In maths, a letter works like an empty box that holds a number we have not fixed yet.
If every box holds 12 pencils, then a boxes hold 12a pencils, and that one expression covers any number of boxes.
Writing 3 × a as 3a and a ÷ b as a fraction, following a few set rules, makes expressions shorter and easier to read.
On this page you put different numbers into x and watch how the value of 2x + 3 changes, then collect the six rules for dropping signs.
A letter is a box for a number
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Two x boxes and the number 3, with the value of 2x + 3 changing as x changes
4
👀 Why use letters?
①Writing an unknown number as a letter lets one expression cover many cases
②In the picture, putting 3 into x gives 2x + 3 = 9, and putting in 5 gives 13. The numbers change, but the shape of the expression, "add x twice, then add 3", stays the same
③In 2x + 3, the parts 2x and 3 are terms. The number multiplying the letter, 2 in 2x, is the coefficient of x, and a term with no letter, like the 3 drawn as a circle, is the constant term
Rules for dropping × and ÷
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Six rules for dropping the multiplication and division signs
🔍 Rules for dropping signs
①Leave out × between a number and a letter, or between two letters
②Write the number before the letter: x × 3 is 3x, never x3
③Write letters in alphabetical order: b × a is ab
④Write 1 × x as x and (−1) × x as −x. In 0.1 × x the 0.1 is not 1, so it stays: 0.1x
⑤Write a letter multiplied by itself as a power: x × x = x²
⑥Write division as a fraction: a ÷ b = a/b (with b ≠ 0)
Finding the value of an expression
Substitution
Putting a number in place of a letter
When x = 3, 2x + 1 = 2 × 3 + 1 = 7
Value of an expression
The result of substituting and calculating
When you substitute, bring back the × signs you dropped and then calculate
💡 Take care when substituting
①Put a negative number in brackets. If x = −2, then 2x + 1 = 2 × (−2) + 1 = −3
②Slide x to a negative number in the picture and brackets appear in the expression at the top
③Brackets matter most with powers. When x = −2, x² = (−2)² = 4, but written without brackets as −2² it becomes −4
④Put a fraction in brackets too, so the order of operations stays clear
Check with examples
Example 1
Each box holds 12 pencils. Write the number of pencils in a boxes as an expression using a letter.
1
(number of pencils) = (pencils per box) × (number of boxes).
pencils = 12 × a
2
It is a number times a letter, so drop the × sign and put the number first.
= 12a (pencils)
▸ 12a pencils
Putting 3 into a gives 36 pencils, and 10 gives 120. However many boxes there are, the one expression 12a is enough.
Example 2
When a = 3 and b = −2, find the value of 2a − 3b.
1
b is negative, so substitute it in brackets.
2a − 3b = 2 × 3 − 3 × (−2)
2
Multiply first: −3 × (−2) = +6.
= 6 + 6 = 12
▸ 12
Without brackets, 3 × −2 puts a minus sign right next to a negative sign, and the signs are easy to misread.
Exam checklist
Key notation
drop × · number first, letters in order · ÷ becomes a fraction
3 × b × a = 3ab, x ÷ 5 = x/5
Grade 7 school test style
When x = −2, what is the value of x² − 3x?
①−2
②2
③6
④10
⑤−10
▸ ④ 10
1
Substitute x = −2 in brackets.
x² − 3x = (−2)² − 3 × (−2)
2
(−2)² = 4 and −3 × (−2) = 6.
= 4 + 6 = 10
🎯 Exam checklist
①When dropping ×, write the number first and the letters in alphabetical order
②Don't write a coefficient of 1, and write −1 as just −. The 1 in 0.1x stays
③Rewrite division as a fraction
④Put negative numbers and fractions in brackets when you substitute
⑤A letter multiplied by itself several times becomes a power: x × x × x = x³