The Numbers give the design

6919 /www/nrich/html/content/id/6919/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <mdo:image height="400" width="566" src="The%20group.jpg" alt="The group"></mdo:image><br></br> <br></br> <br></br> How about making designs like the ones above?<br></br> <br></br> It's just a matter of getting a group of numbers together that repeat and then using a simple drawing rule, then colouring the pattern in.<br></br> <br></br> A good way to start is to take a number sequence that you like, for example square numbers or the $3$ times table.<br></br> Just keep the ones (units) of those numbers, for example $1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4$ etc. or $3, 6, 9, 2, 5, 8, 1, 4, 7, 0, 3, 6$ etc.<br></br> You may notice that with these, and perhaps your own sequences, they go to a $0$ and then repeat.<br></br> <br></br> For this challenge, you just need the numbers that go as far as just before the zero, for example $1, 4, 9, 6, 5, 6, 4, 1$ or $3, 6, 9, 2, 5, 8, 1, 4, 7$.<br></br> <br></br> Now to draw ...<br></br> <br></br> Using squared paper and starting somewhere in the middle, take each of the numbers in turn to tell you how long to draw the line and turn a right angle to the left after you've drawn each line.<br></br> <br></br> This may help you to see what to do, using the example of square numbers and starting with the red $1$. I've coloured each new line with a different colour:<br></br> <br></br> <div style="text-align: center;"><mdo:image height="172" width="279" src="The%20Process%20A.jpg" alt="process A"></mdo:image></div> <br></br> After the final $1$ you start again as shown below:<br></br> <br></br> <div style="text-align: center;"><mdo:image height="311" width="312" src="The%20process%20B.jpg" alt="process B"></mdo:image></div> <br></br> and so on until you get back to the beginning.<br></br> Some squences that you choose may not come back to the start but just keep stepping away!<br></br> <br></br> So now it's your turn ...<br></br> Choose your sequence.<br></br> Pick out the ones (units) until you find it repeating.<br></br> Draw the lines carefully on squared paper turning in the same direction each time by a right angle.<br></br> See what you get and colour it in if you like.<br></br> <br></br> You may like to change a rule or two after a while. (For example, you could change the turn to $60^\circ$ instead of $90^\circ$.)<br></br> <br></br> Please send us your pictures.<br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">We had a collection of responses from Hymers College School. Katie wrote that while doing the challenge she found out a lot of different shapes the different tables made. She added that some people may say "How would you know when to end the sequence?". But the answer was that the shapes will end where you started. On a few of the tables she figured that once you have got the hang of the pattern you would be able to do the sequence backwards like on the $2$ times table. Ishita wrote:-</p> <div>The things I have discovered are in the $7$ times table, in which I experimented on and I found out that in the $7$ times table, if on Microsoft Windows Logo, if you turn left $45$ degrees, it takes $8$ times before it meets with the point were it started and the angles are a lot larger, it is actually $2$ times as large as the $7$ times table in $90$ degrees.</div> <div> </div> <div>When I do the $7$ times table, but then I turn left $90$ degrees it is a totally different shape and it only takes $4$ times to connect to the start from the end, and in comparison, it is a much smaller shape. This means that if I have a different angle on one of the times tables, for example the $3$s, it would be different in comparison of size, quantity in times of having it meet the start and in the style of the shape. </div> <p class="editorial"> <mdo:image width="290" height="200" src="Small%20Ishita%20B.jpg" alt="Ishita b"></mdo:image><mdo:image width="290" height="200" src="Small%20Ishita%20A.jpg" alt="Ishita a"></mdo:image></p> <p class="editorial">From the same school we also had responses from Jessica, Jamie and Quintin. Here are their pictures:-</p> <p class="editorial"> <mdo:image width="290" height="200" src="Small%20J%20%26%20J.jpg" alt="Small J & J"></mdo:image><mdo:image width="290" height="200" src="Small%20Quintin.jpg" alt="Small Quintin"></mdo:image></p> <p class="editorial">A teacher from Stanmore Primary School wrote:-</p> <div>Key Stage 1 maths group have investigated what happens if they use the three times table to draw their pattern. They found that the pattern repeated and that it rotated. The tricky bit was keeping track of which way to turn and making sure that you drew a straight line!