Chinese Counting Board Activity

 

Use the Chinese Counting Board and red and black counting rods for the following computations. Turn the Chinese Counting Board so that it shows Thousands at the top left-hand corner and Units at the top right-hand corner.

 

Example 1: 331-203 (The diagram shows only the top 2rows of the counting board.)

 

Set up the counting board as shown in the diagram on the left, with red rods representing+331 in the first row and black rods representing -203 in the second row. The Chinese worked from left to right. Starting with the hundreds squares, two red rods and two black rods add to zero and are removed from the board. Next, none of the rods are removed in the tens squares since there aren’t any black rods. Next move to the units squares where one red rod and one black rod may be removed.

 

 

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¾

¾

¾

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¾

¾

 

 

 

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Since the answer rods all must lie in a single row, we need to borrow a red rod from the tens square and place its equivalent of 10 red rods in the units square. If were present 10 as the numeral for 8 plus 2 additional rods, we then can remove these two red rods together with the two remaining black rods, leaving the numeral for 8 in the units square. The result is a row of red rods representing the positive number 128.

 

 

 

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¾

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¾

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Example 2: 1,839- 2,853

 

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=

=

 

 

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   ¾

 

 

   ¾

 

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If we follow the steps described in Example 1, the result will be a row of black rods, representing the negative number -1014.

 

Now, use the methods illustrated in the examples above to compute the following sums and differences on the Chinese Counting Board.

 

1.   514+ 1040  (Since both numbers are positive, all rods may be placed in the same row.)

 

2.   3,752- 963

 

3.   561- 89

 

3.     2,777 - 5,050

 

4.     341 + 586 - 623

 

 

 

Chinese Counting Board

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

            Thousands

 

 

 

 

 

 

 

 

            Hundreds

 

 

 

 

 

 

 

 

           

            Tens

 

 

 

 

 

 

 

           

            Units

 

 

 

 

 

 

 

Use red rods for positive numbers and black rods for negative numbers.


Instructor Notes

 

Materials: Make copies of the instructions and of the Chinese Counting Board to distribute to students. Cut 40 1-inch by 1/4-inchstrips from red construction paper and the same number of strips from black construction paper to go with each counting board.

 

Objective: Students will learn an ancient culture’s method of representing numbers and of performing addition and subtraction with those numbers, thereby deepening their understanding of place value and of operations with integers, including negative integers.

 

How to Use: Review how to represent (large) numbers using the vertical and horizontal forms of the Shang numerals and work through Examples 1 and 2 from the instructions. Be sure to work through the details of these examples carefully.

Students may work in groups, pairs, or individually, depending on how much time you have to cut up strips of red and black paper for the counting rods. Red rods are used to represent positive numbers and black rods represent negative numbers. Point out to students that red for positive and black for negative is the same color scheme used for jumper cables for cars, but is the opposite of the scheme used in modern bookkeeping practice. Encourage students to practice carrying and borrowing with the Shang numerals as illustrated in the examples, rather than arriving at the answers using modern techniques.

 

Background: Shang numerals date back to the fourteenth century BCE Shang Dynasty. These numerals are formed by arranging counting sticks or rods within the squares of a counting board. The Shang numeral system is a base-10 place-value system. On the counting board, each square on the far right is a units place; next, going left, are the tens places, hundreds places, etc., until you run out of squares. There was not much need to count beyond 999,999 in ancient China. Therefore, most counting boards had at most 6 squares across. Because of limited space on an 8.5” x11” sheet of paper, the counting board with this activity is only 4squares across.

A counting board master was said to have performed computations in a flurry of waving arms as he quickly removed and replaced counting rods. It was like a dance to numbers.

 

Sources: Activity from The Story of Negative Numbers, by J. Beery, G. Cochell, C. Dolezal, A. Sauk, and L. Shuey, in Historical Modules for the Teaching and Learning of Secondary Mathematics, Mathematical Association of America, Washington, D.C., 2003.

 

Cooke, Roger, The History of Mathematics. Wiley, New York, 1997, pages 223-225.

 

Katz, Victor J.,A History of Mathematics: An Introduction, Addison-Wesley, Reading, Massachusetts, 1998, page 7.

 

Li Yan and Du Shiran, Chinese Mathematics, Clarendon Press, Oxford, 1987, pages 6-11.

 

 

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