1996
/www/nrich/html/content/98/09/bbprob2/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<div style="text-align: center;"><mdo:image src="icon2.png"></mdo:image></div>
<br></br>
<br></br>
"The Pied Piper of Hamelin'' is a story you may have come across. This fellow, who is often dressed in very bright colours, drives the many rats out of town by his playing of pipes. I don't think we know how many rats there were - maybe your story books have said how many. We also do not know how many people lived in the town.
<p>Suppose that there were 100 people and 100 rats. Supposing that all the people and rats have the usual number of legs, there will be 600 legs in the town belonging to People and Rats.</p>
<p>But now, what if you were only told that there were 600 legs belonging to people and rats but you did not know how many people/rats there were?</p>
<p>The first part of this month's challenge is to investigate how many people/rats there could be if the number of legs was 600. To start you off, it is not too hard to see that you could have 150 people and 75 rats; you could have had 250 people and 25 rats. See what other numbers you can come up with. Remember that you have to have 600 legs altogether and rats will have 4 legs and people will
have 2 legs.</p>
<p>I just chose 600 because that lets you have 100 people and 100 rats. You could now extend this idea by having 120 people and 120 rats (just as an example) using 720 legs and see what numbers you can come up with. Again, to start you off, you could have 270 people and 45 rats; or 320 people with 20 rats. Now you carry on.</p>
<p>You could choose any number you like for the total number of legs; try some out of your own. I had a hidden rule that whatever number I chose it meant that we could have the same number of people and rats. (600 legs 100 people and 100 rats; 720 legs 120 people and 120 rats.)</p>
<p>So you could choose the number of legs for yourself using that same rule. (Allowing you to have the same number of people and rats.)</p>
<p>It's probably time to have a look at all the results that you have got and see what things you notice. You can then tell me about them.</p>
<p>When you look at the examples that I did for you, you might notice:-</p>
<table>
<tbody>
<tr>
<td align="right">600 Legs</td>
<td align="left">a) 100 People 100 Rats</td>
</tr>
<tr>
<td> </td>
<td align="left">b) 150 People 75 Rats</td>
</tr>
<tr>
<td> </td>
<td align="left">c) 250 People 25 Rats</td>
<td> </td>
</tr>
<tr>
<td> </td>
</tr>
<tr>
<td> </td>
<td align="left">a) Gives equal numbers of People and Rats</td>
</tr>
<tr>
<td> </td>
<td align="left">b) Gives twice as many People as Rats</td>
</tr>
<tr>
<td> </td>
<td align="left">c) Gives ten times as many People as Rats.</td>
</tr>
</tbody>
</table>
<p>In the next example:-</p>
<table>
<tbody>
<tr>
<td align="right">720 Legs</td>
<td align="left">d) 120 People 120 Rats</td>
</tr>
<tr>
<td> </td>
<td align="left">e) 270 People 45 Rats</td>
</tr>
<tr>
<td> </td>
<td align="left">f) 320 People 20 Rats</td>
</tr>
<tr>
<td> </td>
</tr>
<tr>
<td> </td>
</tr>
<tr>
<td> </td>
<td align="left">d) Gives equal numbers of People and Rats</td>
</tr>
<tr>
<td> </td>
<td align="left">e) Gives six times as many People as Rats</td>
</tr>
<tr>
<td> </td>
<td align="left">f) Gives sixteen times as many People as Rats.</td>
</tr>
</tbody>
</table>
<p>This seems as if it could be worth looking at more deeply. I guess there are other things which will "pop up'' to explore.</p>
<p>Then there is the chance to put the usual question "I wonder what would happen if ...?''</p>
<p>Good luck, please send solutions in and any further ideas which came from asking "I wonder what would happen if ...?''</p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">
<p class="editorial">Thanks to all who sent in solutions. Here are some from Oakley School and St. George's school. This is what Emily sent us;</p>
<br/>
<br/>
If you TAKE 1 rat you have to ADD 2 people
<br/>
E.G
<br/>
2 people + 196 rats 98 people + 101 rats
<br/>
4 people + 195 rats 96 people + 102 rats
<br/>
6 people + 194 rats
<br/>
8 people + 193 rats
<br/>
10 people + 192 rats
<br/>
etc.
