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)
= 2x1x10^{2} + [2x2x10
+ 3x1x10] + 3x2 =
276
The diagrams display actually this multiplication visually.
The method can be generalized to products of 3digit numbers
(or even more) using more sets of parallel lines. It can
also be generalized to products of 3numbers using cubes
of lines rather than squares.
