Building and Using the Amazing Abacus. (A Teacher's Journal).
Elementary school children in areas of Asia still learn arithmetic on the abacus, and shopkeepers still practice the art of calculating with beads in these countries. Many "lightning calculators," people who are able to perform mental calculations at astonishing speeds, manipulate a mental image of an abacus to perform complex calculations.
The abacus is a useful supplement to base-ten blocks for teaching place value. Although number representations on the abacus are not proportional, as they are in base-ten blocks, the rods of the abacus represent places in a base-ten system and the concepts of trading in addition and subtraction still apply.
Our second- and third-grade class at Whatcom Day Academy in Bellingham, Washington, recently studied ancient Japan. We used the opportunity to combine social studies, art, and mathematics by making and learning to use our own abacuses. The following paragraphs outline step-by-step instructions for you to repeat this multifaceted project with your students.
Building an Abacus
Each student will need one sheet of card stock; scissors; a ruler; bamboo skewers (available at grocery stores); and beads or loop cereal. The teacher will need a hot-melt glue gun.
Step 1: Folding the abacus frame
The frame for this project is an origami box. Origami is rich in geometric ideas and helps students develop fine motor skills and practice following directions. Model each step as the class follows along, and take advantage of the opportunity to use geometric terms, such as vertex and midpoint. Discussions about angles, fractions, and parallel and perpendicular lines may be appropriate, depending on the grade level of your students. Take your students through the following steps:
1. Fold the card stock in half horizontally, and crease it; then open it again. Fold the card in half vertically, and crease it; then open it again. The card should now be quartered (see fig. la).
2. With the card stock positioned so that the shorter edges are to the left and right, fold the longer edges on the top and bottom to meet the crease in the center, then unfold (see fig. 1b).
3. Fold the left and right edges in to meet the vertical center crease; this time, do not unfold (see fig. 1c).
4. Fold the top corners down at a 45 degree angle toward the center so that the corners meet the first crease from the top. The corners will not meet each other. Repeat on the bottom (see fig. 1d).
5. Fold the edges that meet in the center outward to the left and right, creasing a line as close as possible to the folded corners without bending the corners (see fig. le).
6. Hook your fingers inside, underneath the left and right flaps, and pull open. The top and bottom edges should be pushed inward, and your box should now have four walls (see fig. 1f).
7. Pinch the edges and corners to get your box to stand upright (see fig. 1g). Your frame is now ready to become an abacus!
Step 2: Preparing the frame for the rods
In this phase of the project, your students practice measurement skills to determine where to place the rods. Sketch the frame on the board with four equally spaced vertical rods and one horizontal rod one-third of the way down from the top, as shown in figure 2. Tell the students to make pencil marks on their frames to show where the rods are to be placed-four marks on the top and bottom edges and one mark on the left and right edges. Ask your students how they can determine the placement of these marks. They need to measure the top or bottom side and divide this length by 5 to find the spacing, then measure the left or right side and divide by 3. Have students make the measurements and place the marks. They should check one another's work because making equally spaced marks is the most difficult part of this project for most children.
When your students have made the marks in the proper places, have them make a 1-centimeter cut through the top edges of the frame at each mark, perpendicular to the edge of the frame. The abacus will still work even if the students make the cuts a little too deep.
Step 3: Placing the beads and rods
The bamboo skewers need to be cut to length. Each student needs four skewers that are 2 centimeters longer than the short vertical side of the frame and one skewer that is 2 centimeters longer than the long horizontal side of the frame. The skewers can be precut, or the students can measure, score the skewers with scissors, and break them off at the proper lengths.
Preheat the hot-melt glue gun. On the board, demonstrate how the beads should be placed on the abacus when it is finished. Each rod has a group of five beads and a group of two beads. If you are using colored cereal loops, such as Froot Loops, your students will need to decide on a color theme. They may wish to have the lower beads be one color and the upper heads, a different color.
Each student should have one long rod and four short rods. Have students place seven beads on each of the shorter rods and push these rods into the slots in the frames so that the rods are parallel. They should then separate the beads so that two are at the top of each rod and the other five are at the bottom. Have them place the longer skewer horizontally in the slots across the other rods (see fig. 3).
As the students finish, have them bring their abacuses to the glue-gun station. Squeeze a small bead of glue on each point of contact between the rods and the frame, and allow the glue to set. If you wish, students can use ordinary white glue, but you will need to allot space for the abacuses to be left undisturbed overnight.
Congratulations--the abacuses are complete!
Using the Abacus
While teaching your students to use the abacus, you may wish to have your own abacus with the bottom cut out so that you can use it as a demonstration tool on the overhead projector.
Numbers on the abacus are arranged using our familiar base-ten system. The rod on the far right represents the ones place; the next rod to the left is tens; then hundreds; and so on. Abacuses commonly have more than ten rods, and some have more than thirty. These extra rods allow more than one number at a time to be represented, but our four-rod abacuses are perfect for learning to display and add numbers.
