Calculating with beads
Tonight's idea
A soroban is a Japanese abacus: a frame of rods with sliding beads. A bar crosses the frame. On every rod, one bead sits above the bar and four sit below.
A bead counts only when it is pushed against the bar. The bead above is worth 5. Each bead below is worth 1. So one rod can show any digit from 0 to 9. For 7, bring the five-bead down and push two one-beads up.
Each rod is one digit of a number: ones on the right, then tens, then hundreds, just as in writing.
Adding is moving beads, and sometimes there are not enough. A rod shows 3 and you want to add 4, but only one one-bead is left. So add 5 and take 1 back: \(3 + 5 - 1 = 7\).
When a rod cannot hold the answer, use the rod to its left. To add 8, add 10 there and take 2 away here. That is a carry, and your fingers do it.
Watch one: 27 + 8
Set 27. The tens rod has two one-beads up. The ones rod has the five-bead down and two one-beads up.
Add 8 to the ones rod. It would pass 9, so add 10 and take away 2 instead, since \(8 = 10 - 2\).
Take 2 here: push the two one-beads on the ones rod back down. It now shows 5.
Add 10 there: push one more one-bead up on the tens rod. It now shows 3.
The soroban reads 35, and \(27 + 8 = 35\).
Your turn
- Read the number on each soroban.
- Draw the beads to show (a) 261 and (b) 509. Shade the beads that count.
- Describe the bead moves on a single rod, and say what the rod shows afterwards. (a) The rod shows 2; add 4. (b) The rod shows 4; add 3. (c) The rod shows 7; take away 4.
- (a) A soroban shows 38. Describe the bead moves that add 4, and say what it shows now. (b) A soroban shows 199. Add 1. Describe what happens on each rod, starting from the ones.
Stretch: a different abacus
A traditional Chinese abacus, the suanpan, has two beads worth 5 above the bar and five beads worth 1 below. What is the largest value one rod can show? Which values can one rod show in more than one way, and why does the soroban never have that problem?
Answers
- (a) 152. (b) 608. (c) 947.
- (a) Hundreds rod: two one-beads up. Tens rod: five-bead down and one one-bead up. Ones rod: one one-bead up. (b) Hundreds rod: five-bead down, no one-beads. Tens rod: nothing touches the bar. Ones rod: five-bead down and four one-beads up.
- (a) Only two one-beads are left, so add 5 and take 1: five-bead down, one one-bead back. The rod shows 6. (b) No one-beads are left, so add 5 and take 2: five-bead down, two one-beads back. It shows 7. (c) 7 is the five-bead and two one-beads, so there are not four ones to remove. Take 5 and give 1: five-bead up, one more one-bead up. It shows 3.
- (a) \(8 + 4\) passes 9, so add 10 and take 6. Ones rod: lift the five-bead away and push one one-bead back, leaving 2. Tens rod: push one more one-bead up, making 4. It shows 42. (b) Ones rod: \(9 + 1\) is 10, so clear the rod to 0 and carry. Tens rod: \(9 + 1\) is 10 again, so clear it and carry again. Hundreds rod: one more one-bead up, making 2. It shows 200. One small push runs along three rods.
Stretch. The largest is \(5 + 5 + 5 = 15\). Two values have two settings: 5 is one five-bead or five one-beads, and 10 is two five-beads or one five-bead with five one-beads. On a soroban the four one-beads reach only 4, so they can never stand in for the five-bead. Every digit from 0 to 9 has exactly one setting.