How to teach long division, with worked examples

What does a child need before long division?

A child is ready for long division when three skills are solid: division facts, place value and subtraction with regrouping. Long division strings them together many times in one problem, so a gap in any of them looks like a long-division problem.

1. Division facts. The child should answer 42 ÷ 7 or 56 ÷ 8 without counting. The Common Core standards expect a child to "know from memory all products of two one-digit numbers" by the end of grade 3 (3.OA.7), one year before long division starts. If facts are slow, the guide on how to learn the times tables fast comes first.
2. Place value. The child should know that the 9 in 952 means 9 hundreds. The place value game drills that reading.
3. Subtraction with regrouping. Each round ends with a subtraction such as 25 − 21 or 144 − 144, and one borrowing slip spoils the rest of the problem.

What are the steps of long division?

The steps of long division are divide, multiply, subtract, bring down, repeated until every digit of the dividend has been used. Some classrooms remember them as "Dad, Mom, Sister, Brother".

1. Divide the current number by the divisor and write the whole-number answer above it.
2. Multiply that quotient digit by the divisor.
3. Subtract the product from the current number.
4. Bring down the next digit of the dividend and start again.

Worked example: 952 ÷ 7

The problem 952 ÷ 7 divides a 3-digit number by a 1-digit number with no remainder.

1. Divide: 9 ÷ 7 = 1. Write 1 above the 9.
2. Multiply: 1 × 7 = 7. Subtract: 9 − 7 = 2.
3. Bring down the 5 to make 25. Divide: 25 ÷ 7 = 3. Write 3 above the 5.
4. Multiply: 3 × 7 = 21. Subtract: 25 − 21 = 4.
5. Bring down the 2 to make 42. Divide: 42 ÷ 7 = 6. Write 6 above the 2.
6. Multiply: 6 × 7 = 42. Subtract: 42 − 42 = 0. No digits are left.

Answer: 952 ÷ 7 = 136.

Worked example with a remainder: 745 ÷ 6

The problem 745 ÷ 6 ends with a number too small to divide, and that number is the remainder.

1. Divide: 7 ÷ 6 = 1. Multiply: 1 × 6 = 6. Subtract: 7 − 6 = 1.
2. Bring down the 4 to make 14. Divide: 14 ÷ 6 = 2. Multiply: 2 × 6 = 12. Subtract: 14 − 12 = 2.
3. Bring down the 5 to make 25. Divide: 25 ÷ 6 = 4. Multiply: 4 × 6 = 24. Subtract: 25 − 24 = 1.
4. No digits are left, and 1 is smaller than 6.

Answer: 745 ÷ 6 = 124 remainder 1.

Worked example with a 2-digit divisor: 864 ÷ 24

The problem 864 ÷ 24 is fifth-grade work, and the new skill is estimating each quotient digit.

1. The first digit, 8, is smaller than 24, so start with 86. Estimate: 24 is close to 20, and 86 ÷ 20 is about 4. Test it: 4 × 24 = 96, which is more than 86, so use 3.
2. Multiply: 3 × 24 = 72. Subtract: 86 − 72 = 14.
3. Bring down the 4 to make 144. Estimate: 144 ÷ 24 is about 6. Multiply: 6 × 24 = 144. Subtract: 0.

Answer: 864 ÷ 24 = 36. An estimate that is one too high is normal; lower the digit and multiply again.

What is the partial quotients method, and why do schools teach it?

The partial quotients method divides by taking away easy multiples of the divisor, such as 10, 20 or 100 of them, and adding up how many were taken.

Schools teach partial quotients because of Common Core standard 4.NBT.B.6, which asks fourth graders to find quotients and remainders "with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division". Grade 5 (5.NBT.B.6) extends the same wording to two-digit divisors. Only grade 6 (6.NS.B.2) asks students to "fluently divide multi-digit numbers using the standard algorithm". Partial quotients is the place-value strategy many schools use for the first two years.

Partial quotients: 952 ÷ 7

The same problem, 952 ÷ 7, takes three chunks with partial quotients.

1. 100 × 7 = 700. Subtract: 952 − 700 = 252. Write 100 at the side.
2. 30 × 7 = 210. Subtract: 252 − 210 = 42. Write 30 at the side.
3. 6 × 7 = 42. Subtract: 42 − 42 = 0. Write 6 at the side.
4. Add the side numbers: 100 + 30 + 6 = 136.

A cautious child can take smaller chunks: 100, then 20 (140, leaving 112), then 10 (70, leaving 42), then 6. The sum is still 136. Smaller chunks only add lines, so the method suits a child who freezes at the first step.

A parent who learned the standard algorithm does not need to switch the child's method. Ask which method the class uses, follow it at home, and show the other one only after the class method is secure.

Common long-division mistakes

Four mistakes cause most wrong long-division answers, and each one has a quick self-check.

1. A missing zero in the quotient. In 612 ÷ 6, the child divides 6 ÷ 6 = 1, brings down the 1, sees that 1 ÷ 6 does not go, and moves on without writing anything. The answer comes out as 12. The correct quotient is 102: when the divisor does not go into the current number, write 0 above the digit and bring down the next one. Self-check: 12 × 6 = 72, nowhere near 612.
2. Misaligned columns. A quotient digit one place off changes the answer by a factor of ten. Squared paper keeps every digit in its column.
3. A remainder bigger than the divisor. In 745 ÷ 6, writing 1 for 14 ÷ 6 leaves 14 − 6 = 8, and 8 is more than 6. Any number left after subtracting must be smaller than the divisor; if it is not, raise the quotient digit.
4. Forgetting to bring down a digit. The quotient ends up one digit short. Self-check: count the digits brought down, and compare the size of the answer with an estimate such as 952 ÷ 7 ≈ 1,000 ÷ 7, which is a little over 140.

How do you check a long division answer?

A long division answer is checked by multiplying the quotient by the divisor and then adding the remainder: the result must equal the dividend.

The check lets a child find the missing-zero mistake without an adult. The standards in England ask for the same habit: the national curriculum for mathematics in England expects Year 6 pupils to divide up to 4-digit numbers by a two-digit number "using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding".

A 15-minute practice routine

A short daily routine builds long division faster than one long weekend session.

1. Five minutes of division facts. Quick recall keeps attention free for the procedure. The What Works Clearinghouse practice guide on assisting students struggling with mathematics recommends that teachers "regularly include timed activities as one way to build fluency in mathematics". The division facts game and the division games for 4th graders cover that part.
2. Two worked problems, said out loud. The child names each step (divide, multiply, subtract, bring down) while writing it. One problem should contain a zero in the quotient, such as 612 ÷ 6 or 824 ÷ 4 = 206.
3. Three independent problems, each checked. The printable division worksheet for 4th graders has an answer key; the division worksheet for 5th graders moves to larger numbers.
4. One minute on the mistakes. Circle the step where each error began; a step that fails twice is tomorrow's practice.

Move from 1-digit to 2-digit divisors only when 3-digit problems come out right with a clean check. The overview of what math each grade should know shows where long division sits next to fractions and decimals.

MathIt's free iOS and Android app has adaptive difficulty, so division practice gets harder only after the easier facts are right, and 22 of its roughly 90 game types play free in a browser with no sign-up.