Order of Operations and Signed Numbers

1.1: Order of Operations and Signed Numbers. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 1 of 53 parts Practice Flashcards

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Signed-number arithmetic

Objective: Add and subtract signed numbers accurately.

Prerequisite skills: integer addition, subtraction, multiplication, and division. Subtracting a number means adding its opposite: a − b = a + (−b). Opposite-signed addends partially cancel, while equal-signed addends combine their magnitudes. Use a number line to interpret direction. Common error: treating subtraction of a negative as another negative. Restrictions and interpretation: Evaluate powers before a leading negative; division by zero is undefined.

Formulas

$a − (−b) = a + b$

Worked examples

Try itEvaluate −29 − (−24).Show answer

Rewrite subtraction as addition: −29 + 24. The first magnitude is 5 larger, so the result is −5. Therefore the requested answer is -5.

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Signed-number arithmetic

Objective: Add and subtract signed numbers accurately.

Formulas

$a − (−b) = a + b$

Worked examples

Try itEvaluate −29 − (−24).Show answer

Rewrite subtraction as addition: −29 + 24. The first magnitude is 5 larger, so the result is −5. Therefore the requested answer is -5.

Grouping and powers

Objective: Evaluate expressions with parentheses and exponents.

Parentheses establish a group; powers act on their bases; multiplication and division come next from left to right; addition and subtraction follow from left to right. A superscript applies only to its stated base. Common error: multiplying before evaluating a grouped power. Restrictions and interpretation: Evaluate powers before a leading negative; division by zero is undefined.

Formulas

$a(b + c)² means a × ((b + c)²)$

Worked examples

Try itEvaluate 3(24 − 27)² + 22.Show answer

The grouped difference is −3. Squaring gives 9. Multiply by 3 to get 27, then add 22: 49. Therefore the requested answer is 49.

Nested operations and signs

Objective: Distinguish a negative base from a negative sign outside a power.

The expressions −a² and (−a)² differ: the former is the opposite of a², while the latter squares a negative base and is positive. Keep intermediate fractions grouped until the final arithmetic. Common error: squaring a leading negative that is outside the base. Restrictions and interpretation: Evaluate powers before a leading negative; division by zero is undefined. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$−a² = −(a²), but (−a)² = a²$

Worked examples

Try itEvaluate −25² + (6 × 25)/3.Show answer

The first term is −(25²) = -625. The quotient is 150/3 = 50. Add to obtain -575. Therefore the requested answer is -575.

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