The abacus is an ancient device used in China and other Asian countries. It is a counting device and can be used to add, subtract, multiply, and divide. There are many types of abaci, but they all work on the same principle. A simple Chinese abacus consists of 13 columns of beads. A horizontal beam separates the frame into two sections. These are called the upper deck and the lower deck. Each bead in the upper deck has a value of FIVE. Each bead in the lower deck has a value of ONE. When you display a number, beads are moved towards the horizontal beam that separates the two decks. The first column on the right is the ONES column and represents numbers 0 to 9. The next column to the left is the TENS column and represents 10 90. The third column is the HUNDREDS column. The place value continues to the last column that represents trillions.

Three of the most used abaci are from China and Japan. They differ in the number of beads that are in the lower deck. The Chinese abacus before 1850 had five beads on the lower deck and two beads on the upper deck (See Figure 1). The Chinese abacus after 1850 has five beads below and one bead on the upper deck (See Figure 2). The Japanese abacus has four beads in the lower deck and one bead on the number deck (See Figure 3).

Figure 1: Chinese Abacus Prior to 1850

Figure 2: Chinese Abacus After 1850

Displaying Numbers on the Japanese Abacus

When you show a number on the abacus, you move beads to the crossbar. When beads are moved away from the crossbar, they are canceled. For example, when a lower bead is canceled, it is lowered from the crossbar and an upper bead is canceled when it is raised from the crossbar. Remember the upper bead represents five units and each lower bead equals one unit.

Note: The Japanese abacus will be used for all activities in this lesson.

Lets show 63 on the abacus.

Go to the tens place. Lower an upper bead to the cross bar. This represents 50. Move one lower bead up to the cross bar. This shows 60.

Move to the ones column and move 3 lower beads up to the cross bar. This shows 63 (60 + 3 = 63).

Lets show 672 on the abacus.

Move to the hundreds column. How many beads should you lower and/or raise to represent 600?

Move to the tens column. How many beads should you lower and/or raise to represent 70?

Move to the ones column. How beads should you lower and/or raise to represent 2?

Does your abacus look like this picture?

1. Using Student Worksheet #1 "Write the Number",identify the number represented on each abacus. Teacher Note: Complete the first several problems together and have students explain how they got their answers. Once students are comfortable with the problems, have them complete the worksheet.

2. Using Student Worksheet #3 "Blank Abacus Solution Sheet", choose new numbers and represent them on the blank abaci.

3. Using Worksheet #2 "Working on the Abacus"illustrate each given number on the blank abaci.

ADDING ON THE ABACUS

The abacus uses the relationship among ones, fives, and tens to do basic computation.

Addition of whole numbers can be solved by using one or more of the following four basic approaches:

Adding-Up

Adding-Up and Taking Off

Taking Off and Place Advancement

Combined Adding-Up, Taking-Off, and Place Advancement

A. Adding-Up

For example:

6 + 3 = ?

Show 6 on the abacus.

Move three more beads to the center.

The following chart shows all the possible combinations that use Adding-Up. (Note: +1 means to move one bead towards the cross bar.)

Given the First

To

Move bead(s)

Number

Add

In lower deck

In upper deck

0,1,2,3,4,5,6,7, or 8

1

+1

0,1,2,5,6, or 7

2

+2

0,1,5, or 6

3

+3

0 or 3

4

+4

0,1,2,3, or 4

5

+1

0,1,2 or 3

6

+1

+1

0,1, or 2

7

+2

+1

0, or 1

8

+3

+1

0

9

+4

+1

B. Adding-Up and Taking-Off

The second approach combines Adding-Up and Taking-Off. Taking-Off is necessary because there are not enough beads on the rod to complete the addition. No regrouping is involved in this type of problem.

For example:

4 + 3 = ?

Show 4 on the abacus.

To add 3, we would move 3 one beads, but there are not 3 one beads to move. However, we do have a five-bead at the top.

If we move the five-bead down, we have added five, but we only want to add 3, so we lower two one-beads to the bottom. Therefore 4 + 3 is shown as 5 + 2 and both equal 7.

The following chart shows all combinations that use both Adding-Up and Taking- Off to arrive at a solution. (Note: +1 means to move one bead towards the cross bar. 1 means to move one bead away from the cross bar.)

