How can I explain my 9 years old brother that $8a\cdot4a \neq 64a$

Question

My youngest brother had a pre-algebra test yesterday and he was asked to tell if two expressions are equal or not.

We agreed on most of the things but on this one I find it hard to make him accept my answer. He was asked to tell if $8a\cdot4a [\space\space\space\space\space\space\space\space] 64a$ are equal or not. He checked for a specific case (one of the only 2 cases where this equation is true) and showed that for $a = 2$ the equation stands, and I find it hard to make him agree with me that if the equation stands only for $a = 0,2$ than the two expression are equal for those $a$'s only and thus the expression can't be called equal.

I think I'm having a hard time explaining him the definition of equality with variables, and here's where you enter the picture.

Answer 1

There are different kinds of so-called "equality". An identity identifies two different forms of the same expression, whether it is $$1/2 ≡ 2/4,\\1/2 ≡ 0.5,\\x+x ≡ 2x, \\(x-1)(x-2) ≡ x²-3x+2, \\f(x) ≡ x+1,$$ and so on.

Equivalence is an essential concept that is found throughout mathematics. We should have been introduced to it in the very beginning; and it would be clearer if we always used the equivalence sign for it. The $≡$ sign means that we are looking at two different forms of the same thing. Forms and equivalence are enormously important concepts.

Then there is conditional equality which would be clearer if we limited use of the equals sign to it. This is an equivalence for only certain values of the variables, if any.

Answer 2

I think he can get this if he knows the commutative and associative properties of arithmetic:

Commutative: $a \cdot b = b \cdot a$.

Associative: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$

Applied to your expression:

$(8 a) \cdot (4 a) \\\underbrace{=}_{associative} 8 \cdot (a \cdot 4) \cdot a \\\underbrace{=}_{commutative} 8 \cdot (4 \cdot a) \cdot a \\\underbrace{=}_{associative} (8 \cdot 4) \cdot (a \cdot a) = 32 a^2$