Shady Symmetry

1868 /www/nrich/html/content/03/10/six5/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Charlie created a symmetrical pattern by shading in four squares on a 3 by 3 square grid: <div style="text-align: center;"><mdo:image height="175" width="178" src="charlie%27s%20pattern.png" alt="pattern with squares"></mdo:image></div> Alison created a symmetrical pattern by shading in two triangles on a 3 by 3 isometric grid: <div style="text-align: center;"><mdo:image height="155" width="178" src="alison%27s%20pattern.png" alt="pattern with triangles"></mdo:image></div> <div>Choose whether you would like to work on <a href="/content/03/10/six5/Squares.pdf">square grids</a> or <a href="/content/03/10/six5/Triangles.pdf">isometric grids</a>.</div> <div> </div> <div style="font-weight: bold;">How many different symmetrical patterns can you make?</div> <div> </div> <div>Here are some questions you might like to consider:</div> <ul> <li>How many different patterns can you make if you are only allowed to shade in one... two... three... four cells?</li> <li>How does the number of patterns with 6 cells shaded relate to the number with 3 cells shaded?</li> <li>Can you make patterns with exactly one... two... three... four lines of symmetry?</li> <li>Can you make patterns with rotational symmetry AND lines of symmetry?</li> <li>Can you make patterns with rotational symmetry but NO lines of symmetry?</li> <li>Can you make patterns using more than one colour?</li> </ul> <div> </div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">There are many possible questions to think about in this problem. A systematic approach is important to avoid counting the same patterns twice, or missing any out! Melissa and Damian looked at the isometric grids, counting patterns with 1, 2, 3, and 4 shaded little triangles. It's important to decide what a different pattern means, for example if you shade the top little triangle at the vertex, and the left-most little triangle at the left vertex, are these different patterns? We'll consider these as the same since they differ only by rotating the picture. Shading 1 little triangle: No matter which you shade there is exactly one line of reflective symmetry. There are only 3 patterns, because any other can be rotated to look like one of these: <mdo:image src="iso-1shade-1.png"></mdo:image><mdo:image src="iso-1shade-2.png"></mdo:image><mdo:image src="iso-1shade-3.png"></mdo:image> 3 ways with 1 triangle shaded. Shading 2 little triangles: To get a line of symmetry you need to choose carefully. To avoid counting patterns which look the same after a rotation we'll just look at the vertical line of symmetry. You can choose two of the triangles on the vertical line in the middle (three ways to do this) or one of the left-hand triangles with it's partner on the right (also three ways to do this). <mdo:image src="iso-2shade-1.png"></mdo:image><mdo:image src="iso-2shade-2.png"></mdo:image><mdo:image src="iso-2shade-3.png"></mdo:image> <mdo:image src="iso-2shade-4.png"></mdo:image><mdo:image src="iso-2shade-5.png"></mdo:image><mdo:image src="iso-2shade-6.png"></mdo:image> 6 ways with 2 triangles shaded. Shading 3 little triangles: Again just look at the vertical line of symmetry. Either take all 3 central triangles, or take just 1 and a symmetrical pair of other ones. There only 1 way of shading all 3 central triangles: <mdo:image src="iso-3shade-1.png"></mdo:image> And 3 ways of taking one of those central ones, with three possible left-right pairs to add to it to make a pattern with 3 shaded little triangles. <mdo:image src="iso-3shade-2.png"></mdo:image> <mdo:image src="iso-3shade-3.png"></mdo:image> <mdo:image src="iso-3shade-4.png"></mdo:image> <mdo:image src="iso-3shade-5.png"></mdo:image> <mdo:image src="iso-3shade-6.png"></mdo:image> <mdo:image src="iso-3shade-7.png"></mdo:image> <mdo:image src="iso-3shade-8.png"></mdo:image> <mdo:image src="iso-3shade-9.png"></mdo:image> <mdo:image src="iso-3shade-10.png"></mdo:image> This makes 1 + 3x3 = 10 ways with 3 triangles shaded. You might like to try this method on shading 4 and 5 little triangles. Shading 6 little triangles: There are 9 triangles in total so when 6 are shaded that leaves 3 unshaded. Can you see that with 6 shaded symmetrically, the remaining 3 