How do you explain to a 5th grader why division by zero is meaningless? I wish to explain my younger brother: he is interested and curious, but he cannot grasp the concepts of limits and integration just yet. What is the best mathematical way to justify not allowing division by zero?

When we first start teaching multiplication, we use successive additions. So, 
3 x 4 = 3               | 3
          + 3           | 6
               + 3      | 9
                    + 3 | 12
=12

Division can be taught as successive subtractions.   So 12 / 3 becomes, 
12 - 3 -> 9 (1)
9 - 3 -> 6 (2)
6 - 3 -> 3 (3)
3 - 3 -> 0 (4)

Now apply the second algorithm with zero as a divisor. Tell your brother to get back to you when he's done.
While this algorithmic approach is not rigorous, I think it is probably a good way of developing an intuitive understanding of the concept.
