Multiplication on Counting Boards (Part 2)

Children can learn multiplication just by using numbers that have only a few digits. But teachers should know how the method that children use works for numbers of any reasonable length.

Here is a verbal description of the method, illustrated by a small example.

Let’s assume that we want to compute 4797 * 137672.

1. The Setting For the Boards
We need to have three counting boards that we call M for "Multiplicand", P for "Product", and S for "Sum".
The M-board should have at least 6 columns because our multiplier has 6 digits.
The P-board needs at least 7 columns because it will hold, one at a time, the product of the multiplicand and one of the digits of the multiplier.
The S-board needs to have at least 10 columns, which is the sum of the lengths of the multiplier and the multiplicand, 4 + 6.

The recommended arrangement of the multiplier m = 4797, which the M-board holding 137672, and P and S, is alternatively

m	M	S	and	m	M	S
	P					P

The reason for this is that the value of the product P is computed when P is aligned with the multiplier M; and to add it to S, we need to align p with S.

2. The General Plan of Action
You can process the multiplier from left to right or from right to left, or in any other order.
Here we go from left to right.

Compute P = 4*M and transform it to S = 1000P

In the leftmost image, 4*M is computed on the P-board. In the rightmost image, the P-board is shifted to the appropriate position, and 4*M is copied onto the S-board. The S-board holds 550,688,000.

Compute P = 7*M and add it to S + 100P → S

In the leftmost image, 7*M is computed on the P-board. In the rightmost image, the P-board is shifted to the appropriate position and 7*M is added to the S-board. The S-board holds 647,058,400.

Because P = 7*M is already computed, add it now to S + P → S

In the image, the P-board is shifted to the appropriate position and 7*M is added to the S-board. The S-board holds 648,022,104.

Finally, compute P = 9*M and add S + 10P → S

In the leftmost image, 9*M is computed on the P-board. In the rightmost image, the P-board is shifted to the appropriate position, and 9*M is added to the S-board. The S-board holds 660,412,584 which shows that the product of the multiplication 4797 * 137672 = 660,412,584.

Remark
The number of 0's in front of P indicate how much to the left P is shifted from the rightmost position under S.

3. Multiplication of M by One-Digit Numbers
Multiplication of a multi-digit number by a one-digit number was called "short multiplication" and was taught as follows.

- Students had to memorize all multiplication "facts" up to 9 times 9.
- They scanned the multiplicand from right to left, multiplying one-digit numbers, and mentally adding the carry to the next product.

Any student who mastered this technique can use it here, but there are better methods.

(1) The only non-zero digits of values of locations on M boards are 1, 2, 3, 4, and 5. Therefore, the only multiplication facts needed are those where at least one digit is at most 5.
(2) Multiplication by 9 reduces to one subtraction, 9 = 10 - 1. Multiplication by 5 reduces to the operation of "halving", and multiplication by 2 is "doubling". And these two operations are even simpler than addition and easy to learn.
(3) Another simplification can be made when using negative digits in the multiplier. For example,
_ _ _ _ _
4797 = 4 8 0 3 = 5 2 0 3, where, 5 2 0 3 = 1000*5 - 100*2 - 3

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