A1, A3, A4AlgebraFoundationGrades 1–4

Algebraic Notation

Q: Do I have to write letters in alphabetical order?

It is convention rather than a rule — ba is not wrong — but alphabetical order makes like terms much easier to spot.

Q: Is x the same as 1x?

Yes. The coefficient 1 is not written, so x - 1x = 0 and 4x + x = 5x.

Algebra uses letters to stand for numbers, and it has its own shorthand. $3 a$ means $3 \times a$ , $a^{2}$ means $a \times a$ , and $\frac{a}{4}$ means $a \div 4$ . Once the notation is automatic, simplifying — collecting like terms and multiplying terms — becomes quick and reliable, and every later algebra topic builds on it.

30 min Video by Maths Genie AQA GCSE Maths

The big picture

A letter in algebra is just a number you do not know yet, so every rule of arithmetic still applies. $3 a + 2 a = 5 a$ for exactly the same reason that 3 apples plus 2 apples is 5 apples.

The vocabulary matters too. AQA papers ask you to identify an expression, an equation, a formula and an identity, and to use the correct one in your working.

Simplifying AlgebraCurated by Kepler, not produced by us. Watch it, then read the notes below.

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What you'll learn

Read and write algebraic notation correctly
Use the words term, expression, equation, formula and identity
Simplify by collecting like terms
Simplify products and quotients such as $3 a \times 4 b$ and $12 x^{2} \div 4 x$

Writing algebra

Leave out the multiplication sign and write the number first: $a \times 3 \times b = 3 ab$ . Letters are usually written in alphabetical order.

A power counts repeated multiplication of the same thing: $y \times y \times y = y^{3}$ , but $3 y = y + y + y$ . Division is written as a fraction: $x \div 5 = \frac{x}{5}$ .

Brackets group terms: $2 (a + b)$ means $2 \times (a + b)$ , whereas $2 a + b$ means only the $a$ is doubled.

1(a) Number first, letters alphabetically:

7 mn

2(b)

p \times p = p^{2}

, so

4 p^{2}

3(c) The whole bracket is divided:

\frac{x + 1}{3}

Answer

7 mn

4 p^{2}

\frac{x + 1}{3}

Your turnSame method, new numbers — try it before moving on.

Write $b \times 5 \times a \times a$ in its simplest form.

Your answer

Type powers with ^, e.g. 3x^2y.

Tip — $3 y$ and $y^{3}$ are different: when $y = 2$ , $3 y = 6$ but $y^{3} = 8$ .

The vocabulary

A $term$ is a single number, letter or product such as $5 x^{2}$ . An $expression$ is a collection of terms with no equals sign, such as $5 x^{2} - 3 x$ .

An $equation$ is true for particular values: $2 x + 1 = 9$ is only true when $x = 4$ . A $formula$ links quantities that stand for different things, such as $A = l w$ .

An $identity$ is true for every value of the letter, and uses the symbol $\equiv$ : $2 (x + 3) \equiv 2 x + 6$ .

1(a) No equals sign — an expression.

2(b) True only when

x = 5

— an equation.

3(c) Links different quantities — a formula.

4(d) True for all

x

— an identity.

AnswerExpression, equation, formula, identity

Spotting an identity

A quick test for an identity: choose any value, say $x = 10$ , and check both sides agree. If one value makes them disagree, it is not an identity. (Agreeing for one value does not prove it is — you need to simplify one side into the other.)

Collecting like terms

$Like terms$ have exactly the same letters raised to the same powers. $4 ab$ and $- ab$ are like terms; $x^{2}$ and $x$ are not.

Add or subtract the numbers in front (the coefficients) of like terms. The sign in front of a term belongs to that term, so move it with the term.

a

terms:

5 a - 2 a = 3 a

b

terms:

3 b + b = 4 b

3The

- 7

has no like term, so it stays.

Answer

3 a + 4 b - 7

Your turnSame method, new numbers — try it before moving on.

Simplify $6 x - 4 y - 2 x + 9 y$ .

Your answer

Your turnSame method, new numbers — try it before moving on.

Simplify $2 ab + 7 a - 5 ab + 3 a$ .

