Cyufjy

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 $3cdot 7 = 21$.

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

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

(I hope this is better explanation?)

Older explanation:

Suppose that $1over 0$ is some number $a$. So $$1over 0 =a.$$ Remember that $$boxedbover c = diff b = ccdot d$$ So we get $$1= acdot 0=0$$ a contradiction. So $1over 0$ doesn't exist.

An explanation that might make sense to a fifth grader is one that gets to the heart of why we have invented these operations in the first place.

Multiplication is a trick we use to add similar things to form a sum. When we say 5 x 3, what we really mean is take five things of size three each and add them all together. We invented this trick because we are frequently in the situation where we have many of a similar thing, and we wish to know their sum.

Division is the same trick but the other way. When we say 15 / 3, we are asking the question "how many times would we have to add a thing of size three starting from nothing to make a thing of size fifteen?" We'd have to add five things of size three together to make a thing of size fifteen. Again, division is just a trick we use to answer questions about sums.

Now it becomes clear why division by zero is not defined. There is no number of times you can add zero to itself to get a non-zero sum.

A sophisticated fifth grader would then note that 0 / 0 is by this definition defined as zero. Going into why 0 / 0 is not defined would require more work!

For non-zero divided by zero, there is no number at all of times that you can add zero to itself to get non-zero. For zero divided by zero, every number of times you add zero to itself, you get zero, so the solution is not unique. We like our mathematical questions to have unique answers where possible and so we by convention say that 0 / 0 is also not defined.

The Wikipedia article Division by zero lists the usual arguments why there is no good choice for the result of such an operation.

I prefer the algebraic argument, that there is no multiplicative inverse of $0$,
this would need you to explain a bit about algebra.

The argument from calculus, looking at limits of $1/x$, I find also useful, but perphaps harder to explain.

How many nothings do you need to add together to get 12?

Division by zero is meaningless because that's what we decided division means. All you can do is explain why such a convention is a useful one for ordinary arithmetic.

It might even help to demonstrate some other context (e.g. arithmetic in the projectively extended number line) where it can be useful to define division by zero, so that the student is able to compare and contrast the reasons why we might or might not like to define something.

Your question might be better placed on https://matheducators.stackexchange.com/

Ask Siri.

Imagine that you have zero cookies and you split them evenly among
zero friends. How many cookies does each person get? See? It doesn't
make sense. And Cookie Monster is sad that there are no cookies, and
you are sad that you have no friends.

You shouldn't try to do that. Instead make counter question.

"What should it be, then?" and let them think about it.

I don't have kids (my wife says one 3-year-old in the house is enough for her) and it's been a while since I was in the 5th grade (although at work sometimes...), but I'll give it a go.

I know you're too old to play with blocks, but lets start with 12 blocks.

Let's start with $12/6$ - that's $2$, right? Take $6$ at a time and there are two "sets". There are $2$ sets of $6$ in $12$.

Then $12/4$ is $3$ - $3$ sets of $4$ in $12$.

Then $12/3$ is $4$ - $4$ sets of $3$ in $12$ (commutation of the last case).

Then $12/2$ is $6$ - $2$ sets of $6$ in $12$ (commutation of first case).

Then $12/1$ is $12$ - $1$ set of $12$ in $12$ (degenerate case).

Notice the size of the result set is getting bigger as the denominator (the number on the bottom) gets smaller.

Before we go to $0$ let's try something between $1$ and $0$ - $1/2$ or $0.5$. Think of just splitting each block into two (take a hatchet to the wooden blocks blocks, or just imagine it if mom doesn't want you handling a hatchet).

$12/0.5$ is $24$ - $24$ sets of $0.5$ (half-pieces) in $12$

$12/0.25$ is $48 - 48$ sets of $0.25$ (quarter-pieces) in $12$

$12/0.125$ is $96 - 96$ sets of $0.125$ (pieces of eight**) in $12$

$12/0.0625$ is $192 - 192$ sets of $0.0625$ (pieces of 16) in $12$

The close you get to zero, the larger the set you get gets.

$12/0.000000001$ (a billionth) is $12$ billion sets of a billionth of a block (aka, sawdust)

The as you approach zero, the resulting set size is too large to represent (not enough paper in this room, not enough memory on this computer) and the size of the pieces approach zero.

A cheat for "Too large to represent" is "infinity".

** pirate reference - do 5th graders still like pirates these days?

One would need to first explain what we mean by division. That is, what does $/$ mean in the expression $a/b,$ where $a$ and $b$ are integers?