Given the First

To

Equivalent to

Move Bead(s)

Number

Add

Computation

In lower deck

In upper deck

4

1

(+5 4)

-4

+!

4 or 3

2

(+5 3)

-3

+1

4, 3, or 2

3

(+5 2)

-2

+1

4, 3, 2, or 1

4

(+5 1)

-1

+1

C. Taking-Off and Place Advancement

The third approach combines Taking-Off and Place Advancement. All problems that require the use of Taking-Off and Place Advancement are sums that are greater than ten and therefore require regrouping. Once again, we dont have enough beads on a given rod to complete the addition. We will take some off and add beads to the next place value.

For example:

7 + 4 = ?

Show 7 on the abacus.

There are not enough unused beads on this rod to add 4 so we will need to use beads from the tens rod (second rod from the right). Once again we look for compatibles for ten. Adding 4 is the same as subtracting 6 and adding ten (10 6 = 4).

To subtract 6, move a five-bead up and a one-bead down. Then slide one ten bead up (from the lower deck) on the next rod to add 10.

So 7 + 4 = 7 + (10 6) = 10 + 1 = 11.

The following chart shows all combinations that use both Taking Off and Place Advancement. (Note: +1 means to move one bead towards the cross bar. 1 means to move one bead away from the cross bar.)

Given the

To

Equivalent

Move bead(s)

First

Add

To

In the

In the

Lower deck

Number

Computation

Lower deck

Upper deck

Adjacent(left)

(ones digit)

(ones digit)

(tens digit)

9

1

(-9 + 10)

-4

-1

+1

8 or 9

2

(-8 + 10)

-3

-1

+1

7,8, or 9

3

(-7 + 10)

-2

+1

+1

6,7,8, or 9

4

(-6 + 10)

-1

-1

+1

5,6,7,8, or 9

5

(-5 + 10)

-1

+1

4 or 9

6

(-4 + 10)

-4

+1

3,4,8, or 9

7

(-3 + 10)

-3

+1

2,3,4,7,8, or 9

8

(-2 + 10)

-2

+1

1,2,3,4,6,7,8, or 9

9

(-1 + 10)

-1

+1

D. Combined Adding-Up, Taking-Off and Place Advancement

The fourth approach is Combined Adding-Up, Taking-Off, and Place Advancement.

All of the sums are greater than ten and therefore involve regrouping. In this situation,

we need to look for compatibles for ten and five to complete the addition.

For example:

8 + 6 = ?

Show 8 on the abacus. There are not enough unused beads on this rod to add 6 so we will need to use beads from the tens rod.

Adding a ten-bead means that we need to subtract 4 in order to make 6. (10 4 = 6) The ones rod only has 3 one-beads so we dont have enough one beads to subtract 4. We need to use a five-bead to complete the addition.

When we cancel a five-bead and add a one-bead that is the same as subtracting 4 (-5 + 1 = -4).

8 + 6 = 8 + (10 5 + 1) = 8 + 6 = 14

The following chart shows all combinations that use Combined Adding-Up, Taking-Off, and Place Advancement. (Note: +1 means to move one bead towards the cross bar. 1 means to move one bead away from the cross bar.)

Given the

To

Equivalent

Move bead(s)

First

Add

To

In the

In the

Lower deck

Number

Computation

Lower deck

Upper deck

Adjacent (left)

(ones digit)

(ones digit)

(tens digit)

5,6,7, or 8

6

( +1 5 + 10)

+1

-1

+1

5,6, or 7

7

(+2 5 + 10)

+2

-1

+1

5 or 6

8

(+3 5 + 10)

+3

-1

+1

5

9

(+4 5 + 10)

+4

-1

+1

Now lets try some simple addition problems. Use the charts and an abacus or yarn and counters to make an abacus. Record your solutions on the blank Addition Chart.

Additional Activities:

Research: Encourage students to search the Web for more information on how to use the abacus for subtraction, multiplication, and division.

Challenge: In groups of four, have the students take the roles of the Abacus Adder, the Calculator and the Mental Computer and a score keeper. The score keeper will give the group an addition problem to solve. The Abacus Adder will find the sum using an abacus. The Calculator will find the sum using a calculator. The Mental Computer will find the sum mentally without paper and pencil. The first person to get the correct sum wins a point. Play several rounds and have students switch roles. The discussion could take many different turns, depending on who is winning.