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Another approach: http://en.wikipedia.org/wiki/Grid_method_multiplication

Of course it's all multiplication, and all essentially the same thing. I find the grid method the least notation-focused of at least these three, and so I think more accessible to assimilation. Maybe I'm biased, I didn't learn multiplication this way but found myself naturally doing it like this in my head and then was pleasantly surprised to find it had a name. It's also amenable to estimation.

Thanks for the link. The grid is also helpful as a way to visualize things like:

  (1 + a)^2 = 1 + 2a + a^2
There is a square with side 1+a, and the grid consists of four pieces, sizes 1, a, a, and a^2.

Thinking about approximations when a << 1, you can motivate why

  (1 + a)^2 ~= 1 + 2a 
really easily by imagining the grid you refer to.

In other words, keeping this grid picture in mind can be helpful to more mature mathematicians/physicists/engineers...probably more useful than the standard multiplication algorithm is.

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