Algebra basics: letters, expressions and like terms

What is algebra?

Algebra is maths with letters standing for numbers. Instead of saying "think of a number and add 4", you can write n + 4, where n is the number. A letter can stand for an unknown number you want to find, or for any number at all, so one line of algebra can describe a whole pattern.

Children use the idea long before they see a letter. A Year 1 missing number question like 7 = ? − 9 is algebra in disguise. In Year 6 maths, the national curriculum asks children to write missing number problems with letters, use simple formulae and find pairs of numbers that fit an equation with two unknowns, such as a + b = 10. The curriculum gives a perimeter example: a rectangle with sides of 2 cm and b cm and a perimeter of 20 cm gives 4 + 2b = 20.

At secondary school, algebra becomes a big part of KS3 maths. The programme of study for key stage 3 covers Years 7 to 9 together and doesn't say which year each part is taught in, so schools set their own order. Your child's school can tell you what its scheme of work covers each year.

What are algebraic terms?

An algebraic term is one part of an expression, separated from the next part by a + or − sign. In 4x + 3y − 7 there are three terms: 4x, 3y and −7.

• 4x is a term with a letter in it. The 4 is the coefficient: the number the letter is multiplied by.
• −7 is a term with no letter, often called a constant, because its value never changes.
• The sign in front belongs to the term, so the last term is −7, not 7.

What is an algebraic expression?

An algebraic expression is a group of terms with no equals sign, such as 3a + 5 or 2x − y. It doesn't say anything is true; it's like a phrase rather than a sentence. If you know what the letters stand for, you can work out its value.

Children learn to tell apart words that sound alike:

Word	What it is	Example
Term	One part of an expression	5x
Expression	Terms joined by + or −, no equals sign	5x + 2
Equation	Two things that are equal, true for particular values	5x + 2 = 17
Formula	A rule linking quantities	P = 2l + 2w
Inequality	One side bigger or smaller than the other	5x + 2 > 17

At GCSE, children also meet identities: statements that are true for every value of the letter, such as 2(x + 3) = 2x + 6.

Algebraic notation: how algebra is written

Algebraic notation is a set of short ways of writing things, and the key stage 3 curriculum lists them:

• No times sign. ab means a × b, and 3y means 3 × y.
• The number goes first. 3y, not y3. It also means y + y + y.
• Powers. a² means a × a ("a squared"), and a³ means a × a × a ("a cubed").
• Fractions for dividing. a ÷ b is usually written as a fraction, a over b.
• Brackets group things together: 2(a + b) means 2 lots of a + b. The expanding brackets guide shows how to multiply them out.

Simplifying algebraic expressions: collecting like terms

To simplify an algebraic expression, collect the like terms: add or subtract the terms that have exactly the same letters. 3a and 5a are like terms, so they make 8a. 3a and 5b are not, so 3a + 5b stays as it is.

Worked example: collecting like terms

Simplify 4x + 3y + 2x − y + 5.

1. Find the x terms: 4x and 2x. Together they make 6x.
2. Find the y terms: 3y and −y (which means −1y). Together they make 2y.
3. The 5 has no letter, so it stays.

The answer is 6x + 2y + 5. It's the same expression, just tidier. A quick check: put in x = 1 and y = 1. The first version gives 4 + 3 + 2 − 1 + 5, which is 13. The simplified one gives 6 + 2 + 5, which is also 13.

Terms with different powers aren't like terms either: x² and x are different, so x² + 3x can't be simplified.

Substitution: putting numbers into expressions

Substitution means swapping each letter for a number and working out the answer. It's how a formula gets used, and it's a good way to check work.

Worked example: substituting into an expression

Work out 3a + 2b when a = 5 and b = 4.

1. 3a means 3 × a, and 3 × 5 = 15.
2. 2b means 2 × b, and 2 × 4 = 8.
3. Add the two parts: 15 + 8 = 23.

Worked example: a formula with a square

The formula for the area of a square is A = s², where s is the length of a side. When s = 6 cm, A = 6 × 6, so the area is 36 cm².

Substituting negative numbers works the same way, with brackets to keep the signs clear. If n = −3, then 2n is 2 × (−3), which is −6.

How to factorise in algebra

Factorising is expanding brackets in reverse: you take out a common factor and put brackets back in. 6x + 9 = 3(2x + 3), because 3 goes into both terms. The factorising guide takes it step by step. Once children can simplify, expand and factorise, they use all three to solve equations.

Common mistakes in basic algebra

• Reading 3a as 30-something. If a = 4, 3a means 3 × 4, which is 12, not 34.
• Mixing up a² and 2a. If a = 5, a² is 25 but 2a is 10.
• Collecting unlike terms. 2x + 3y is not 5xy. Only matching letters can be added.
• Losing the sign. In 7x − 2x + 4, the 2x is being taken away, so it simplifies to 5x + 4.
• Thinking x stays the same everywhere. In one question, x has one value. In the next question, it can be a different number.

How to help at home

• Play "think of a number". "I think of a number, double it and add 1. I get 15. What was my number?" That's 2n + 1 = 15 without the letters.
• Use letters for real things. If a cinema ticket costs t pounds, what do 4 tickets cost? 4t.
• Check by substituting. When your child simplifies an expression, pick a small number for the letter and see whether both versions give the same answer.
• Keep the times tables sharp. Algebra uses multiplication all the time, and times tables that come straight from memory leave more room for thinking about the letters.

After the algebra basics: what comes next

Basic algebra leads on to expanding brackets, factorising, solving equations, inequalities and finding the nth term of a sequence, all part of the key stage 3 programme of study. At GCSE, algebra is also used to show that two expressions are always equal, and the strongest students go on to write algebraic proofs.

Algebra questions to try

1. How many terms are in 5p − 2q + 8? What is the coefficient of q?
2. Simplify 3m + 4n + 2m − n.
3. Simplify 6x + 2 − 4x + 7.
4. Work out 4a − b when a = 3 and b = 5.
5. Work out x² + 3 when x = 4.
6. A taxi charges 3 pounds plus 2 pounds a mile. Write an expression for the cost of m miles, then work out the cost of 8 miles.

Answers

1. Three terms: 5p, −2q and 8. The coefficient of q is −2.
2. 5m + 3n
3. 2x + 9
4. 4a is 12, then take away 5: 7
5. 4 × 4 = 16, then add 3: 19
6. 2m + 3 pounds. For 8 miles: 2 × 8 = 16, and 16 + 3 = 19 pounds.

For the rest of what children learn in Years 7 to 9, see the KS3 maths guide.