Division Makes Sense

Why division feels harder than the other three

Addition, subtraction and multiplication all have an obvious picture behind them. Division usually arrives as a procedure — a staircase of digits, bring down the next one, repeat — and the picture never gets drawn.

So people end up able to run the procedure without being able to say what the answer means. That gap is where most of the trouble lives.

Division asks one of two questions

Take 12 ÷ 3. There are two different things you might be asking, and school usually teaches only the first.

Sharing. You have 12 sweets and 3 children. How many does each child get? Four.

Measuring. You have a 12-metre rope and you cut 3-metre pieces. How many pieces do you get? Also four.

Different pictures, same answer. The second one is the more useful of the two, and it is the one that makes the rest of this page easy — so keep it: how many of these fit into that?

Division is multiplication read backwards

a ÷ b = c is another way of writing c × b = a. That is not a trick for checking your work; it is the definition.

It means every division fact is a multiplication fact you already know, seen from a different side. 12 ÷ 4 = 3, 12 ÷ 3 = 4 and 3 × 4 = 12 are one fact wearing three hats.

It also means order matters in a way it does not for multiplication. 3 × 12 and 12 × 3 are the same; 12 ÷ 3 and 3 ÷ 12 are not remotely the same. Nor can you regroup: (24 ÷ 4) ÷ 2 is 3, while 24 ÷ (4 ÷ 2) is 12.

Why you cannot divide by zero

This is usually taught as a rule to memorise, and it does not have to be. Use the definition.

6 ÷ 0 would be asking: what number, multiplied by 0, gives 6? Nothing does — anything times zero is zero. There is no answer, so the question has none. Not infinity; just nothing that fits.

0 ÷ 0 fails for the opposite reason. What number times 0 gives 0? Every number does. An answer that could be anything is no answer either.

Dividing does not always make things smaller

Almost everyone carries the belief that division shrinks things. It holds only while you divide by numbers bigger than 1.

Ask the measuring question about 6 ÷ 0.5: how many halves fit into 6? Twelve. The answer is double what you started with, and nothing strange happened — you were counting small things, so there were a lot of them.

That is also the whole reason dividing by a fraction means multiplying by the flipped fraction. Dividing by 1/2 asks how many halves fit, and halves fit twice as often, so you multiply by 2.

What long division is actually doing

Every step of long division asks the same measuring question, one digit at a time: how many times does the divisor fit into what is left over so far? You write that down, take it away, and bring down the next digit to see what is still unclaimed.

The remainder is simply what never managed to fit. It is always smaller than the divisor — if it were not, another whole one would have fitted, and you would not have finished.