What Is Algebra? A KS3 Introduction for Year 7 and 8 Students

Algebra is the branch of mathematics that uses letters and symbols to represent unknown numbers and general relationships. Instead of writing "a number multiplied by 3 equals 12," algebra writes 3x = 12. The letter x stands for the unknown. Algebra lets us describe rules that work for every number, not just one specific case.

Why do we use letters in maths?

Consider this situation: a box contains some apples. You do not know how many, so you call the number n. If someone adds 5 more apples, the box now contains n + 5 apples. You can write that statement and work with it without ever knowing the exact value of n.

Letters in algebra serve two purposes:

Unknowns — a specific number you need to find (e.g. solve x + 3 = 10 to find that x = 7).

Variables — a quantity that can take many values (e.g. the formula A = lw describes the area of any rectangle, whatever l and w happen to be).

Key vocabulary

Understanding algebra starts with knowing the correct words.

Term | Definition | Example
Variable | A letter representing an unknown or changing number | x, y, n
Constant | A fixed number | 5, −3, 100
Term | A single number, variable, or their product | 4x, 7, −2y
Expression | A collection of terms combined by + or − | 3x + 2y − 5
Equation | Two expressions set equal to each other | 2x + 1 = 9
Formula | An equation linking two or more variables | A = lw
Coefficient | The number in front of a variable | In 4x, the coefficient is 4

Terms in detail

A term can be:

A number alone: 7

A letter alone: x (which means 1 × x)

A number × letter: 3y

A number × letter × letter: 5xy

The terms in the expression 4x + 2y − 5 are 4x, 2y, and −5. Notice that −5 is a term — the minus sign belongs to it.

What is an expression?

An algebraic expression contains variables but no equals sign. You can simplify an expression by collecting like terms, but you cannot "solve" it because there is nothing to solve for.

Like terms share exactly the same variable(s) to the same power(s). For example, 3x and 7x are like terms (both are multiples of x); 3x and 3y are not (different variables).

Worked example 1 — simplifying an expression

Simplify 4x + 2y + 3x − y.

Collect like terms:

x-terms: 4x + 3x = 7x

y-terms: 2y − y = y (since 1y = y)

Answer: 7x + y

Worked example 2 — an expression with numbers and letters

Simplify 5 + 3n − 2 + n.

Number terms: 5 − 2 = 3

n-terms: 3n + n = 4n

Answer: 4n + 3

What is an equation?

An equation has an equals sign and states that two expressions have the same value. You solve an equation to find the value of the unknown.

Worked example 3 — reading an equation

In the equation 2x + 4 = 10:

The left-hand side (LHS) is 2x + 4.

The right-hand side (RHS) is 10.

The equation tells us there is a value of x that makes both sides equal. That value is x = 3 (since 2 × 3 + 4 = 10 ✓).

Worked example 4 — writing an equation from words

"Five more than three times a number equals 17. What is the number?"

Let the number be n. Write: 3n + 5 = 17.

Solve: subtract 5 from both sides: 3n = 12. Divide both sides by 3: n = 4.

Answer: 4 (Check: 3 × 4 + 5 = 17 ✓)

What is a formula?

A formula is a fixed rule that connects two or more variables. You substitute known values into the formula to calculate an unknown.

Example: The perimeter of a rectangle is P = 2(l + w).

If l = 5 cm and w = 3 cm, then P = 2(5 + 3) = 2 × 8 = 16 cm.

Formulae appear everywhere in KS3: area, speed, temperature conversion, and more.

The difference between an expression, an equation, and a formula

Type | Has equals sign? | Can you solve it? | Example
Expression | No | No (simplify only) | 3x + 2
Equation | Yes | Yes (one answer) | 3x + 2 = 11
Formula | Yes | Substitute values | P = 2(l + w)

How algebra connects to number rules

Algebra makes number patterns precise. Consider the rule "any number added to itself equals double that number." In arithmetic you might verify: 3 + 3 = 6, 7 + 7 = 14. Algebra states it once for all numbers: n + n = 2n. This is why algebra is described as generalised arithmetic.

A second example: you probably know that (3)² = 9 and (−3)² = 9. Algebra expresses the general rule: (−x)² = x² for any value of x.

How algebra fits the KS3 national curriculum

The Department for Education's KS3 mathematics programme of study introduces algebra in Year 7 and expects pupils to "use and interpret algebraic notation, including ab in place of a × b, 3y in place of y + y + y and 3 × y, and coefficients written as fractions rather than as decimals." The AQA GCSE Mathematics specification builds from this foundation through substitution, solving equations, and graphing linear functions across Years 7, 8, and 9.

Mastery of the basics — terms, expressions, and equations — is essential preparation for GCSE algebra, which covers quadratics, simultaneous equations, and algebraic proof.

Common misconceptions at Year 7

Misconception 1 — "3n means 3 + n." No. 3n = 3 × n. In algebra, a number written immediately before a letter always means multiplication.

Misconception 2 — "x always equals 24 (or some default value)." The letter x has no fixed value; it represents whatever number makes the equation true in that particular problem.

Misconception 3 — "You can only use x." Any letter works. t, n, h, a are equally valid. Choose letters that remind you of what they stand for (e.g. t for time, h for height).

Misconception 4 — "Expressions and equations are the same." An expression is a mathematical phrase with no equals sign; an equation asserts two things are equal and has a specific solution.

Frequently asked questions

What is the difference between algebra and arithmetic?

Arithmetic works with specific numbers: 3 + 4 = 7. Algebra works with general rules by using letters: a + b = c (true for any values of a, b, and c where that relationship holds). Algebra lets you describe patterns, make formulae, and solve problems where you do not yet know all the numbers.

Do letters in algebra always have to be lowercase?

By convention, KS3 and GCSE maths uses lowercase letters for variables (x, y, n). Uppercase letters sometimes denote a specific point or area (e.g. A for area). In science, uppercase letters can appear in formulae (F = ma). You will be told which letter to use in exam questions, so follow the question's notation exactly.

Why does 3n mean 3 times n rather than 3.n or 3 and n?

Because writing 3 × n risks confusion with the letter x (a multiplication sign looks like the variable). Algebra adopted the convention of placing a number directly against a letter to indicate multiplication. So 3n, 3(n), and 3 × n all mean the same thing; 3n is the standard notation.

Can a variable take any value, including fractions or negatives?

Yes. Unless a problem specifically restricts the domain (for example, "n is a positive whole number"), a variable can be any real number — negative, fractional, zero, or large. This generality is what makes algebra so powerful.

For Socratic algebra tutoring that asks you the right question at the right moment, visit aitutors.me.