Negative Digits in Multiplication
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The difficulty of short multiplication that is being carried out on a counting board or with pencil and paper depends on both the one-digit multiplier and the multi-digit multiplicand. For example, it is much easier to compute 6*203 than 6*897. A very easy way to simplify difficult cases is to use positive and negative digits in writing, and also to represent positive numbers on the counting board by mixing white and red tokens. Negative digits are written by putting the minus sign above the decimal digit, 1, ... , 9. There is no negative 0, because the number 0 is neither negative nor positive. An Example 897 = 1 1 0 3, can be read, “one thousand minus one hundred three”. 
The left board holds 897. The right board holds 1 1 0 3.
Students should learn to change representations of numbers between these representations mentally. For example: 897 → 1 2 9 7 → 1 1 1 7 → 1 1 0 3. After they learn this technique, students should decide for themselves which digit(s) to change and which to leave unchanged, because that would depend on their skills in mental computation. Additional Examples Note that in each example, it is more simple to use negative digits than only white tokens.
Example
3*779 Using Only White Tokens 
3*M: 2*M → P; P+M → P The top board holds 779 in each image. In the leftmost image, the bottom board holds 2*779 = 1558. In the center image, the bottom board holds 1558+779=2337. In the rightmost image, the bottom board holds 2337 after regrouping. Using Negative Digits 
3*M: 2*M → P; P+M → P The top board holds 779 = 1 2 2 1 in each image. In the leftmost image, the bottom board holds 2 * 1 2 2 1. In the center image, 1 2 2 1 is added to the bottom board. In the rightmost image, the bottom board holds 2337 after regrouping.
Example
2*867 Using Only White Tokens 
The top board holds 867 in each image. In the leftmost image, the bottom board holds 2*867 = 1734. In the rightmost image, the bottom board holds 1734 after regrouping. Using Negative Digits 
The top board holds 867 = 1 1 3 3 in each image. In the leftmost image, the bottom board holds 2 * 1 1 3 3. In the rightmost image, the bottom board holds 1734 after regrouping.
Example
4*958 Using Only White Tokens 
4*M: 2*M → P; 2*P → P In the leftmost image, the top board holds 958 and the bottom board holds 2*958 = 1916. In the center image, the top board holds 1916 and the bottom board holds 2*1916 = 3832. In the rightmost image, the top board remains the same and the bottom board holds 3832 after regrouping. Using Negative Digits 
4*M: 2*M → P; 2*P → P In the leftmost image, the top board holds 958 = 1 0 4 2 and the bottom board holds 2 * 1 0 4 2. In the center image, the top board holds 2 * 1 0 4 2 and the bottom board holds 2 * the top board. In the rightmost image, the top board remains the same and the bottom board holds 3832 after regrouping.
Example
3*692 Using Only White Tokens 
3*M: 2*M → P; P+M → P The top board holds 692 in each image. In the leftmost image, the bottom board holds 2*692 = 1384. In the center image, the bottom board holds 1384+692=2076. In the rightmost image, the bottom board holds 2076 after regrouping. Using Negative Digits 
3*M: 2*M → P; P+M → P The top board holds 692 = 1 3 0 8 in each image. In the leftmost image, the bottom board holds 2 * 1 3 0 8. In the center image, 1 3 0 8 is added to the bottom board. In the rightmost image, the bottom board holds 2076 after regrouping. Number Board index |