2.1 Numbers and Operations

Numbers and Operations

The Praxis Elementary Education: Mathematics (5003) subtest contains 50 selected-response questions answered in 65 minutes, scored on a 100-200 scale with a passing score set by each state (most commonly 157). Numbers and Operations is the heaviest category, contributing 20 of the 50 questions (40%). An on-screen four-function calculator is available, so the test rewards conceptual reasoning over arithmetic speed.

Place Value

The base-ten system assigns each digit a value based on position. In 5,432.17, the 5 means 5,000, while the 7 means seven hundredths (0.07).

Core relationship: each place is 10 times the place to its right and 1/10 the place to its left. This is the conceptual backbone the test wants future teachers to explain, not just compute. A common trap: confusing the digit (5) with its value (5,000).

Properties of Operations

The 5003 frequently asks you to identify a property by example:

• Commutative: a + b = b + a; a × b = b × a (order does not matter)
• Associative: (a + b) + c = a + (b + c); grouping does not matter
• Distributive: a × (b + c) = (a × b) + (a × c) — the most-tested property
• Identity: a + 0 = a; a × 1 = a
• Zero property: a × 0 = 0

Note subtraction and division are neither commutative nor associative; 8 − 3 ≠ 3 − 8.

Factors, Fractions, Decimals, and Percents

Factors and Multiples

Use prime factorization to find the GCF (greatest common factor) and LCM (least common multiple). For 12 and 18: 12 = 2² × 3, 18 = 2 × 3². GCF = product of shared primes at the lowest power = 2 × 3 = 6. LCM = highest power of each prime = 2² × 3² = 36. GCF problems often hide in "largest equal groups" word problems; LCM in "events that repeat together."

Fractions

Operation rules:

• Add/subtract: convert to a common denominator, combine numerators. 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
• Multiply: multiply across; 2/3 × 4/5 = 8/15.
• Divide: multiply by the reciprocal (keep-change-flip). 2/3 ÷ 1/4 = 2/3 × 4/1 = 8/3.

The most common error the test probes: students who "flip the wrong fraction" — only the divisor (second fraction) is inverted.

Decimals and Percents

The fraction–decimal–percent triangle is essential. To convert:

• Fraction → decimal: divide numerator by denominator (3/4 = 0.75).
• Decimal → percent: multiply by 100 (0.75 = 75%).
• Percent → fraction: write over 100 and simplify (75% = 75/100 = 3/4).

Percent worked example: A $40 jacket is 25% off. Discount = 0.25 × 40 = $10, so the sale price is $30. A percent-increase trap: a 20% raise on $50 adds $10, giving $60 — not 20 added directly.

Integers and Order of Operations

Integer Rules

Elementary teachers must reason about signed numbers on a number line:

• Adding a negative moves left: 3 + (−5) = −2.
• Subtracting a negative adds: 4 − (−6) = 4 + 6 = 10.
• Same signs multiply/divide to positive; different signs to negative: (−3)(−4) = 12, but (−3)(4) = −12.

Order of Operations (PEMDAS)

1. Parentheses
2. Exponents
3. Multiplication and Division — left to right (equal priority)
4. Addition and Subtraction — left to right (equal priority)

Trap: PEMDAS does not mean all multiplication before all division. In 8 ÷ 2 × 3, work left to right: 8 ÷ 2 = 4, then 4 × 3 = 12.

Worked example: 3 + 4 × 2² − (6 − 2) = 3 + 4 × 4 − 4 = 3 + 16 − 4 = 15. Resolve the parentheses (4), the exponent (4), then multiplication (16), then add and subtract left to right.

Rounding and Estimation

The 5003 includes estimation to check reasonableness. Round 4,728 to the nearest hundred: the tens digit is 2 (< 5), so round down to 4,700. Estimation also catches absurd answer choices — if a discounted price exceeds the original, eliminate it instantly.

Exponents, Squares, and Roots

An exponent counts repeated multiplication: 2⁴ = 2 × 2 × 2 × 2 = 16. Perfect squares appear in area and Pythagorean items, so know 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. The square root undoes squaring: √49 = 7 because 7² = 49. The test also uses scientific notation for very large or small numbers: 3.4 × 10³ = 3,400, and 5 × 10⁻² = 0.05. Negative exponents indicate reciprocals: 2⁻³ = 1/2³ = 1/8.

Comparing and Ordering Rational Numbers

To order fractions, convert to a common denominator or to decimals. Comparing 2/3, 3/5, and 5/8 as decimals (0.667, 0.600, 0.625) shows 3/5 < 5/8 < 2/3. A benchmark strategy compares each fraction to 1/2: 3/8 < 1/2 because 3 is less than half of 8. On a number line, values farther right are larger, so −2 > −5 even though 5 > 2 in absolute size — a sign-reasoning trap the 5003 deliberately sets for negative integers.