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This topic consolidates prealgebra skills with variables: evaluating expressions by substitution, simplifying by combining like terms, and applying the distributive property. In algebra 1, these skills must be automatic because they are used in nearly every subsequent topic — from solving equations to manipulating polynomials. Students should be fluent with multi-variable expressions, nested parentheses, and expressions involving rational numbers. This review also formalizes the distinction between expressions (no equals sign, cannot be "solved") and equations (has an equals sign, can be solved for a variable).
Use a diagnostic assessment to identify gaps from prealgebra. Practice problems that combine multiple skills: distribute, then combine like terms, then evaluate for given variable values. Emphasize that simplification means writing an equivalent expression with fewer terms. Include expressions with fractions and negative coefficients.
An expression is a mathematical phrase built from numbers, variables, and operations — but with no equals sign. It has a value (which may depend on the variables), but it cannot be "solved." The right word is simplified: rewriting it in an equivalent form with fewer terms or a cleaner structure. Contrast this with an equation, which has an equals sign and can be solved for a variable. This distinction matters every time you encounter a new problem: your first move is always to ask, "Is this an expression to simplify, or an equation to solve?"
You already know the two main simplification tools. Combining like terms groups terms that have identical variable parts: 5x² − 2x + 3x² + 7 simplifies to 8x² − 2x + 7. The rule is strict — terms are "like" only if every variable and every exponent match exactly. So 3x and 3x² are not like terms, and 3x and 3y are not like terms. Only the coefficients (the numerical parts) change when you combine; the variable part stays the same. The distributive property lets you remove parentheses: 2(3x − 4) = 6x − 8, because 2 multiplies every term inside. The most common error is distributing a negative: −(3x − 4) = −3x + 4, not −3x − 4. The negative multiplies both terms, flipping the sign of each.
Evaluation means substituting specific numbers for variables and computing. If f = 3x² − 2x + 1 and x = −2, substitute: 3(−2)² − 2(−2) + 1 = 3(4) + 4 + 1 = 17. Two habits prevent errors: always write parentheses around substituted values (especially negatives), and follow order of operations carefully — exponents before multiplication before addition. Using parentheses around −2 before squaring ensures you compute 3 × 4, not (3)(−2)(2) = −12.
These skills need to be automatic because algebra 1 builds on them everywhere. Solving equations requires simplifying both sides before isolating the variable. Factoring polynomials requires recognizing patterns in expressions. Graphing functions requires evaluating them at many input values. The goal of this review is not the skills themselves but what they enable: once combining like terms and distributing feel effortless, cognitive attention is free to focus on new ideas rather than algebraic bookkeeping.