The first mathematician study we covered was Archimedes. Archimedes was a well known mathematician and inventor, alive during 287-212 BC in Ancient Greece. Well known for the Archimedes’ Principle and the Archimedes’ Screw, he was most interested in catoptrics, mechanics and pure maths.

## Archimedes Resources

First, we learnt all about Archimedes by first reading the chapter about his life in the following book:

From there we watched a couple of videos on YouTube:

Archimedes was a great mathematician of his time and is well-known for many discoveries and theories. There were two things I wanted to focus on this month:

- Measuring something of a huge number
- Archimedes’ Law of Buoyancy

### Measuring Large Quantities

Archimedes set himself the (not very easy) goal of figuring out how much sand could fit into the universe. I wanted to use the very general principle that if one knows how much of a certain substance fits into a small container one can estimate fairly accurately how much of the same substance will fit into a much larger container. We used something accessible and attractive to children. Chocolate. I filled a one cup measuring bowl full of tiny M’n’Ms:

The girls measured out one tablespoon of the same sweets:

A tablespoon contained ten sweets:

I asked the girls how we could work out how many the cup held now we know how many one tablespoon holds. I was blown away by my six-year-old when she answered completely accurately that we needed to find out how many tablespoons the cup held! So they went about measuring:

They found that the cup held thirteen tablespoons of sweets plus five left over. From this they were able to calculate that the cup would hold approximately 135 sweets. They counted and you’ll never guess how many it held? 133!! This was really very cool for all of us to see 🙂

### Archimedes’ Law of Buoyancy

Oh. My. Goodness. It has taken me three weeks of reading and rereading to get my head around this, and it is sooo simple! I am wondering if I may be intellectually challenged….Hmmm.

Okay, so I wanted to teach the girls about Archimedes’ Law of Buoyancy. They already knew the story of ol’ Archimedes running through the streets naked having figured out that the body displaces the same volume of water as the volume it takes up. That part is simple, not just because the very idea of a man running naked through the streets is memorable, but also we have probably all had the experience of running too much water in the bath, getting in and a whole heap of water spilling over the sides….or maybe it is just my very own elephantine bottom which causes bathroom tidal waves 🙂 Whatever the case, I always understood the volume displacement. That makes sense. But the reason why certain things floated and others didn’t – that I struggled a bit with.

#### So what types of things float, and what types of things sink?

I asked the girls, and they answered as most young children would – heavy items sink and light items float. So I had them experiment to prove or disprove their theory.

I had the girls experiment to their heart’s content, filling up a bowl and letting them chuck in all manner of things, large and small, heavy and light. They filled in their sheet as they went along:

Between us we figured out that * in general* heavy sank and light floated. But was that always the case? We used some simple balance scales to see which was heavier – the lens cap (which had floated) or the plastic Playmobil helmet (which had sunk):

The lens cap was heavier. Oh. So I asked them to pick ten items, which they weighed, and filled in the table I had made up for them:

Oh dear. We had just disproved our own theory.

#### Learning about Density from Marsh Mallows

We did a second experiment which, having filled up my camera’s storage space, I deleted by mistake to make room for more photos. This was a simple experiment which tested a normal (and therefore air-filled) marshmallow against a kneaded (and therefore airless) marshmallow:

#### Learning more about Density from Raisins

We replicated this experiment using raisins. This is SUCH a great visual of what density is actually all about. We took two glasses. In one we placed water, in the other we placed fizzy water. We added five raisins to each and watched. All the raisins sank straight away:

But then, after a few moments, the raisins in the fizzy water began to rise to the top. It was like they had armbands on. And in a way they did. The air bubbles in the fizzy water attached themselves to the raisins and rose them up:

The girls had no problem explaining that it was the air bubbles that was making the raisins rise up:

And they concluded quite nicely that the more air something has, the more likely it is to float:

Cooooel!

#### Experimenting with Buoyancy using Aluminium foil

In the last experiment, we used aluminium foil to demonstrate that it was density rather than weight which predicted whether something would sink or float. For this experiment we gathered three sheets of aluminium which were exactly the same size and therefore exactly the same weight. The first we scrumpled up lightly, the second a little more and the third we took a hammer to, banging it into a tight ball:

The balls were now no longer the same size, although their weight remained the same. We chatted about which one might float and which might sink:

It was unanimous! Both girls thought the smaller ball would sink due to the fact that they had knocked all the air out of it 🙂

And, of course, they were right:

This was great demonstration that density is related to how close together the material is packed, or how much air the material contains. I extended it slightly by explaining that Archimedes had discovered that an item, be it a ball or a huge cargo ship, will only float if the *weight* of the water displaced by the item is greater than the weight of the *submerged* part of the item.

So weight does have something to do with it, but it is the relationship between the weight of the displaced water in comparison to the material doing the displacing (hence the reason that boats are filled on top rather than down below and why usually the hulls of boats are filled with air). So long as the hull is lighter than the water it displaces the boat will float.

But what happens if you change the density of the water? If we made the water increase in density, would something that had previously sunk begin to float? Well, that’s what we did next.

We collected two small bowls filled with equal amounts of water. We dissolved as much salt as we could in one of them and left the other as plain water. We used two Playmobil bits – a helmet and a jug. First we placed both in the bowl containing the plain water:

They both sank. We then dried them off and placed them in the bowl which had the salty water. They both floated!

This shows that both the density of the liquid and the density of the object will affect whether the object will float or sink.

We have had great fun with our maths over the past couple of months, making it A LOT more hands on. I have not had even one complaint, and I think they are learning heaps 🙂

For more of our living maths this term see the posts below (we have done all this concurrently):