are also arranged symmetrically? The same is true when shading 3 triangles, the remaining 6 are also arranged symmetrically. That means that each pattern we found with 3 shaded triangles gives rise to exactly 1 pattern with 6 shaded triangles. Think of swapping the purple and white colours in the pictures above! This means there are also 10 patterns with a line of symmetry when shading 6 of the triangles. More lines of symmetry: Notice that the patterns with 1 and 2 shaded triangles always have only 1 line of symmetry. Some of the patterns with 3 shaded triangles have 3 lines of symmetry and rotational symmetry. Can you find any patterns with exactly 2 lines of symmetry? If so, how? If not, why not? Christian investigated the square grid and found many patterns like the ones below. Notice that we can use a similar method for finding patterns exploiting symmetry. Things seem a bit more complicated with the square though... Shading 1 little square: The square has two distinct types of reflectional symmetry; in diagonal and in side-bisecting lines. No amount of rotating the square makes these symmetries look the same. Like with the isometric grid, there are 3 ways of shading 1 square: <mdo:image src="sqr-1shade-1.png"></mdo:image><mdo:image src="sqr-1shade-2.png"></mdo:image><mdo:image src="sqr-1shade-3.png"></mdo:image> Shading 2 little squares: We'll need to shade two squares along a line of symmetry or a pair either side of a line of symmetry. Remember the diagonal and the side-bisecting lines! Be careful not to double count some patterns because opposite corners can be made to look the same by 180 degree rotations! <mdo:image src="sqr-2shade-1.png"></mdo:image><mdo:image src="sqr-2shade-2.png"></mdo:image><mdo:image src="sqr-2shade-3.png"></mdo:image> <mdo:image src="sqr-2shade-4.png"></mdo:image><mdo:image src="sqr-2shade-5.png"></mdo:image><mdo:image src="sqr-2shade-6.png"></mdo:image> There are 6 different patterns with 2 shaded squares. Shading 3 little squares: Use the same method as before - work systematically along and either side of lines of symmetry. <mdo:image src="sqr-3shade-1.png"></mdo:image> <mdo:image src="sqr-3shade-2.png"></mdo:image><mdo:image src="sqr-3shade-3.png"></mdo:image><mdo:image src="sqr-3shade-4.png"></mdo:image> <mdo:image src="sqr-3shade-9.png"></mdo:image> <mdo:image src="sqr-3shade-5.png"></mdo:image> <mdo:image src="sqr-3shade-6.png"></mdo:image><mdo:image src="sqr-3shade-7.png"></mdo:image><mdo:image src="sqr-3shade-8.png"></mdo:image><mdo:image src="sqr-3shade-10.png"></mdo:image> There are 10 different patterns with 3 shaded squares. Do you notice anything similar to the isometric case? What about 4, 5, 6, 7 or 8 shaded squares? Do you need to do any more work? There are many more questions that you could ask about these patterns. Have fun investigating further. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <h3>Why do this problem?</h3> <a href="http://nrich.maths.org/public/viewer.php?obj_id=1868">This problem</a> is a good activity for the visualisation of symmetry, and for encouraging students to work systematically. There are many different avenues for exploration and extension, and learners' work can be used to brighten up the classroom walls. <h3>Possible approach</h3> <div>As the class come into the room, display the two patterns from the problem at the front for everyone to see. The patterns are available on <a class="powerpoint" href="/content/03/10/six5/Shady.ppt">this PowerPoint Slide.</a> <mdo:image src="alison%27s%20pattern.png"></mdo:image> <mdo:image src="charlie%27s%20pattern.png"></mdo:image> <div>Ask students to discuss the two images in pairs, focusing on what they notice about the two pictures, what is the same and what is different. Then bring the class together to share their ideas.</div> <div> </div> <div>Explain that the challenge will be to explore symmetrical patterns drawn on grids of triangles or squares, and give students a little longer with their partners to come up with some lines of enquiry to explore. Collect their ideas together on the board at the front (some suggestions are made in the problem if more ideas are needed).