Good way to convince a young kid that $0*0 = 0$?

Question

My little brother (6 years old) asked me a question ("What is $0*0$?") and gave an answer to his own question which I found ridiculous so I refuted it but he still thinks he is right.

He says that the first $0$ in $0*0$ means "no". He argues that $0$ multiplied by $6$ is $0$ because there is no 6. Then he argued that $0$ multiplied by $anything$ is $0$ because there is no $anything$. Since there is "no anything", it's nothing and thus 0. I told him it doesn't help his case of $0*0 \ne 0$ 0 that he just said $0*(anything) = 0$. Then he told me to be quiet and said that the 2nd zero is more important. Zero literally means "nothing" to my brother so he said $0*0$ means there is no "nothing" which means that it is "something" so $0*0 = something$. Since $0$ is "nothing" and he concluded that $0*0 = something$, he says my argument of $0*0 = 0$ is flawed because L.H.S. = $something$ and R.H.S. = $nothing$ which aren't equal to each other. It's good that he's questioning why things work they are but I feel like it's just wordplay and I told him that he shouldn't confuse Math and English and that he is playing with words. He is pretty stubborn and still thinks he's right. For multiplication, visual representations of what is going on is helpful to young children but $0*0$ isn't exactly the best thing to visualize. What is the best way to convince him that $0*0 = 0$, at least in the arithmetic we're dealing with in this context?

Answer 1

Try this. Let $a$ be any number. Then $a\cdot 0 = a(0 + 0) = a\cdot 0 + a\cdot 0$. Subtract to get $a\cdot 0 = 0$ for any number $a$. Hence it works for $a = 0$.

Answer 2

In my opinion, you should change his terminology away from "(any/no)-thing" and into any number.

A more correct sentence would be "any number times zero equals zero".

Then remind him that zero is a number.

Answer 3

Let's say I have a school. The school has three classrooms, with two kids in each room. (Clearly an underfunded school.) Hopefully you can convince him — perhaps drawing it might help? — that there must be $3\times2=6$ kids in the school.

Now, let's say that there is another school, but with no rooms and no kids in each room. (Horribly underfunded.) How many kids are in the school? $0$. So $0\times0=0$.

(A problem might arise if he would ask for a similar proof that $0\times2=0$, since that corresponds to "a school has has no rooms, and two kids in each room." How can there be two kids in each room if there are no rooms? I'm sure you'll manage.)

Answer 4

Try framing multiplication $a \times b$ as the area of a rectangle with sides of length $a$ and $b$.

So a $2 \times 3$ box has area 6:

A $2 \times 0$ box has area 0, as does a $0 \times 3$ box.

And a $0 \times 0$ box has area…?