Well, whatever it is, it is a way of combining two numbers. Now recall that every time we defined an operation (say addition), we always had a unique result as the product of the combination, so that we would like this to continue to hold. What else? We define $/$ indirectly, by looking at what we want $a/b$ to mean. Well, we want it to stand for the number $c$ which when multiplied together with $b$ recovers $a.$ (Recall how we similarly defined subtraction as the inverse operation of $+.$)

Therefore, in summary, if we let $a/b=c,$ then by definition this equality is equivalent to $c×b=a.$ Also, we want $c$ to be unique for all possible integers $a$ and $b.$

Now consider the expression $a/0.$ First let us take $ane0.$ Then if we let $a/0=c,$ it follows by definition that $c×0=a.$ But with the way we defined multiplication (remind him of this), we required that $0$ must make any number vanish, so that there simply is no such $c$ as we seek. If now we let $a=0,$ then we want a unique $c$ such that $c×0=0.$ But again, by the property $r×0=0,,,forall r$ which we've previously allowed in defining $×,$ we have infinitely many candidates for $c$ and there is no other condition we can impose to select one uniquely. We therefore do not allow ourselves to divide by $0$ in any case, in order to avoid all that mess.

@Jack M and and @greedoid probably highlight a good point: division does not exist. It's only the inverse operation of multiplication.

You could explain your brother the complete truth: dividing 20 by 5 is about finding the only answer (if it exists) to this question: what number can be multiplied by 5 to give 20?. The unique answer is easy: 4 times 5 is 20.

And the division is only another phrasing to say the exact same thing: 20 divided by 5 is 4.

Can you always find one and only one answer? Yup, almost always...

There's only one exception...

What number, multiplied by 0, gives 20? There's none.

So "division" by 0 has no meaning, since we cannot find any number that satisfies our definition.

You could even draw his attention by mentioning that most grown-ups don't know there's no such thing as "division", and that's the first step to learn about "E-vector spaces", "rings" and other funny-named artefacts when he's in college... or before that!

Note: what if he raises a question about "0/0"?

OK, let's try: "what number, multiplied by 0, gives 0?" All of them! We cannot find one and only one answer, so, it's still impossible to divide 0 by 0!

Division is sharing:

1 / 10:

10 boys in a class grab at a toy -- they rip the toy to tiny bits!

1 / 2:

2 boys fight for a toy -- they rip the toy in half!

1 / 0:

A different toy is alone -- he is a special boy!

Because before you think about dividing something, it is more important to consider if you have someone to divide it for (he/she/it must be present, exist, etc). If you do not have anyone who can 'benefit' from the division, no point in dividing. Non rigorous, pragmatic, heuristic approach. It might pave the way for more reasoned proofs and demonstrations.

The way I taught it, even to junior college students who were taking elementary mathematics courses, was with a calculator.

I would show them that 1/1 = 1, 1/0.1 = 10, 1/0.01 = 100, and so on. I would ask them if they saw how the numbers kept getting bigger as we divided by smaller and smaller numbers. Then I would ask them what they thought would happen when we hit zero. "We would get the biggest possible number that exists, right? But there is no biggest number. So dividing by zero gives you a number that doesn't exist. Does that make any sense? No. So we say that dividing by zero is undefined."

To divide means to subtract many times.

It might be a duplicated answer and I apologize, in case. But, according to my experience as a teacher, this worked well.

The point, as others had observed, is what does "to divide" mean. This sometimes looked obscure to the students, whereas the concept of subtraction was more clear.

Thus, once you convey the message that "to divide" means "to subtract many times", everything becomes more clear.

How many times can we subtract $3$ from $10$? Well, usually my students got this.

How many times can we subtract $0$ from $10$? Well, how many times we want!

So there is not a precise answer, because any answer is good. This made more clear the sense of "not defined", at least to my students.

Hope it helps!

Explain him the problems, don't enforce him as an "official view".

Explain him, what are the problems of the division by zero.

Let him to think about a possible solution.

You might also explain, that also the negative numbers don't have a suqare root, but this problem had a solution, the imaginary numbers. Let him try to think about a similar solution for the division by zero.

Number of marbles : Number of boxes = Number of marbles in each box.

20 marbles : 4 boxes = 5 marbles per box

0 marbles : 4 boxes = 0 marbles per box

20 marbles : 0 boxes = "how many marbles in each box while no box?" ---> undefined!