Water Displacement Observation

Many of our best learning moments start with simple conversations.

Today during lunch, N was (per usual) not focusing on his food. Instead of eating, he was dipping his hand in and out of his water cup. As I started explaining explain that he was making his water dirty, he interrupted me.

“Mommy, look! When I put my hand in the cup, the water gets higher!” His look of pure astonishment caught me off guard. I stopped, mid-lecture, and thought about it. It was hard for me to think back to a time where these kinds of simple observations were new and exciting … but then I remembered the story of Archimedes in the bathtub and realized that even the simplest observations can have profound implications.

Archimedes’ discovery

I told N a little story about an ancient Greek mathematician. (I had to do a quick Google search to frame the basic story line ;). The following is my own version of events based loosely on actual historical accounts.)

Once upon a time, in a faraway land named Syracuse, a King wanted a crown. He found the best crown-maker in all the land and gave him a large piece of gold. The King asked him to make a brilliant crown with the gold.

The crown-maker worked hard for a long time, carefully crafting a beautiful crown from the lump of gold.

Finally, his hard work was finished. He presented the crown to the King.

The King held the crown in his hands, turning it this way and that to admire all the intricate details. He loved the way that it sparkled in the sun.

The King weighed the crown, just to make sure that it was the same weight as the piece of gold he had given the crown-maker. He was pleased to see that it was the correct weight, so he thanked the crown-maker, paid him many monies, and proudly wore the crown.

Later that day, however, the King began to wonder …

He wondered if maybe … just maybe … he had been tricked.

Had the crown-maker truly used all the gold that the King had given him? Or had he used something else instead of pure gold, perhaps some metals that were not quite as precious, and stolen the extra gold for himself?

Once the King started wondering, he couldn’t stop wondering. He thought about it night and day, but he didn’t know what to do.

Eventually the King put out a royal decree.

The Mystery of the Golden Crown

“If anyone can determine for certain that this crown is made of pure gold and is made from exactly the same volume of gold that I gave to the goldsmith — without harming the crown in any way — he shall receive a royal reward.”

Now, there was a very smart man, named Archimedes, who heard the decree. He was a mathematician and he loved a good challenge. Once Archimedes started wondering about this mystery, he too couldn’t stop wondering. He thought about it night and day, and he was determined to solve the mystery.

One day, as he was taking a bath, he suddenly noticed something. When he sat down in the tub, the water level got higher!

“Eureka!” he shouted, (which means, “I found it!”) jumping out of the bathtub and running naked through the streets. He was so excited about his discovery that he wanted to tell everybody right away — and he even forgot to stop and get dressed first!

Why was this so exciting?

Archimedes had just realized a very basic and important scientific principle.

{“Mommy, what is a scientific principle?”

“It’s something that we think through very carefully until we truly understand it — and then we use that idea in the future to help us understand lots of more complicated things.”

When he sat down in the water, all the space that his body took up forced the water to move out of its place to make way for him — the water had nowhere else to go but up, so the water level got higher and splashed out over the edge of the bathtub. The amount of water that moved out of the way — or, got displaced — was exactly the same as the amount of his body that was under the water.

Archimedes realized that he could measure the volume (size) of anything by placing it in water and measuring the amount of water that got displaced.

This was the missing piece to the puzzle. Archimedes went to the King (after he had calmed down a bit and gotten dressed) and asked him for the crown and a piece of gold that was the same size he had given to the crown-maker.

The crown should have been the same weight and size as the original piece of gold, if the crown-maker had used every bit of gold the King had provided … but size was a tricky thing to measure since the crown had been formed into a new shape. If the crown had been in the shape of a box (a rectangular prism, to be precise), he could have easily measured the volume (length times width times height) and compared it to the original piece of gold. But the crown was a delicate and intricate wreath-shaped piece of art, and it was difficult to say whether it was exactly the same volume as the original lump of gold. Now, however, Archimedes knew that he could test it by putting it under water.

If both the crown and a piece of gold (that was the same size as the original piece of gold the King gave to the crown-maker) displaced the same amount of water, then Archimedes would know that the crown was made from that exact amount of pure gold.

But … it turned out that the King’s suspicions were correct — he had indeed been tricked. The crown weighed the same amount as the original amount of gold, but, when put under the water, it displaced a larger amount of water than the piece of gold did. This meant that the crown-maker had mixed in some other metals to make the crown as heavy as gold (to try to trick the King), but this resulted in a crown that was slightly larger than the original lump of gold — and the crown-maker had stolen the extra, leftover gold for himself.

Archimedes received a royal reward. The crown-maker … probably went out of business. (Unless the bad publicity worked to his advantage and got him a load of side hustles. History doesn’t tell us much about that part, so we’ll never know.)

Our volume experiment

After the impromptu story I gave N a plum and an expo marker. We marked the level of water in his cup, then put the plum in and marked the new level.

I explained that the plum moved the water out of the way, and we could find out the size (volume) of the plum by measuring how much the water raised.

He was not as impressed by this as I had hoped. We cleaned up lunch and I put the boys down for a nap while I thought about this. From a preschooler’s perspective, it was interesting that an object displaces water, but the novelty of this observation soon wore off and left him with no relevant conclusion.

I knew I couldn’t stretch him so far as to try to explain how to use the mathematical formulas to compare densities of different objects, but I did want to extend this learning moment a little bit to give him something more to think about.

While the boys were napping, I pulled together some ideas and supplies. I actually ran to get the boys, shortly after putting them down to bed, because I wanted to try the experiments right away. (Eureka! Wake up! Science experiment In The HOUSE!)

This post would get way too long, though, if I wrote the whole thing here, so I’ve written a separate post detailing our experiment and lessons learned. That post also includes a free printable with the story of Archimedes and instructions for a simple science experiment.

Learning Resources for Teachers, Tutors, and Parents

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