Your answer

Tip — $ab$ and $ba$ are like terms — multiplication can be done in any order.

Multiplying and dividing terms

To multiply terms, multiply the numbers, then multiply the letters. Repeated letters become powers: $3 a \times 4 a = 12 a^{2}$ .

To divide, write it as a fraction and cancel common factors from the numbers and the letters: $\frac{12 x ^{2}}{4 x} = 3 x$ .

1(a) Numbers:

2 \times 5 = 10

. Letters:

p \times p \times q = p^{2} q

2(b) Numbers:

18 \div 6 = 3

. Letters:

\frac{a ^{2} b}{ab} = a

Answer(a)

10 p^{2} q

(b)

3 a

Your turnSame method, new numbers — try it before moving on.

Simplify $4 m \times 3 n \times 2 m$ .

Your answer

Type powers with ^, e.g. 6a^2b.

Think like an examiner

When asked to "simplify fully", check no like terms remain and numbers are multiplied out.
Keep each sign attached to the term after it: in $5 a - 2 b + 3 a$ , the $-$ belongs to $2 b$ .
Classification questions are about the equals sign: none means expression; $=$ means equation or formula; $\equiv$ means identity.

Notation to know

3 a = a + a + a

A coefficient counts how many.

a^{3} = a \times a \times a

A power counts repeated multiplication.

2 (x + 3) \equiv 2 x + 6

\equiv

means true for every value: an identity.

Watch out for these

Writing

3 x + 2 y = 5 x y

x

and

y

are unlike terms, so

3 x + 2 y

is already simplified.

Writing

a \times a = 2 a

a \times a = a^{2}

;

a + a = 2 a

Simplifying

x^{2} + x^{2}

x^{4}

Adding like terms adds coefficients:

2 x^{2}

Losing the sign:

7 y - 3 x + 2 y = 9 y + 3 x

The

-

stays with

3 x

9 y - 3 x

Stretch yourself

The perimeter of this shape is written as $(4 a + 3 b) + (2 a - b) + (a + 5 b) + P$ , where $P$ is the fourth side. The total perimeter is $10 a + 9 b$ . Find an expression for $P$ , and decide whether $P \equiv 3 (a + b) - b$ .

Hint — Collect the three known sides first, then subtract them from the total. Then expand the bracket.

Questions students ask

Do I have to write letters in alphabetical order?

It is convention rather than a rule —

ba

is not wrong — but alphabetical order makes like terms much easier to spot.

x

the same as

1 x

Yes. The coefficient 1 is not written, so

x - 1 x = 0

and

4 x + x = 5 x

What is the difference between a formula and an equation?

A formula links different quantities and works for any values (

A = l w

). An equation is a statement you solve to find particular values.

Key takeaways

Write numbers first and leave out the $\times$ sign: $3 ab$ .
$a^{2} = a \times a$ , which is different from $2 a = a + a$ .
Expression: no equals. Equation: true for some values. Formula: links quantities. Identity: true for all values.
Collect like terms by adding coefficients of identical letter parts.
Multiply numbers and letters separately; divide by cancelling.

Exam-style questions

Practise it the way it is examined

3 original questions · 9 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 14 marks

(a)Simplify $w + w + w + w$ .[1]
(b)Simplify $5 \times p \times q$ .[1]
(c)Simplify $6 c - 2 d + c + 5 d$ .[2]

Question 2Grade 32 marks

Here are four statements. $3 (n + 2) \equiv 3 n + 6 A = π r^{2} 5 n - 1 = 194 n^{2} - n$

Write down which one is an expression, which is an equation, which is a formula and which is an identity.[2]

Question 3Grade 43 marks

(a)Simplify $3 x^{2} y \times 4 x y$ .[2]
(b)Simplify $\frac{20 a ^{3} b}{5 ab}$ .[1]

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Algebraic Notation

Writing algebra

The vocabulary

Collecting like terms

Multiplying and dividing terms

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Algebraic Notation

Writing algebra

The vocabulary

Collecting like terms

Multiplying and dividing terms

Practise it the way it is examined

Skill drill

Practise this topic

Related topics