</div> <div> </div> <div>Next week they are going to investigate what happens if they use a different times table to create their pattern. They made some predictions: 'it might be the same pattern but a different size' and 'it might be a different pattern but it will still repeat and still rotate'.</div> <div> </div> <div>They thought that it was amazing that using numbers could create a repeating pattern. </div> <div> </div> <p><span class="editorial">Here are some images they sent in:</span></p> <p> <mdo:image width="360" height="390" src="KS%21a.jpg" alt="KS1 a"></mdo:image></p> <div> </div> <div> <mdo:image width="361" height="394" src="KS1b.jpg" alt="KS1b"></mdo:image></div> <div> </div> <p class="editorial"> <mdo:image width="360" height="390" src="KS1c.jpg" alt="KS1c"></mdo:image></p> <p class="editorial">Nina from Wolfson Hillel School sent us the following notes. These impressed me; particularly her ideas which led to the picture.</p> <div>For my design I chose to use prime numbers. I made a pattern which went $3,2,7,5,1,5,7,2,3$; once I had gone round four times and had made a full shape I decided to make smaller version of it inside it. I knew that for my full shape I needed a square that measured $21$cms by $21$cms so I measured inside the shape and I found I could make four squares that measured $7$cms by $7$cms so I divided all the numbers by three as $21$ divided by $3$ equals $7$. Then I coloured all the bits in between $5,1,5$ red, all the corners blue and the rest I just chose nice colours, then I went over the regular sized shape's outline in brown. </div> <div style="text-align: center;"> <mdo:image width="236" height="218" alt="nina" src="Nina%27s%20pic.jpg"></mdo:image></div> <div> </div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>The Numbers Give the Design</h2> <br></br> <mdo:image alt="The group" height="400" src="The%20group.jpg" width="566"></mdo:image><br></br> <br></br> <br></br> How about making designs like the ones above?<br></br> <br></br> It's just a matter of getting a group of numbers together that repeat and then using a simple drawing rule, then colouring the pattern in.<br></br> <br></br> A good way to start is to take a number sequence that you like, for example square numbers or the $3$ times table.<br></br> Just keep the ones (units) of those numbers, for example $1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4$ etc. or $3, 6, 9, 2, 5, 8, 1, 4, 7, 0, 3, 6$ etc.<br></br> You may notice that with these, and perhaps your own sequences, they go to a $0$ and then repeat.<br></br> <br></br> For this challenge, you just need the numbers that go as far as just before the zero, for example $1, 4, 9, 6, 5, 6, 4, 1$ or $3, 6, 9, 2, 5, 8, 1, 4, 7$.<br></br> <br></br> Now to draw ...<br></br> <br></br> Using squared paper and starting somewhere in the middle, take each of the numbers in turn to tell you how long to draw the line and turn a right angle to the left after you've drawn each line.<br></br> <br></br> This may help you to see what to do, using the example of square numbers and starting with the red $1$. I've coloured each new line with a different colour:<br></br> <br></br> <div style="text-align: center;"><mdo:image alt="process A" height="172" src="The%20Process%20A.jpg" width="279"></mdo:image></div> <br></br> After the final $1$ you start again as shown below:<br></br> <br></br> <div style="text-align: center;"><mdo:image alt="process B" height="311" src="The%20process%20B.jpg" width="312"></mdo:image></div> <br></br> and so on until you get back to the beginning.<br></br> Some squences that you choose may not come back to the start but just keep stepping away!<br></br> <br></br> So now it's your turn ...<br></br> Choose your sequence.<br></br> Pick out the ones (units) until you find it repeating.<br></br> Draw the lines carefully on squared paper turning in the same direction each time by a right angle.<br></br> See what you get and colour it in if you like.<br></br> <br></br> You may like to change a rule or two after a while. (For example, you could change the turn to $60^\circ$ instead of $90^\circ$.)<br></br> <br></br> Please send us your pictures.<br></br> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div>This <a href="http://nrich.maths.org/6919&part=">activity</a> engages the pupils in a simple geometric construction that has the potential to generate enthralling designs. Children who use Logo or control robots like Roamer or Pixie might like to explore in a different way without the need for geometric drawings. This task offers the opportunity for learners to vary their input, observe the effects, begin to make predictions and perhaps generalisations. You could think of the sequence of instructions both as a sequence of actions (turn $90^\circ$ and then go forward the specified amount) as a compound action, and also as a single action that is to be repeated over and over.