3.4 Algebra Basics

From Arithmetic to Algebra

Algebra is where many students first meet abstract symbols, so the ParaPro checks both that you can do the algebra and that you can ease students into it. The bridge from arithmetic is the variable — a letter that stands for an unknown or changing number. In the expression 3x + 5, the letter x is the variable, 3 is its coefficient (the number multiplying it), and 5 is a constant.

Expressions vs. Equations

The single most important distinction is whether an equals sign is present.

Evaluating Expressions

To evaluate an expression, substitute a given value for the variable and apply order of operations.

Worked Example: Evaluate 2x + 3 when x = 4. Replace x: 2(4) + 3 = 8 + 3 = 11. When x = 7, the same expression gives 2(7) + 3 = 17 — the value changes because x changed.

Combining Like Terms

Like terms have the same variable raised to the same power. You can only combine like terms, and you do so by adding or subtracting their coefficients.

So 4x + 3 + 2x − 1 simplifies to (4x + 2x) + (3 − 1) = 6x + 2.

Solving Equations with Inverse Operations

The golden rule is balance: whatever you do to one side of an equation, do to the other. To isolate the variable, undo operations using their inverses — addition undoes subtraction, and multiplication undoes division.

One-step equations:

Two-step equations undo addition/subtraction first, then multiplication/division — the reverse of PEMDAS.

Worked Example: Solve 2x + 5 = 13.

1. Subtract 5 from both sides: 2x = 8.
2. Divide both sides by 2: x = 4.
3. Check by substituting: 2(4) + 5 = 8 + 5 = 13 ✓.

Always check — substituting the answer back catches most arithmetic slips and is a habit you model for students.

Translating Words into Algebra

Word problems hinge on turning phrases into symbols. Memorize the keyword-to-operation map.

Worked Example: "Seven less than three times a number is 14." Translate: 3x − 7 = 14. Solve: add 7 → 3x = 21; divide by 3 → x = 7. (Note that "less than" reverses order: it is 3x − 7, not 7 − 3x.)

Patterns and the Coordinate Plane

Patterns are the entry point to algebraic thinking. A function table lists inputs and outputs; finding the rule that links them is early algebra.

The output rises by 2 each time and starts 3 above twice the input, so the rule is output = 2n + 3. Recognizing such rules prepares students for variables.

The coordinate plane is formed by a horizontal x-axis and a vertical y-axis that cross at the origin (0, 0). A point is an ordered pair (x, y): move right/left by the x-value, then up/down by the y-value. The plane has four quadrants, numbered I–IV counterclockwise from the upper right.

Worked Example: To plot (3, −2), start at the origin, move 3 units right (positive x), then 2 units down (negative y). The point lands in Quadrant IV. The order is fixed: x always comes first.

Supporting Algebra Readiness

Application items reward building from the concrete to the symbolic:

1. Use manipulatives (counters, algebra tiles) so a variable becomes "a hidden number of objects" before it is a letter.
2. Use a balance scale so students feel that both sides must stay equal — adding to one side requires adding to the other.
3. Start equation-solving with patterns and function tables, then introduce a variable for the rule.
4. Teach the check-by-substitution habit so students self-verify.
5. Make word-problem translation a two-column activity (phrase ↔ symbol) and watch for the "less than" reversal.

Avoid answers that present rules as tricks with no meaning — ETS rewards understanding the equality and inverse-operation logic.

Recap

Variables stand for unknowns; expressions are simplified or evaluated while equations are solved; isolate the variable with inverse operations and always check; translate keywords carefully (mind "less than"); read patterns as rules; and plot ordered pairs x-then-y. Effective readiness work is concrete-first and pattern-driven.