2.2 Algebraic Thinking

Algebraic Thinking

Algebraic Thinking supplies 15 of the 50 questions (30%) on the 5003 subtest. The emphasis is on recognizing structure — patterns, relationships between quantities, and how a change in one variable affects another — because these are the early-algebra skills elementary teachers build.

Patterns and Sequences

To extend a sequence, first determine whether the change is additive or multiplicative. For 3, 6, 12, 24, the values double, so it is geometric (ratio 2); the next term is 48. The nth-term idea also appears: in 2, 5, 8, 11 the rule is 3n − 1, so the 10th term is 3(10) − 1 = 29. A trap is assuming a sequence is arithmetic when consecutive differences are not equal.

Variables and Expressions

A variable is a symbol standing for an unknown quantity; an expression combines numbers, variables, and operations without an equals sign (for example, 3x + 5). Translating English to algebra is heavily tested:

Order trap: "6 less than a number" is n − 6, not 6 − n. The phrase reverses the written order. To evaluate, substitute and follow order of operations: if x = 4, then 3x + 5 = 3(4) + 5 = 17.

Equations, Inequalities, and Tables

Solving Equations

An equation states two expressions are equal. Solve by performing inverse operations on both sides to isolate the variable, undoing operations in reverse PEMDAS order.

Worked example: 2x + 3 = 11.

1. Subtract 3 from both sides: 2x = 8.
2. Divide both sides by 2: x = 4.

Check by substituting: 2(4) + 3 = 11. ✓ Always verify; the test includes near-miss distractors that result from skipping a step.

Inequalities

Inequalities use <, >, ≤, ≥, or ≠ and are solved like equations with one critical rule: when you multiply or divide both sides by a negative number, reverse the inequality symbol. Example: −2x < 6 → divide by −2 and flip → x > −3. Forgetting to flip is the single most common inequality error the 5003 targets.

Function (Input/Output) Tables

A function table encodes a rule mapping each input x to one output y.

The output rises by 3 for each step in x (slope 3), and when x = 0 the pattern gives 1, so the rule is y = 3x + 1. Test it: 3(2) + 1 = 7. ✓ Strategy: find the constant change (the rate), then back out the starting value.

The Coordinate Plane

The coordinate plane is formed by a horizontal x-axis and vertical y-axis meeting at the origin (0, 0). A point is named by an ordered pair (x, y): move x units horizontally first, then y units vertically.

Elementary-grade standards emphasize Quadrant I (whole-number coordinates), so many 5003 items stay positive, but you should know all four quadrants and that a point on an axis (such as (4, 0)) lies on no quadrant.

Graphing Linear Relationships

Plotting a function table produces a straight line for linear rules. For y = 2x, the points (1, 2), (2, 4), (3, 6) climb steadily — a proportional relationship passing through the origin. The test connects tables, graphs, and equations as three representations of the same pattern, asking which graph matches a given table.

Common trap: reversing the coordinates. Plotting (3, 5) means right 3, up 5 — not right 5, up 3. Always read x before y. Distance along an axis between two points sharing a coordinate (such as (2, 1) and (2, 6)) is found by subtracting the differing coordinate: 6 − 1 = 5 units.

Properties and Proportional Reasoning in Algebra

The 5003 blends algebra with proportional reasoning. A proportion sets two ratios equal — x/4 = 6/8 — solved by cross-multiplying: 8x = 24, so x = 3. This underlies scaling, rate, and unit-price problems: if 3 notebooks cost $4.50, then 1 costs $1.50 and 7 cost $10.50.

Watch for two-step word problems that require building the equation first. "A taxi charges a $3 base fee plus $2 per mile; the fare was $17" becomes 2m + 3 = 17, so m = 7 miles. The algebraic-thinking category rewards modeling a situation symbolically, then solving — not just computing a given equation. A frequent distractor reflects the answer to the wrong step, such as 2m = 14 returned as m = 14.

Properties Applied to Variables

The distributive property simplifies expressions: 3(x + 4) = 3x + 12. Combining like terms is also tested: 5x + 2 + 3x − 1 = 8x + 1; only terms with the same variable part combine, so 8x and 1 cannot merge further. Recognizing that 8x + 1 is fully simplified, while 8x and 1 stay separate, is a structural insight the exam expects of future teachers.