L4 · Why division by 0 is impossible

Learning goal

“I can explain why dividing by 0 has no answer - using grouping and the multiplication check - and I can explain why 0 divided by a number is different: 0 / n = 0.”

Dividend — the number being divided up: in 12 / 3, the dividend is 12.

Divisor — the number you divide by: in 12 / 3, the divisor is 3.

Quotient — the answer to a division: in 12 / 3 = 4, the quotient is 4.

Undefined — an expression with no possible value. Division by 0 is undefined.

Worked example

What is 12 / 3? What about 12 / 0? Step 1: 12 / 3 asks: how many groups of 3 can I make from 12? Build them: 4 groups, no counters left. Quotient: 4. Step 2: 12 / 0 asks: how many groups of 0 can I make from 12? Every group you "make" contains nothing, so the pile of 12 counters never shrinks. You could build groups forever and never finish. Step 3: Sharing says the same thing: sharing 12 cookies among 0 people - there is nobody to hand a cookie to, so "how many does each person get?" has no answer. Say it plainly: division by 0 is not hard or tricky - it is meaningless. There is no quotient, and mathematicians call it undefined.

Check yourself — explain it

12 / 0 has no answer because...

Paired problem — example still visible

A teacher tries to sort 20 pencils into bundles of 0. What goes wrong? -> Each bundle holds nothing, so the 20 pencils are never used up - bundling never ends, and there is no number of bundles.

Faded completion — last step is yours

"7 / 0: making groups of ___ from 7 removes ___ counters each time, so the quotient is ___" -> 0; 0; undefined (there is none)

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