I did a couple of worksheets on Multiplication, to test out the GridMaths tools.

My feeling is that multiplication should be taught with Rectangles.  So once students are familiar with counting in groups, they can play around with something like this interactive demo to get a basic feel for the distributive rule.

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[ The above animation is a custom interactive widget created in javascript, using Raphael and HTML5 divs for the draggable UI.  You drag the red dots and the diagram resizes and recalculates in realtime.  Other screenshots in this post are all done using MathsGrid itself ]

The distributive rule says we can divide up A and divide up B any way we like, and the product is just all the pieces of A times all the pieces of B, summed up together.

For humans with 10 fingers, its easy to break A and B up by 1s,10s,100s, which means we need to multiply each digit of A by each digit of B, then sum them up.

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So long multiplication kind of falls into place, as a way of keeping the pieces straight :

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Then the Lattice method can be seen as just another convenient system for keeping all the pieces organised and lining up the 1s,10s,100s etc :

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The first time I saw this Lattice method on Hacker News it looked like Magic.. but after you do one you realize it really is the same thing, just a cool way of keeping track of the pieces.