</div> <br></br> <h3>Possible approach</h3> <div>Show the designs and ask the children what it is that they see or what strikes them about the images.</div> <br></br> <div>Reveal how they are constructed so simply from lines and turning right angles. Discussion can then focus on the choice of numbers that repeat themselves. What sequences might they like to try? </div> <div> </div> <div>Once everyone has tried out a number of different sequences, you could encourage children to 'tweak' one of their sequences. Can they predict what difference their 'tweaking' will make to their design? They could then test their ideas. </div> <br></br> <div>The resulting designs would make a lovely display and children could be involved in writing the description of how the patterns were produced. It may also be possible for learners to make some general statements about the designs.</div> <br></br> <h3>Key questions</h3> <div>What can you see?</div> <div>Tell me about your sequence of numbers.</div> <div>Which parts do you like/dislike?</div> <div>What could you change?</div> <div>Can you make any predictions about your new pattern before you draw it?</div> <br></br> <h3>Possible extension</h3> <div>Pupils can select a random set of numbers to use and see their effect, then adjust the numbers in a particular way for the next design they try.</div> <br></br> <h3>Possible support</h3> <div>Some pupils may need some help with the drawing of straight lines accurately. Those that want to pursue turning by different amounts might find isometric paper helpful.</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Here are some sequences that I've tried which might give you some insipiration:<br></br> <mdo:image height="427" width="394" alt="Fib Tri" src="Fibon%20Tri%20Sm.jpg"></mdo:image><br></br> <br></br> <mdo:image height="276" width="254" src="7x%20Sq%20Sm.jpg" alt="7xSq"></mdo:image><mdo:image height="349" width="310" alt="3xSq" src="3x%20Sq%20Sm.jpg"></mdo:image><br></br> <mdo:image height="380" width="358" src="2X%20Tri%20Sm.jpg" alt="2xTri"></mdo:image><mdo:image height="348" width="369" src="Pat%20A%20sm.jpg" alt="Pat A"></mdo:image><mdo:image height="326" width="389" src="pat%20B%20Sm.jpg" alt="Pat B"></mdo:image><mdo:image height="301" width="277" src="sq%20Nos%20Sq%20sm.jpg" alt="Sq Nos Sq"></mdo:image><br></br> <br></br> <br></br> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">THESE ARE THE SOLUTIONS FOR JUNE 2010<br></br> <br></br> Many thanks for the great pictures that were sent in. It seems you enjoyed the activity as well as drawing some conclusions. Here is Maisy's comment who learns at home.<br></br> I have been investigating solutions for the numbers give the design I loved making different shapes out of number patterns such as square numbers, $7$x table and triangular numbers etc. I started off doing them by hand but after a bit I decided to do it on textease logo I found that the $3$x table was a mirror image of the $7$x table. I liked the triangular numbers best as they made beautiful patterns.<br></br> <mdo:image width="257" height="356" alt="maisy" src="maisy.jpg"></mdo:image><br></br> Emma and Lily From Kings Sutton Primary School send in a Word document;<br></br> We wanted to see what the differences would be if we kept the same times table but changed the angle.<br></br> So here are their lovely pictures for $90$, $60$ and $45$ degrees.<br></br> <mdo:image width="505" height="155" alt="emma whi" src="Emma%20Whi.jpg"></mdo:image><br></br> We had a hypothesis that as the angle got smaller the shape would be more complicated. When we did a $60$ degree angle this theory changed because the $60$ degree one was very simple whereas the $90$ degree angle picture was more confusing. The $45$ degree picture matched with our hypothesis and the shape is very confusing compared to the $60$ or the $90$ degree shape.<br></br> <br></br> Emma and Angel from the same school also sent in a document as follows;<br></br> After many tries and dead ends we eventually found out the solution to the problem! We found out that if the times table didn't have a odd number as a unit then when you repeated the pattern it wouldn't link up. However if you used a times table with a odd number as a unit the lines would make a pattern. These are some examples, first $11$ then $13$ and last is the $2$ times table.<br></br> <mdo:image width="591" height="167" alt="angel and emma" src="angel%20and%20Emma.jpg"></mdo:image><br></br> <br></br></mdoxml> 2 3 0 1 0 0 0 The Numbers give the design Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot. 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