Making Spirals

8294 /www/nrich/html/content/id/8294/ 1 2012-07-01T00:00:00 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">There are lots of different spirals and lots of different ways of creating them. Here are two different ways. Try them out and see whether you can create some more of your own. Here is a link to another NRICH activity with instructions to create <a href="http://nrich.maths.org/5375">Archimedes' Spiral</a>. And now for another one: a Golden or Fibonacci Spiral. <mdo:image alt="" src="Spiral-squre.jpg" style="width: 500px; height: 319px;"></mdo:image> Here are the instructions: <div class="toggle">Take a piece of A$4$ squared or graph paper. The best one to use is one that is divided into 5mm squares. Put the paper on your desk so that its longest side is horizontal. Start by drawing a square with a side of one about $10$ squares up from the bottom edge of the paper and $15$ squares in from the right hand side. Draw another square with a side of one above it. Now draw a square of side two to the right of your first two squares and then a square of side $3$ above that. You can now start to draw your spiral. Each square has a quarter of a circle in it which joins one corner of the square to its opposite corner. Can you see where to draw your next square and curve? The sizes of your squares follow the Fibonacci sequence: $1$, $1$,$2$, $3$, $5$, $8$, $13$, $21$, $34$. Can you see why? You will run out of space on your paper when you get to $21$ or $34$ - it just depends on exactly where you positioned your first square on the paper.</div> What do you notice about these spirals? Are they similar? <div class="toggle">These two spirals are made by starting in the centre and building outwards.</div> Can you create any more rules for making your own spirals? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">I imagine that many of you explored making spirals but nothing arrived on our desks! Remember that you can also email us if you produce something that can be photographed. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Making Spirals</h2> There are lots of different spirals and lots of different ways of creating them. Here are two different ways. Try them out and see whether you can create some more of your own. Here is a link to another NRICH activity with instructions to create <a href="http://nrich.maths.org/5375">Archimedes' Spiral</a>. And now for another one: a Golden or Fibonacci Spiral. <mdo:image alt="" src="Spiral-squre.jpg" style="width: 500px; height: 319px;"></mdo:image> Here are the instructions: <div class="toggle">Take a piece of A$4$ squared or graph paper. The best one to use is one that is divided into 5mm squares. Put the paper on your desk so that its longest side is horizontal. Start by drawing a square with a side of one about $10$ squares up from the bottom edge of the paper and $15$ squares in from the right hand side. Draw another square with a side of one above it. Now draw a square of side two to the right of your first two squares and then a square of side $3$ above that. You can now start to draw your spiral. Each square has a quarter of a circle in it which joins one corner of the square to its opposite corner. Can you see where to draw your next square and curve? The sizes of your squares follow the Fibonacci sequence: $1$, $1$,$2$, $3$, $5$, $8$, $13$, $21$, $34$. Can you see why? You will run out of space on your paper when you get to $21$ or $34$ - it just depends on exactly where you positioned your first square on the paper.</div> What do you notice about these spirals? Are they similar? <div class="toggle">These two spirals are made by starting in the centre and building outwards.</div> Can you create any more rules for making your own spirals? </div> <h3>Why do this problem?</h3> <a href="http://nrich.maths.org/8294">This problem</a> provides opportunities for children to develop their drawing skills and to learn to follow detailed instructions. It also offer opportunities to explore a number of different ways of creating spirals. <h3>Possible approach</h3> For a whole class activity you could ask different groups to work on creating the different spirals and compare their results. Group collaboration and discussion will help them to make sense of the instructions and to follow them accurately. During the holidays investigating and spotting spirals might be a family project as you travel around or explore the internet for images and information. <h3>Key questions</h3> Where should you start? What do you need to do next? Have you counted/measured accurately? <h3>Possible extension</h3> Children could take photographs of spirals they notice in their surroundings and compare these with the ones they have created. They could investigate rules for making spirals. The analysis of spirals and their equations takes you into some pretty tricky mathematics that is likely to be beyond most children at this level but even looking at the equations can be interesting. <h3>Possible support</h3> It would be possible to create some supporting resources such as sheets with the squares started on them. Children could then engage with creating images and spotting similar spirals around them or in pictures.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Follow the instructions very carefully taking one step at a time. It is easy to get in a muddle so it may be helpful to work in pairs so that you check what you are doing each time.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Here is another one made by starting at the outside and working inwards: <mdo:image src="Spiral-fold.jpg"></mdo:image> Here are the instructions to draw it: Take an A$4$ sheet of plain paper and put it on your desk so that the longest side is horizontal. Fold it in half with a vertical fold. Draw a curve from the bottom left had corner to the fold line. Halve the right hand half horizontally and continue the curve from your top to the new fold line. Conitue in the same way repeating the halving and folding exercise so that your curve spirals in to its centre. This spiral is different from the Golden Spiral and Archimedes' Spiral. </mdoxml> 2 5 0 1 0 0 0 Making Spirals Can you make a spiral for yourself? Explore some different ways to create your own spiral pattern and explore differences between different spirals. 2D Geometry, Shape and Space Locus/loci in 2D Admin Upper primary mapping document