What this experiment is really all about is area, and showing that it's always the same and a useful tool that helps us work our way out of variuos paradoxes.

I start by arranging the two sets in a “paradox pair” to grab attention and ask them what is going on. They tend to suggest that one piece is missing or that the pieces are different. Ask them to rearrange them to make them into a different shape, or to make the two sets identical - just let them have a play around with the pieces. You could try challenging them to recreate various shapes, or seeing if they can reform it into a perfect square. Much of this experiment is less about you explaining things and instead them exploring them physically, getting a sense of shape and area.

The real fun though is showing them some of the "paradoxes" that can be made with tangrams. A couple of these are shown in the images, but feel free to make up your own. Most of the paradoxes stem from the fact that some of the side lengths almost match up, but not quite, as well as some liberal use of empty space between pieces. The two squares paradox shows this the most strongly, where all the small triangles are rotated by 45 degrees, and therefore their side length changes (as the length of the longest side is root two the other sides) and thus there's very little chance of it all fitting together perfectly. Thus some gaps are left whilst other pieces are squished together, which can lead to the whole thing looking very different.

However we might rearrange the pieces though, the area will always remain the same, showing that we haven't actually lost any area, just given the illusion it's gone by rearranging it (and introducing some extra space between pieces). This is perhaps best shown in the fat monk paradox where he seems to literally lose a foot (however it takes a lot of table space to set it up, maybe use the floor?).

Another very effective one is two “squares” where one is missing a corner. With this one, you can start by asking them about what happened to the corner: is a piece missing? Are the pieces different. Try to get them to rearrange one of the shapes into the other (e.g. both perfect squares). They'll see that they're identical and at this point they're usually really interested. You can ask them what happened to the area, or where the corner went, or what shape the irregular polygon is (some realise that even with the corner it wouldn't be a square). Then, to make it clear, put the two shapes on top of one another. This works best if the square is yellow and the hexagon is orange, so you can see the orange has a longer side and the yellow fills some of the missing corner.

You can explain briefly how to add up the area of the tiles. Both using formulas for area of common shapes (n.b. the trapezium is really just a rectangle with a triangle cut off one end and moved to the other) and by counting the number of triangles that add up to each shapes total area. The second may be easier with younger kids.

You could think about discussing tessellation with them, how repeating patterns of shapes that fit together can be extended to infinity.

As an aside, there is also a kind of an illusion at play here, where the missing corner in the square appears bigger than the corner that is actually “missing” from the square.

Real world applications:

I'm a bit stumped here... Of course there's lot's of discussion to be had about geometry and areas, and even some stuff about tiling (could expand with some Penrose tiles or even some Islamic art) but as for where you see this in real life I'll leave you to think of your own examples. And then tell us because clearly we need some help...