1886 /www/nrich/html/content/03/10/penta2/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p align="left">Systematically explore the range of symmetric designs that can be created by shading whole squares of the grid below.<br></br> <br></br> Use could use <a href="/content/03/10/penta2/SymmChallBlankGrids.doc">this sheet of blank grids</a> to record your results, or simply squared paper.<br></br> <br></br> <br></br> <mdo:image width="200" height="200" alt="Square grid formed from 9 smaller squares" src="sqgrid1.gif"></mdo:image><br></br> How many can you find?</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Christina from Marborough Primary, London has given this one some thought and made the sensible suggestion of trying to find all the symmetrical patterns with one coloured square, then with two squares, then three, then four. That's just what Tom has done below.<br></br> <br></br> Tom was very careful to make sure he found them all. First, he looked for patterns with no coloured squares. Of course, there's only one of those:<br></br> <br></br> <div style="text-align: center;"><mdo:image height="150" width="150" alt="No shaded squares" src="symmetry0.gif"></mdo:image></div> <div style="text-align: left;">Then he looked for patterns with one coloured square. He decided to count two patterns as the same if they were just rotations of each other, as otherwise there would be too many. Here are the patterns he found:</div> <div style="text-align: center;"><mdo:image height="152" width="150" alt="Top left square shaded" src="symmetry1a.gif"></mdo:image><mdo:image height="153" width="123" alt="Top centre square shaded" src="symmetry1b.gif"></mdo:image><mdo:image height="153" width="153" alt="Centre square shaded" src="symmetry1c.gif"></mdo:image></div> <div>Next he looked for patterns with two coloured squares. He had to be a bit more careful here, to make sure that he didn't miss any. First of all, he coloured in the top left square. Then he wondered whether he could find any patterns with this square coloured where the line of symmetry was vertical. He found this one:</div> <br></br> <div style="text-align: center;"><mdo:image height="152" width="124" alt="Top left and top right squares shaded" src="symmetry2a.gif"></mdo:image></div> <div>but that was the only one. Then he looked for patterns with this square coloured where the line of symmetry was horizontal, but he came up with the same one again (rotated, of course). Then he looked for ones where the line of symmetry was diagonal. Here are the two patterns he found:</div> <br></br> <div style="text-align: center;"><mdo:image height="153" width="153" alt="Top left and centre squares shaded" src="symmetry2b.gif"></mdo:image><mdo:image height="151" width="153" alt="Top left and bottom right squares shaded" src="symmetry2c.gif"></mdo:image></div> <div>These were the only symmetrical patterns with a corner square shaded. Next he shaded the top centre square and looked for patterns with a vertical line of symmetry. Here's what he found:</div> <br></br> <div style="text-align: center;"><mdo:image height="153" width="125" alt="Centre top and centre squares shaded" src="symmetry2d.gif"></mdo:image><mdo:image height="156" width="156" alt="Centre top and centre bottom squares shaded" src="symmetry2e.gif"></mdo:image></div> <div style="text-align: left;">He noticed that he wouldn't get any new patterns by looking for ones with horizontal lines of symmetry, so he looked for patterns with a diagonal line of symmetry. This is the only one he found:</div> <div style="text-align: center;"><mdo:image height="153" width="153" alt="Middle left and top centre squares shaded" src="symmetry2f.gif"></mdo:image></div> <div style="text-align: left;">He used a similar system to find all the symmetrical patterns with three or four shaded squares. Here's what he found:</div> <div style="text-align: center;"><mdo:image height="155" width="122" src="symmetry3a.gif" alt="Top row shaded"></mdo:image><mdo:image height="153" width="122" src="symmetry3b.gif" alt="Top left, top right and centre squares shaded"></mdo:image><mdo:image height="153" width="121" src="symmetry3c.gif" alt="Top left, top right and bottom centre squares shaded"></mdo:image><mdo:image height="153" width="153" src="symmetry3d.gif" alt="Top left, top right and bottom right squares shaded"></mdo:image><mdo:image height="151" width="152" src="symmetry3e.gif" alt="Top left, centre and bottom right squares shaded"></mdo:image><mdo:image height="151" width="151" src="symmetry3f.gif" alt="Top left, top centre and middle left squares shaded"></mdo:image><mdo:image height="153" width="153" src="symmetry3g.gif" alt="Top left, middle