Eureka!

To celebrate a famous mathematician and the thrill of problem-solving.

by The Revd Alan M. Barker

Suitable for Key Stage 2

Aims

To celebrate a famous mathematician and the thrill of problem-solving.

Preparation and materials

• You will need a straight-sided Pyrex pie-dish, filled with water and standing on a baking tray.
  Two identical tall glass tumblers.
  Some bottles of tap water for topping up.
  A bucket for waste water.
  Put all this on a level waterproof surface, well clear of electrical apparatus!

• One heavy-duty 6-inch rubber ring – obtainable from all good pet shops! (school quoits will probably float).

• A large flat pebble, somewhat less in volume.

• 500g packets of plain flour and cornflour or other suitable products.

• You will need to rehearse this presentation but, when done with enthusiasm, it is worth the trouble.

Assembly

1. Begin with enthusiasm: Today we are going to celebrate the work of a great mathematician, the fun of asking questions, and the thrill of problem-solving!
   
   Ask: Where do you get your best ideas? Perhaps while thinking in bed … or even in the bath!
   
   Does anyone count the pattern on the curtains or the tiles on the bathroom wall? And do everyday things sometimes lead you to ask the question ‘Why?’?
   
   For instance, why does soap get lost under the water, while a bath duck bobs on top? Why does a small coin sink, but a huge ocean liner float?
2. A famous mathematician called Archimedes can help us with such questions. He was born in Ancient Greece in 287 BC. Archimedes was a great thinker, and a brilliant problem-solver, and because of this the king often turned to him for help.
3. One day the king called for Archimedes and showed him a new crown. The king had instructed that it should be made of pure gold, but he suspected that the goldsmith had dishonestly kept some of the gold for himself, and added silver instead. (The crown would look the same, but silver was a lighter metal that didn’t cost nearly as much.) Had the king been tricked?
   
   He asked whether Archimedes could find out whether the crown really was made of pure gold and nothing else. He also instructed that the crown should not be damaged in any way!
4. Archimedes thought long and hard about this problem. The crown was a complicated shape – round and pointed – and seemed impossible to measure. Archimedes thought so long and hard that nothing else seemed to matter. He was such a deep thinker that sometimes he missed meals and forgot to change his clothes! Eventually his servants persuaded him to relax at the public baths. Archimedes lay in the hot water with the problem still going round and round in his mind. He looked at the puddles of water all around him on the floor. The bath had been filled to the brim and, as he had stepped into it and sat down, the water had spilled over the edge.
5. That was the answer! Suddenly, Archimedes could see the solution to his problem. ‘Eureka!’ he shouted (which is Greek for ‘I’ve found it!’ – invite everyone to repeat the word). Archimedes leapt out of the bath and without even bothering to get dressed ran home to try out his idea.
   
   ‘Eureka! Eureka!’ he cried. He was so excited his servants heard him coming. ‘What’s Archimedes discovered now?’ they asked. ‘And where are his clothes?’
6. What Archimedes had realized was this: different substances of the same weight (mass) take up a different space, or volume. Compare a 500g packet of ordinary flour with 500g of lighter cornflour – both look very similar but are not exactly the same size. Archimedes realized that he could compare the volume of the crown (how big is it?) with the volume of the same weight of pure gold.
7. Sitting in his bath, Archimedes had thought of the way to do this. Invite everyone to watch a re-enactment of what Archimedes did.
   
   First, Archimedes filled a bowl of water to the brim and carefully lowered the crown into it – just as he had got into the bath. He measured the volume of water that overflowed. He then took exactly the same weight of pure gold and did the same. The volume of water was slightly different: that meant that the metal in the crown was not the same as pure gold. It had been ‘filled out’ with silver. Archimedes’ experiment showed that the king had been cheated.
   
   As you speak, enact Archimedes’ experiment. Place the ‘crown’ (the ring/quoit) into a brimming bowl of water standing on the tray. Carefully lift the bowl to one side and decant the displaced water into the first tumbler. Replace and refill the bowl and repeat the procedure using ‘pure gold’ – the pebble). Compare the levels in the two tumblers.
8. Reassure any children who might have found the story difficult to follow. You do need to think about it! Most important, the story tells us that new ideas can be exciting and fun.
   
   As a mathematician and scientist, Archimedes carefully observed the world around him and thought about what he saw. He also knew how vital it is to keep on trying to solve a difficult problem. He didn’t give up and eventually everything became clear. That was the thrilling ‘eureka’ moment!’
   
   Conclude with this advice: So don’t ever become discouraged if you find a maths problem hard to understand. The moment will come when you eventually exclaim (all together) ‘Eureka!’ –‘I’ve got it!’  
9. Observe that Archimedes had many other exciting ideas. The mathematical discoveries he made so long ago are still used by engineers today as they design boats and buildings, tools and machinery.

Time for reflection

In this prayer, invite everyone to exclaim ‘Eureka!’

Lord God,

Thank you for the order of your creation,

and for our ability to measure and reason things out.

Give us enquiring minds

and help us to work patiently

towards the thrill of fresh discoveries

when we too can shout: ‘Eureka!’

‘I’ve found it!’

Amen.

Song/music

‘It’s a new day there’s hope’, verses 1–3 (Sing to God, 106)