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"The Pied Piper of Hamelin'' is a story you may have heard or read. This man, who is often dressed in very bright colours, drives the many rats out of town by his pipe playing - and the children follow his tune.
Suppose that there were $100$ children and $100$ rats. Supposing they all have the usual number of legs, there will be $600$ legs in the town belonging to people and rats.
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 children/rats there were?
The challenge is to investigate how many children/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 $100$ children and $100$ rats; or you could have had $250$ children 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 children will have $2$ legs.
When it's time to have a look at all the results that you have got and see what things you notice you might write something like this:
a) $100$ Children and $100$ Rats - the same number of both,
b) $150$ Children and $75$ Rats - twice as many Children as rats,
c) $250$ Children and $25$ Rats - ten times as many Children as Rats.
This seems as if it could be worth looking at more deeply. I guess there are other things which will "pop up'', to explore.
Then there is the chance to put the usual question "I wonder what would happen if ...?''
How many legs do your rats have?
Setting different target numbers of legs offers the chance to explore multiples of 2 and 4 and how they are related. Each target number will have a range of possible solutions. Encourage the children to generalise about how the numbers of rats and people are related.
Another avenue for extension woul be to 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. This option links with Noah.
Some children may find the large numbers being considered in the presentation of the problem too high to make sense of so start them off with lower targets such as 20 or 30 legs. Noah is a similar problem involving fewer legs. Some toys or pictures representing the different animals may help some pupils to get started. Modelling clay bodies with straw legs can also be very helpful. Children could be given 20 lengths of straw and work on sharing them between people and rats as a way in to dealing with the larger numbers in a more abstract way.
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
This activity investigates how you might make squares and pentominoes from Polydron.
If you had 36 cubes, what different cuboids could you make?