Chain of Changes

221 /www/nrich/html/content/02/05/letme1/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Here is a set of shapes. For this problem we will call them pieces . <div style="text-align: center;"><mdo:image src="shapes.gif" alt="Set of Shapes: circles, squares, oblongs and triangles of colours red yellow and blue."></mdo:image> </div> Each of these pieces is either a different colour or a different shape from all the others. These problems ask you to arrange the pieces in a line so that you change either colour or shape in the next piece along. If we start with a blue triangle the next shape has to be either another triangle or another blue shape. The first puzzle is to arrange all the shapes in such a line starting with the blue triangle and ending with the red circle. There are lots of different ways of doing it! <div style="text-align: center;"><mdo:image width="554" height="58" src="221.gif" alt=""></mdo:image></div> The second problem is to arrange the pieces in the same way, starting with the blue triangle and ending with the red circle, but to change first colour, then shape or vice-versa. If you put a yellow triangle after the blue one and so change the colour, next you must put another yellow piece and so change the shape. You will not be able to use all the pieces in this way but the problem is to see how many you can use. Why do you think you cannot use all the pieces? You might like to try using this interactivity. <a href="/content/02/05/letme1/LogiB.swf">Full screen version</a> <mdo:flash height="400" width="550"><param value="/content/02/05/letme1/LogiB.swf" name="movie" ></param><param value="7" name="flashplayerversion" ></param></mdo:flash> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Taylor and Jessica from Stourport Primary sent this solution to the first part of the problem: Blue Triangle Blue Square Blue Rectangle Blue Circle Yellow Circle Yellow Square Yellow Rectangle Yellow Triangle Red Triangle Red Rectangle Red Square Red Circle We thought this was correct because it is a very easy way to order these shapes it is also very quick and efficient. We knew that the sequence had to have the same colour or same shape after one another so we ordered them in groups of colour. Which we thought was very clever. After every group of colour it still worked because it was the same shape as the one before for example, Yellow Square, Yellow Circle, Red Circle. After we had finished we checked it through carefully to make sure everything was correct. As we went through we realised that we had made a small mistake but that was soon fixed. Thank you both. Not many of you tried to find more than one solution, but James from Haytor View Primary sent us two different chains, both of which are different from Taylor and Jessica's: <mdo:image height="248" width="600" src="Jamessol.jpg" alt=""></mdo:image> Oliver, also from Haytor View Primary, sent in two more solutions: <mdo:image height="159" width="600" alt="" src="Olliversol.jpg"></mdo:image> Jed, Jordan, Cameren, Ashley, Charlotte and Lewis, who also go to Haytor View Primary sent in solutions too. Here are their various ideas which are different to those above: blue triangle blue square yellow square red square red rectangle yellow rectangle blue rectangle blue circle yellow circle yellow triangle red triangle red circle blue triangle blue oblong yellow oblong red oblong red square yellow square blue square blue circle yellow circle yellow triangle red triangle red circle Jack and Charlie from Stourport Primary found another way: blue triangle blue circle blue rectangle blue square red square red rectangle red triangle, ellow tringle yellow rectangle yellow square yellow circle red circle And here is what was sent by Devansh from Delhi Public School, Vasant Vihar: To place the same shapes or same colour as per the question we should start first with the Blue triangle then place the Blue Circle then the Yellow Circle, the Yellow Square, the Red Square, the blue Square, the Blue rectangle, the Red rectangle, the Yellow Rectangle, the Yellow Triangle, the Red Triangle and lastly place the Red Circle to complete the sequence. Do you think we have found all the ways? How will we know? Lucy, Jodie, Josef, Jack and Jordan from Haytor View looked at the second part and decided that you can't use them all because you run out of pieces to use. Alistair was able to explain this second part of the problem: Each piece has to have a pair of the same colour next to it in the line, except for the end and the start. Also each piece has to have a pair of the same shape next to it in the line. To get the longest line, there are exactly ten pieces in the line with two left over. The ten pieces include four shape pairs plus two odd shapes at the ends. <mdo:image src="solution.gif" alt="solution"></mdo:image> James from Haytor View found another way to do it: <mdo:image height="171" width="600" src="Jamessol2.jpg" alt=""></mdo:image> Maybe you can find another way. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Chain of Changes</h2> Here is a set of shapes. For this problem we will call them pieces . <div style="text-align: center;"><mdo:image alt="Set of Shapes: circles, squares, oblongs and triangles of colours red yellow and blue." src="shapes.gif"></mdo:image></div> Each of these pieces is either a different colour or a different shape from all the others. These problems ask you to arrange the pieces in a line so that you change either colour or shape in the next piece along. If we start with a blue triangle the next shape has to be either another triangle or another blue shape. The first puzzle is to arrange all the shapes in such a line starting with the blue triangle and ending with the red circle. There are lots of different ways of doing it! <div style="text-align: center;"><mdo:image alt="" height="58" src="221.gif" width="554"></mdo:image></div> The second problem is to arrange the pieces in the same way, starting with the blue triangle and ending with the red circle, but to change first colour, then shape or vice-versa. If you put a yellow triangle after the blue one and so change the colour, next you must put another yellow piece and so change the shape. You will not be able to use all the pieces in this way but the problem is to see how many you can use. Why do you think you cannot use all the pieces? You might like to try using this interactivity. <a href="/content/02/05/letme1/LogiB.swf">Full screen version</a> <mdo:flash height="400" id="/content/02/05/letme1/LogiB.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/02/05/letme1/LogiB.swf"></param> <param name="flashplayerversion" value="7"></param> </mdo:flash></div> <h3>Why do this problem?</h3> <a href="http://nrich.maths.org/public/viewer.php?obj_id=221&part=index">This problem</a> will help children to refer to the shapes by name and visualise the next shape to place in the pattern. They will need to look carefully at the properties of each shape. It will also encourage them to use a trial and improvement approach in solving problems. <h3>Possible approach</h3> <div>You will need plenty of Logic Blocks or of shapes cut out from <a href="/content/02/05/letme1/221.pdf">this sheet</a> which has two copies of each of the shapes. A full set of Logic Blocks can provide enough for four children, pairs or small groups depending on how the children are working. (One group has the large, thick pieces, one the large, thin pieces and so on).</div> <div>You could start with one of the pieces and ask children to describe it. Ask if they can find one which is the same shape but a different colour. Then you could go on either using the large, thick blocks or <a href="/content/02/05/letme1/LogiB.swf">this interactivity</a> to make sure that all in the group really understand the problems to be solved. After this the children could work in pairs or threes on the actual problem so that they are able to talk through their ideas with the others.</div> <div>At the end you could discuss the different ways that the children found of doing the first part of the problem. Then discuss why there was not a way of doing the second part using all the pieces. Children could also discuss the pattern that results in changing shape and colour alternately.</div> <div>The work makes a good display using either the children's own drawings or paper copies of the pieces using <a href="/content/02/05/letme1/221.pdf">this sheet</a> .</div> <h3>Key questions</h3> <div>Are you going to change the colour or the shape this time?</div> <div>Which shape are you going to use next?</div> <div>Can you find another way of doing it?</div> <div>Why can't you use all the shapes this time?</div> <h3>Possible extension</h3> Those who find these tasks straightforward could use a full set of Logic Blocks and also change the size and thickness of the pieces. <h3>Possible support</h3> It might help some children to make their own chains which started with the blue triangle, but without specifying the end point. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Are you going to change the colour or the shape first? Which shape could you use next? Can you find another way of doing it? </mdoxml> 2 4 1 0 0 0 0 Chain of Changes Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle? Sequences, Functions and Graphs Sequences Using, Applying and Reasoning about Mathematics Working systematically 2D Geometry, Shape and Space Squares Using, Applying and Reasoning about Mathematics Trial and improvement 2D Geometry, Shape and Space Rectangles 2D Geometry, Shape and Space Equilateral triangles 2D Geometry, Shape and Space Circles Admin Lower primary mapping document