Ingrid

Teach this every autumn and the sentence that lands best is: division asks a question, and dividing by zero asks a question with either no answer or every answer.

Ten divided by zero asks what times zero gives ten. Nothing does, so there is no answer. Zero divided by zero asks what times zero gives zero, and everything does, so there is no single answer. Two different failures, both called undefined, and separating them is what makes it click.

Undefined means no answer exists, not that the answer is too big to write down. Those get confused constantly.

Approaching from the left goes to minus infinity and from the right to plus infinity. One symbol cannot be both.