This morning, as we were driving to church, T piped up from the back seat, "How long until S's birthday?"

Simultanously, my wife said, "One week," and I said, "Seven days."

T thought about this for a few minutes, and then asked, "How many days are there in a week?"

"Seven," I said.

He thought for a little bit longer, then judiciously commented, "Then you and mama were both right!"

"That's right," I said. "We're both right, because 'one week' and 'seven days' mean exactly the same thing. They're just different ways of saying the same thing."

This idea of "two things that look different but really aren't" is a central concept of algebraic manipulation. It's also a concept that Algebra students sometimes miss out on, or easily forget. I find sometimes that students are going through motions of simplifying expressions, or solving equations, without really understanding that what they're doing is writing exactly the same thing in different ways. This leads to them doing strange things that change the value of the expression. 

Students should be taught and reminded repeatedly that our properties of arithmetic and algebra are simply ways of writing the same thing in different ways.

The simplest example of this is to recognize that 1 + 3 and 4 are two expressions that look very different, but have exactly the same meaning. In the same way, algebra students will learn that like terms can be combined. What does that mean? It means that 1x + 3x and 4x are exactly the same thing, even though they look different.

What is the distributive property? It is a rule that allows you to take an unfactored expression and write it as a factored expression that means the same thing, but just looks different. If you have the expression 3x + 5x² the distributive property lets you rewrite this as x(3 + 5x). This is not a different expression; its the same expression written in a different way. It's like saying "seven days" instead of "one week." The two phrases look and sound very different, but they have the same meaning.

And since this is an equivalence, it also works the other way: if we start with x(3 + 5x), we can rewrite it as 3x + 5x².

The same is true for every rule of Algebraic manipulation. The additive property of equality is really saying that 

x - 5 = 7 is exactly the same equation as x = 12. They look very different, but they mean exactly the same thing.

Finally, all of algebra is built (in a sense) on the concept of substitution. The substitution property tells us that if we have two expressions that mean the same thing, then we can write one in place of the other. If "one week" and "seven days" mean the same thing, I can replace one with the other in a sentence. If x and y + 2 mean the same thing, I can replace x with (y + 2) in any expression.

With this in mind, the properties we introduce should not be seen as restrictions on what we can do - they are open doors that give us permission to do something. Knowing that "one week" means the same thing as "seven days" gives us permission to replace "one week" with "seven days" or vice-versa. Knowing the destributive property gives us the freedom to replace 3x + 6y with 3(x + 2y) - and vice-versa.

This should be emphasized with every rule that crops up along the way. I'm going to start using the "one week" is the same as "seven days" illustration every time I introduce a new rule. My students will probably get sick of hearing it, but hopefully it'll help cement in their minds the concept that we're not changing the meaning of an expression by following the rules.