The top portion of the abacus that contains two beads per rod is called heaven; the lower portion is called earth. The beads in earth are each worth one unit; the beads in heaven, five units.
Numbers are formed at the crossbeam. The abacus shown in figure 4 is in the ready position, displaying the number 0. Each bead that is brought up to the crossbar from earth is worth one unit, whereas each bead that is brought down to the crossbar from heaven is worth five units.
Have your students clear their abacuses by sliding the beads in heaven to the top and the beads in earth to the bottom. Sliding one bead at a time upward in the ones column, have them count with you to 5. When all five beads in earth are at the crossbar, trade them for 1 five by sliding all five downward, then sliding one bead from heaven down to the crossbar. Each student should have only one bead at the crossbar; because this bead is in heaven, the abacus now shows 5.
Continue counting up to 9 by bringing one bead at a time up from earth. Stop when you have four beads at the crossbar in earth and one bead at the crossbar in heaven. The even numbers from 0--9 are shown in figure 5.
Next have your students clear their abacuses and enter 10 on the rods by sliding one bead on the tens rod up to the crossbar. Have them count by tens up to 90 in exactly the same way that they counted to 9 on the ones rod. Continue by having them count by hundreds to 900 and by thousands to 9000.
Students should now have an understanding of what digits look like on the abacus. Enter a few numbers on your abacus, such as 34, 52, 49, and 108, and challenge your students to name the number that the abacus is displaying. Pick numbers for students to enter on their abacuses, give them a moment to make the numbers, then show them the correct bead placement on the overhead projector to allow them to check their answers. Our students loved displaying the numbers, and although they needed a few minutes to get the knack, they were soon begging for larger numbers. Be sure to have them display 9999, and ask them to enter their birthdays or birth years.
Your students have already learned how to count to 9. Have them do this activity again, bringing up one bead at a time to count to 5, then trading all five beads in earth at the crossbeam for one bead in heaven. Continue bringing up one bead at a time, counting 6, 7, 8, 9. When the abacus shows 9, have them perform the following steps to add 1 to 9 to make 10 (see fig. 6):
1. The abacus displays the number 9.
2. Slide the last bead from earth up to join the others at the crossbar. The abacus now shows 10 as 1 five plus 5 ones.
3. The five beads in earth at the crossbar can be traded for 1 five. Slide all five beads in earth to the bottom, and slide the remaining bead in heaven down to the crossbar. The abacus still shows 10, but this time, it is represented as 2 fives.
4. Finally, trade the 2 fives for 1 ten.
Your students can now continue counting. Each time that a 10 shows on the far-right rod, the beads should be exchanged for an additional bead on the tens rod. The tens rod behaves exactly the same way as the ones rod: when five beads from earth are at the crossbeam, they should be traded for one bead in heaven; in other words, 5 tens are traded for 1 fifty. When the rod has two beads in heaven, they should be traded for one bead on the next rod to the left; 2 fiftys are traded for 1 hundred. How high can your students count? For an additional challenge, your students can try to count by twos, threes, or other multiples.
For further study
Adding and subtracting are easy on the abacus, and you might consider allowing your students to invent their own methods for these operations. The Internet is an abundant source of information on using the abacus; perhaps your students can research the abacus to see what other information they can find about this ancient calculating device. Many books have been written on the subject, some of which are listed in the bibliography.
This project was very popular with our students and allowed us to use and explore a number of mathematics vocabulary words and concepts. Our students also furthered their understanding and appreciation of Asian cultures and were amazed to learn of other ways to represent numbers. We wish you guddorakku (good luck) with the project!
Michael Naylor, firstname.lastname@example.org, is a mathematics educator at Western Washington University in Bellingham Washington. His interests include elementary, mathematics education, geometry, recreational mathematics, and music. Pamela Naylor email@example.com, teaches at Whatcom Day Academy in Bellingham Washington. She is interested in cross-curricular activities, special education, and art.
Note: The authors' photograph on the first page of this article is displaying pi's first twenty-one digits: 3.14159265358979323846.
Dilson, Jesse. The Abacus. New York: St Martin's Press, 1968.
Kojima, Takashi. The Japanese Abacus: Its Use and Theory. Tokyo: Charles E. Tuttle Co., 1954.
Needham, Kate The Usborne Book of Origami. London: Usborne Publishing, 1991.
Pullan, J. M. The History of the Abacus. New York: Frederick A. Praeger, 1969.
Tani, Yukio. The Magic Calculator: The Way of the Abacus. Tokyo: Japan Publications Trading Co., 1964.
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|Author:||Naylor, Michael; Naylor, Pamela|
|Publication:||Teaching Children Mathematics|
|Date:||Dec 1, 2001|
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