right and bottom centre squares shaded"></mdo:image><mdo:image height="120" width="152" src="symmetry3h.gif" alt="Top centre, middle right and bottom centre squares shaded"></mdo:image><mdo:image height="154" width="155" src="symmetry3i.gif" alt="Middle column shaded"></mdo:image><mdo:image height="153" width="153" src="symmetry3j.gif" alt="Top centre, centre and middle right squares shaded"></mdo:image></div> <div style="text-align: center;"><mdo:image height="155" width="121" alt="Top row and bottom centre square shaded" src="symmetry4a.gif"></mdo:image><mdo:image height="152" width="120" alt="Top row and bottom centre square shaded" src="symmetry4b.gif"></mdo:image><mdo:image height="152" width="121" alt="Top left, top right, centre and bottom centre squares shaded" src="symmetry4c.gif"></mdo:image><mdo:image height="154" width="122" alt="Top left, top right, middle left and middle right squares shaded" src="symmetry4d.gif"></mdo:image> <mdo:image height="153" width="153" alt="Top left, top right, bottom left and bottom right squares shaded" src="symmetry4e.gif"></mdo:image> <mdo:image height="151" width="152" alt="Top left, top centre, middle left and middle centre squares shaded" src="symmetry4f.gif"></mdo:image> <mdo:image height="154" width="154" alt="Top left, top centre, middle left and bottom right squares shaded" src="symmetry4g.gif"></mdo:image> <mdo:image height="154" width="153" alt="Top left, top right, centre and bottom left squares shaded" src="symmetry4h.gif"></mdo:image><mdo:image height="153" width="153" alt="Top left, centre, middle right and bottom centre squares shaded" src="symmetry4i.gif"></mdo:image> <mdo:image height="153" width="153" alt="Top left, top centre, middle right and bottom right squares shaded" src="symmetry4j.gif"></mdo:image> <mdo:image height="148" width="150" alt="Top centre, middle left, middle right and bottom centre squares shaded" src="symmetry4k.gif"></mdo:image></div> <div style="text-align: left;">Finally, Tom noticed that really these told him all of the symmetrical patterns, because he could imagine a coloured grid where the white squares were the ones that had been shaded, and these would give the patterns with five, six, seven, eight or nine shaded squares.</div> <div style="text-align: left;">So Tom found 62 symmetrical patterns in all.</div> <div style="text-align: left;"> </div> <div style="text-align: left;">Children from Kellett School in Hong Kong, noticed that Tom had missed out one of the patterns.  Here is the image they sent:</div> <div style="text-align: left;"> </div> <div style="text-align: left;"> <mdo:image height="200" width="200" alt="" src="FinalSol.jpg"></mdo:image></div> <div style="text-align: left;"> </div> <div style="text-align: left;">So that makes a total of 64 possible solutions.  Well done!</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Symmetry Challenge</h2> <br></br> <p align="left">Systematically explore the range of symmetric designs that can be created by shading whole squares of the grid below.<br></br> <br></br> Use could use <a href="/content/03/10/penta2/SymmChallBlankGrids.doc">this sheet of blank grids</a> to record your results, or simply squared paper.<br></br> <br></br> <br></br> <mdo:image alt="Square grid formed from 9 smaller squares" height="200" src="sqgrid1.gif" width="200"></mdo:image><br></br> How many can you find?</p> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=1886&amp;part=index">The problem</a> requires learners to recognise and visualise the transformation of a 2D shape, and invites them to work systematically in a spatial environment. It is a problem that is accessible to most pupils even if they need support in organising and presenting their ideas and ensuring the completeness of their solution.</div> <br></br> <h3>Possible approach</h3> <div>You could start by displaying <a href="/content/03/10/penta2/SymmChallStarter.doc">these two shaded grids</a> on the board to simulate a discussion about reflection symmetry. It might also help to have some blank $3$ by $3$ grids on the board for learners to shade as they talk.</div> <br></br> <div>Invite each pupil to produce one symmetrical shading on a copy of <a href="/content/03/10/penta2/SymmChallBlankGrids.doc">this sheet of blank grids</a> . Ask the pupils to put themselves into groups according to the number of squares they have shaded on their grid and answer the following questions:</div> <ul> <li>Are the patterns you have all symmetrical?</li> <li>Which are the same and which are different?</li> <li>Are there any more with the same number of squares shaded?