Mathematics from zero
Division
You have 12 apples and 4 friends. You hand them out fairly — one each, again, again — until they are gone. Each friend ends up with 3. Splitting an amount into equal shares is division.
After this lesson you can say what division means, name its result, see the two questions it answers, explain why it undoes multiplication, say what a remainder is, and explain why dividing by zero is not allowed.
Division splits an amount into equal groups. We write it with the division
sign, ÷. 12 ÷ 4 = 3 is read “twelve divided by four makes three”. You start with
a total, split it into equal parts, and the answer tells you the size of each part — or
how many parts there are.
The result of a division is the quotient. In 12 ÷ 4 = 3, the 3 is the
quotient. The number you start with — the 12 — is the amount being split, and
the 4 is how you are splitting it. The quotient is what each fair share comes to.
Division answers two questions. Sharing: “12 apples among 4 friends — how many
each?” Grouping: “12 apples into bags of 4 — how many bags?” Both are 12 ÷ 4 = 3.
The grouping question is easy to see on a number line: count how many jumps of 4 it
takes to reach 12.
Division undoes multiplication. 12 ÷ 4 = 3 because 3 × 4 = 12. To divide is to
ask “what number, times the divisor, gives this total?” That makes checking easy:
multiply the quotient by what you divided by, and you should get back the number you
started with. It also explains a rule: any number ÷ 1 is itself, since one group
holds the whole amount.
Sometimes the split is not even — that leftover is the remainder. 13 ÷ 4 does not
land cleanly: 3 jumps of 4 reach 12, and 1 is left over. We say 13 ÷ 4 = 3 remainder 1. The remainder is always smaller than the divisor — if it were 4 or more, you could
fit another whole group. And you can never divide by 0: “split 12 into groups of 0”
has no answer, because groups of nothing never add up to 12.
Divide 13 ÷ 4.
Think of it as grouping: how many groups of 4 fit inside 13? Count jumps of 4: one jump reaches 4, two reach 8, three reach 12. A fourth jump would reach 16 — too far. So 3 whole groups fit.
After 3 groups you have used 12. The amount left is 13 − 12 = 1. That leftover 1 is
smaller than the divisor 4, so it cannot form another group.
The answer is 13 ÷ 4 = 3 remainder 1. Check it: 3 × 4 = 12, and 12 + 1 = 13. The
quotient times the divisor, plus the remainder, returns the number you started with.
Why this works
Why is dividing by zero not allowed? Division asks “how many of the divisor make the
total?” With a divisor of 0, you are asking how many groups of nothing make 12 — and no
amount of empty groups will ever reach 12. The question has no answer, so 12 ÷ 0 is
left undefined. This is not a missing rule; it is a question that cannot be answered.
Common mistake
A common mistake is letting the remainder be as large as, or larger than, the divisor —
writing 13 ÷ 4 = 2 remainder 5. But 5 is bigger than 4, so another whole group of 4
still fits. Always keep dividing until the leftover is smaller than the divisor. The
remainder is the part too small to split any further.
Share 12 equally among 3. How many each? Type the quotient.
Divide: 20 ÷ 5. Type the quotient.
Divide: 7 ÷ 1. Type the quotient.
Divide 15 ÷ 4. Type the quotient (ignore the remainder).
For 15 ÷ 4, type the remainder.
How can you check that 12 ÷ 4 = 3 is correct?
Division splits an amount into equal groups, and its result is the quotient. It answers both “how many in each share” and “how many groups”, and it undoes multiplication — so check a division by multiplying the quotient by the divisor. When the split is not even, the leftover is the remainder, always smaller than the divisor. Dividing by zero is undefined, because no number of empty groups can ever reach the total.