Use properties of equality to justify each step of an algebraic proof, written in two-column form.
A proof is simply a sequence of logical steps, each one justified by a definition, postulate, theorem, or property, that leads from what is given to what you want to show. Before tackling geometric proofs, we practice the format on algebra you already know how to do, since the steps are familiar and all the attention can go to the JUSTIFICATIONS. A two-column proof lists statements on the left and the reason for each statement on the right, starting from the given information and ending at the goal.
The reasons come from the properties of equality: the Addition and Subtraction Properties (adding or subtracting the same number from both sides), the Multiplication and Division Properties (doing the same to both sides), the Distributive Property, and the Substitution Property (replacing a quantity with something equal to it). These are exactly the moves you already make solving an equation -- a proof just asks you to name the move every single time instead of doing it silently in your head.
Mastery looks like: They write a complete two-column algebraic proof, correctly citing the property of equality that justifies each step.
Common stumbles: Skipping a step because it "seemed obvious"; citing the wrong property (especially confusing Addition/Subtraction Property with Multiplication/Division Property).