Here is an interesting method to visualize multiplication that reduces it to simple counting!

Draw sets of parallel lines representing each digit of the first number to be multiplied (the multiplicand, see figs. 1 and 2 further below).
Draw sets of parallels, perpendicular to the first sets of parallels, corresponding to each digit of the second number (the multiplier).
Put dots where each line crosses another line.
On the left corner, put a curved line through the wide spot with no points. Do the same with the right.
Count the points in the right corner.
Count the points in the middle.
Count the ones in the left corner.
If the number on the right is greater than 9, carry and add the number in the tens place to the number in the middle (see fig. 2). If the number in the middle is greater than 9, do the same thing except add it to the number from the left corner.
Write all those numbers down in that order and you will have your answer (see products in figs. 1 and 2).

This visual method is very valuable to teach the basis of multiplication to children. However, it isn’t very useful when handling large numbers.

The Math Behind the Fact: The Distributivity of Multiplication
The method works because the number of parallel lines are like decimal placeholders and the number of dots at each intersection is a product of the number of lines. You are then summing up all the products that are coefficients of the same power of 10. Thus in the example shown in fig. 1:
23 x 12 = (2x10 + 3)(1x10 + 2) = 2x1x10² + [2x2x10 + 3x1x10] + 3x2 = 276
The diagrams display actually this multiplication visually. The method can be generalized to products of 3-digit numbers (or even more) using more sets of parallel lines. It can also be generalized to products of 3-numbers using cubes of lines rather than squares.

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