put together symmetrically in <span style="text-decoration: underline;">more</span> ways than the original
problem? i.e. we're looking for a 7-ways-set, an 8-ways-set... (If
anyone finds more symmetrical combinations for the original
problem, then this task becomes even more challenging!)<br></br>
<br></br>
"Between us, can we find a complete collection: a 0-ways-set, a
1-ways-set, a 2-ways-set... ?"<br></br>
"When you have designed a set, and think you have found all the
symmetrical arrangements, draw them clearly and stick your work to
the board, for others to check." The board could be prepared with
headings: 0-ways-sets, 1-ways-sets, 2-ways-sets... Ask all students
to take responsibility for checking at least one displayed solution
and confirm that it is in the right category.<br></br>
<br></br>
Keep the work on display at the end of the lesson, so that students
(from this class or another) can add to it over the next couple of
weeks.<br></br>
<br></br>
<h3>Key questions</h3>
Where can the mirror lines be?<br></br>
Is there a systematic way of checking that you've found all the
arrangements?<br></br>
<h3><br></br>
Possible extension</h3>
<div>Students might like to look at <a href="http://nrich.maths.org/public/viewer.php?time=1240635490&obj_id=1840&part=solution">
Andrei's solution</a> and try to understand the logic behind his
approach.</div>
<br></br>
<div>Encourage students to find three shapes that have very few
possible arrangements (or none), and/or more than anything found so
far.</div>
<br></br>
<h3>Possible support</h3>
Provide mirrors, and/or scissors so students can cut out their
arrangement of shapes and fold them to check potential mirror
lines.<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><p>Try using a mirror to check your
symmetries.</p>
<p>Try keeping the biggest shape fixed in one position, place the
rectangle systematically in all possible positions, and see whether
the square can be added to make a symmetrical shape.</p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">Taken from Possible Extension in the Teachers' Notes:<br></br>
Students might like to look at Andrei's solution , and try to
understand the logic behind his approach. <br></br>
<p class="editorial">An excellent solution from Andrei of School
205 Bucharest. I like this very much because of the effective and
systematic approach he took to investigating the problem.</p>
<p class="editorial">Here is Andrei's solution:</p>
<p>I worked systematically. I took the big (yellow) piece and I
placed the rectangle in all the possible locations around it. Then
I looked if the figure admitted a symmetry line (vertical,
horizontal, or at a 45$^{\circ}$ angle) adding a small square
(blue). Here is all my work and the solutions I found, with the
corresponding symmetry line:</p>
<table width="95%" cellspacing="0" cellpadding="0" border="0" align="center">
<tbody>
<tr>
<td width="34%">
<div align="center"><mdo:image height="100" width="165" src="square1.gif"></mdo:image></div>
</td>
<td width="34%">
<div align="center"><mdo:image height="100" width="198" src="square2.gif"></mdo:image></div>
</td>
<td width="34%">
<div align="center"><mdo:image height="99" width="132" src="square3.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image height="100" width="165" src="square4.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="100" width="165" src="Square5.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="133" width="133" src="square6.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image height="133" width="174" src="square7.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="166" width="133" src="square8.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="132" width="133" src="square9.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image height="166" width="133" src="square10.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="132" width="133" src="square11.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="166" width="133" src="square12.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image height="133" width="166" src="square13.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="133" width="166" src="square14.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="100" width="198" src="square15.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image height="100" width="165" src="sq16.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="100" width="198" src="sq17.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="100" width="165" src="sq18.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image height="160" width="165" src="sq19.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="131" width="165" src="sq20.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="133" width="165" src="sq21.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image height="147" width="147" src="sq22.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="155" width="157" src="sq23.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="148" width="184" src="sq24.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image height="140" width="140" src="sq25.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="100" width="168" src="sq26.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="100" width="133" src="sq27.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image height="133" width="133" src="sq28.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="133" width="133" src="sq29.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="166" width="133" src="sq30.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image height="133" width="133" src="sq31.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="165" width="132" src="sq32.gif"></mdo:image></div>
</td>
<td>
<div align="center"><mdo:image height="133" width="165" src="sq33.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
<tr>
<td>
<div align="center"><mdo:image height="133" width="166" src="sq34.gif"></mdo:image></div>
</td>
<td>
<div align="center"></div>
</td>
<td>
<div align="center"></div>
</td>
</tr>
</tbody>
</table>
<br></br>
<div>Sara spotted that Andrei had missed a solution. <a href="/content/03/07/six1/Reflecting%20Squarely%20Solutions.xls">Here</a>
is Sara's complete set - her last solution is the one missing in
the list above.</div>
<br></br></mdoxml>
2
3
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Reflecting Squarely
In how many ways can you fit all three pieces together to make
shapes with line symmetry?
Transformations and their Properties
Reflections
Using, Applying and Reasoning about Mathematics
Visualising
Transformations and their Properties
Symmetry
Secondary Mapping Document
Transformations