Great Squares

72 /www/nrich/html/content/99/11/bbprob1/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> I was doodling the other day and drew a little square like this:- <mdo:image alt="" src="nov-dra1.jpg"></mdo:image> I supposed that the side was one (something) long. Well I wondered what would happen if I drew the four lines a bit longer, in fact twice as long so that the extra bits stuck out. <mdo:image alt="" src="nov-dra2.jpg"></mdo:image> This was quite a nice little design, I thought, and then I noticed that it looked at though the ends of these lines looked as though they could make a square. So I drew one! I've used a different colour to show this new square. <mdo:image alt="" src="nov-dra3.jpg"></mdo:image> Now, mathematical patterns usually go on repeating themselves so I used that idea to pretend that this new green square was my first one and so I drew the extra bits again, so that the lines were twice as long as the square. <mdo:image alt="" src="nov-dra4.jpg"></mdo:image> So I went on! A new square appeared, now red! <mdo:image alt="" src="nov-dra5.jpg"></mdo:image> Extend that one. . . <mdo:image alt="" src="nov-dra6.jpg"></mdo:image> I really liked what was happening here! <mdo:image alt="" src="nov-dra7.jpg"></mdo:image> and so on. . . . <mdo:image alt="" src="nov-dra8.jpg"></mdo:image> <mdo:image alt="" src="nov-dra9.jpg"></mdo:image> I had to stop there because it had up to the size of the paper. I suggest that you print these pages out so far and have a good look at the way that the pattern has grown and see what things you notice in this last picture. <comment> #set var="roll-text" value="Send us your findings if you get this far" --&gt;&lt;/p&gt; &lt;p&gt;I found myself looking at a square and the extra lines and just one more square :-&lt;/p&gt; &lt;img alt="" src="/content/99/11/bbprob1/nov-dra3.jpg" /&gt;&lt;br /&gt; &lt;br clear="all" /&gt; &lt;br /&gt; &lt;p&gt;But I wanted it bigger and I thought about putting in some extra lines that again were just extensions of lines already there!&lt;/p&gt; &lt;img alt="" src="/content/99/11/bbprob1/nov-sqrpa4.jpg" /&gt;&lt;br /&gt; &lt;br clear="all" /&gt; &lt;br /&gt; &lt;p&gt;Perhaps it would be good to print this out, if you have not done so already and explore the shapes, areas, lengths, angles etc.&lt;/p&gt; &lt;p&gt;It would be really good to cut along the lines and see what happens. &lt;!-- #set var="roll-text" value= "You can scan, photocopy, draw - then cut, rearrange and paste. Messy!" --&gt;&lt;!-- #set var="roll-text" value="" </comment> When I looked at this shape I could imagine that I was looking through a square window at a pattern of squares but could only see the one square in the middle. So I printed out these:- <mdo:image alt="" src="nov-squares.jpg"></mdo:image> Then I printed out the bigger square . . . . <mdo:image alt="" src="nov-sqrtsquare.jpg"></mdo:image> This one I traced onto a "see through" sheet and placed it in different places over the grid of smaller squares. This was great fun and led to some interesting conversations. Have a GO! I think that this is one of the most exciting shape investigations that I've put on the NRICH site, so let's have a lot of workings sent to Cambridge U.K. from all over the world so that we can show how people from different countries are thinking and working in their maths. Pester your teachers to collect some results and send them off. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> There are lots of answers to this problem, depending on what questions you choose to ask. Have a go yourself, and if you discover anything interesting, <a href="mailto:%20nrich@damtp.cam.ac.uk"></a> <a href="mailto:nrich@damtp.cam.ac.uk">e-mail</a> us to tell us what you've done! Please don't worry that your solution is not "complete" - we'd like to hear about anything you have tried. Teachers - you might like to send in a summary of your children's work. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Great Squares</h2> I was doodling the other day and drew a little square like this:- <mdo:image alt="" src="nov-dra1.jpg"></mdo:image> I supposed that the side was one (something) long. Well I wondered what would happen if I drew the four lines a bit longer, in fact twice as long so that the extra bits stuck out. <mdo:image alt="" src="nov-dra2.jpg"></mdo:image> This was quite a nice little design, I thought, and then I noticed that it looked at though the ends of these lines looked as though they could make a square. So I drew one! I've used a different colour to show this new square. <mdo:image alt="" src="nov-dra3.jpg"></mdo:image> Now, mathematical patterns usually go on repeating themselves so I used that idea to pretend that this new green square was my first one and so I drew the extra bits again, so that the lines were twice as long as the square. <mdo:image alt="" src="nov-dra4.jpg"></mdo:image> So I went on! A new square appeared, now red! <mdo:image alt="" src="nov-dra5.jpg"></mdo:image> Extend that one. . . <mdo:image alt="" src="nov-dra6.jpg"></mdo:image> I really liked what was happening here! <mdo:image alt="" src="nov-dra7.jpg"></mdo:image> and so on. . . . <mdo:image alt="" src="nov-dra8.jpg"></mdo:image> <mdo:image alt="" src="nov-dra9.jpg"></mdo:image> I had to stop there because it had up to the size of the paper. I suggest that you print these pages out so far and have a good look at the way that the pattern has grown and see what things you notice in this last picture. <comment> #set var="roll-text" value="Send us your findings if you get this far" --&gt;&lt;/p&gt; &lt;p&gt;I found myself looking at a square and the extra lines and just one more square :-&lt;/p&gt; &lt;img alt="" src="/content/99/11/bbprob1/nov-dra3.jpg" /&gt;&lt;br /&gt; &lt;br clear="all" /&gt; &lt;br /&gt; &lt;p&gt;But I wanted it bigger and I thought about putting in some extra lines that again were just extensions of lines already there!