We'll bang the drum

5493 /www/nrich/html/content/id/5493/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Here is a pile of drums that will each give a beat. Start off by placing two drums on the wheel. As the wheel turns, the drums will sound. How many different rhythms can you make with just two drums? Now try with 3 drums. As the wheel turns so the drums will play. What different rhythms can you make now? Are you sure they are all different from each other? <a href="/content/id/5493/BangTheDrum.swf" onclick="mediaSave()">Full Screen Version</a> <mdo:flash height="400" width="550"><param name="movie" value="/content/id/5493/BangTheDrum.swf" ></param><param name="flashplayerversion" value="7" ></param><param name="height" value="400" ></param><param name="width" value="550" ></param></mdo:flash> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Genia from Deer Park School thought hard about this problem: For this problem there are 3 possibilites to put two drums on the wheel with 6 circles on it. <div style="clear: both;">One of them is putting 2 drums on first and second circles. <div style="clear: both;">The second one is putting 2 drums on first and third circles. <div style="clear: both;">And the last one is putting 2 drums on second and fifth circles.</div> <div style="clear: both;">But there are also three speeds. So we will take all three of our rhythms and multiply it by three different speeds = nine (that's how many rhythms you can make with just two drums and a wheel with six circles on it)</div> <div style="clear: both;"></div> Well done, Genia. Perhaps you have described your ways slightly differently to Genia but you should find if you are looking for DIFFERENT possibilities, then there are only three. Matt from Lincoln looked at having three drums on the wheel. He says: <div style="clear: both;">I simply put a drum in one spot and kept it there while I changed the position of the second. Then after I figured out all of the solutions for one being in that spot I moved the first one over and repeated the process. However, I took care not to repeat the same rhythms.</div> <div style="clear: both;"></div> You've described a very good way of approaching the problem, Matt. It's good to have some sort of system for finding all the ways so that you can check whether you've got them all. Matt thinks there are 15 different ways, but unfortunately you didn't say what they are, Matt. Perhaps you coud list them for us? Does anyone agree or disagree? </div> </div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>We'll Bang the Drum</h2> Here is a pile of drums that will each give a beat. Start off by placing two drums on the wheel. As the wheel turns, the drums will sound. How many different rhythms can you make with just two drums? Now try with 3 drums. As the wheel turns so the drums will play. What different rhythms can you make now? Are you sure they are all different from each other? <a href="/content/id/5493/BangTheDrum.swf" onclick="mediaSave()">Full Screen Version</a> <mdo:flash height="400" id="/content/id/5493/BangTheDrum.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/5493/BangTheDrum.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="400"></param> <param name="width" value="550"></param> </mdo:flash></div> Putting the drums in position and then turning the wheel can lead to some good discussions about whether there is a need to record, and if so how. Ideally, allow the children to come up with their own suggestions. One way is in the form of a table putting the 6 places around the wheel across the table and marking each place where a drum beat would go. This might lead to a table like this for three drums then four drums: <mdo:image alt="35possibilities" height="511" src="35possibilities4%20Idea4.jpg" width="284"></mdo:image> The children are likely to come up with more because of those that are actually equivalent to others. Again a lot of possibilities for discussion will arise about how they will decide which are the same and which are different. Is the first row actually the same as the fourth row in the table if the wheel spins for several turns? They can use the interactivity to listen to the rhythms carefully. The group will need to decide on their own "rules" about what is the same and what different. This is a good opportunity to encourage children to find a system to work out the different combinations. For example, in the table above we have kept the first two drums in the same place and then found all the ways to put the third drum on the wheel. Then we have kept the first drum in the same place but altered the second drum's place and again found all the different positions for the third drum. Children might find this way of working (systematically) rather tricky, depending on their previous experience, so you will perhaps need to model a system for them in the first instance. Finding all the possibilities for three drums is quite challenging but some pupils could go on to look at four drums. Another way to record might be more like how the wheel looks, in a pictorial representation. Here are some examples for three and four drums: <mdo:image alt="recording" height="433" src="Recordsing4Idea4.jpg" width="419"></mdo:image> This way of recording might make it easier to spot which are the same and which different. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> How will you record what you have done? You might like to draw pictures, make lists or make a table. Listen carefully - how will you decide which rhythms are the same and which are different? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Imagine places numbered 1-6 then for two drums: 1 and 2 (ie no gaps) 1 and 3(ie one gap) 1 and 4 (ie two gaps) Others are the same For three drums see Bernrad's table although some might be though of as equivalent. </mdoxml> 2 5 1 0 0 0 0 We'll bang the drum How many different rhythms can you make by putting two drums on the wheel? Information and Communications Technology Interactivities Using, Applying and Reasoning about Mathematics Working systematically Decision Mathematics and Combinatorics Combinations Applications Music