</li> </ul> <div>Bring the group together to talk about some of the issues, which may include talking about what counts as the same and how they went about finding all the possibilities. Introduce the task and encourage each group to work on it as a whole.</div> <br></br> <div>The plenary should focus on persuasion that the approach adopted by particular groups will yield all solutions. It is important not to get too &#39;bogged down&#39; with listing all the possibilities. Pupils could share out all the possibilities between the group and produce one or more image each which could then be displayed in an organised fashion to emphasise a system that has been adopted.</div> <div> </div> <div><em>You can read about <a href="https://www.ncetm.org.uk/resources/34152">one teacher&#39;s experience</a> of using this task in the classroom.</em></div> <br></br> <h3>Key questions</h3> <div>Try shading $1$ square only. What symmetries are possible?</div> <div>If you try shading $2, 3, 4 \ldots$squares, what symmetries are possible now?</div> <div>How will you record your findings?</div> <div>The $3$ by $3$ grid has four lines of symmetry and rotational symmetry of order $4$. How might this help?</div> <br></br> <h3>Possible extension</h3> <div>The problem can be extended to discuss larger square lattices, e.g. $4$ by $4$ and whether there are any differences between even and odd lengths of side. The activity <a href="http://nrich.maths.org/public/viewer.php?obj_id=1868&amp;part=index">Shady Symmetry</a> is also an extension possibility.</div> <h3>Possible support</h3> <div>Some children might find it helpful to start with a $2$ by $2$ grid so that they feel confident that they understand the requirements of the problem.</div> <p><br></br> <br></br> <br></br>  </p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>The $3$ by $3$ grid has $4$ lines of symmetry.<br></br> How might this help?<br></br> Flips or Turns / Reflections-- are they the same or different?<br></br> Try shading $1$ square only. What symmetries are possible?<br></br> Try shading $2, 3, 4 \ldots$ squares only. What symmetries are now possible?<br></br> <br></br> How helpful would this table be?</p> <table cellspacing="0" cellpadding="0" bordercolor="#000000" border="1" width="95%"> <tbody> <tr bordercolor="#000000"> <td width="13%"></td> <td bgcolor="#CCCCCC" colspan="6"> <div align="center">Lines of symmetry</div> </td> </tr> <tr bordercolor="#000000"> <td width="14%"></td> <td width="14%"> <div align="center"></div> </td> <td width="14%"> <div align="center">0</div> </td> <td width="14%"> <div align="center">1</div> </td> <td width="14%"> <div align="center">2</div> </td> <td width="14%"> <div align="center">3</div> </td> <td width="14%"> <div align="center">4</div> </td> </tr> <tr bordercolor="#000000"> <td width="14%" rowspan="4"> <div align="center"> <p>Shaded<br></br> Squares</p> </div> </td> <td height="21" width="14%"> <div align="center">1</div> </td> <td width="14%"> <div align="center"></div> </td> <td width="14%"> <div align="center"></div> </td> <td width="14%"> <div align="center"></div> </td> <td width="14%"> <div align="center"></div> </td> <td width="14%"> <div align="center"></div> </td> </tr> <tr> <td width="14%"> <div align="center">2</div> </td> <td width="14%"> <div align="center"></div> </td> <td width="14%"> <div align="center"></div> </td> <td width="14%"> <div align="center"></div> </td> <td width="14%"> <div align="center"></div> </td> <td width="14%"> <div align="center"></div> </td> </tr> <tr> <td width="14%"> <div align="center">3</div> </td> <td width="14%"> <div align="center"></div> </td> <td width="14%"> <div align="center"></div> </td> <td width="14%"> <div align="center"></div> </td> <td width="14%"> <div align="center"></div> </td> <td width="14%"> <div align="center"></div> </td> </tr> <tr> <td width="14%"> <div align="center">4</div> </td> <td width="14%"> <div align="center"></div> </td> <td width="14%"> <div align="center"></div> </td> <td width="14%"> <div align="center"></div> </td> <td width="14%"> <div align="center"></div> </td> <td width="14%"> <div align="center"></div> </td> </tr> </tbody> </table> <p>What omissions do you notice?</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br>One with no squares shaded Three lattices with one square shaded Seven with two Ten with three Seven with four Seven with five (complements four) Ten with six (complements three) Seven with seven (complements two) Three with eight (complements one) One with all nine shaded Total 42<br></br></mdoxml> 2 5 0 1 0 0 0 Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results. Transformations and their Properties Symmetry Using, Applying and Reasoning about Mathematics Working systematically Admin Upper primary mapping document