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

“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.”

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.

New story

Suppose that we can divide numbers with $0$. So if I would divide $1$ with zero i would get some new number name it $a$. Now what can we say about this number $a$?

Remember:

If I divide say $21$ with $3$ we get $7$. Why? Because $3\cdot 7 = 21$.

And similiary if I divide $36$ with $9$ we get $4$. Why? Because $9\cdot 4 = 36$.

So if I divide $1$ with $0$ and we get $a$ then we have $a\cdot 0 =1$ which is clearly nonsense since $a\cdot 0 =0$.

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Old explanation:

Suppose that ${1\over 0}$ is some number $a$. So $${1\over 0} =a.$$ Remember that $$\boxed{{b\over c} = d\iff b = c\cdot d}$$ So we get $$1= a\cdot 0=0$$ a contradiction. So ${1\over 0}$ doesn't exist.