Chapter 1: Number and Operations

Competency 0001 | 10 selected-response items | 20% of total score | Calculator: 4-function provided

Number and operations is the mathematical foundation that everything else rests on. Before a student can write an algebraic equation or measure a shape, they need to understand what numbers are, how our place-value system works, how fractions behave, and how operations connect. This chapter covers every indicator in Competency 0001 — from the prenumeracy milestones of a toddler to the rational number reasoning required of a grade 2 teacher. Read actively: sketch the diagrams, work the examples, and let the visuals build your mental model.

(1) PRENUMERACY CONCEPTS AND MATHEMATICAL DEVELOPMENT OF CHILDREN

(A) The Five Counting Principles

Gelman and Gallistel's counting principles describe what a child must understand before formal number operations can develop. The exam expects you to identify which principle a child is applying — or violating — in a given scenario.

1. One-to-One Correspondence

Each object counted gets exactly one counting word. A child who counts a row of 5 blocks saying "1, 2, 3, 4, 5" while touching each block once is applying this principle correctly.

2. Stable Order

Number words must always be said in the same sequence: 1, 2, 3, 4, 5 — never 1, 3, 2, 5, 4. This must be memorized before counting becomes reliable.

3. Cardinality

The last number word said represents the total quantity of the set. If a child counts 5 bears and you ask "How many?", a child with cardinality understanding says "5" — not counting again from the start.

4. Abstraction

Any collection of objects — physical or imagined, same or different — can be counted. A child can count 3 dogs, 2 cats, and 1 bird as a set of 6 animals.

5. Order Irrelevance

Objects in a set can be counted in any order — left to right, right to left, random — and the total stays the same. This is NOT the commutative property; it's about counting flexibility.

TEST READY TIP Subitizing — recognizing a small quantity instantly without counting — is a key prenumeracy skill. A child who sees ••• and instantly says "three" without counting is subitizing. Perceptual subitizing works for 1–4 objects; conceptual subitizing works for larger groups by recognizing sub-groups (e.g., seeing 6 as two groups of 3). Questions may ask you to identify an activity that develops subitizing: dot cards, dice patterns, and ten-frames all develop this skill.

(B) Conservation of Number

Conservation of number is the understanding, described by Piaget, that the quantity of a set remains the same regardless of how the objects are physically arranged. A child who has not yet conserved number will say a spread-out row of 5 coins has "more" than a bunched-up row of 5 coins. Most children achieve conservation of number between ages 5 and 7.

(2) THE REAL NUMBER SYSTEM — CLASSIFICATION AND HIERARCHY

(A) The Nested Number System

Your exam will ask you to classify a given number within the real number hierarchy. You need to know which sets are subsets of which — think of these as nested rings, where every natural number is also a whole number, every whole number is also an integer, and so on.

Real Numbers — all rational and irrational numbers combined; everything on the number line

Rational Numbers — any number expressible as a/b where a and b are integers and b ≠ 0; includes terminating and repeating decimals (e.g., 3/4, −5, 0.333…, 2.5)

Integers — whole numbers and their negatives: …−3, −2, −1, 0, 1, 2, 3…

Whole Numbers — 0, 1, 2, 3, 4… (non-negative integers)

Natural Numbers (Counting Numbers) — 1, 2, 3, 4… (no zero)

Note on Fractions and Decimals: Fractions like 2/3 and repeating decimals like 0.666… are rational but are NOT integers. They live in the rational layer but outside the integer ring.

Irrational Numbers — cannot be expressed as a/b; decimals that never terminate and never repeat (e.g., π ≈ 3.14159…, √2 ≈ 1.41421…, e). These are real but NOT rational.

On the Exam: You will see classification questions like "Which of the following is rational but not an integer?" The answer is any fraction or repeating decimal (e.g., 1/3, −0.25, 5/7). You will also see "Which is irrational?" — look for square roots of non-perfect-squares (√3, √5, √7) and constants like π.

TEST READY TIP A decimal that terminates (e.g., 0.75) or repeats (e.g., 0.333…) is always rational — you can convert it to a fraction. A decimal that goes on forever without repeating is always irrational. Use this test every time: can I write it as a fraction with integer numerator and denominator? If yes → rational. If no → irrational.

