Counting & Whole Number Operations
Whole number operations form the bedrock of elementary mathematics, and the Praxis 5008 tests not just your ability to compute—but your ability to teach computation in ways that build lasting conceptual understanding. You need to know why regrouping works, how to model multiplication with arrays, and what to say when a student insists "multiplying always makes numbers bigger." This lesson covers every major whole-number concept tested on the exam, with attention to student misconceptions and how to address them in instruction.
Learning Outcomes
- Distinguish cardinal, ordinal, and nominal uses of numbers and explain their classroom significance
- Use expanded form, standard form, and word form interchangeably for whole numbers through the millions
- Apply and explain the commutative, associative, distributive, identity, and zero properties of operations
- Model multi-digit addition, subtraction, multiplication, and division using multiple representations
- Find GCF and LCM using prime factorization and explain why each is useful
- Identify and address the most common student misconceptions about whole number operations
- Apply order of operations correctly and diagnose PEMDAS errors in student work
(1) Number Sense and Types of Numbers
(A) Cardinal, Ordinal, and Nominal Numbers
Cardinal numbers tell how many — they answer the question "how many objects are there?" and represent quantity (1, 2, 3, 47, 1,000). Ordinal numbers communicate position or rank in a sequence — first, second, third, 21st. Nominal numbers act as labels or identifiers with no quantitative meaning — a jersey number 23, a zip code 90210, or a bus route 7. Students intuitively understand cardinal numbers first; ordinal language is often acquired through classroom routines like "line up first, second, third."
- Cardinal example: "There are 8 crayons in the box." — quantity
- Ordinal example: "Marcus finished 3rd in the spelling bee." — rank
- Nominal example: "My locker number is 247." — label, not quantity
On the Exam: Questions may present a scenario and ask which type of number is being used, or ask which instructional context introduces ordinal numbers. Watch for "label" scenarios — phone numbers, social security numbers, and bus routes are nominal even though they contain digits.
(B) Skip Counting
Skip counting means counting forward or backward by a fixed interval rather than by ones, building multiplicative reasoning before students learn formal multiplication tables. Counting by 2s (2, 4, 6, 8…), 5s (5, 10, 15…), 10s (10, 20, 30…), and 100s (100, 200, 300…) develops number pattern recognition and is a bridge to understanding multiplication as repeated addition.
- Pedagogical value: Students who skip-count fluently by 5s understand that 7 × 5 = 35 long before they memorize the fact — they have a counting strategy to fall back on
- Hundred chart connection: Shading every 5th number on a 100-chart makes the visual pattern of multiples concrete
- Number lines: Marking equal jumps on a number line makes the interval explicit and visual
COMMON TRAP: Students sometimes shift intervals mid-count (e.g., 5, 10, 15, 25…). A teacher named Ms. Rivera addresses this by having students tap a steady beat while counting — the rhythm reinforces the constant interval.
(2) Place Value in Whole Numbers
(A) The Place Value Chart
Place value is the system by which a digit's position in a number determines its value — the digit 4 in 4,000 represents four thousands, while the same digit in 40 represents four tens. Understanding place value is the single most important whole-number concept because it underlies every algorithm and estimation strategy students will use.
| Period | Place | Value | Example digit = 3 |
|---|---|---|---|
| Millions | Millions | 1,000,000 | 3,000,000 |
| Thousands | Hundred-thousands | 100,000 | 300,000 |
| Thousands | Ten-thousands | 10,000 | 30,000 |
| Thousands | Thousands | 1,000 | 3,000 |
| Ones | Hundreds | 100 | 300 |
| Ones | Tens | 10 | 30 |
| Ones | Ones | 1 | 3 |
- Standard form: 4,382 — digits written in their place positions
- Expanded form: 4,000 + 300 + 80 + 2 — each digit shown as its place-value quantity
- Word form: "four thousand, three hundred eighty-two" — commas signal the periods (millions, thousands, ones)
On the Exam: You may see a number written in one form and asked to convert it to another, or a scenario where a student writes 305 as "3005" (confusing hundreds and thousands). Recognize that the student has placed a 0 in thousands when there is none — a place-value gap error, best addressed with base-ten blocks.
