Small Friends and Big Friends: the abacus formulas explained
Every abacus tradition — Japanese soroban, and every modern programme built on it — rests on two families of formulas. Teachers call them Small Friends and Big Friends. If a student is stuck, it is almost always one of these two that hasn't clicked yet.
First: why formulas exist at all
An abacus rod can only show 0–9. The lower ("earth") beads are worth 1 each; the upper ("heaven") bead is worth 5. So a rod showing 4 has all four earth beads up — and there is physically no fifth earth bead to add.
So what does a child do when they need 4 + 3? They can't just "add 3 more." The beads aren't there. That's the entire reason the formulas exist: they are the moves that let you add a number you don't have room for, by using the beads you do have.
Small Friends — the 5's complements
Small Friends use the heaven bead (worth 5). The idea: to add 3 when you can't, add 5 and take away 2 — because 3 = 5 − 2.
| To add | Do this | Because |
|---|---|---|
| +4 | −1 +5 | 4 = 5 − 1 |
| +3 | −2 +5 | 3 = 5 − 2 |
| +2 | −3 +5 | 2 = 5 − 3 |
| +1 | −4 +5 | 1 = 5 − 4 |
And every one reverses for subtraction: −4 becomes +1 −5, −3 becomes +2 −5, and so on. Two numbers are "small friends" if they add to 5: 1&4, 2&3.
Big Friends — the 10's complements
Big Friends carry to the next rod. To add 9 when the rod is full, add 10 (one bead on the next rod left) and take away 1 — because 9 = 10 − 1.
| To add | Do this | Because |
|---|---|---|
| +9 | −1 +10 | 9 = 10 − 1 |
| +8 | −2 +10 | 8 = 10 − 2 |
| +7 | −3 +10 | 7 = 10 − 3 |
| +6 | −4 +10 | 6 = 10 − 4 |
| +5 | −5 +10 | 5 = 10 − 5 |
Two numbers are "big friends" if they add to 10: 1&9, 2&8, 3&7, 4&6, 5&5.
Double Combination — when you need both
Sometimes the next rod is full too, and a child needs a Small Friend and a Big Friend in the same move. That's Double Combination: +6 = +1 −5 +10, +7 = +2 −5 +10, +8 = +3 −5 +10, +9 = +4 −5 +10. This is the step where most students slow down — not because it's new maths, but because it's two learned moves fused into one.
How to teach them (what actually works)
- Teach the pairs before the formulas. A child who can answer "what makes 10 with 7?" instantly will learn Big Friends in days. One who can't will struggle for weeks. Drill the pairs first.
- Never let them memorise the table as words. "+4 is minus one plus five" recited from memory collapses under speed. They must see why 4 = 5 − 1 on the beads.
- Small Friends fully, before Big Friends at all. Mixing them early is the single most common cause of a stuck student.
- Slow the numbers down, not the child. When a student stumbles, the instinct is more questions. Usually the fix is more time per question — drop the display speed until they're right, then climb.
- Expect Double Combination to take longer. It's normal. It isn't a sign they've fallen behind.
The order that works
Foundations → Small Friends → Big Friends → Double Combination → advanced add/subtract → multiplication → division → anzan. That's the classical order for a reason: every step depends entirely on the one before it. When a student is stuck at multiplication, the cause is nearly always an unfinished Big Friend.
Our Levels 2, 3 and 4 cover exactly these three families — 382, 580 and 412 questions respectively, so students get the volume the formulas really need. There's also a free formula reference with every rule.
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