PlusyLong division step by step: a parent's guide
Long division is four small steps repeated until the digits run out: divide, multiply, subtract, bring down. This guide works two examples digit by digit, shows the bus stop layout used in the UK, and gives you the question to ask for each common slip.
You may recognise this: your child gets the first digit of the answer, brings a number down, and then stops and looks at you. Or the whole page is done and the answer is 26 when it should be 206. Long division is rarely one big thing a child cannot do. It is four small things, and somewhere in the loop one of them drops out.
What are the steps of long division?
The steps of long division are divide, multiply, subtract and bring down, repeated for each digit of the number being divided. Divide to find one digit of the answer, multiply to see how much that digit uses up, subtract to find what is left, and bring down the next digit to join what is left. Then start again.
Three words help. The dividend is the number being shared out (96 in 96 ÷ 4). The divisor is what you divide by (4). The quotient is the answer (24).
Divide
Look at the first digit of the dividend. How many times does the divisor fit into it? Write that digit on top, directly above the digit you were looking at. If it does not fit at all, look at the first two digits together.
Multiply
Multiply the digit you just wrote by the divisor. Write the result underneath, lined up on the right.
Subtract
Take that away from the number above it. What is left must be smaller than the divisor. If it is not, the digit on top was too small.
Bring down
Bring the next digit of the dividend down beside what is left. That makes a new number. Go back to the first step and divide again.
The step children lose is the one after "bring down". Bringing down is not the end of a turn, it is the start of the next one. The question to keep asking is: "You brought a digit down. What do we do with a new number?" Divide again.
How do you work out 96 ÷ 4?
To work out 96 ÷ 4 by long division, divide 4 into 9 to get 2, multiply 2 by 4 to get 8, subtract 8 from 9 to leave 1, and bring down the 6 to make 16. Then 4 goes into 16 exactly 4 times with nothing left, so 96 ÷ 4 = 24.
24
--
4 ) 96
8
-
16
16
--
0- Divide. 4 into 9 goes 2 times (2 × 4 = 8, and 3 × 4 = 12 is too many). Write 2 on top, above the 9.
- Multiply. 2 × 4 = 8. Write 8 under the 9.
- Subtract. 9 - 8 = 1. One is less than 4, so the 2 was right.
- Bring down. Bring the 6 down beside the 1. The new number is 16.
- Divide again. 4 into 16 goes 4 times. Write 4 on top, above the 6.
- Multiply. 4 × 4 = 16. Write 16 underneath.
- Subtract. 16 - 16 = 0. No digits are left to bring down, so the work is finished.
How do you divide by a two-digit number?
Dividing by a two-digit number uses the same four steps, with one extra skill: estimating each digit of the answer, because nobody knows the 24 times table by heart. For 1248 ÷ 24, the answer is 52. You find the 5 by asking how many 24s fit into 124, and the 2 by asking how many fit into 48.
52
----
24 ) 1248
120
---
48
48
--
0- Divide. 24 does not fit into 1. It does not fit into 12. So look at 124. 24 into 124 goes 5 times. Write 5 on top, above the 4, because 124 ends at the 4.
- Multiply. 5 × 24 = 120. Write 120 under 124.
- Subtract. 124 - 120 = 4. Four is less than 24, so the 5 was right.
- Bring down. Bring the 8 down beside the 4. The new number is 48.
- Divide again. 24 into 48 goes 2 times. Write 2 on top, above the 8.
- Multiply. 2 × 24 = 48. Write 48 underneath.
- Subtract. 48 - 48 = 0. Nothing is left to bring down. The answer is 52.
How do you estimate each digit of the answer?
Estimate a digit of the answer by rounding the divisor to a friendly number, guessing, and then testing the guess by multiplying. If the product is bigger than the number you are dividing into, the guess is too big. If what is left after subtracting is as big as the divisor or bigger, the guess is too small.
