How do you explain to a 5th grader why division by zero is meaningless?

Solution 1:

“One of the ways to look at division is as how many of the smaller number you need to make up the bigger number, right? So 20/4 means: how many groups of 4 do you need to make 20? If you want 20 apples, how many bags of 4 apples do you need to buy?

So for dividing by 0, how many bags of 0 apples would make up 20 apples in total? It’s impossible — however many bags of 0 apples you buy, you’ll never get any apples — you’ll certainly never get to 20 apples! So there’s no possible answer, when you try to divide 20 by 0.”

Solution 2:

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.