An expression: calculating with letters

Why letters show up in maths, what a variable is, and how to read the expression 3x + 2.

A letter is a box for a number

Someone once told you maths is numbers. And now 3x + 2 shows up in your notebook and you ask: what is a letter doing there?

A letter is a box for a number. You do not know yet which number is in it — maybe you will find out later, maybe it will change. That is why it is called a variable.

An example from life: one tram ride costs 20 Kč. How much do you pay for x rides? 20 · x, shortened to 20x. You do not know how many rides you will take — but the formula works for every number. That is the whole power of letters: one notation instead of a thousand questions.

The whole writing of numbers, letters and operations is called an expression. 3x + 2 is an expression. 5a − b too. An expression is not an equation — it has no equals sign, nothing in it gets solved. It is just a recipe for a calculation.

Today you will learn to read expressions, write them and understand their parts. Calculating with them starts next time.

Ema is selling muffins at the school fair for 15 Kč. Adam is running the till and keeps calculating: 3 muffins 45, then 5 muffins 75… Ema writes on paper: 15m. “What is that?” — “A recipe. m is the number of muffins. You sell seven? 15 · 7 = 105. You sell twenty? 15 · 20 = 300. One notation, every sale.” Adam then adds his own upgrade: 15m + 10l, where l is lemonade for ten crowns. At the end of the fair they substitute the numbers and the till matches to the crown.

💡

Between a number and a letter the times sign is not written: 3x means 3 · x. Between numbers it stays: 3 · 5 cannot be written as 35.

How to read and write expressions

Parts of an expression. In the expression 3x + 2 there is: the variable x (a box for a number), the coefficient 3 (the number the variable is multiplied by) and the term 2 (a number on its own, called the constant term).

Reading. 3x + 2 you read “three x plus two” and it means: take the number from the box, multiply by three, add two.

Writing by the agreed rules:

Times between a number and a letter is left out: 5 · a = 5a. You always write the number before the letter: a5 is not written.
1x is written just as x. The one hides.
x · x = x². Powers work with letters too.
規約 are separated by plus and minus: the expression 4a + 2b − 7 has three terms.

Translating from English into an expression — you will do this all the time:

“a number 5 bigger than n” → n + 5. “double the number t” → 2t. “half of the number k” → k/2. “3 less than double x” → 2x − 3.

An expression is a sentence in the language of maths. Learning to read it is like learning a few words — then you just put them together.

Example 1: break down the expression 5a − 3b + 8

How many terms does it have and what is in them?

Plus and minus separate the terms: 5a, −3b and +8. Three terms.

First term: coefficient 5, variable a. Second: coefficient −3 (the minus belongs to it!), variable b. Third: the number 8 with no variable — the constant term.

That the minus belongs to the term after it is important. You can read the expression as a sum: 5a + (−3b) + 8. Same value, but you can see who owns the sign.

Example 2: translate a sentence into an expression

Tereza is 3 years older than Jonáš. Jonáš is j years old. How old is Tereza?

“3 older” means add 3: Tereza is j + 3.

Second round: mum is twice as old as Tereza. Mum is 2 · (j + 3) = 2(j + 3) years old.

The brackets are necessary — double Tereza’s whole age, not just Jonáš’s. Without brackets, 2j + 3 would mean “double Jonáš plus three”, and that is a different number.

Example 3: the fair in one expression

A muffin costs 15 Kč, lemonade 10 Kč, table hire is 50 Kč. Write an expression for the profit.

Takings from muffins: 15m. From lemonades: 10l. Minus hire: −50.

Profit = 15m + 10l − 50.

Sense check: you sell 10 muffins and 5 lemonades → 150 + 50 − 50 = 150 Kč. The expression is a calculator into which you just substitute the sales. That is exactly how formulae in spreadsheets work, which you may know from Sheets.

⚠️

3x does not mean “thirty-something” or “3 and x next to each other”. It is multiplication: 3 · x. When you substitute 12 for x, 3x = 36, not 312.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. スタート with ten. When it goes well, add more.

Practise calculating with negative numbers — in expressions the signs swarm and they must not slow you down.

Practise the questions

On paper

Translate. From English into maths and back.

Write as an expression: a number 7 bigger than x; triple the number y; half of the number z reduced by 4.
Break down the expression 7k − 2m + 15: how many terms it has, what the coefficients are, which term is the constant.
A cinema ticket costs 150 Kč and popcorn 80 Kč. Write an expression for the price for d children, when each has a ticket and together they buy p popcorns.
Translate back into English: 2n + 1. Invent a situation it fits.

Now you 💪

I know a letter is a variable — a box for a number.
I can count the terms of an expression and find the coefficients.
I translate the sentence “5 more than double” into 2x + 5.

✅

Done when: You have expressions translated both ways, an expression broken down into terms, and a run of five correct in practice.

What to take from this lesson

A variable is a box for a number. An expression is a recipe for a calculation.
3x means 3 · x — times between a number and a letter is not written.
規約 are separated by plus and minus; the sign belongs to the term after it.
An expression has no equals sign. Nothing in it gets solved, it only describes a calculation.
Translation from English: “3 more” = +3, “double” = 2·, “half” = :2.