For 24 into 124: 24 is close to 25, and four 25s make 100, so five is a sensible guess. Then test it:
| Guess | Multiply | What it tells you |
|---|---|---|
| 4 | 4 × 24 = 96, leaves 28 | 28 is bigger than 24, so another 24 fits. Too small. |
| 5 | 5 × 24 = 120, leaves 4 | Fits, and 4 is less than 24. Right. |
| 6 | 6 × 24 = 144 | 144 is bigger than 124. Too big. |
A second way, for a child who finds guessing stressful: first write the multiples of the divisor down the side of the page. For 24 that is 24, 48, 72, 96, 120, 144. Now every "how many fit?" is answered by looking at the list.
What happens when there is a remainder?
A remainder is what is left when the last subtraction does not come to zero and there are no more digits to bring down. For 97 ÷ 4, the steps run exactly as for 96 ÷ 4, but the last subtraction is 17 - 16 = 1. The answer is 24 remainder 1, often written 24 r 1.
24 r 1
--
4 ) 97
8
-
17
16
--
1The remainder must always be smaller than the divisor: a remainder of 5 when dividing by 4 means another 4 would have fit. And the check is always the same: answer times divisor, plus the remainder, gives back the number you started with. For 1250 ÷ 24 the answer is 52 r 2, because 52 × 24 = 1248 and 1248 + 2 = 1250.
What are the commonest slips, and what should I ask?
The slips to look for in long division are stopping after bringing down, leaving out a zero in the answer, writing a digit in the wrong place, choosing a digit that is too small, and a subtraction error in the middle. For each one, ask a question so that your child finds the step themselves.
| The slip | What you see | The question to ask |
|---|---|---|
| Stops after bringing down | A digit is brought down and the work ends, or the answer is one digit short | "You have a new number. What is the first step with a new number?" |
| Leaves out a zero | 824 ÷ 4 = 26 instead of 206 | "How many 4s fit into 2? If the answer is none, what do we write on top?" |
| Digit in the wrong place | The first digit sits above the wrong column, so the answer is ten times too big or too small | "Which number did you divide into? Where does it end?" |
| Digit too small | What is left after subtracting is as big as the divisor | "Is what is left smaller than the divisor? Would one more fit?" |
| Subtraction error | Every step is right in order, but the final answer fails the check | "Multiply your answer back. Does it give the number we started with?" |
Take the zero. In 824 ÷ 4, the first step gives 2, with nothing left. Bring down the 2. Four does not fit into 2, and it is tempting to bring down the next digit and move on. But every digit brought down needs a digit on top, even when that digit is 0. Write 0, then bring down the 4 to make 24, and 4 into 24 is 6. The answer is 206, and 206 × 4 = 824.
If the same slip keeps returning, it is a pattern in the method and not carelessness: see careless mistake or real gap?
What is the bus stop method?
The bus stop method is the UK name for short division, set out under a bracket that looks like a bus shelter. It is the same divide-and-carry idea as long division, but the multiplying and subtracting are done in the head, and what is left over is written as a small digit beside the next one.
1 4 2 --------- 6 ) 8 ²5 ¹2
Long division writes every multiplication and subtraction out, which is slower but easier to follow and to check when the divisor has two digits. A child who learned the bus stop first already knows the loop.
Your child may also bring home "chunking" (UK) or "partial quotients" (US). That is division by taking away easy groups: for 96 ÷ 4, take away ten 4s (40), another ten 4s (40), then four 4s (16), and add up the groups: 10 + 10 + 4 = 24. Other unfamiliar methods are in new math methods explained.
When is long division taught?
Long division is expected by the end of 6th grade in the US Common Core, where standard 6.NS.2 asks children to "fluently divide multi-digit numbers using the standard algorithm". England's national curriculum expects it in Year 6, with short division in Year 5. States and schools vary.
Before that, the Common Core has 4th and 5th graders divide using strategies based on place value, so a 5th grader solving 1248 ÷ 24 in chunks is doing what the grade asks. The full ladder is in math by grade.
What does a child need to know first?
Before long division, a child needs the times tables from memory and subtraction with borrowing (also called regrouping or exchanging). Every turn of the loop uses one multiplication fact and one subtraction. A child who has to work those out each time runs out of attention for the method itself.