COMMON TRAP The exam exploits the fact that √4 = 2, which IS an integer — so not every square root is irrational. Only the square roots of non-perfect-squares are irrational. Also watch for 0: it is a whole number and an integer, but it is NOT a natural (counting) number. Candidates routinely misplace 0.

(B) Rational vs. Irrational — Classroom Connection

In a Grade 4 classroom, your students are working entirely within rational numbers — they encounter fractions, decimals, and negative integers on the number line. You won't introduce irrational numbers at that level, but your exam tests whether you can classify them correctly and explain why the real number system is structured the way it is. When a 4th grader asks "Are there numbers between 1 and 2?", your answer draws on this hierarchy: yes, infinitely many — including fractions like 3/2, decimals like 1.7, and irrational numbers like √2.

(3) PLACE VALUE AND THE BASE-TEN SYSTEM

(A) How Place Value Works — The Visual Structure

Our number system is base-ten: each position is worth 10 times the position to its right. This is the most powerful idea in elementary mathematics — it lets 10 digits represent any number imaginable.

Millions 1,000,000	Hundred Thousands 100,000	Ten Thousands 10,000	Thousands 1,000	Hundreds 100	Tens 10	Ones 1	• Tenths 1/10	Hundredths 1/100	Thousandths 1/1000
—	—	—	4	3	7	2	.	8	5

The number 4,372.85 shown in the place value chart. Each position is worth exactly 10× the position to its right.

Key relationships to master:

Moving one place to the left multiplies the value by 10: 3 tens = 30 ones
Moving one place to the right divides the value by 10: 3 tens = 0.3 hundreds
The number 4,372 means: (4 × 1,000) + (3 × 100) + (7 × 10) + (2 × 1) — this is expanded form
Word form: "four thousand, three hundred seventy-two"

(B) Rounding — The Number Line Method

Rounding asks: which benchmark is this number closer to? Use a number line to see it visually.

Round 347 to the nearest hundred.

300

350

▾

347

400

347 is between 300 and 400. The midpoint is 350. Since 347 < 350, it is closer to 300. Rounded to the nearest hundred: 300.

COMMON TRAP Students (and sometimes teachers) say "if it's 5, round up." The rule is: if the digit in the rounding position is 5 or greater, round up. So 350 rounds to 400 — not 300. The 5 is AT the midpoint, and convention says round up. On the exam, you may see a question about a student who rounds 350 to 300 — the student is applying the rule incorrectly by ignoring the 5.

(C) Properties of Operations — The Four Foundational Laws

There are four named properties of operations that your exam will test directly. These properties are what justify every algorithm you teach — when a Grade 3 student uses a mental math shortcut, they are intuitively applying one of these laws. Know each one cold: its name, its symbolic form, which operations it applies to, and a classroom-level example.

Property	What it says	Symbolic Form	Applies to	Grade 2–3 Classroom Example
Commutative	Order doesn't matter	a + b = b + a a × b = b × a	+ and × NOT − or ÷	A Grade 2 student counting 3 + 5 by starting from 5 and counting up 3 more is applying the commutative property to make counting easier.
Associative	Grouping doesn't matter	(a + b) + c = a + (b + c)	+ and × NOT − or ÷	A Grade 3 student computing 7 + 8 + 2 rearranges to (7 + 3) + 8 — wait, that uses commutativity too — or groups (8 + 2) + 7 = 17. Regrouping to make a ten is associativity in action.
Distributive	Multiply across a sum or difference	a × (b + c) = (a×b) + (a×c)	× over + and −	A Grade 3 student who doesn't know 6 × 7 decomposes it as 6 × (5 + 2) = 30 + 12 = 42. The area model of multiplication makes distributivity visible as separate rectangles.
Identity	Adding 0 or multiplying by 1 leaves the number unchanged	a + 0 = a a × 1 = a	+ and ×	A Grade 2 student learns that adding 0 to any number leaves it unchanged — useful for establishing that 0 is the additive identity. This also underpins why multiplying a fraction by 2/2 (= 1) creates an equivalent fraction.

