Use and interpret algebraic notation in Year 7 Maths

Algebraic notation KS3: the core conventions

Algebraic notation replaces repeated words with agreed symbols. Multiplication signs are usually omitted when a number or letter is written beside a variable: 5 multiplied by x becomes 5x, and a multiplied by b becomes ab. The number is written before the letter, so x multiplied by 7 is normally written 7x. A repeated factor can use an index: x multiplied by x is x^2, while x + x is 2x. These statements mean different things and should not be confused.

Division is normally written as a fraction, such as a/4, rather than a division sign. Brackets show that an operation applies to a complete expression: 3(x + 2) means three lots of x + 2, while 3x + 2 adds two only after multiplying x by three. Standard notation matters because it lets another reader reconstruct the relationship without guessing.

Terms, coefficients, expressions, equations and identities

A term is a single algebraic part separated by addition or subtraction. In 4x + 7 - 2y, the terms are 4x, 7 and -2y. A coefficient is the numerical factor multiplying a variable, so the coefficient of x is 4 and the coefficient of y is -2. A factor is something multiplied to make a term or product. Like terms have exactly the same variable part: 3x and 8x are like terms, but 3x and 3x^2 are not.

An expression represents a value but does not assert equality. An equation includes an equals sign and is true only for values that satisfy it. A formula links quantities, such as P = 2l + 2w for the perimeter of a rectangle. An identity is true for every allowed value, such as 2(a + 3) = 2a + 6. Naming the object correctly helps decide what can be simplified, substituted, solved or proved.

Writing algebraic expressions from words and geometry

Translate a statement in small steps. Five more than n is n + 5, while five times n is 5n. The phrase five less than n means n - 5; reversing it to 5 - n changes the value. The sum of x and 6 means x + 6, the product of p and q means pq, and a number shared equally between four people can be written n/4. Read the result back aloud to check that every operation keeps its original order.

Geometry gives a useful check because each letter has a physical meaning. A rectangle with length x + 3 and width x has perimeter 2(x + 3) + 2x, which simplifies to 4x + 6. Its area is x(x + 3), not 2x + 3. Include units after substituting numerical values: perimeter uses length units and area uses square units. A diagram should label which dimension each variable represents before an expression is formed.

Checking equations, multiple solutions and operation laws

To check whether a value satisfies an equation, substitute it everywhere the variable appears and calculate both sides independently. For 3x + 2 = 14, x = 4 works because both sides become 14. An equation with two variables can have many solutions: x + y = 10 is satisfied by 1 and 9, 2 and 8, and other pairs. Record pairs carefully and test each one rather than assuming there must be one answer.

Addition and multiplication are commutative: a + b = b + a and ab = ba. Subtraction and division are not: a - b does not generally equal b - a, and a/b does not generally equal b/a. Associative and distributive laws explain valid regrouping and expansion, but they do not permit arbitrary reordering. When simplifying, name the law or test the step with easy numbers so that a familiar-looking but invalid transformation is caught.

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