(B) Comparing and Ordering Whole Numbers
Comparing whole numbers requires looking left-to-right through place values — the number with more digits is always larger; for equal-length numbers, compare digit-by-digit from left to right. The symbols < (less than), > (greater than), and = (equal) communicate the relationship.
- Number line strategy: Place both numbers on a number line — the number farther right is always greater. This spatial representation helps visual learners and concretizes the abstract comparison.
- Aligning by place value: Writing numbers vertically aligned to their ones place before comparing prevents digit-count confusion (e.g., comparing 999 vs. 1,001).
- Classroom example: Mr. Torres asks second-graders to use base-ten blocks to build 342 and 324, then determine which pile has more — students see that the hundreds match, so they count tens and find 4 tens > 2 tens.
(3) Properties of Operations
(A) The Five Core Properties
The properties of operations are mathematical laws that govern how numbers behave under addition and multiplication — they justify every flexible computation strategy and mental math shortcut students will ever use.
| Property | Definition | Example | Applies To |
|---|---|---|---|
| Commutative | Order doesn't change the result | 3 + 5 = 5 + 3 ; 4 × 6 = 6 × 4 | Addition, Multiplication |
| Associative | Grouping doesn't change the result | (2 + 3) + 4 = 2 + (3 + 4) | Addition, Multiplication |
| Distributive | Multiply across a sum or difference | 3 × (4 + 5) = 3×4 + 3×5 = 27 | Multiplication over + or − |
| Identity | Operating with the identity leaves the number unchanged | 7 + 0 = 7 ; 9 × 1 = 9 | Add (0), Multiply (1) |
| Zero Property | Any number multiplied by 0 is 0 | 5,000,000 × 0 = 0 | Multiplication only |
COMMON TRAP: The commutative and associative properties do NOT apply to subtraction or division. Students (and some test-takers) assume 8 − 3 = 3 − 8. It does not. 10 ÷ 2 ≠ 2 ÷ 10. Always verify which operations each property governs before answering a scenario question.
On the Exam: The distributive property is tested heavily in the context of mental math and area models. Know that 6 × 13 = 6 × 10 + 6 × 3 = 60 + 18 = 78. This exact decomposition is what the area model makes visible. Exam questions often ask which property justifies a computational shortcut.
(4) Multi-Digit Addition and Subtraction
(A) Regrouping (Carrying and Borrowing)
Regrouping is the process of exchanging equivalent groups across adjacent place values — trading 10 ones for 1 ten when adding, or trading 1 ten for 10 ones when subtracting. The traditional terms "carrying" and "borrowing" are discouraged in modern math education because they obscure the mathematical meaning; "regrouping" makes the place-value exchange explicit.
- Addition with regrouping: 47 + 38 — ones: 7 + 8 = 15, write 5 in ones place, regroup 1 ten to the tens column; tens: 4 + 3 + 1 (regrouped) = 8; result: 85
- Subtraction with regrouping: 52 − 28 — ones: 2 < 8, regroup 1 ten from tens place, making 12 ones and 4 tens; ones: 12 − 8 = 4; tens: 4 − 2 = 2; result: 24
- Base-ten blocks: The most powerful concrete model — physically swapping a rod (ten) for 10 unit cubes makes regrouping visible and tangible before moving to paper algorithms
- Open number lines: Useful for flexible, left-to-right addition/subtraction without the column algorithm — students see a running total and choose their own decompositions
COMMON TRAP — Student Misconception: When subtracting across a zero (e.g., 300 − 147), students often subtract the bottom digit from the top regardless of which is larger: they write 300 − 147 = 247 by computing 3 − 1 in hundreds, 0 − 4 = "can't do it so write 4," 0 − 7 = "can't do it so write 7." The fix: use base-ten blocks to physically regroup 1 hundred → 10 tens → 10 ones before subtracting.
(5) Multiplication
(A) Conceptual Models for Multiplication
Multiplication is repeated addition of equal groups, but it also represents area, combinations, and scaling — and research shows that students who understand multiple representations of multiplication are more flexible problem-solvers. The Praxis 5008 expects you to know which model is best suited to a given learning goal.
3 × 4 = three groups of four. Best for word problems about bags of apples, rows of chairs. Emphasizes the multiplicative structure of repeated addition.
3 rows × 4 columns = 12 total. Visually shows commutativity — rotate the array 90° and you see 4 × 3. Used heavily in grades 2–3.