COMMON TRAP — The "Zero Property" Confusion You may recall seeing a "zero property of multiplication" (any number times 0 equals 0) listed as a fifth property in some elementary textbooks. Your exam will not test it as a standalone named property at the same level as the four above — and for good reason. The result that a × 0 = 0 is actually a consequence of the distributive and additive identity properties, not an independent axiom: a × 0 = a × (0 + 0) = (a × 0) + (a × 0), which forces a × 0 = 0. Know the four core properties cold; if a × 0 = 0 appears in a question, it will be presented as an application of the distributive or identity property, not as its own named law.

TEST READY TIP — Additive Inverse A fifth concept you should know is the additive inverse (opposite) property: for every number a, there exists −a such that a + (−a) = 0. This is what makes subtraction possible in the integer system and is directly tested when the exam asks why 5 − 5 = 0 or how you justify the rule for subtracting negative numbers. Every number's opposite is its additive inverse; together they sum to the additive identity (0).

(4) JUSTIFYING COMPUTATIONAL ALGORITHMS

(A) The Standard Addition Algorithm — Why It Works

The standard algorithm works because of place value and the associative/commutative properties of addition. When you "carry" a 1 to the tens column, you are regrouping 10 ones into 1 ten — a place-value trade.

Standard Algorithm: 247 + 185

H   T   O
  1   1   
  2   4   7
+ 1   8   5
  4   3   2

What's actually happening:

Ones: 7 + 5 = 12 → write 2, carry 1 ten
Tens: 4 + 8 + 1(carried) = 13 → write 3, carry 1 hundred
Hundreds: 2 + 1 + 1(carried) = 4 → write 4
Result: 432 ✓

The "carry" is a physical regrouping: 10 ones become 1 ten. The algorithm is justified by place value. In a Grade 2 classroom, use base-ten blocks to make this trade concrete before introducing the written procedure.

(B) Multiplication — The Area Model Makes It Visible

The area model shows why the standard multiplication algorithm works: you are computing the area of a rectangle where each side represents a factor. Every partial product is a visible rectangle. This is the Grade 3–4 bridge between concrete multiplication and the abstract standard algorithm.

34 × 27 using the area model

30 × 20 600

4 × 20 80

30 × 7 210

4 × 7 28

↑ Width = 34 = 30 + 4

Add all four partial products:

The standard algorithm compresses these four partial products into two rows. The area model makes each partial product visible and is the recommended Grade 4 representation before students transition to the standard algorithm.

(C) Division — Interpreting the Standard Algorithm

Long division implements repeated subtraction of equal groups. The algorithm works from the largest place value down, distributing the dividend into equal groups.

752 ÷ 4 — What the algorithm actually means

Step-by-step meaning:

How many 4s in 7 hundreds? 1 (1 × 4 = 4 hundreds). Remainder: 3 hundreds = 30 tens → "bring down" to get 35 tens.
How many 4s in 35 tens? 8 (8 × 4 = 32). Remainder: 3 tens = 30 ones → bring down the 2 to get 32 ones.
How many 4s in 32 ones? 8 (8 × 4 = 32). Remainder: 0.

Answer: 188. Check: 188 × 4 = 752 ✓

In a Grade 4 classroom, connect "bring down" to the place-value trade: 3 remaining hundreds become 30 tens, which combine with the 5 tens already there — that's why you bring down the next digit.

(5) FRACTIONS — CONCEPTS, OPERATIONS, MIXED NUMBERS, AND IMPROPER FRACTIONS

(A) What a Fraction Actually Means — Three Models

A fraction represents a relationship. The exam tests whether you understand all three standard interpretations:

Part-Whole Model

3/4 = 3 of 4 equal parts are shaded. The whole is divided into 4 equal parts; 3 are selected.

Number Line Model

●

3/4 is a point 3/4 of the way from 0 to 1. It represents a quantity, not just a part of a shape.

Division / Ratio Model

3 ÷ 4 = 3/4

3 pizzas shared equally among 4 people. Each person gets 3/4 of a pizza. Fraction = division problem.

