LET Elementary Mathematics — Numbers, Number Sense and the Four OperationsSummary

All numbers encountered in K–12 mathematics belong to the real number system, which is organized as nested sets. Natural numbers (1, 2, 3, ...) form the smallest set; every natural number is also whole (0, 1, 2, 3, ...), every whole number is an integer (..., -2, -1, 0, 1, 2, ...), and every integer is rational (since any integer n can be written as n/1). Rational numbers (Q) include all fractions a/b where b ≠ 0, as well as terminating decimals (0.5 = 1/2) and repeating decimals (0.333... = 1/3). Irrational numbers are non-terminating, non-repeating decimals such as π, √2, and √3; they cannot be written as fractions. Together, rational and irrational numbers form the real numbers (R). This hierarchy is crucial because it determines which operations are 'closed' within each set—for example, the integers are closed under addition and multiplication but not under division (3 ÷ 2 = 1.5 is not an integer).

Real numbers obey fundamental properties that justify every step in a calculation. The commutative property states that order does not matter for addition and multiplication: a + b = b + a and a × b = b × a. However, subtraction and division are not commutative (5 - 3 ≠ 3 - 5). The associative property permits regrouping for addition and multiplication: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c), but again not for subtraction or division. The distributive property links addition and multiplication: a(b + c) = ab + ac; this property is the single most tested justification on the LET and is vital for teaching factoring and simplification in Grades 3–6. The identity property establishes that 0 is the additive identity (a + 0 = a) and 1 is the multiplicative identity (a × 1 = a). The inverse property states that every real number a has an additive inverse -a such that a + (-a) = 0, and every non-zero number a has a multiplicative inverse 1/a such that a × (1/a) = 1. Closure means that a set is closed under an operation if the result stays in the set; for instance, the integers are closed under addition but not division.

The order of operations is a fixed sequence that ensures everyone solves the same expression the same way. GEMDAS stands for Grouping, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). Grouping symbols include parentheses ( ), brackets [ ], fraction bars, and radicals. The critical insight is that multiplication and division share the same rank and are evaluated left to right—whichever appears first is done first. Similarly, addition and subtraction share a rank and are evaluated left to right. For example, 20 ÷ 4 × 5 is evaluated as (20 ÷ 4) × 5 = 5 × 5 = 25, not 20 ÷ (4 × 5) = 1. The most common LET error trap is treating one operation as always before another when they rank equally. Students at Grades 1–3 begin with simple operations; by Grade 5, they encounter mixed operations and exponents.

Integers include positive numbers, negative numbers, and zero. In addition, if the signs are the same, add the absolute values and keep the sign (e.g., -8 + (-3) = -11, and 5 + 7 = 12); if the signs are different, subtract the smaller absolute value from the larger and take the sign of the larger (e.g., -8 + 3 = -5, and 12 + (-7) = 5). In subtraction, change the second number to its opposite and add: a - b = a + (-b). So 5 - (-3) = 5 + 3 = 8, and -6 - 4 = -6 + (-4) = -10. In multiplication and division, like signs (both positive or both negative) yield a positive result, and unlike signs yield a negative result. Thus (-6) × (-4) = 24, (-6) × 4 = -24, and (-24) ÷ 6 = -4. A useful rule: an even number of negative factors produces a positive product, and an odd number of negative factors produces a negative product. Negative exponents also follow this rule: (-2)⁴ = 16 (even power), and (-2)³ = -8 (odd power). Note that -2² = -4 (only the 2 is squared), while (-2)² = 4 (the negative sign is included).

A factor (or divisor) of a number n is any integer that divides n exactly, leaving no remainder. A multiple of n is any number that n divides into exactly—that is, the product of n and another integer. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12; the multiples of 3 are 3, 6, 9, 12, 15, .... Divisibility rules are shortcuts for checking whether one number divides another without performing the full division: a number is divisible by 2 if its last digit is even (e.g., 138); by 3 if the sum of its digits is divisible by 3 (231: 2 + 3 + 1 = 6, divisible by 3); by 5 if it ends in 0 or 5; by 10 if it ends in 0. A prime number has exactly two factors—1 and itself (examples: 2, 3, 5, 7, 11, 13). The number 2 is the only even prime; all other primes are odd. A composite number has more than two factors (4, 6, 8, 9, 10, 12, ...). The number 1 is neither prime nor composite by definition. Every composite number can be factored into a unique product of primes, called its prime factorization (Fundamental Theorem of Arithmetic). For example, 60 = 2² × 3 × 5.

The greatest common factor (GCF) of two or more numbers is the largest number that divides all of them exactly. It is used when you need to split items or people into equal groups with no remainder. For example, if you have 48 pencils and 36 notebooks to distribute equally to a class with no leftover items, the maximum number of students is the GCF of 48 and 36. To find the GCF using prime factorization, factor each number, identify the common prime factors, and multiply them using the smallest exponent. For 18 = 2 × 3² and 24 = 2³ × 3, the common primes are 2 and 3; the GCF = 2¹ × 3¹ = 6. The least common multiple (LCM) of two or more numbers is the smallest number that all of them divide into. It is used when events repeat at different intervals and you want to know when they coincide—for example, when two school bells toll together, or when adding fractions with different denominators. To find the LCM, factor each number and multiply every prime that appears, using the highest exponent. For 18 and 24, the LCM = 2³ × 3² = 8 × 9 = 72. A useful check: for any two numbers a and b, GCF(a, b) × LCM(a, b) = a × b. So 6 × 72 = 432 = 18 × 24.

