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. $3a$ 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. $3a + 2a = 5a$ 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.

Open on YouTube

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 $3a \times 4b$ and $12x^2 \div 4x$

Writing algebra

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

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

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

Example 1Write in simplest notation: (a)

m \times 7 \times n

(b)

p \times p \times 4

(c)

(x + 1) \div 3

1(a) Number first, letters alphabetically:

7mn

2(b)

p \times p = p^2

, so

4p^2

3(c) The whole bracket is divided:

\frac{x + 1}{3}

Answer

7mn

4p^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.

Show worked solution

Solution

1Number first: $5$ .
2 $a \times a = a^2$ , then $b$ : $5a^2b$ .

Answer: $5a^2b$

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

The vocabulary

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

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

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

Example 1Classify each: (a)

4x - 7

(b)

4x - 7 = 13

(c)

v = u + at

(d)

x + x \equiv 2x

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.

Answer

Expression, 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

$\textbf{Like terms}$ have exactly the same letters raised to the same powers. $4ab$ 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.

Example 1Simplify

5a + 3b - 2a + b - 7

a

terms:

5a - 2a = 3a

b

terms:

3b + b = 4b

3The

-7

has no like term, so it stays.

Answer

3a + 4b - 7

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

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

Your answer

Show worked solution

Solution

1 $x$ terms: $6x - 2x = 4x$ .
2 $y$ terms: $-4y + 9y = 5y$ .

Answer: $4x + 5y$

Example 2Simplify

3x^2 + 5x - x^2 - 8x

x^2

terms:

3x^2 - x^2 = 2x^2

x

terms:

5x - 8x = -3x

Answer

2x^2 - 3x

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

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

Your answer

Show worked solution

Solution

1 $ab$ terms: $2ab - 5ab = -3ab$ .
2 $a$ terms: $7a + 3a = 10a$ .

Answer: $10a - 3ab$

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: $3a \times 4a = 12a^2$ .

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

Example 1Simplify (a)

2p \times 5q \times p

(b)

18a^2b \div 6ab

1(a) Numbers:

2 \times 5 = 10

. Letters:

p \times p \times q = p^2q

2(b) Numbers:

18 \div 6 = 3

. Letters:

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

Answer

(a)

10p^2q

(b)

3a

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

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

Your answer

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

Show worked solution

Solution

1Numbers: $4 \times 3 \times 2 = 24$ .
2Letters: $m \times m \times n = m^2n$ .

Answer: $24m^2n$

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 $5a - 2b + 3a$ , the $-$ belongs to $2b$ .
Classification questions are about the equals sign: none means expression; $=$ means equation or formula; $\equiv$ means identity.

Notation to know

3a = a + a + a

A coefficient counts how many.

a^3 = a \times a \times a

A power counts repeated multiplication.

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

\equiv

means true for every value: an identity.

Watch out for these

Writing

3x + 2y = 5xy

x

and

y

are unlike terms, so

3x + 2y

is already simplified.

Writing

a \times a = 2a

a \times a = a^2

;

a + a = 2a

Simplifying

x^2 + x^2

x^4

Adding like terms adds coefficients:

2x^2

Losing the sign:

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

The

-

stays with

3x

9y - 3x

Stretch yourself

The perimeter of this shape is written as $(4a + 3b) + (2a - b) + (a + 5b) + P$ , where $P$ is the fourth side. The total perimeter is $10a + 9b$ . 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.

Reveal solution

SolutionThe three known sides total

7a + 7b

, so

P = (10a + 9b) - (7a + 7b) = 3a + 2b

. Expanding,

3(a + b) - b = 3a + 3b - b = 3a + 2b

. The two sides simplify to the same expression, so yes:

P \equiv 3(a + b) - b

is an identity.

\blacksquare

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

1x

Yes. The coefficient 1 is not written, so

x - 1x = 0

and

4x + x = 5x

What is the difference between a formula and an equation?

A formula links different quantities and works for any values (

A = lw

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

Key takeaways

Write numbers first and leave out the $\times$ sign: $3ab$ .
$a^2 = a \times a$ , which is different from $2a = a + a$ .
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]
Show mark scheme (a)
- B1 $4w$
Answer: $4w$
(b)Simplify $5 \times p \times q$ .[1]
Show mark scheme (b)
- B1 $5pq$ (accept $5qp$ )
Answer: $5pq$
(c)Simplify $6c - 2d + c + 5d$ .[2]
Show mark scheme (c)
- B1One term correct: $7c$ or $+3d$
- B1 $7c + 3d$
Answer: $7c + 3d$

Question 2Grade 32 marks

Here are four statements. $\quad 3(n + 2) \equiv 3n + 6 \quad\; A = \pi r^2 \quad\; 5n - 1 = 19 \quad\; 4n^2 - n$

Write down which one is an expression, which is an equation, which is a formula and which is an identity.[2]
Show mark scheme
- B1Two correctly identified
- B1All four correct: identity $3(n + 2) \equiv 3n + 6$ ; formula $A = \pi r^2$ ; equation $5n - 1 = 19$ ; expression $4n^2 - n$

Question 3Grade 43 marks

(a)Simplify $3x^2y \times 4xy$ .[2]
Show mark scheme (a)
- B1Numbers and one letter correct, e.g. $12x^3$ …
- B1 $12x^3y^2$
Answer: $12x^3y^2$
(b)Simplify $\dfrac{20a^3b}{5ab}$ .[1]
Show mark scheme (b)
- B1 $4a^2$
Answer: $4a^2$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

Simplify $a + a + a$ .
- $3a$
- $a^3$
- $3 + a$
- $aaa$
Show answer and explanation
Answer: $3a$
Three lots of $a$ .
Simplify $y \times y \times y$ .
- $y^3$
- $3y$
- $y + 3$
- $3y^3$
Show answer and explanation
Answer: $y^3$
Repeated multiplication is a power.
Simplify $4x + 3y - x + 2y$ .
- $3x + 5y$
- $5x + 5y$
- $8xy$
- $3x + y$
Show answer and explanation
Answer: $3x + 5y$
$4x - x = 3x$ and $3y + 2y = 5y$ .
Simplify $3a \times 5b$ .
- $15ab$
- $8ab$
- $15a + b$
- $35ab$
Show answer and explanation
Answer: $15ab$
Multiply the numbers, then write the letters together.
Which of these is an identity?
- $2(x + 4) \equiv 2x + 8$
- $2x + 4 = 10$
- $P = 2l + 2w$
- $2x + 8$
Show answer and explanation
Answer: $2(x + 4) \equiv 2x + 8$
True for every value of $x$ , shown by $\equiv$ .
Simplify $5x^2 - 2x + x^2 + 7x$ .
- $6x^2 + 5x$
- $11x^3$
- $4x^2 + 5x$
- $6x^4 + 5x^2$
Show answer and explanation
Answer: $6x^2 + 5x$
$x^2$ and $x$ terms are collected separately.
Simplify $2m \times 3m \times m$ .
- $6m^3$
- $6m^2$
- $5m^3$
- $6m$
Show answer and explanation
Answer: $6m^3$
$2 \times 3 = 6$ and $m \times m \times m = m^3$ .
Simplify $\dfrac{24p^2q}{8pq}$ .
- $3p$
- $3pq$
- $16p$
- $3p^2$
Show answer and explanation
Answer: $3p$
$24 \div 8 = 3$ ; $p^2 \div p = p$ ; $q \div q = 1$ .

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