Kani Math
Cosmic Abacus
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Free printable chart

Small friends and big friends chart

Every complement rule the soroban uses, on one page you can print and put on the wall. Small friends make five, big friends make ten.

Small friends — adding

When there are not enough earth beads free, add the heaven bead and take the friend away.

Small friends — adding
You wantSo you doBecause
+1+5 − 41 + 4 = 5
+2+5 − 32 + 3 = 5
+3+5 − 23 + 2 = 5
+4+5 − 14 + 1 = 5

Small friends — subtracting

The same pairs in reverse: take the heaven bead off and put the friend back.

Small friends — subtracting
You wantSo you doBecause
−1−5 + 41 + 4 = 5
−2−5 + 32 + 3 = 5
−3−5 + 23 + 2 = 5
−4−5 + 14 + 1 = 5

Big friends — adding

When the rod itself is full, add one to the next rod and take the friend off this one.

Big friends — adding
You wantSo you doBecause
+1+10 − 91 + 9 = 10
+2+10 − 82 + 8 = 10
+3+10 − 73 + 7 = 10
+4+10 − 64 + 6 = 10
+5+10 − 55 + 5 = 10
+6+10 − 46 + 4 = 10
+7+10 − 37 + 3 = 10
+8+10 − 28 + 2 = 10
+9+10 − 19 + 1 = 10

Big friends — subtracting

Borrow one from the next rod, then add the friend back on this one.

Big friends — subtracting
You wantSo you doBecause
−1−10 + 91 + 9 = 10
−2−10 + 82 + 8 = 10
−3−10 + 73 + 7 = 10
−4−10 + 64 + 6 = 10
−5−10 + 55 + 5 = 10
−6−10 + 46 + 4 = 10
−7−10 + 37 + 3 = 10
−8−10 + 28 + 2 = 10
−9−10 + 19 + 1 = 10

How to use this chart

Print it and keep it beside the abacus for the first few weeks. A child looks up a rule two or three times, then stops needing it — the point is to make the substitution automatic in the fingers, not to memorise a table.

Every rule here is the same idea: the bead you want is not free, so you use a bigger bead and give back the difference. That single substitution is what turns the abacus from a counting device into a calculating one, and it is why abacus training transfers to mental arithmetic at all.

Practise small friends →Practise big friends →Try it on the free abacus →How to use an abacus →Printable soroban flash cards →

Common questions

What is a small friend on the abacus?
A small friend is the number that pairs with yours to make 5 — 1 with 4, 2 with 3. It matters when a rod has no free earth beads left: to add 4, push the heaven bead down for 5 and take the small friend 1 straight back off. The addition table above lists all four rules. Elementary A is where this is taught.
What is a big friend on the abacus?
A big friend is the number that pairs with yours to make 10 — 3 with 7, 4 with 6, up to 9 with 1. You use it when the rod cannot take the digit at all: add 10 as one bead on the rod to the left, then take the big friend off the rod you are on. That carry is Elementary B, the level after small friends.
Do the same pairs work for subtraction?
Yes — the same pairs, run backwards. To subtract 4 when the rod has too few earth beads, take 5 off and add the small friend 1 back. To subtract 7, borrow 10 from the rod on the left and add the big friend 3 here. The chart prints all four tables — small and big, adding and subtracting — so nothing has to be reversed in your head.
Do I have to memorise the complements?
The pairs, yes; the table, no. Four small-friend pairs and the ten-complements should come back instantly, because a pause mid-sum costs more than a wrong digit. The rows are not learned by heart: a child looks one up two or three times and the substitution moves into the fingers. That is what the chart is for — a lookup for the first weeks, not a revision list.
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