Guess, multiply back, take away, move along. Division is multiplication in a mirror.
Updated · 2 min read
Setting up the board
Set the number being divided (the dividend) towards the right, its last digit on a unit rod.
Set the number you are dividing by (the divisor) to its left, leaving four empty rods between them. Four, not fewer: with a smaller gap the answer is hard to tell apart from the divisor.
The answer (the quotient) grows in the gap.
The loop: compare, guess, multiply, take away
Compare the divisor with the first digit of what is left of the dividend. If the divisor is bigger, compare it with the first two digits instead.
Guess how many times it goes in. That is the next answer digit; set it in the gap.
Multiply the divisor by that digit and take it away from the digits you compared.
Move one rod to the right and repeat until the dividend is used up.
With a one-digit divisor, each answer digit goes two rods to the left of the last dividend digit you compared.
Worked example: 96 ÷ 3
Worked example: 96 ÷ 3 = 32
1Move 1. Set 3 on the left, leave four empty rods, then set 96. How many 3s fit inside 9? 3. Put it two rods to the left. Thumb up 3 beads. 👍 thumb2Move 2. Now take 9 back off the dividend. 9 = 5 + 4. Big bead up and 4 down together. ☝️ index finger3Move 3. The divisor is bigger than 0, so look at 6. How many 3s fit? 2. Thumb up 2 beads. 👍 thumb4Move 4. Take 6 off these two rods. 6 = 5 + 1. Big bead up and 1 down together. ☝️ index finger✓Answer: 32, read from the gap between the divisor and the dividend
Worked example: 837 ÷ 3
Here the divisor is bigger than what is left twice in a row, so those steps compare two digits at a time.
Worked example: 837 ÷ 3 = 279
1Move 1. Set 3 on the left, leave four empty rods, then set 837. How many 3s fit inside 8? 2. Put it two rods to the left. Thumb up 2 beads. 👍 thumb2Move 2. Now take 6 back off the dividend. 6 = 5 + 1. Big bead up and 1 down together. ☝️ index finger3Move 3. The divisor is bigger than 2, so look at 23. How many 3s fit? 7. 7 = 5 + 2. Big bead down and 2 up together. 👍 thumb + ☝️ index finger, together4Move 4. Take 21 off these two rods. Index finger down 2 beads. ☝️ index finger5Move 5. Index finger down 1 bead. ☝️ index finger6Move 6. The divisor is bigger than 2, so look at 27. How many 3s fit? 9. 9 = 5 + 4. Big bead down and 4 up together. 👍 thumb + ☝️ index finger, together7Move 7. Take 27 off these two rods. Index finger down 2 beads. ☝️ index finger8Move 8. 7 = 5 + 2. Big bead up and 2 down together. ☝️ index finger✓Answer: 279, read from the gap between the divisor and the dividend
When your guess is too big
Guess quickly from the first digits and fix it if needed — that really is faster than working each digit out carefully. You find out straight away, because the amount you are about to take away is bigger than what is on the board.
For example, in 4,698 ÷ 54, comparing 5 with 46 suggests 9. Set 9, take away 5 × 9 = 45 from 46, leaving 1. Now 4 × 9 = 36 is needed, but only 19 is there — so 9 was too big. Lower the answer digit to 8, add one 5 back onto the remainder to undo the extra, and take away 4 × 8 = 32 instead. Then carry on as normal.
If a guess is too small (rarer), do the opposite: raise the digit by 1 and take one more divisor away.
Checking your answer
Multiply the quotient by the divisor. You should land back on the original dividend, on the same rods it started on.
The quotient grows in the gap between the divisor and the dividend. With fewer than four rods it becomes hard to tell where the divisor ends and the answer begins.
Do I have to get each guess right first time?
No. Guess from the leading digits and correct it on the next step if it was too big. This is part of the method, not a workaround.
What about remainders?
Whatever is left on the dividend rods when you run out of digits is the remainder. You can also keep going onto the rods to the right to get decimal places.