4771 /www/nrich/html/content/id/4771/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Pick two cogs in the interactivity below, place them near each other and turn one of them.<br></br> What happens?<br></br> (You can turn a selected cog by using the arrow keys on your keyboard or by using the green arrows on the interactivity.)<br></br> <br></br> <a href="/content/id/4771/cogs.swf">Full Size Version</a> <br></br> <mdo:flash height="400" width="530"><param name="movie" value="/content/id/4771/cogs.swf" ></param><param name="flashplayerversion" value="7" ></param><param name="height" value="400" ></param><param name="width" value="530" ></param></mdo:flash><br></br> <br></br> <br></br> Now pick a different pair and do the same thing again.<br></br> <br></br> What do you notice about the direction in which the cogs turn each time?<br></br> <br></br> Now pick two cogs with 5 &quot;teeth&quot;. <br></br> <div style="clear: both;">Mark a &quot;tooth&quot; on each cog by selecting it and then pressing the blue button. You may like to line up the marked teeth so that they start next to each other.</div> <div style="clear: both;">Watch what happens to the teeth that are marked as the two cogs turn.</div> <div> What can you say about the way that the marked tooth of one cog fits into the gaps of the second cog?</div> <div></div> <div>Will this always be the case whichever two cogs you pick?</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p><span class="editorial">Kathryn from Ardingly College Junior School described the way the cogs move:</span></p> If you turn the cogs to the right they will move away from each other, but if you turn them to the left they will move towards each other. This is the same no matter which sized cogs you have.<br></br> <br></br> <p class="editorial">Ollie and Tom, also from Ardingly College Junior School, put this a bit differently:</p> One cog moves one way and the other the other way.<br></br> <br></br> Alternatively, you could say that one moves clockwise and the other anticlockwise.<br></br> <br></br> <p class="editorial">Unfortunately we didn't have any solutions for the next part of the question. Have a look at our solution and test it for yourself:</p> <p class="editorial"> </p> <p class="editorial"> </p> <div>With two identical cogs, the marked tooth on one cog will always go in the same gap on the other cog.</div> <br></br> <div>If the cogs differ in size by one tooth, then eventually, the marked tooth on the smaller cog will fit into each gap in the larger cog in turn.</div> <br></br> <div>If the number of teeth on each cog share a common factor, then a marked tooth on one cog will never go in all the gaps of the other cog.</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Turning Cogs</h2> <br></br> Pick two cogs in the interactivity below, place them near each other and turn one of them.<br></br> What happens?<br></br> (You can turn a selected cog by using the arrow keys on your keyboard or by using the green arrows on the interactivity.)<br></br> <br></br> <a href="/content/id/4771/cogs.swf">Full Size Version</a><br></br> <mdo:flash height="400" id="/content/id/4771/cogs.swf" width="530"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/4771/cogs.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="400"></param> <param name="width" value="530"></param> </mdo:flash><br></br> <br></br> <br></br> Now pick a different pair and do the same thing again.<br></br> <br></br> What do you notice about the direction in which the cogs turn each time?<br></br> <br></br> Now pick two cogs with 5 "teeth".<br></br> <div style="clear: both;">Mark a "tooth" on each cog by selecting it and then pressing the blue button. You may like to line up the marked teeth so that they start next to each other.</div> <div style="clear: both;">Watch what happens to the teeth that are marked as the two cogs turn.</div> <div>What can you say about the way that the marked tooth of one cog fits into the gaps of the second cog?</div> <div> </div> <div>Will this always be the case whichever two cogs you pick?</div> </div> <br></br> This challenging problem can be taken to different stages by different children, but offers a starting point for all. Pupils should be encouraged to play with the interactivity and explore what happens. Drawing their ideas together will enable the class to offer some hypotheses and then they can test out their ideas. The problem is in fact about common factors (in the number of "teeth"), but some children may be happy to stick to cogs of the same size.<br></br> <br></br> If the interactivity is not available, you might like to print off this <a href="/content/id/4771/cogs.doc">sheet of cogs</a> and cut them out for the pupils to use practically, or for you to demonstrate on an OHP.<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Do both cogs turn in the same direction?<br></br>With two cogs that have five teeth, will the marked tooth on one cog eventually fit in every gap of the other cog as they turn? Why or why not?<br></br>What happens if you pick a cog with five teeth and a cog with 6 teeth?<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">With two identical cogs, the marked tooth on one cog will always go in the same gap on the other cog.<br></br><br></br>If the cogs differ in size by one tooth, then eventually, the marked tooth on the smaller cog will fit into each gap in the larger cog in turn.<br></br><br></br>If the number of teeth on each cog share a common factor, then a marked tooth on one cog will never go in all the gaps of the other cog. <br></br><br></br></mdoxml> 2 5 0 1 0 0 0 What happens when you turn these cogs? Investigate the differences between turning two cogs of different sizes and two cogs which are the same. Information and Communications Technology Interactivities Numbers and the Number System Common factors Using, Applying and Reasoning about Mathematics Visualising