Long division is division of two multi-digit decimals. The algorithm for doing it with paper and pencil is now as obsolete as starting a fire with flint and iron. And mastering it does not contribute to one's understanding of many uses of the arithmetic operation of division. But the history of arithmetic is an important part of the history of thought. So some teachers may want to show their students how accountants were doing arithmetic on paper, using a steel pen and inkpot, during their daily work. The basic step of the long division algorithm is "short division", which is finding a one-digit quotient of two multi-digit numbers. For example, the first step in dividing 47 into 373 ..., is to find the biggest one-digit number d such that 47*d ≤ 373, and to compute the difference 373 - 47*d. Here, the correct answer is d = 7 and the written work looks like this:
3 7 3 ...
But if you made a wrong guess or chose d = 8 or 6, the work would look like this:3 2 9 4 4
Explaining arithmetic algorithms on counting boards is easier than explaining them in writing because changing configurations of counters is easier than editing written text. Example of 47 Divided Into 373 You need only two boards, one for recording the quotient, and one for holding the dividend, which is decreased until only a remainder is left. The divisor is not changed, so it can be written down or remembered during the computation. We assume that the short multiplication that is required is done mentally. Otherwise we would use one more board to compute the product. (Using red counters to represent negative digits is essential.) (1) Correct guess 7
The mult board (shown on the left) holds 373 and the quot board (shown on the right) holds 0. In the left image, 47*7 = 329 is subtracted from the mult board. The quot board holds 7. After regrouping is performed, the mult board holds 44 as shown in the right image.
The mult board holds 373 and the quot board holds 0. In the left image, 47*8 = 376 is subtracted from the mult board. The quot board holds 8. After regrouping is performed, the mult board holds -3 as shown in the right image. In the left image, 47 is added to the mult board and 1 is subtracted from the quot board. After regrouping is performed, the mult board holds 44 and the quot board holds 7 as shown in the right image.
The mult board holds 373 and the quot board holds 0. In the left image, 47*6 = 282 is subtracted from the mult board. The quot board holds 6. After regrouping is performed, the mult board holds 91 as shown in the right image. In the left image, 47 is subtracted from the mult board and 1 is added to the quot board. After regrouping is performed, the mult board holds 44 and the quot board holds 7 as shown in the right image. On a counting board, when you have made a wrong guess, you don't undo what you have already done. Instead, you correct the current result. Number Board index |