&lt;/p&gt; &lt;img alt="" src="/content/99/11/bbprob1/nov-sqrpa4.jpg" /&gt;&lt;br /&gt; &lt;br clear="all" /&gt; &lt;br /&gt; &lt;p&gt;Perhaps it would be good to print this out, if you have not done so already and explore the shapes, areas, lengths, angles etc.&lt;/p&gt; &lt;p&gt;It would be really good to cut along the lines and see what happens. &lt;!-- #set var="roll-text" value= "You can scan, photocopy, draw - then cut, rearrange and paste. Messy!" --&gt;&lt;!-- #set var="roll-text" value="" </comment> When I looked at this shape I could imagine that I was looking through a square window at a pattern of squares but could only see the one square in the middle. So I printed out these:- <mdo:image alt="" src="nov-squares.jpg"></mdo:image> Then I printed out the bigger square . . . . <mdo:image alt="" src="nov-sqrtsquare.jpg"></mdo:image> This one I traced onto a "see through" sheet and placed it in different places over the grid of smaller squares. This was great fun and led to some interesting conversations. Have a GO! I think that this is one of the most exciting shape investigations that I've put on the NRICH site, so let's have a lot of workings sent to Cambridge U.K. from all over the world so that we can show how people from different countries are thinking and working in their maths. Pester your teachers to collect some results and send them off. </div> <h3>Why do this problem?</h3> <div>The <a href="http://nrich.maths.org/public/viewer.php?obj_id=72&part=">activity</a> is good for pupils working with squares and triangles. It will increase their familiarity with the properties of these shapes and is a good introduction to 'thinking outside the box'.</div> <h3>Possible approach</h3> <div>Working with the groups of pupils I've found it good to slowly present the first few stages of the growth of the squares. In this way I made sure that they understood each step.</div> <h3>Key questions</h3> <div>What do you notice?</div> <div>How do you know ...? (Following something they've said or recorded.)</div> <h3>Possible extension</h3> <div><a href="/content/99/11/bbprob1/GreatSqHelp.doc">This sheet</a> gives some ideas for how to take the activity further.</div> <div> </div> <h3>For the exceptionally mathematically able</h3> Go to Extending Great Squares <a href="http://nrich.maths.org/7237&part=">here</a> for ideas for this pupil. <h3>Possible support</h3> <div>Some pupils will need help with accurate drawing, Squared paper is useful. Some pupils will benefit from using a computer drawing program.</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> If we look at the time that the first square ABCD has been enlarged to EFGH we get this picture:- Square Pattern sqrpatTe3 <mdo:image width="361" height="363" src="sqrpatTe4.jpg" alt=""></mdo:image> We have some geometric facts which can be derived. I'm putting these into the notes because if you are working with particularly able pupils you may need to get a glimpse of these relationships. 1. Pythagoras gives us the relationship at AF squared added to AE squared will give us EF squared. 2. AB has been extended by the same length again so that AF is twice as long as AB. 3. AE is the extension of AD and so the two are the same length. It's nice to take the lengths of the sides of the inner square as 1. So we see that AF is 2, AE is 1 and using that realization of Pythagoras be have: 1^2 + 2^2 = EF^2 1 + 4 = 5 So the length of EF is the square root of 5 which is 2.236 ish. But we can just call it "root 5" all the while. You may find that some of these pupils look at areas. The area of ABCD is 1 sq unit The area of EFGH is root 5 multiplied by root 5 so is 5! The areas of subsequently larger squares increases by powers of 5 as the lengths of the sides increase by powers of root 5. When we come to the following diagram:- Square pattern sqrpatTe4 <mdo:image width="361" height="363" src="sqrpatTe4.jpg" alt=""></mdo:image> Interesting discussions are to be had with pupils who have some idea that Triangle BFW will swing around point W to fit on top of EABW so as to form a square EAB(NewF). Some pupils can treat this very usefully as a jigsaw and cut it out carefully. Others may well use their intuition and others their knowledge to get there. When they have got there you can see the fact that the larger square is equivalent in area to 5 of the smaller squares. So we have done a construction that is fairly simple. Now look at it the other way round and we have a large square in which 4 lines have been drawn from the centre of each side to the appropriate opposite corner. Now we also know that the area of the smaller square is 1/5 of the larger. So, we can now be presented with a square and be able to generate the square that has an area 1/5 of the original! Finally, I have not actually done the last bit with pupils but I see that there can be great fun and discovery with acetate sheets of the root 5 sided square placed over a whole array of smaller squares of size 1. This really has great potential for KS 2 3 4 and my undergraduate students here at Cambridge! </mdoxml> 2 3 0 1 1 0 0 Great Squares Investigate how this pattern of squares continues. You could measure lengths, areas and angles. Using, Applying and Reasoning about Mathematics Investigations Measures and Mensuration Length/distance 2D Geometry, Shape and Space Squares Mathematics Tools Measuring equipment Measures and Mensuration Area