How to Teach Abacus Formulas at Home (Even If You Never Learned Them)

Most parents who help their child with the abacus never learned it themselves. That turns out to be fine — you don't need to be faster than your child, you need to know what comes next and what to say when they get stuck. This article gives you both.

First, the one idea behind every formula

Every abacus formula exists for one reason: you tried to move beads and there weren't enough.

• If you need to add 3 and only two earth beads are free, you can't do it directly.
• If you need to add 5 and the heaven bead is already down, you can't do it directly.

The formulas are simply the two ways around that: borrow from 5 (the Small Friend) or borrow from 10 (the Big Friend). That is the whole theory. Everything else is practice.

500

50

5

100Hundreds

10Tens

1Ones

4

The order to teach them

Teach in this order and nothing will feel like a leap:

1. Direct addition — pushing earth beads up, then using the heaven bead for 5–9. No formulas.
2. Direct subtraction — pushing earth beads down, then releasing the heaven bead.
3. Small Friend addition — when earth beads run out: add 5, take away the friend.
4. Small Friend subtraction — when you can't take away enough earth beads: take away 5, add the friend back.
5. Big Friend addition — when the rod is full: add 10 on the next rod, take away the friend.
6. Big Friend subtraction — when the rod is too small: take 10 from the next rod, add the friend back.
7. Family formulas — when a Big Friend move itself needs a Small Friend.

This is exactly the chapter order of Level 2 in Abacus Kids, and it is the order abacus centres use. The first three chapters are free.

The friends, in one place

Small Friends add up to 5:

• 1 ↔ 4
• 2 ↔ 3

Big Friends add up to 10:

• 1 ↔ 9
• 2 ↔ 8
• 3 ↔ 7
• 4 ↔ 6
• 5 ↔ 5

That is everything your child has to memorise. Seven pairs.

The one question that decides which formula to use

When your child is stuck, don't tell them the formula. Ask:

"Can you move the beads directly?"

• If yes — move them. No formula.
• If no, but the heaven bead is free — Small Friend. "Add five, take away the friend."
• If no, and the heaven bead is already used — Big Friend. "Add ten next door, take away the friend."

Children who learn to ask this question themselves stop needing you within a few weeks.

Worked example to practise together: 4 + 3

1. Set 4 — four earth beads up.
2. Ask: can we add 3 directly? No — no earth beads left.
3. Is the heaven bead free? Yes → Small Friend. The friend of 3 is 2.
4. Push the heaven bead down (+5). Push two earth beads down (−2).
5. Read the rod: 7.

Now try 7 − 4 the same way: can't take four earth beads (only two are up), so release the heaven bead (−5) and add one earth bead (+1). Answer 3. The full explanation with more examples is in our Small Friend guide.

Then 8 + 5, for the Big Friend

1. Set 8 — heaven bead down, three earth beads up.
2. Can we add 5 directly? No — the heaven bead is already used.
3. Big Friend. The friend of 5 is 5.
4. Push one earth bead up on the tens rod (+10). Release the heaven bead on the ones rod (−5).
5. Read: 13.

That move — putting the 1 on the tens rod first, on purpose — is why abacus children rarely make carrying mistakes in written math later. Details and the Family formulas are in the Big Friend guide.

Four mistakes parents make (and easy fixes)

• Explaining instead of asking. Ask "can you move them directly?" and wait. Silence is fine.
• Moving to formulas before bead-reading is instant. If your child still counts beads to read a rod, go back to Level 1 for a week. It will feel like a step back and save a month.
• Long sessions. Ten minutes. One chapter's practice set. Stop while it is going well.
• Correcting the answer instead of the move. If the answer is wrong, don't say the right number. Say "show me the beads for 4 again" and let them find the wrong move themselves.

How the app helps you

You don't have to hold the formulas in your head. In Step by Step mode the app shows the formula hint and bead-direction arrows at the exact moment they are needed, and the quiz at the end of every chapter tells you both whether it has stuck. Your job is the question — "can you move them directly?" — and the ten minutes a day.