An array arranges things in neat rows and columns. The moment things line up in a tidy rectangle, you can read off a multiplication at a glance: count the rows, count the columns, and multiply.

So an array with a rows of b dots holds a \times b dots in all — no counting one by one.

See it built

Watch an a \times b rectangle fill in, square by square. First the rows, then the columns — and the total is just rows times columns. Step through it.

Here is the lovely part. Count the same rectangle by rows and you get a rows of b, which is a \times b. Count it by columns and you get b columns of a, which is b \times a. It is the very same rectangle, so the total cannot change:

That is a picture of why the order of a multiplication never matters — turn the array on its side and the squares are all still there.

This way of seeing a product as the area of a rectangle is called the area model, and it is the trick that will later let us multiply much bigger numbers by chopping a big rectangle into easy pieces.

See it explained

Khan Academy builds multiplication up from area models: