Notation and substitution

1. Writing algebra the short way

Key idea A letter stands for a number. Algebra leaves out the × sign and writes everything as short as it can.

These are the rules. The left column is what a calculation means; the middle column is how you write it.

It means	Write it as	The rule
3 × y	3y	Leave out the ×. The number goes first.
y × 3	3y	The number still goes first, never y3.
y + y + y	3y	Three lots of y.
a × b	ab	Letters side by side are multiplied. Write them in alphabetical order.
1 × x	x	Don’t write the 1.
−1 × x	−x	Just the minus sign.
a × a	a²	“a squared”
a × a × a	a³	“a cubed”
a × a × b	a²b	Only the a is squared.
a ÷ b	ab	Division is written as a fraction.
x ÷ 2, or half of x	x2 or 12x	Write a fraction, not 0.5x.
3 × (x + 2)	3(x + 2)	A number next to a bracket multiplies everything inside it.
(x + 2) ÷ 5	x + 25	The fraction line acts like a bracket: all of x + 2 is divided by 5.

Watch out 3a and a³ are different. 3a means 3 × a. a³ means a × a × a. If a = 2, then 3a = 6 but a³ = 8.

Watch out In 2x², only the x is squared: it means 2 × x × x. If you want the whole of 2x squared, you need a bracket: (2x)² = 2x × 2x.

Example 1: write these the short way

Question	Answer	Why
5 × p × q	5pq	Leave out both × signs.
m × 7	7m	The number goes first.
b × a × b	ab²	Two b’s make b². Letters in alphabetical order.
n ÷ 6	n6	Division is a fraction.
2 × (t − 1)	2(t − 1)	Keep the bracket: the 2 multiplies all of t − 1.

Example 2: write these the long way

Question	Answer
4x²y	4 × x × x × y
3(a + b)	3 × (a + b): three lots of a + b
m5	m ÷ 5
−2c³	−2 × c × c × c

2. The words of algebra

Key idea Terms are the pieces of an expression, split up by + and − signs. The sign in front of a term belongs to that term.

A variable is a letter that stands for a number that can change.
A term is a number, a letter, or numbers and letters multiplied together: for example 5, x, 3y or −4ab.
The coefficient is the number in front of the letters, with its sign. In −4ab the coefficient is −4.
A constant is a term with no letter: just a number.

Example 3: the parts of 5x² − 3xy + y − 8

This expression has four terms.

Term	Coefficient	Notes
5x²	5	5 × x × x
−3xy	−3	The minus sign in front belongs to this term.
y	1	y means 1y. The 1 is not written.
−8	none	The constant term: it has no letter.

Fractions work the same way: in x4 the coefficient of x is 14, because x4 = 14x.

Then there are four words for whole pieces of algebra. You need to tell them apart.

Word	What it is	Examples
Expression	Terms joined by + and −, with no equals sign. You can work out its value, but you can’t solve it.	4n + 1 a²b 3(x − 2)
Equation	Has an equals sign, and is true for only certain values of the letter.	4n + 1 = 13 is only true when n = 3
Formula	A rule that links two or more quantities. Each letter stands for something real.	A = lw v = u + at
Identity	True for every value of the letters. It is written with the sign ≡.	y + y + y ≡ 3y a × b ≡ ab

In the formula A = lw, A is on its own on one side. It is called the subject of the formula.

3. Turning words into expressions

Key idea Read the words in order and write down what happens to the number. If a whole amount is then multiplied, divided or squared, it needs a bracket.

Here are the ways a question can say it. Each one uses n for the number.

The words	The expression
7 more than n / n increased by 7 / the sum of n and 7	n + 7
7 less than n / n decreased by 7 / subtract 7 from n	n − 7
n less than 7 / n subtracted from 7	7 − n
4 times n / the product of 4 and n / 4 lots of n	4n
twice n / double n	2n
n divided by 3 / a third of n / n shared equally between 3	n3
half of n	n2 or 12n
n squared / n multiplied by itself	n²
n cubed	n³

Watch out “7 less than n” is n − 7, even though the 7 is said first. Start with n and take 7 away. Test it with a number: 7 less than 10 is 3, and 10 − 7 = 3.

Example 4: the order changes the expression

The words	The expression	Test with n = 4
Double n, then add 5	2n + 5	2 × 4 + 5 = 13
Add 5 to n, then double	2(n + 5)	2 × (4 + 5) = 18
Multiply n by 3, then subtract 4	3n − 4	3 × 4 − 4 = 8
Subtract 1 from n, then square	(n − 1)²	(4 − 1)² = 9
Add 6 to n, then divide by 2	n + 62	(4 + 6) ÷ 2 = 5

The first two use the same numbers but give different answers. The bracket shows that you add before you double.

Example 5: a bag of sweets

A bag has n sweets. Mo eats 3 of them, then shares the rest equally between 4 friends. Write an expression for the number of sweets each friend gets.

Mo eats 3

n − 3 sweets are left
All of what is left is shared between 4

the whole of n − 3 goes on top of the fraction

Answern − 34

Test it: with 23 sweets, 23 − 3 = 20, and 20 shared between 4 is 5 each.

Example 6: change from a £5 note

Pens cost x pence each. Lily buys 6 pens and pays with a £5 note. Write an expression for her change, in pence.

Put everything in the same units

£5 = 500p
6 pens at x pence each

6x pence
Change is what she paid, take away what it cost

Answer500 − 6x

Test it: if pens cost 45p, 500 − 6 × 45 = 500 − 270 = 230p change.

Example 7: a square

Each side of a square is s + 2 centimetres long. Write an expression for its perimeter.

A square has 4 equal sides, so the perimeter is 4 lots of the whole side.
The whole of s + 2 is multiplied by 4, so it needs a bracket.

Answer4(s + 2) cm

4s + 2 would be wrong: that only multiplies the s by 4.

4. Writing a formula

Key idea A formula has a letter for the answer, an equals sign, and an expression. “3h + 8” is an expression. “C = 3h + 8” is a formula.

Method

Find the part that changes

Something “each” or “per” gets multiplied by its letter.
Find the part that stays the same

A fixed charge or a starting amount is added on (or is where you start taking away from).
Write the letter for the answer, then =, then the expression
Test it with a small number

Example 8: bike hire

A shop charges £8 to hire a bike, plus £3 for every hour. Write a formula for the total cost, C pounds, of hiring a bike for h hours.

The part that changes: £3 for each hour

3h
The part that stays the same: £8

3h + 8

AnswerC = 3h + 8

Test it: 2 hours is £8 + £3 + £3 = £14, and 3 × 2 + 8 = 14.

Example 9: something that goes down

A candle is 30 cm tall. It burns down 2 cm every hour. Write a formula for its height, H cm, after t hours.

It starts at 30 cm
It loses 2 cm for each hour, so take away 2t

AnswerH = 30 − 2t

Test it: after 5 hours it has lost 10 cm, so it is 20 cm tall. 30 − 2 × 5 = 20.

Example 10: two letters

Cinema tickets cost £12 for an adult and £7 for a child. Write a formula for the total cost, T pounds, for a adults and c children.

AnswerT = 12a + 7c

Test it: 2 adults and 3 children cost 24 + 21 = £45, and 12 × 2 + 7 × 3 = 45.

Watch out Check the units. If apples cost 45p each, the cost of n apples is 45n pence. If the question wants the cost in pounds, the formula is C = 0.45n.

5. Substitution

Key idea To substitute means to swap each letter for its number. Then work out the answer using BIDMAS.

Method

Write the expression again with each letter swapped for its number

Put the × signs back in: 4m becomes 4 × 6.
Put any negative number in brackets
Work it out with BIDMAS, one step at a time

Brackets, then indices (powers and roots), then × and ÷, then + and −.

Example 11: one letter

Work out the value of 4m + 9 when m = 6.

4 × 6 + 9

=24 + 9

=33

Answer33

Example 12: a division

Work out the value of 30 − k4 when k = 12.

30 − 124

=30 − 3

=27

Divide before you subtract.

Answer27

Example 13: two letters

Work out the value of 4x − 7y when x = 3 and y = 2.

4 × 3 − 7 × 2

=12 − 14

=−2

Both multiplications happen first. Answers can be negative even when the letters are positive.

Answer−2

Example 14: letters side by side

Work out the value of ab when a = 3 and b = 7.

3 × 7

=21

Answer21

Watch out ab means a × b. Writing the numbers side by side to get 37 is wrong. In the same way, 3x when x = 4 is 12, not 34.

Watch out BIDMAS still applies after you substitute. 2 + 3x when x = 4 is 2 + 3 × 4 = 2 + 12 = 14. Working left to right to get 5 × 4 = 20 is wrong.

6. Substituting negative numbers

Key idea Always put a negative number in brackets when you substitute it. The bracket keeps the minus sign with its number.

The rules you need:

Rule	Example
Negative × positive is negative	(−3) × 5 = −15
Negative × negative is positive	(−4) × (−3) = 12
Subtracting a negative is the same as adding	12 − (−5) = 12 + 5 = 17
Squaring a negative gives a positive	(−6)² = (−6) × (−6) = 36
Cubing a negative gives a negative	(−2)³ = (−2) × (−2) × (−2) = −8

Example 15: multiplying by a negative

Work out the value of 5y − 2 when y = −3.

5 × (−3) − 2

=−15 − 2

=−17

Answer−17

Example 16: subtracting a negative

Work out the value of 12 − x when x = −5.

12 − (−5)

=12 + 5

=17

Answer17

Example 17: squaring a negative

Work out the value of x² + 4 when x = −6.

(−6)² + 4

=36 + 4

=40

Answer40

Watch out On a calculator, type (−6)² with the brackets. Without them, −6² gives −36, because the calculator squares the 6 first and then puts the minus sign in front.

Example 18: a number in front of a square

Work out the value of 3x² when x = −2.

3 × (−2)²

=3 × 4

=12

Square first (indices come before multiplying). Only the x is squared, so (3 × −2)² = 36 is wrong.

Answer12

Example 19: two negatives

Work out the value of pq + 10 when p = −4 and q = −3.

(−4) × (−3) + 10

=12 + 10

=22

Answer22

Example 20: harder: a minus sign in front of a square

Work out the value of −x² when x = −5.

−(−5)²

=−(25)

=−25

−x² means −(x²). Square first, then the minus sign in front makes it negative.

Answer−25

7. Brackets, powers, roots, fractions and decimals

Key idea Substitute first, then BIDMAS does the rest. A fraction line works like a bracket: work out the whole top and the whole bottom, then divide.

Example 21: a bracket

Work out the value of 3p(q − r) when p = 2, q = 9 and r = 4.

3 × 2 × (9 − 4)

=3 × 2 × 5

=30

Answer30

Example 22: a power and a square root

Work out the value of a²(10 − √b) when a = 3 and b = 16.

3² × (10 − √16)

=9 × (10 − 4)

=9 × 6

=54

The bracket first: inside it, the square root comes before the subtraction.

Answer54

Example 23: a fraction

Work out the value of 5c + d4 when c = 2 and d = −6.

5 × 2 + (−6)4

=10 + (−6)4

=44

=1

Answer1

Example 24: decimals

Work out the value of 6g − 4h when g = 1.5 and h = 0.25.

6 × 1.5 − 4 × 0.25

=9 − 1

=8

Answer8

Example 25: a fraction as the value

Work out the value of 8x² when x = 12.

8 × (12)²

=8 × 14

=2

Square the fraction first: 12 × 12 = 14.

Answer2

8. Substituting into formulae

Key idea A formula works the same way. Put the numbers in, work out the subject, and give the units.

The formula is usually given in the question, often from science. You don’t need to know what it means to use it: just match each letter to its number.

Example 26: speed

v = u + at. Work out v when u = 4, a = 3 and t = 6.

v = 4 + 3 × 6

=4 + 18

=22

Answerv = 22

at means a × t, and it is worked out before the adding.

Example 27: kinetic energy

E = 12mv². Work out E when m = 6 and v = 5.

E = 12 × 6 × 5²

=12 × 6 × 25

=3 × 25

=75

Only the v is squared. Half of 6 is 3, then 3 × 25 = 75.

AnswerE = 75

Example 28: a negative value

F = 1.8C + 32 changes a temperature in °C into °F. Work out F when C = −15.

F = 1.8 × (−15) + 32

=−27 + 32

=5

Answer5°F

Example 29: with a calculator, then round

A = πr². Work out A when r = 4.2 cm. Give your answer to 1 decimal place.

A = π × 4.2²

=π × 17.64

=55.417…

=55.4 (to 1 d.p.)

Answer55.4 cm² (to 1 d.p.)

Watch out Round only at the end. Write down the full calculator answer first, so the examiner can see it, then round.

Sometimes a question gives you the subject and asks for a different letter, for example “find u when v = 20”. Substitute in the same way, then solve the equation that is left. Solving is in the Linear equations lesson, and changing the subject is in Rearranging formulae.

9. Practice: pick your level

A mix of everything on this page. Write your working on paper, then pick an answer. Move up a level when you are getting most of them right.

The mistakes that cost marks

Writing the number after the letter. y × 8 written as y8. the number goes first: 8y.
Mixing up 3a and a³. 3a is 3 × a. a³ is a × a × a.
Turning “less than” round the wrong way. “5 less than n” written as 5 − n. start with n and take 5 away: n − 5. Test it with a number.
Leaving out a bracket. “Add 3 to n, then double” written as 2n + 3. the whole of n + 3 is doubled: 2(n + 3).
Putting numbers side by side. 3x when x = 4 worked out as 34. 3x means 3 × x, so 3 × 4 = 12.
Forgetting BIDMAS after substituting. 2 + 3x when x = 4 worked out as 5 × 4 = 20. multiply first: 2 + 12 = 14.
Squaring a negative and getting a negative. (−3)² = −9. (−3) × (−3) = 9. Use brackets on a calculator too.
Squaring the number in front as well. 2x² when x = 3 worked out as 6² = 36. only the x is squared: 2 × 9 = 18.
Losing a minus sign when subtracting a negative. 10 − (−2) = 8. subtracting a negative is adding: 10 + 2 = 12.
Rounding too early, or leaving out the units. write the full answer, round at the end, and give the units the question uses.

Quick checklist

Tick each one when you can do it without looking back.

I can write algebra the short way: 3y, ab, a²b, ab and 3(x + 2).
I know the difference between 3a and a³, and between 2x² and (2x)².
I can pick out the terms, coefficients and constant in an expression, with their signs.
I can tell an expression, an equation, a formula and an identity apart.
I can turn words into an expression, putting brackets where a whole amount is multiplied, divided or squared.
I can write a formula from a real-life situation, with the right units.
I can substitute positive and negative numbers, decimals and fractions into an expression.
I put negative numbers in brackets and know that (−3)² = 9.
I use BIDMAS after substituting, including powers, roots, brackets and fraction lines.
I can substitute into a formula, then give the units and round at the end.

1. Writing algebra the short way

Example 1: write these the short way

Example 2: write these the long way

The term builder: tap to multiply

2. The words of algebra

Example 3: the parts of 5x2 − 3xy + y − 8

Pick it apart: tap the term you are asked for

Sort the cards: expression, equation, formula or identity?

3. Turning words into expressions

Example 4: the order changes the expression

Example 5: a bag of sweets

Example 6: change from a £5 note

Example 7: a square

The number machine: each step changes the expression

4. Writing a formula

Method

Example 8: bike hire

Example 9: something that goes down

Example 10: two letters

Build the formula: slide, look at the bar, then pick

5. Substitution

Method

Example 11: one letter

Example 12: a division

Example 13: two letters

Example 14: letters side by side

Substitution explorer: move the sliders and watch the working

6. Substituting negative numbers

Example 15: multiplying by a negative

Example 16: subtracting a negative

Example 17: squaring a negative

Example 18: a number in front of a square

Example 19: two negatives

Example 20: harder: a minus sign in front of a square

Who is right? Check their working

7. Brackets, powers, roots, fractions and decimals

Example 21: a bracket

Example 22: a power and a square root

Example 23: a fraction

Example 24: decimals

Example 25: a fraction as the value

Substitution workshop: work it out, then type your answer

8. Substituting into formulae

Example 26: speed

Example 27: kinetic energy

Example 28: a negative value

Example 29: with a calculator, then round

The formula lab: pick a formula and change the values

9. Practice: pick your level

The mistakes that cost marks

Quick checklist

Download

Stuck on this topic?

Example 3: the parts of 5x² − 3xy + y − 8