Estimation is a critical thinking skill that allows you to predict the approximate size of an answer before calculating exactly, serving as a check against errors. The main techniques are rounding (adjusting each number to a nearby 'round' number), using compatible numbers (numbers that are easy to compute with, like 25 × 4 = 100), and front-end estimation (using only the leading digits). For example, to estimate 4,987 × 21, round 4,987 to 5,000 and 21 to 20, giving 5,000 × 20 = 100,000. The exact answer is 104,727, which is close, confirming no major error occurred. Mental computation shortcuts include multiplying by 10, 100, or 1,000 (which shifts the decimal point), multiplying by 5 (multiply by 10 and halve), and using 25 × 4 = 100 to simplify. These skills are taught from Grade 1 onward and are explicitly required by the DepEd for mathematical fluency.

The absolute value of a number is its distance from zero on the number line; it is always non-negative. Notation: |a| = a if a ≥ 0, and |a| = -a if a < 0. For example, |-7| = 7 and |7| = 7. Importantly, absolute value is applied after operations inside the bars, so |3 - 8| = |-5| = 5, not |3| - |8| = -5. On the number line, numbers increase from left to right, so any negative number is less than any positive number (-10 < 0 < 5). Among negative numbers, the one closer to zero is larger (-3 > -8, even though 8 > 3 in absolute value). To compare fractions with different denominators, convert to decimals or use cross-multiplication: for 3/8 versus 5/12, compute 3 × 12 = 36 and 5 × 8 = 40. Since 40 > 36 and 40 is paired with the second fraction, 5/12 > 3/8. When comparing a mixed list of fractions, decimals, and percentages, convert all to the same form (usually decimals) before ordering.

Scientific notation is a compact way to write very large or very small numbers as a product of a number between 1 and 10 and a power of 10. To convert a standard number to scientific notation, move the decimal point so that exactly one non-zero digit appears to its left, and count how many places it moved: that count is the exponent. For example, 45,000 becomes 4.5 × 10⁴ (decimal moved 4 places left), and 0.0032 becomes 3.2 × 10⁻³ (decimal moved 3 places right; negative exponent signals a number less than 1). This notation links directly to metric prefixes: kilo (k) = 10³, centi (c) = 10⁻², milli (m) = 10⁻³. Place value refers to the value a digit holds based on its position in a number. In 4,725, the digit 7 is in the hundreds place and represents 700. In 27.4, the 4 is in the tenths place and represents 4/10 = 0.4. Place value is taught from Grade 1 and is foundational to understanding how to compare, round, and compute with multi-digit numbers. Rounding rules apply place value: identify the target place, look at the digit immediately to its right, and round up if that digit is 5 or more (round down if 4 or less).

Roman numerals use letters to represent values: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1,000. Symbols are added when they are equal or decrease in value from left to right (VIII = 8, XVI = 16). A smaller symbol before a larger one is subtracted (IV = 4, IX = 9, XL = 40, XC = 90, CD = 400, CM = 900). So 49 = XLIX (40 + 9), and 2024 = MMXXIV (1,000 + 1,000 + 20 + 4). Roman numerals are used in Grades 3–4 to mark time on analog clocks, in historical contexts, and on formal documents—hence their inclusion in the K-12 BEC.

The nested structure of the real number system (natural ⊂ whole ⊂ integer ⊂ rational ⊂ real) determines which operations are closed within each set. Understanding this hierarchy explains why 3 ÷ 2 = 1.5 is not an integer, and why we can only take the square root of non-negative numbers within the reals. Properties like closure are defined relative to the set in question. This structure frames all subsequent topics in the chapter.

GEMDAS is the rule that ensures all calculations are done in the same order, preventing ambiguity. Every worked example in this chapter applies GEMDAS. Errors in order of operations lead to wrong answers; mastery of GEMDAS is a prerequisite for correct arithmetic and algebra. Teaching pupils the sequence starting in Grade 3 builds computational fluency and confidence.

Once you find the prime factorization of two numbers, you can calculate both GCF (common primes, lowest exponents) and LCM (all primes, highest exponents) directly. This connection shows that GCF and LCM are not separate topics but two applications of the same underlying structure. The relationship GCF × LCM = product of the numbers is a powerful check on correctness.

Rounding relies on understanding place value (which place are we rounding to?). Estimation, in turn, relies on rounding. All three concepts work together to develop number sense and allow pupils to predict answer sizes. Teaching place value thoroughly in Grades 1–3 makes rounding and estimation much easier in Grades 4–6.

Absolute value tells you the distance of a number from zero, which is the key to comparing integers correctly on a number line. Understanding that |-8| = 8 and |-3| = 3 helps you see that -3 is closer to zero and thus larger than -8, even though 8 > 3 in absolute terms. This relationship is often misunderstood by pupils and requires careful teaching with concrete number lines.

The sign rules for multiplication (like signs → positive, unlike → negative) extend to exponents: an even power of a negative base is positive, an odd power is negative. So (-2)² = 4 but (-2)³ = -8. This relationship helps pupils remember exponent rules and see them as extensions of simpler multiplication rules.

The distributive property a(b + c) = ab + ac is not just a computational rule; it is the foundation for factoring, expanding brackets, and simplifying algebraic expressions. This property is tested repeatedly on the LET because it is so central to all algebra work in Grades 5–6 and beyond.

Scientific notation expresses numbers as powers of 10, which directly connects to metric prefixes. Kilo (10³), milli (10⁻³), and centi (10⁻²) are expressions of these powers. This relationship makes scientific notation relevant to measurement topics in Grades 5–6 and helps pupils see unity across the mathematics curriculum.