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 arenÕt any black rods. Next move to
the units squares where one red rod and one black rod may be removed.
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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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Example 2: 1,839- 2,853
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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
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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 cultureÕs
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
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,