Justifying Each Step

In this lesson you will: Name the property that makes each step in a solution valid; Write a short justification for a solution method; Judge whether someone else's step is legal, and say why if it is not.

Every step has a name

A.REI.1 asks you to explain each step in solving an equation as following from the equality asserted at the previous step. In practice that means: for every line you write, you should be able to say which rule let you write it.

There are only a handful of rules, and you already use all of them.

Two kinds of move, and the difference matters

Look at the caption above again. The first four properties do something to both sides of the equation. The last three rewrite one side into an equivalent form without touching the other.

That distinction is what most justification questions are really testing. Distributing is not \"doing the same thing to both sides\" — it is rewriting one side, and the equation stays true because the rewritten side equals what it replaced.

Common mistake — Calling every step "the distributive property"

Asked to justify 3x+5x=8x, a student writes \"distributive property.\"

It is an understandable guess, because the distributive property is the one with a name everybody remembers, and combining like terms does feel related to it.

But this step did not distribute anything — nothing was multiplied across a sum. Two like terms were added, using the fact that 3x+5x=(3+5)x. Naming it correctly is combining like terms.

There is a second version of this error: calling x+0=x the identity property when the step actually added zero to both sides.

How to catch it: Point at the parentheses. If there are none in the line you are justifying, and none disappeared, the distributive property was not what you used.