Essential Algebra Rules for Secondary Students: Why Learning Them Matters
27 May 2025
BY: matrixmath
Learning algebra is a core foundation for secondary students as they develop a deeper understanding of mathematics, and it will be necessary for students keen on careers that make significant use of science and mathematics. With various topics to learn, some students may find themselves confused by the seemingly abstract nature of algebra’s rules. Learning these fundamental rules is essential for any child’s later success in mathematics, even after they complete secondary school. Core Algebra Rules to Learn Variables and Constants Algebra solves mathematical problems using variables, constants, and other key rules. Variables, represented by letters like x and y, can change value based on the context of the mathematical problem, while constants (represented by numbers) are fixed values that do not change. Expressions and Equations A mathematical expression is a combination of variables, numbers, and operations (e.g., addition or multiplication) that forms a value. The sample above (x + 2y – 5) is an expression. An equation, on the other hand, contains an equal sign and shows how two expressions are related. For example, x + 2y – 5 = 4 is an equation. Like and Unlike Terms Variables with similar terms, like a and 2a, are known as like terms. Like terms can be added, subtracted, multiplied, and divided with one another. a + 3a = 4a x × 2x = 2x² Different terms, like 2x and 3y, are called unlike terms and cannot be added, subtracted, multiplied, or divided by other unlike terms. Order of Operations When solving algebraic problems, you need to abide by the BODMAS order of operations. When solving an equation, you must follow this specific order of operations: Brackets, Orders (like powers and roots), Division/Multiplication, Addition/Subtraction. Commutative Rule of Addition & Multiplication When two terms are added or multiplied, the order of operations is not relevant: (a + b) = (b + a) (a × b) = (b × a) This rule applies to both variables and constants. (4x + 3y) = (3y + 4x) (y + 3) × 2y = 2y × (y + 3) Associative Rule of Addition & Multiplication The associative rule is much like the commutative rule, but covers expressions or equations with three or more terms. a + (b + c) = (a + b) + c a × (b × c) = (a × b) × c To illustrate: x + (2y + 5) = (x + 2y) + 5 2x × (3 × 4y) = (2x × 3) × 4y Distributive Rule of Multiplication Consider the following equation: a × (b + c) = (a × b) + (a × c) = ab + ac When a variable or constant (a) is multiplied by an expression in brackets (b + c), you distribute the multiplication of (a) to each term inside the brackets. Thus: 2 × (2x + 3y) = (2 × 2x) + (2 × 3y) = 4x + 6y x × (3x + 2y) = (x × 3x) + (x × 2y) = 3×2 + 2xy The rule also applies to expressions with more than three terms. Balancing of Equations A mathematical equation can be considered a weighing scale, so any change to one side of the equation also affects the other side. Take this simple equation: x + 7 = 12 To find x, you need to balance the equation by subtracting seven from both sides. x + 7 – 7 = 12 – 7 x = 5 Algebraic Identities Algebraic identities are equations where the left-hand side of the equation is identically equal to the right-hand side. (a + b)² = a² + 2ab + b² (a − b)² = a² − 2ab + b² a² − b² = (a + b)(a − b) (x + a)(x + b) = x² + x(a + b) + ab When the variables a, b, and x are assigned numbers, both sides of the equation will become balanced. If not, the equation is not an identity. Why Learn These Algebra Rules? Learning these rules is vital in helping students understand how to apply them correctly to solve various algebraic problems. But equally important is their significance in their later years, whether math-related or otherwise. Additional Preparations The secondary school Maths syllabus comprises intermediate and advanced Maths concepts that require students to understand algebra and how to apply its rules and concepts to the new lessons they will learn. Even some science subjects, like Physics, require algebraic knowledge for various topics, while half of the Additional Mathematics syllabus requires a student’s understanding of algebraic equations and expressions to solve problems. Develop Cognitive Skills Algebra knowledge is not only used for solving complex Maths questions but also empowers a student’s cognition and encourages forward-thinking behaviours, especially in finding solutions to complex algebraic equations. It helps to develop a student’s mathematical reasoning, allowing them to identify variables and constants from a given problem and then apply the correct rules to find the right solution. The helpful boost to cognitive ability and critical thinking ensures that students can take on various interesting challenges and questions that encourage further self-development, which aids them well into their future. Foundation for University Education In fact, algebra remains relevant once students begin their higher education journey. Just as it applies in several other secondary-level subjects, algebra is also essential for various majors that students are keen to pursue, ranging from engineering to economics. Maintaining one’s algebraic knowledge will ensure they have the fundamentals to thrive in these more challenging subjects and use what they learned to succeed in their careers. Real-Life Applications After all, algebra has many real-life applications, as it can reliably help to chart various changes across nearly all human activities. 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