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I have a small child and I am always trying to find the best way to explain new things to her. The area of a square is a very basic concept that is related to the idea of area unity and can be used to explain a lot of facts in basic geometry. Is there a good way to explain this to a small kid? I really dont remember the first time that I learned that. However, the things in math become very easy if you get correctly the flavor of important concepts.

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Try dividing a line into equal halves. Ask the kid to count the number of parts. Now, divide a square into smaller squares of side length half the original one. Ask again to count the number of squares that you obtain. Do this for three equal divisions. Four ... – Singhal Feb 23 '14 at 12:02
Does the child already understand multiplication? If not, then it is probably not yet time to introduce $a^2$. If so, how did you introduce multiplication, if not with rectangular grids of dots or other objects? To talk about area, just tile your rectangles with unit square tiles (and only use integral sides!). – Matt Feb 23 '14 at 12:03
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If $a=1$, then there's nothing to prove. The square with side length $1$ just happens to be what we choose to measure areas in.

For an integral $a>1$, you can divide the large square into unit squares, and if the child understands multiplication at all (otherwise all is lost), it should be clear that the area of the large square is $a\cdot a$.

If $a$ is rational but not an integer, it gets more difficult. The child will have to understand fractions, of course, because that's what the area is going to be. I suppose it might be possible to get through with an argument that sets $a=p/q$ and then divides the square into $p^2$ little squares of side length $1/q$, and then asks how many of those little squares make up an unit square. Then the total area follows by division and all you have to explain then is that $(p/q)^2=p^2/q^2$. That will be something of a trip, I suppose.

If $a$ is not even rational, we get into real analysis, and there probably won't be anything to do but simply assert the result as a definition -- unless the child in question happens to be a complete prodigy.

If $a$ is a measured decimal fraction, then the way forward is probably to view it as an abbreviation of a fraction with a power of 10 in the denominator and proceed like for rationals. The $(p/q)^2=p^2/q^2$ step might be easier to motivate here if you can appeal to moving-the-decimal-point.

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Area might better be introduced with rectangles and $a \times b$ before you move on to squares, especially if you start with $2 \times b$.

I would suggest wooden cubes: the ones with letters of the alphabet are probably ideal.

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Your question suggests she understands not just multiplication but exponents. If that's not the case, you might want to start with the former. (The latter you can think of as a special case of the former.) In that light, one way to teach multiplication — though I imagine the multiplication table should come at least as early as this — that I think looks rather intuitive is to imagine a bird's-eye view of a case of bottles. One side will have $m$ bottles; the other, $n$. We can count up all the bottles in the case to get $mn$. I remember a few years back hearing about this on an episode of "More or Less", a radio show hosted by Tim Harford of the Financial Times; I'm having trouble finding it now, but if you can find it, the show does a better job of explaining it.

Edit: Found it: It's the first segment in the show.

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