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Division

Division is the one operation most children meet twice: first as sharing, where 12 sweets go between 4 people, and later as an algorithm with a bracket and a bring-down. The bridge between them is the remainder — the moment a child sees that not everything shares out evenly, and that what is left over is part of the answer rather than a mistake.

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Short division with a bring-down

92 / 4

  1. How many fours in 9? Two, with 1 left over. Write 2 above the 9.
  2. That leftover 1 is a whole ten, so bring it down in front of the 2 to make 12.
  3. How many fours in 12? Exactly three. Write 3 above the 2.
  4. The answer is 23.

Worth knowing. The bring-down is the step that makes long division long, and it is the one worth saying out loud. The 1 that moves across is ten ones, not a stray digit.

A division with a remainder

17 / 5

  1. How many fives fit inside 17? Three, because 3 x 5 is 15.
  2. Four fives would be 20, which is too many.
  3. 17 take away 15 leaves 2, and 2 is smaller than 5, so it cannot make another group.
  4. The answer is 3 remainder 2, written 3 r 2.

Worth knowing. The remainder is always smaller than the divisor. A remainder the same size as the divisor, or bigger, means another whole group was missed.

Where it usually goes wrong

  • Writing a remainder that is bigger than the divisor, which means a group was missed.
  • Dropping the remainder entirely and giving only the quotient, so 17 / 5 becomes 3.
  • Forgetting to write a zero in the quotient when a digit of the dividend is too small to divide, so 618 / 6 comes out as 13 instead of 103.

The Professor checks a division by multiplying back. If the quotient times the divisor plus the remainder is not the number you started with, something went astray.

Practise it

All level paths

Questions teachers and parents ask

When do children learn division with remainders?

Common Core puts whole-number quotients within 100 in grade 3 and four-digit dividends with remainders in grade 4 (4.NBT.B.6). In practice remainders often turn up earlier through sharing problems, because a child dividing 13 sweets between 4 people meets one whether the curriculum is ready or not.

Should a remainder be written as a fraction or a decimal instead?

Not at this stage. "17 divided by 5 is 3 remainder 2" comes years before 3.4 does, and a child who writes 3.4 in grade 3 will be marked wrong by their teacher whatever a website says. Fractional and decimal quotients come with the fractions and decimals work later.

Is short division or long division better to teach?

They are the same algorithm written at different lengths. Short division keeps the working in the head and suits single-digit divisors; long division writes each subtraction down and is what a child needs once the divisor has two digits. Most children want short division for the 2s to 9s and long division after that.

Why does division by zero have no answer?

Because the question has no answer to give. "How many zeros make 12" cannot be satisfied by any number of them, so there is nothing to write down. It is worth saying plainly rather than calling it a rule, because a child who hears it as a rule tends to assume the answer is zero.

Where this shows up

Standards

  • 3.OA · Operations and algebraic thinking
  • 4.OA · Operations and algebraic thinking
  • 4.NBT · Number and operations in base ten
  • 5.NBT · Number and operations in base ten

Formats