</div> <div> </div> <div>Now allow pairs or small groups to choose one of the ideas to work on, and hand out some of these <a href="/content/03/10/six5/Squares.pdf">square</a> and <a href="/content/03/10/six5/Triangles.pdf">triangular</a> grids. Make the class aware that at the end of the time spent on this (it could be over several lessons) they will be expected to display their work in a way that will convince others that they have considered every possible symmetrical pattern for their chosen question. While students are working on the task, there may be opportunities to share what people are thinking about through mini-plenaries, particularly to draw attention to those who are working in a systematic way.</div> <h3>Key questions</h3> <ul> <li>What different types of symmetry do the initial grids exhibit?</li> <li>If you colour a triangle or square here, what else must be coloured in to keep it symmetrical?</li> <li>What are the possible symmetries of a finished pattern? </li> <li>How can you be sure you have found all the symmetric patterns?</li> </ul> <h3>Possible extension</h3> <div>The problem can be extended to be done on these 4 by 4 <a href="/content/03/10/six5/Squares2.pdf">square</a> and <a href="/content/03/10/six5/Triangles2.pdf">triangular</a> grids, and of course there are opportunities to extend into three dimensions...</div> <h3>Possible support</h3> <div>Encourage students to begin by looking at all the patterns that can be made by first colouring in just one cell, then two, then three and so on.</div> <div> </div> <div>Have tracing paper available if required.</div> <div> </div> <div>Encourage students to number the cells of their grid to help them to list shadings in a systematic way.</div> <div>Draw a big table on the board with</div> <div style="margin-left: 40px;">column headings: 0, 1, 2, 3, more lines of symmetry,</div> <div style="margin-left: 40px;">row headings: order 1, 2, 3, more for rotational symmetry.</div> <div>Fill in "not symmetrical" in the top left cell, then ask students to stick their solutions up in the correct cell on the table as they complete them.</div> </div></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Begin by looking at all the patterns that can be made by first colouring in just one cell, then two, then three and so on. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> We received a couple of very good solutions from students who considered the number of different patterns that could be created from shading four triangles on a 4 by 4 isometric grid. Mary sent us her work: Think about the four little triangles in a vertical line in the middle of the triangle. We must have an even number of them in our pattern. We can count the different possibilities separately. All four of the vertical ones: $1$ possibility (we don't need to colour in any more triangles). Two of the vertical ones: there are $6$ possible pairs that don't include the middle ones, each of which can go with any of the $6$ vertical pairs, so there are $6 \times 6=36$ possibilities. None of the vertical ones: there are $6$ pairs that don't include the middle ones. Once we've picked one of them, there are $5$ others left to choose from. But this counts every possibility twice (A with B and B with A), so there are $\frac{6 \times 5}{2}=15$ possibilities. So there are $1+36+15=52$ total possible patterns. Below are the twelve symmetrical pairs. <mdo:image alt="First pair" height="267" src="filled.GIF" width="304"></mdo:image><mdo:image alt="Second pair" height="267" src="filled1.GIF" width="304"></mdo:image><mdo:image alt="Third pair" height="267" src="filled2.GIF" width="304"></mdo:image><mdo:image alt="Fourth pair" height="267" src="filled3.GIF" width="304"></mdo:image><mdo:image alt="Fifth pair" height="267" src="filled4.GIF" width="304"></mdo:image><mdo:image alt="Sixth pair" height="267" src="filled5.GIF" width="304"></mdo:image> <mdo:image alt="Seventh pair" height="267" src="filled6.GIF" width="304"></mdo:image><mdo:image alt="Eighth pair" height="267" src="filled7.GIF" width="304"></mdo:image> <mdo:image alt="Ninth pair" height="267" src="filled8.GIF" width="304"></mdo:image><mdo:image alt="Tenth pair" height="267" src="filled9.GIF" width="304"></mdo:image><mdo:image alt="Eleventh pair" height="267" src="filled10.GIF" width="304"></mdo:image><mdo:image alt="Twelfth pair" height="267" src="filled11.gif" width="304"></mdo:image> Neil