GLOSSARY.PLAIN + SHORT.

A letter that holds a number you do not know yet. A placeholder, not a mystery.

A math phrase with no equals sign. Something to read, not something to balance.

One piece of an expression, separated from the others by + or −.

A term with no variable attached. Its value never changes.

The number multiplying a variable. It tells you how many of that variable you have.

Anything being multiplied. In 3x the 3 and the x are both factors, which is why 3x means 3 × x and not 3 and then x.

Replacing a variable with a specific number, without changing the operation around it.

Substitute the numbers first, then do the arithmetic until one value is left. Evaluating always ends with a number.

An expression built only from numbers — no variable anywhere in it. It has one value, and order of operations is how you find it.

The agreed order everyone follows so an expression has exactly one value: grouping, then exponents, then multiply and divide, then add and subtract.

Rewrite an expression in a shorter equal form. The value stays the same — only the way it looks changes.

Parentheses, brackets, and the fraction bar. They say “do me first,” whatever the operations inside them happen to be.

The small raised number that says how many times to multiply the base by itself. It is a count of factors, not a multiplier.

The number or variable that an exponent acts on — the thing being multiplied by itself.

Adding 0, or multiplying by 1, leaves a number exactly as it was. Nothing changes, so nothing has to be rewritten.

Multiplying anything by 0 gives 0. The variable does not survive the trip.

Order does not matter for addition or multiplication. It does matter for subtraction and division, every single time.

Regrouping with parentheses does not change the result — as long as the expression is all addition or all multiplication.

Multiply the outside number by every term inside the parentheses — every one, including the second one.

Two expressions that give the same value for every number you put in. They look different; they are not different.

Terms with the exact same variable part. Only like terms can be combined.

Terms whose variable parts do not match. They can sit in the same expression forever without ever combining.

A claim that two expressions are equal. Because it is a claim, you can check it — and balance it.

The number that makes an equation true. Put it back in and both sides name the same number.

Find every value that makes the equation true. Solving ends with the variable alone on one side.

The opposite move that undoes an operation. Add undoes subtract, multiply undoes divide.

Both sides of an equation name the same number, so any move you make has to happen on both sides.

Get the variable alone on one side with nothing attached to it. That is the whole goal of solving.

One thing is attached to the variable, so one inverse operation solves it.

Two things are attached to the variable. Undo them in reverse order — the constant first, then the coefficient.

An equation you tidy before you undo anything: distribute, then combine like terms, then solve.

Substitute your answer back into the original equation and confirm both sides match. It is the last step, not an optional one.

Equations with exactly the same solution. Every legal move you make creates another one.

The variable appears on the left and the right. Collect the variable terms on one side first, then solve as usual.

The variables cancel and what is left is false, so no number can make the equation true.

The variables cancel and what is left is true, so every number works.

An equation that is true for whatever the variable is. Both sides are the same expression underneath, so it has infinitely many solutions.

Say in words what the letter stands for before you write the equation. A letter with no job is how a word problem goes wrong.

A comparison that says one side is bigger or smaller, not equal. It has a range of answers rather than one.