This Pied Piper of Hamelin

'This Pied Piper of Hamelin' printed from https://nrich.maths.org/

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This Pied Piper of Hamelin

"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 ...?''

Why do this problem?

This activity, based on the well-known story, opens the door to opportunities for doing mathematical calculations that can be explored with or without a spreadsheet. The story scenario is motivating and gives the children a meaningful context in which to make sense of these calculations. It can be extended by allowing pupils to create further questions to answer.

Possible approach

Reading a version of The Pied Piper of Hamlin with the children so that they are familiar with the story before starting this investigation is a good way to start.

Then 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 general conversations about the number of animals/people and how the number of each affects the other.

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.

Key questions

How many legs do your rats have?

What could you replace a rat with?

Can you tell me about the way you are working out so many possibilities?

(And for the pupils who have gone much further)

What have you noticed about all your results so far?

Can you explain why . . . . . has happened?

Possible extension

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.

Possible support

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.