5483
/www/nrich/html/content/id/5483/
1
2011-02-01T00:00:01
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You could have a go at <a href="http://nrich.maths.org/public/viewer.php?obj_id=5482&part=index">
Clapping Times</a> before trying this problem.<br></br>
<br></br>
This is a very practical activity - you might like to use some
musical instruments, for example a drum or a triangle, rather than
using your hands and parts of your body.<br></br>
<br></br>
Begin a rhythm: clap, clap, click (your fingers), clap, clap,
click, clap, clap, click, clap, clap, click ...<br></br>
<br></br>
What will you be doing on the $15$th beat? <br></br>
How do you know this without actually doing it?<br></br>
What will you be doing on the $20$th beat? <br></br>
Again, explain how you can predict this.<br></br>
How about on the $99$th beat?<br></br>
What would you be doing on the $100$th beat?<br></br>
<br></br>
If there is someone else with you, ask them to come and join in. If
you're on your own, it doesn't matter, you'll just have to imagine
that someone else is there.<br></br>
<br></br>
You and your friend are going to both start a different rhythm at
the same time. <br></br>
You will do clap, clap, click, clap, clap, click ... as you did
before.<br></br>
Ask your friend to do click, clap, clap, click, clap, clap, click,
clap, clap ... <br></br>
<br></br>
Have a go so that you get a steady rhythm going.<br></br>
If you both start at the same time, when will you both click your
fingers at the same time? <br></br>
Why?<br></br>
Are there other ways that you could have clapped and clicked for
this to be the case? <br></br>
<br></br>
How could you change your rhythms so that you do click at the same
time? <br></br>
How could you predict when this was?<br></br></mdoxml>
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<p><span class="editorial">Rhiannon from St Mary Redcliffe Primary
School sent a very well-explained solution to this problem. She
wrote:</span></p>
On the 15th beat, you will be doing a click. I know this because 15
is in the three times table and there is a click every three
beats.<br></br>
<div style="clear: both;">On the 20th beat, you will be doing a
clap, because 20 is not in the three times table.<br></br>
<div style="clear: both;">On the 99th beat you will be doing a
click, because 99 is in the three times table. I know this because
9+9 = 18 and 1+8=9 and 9 is in the three times table.<br></br>
<div style="clear: both;">On the 100th beat you will be doing a
clap because you did a click on the 99th beat.<br></br>
<br></br>
<div style="clear: both;">If two people do the following
rhythms:<br></br>
<div style="clear: both;">person 1: clap clap click<br></br>
<div style="clear: both;">person 2: click clap clap<br></br>
<div style="clear: both;">then you won't click at the same
time.<br></br>
<div style="clear: both;">On person 1, you click if the number is
dividable by three <span class="editorial">(we can say "divisible
by 3")</span> .<br></br>
<div style="clear: both;">On person 2, you click if the number has
remainder 1 when divided by three. Since a number can never ever be
dividable by three and remainder 1, you won't click at the same
time.<br></br>
<div style="clear: both;">To have the same result, person 1 could
go "clap click clap", person 2 could go "clap clap click".<br></br>
<div style="clear: both;">So that you do click at the same time you
could:<br></br>
<div style="clear: both;">person 1: clap clap click<br></br>
<div style="clear: both;">
<div style="clear: both;">person 2: clap clap clap click</div>
You would both click at the same time every 12 beats.</div>
<div style="clear: both;"></div>
<p class="editorial" style="clear: both;">Fantastic, Rhiannon!
Rohan from Longbay Primary School suggested an alternative:</p>
<div style="clear: both;">Change your friend's rhythm to: click
clap click etc. and keep your rhythm the same. 9.</div>
<div style="clear: both;">A click on your rhythm occurs every 3rd
beat and a click on your friend's rhythm will occur every 1st and
3rd beat so you will both click at the same time on the 3rd beat.
<div style="clear: both;"></div>
<p class="editorial" style="clear: both;">Well done to everyone who
sent in solutions.</p>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
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<h2>Music to My Ears</h2>
<br></br>
You could have a go at <a href="http://nrich.maths.org/public/viewer.php?obj_id=5482&part=index">Clapping Times</a> before trying this problem.<br></br>
<br></br>
This is a very practical activity - you might like to use some musical instruments, for example a drum or a triangle, rather than using your hands and parts of your body.<br></br>
<br></br>
Begin a rhythm: clap, clap, click (your fingers), clap, clap, click, clap, clap, click, clap, clap, click ...<br></br>
<br></br>
What will you be doing on the $15$th beat?<br></br>
How do you know this without actually doing it?<br></br>
What will you be doing on the $20$th beat?<br></br>
Again, explain how you can predict this.<br></br>
How about on the $99$th beat?<br></br>
What would you be doing on the $100$th beat?<br></br>
<br></br>
If there is someone else with you, ask them to come and join in. If you're on your own, it doesn't matter, you'll just have to imagine that someone else is there.<br></br>
<br></br>
You and your friend are going to both start a different rhythm at the same time.<br></br>
You will do clap, clap, click, clap, clap, click ... as you did before.<br></br>
Ask your friend to do click, clap, clap, click, clap, clap, click, clap, clap ...<br></br>
<br></br>
Have a go so that you get a steady rhythm going.<br></br>
If you both start at the same time, when will you both click your fingers at the same time?<br></br>
Why?<br></br>
Are there other ways that you could have clapped and clicked for this to be the case?<br></br>
<br></br>
How could you change your rhythms so that you do click at the same time?<br></br>
How could you predict when this was?</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=5483&part=index">This activity</a> centres on factors and multiples in a very practical context, and introduces the idea of common multiples.</div>
<br></br>
<h3>Possible approach</h3>
<div>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!).</div>
<br></br>
<div>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.</div>
<br></br>
<div>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.</div>
<br></br>
<h3>Key questions</h3>
<div>How do you know what you will be doing on this beat?</div>
<div>How do you know when you will be clapping/clicking together?</div>
<div>Tell me about what you have written down.</div>
<br></br>
<h3>Possible extension</h3>
<div>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.</div>
<br></br>
<h3>Possible support</h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=5482&part=index">Clapping Times</a> makes a good introduction to this problem which many learners would benefit from doing first.</div>
<br></br>
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If you count "$1$" as you do your first clap, then "$2$" when you
clap for the second time, then "$3$" when you click, then "$4$" as
you clap again etc, this might help.<br></br>
How many beats are there in one repetition of the rhythm?<br></br>
You could write down numbers, say from $1$ to $12$ and underneath
each number write whether you are clapping or clicking. Do you
notice anything that the clicking numbers have in common?<br></br>
How about writing down both rhythms with the beat numbers for the
second part of the problem? This might help you spot some patterns
again. <br></br></mdoxml>
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15th beat - click<br></br>
20th - clap<br></br>
99th - click<br></br>
100th - clap<br></br>
<br></br>
multiples of 3 are clicks, others are claps<br></br>
<br></br>
two will never click at same time -one is clicking on multiples of
3, the other on one more than a multiple of 3<br></br>
other ways not click at same time e.g. clap, clap, click and clap,
click, clap <br></br>
<br></br>
to click at same time, either change so that the repeat length is
different or alter the number of claps and taps<br></br>
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Music to my Ears
Can you predict when you'll be clapping and when you'll be clicking
if you start this rhythm? How about when a friend begins a new
rhythm at the same time?
Applications
Music
Using, Applying and Reasoning about Mathematics
Visualising
Using, Applying and Reasoning about Mathematics
Practical Activity
Numbers and the Number System
Factors and multiples
Admin
Upper primary mapping document