3.1 Number Sense and Basic Operations

Why Number Sense Anchors the Math Domain

The Mathematics domain is one of three equally weighted parts of the ParaPro Assessment (1755): about 30 questions, roughly one-third of the 90-question test. Within math, ETS splits items into two kinds — content knowledge (can you do the math?) and classroom application (can you help a student do the math?). Number sense is the foundation both halves build on, so it appears constantly. The content stays at a K-8 level, but the application questions expect you to reason like a teaching assistant supporting a struggling learner.

Number sense means understanding what numbers represent and how they relate, not just memorizing procedures. A paraeducator with strong number sense can spot that a student who writes 0.3 > 0.25 because "three is bigger than twenty-five" has a place-value gap, not a calculation error.

Place Value

Every digit's worth depends on its position. Reading place value correctly is tested directly and underlies decimals, rounding, and estimation.

In 4,523, the 4 sits in the thousands place, so its value is 4,000. Place value continues to the right of the decimal point, where each position is ten times smaller than the one before it.

In 3.45, the 5 is in the hundredths place (value 0.05) and the 4 is in the tenths place (value 0.4).

The Four Operations and Their Vocabulary

ParaPro word problems hinge on translating language into an operation, so know the result words for each:

Order of Operations (PEMDAS)

When an expression mixes operations, a fixed order prevents two people from getting different answers. The mnemonic is PEMDAS — "Please Excuse My Dear Aunt Sally."

The most-tested trap is treating multiplication and division (or addition and subtraction) as separate ranked steps. They are not — you work left to right within each pair. So 12 ÷ 4 × 3 = 3 × 3 = 9, not 12 ÷ 12 = 1.

Worked Example: Evaluate 3 + 4 × 2² − (6 − 1).

1. Parentheses: (6 − 1) = 5 → 3 + 4 × 2² − 5
2. Exponents: 2² = 4 → 3 + 4 × 4 − 5
3. Multiply: 4 × 4 = 16 → 3 + 16 − 5
4. Add/subtract left to right: 3 + 16 = 19, then 19 − 5 = 14.

Integers: Working with Negative Numbers

An integer is any whole number, positive, negative, or zero. The ParaPro tests both adding/subtracting and multiplying/dividing signed numbers.

Adding and subtracting:

• Same signs → add the values, keep the shared sign: (−3) + (−5) = −8
• Different signs → subtract the smaller value from the larger, keep the sign of the larger: (−7) + 4 = −3

Multiplying and dividing:

• Same signs → positive: (−3) × (−4) = 12
• Different signs → negative: (−3) × 4 = −12

A reliable mental shortcut: an even number of negative signs gives a positive result; an odd number gives a negative result.

Factors, Multiples, and Estimation

Factors of a number divide into it evenly: the factors of 12 are 1, 2, 3, 4, 6, and 12. Multiples are what you get by multiplying: the multiples of 4 are 4, 8, 12, 16, 20, … The greatest common factor (GCF) is the largest factor two numbers share (GCF of 12 and 18 is 6); the least common multiple (LCM) is the smallest multiple they share (LCM of 4 and 6 is 12). These ideas resurface in the next section when you find common denominators.

Rounding simplifies numbers for estimates. To round, find the target place, look one digit to its right, and round up if that digit is 5 or more, down if 4 or less.

Worked Example: Round 3,847 to the nearest hundred. The hundreds digit is 8; the digit to its right is 4. Because 4 < 5, round down → 3,800.

Estimation is its own tested skill and a habit you model for students. Before computing 48 × 21, a student should estimate 50 × 20 = 1,000 so they can reject an answer like 108 as unreasonable.

Supporting Number Sense in the Classroom

Application items reward concrete-to-abstract teaching. The best answer choices usually:

1. Use base-ten blocks or place-value charts to make grouping visible.
2. Build number lines so integers and ordering become spatial.
3. Ask students to estimate first, then compute, then compare.

Avoid answers that jump straight to abstract rules, single out a student, or simply give the answer — ETS rewards scaffolding and visualization.

Recap

Position gives a digit its value; PEMDAS sets the order of operations and you move left to right within multiply/divide and add/subtract; integer signs follow odd/even rules; rounding looks one place right; and good number-sense instruction stays visual and estimate-driven.