(B) Equivalent Fractions — The Multiply-Both Rule, Justified

Equivalent fractions name the same quantity using different numerators and denominators. The rule: multiply (or divide) both numerator and denominator by the same nonzero number. This works because you are multiplying the fraction by 1 (in disguise).

Why does 2/3 = 4/6?

2/3 × (2/2) = 4/6

Multiplying by 2/2 is multiplying by 1. The value doesn't change. Visually: if you cut each third into 2 equal pieces, you get 6 pieces total, and the same area is covered by 4 of those 6 pieces.

2/3

4/6

(C) Mixed Numbers and Improper Fractions

A mixed number combines a whole number and a proper fraction: 2¾ means "two wholes and three-fourths." An improper fraction has a numerator greater than or equal to its denominator: 11/4. Both represent the same quantity. Your exam tests conversion in both directions and operations involving mixed numbers.

Mixed Number → Improper Fraction

Procedure: Multiply the whole number by the denominator, then add the numerator. Keep the same denominator.

      2¾ = (2 × 4) + 3 / 4 = 8 + 3 / 4 = 11/4
    

Think of it as: 2 whole pies (each cut into 4 slices) = 8 slices, plus 3 more = 11 slices total, each being 1/4 of a pie.

Improper Fraction → Mixed Number

Procedure: Divide numerator by denominator. The quotient is the whole number; the remainder is the new numerator; keep the same denominator.

      17/5 → 17 ÷ 5 = 3 remainder 2 → 3⅖
    

17 fifths = 15 fifths (= 3 wholes) + 2 fifths left over. The fraction strip below 1 is 2/5.

Operations with mixed numbers appear on the exam in two forms: adding/subtracting (which may require borrowing) and multiplying/dividing (where converting to improper fractions first is almost always the cleanest strategy).

Example: 3½ + 2¾

Method 1 — Keep mixed numbers:

Add whole parts: 3 + 2 = 5
Add fraction parts: 1/2 + 3/4 = 2/4 + 3/4 = 5/4 = 1¼
Combine: 5 + 1¼ = 6¼

Method 2 — Convert to improper fractions first:

3½ = 7/2 2¾ = 11/4
LCD = 4: 7/2 = 14/4
14/4 + 11/4 = 25/4 = 6¼ ✓

Example: 2⅓ × 1½

Convert: 2⅓ = 7/3 1½ = 3/2
Multiply: (7/3) × (3/2) = 21/6 = 7/2 = 3½

For multiplication and division with mixed numbers, always convert to improper fractions first — trying to multiply mixed numbers directly leads to errors. This is the strategy you'll want to model for Grade 5 students.

COMMON TRAP — Borrowing with Mixed Number Subtraction When subtracting mixed numbers and the fraction part of the subtrahend is larger than the fraction part of the minuend, you must borrow. For example: 5¼ − 2¾. You can't do ¼ − ¾ directly, so borrow 1 whole from the 5: rewrite as 4 + 1¼ = 4 + 4/4 + 1/4 = 4 + 5/4. Now: 4 5/4 − 2¾ = 4 5/4 − 2 3/4 = 2 2/4 = 2½. The exam will present a student error where the student simply subtracts the smaller fraction from the larger regardless of position — your job is to identify that error.

TEST READY TIP On the exam, you will encounter mixed number questions framed as "which representation is equivalent to…" or "a student converted 3⅖ to an improper fraction and got 17/5 — is that correct?" Work both conversions fluently. Also know that improper fractions are not "wrong" — they are simply a different, equally valid representation, and the exam may ask which form is most useful in a given context (improper fractions are easier for operations; mixed numbers are more intuitive for interpreting size).

(D) Adding and Subtracting Fractions — Common Denominator, Visually

Compute: 1/4 + 2/3

Step 1: Find the LCD (least common denominator): LCD(4, 3) = 12

Step 2: Convert: 1/4 = 3/12 and 2/3 = 8/12

Step 3: Add numerators, keep denominator: 3/12 + 8/12 = 11/12

1/4 = 3/12 (blue)

2/3 = 8/12 (blue)

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