A rectangle with dimensions equal to the factors — area equals the product. Directly connects to the distributive property and extends to multi-digit multiplication: 23 × 14 becomes (20 + 3)(10 + 4) = 200 + 80 + 30 + 12 = 322.
Break each factor by place value and multiply each part: 23 × 14 → (20 × 10) + (20 × 4) + (3 × 10) + (3 × 4) = 200 + 80 + 30 + 12 = 322. Bridges the area model to the standard algorithm.
COMMON TRAP — "Multiplication always makes bigger": This is one of the most pervasive student misconceptions in elementary math. It is true for whole numbers greater than 1 — but when multiplying a positive number by a positive fraction or decimal less than 1, the product is smaller than the original positive number. While Praxis 5008 tests whole-number operations specifically, expect scenario questions where a student generalizes incorrectly, and know how to address it: use the area model with unit fractions to show shrinking area.
(6) Division
(A) Division Concepts and Models
Division has two distinct interpretations that students must understand separately: partitive division (sharing into equal groups — "24 cookies shared among 6 children, how many each?") and measurement/quotitive division (repeated subtraction — "24 cookies packed in bags of 6, how many bags?"). Both interpretations yield the same equation 24 ÷ 6 = 4, but they represent different real-world structures.
- Partitive (fair sharing): The divisor tells the number of groups; the quotient tells the size of each group
- Quotitive (measurement): The divisor tells the size of each group; the quotient tells the number of groups
- Repeated subtraction: 24 − 6 − 6 − 6 − 6 = 0 (subtracted 4 times) — connects division to subtraction and skip counting
- Long division algorithm: Divide → Multiply → Subtract → Bring down (DMSB) — the standard algorithm for multi-digit division; should only be taught after students understand the concept through models
- Remainders: What remains after equal sharing; in context, the remainder may be dropped (how many full buses?), rounded up (how many buses needed?), or expressed as a fraction/decimal
COMMON TRAP — "Division always makes smaller": Like the multiplication misconception, this is only true when dividing a whole number by a whole number greater than 1. Dividing by a fraction yields a larger quotient (e.g., 4 ÷ ½ = 8). Exam scenarios may ask how to address this when students overgeneralize.
On the Exam: Questions frequently ask which model (partitive vs. quotitive) matches a given word problem. "Ms. Chen has 36 stickers to share equally among 9 students" — partitive. "Ms. Chen wants to give each student 9 stickers and has 36 total — how many students can she give to?" — quotitive. The equation is the same; the meaning differs.
(7) Factors, Multiples, GCF, and LCM
(A) Factors and Multiples
Factors are whole numbers that divide evenly into a given number — 1, 2, 3, 4, 6, and 12 are all factors of 12. Multiples are the products of a number and any whole number — multiples of 4 are 4, 8, 12, 16, 20…. Students confuse these two constantly, so clear vocabulary instruction matters.
The largest factor shared by two or more numbers. GCF(12, 18) = 6. Used for simplifying fractions and solving "how to divide into equal groups" problems. Methods: list all factors, use prime factorization, or the Euclidean algorithm.
The smallest positive multiple shared by two or more positive whole numbers. LCM(4, 6) = 12. Used for adding/subtracting fractions with unlike denominators and solving "when will events coincide?" problems. Methods: list multiples, use prime factorization.
Prime factorization is the most reliable method for both GCF and LCM:
- GCF: write prime factorizations of both numbers; multiply the shared prime factors (lowest powers)
- LCM: write prime factorizations; multiply all prime factors at their highest powers
- Example — GCF(12, 18): 12 = 2² × 3; 18 = 2 × 3². Shared: 2¹ × 3¹ = 6. LCM: 2² × 3² = 36
On the Exam: Real-world GCF problems involve "splitting into equal groups" — "Jada has 24 red and 36 blue marbles and wants to sort them into identical bags with no leftovers; what is the greatest number of bags?" (GCF = 12). Real-world LCM problems involve "when do cycles align?" — "Buses on Route A come every 4 minutes, Route B every 6 minutes; when do they both arrive together?" (LCM = 12).
(B) Prime and Composite Numbers
A prime number has exactly two distinct factors: 1 and itself (2, 3, 5, 7, 11, 13, 17, 19, 23…). A composite number has more than two factors (4, 6, 8, 9, 10…). The number 1 is neither prime nor composite — it has exactly one factor.