<br/>
Hope this helps!
<br/>
<br/>
<p class="editorial">Ben, Hannah & Scott sent in these thoughts;</p>
<br/>
We figured that if you add 1 rat and take 2 people. For example 108 rats and 84 people, 109 rats and 82
people and 110 rats and 80 people.
<br/>
<br/>
<p class="editorial">Otis, Jason & Sophis sent their solutions for 600 legs;</p>
<br/>
our solutions are:
<br/>
298 people, 1 rat
<br/>
298x2 = 596 legs
<br/>
1 rat = 4 legs
<br/>
298 people(596 legs) + 1 rat(4 legs)
<br/>
<br/>
150x2 = 300 legs
<br/>
75x4 = 300 legs
<br/>
150 people(300 legs) + 75 rats(300 legs)
<br/>
<br/>
</mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<h3><span style="font-weight: bold;">Why do this
problem?</span></h3>
<div>This <a href="http://nrich.maths.org/public/viewer.php?obj_id=1996&part=">activity,</a>
based on the well-known story, opens the door to a whole realm of
mathematical calculations that can be explored with or without a
spreadsheet. It also gives opportunities for pupils to create
further questions to answer.</div>
<br></br>
<h3>Possible approach</h3>
<div>If possible, it would be good to read a version of The Pied
Piper of Hamlin with the children so that they are familiar with
the story before starting this investigation.</div>
<br></br>
<div>On a second reading, you could use the story to talk about the
number of legs at particular times. You could also pose some
theoretical questions, such as asking the children to imagine
you've opened the book at a page which had 10 legs on it in total.
How many people and how many rats could there have been? Learners
could work on this in pairs using mini-whiteboards and then you can
talk about the possiblities as a whole group. This will lead into a
general chat about the number of animals/people and how the number
of each affects the other.</div>
<br></br>
<div>You might also want to spend some time sharing ways of
recording what the children are doing. Some might be drawing
pictures or symbols for the rats/people, others might be recording
numbers only. It is worth talking about the different ways and the
advantages/disadvantages of each. You may find that after some
discussion, a few children adopt a different way of recording to
the one they started with.</div>
<br></br>
<h3>Key questions</h3>
<div>How many legs do your rats have?</div>
<div>What could you replace a rat with?</div>
<div>Can you tell me about the way you are working out so many
possibilities?</div>
<br></br>
<div>(And for the pupils who have gone much further)</div>
<div>What have you noticed about all your results so far?</div>
<div>Can you explain why . . . . . has happened?</div>
<br></br>
<h3>Possible extension</h3>
<div>Look at animals with other numbers of legs and perhaps three
types of different-legged animals at the same time - eg. birds,
spiders and pigs.</div>
<br></br>
<h3>Possible support</h3>
<div>Some models,toys or pictures representing the different
animals may help some pupils to get started.</div>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">If you had one less rat, what could you replace it with to keep the
number of legs the same?<br></br>
How are you keeping track of what you have done?<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
Here is just a taste of the results with 720 qne 600 legs showing
the ratios of rats to humans nad humans to rats.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image width="372" height="423" src="Picture%201.jpg" alt="pic 1"></mdo:image></div>
<br></br>
Lara, John-Anthony, Harry and Richard of Loretto Junior say: If you
start with 100 rats and 100 humans and then do the following:
Humans 98 and rats 101 Humans 96 and rats 102 And so on -- then you
get a lot of combinations!! Yes - you're right, well done. That's a
great way to start. So, everytime you have one more rat, you need
to lose two humans so that the total number of legs stays at 600.
Is it possible to have more than 100 humans then and fewer than 100
rats? We'd love to hear from you if you've investigated this
problem further.Please don't worry that your solution is not
"complete" - we'd like to hear about anything you have tried.<br></br></mdoxml>
2
4
0
1
0
0
0
The Pied Piper of Hamelin
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
Using, Applying and Reasoning about Mathematics
Working systematically
Calculations and Numerical Methods
Multiplication & division
Using, Applying and Reasoning about Mathematics
Investigations
Calculations and Numerical Methods
Addition & subtraction