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.

  • $\begingroup$ "=" means "always equal", not "sometimes equal". Try literally any other value of $a$. $\endgroup$ – vadim123 Jan 6 '14 at 16:29
  • $\begingroup$ I tried always, I also tried to demonstrate the definition as a scale where the two sides of the scale must always weight the same, no matter what weight they carry. $\endgroup$ – Georgey Jan 6 '14 at 16:32
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    $\begingroup$ Try this: $a[ ~~~]5$. Would he say "equal" for this one too? $\endgroup$ – vadim123 Jan 6 '14 at 16:33
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    $\begingroup$ @vadim123 haha that actually worked! (he's sitting right next to me right now) $\endgroup$ – Georgey Jan 6 '14 at 16:35
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    $\begingroup$ I think your little brother is perplexed by the subtle difference between an equation and an identity. $\endgroup$ – Lucian Jan 6 '14 at 20:38

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.


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:

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

  • $\begingroup$ I'm not sure he does but that might be a great opportunity to teach him those cornerstones of mathematics and algebra especially. $\endgroup$ – Georgey Jan 6 '14 at 16:33
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    $\begingroup$ Really? For a 9 years old? $\endgroup$ – JPLF Jan 6 '14 at 16:36
  • $\begingroup$ @JPLF, I can give it a shot, doesn't it worth a try? $\endgroup$ – Georgey Jan 6 '14 at 16:36
  • $\begingroup$ @JPLF: I saw "pre-algebra test." My kids learned these properties in pre-algebra. $\endgroup$ – Ron Gordon Jan 6 '14 at 16:36
  • $\begingroup$ People learn these properties in pre-algebra, but I think most of them fail to get understanding out of them until much later. Perhaps that's OK-surely it's better for him to get the right answer, and later on start getting it for the right reasons. $\endgroup$ – Kevin Carlson Jan 10 '14 at 23:50

This is a late answer, but I think none of the others address the real issue. Basically, your brother agrees that if $a=2$, then $4a \times 8a = 64a$. But when people write "$4a \times 8a = 64a$" all by itself, they implicitly are stating "$4a \times 8a = 64a$ for every $a$", where the type of $a$ is inferred from the context. In this case every object is assumed to be a (real) number if not specified. It will no longer be the case when he learns about other kinds of mathematical objects, so it would be good to point out these kinds of implicit assumptions. In other words, it is a matter of convention in what it means rather than something you can ever convince someone of. As for formalism, I hope he understands that mathematics is about logical reasoning and not guessing and so he would in time appreciate formal manipulation using the field axioms.


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