Ofsted, England's schools inspectorate, describes the mechanism in its research review of mathematics: pupils who cannot quickly recall math facts "struggle with calculations due to their working memory being overloaded".
- Times tables, both ways. Not only 6 × 7 = 42 but "how many 6s in 45?" Seven, with 3 left over.
- Subtraction with borrowing. 124 - 96 has to be reliable, because one slip there spoils everything after it.
- Lining up. Squared paper, one digit to a square, removes a whole class of errors.
Check these three before doing more long division. Our suggestion: a week on the 6, 7 and 8 tables before another week of division. See why times tables don't stick and carrying and borrowing mistakes.
How should we practice long division at home?
Practice long division with one fully worked example, then examples with steps left out, using easy numbers, a few at a time on several days. A US federal practice guide recommends worked-out examples, then excluding steps and asking students to provide them, and using easy-to-process numbers while a procedure is new.
Day one: watch one, finish one
Work 96 ÷ 4 aloud while your child watches. Then set out 84 ÷ 3 and do the first turn yourself (3 into 8 goes 2, 2 × 3 = 6, 8 - 6 = 2). Your child brings down the 4 and finishes: 3 into 24 goes 8. Answer: 28.
Day two: you start less
Set out two questions with a one-digit divisor. Write only the first digit of the answer. Your child does the rest and checks by multiplying back.
Day three: add one hard thing
A zero in the answer (824 ÷ 4), or a remainder (97 ÷ 4). One new difficulty at a time.
The next week: two digits
Start with a friendly divisor such as 25 or 20, write its multiples down the side, and work one example together before your child tries alone.
Two or three questions a day is our suggestion: no study sets a number for home practice. One review of 317 experiments found that practice spread over time beat the same practice in one block, though on recalling words and facts, not arithmetic.
Where does Plusy fit?
Plusy is a math practice app for children 6 to 12 whose Division road runs from dividing by 2 up to dividing by two-digit numbers, with long division worked in columns the way it is written on paper. The answer goes on top, and the working rows appear one at a time as each is filled in.
When your child is stuck, Hint asks a guiding question first. Show me fills in that one step and explains it, then hands the question back. It never solves the whole question in one go. See column questions and hints and steps.
The roads have no remainders. A list you make yourself can, and Plusy walks through them the same way. Plusy has no decimals: what is left over is written as a remainder.
Every road, world and game is free, in a browser on a phone, tablet or computer. Start free.
Questions
What are the four steps of long division?
Divide, multiply, subtract, bring down. You divide to find one digit of the answer, multiply that digit by the divisor, subtract to see what is left, and bring down the next digit. Then you repeat the four steps until no digits are left.
What is the difference between short division and long division?
They are the same method written two ways. Short division, called the bus stop method in the UK, does the multiplying and subtracting in the head and writes only small carried digits. Long division writes every multiplication and subtraction underneath, which is easier to follow with a two-digit divisor.
What grade do children learn long division?
The US Common Core expects fluent long division with the standard algorithm by the end of 6th grade (standard 6.NS.2), after place value strategies in 4th and 5th grade. England's national curriculum expects long division in Year 6 and short division in Year 5.
How do you check a long division answer?
Multiply the answer by the divisor and add any remainder. The result should be the number you started with. For 1248 ÷ 24 = 52, check that 52 × 24 = 1248. For 97 ÷ 4 = 24 r 1, check that 24 × 4 + 1 = 97.
Why does my child keep forgetting a zero in the answer?
Because when the divisor does not fit into the new number, it feels as if nothing happened, so nothing gets written. The rule is that every digit brought down needs a digit on top, even if that digit is 0. In 824 ÷ 4, the 0 in 206 is the answer to "how many 4s fit into 2?"
My child knows the steps but still gets wrong answers. What should I check?
Check the two skills inside every step: times tables and subtraction with borrowing. If either is slow or unreliable, the long division will be too. Then have your child multiply the answer back to find out for themselves whether it is right.