Beat the drum beat!

5495 /www/nrich/html/content/id/5495/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> This problem follows on from <a href="http://nrich.maths.org/public/viewer.php?obj_id=5493&part=index">We'll Bang the Drum</a> so you may like to try that one first if you haven't already. Here are drums that can be placed on a wheel. As the wheel goes round, each drum beats. Before you read any further, explore the interactivity by making some different rhythms. In how many ways can you create a rhythm with a steady beat? Explain how you have done this. If there were 12 positions on the wheel, in how many ways would you be able to create a steady beat? How do you know? Now place just three drums on the wheel to create a new rhythm. What would happen if you played this rhythm backwards? Would it sound different? Using three drums, make a rhythm which would sound the same forwards as it does backwards. How many different ways can you do this using just three drums? Now try creating rhythms which sound the same forwards and backwards with four drums. How about five drums? How could you tell just by looking at the arrangement of drums whether it would sound the same when the wheel rolled forwards or backwards? <a href="/content/id/5495/BangTheDrum.swf">Full Screen Version</a> <mdo:flash height="400" width="550"><param name="movie" value="/content/id/5495/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"> Ben from Victoria College, Jersey sent this solution to the first part of the problem: For a steady drum beat there are four possible ways: 1, 2, 3 or 6 drums equally spaced. That's a very clear way of describing it, Ben, thank you. For the second part of the problem, some of you may have worked out that a rhythm will sound the same forwards as backwards only if you arrange the drums symmetrically. There are only two ways of placing 3 drums symmetrically on 6 points: <mdo:image width="500" height="240" alt="3 drums" src="3%20drums.gif"></mdo:image> <div class="editorial">Try these out on the interactive so that you are sure that they work both ways. With four drums there also two arrangements which will work both backwards and forwards:</div> <div><mdo:image width="500" height="240" alt="4 drums" src="4%20drums.gif"></mdo:image></div> Finally with 5 drums, there is only one rhythm that you can make, as moving the drums around just creates the same sequence with only one empty space. However this arrangement is symmetrical so it does sound the same forwards as backwards: <mdo:image width="250" height="240" src="5%20drums.gif" alt="5 drums"></mdo:image> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Beat the Drum Beat!</h2> This problem follows on from <a href="http://nrich.maths.org/public/viewer.php?obj_id=5493&part=index">We'll Bang the Drum</a> so you may like to try that one first if you haven't already. Here are drums that can be placed on a wheel. As the wheel goes round, each drum beats. Before you read any further, explore the interactivity by making some different rhythms. In how many ways can you create a rhythm with a steady beat? Explain how you have done this. If there were 12 positions on the wheel, in how many ways would you be able to create a steady beat? How do you know? Now place just three drums on the wheel to create a new rhythm. What would happen if you played this rhythm backwards? Would it sound different? Using three drums, make a rhythm which would sound the same forwards as it does backwards. How many different ways can you do this using just three drums? Now try creating rhythms which sound the same forwards and backwards with four drums. How about five drums? How could you tell just by looking at the arrangement of drums whether it would sound the same when the wheel rolled forwards or backwards? <a href="/content/id/5495/BangTheDrum.swf">Full Screen Version</a> <mdo:flash height="400" id="/content/id/5495/BangTheDrum.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/5495/BangTheDrum.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="400"></param> <param name="width" value="550"></param> </mdo:flash></div> Like <a href="http://nrich.maths.org/public/viewer.php?obj_id=5493&part=index">We'll Bang the Drum</a> , the interactivity in this problem has plenty of scope to be used to explore areas of mathematics not covered by these particular questions. The way the problem has been written allows opportunites for you to highlight ideas about factors, multiples and symmetry with the children. You might like to introduce the problem by "playing" two examples of rhythms on the wheel, one with a steady beat and one with no steady beat. Asking for differences between the two will help pupils understand what we mean by "steady" in this context and they should be able to explore the first parts of the problem in pairs using the interactivity. After a few minutes, it would be worth bringing the whole group together to talk about how they are remembering what rhythms they have tried. In this way, they will be encouraged to think about how to record what they are doing. These recordings of ideas can lead to interesting discussions as to the sameness/difference of the rhythms. For example: <mdo:image alt="same rhythms but different positions" height="173" src="Same%3ADiff1%3F.jpg" width="562"></mdo:image> Similarly having discussions about these further three examples could prove very valuable: <mdo:image alt="same rhythms with four drums, but different positions" height="173" src="Same%3ADiff2%3F.jpg" width="562"></mdo:image> You could also make more of a specific link to musical notation. Placing 3 - 8 drum beats within one turn of the wheel can easily be related to having a bar of music available for a drum beat rhythm that equates to four-four time (four crotchet beats in a bar). Since we have 8 spaces, each one would be equivalent to a quaver in length. Listening to the beat of the drum can give an illusion that when there is one space on the wheel after a drum beat, the beat length increases by a quaver. Here are twelve examples of putting either 3 or 4 beats for different effects: <mdo:image alt="12examples" height="354" src="1st%2012%20Ferris.jpg" width="280"></mdo:image> So for example in number 6 when the "bottom" drum sounds it appears to last for 4 quaver beats before the next drum beat sounds. That second beat appears to last for 2 quavers, as does the last beat. In example 9, the first appears to last for 4 quavers, the second and third beats last just 1 quaver each and the last beat lasts for 2 before the rhythm starts again. If you want to take the note values further you might look at the eight quaver values in one complete turn. Then the twelve examples shown above include notes of value 1, 2, 3, 4, 5 and 6. These could be recorded as follows: <mdo:image alt="details of 12" height="340" src="ferris1st12Details.jpg" width="432"></mdo:image> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> If you want a steady beat, what can you say about the gaps between all the drums on the wheel? How will you record what you have tried? You might like to use the "save" button on the interactivity too. Probably one of the best things you can do is try your ideas out on the interactivity and tweak what you have until you find what you want! Then, when you have a few examples, you can begin to look for patterns. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> If number positions as 1-8 then steady beats: 1, 2, 3, 4, 5, 6, 7, 8 1, 3, 5, 7 or starting anywhere but missing out one place each time 1, 4 or again, starting anywhere but missing out three places each time If 12 places, could do: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 1, 3, 5, 7, 9, 11 1, 4, 7, 10 1, 5, 9 ie step in factors </mdoxml> 2 4 0 1 0 0 0 Beat the drum beat! Use the interactivity to create some steady rhythms. How could you create a rhythm which sounds the same forwards as it does backwards? Applications Music Information and Communications Technology Interactivities Transformations and their Properties Symmetry Numbers and the Number System Factors and multiples