- Why 2 is special: 2 is the only even prime number. All other positive even integers greater than 2 are composite because they share the factor 2.
- Sieve of Eratosthenes: A method for finding all primes up to a given number — cross out multiples of each prime starting from 2; what remains uncrossed are primes. Great visual activity for grades 4–5.
- Prime factorization: Every composite number can be expressed as a unique product of primes (Fundamental Theorem of Arithmetic). Factor trees make this concrete: 36 = 2 × 2 × 3 × 3 = 2² × 3².
COMMON TRAP: Students frequently classify 1 as prime (it has only "one" and "itself," which are the same). The correct answer is that 1 is neither prime nor composite. Exam questions sometimes embed this in a scenario where a student lists 1 as a prime — know how to correct this misconception.
(8) Order of Operations
(A) PEMDAS / GEMDAS
The order of operations is a convention that establishes which calculations to perform first when an expression contains multiple operations — without it, the same expression could yield different results depending on who evaluates it. PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) is the standard mnemonic; GEMDAS replaces P with G for "Grouping symbols" to include brackets and fraction bars.
| Step | Symbol | What it includes | Priority |
|---|---|---|---|
| Parentheses/Grouping | ( ) [ ] { } | Innermost first, then outward | 1st |
| Exponents | x² | Powers and roots | 2nd |
| Multiplication & Division | × ÷ | Left to right — equal priority | 3rd |
| Addition & Subtraction | + − | Left to right — equal priority | 4th |
Worked example: Evaluate 3 + 4 × 2 − (5 − 1)²
- Step 1 — Parentheses: (5 − 1) = 4 → expression becomes 3 + 4 × 2 − 4²
- Step 2 — Exponents: 4² = 16 → expression becomes 3 + 4 × 2 − 16
- Step 3 — Multiplication: 4 × 2 = 8 → expression becomes 3 + 8 − 16
- Step 4 — Left to right: 3 + 8 = 11; 11 − 16 = −5
COMMON TRAP — M before D, A before S: Students treat PEMDAS as a strict left-to-right hierarchy and always multiply before dividing. In reality, multiplication and division are performed left to right with equal priority. So 12 ÷ 4 × 3 = (12 ÷ 4) × 3 = 3 × 3 = 9, not 12 ÷ 12 = 1. Same rule applies to addition and subtraction. The Praxis WILL test this.
Chapter Summary
- Cardinal, ordinal, nominal — Cardinal = how many; ordinal = position; nominal = label with no quantitative meaning
- Place value is everything — A digit's position determines its value; every algorithm depends on this understanding; base-ten blocks make it concrete
- Properties justify mental math — Commutative and associative apply to addition and multiplication only; distributive property underpins area models and partial products
- Regrouping, not borrowing — Frame column arithmetic as place-value exchange to preserve mathematical meaning
- Four multiplication models — Equal groups, arrays, area model, partial products; each targets a different aspect of multiplicative structure
- Two division interpretations — Partitive (fair sharing) and quotitive (measurement); same equation, different meanings; know how to identify each from a word problem
- GCF vs. LCM — GCF for equal-group splitting; LCM for cycle alignment; prime factorization finds both reliably
- 1 is neither prime nor composite — Two distinct factors required for prime; 1 has only one
- PEMDAS left-to-right rule — Multiplication and division equal priority left-to-right; same for addition and subtraction; order within each tier matters
- Address "always bigger / always smaller" early — These multiplication and division misconceptions cause persistent errors; use area models and real-world contexts to correct them
Test Ready Tips
- When a scenario describes a student error, always ask "what does this error reveal about the student's understanding?" — then match the error to a specific misconception and an appropriate intervention model
- Praxis 5008 favors questions that show MULTIPLE representations — if you can connect a concept to a concrete model (blocks), a pictorial model (arrays), and an abstract model (equation), you are thinking like the test writers want
- For GCF/LCM word problems, key language: "split equally" or "divided evenly" → GCF; "when will they coincide" or "how often together" → LCM
- For order of operations, always check if M and D appear together in the same expression — evaluate left to right when they do
- Know that the distributive property is the mathematical foundation behind every multi-digit multiplication strategy — area model, partial products, and the standard algorithm all rely on it