sent in this solution. It's not quite right, but with just a small correction the reasoning can be used to arrive at the solution: There are $12$ different "symmetrical pairs" in the big triangle (shown above), and $2$ different pairs must be used on each pattern. So pair number $1$ can go with any one of pairs $2$ through to $12$. This gives us $11$ patterns. Continuing in the same way: $1$ goes with $2$ through to $12$ making $11$ patterns. $2$ goes with $3$ through to $12$ making $10$ patterns. $3$ goes with $4$ through to $12$ making $9$ patterns. ...up until... $11$ goes with $12$ through to $12$ making $1$ pattern. We need to add all integers $1$ to $11$, for which we can use the formula: <h3 style="text-align: center;">$\frac{n^2+n}{2}$</h3> $\frac{11^2+11}{2}=\frac{121+11}{2} = \frac{132}{2} = 66$ So the answer is $66$ patterns. In fact, this isn't quite right, because not all of the pairs can go together. For example, we can't combine both pairs shown in the fourth row above. So there aren't quite as many as $66$. Taking this into account we can use Neil's approach and work out that there are $11+10+9+8+7+6+1=52$ possibilities. Neil sent in this solution. It's not quite right, but it makes a good start. There are $12$ different "symmetrical pairs" in the big triangle (shown below), and $2$ different pairs must be used on each pattern. So pair number $1$ can go with any one of pairs $2$ through to $12$. This gives us $11$ patterns. Continuing in the same way: $1$ goes with $2$ through to $12$ making $11$ patterns. $2$ goes with $3$ through to $12$ making $10$ patterns. $3$ goes with $4$ through to $12$ making $9$ patterns. ...up until... $11$ goes with $12$ through to $12$ making $1$ pattern. We need to add all integers $1$ to $11$, for which we can use the formula: <h2 style="text-align: center;">$\frac{n^2+n}{2}$</h2> $\frac{11^2+11}{2}=\frac{121+11}{2} = \frac{132}{2} = 66$ So the answer is $66$ patterns. In fact, this isn't quite right, because not all of the pairs can go together. For example, we can't combine both pairs shown in the fourth row below. So there aren't quite as many as $66$. Taking this into account we can use Neil's approach and work out that there are $11+10+9+8+7+6+1=52$ possibilities. Mary sent us her work: Think about the four little triangles in a vertical line in the middle of the triangle. We must have an even number of them in our pattern. We can count the different possibilities separately. All four of the vertical ones: $1$ possibility (we don't need to colour in any more triangles). Two of the vertical ones: there are $6$ possible pairs that don't include the middle ones, each of which can go with any of the $6$ vertical pairs, so there are $6 \times 6=36$ possibilities. None of the vertical ones: there are $6$ pairs that don't include the middle ones. Once we've picked one of them, there are $5$ others left to choose from. But this counts every possibility twice (A with B and B with A), so there are $\frac{6 \times 5}{2}=15$ possibilities. So there are $1+36+15=52$ total possible patterns. Below are the twelve symmetrical pairs. <mdo:image alt="First pair" height="267" src="filled.GIF" width="304"></mdo:image><mdo:image alt="Second pair" height="267" src="filled1.GIF" width="304"></mdo:image><mdo:image alt="Third pair" height="267" src="filled2.GIF" width="304"></mdo:image><mdo:image alt="Fourth pair" height="267" src="filled3.GIF" width="304"></mdo:image><mdo:image alt="Fifth pair" height="267" src="filled4.GIF" width="304"></mdo:image><mdo:image alt="Sixth pair" height="267" src="filled5.GIF" width="304"></mdo:image> <mdo:image alt="Seventh pair" height="267" src="filled6.GIF" width="304"></mdo:image><mdo:image alt="Eighth pair" height="267" src="filled7.GIF" width="304"></mdo:image> <mdo:image alt="Ninth pair" height="267" src="filled8.GIF" width="304"></mdo:image><mdo:image alt="Tenth pair" height="267" src="filled9.GIF" width="304"></mdo:image><mdo:image alt="Eleventh pair" height="267" src="filled10.GIF" width="304"></mdo:image><mdo:image alt="Twelfth pair" height="267" src="filled11.gif" width="304"></mdo:image> </mdoxml> 2 3 0 0 1 0 0 Shady Symmetry How many different symmetrical shapes can you make by shading triangles or squares? 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