Copyright © University of Cambridge. All rights reserved.
'Music to My Ears' printed from http://nrich.maths.org/
Why do this problem?
centres on factors and multiples in a very practical context, and introduces the idea of common multiples.
You could introduce the first rhythm yourself by clapping and clicking, and asking the questions orally. This will encourage children to listen carefully and think about how the beat number connects to the different actions. The important point here is that pupils will understand how the repeated pattern links with factors and multiples, and this will enable them to predict where certain
actions will occur. Learners might articulate this in different ways, for example, by referring to numbers which are in certains 'times tables'. Of course, you can invite some children to physically demonstrate the rhythms so that the group's hypotheses are checked (at least for lower numbers of beats!).
You may wish to encourage children to jot things down to help them predict the sounds they will hear. This recording would be good to share in its own right as again, pupils will have found different representations. For example, some may use a $100$ square with highlighted or annotated numbers, some may draw a number line of sorts with abbreviations or symbols for claps and clicks. You
could talk about the advantages of each method and you could discuss how they would record differently if someone else needed to understand their work.
There are lots of variations on this idea with more than two different sounds or movements and a repeat pattern involving more than three beats, which you can go on to once the children are more confident.
How do you know what you will be doing on this beat?
How do you know when you will be clapping/clicking together?
Tell me about what you have written down.
As an extension you could ask pupils to investigate and prepare an example of two rhythms to bring back as a challenge for the rest of the class, for example CCCT, CCCT, CCCT ... and CCT, CCT, CCT ... (where C is clap and T is tap). Here the Ts coincide on all multiples of $12$. This problem has kept the ideas quite simple by only allowing one tap in a sequence and always at the end. This
could be made more difficult by allowing the tap anywhere or having more than one tap.
makes a good introduction to this problem which many learners would benefit from doing first.