Brackets go first · Lập trình với Martin

A bracket shouts: work me out first!

Look at two almost the same write-ups:

2 + 3 × 4 and (2 + 3) × 4

They differ by only two little curves. And yet the first gives 14 and the second 20. Those curves are called brackets and in maths they have one job: to say what is worked out first.

What is in the brackets, you work out before everything else. Always. No exception. A bracket is like a VIP ticket — whoever has it goes first, even if they stand at the end.

Why is that needed? Because a long question can be read in more than one way and each way gives a different number. Maths needs everyone in the world to get the same result. Brackets are the agreement on how to read it clearly.

Today you will learn to read brackets and to write them. And next time you will find out what happens when there is no bracket in the question.

Tereza reads a pancake recipe with mum: “Mix 2 eggs and 3 spoons of sugar, then whisk everything four times.” Mum laughs that the recipe is like maths: first mix, then four times — that is (2 + 3) × 4. If you whisked first and then added sugar, it would turn out differently. Tereza realises she already knows brackets from life: “first you do this, then that”. Only in maths instead of sentences you write little curves.

💡

Read a bracket as the word first: (2 + 3) × 4 = “first add 2 and 3, then times the result by four”.

How to work out a question with a bracket

The method has three steps and it works on every question with a bracket.

Step 1: find the bracket and work out what is inside.

(2 + 3) × 4 → inside is 2 + 3 = 5.

Step 2: rewrite the question — instead of the bracket write the result.

(2 + 3) × 4 changes to 5 × 4.

Step 3: finish the rest.

5 × 4 = 20.

The important bit is the rewriting. Do not try to hold everything in your head — write the middle step on paper. Each line is the same question, just one step simpler:

(2 + 3) × 4 = 5 × 4 = 20

A method written like this a teacher can read, a friend can check, and you can find a mistake in it even in a week.

When there are more brackets — say (10 − 4) × (1 + 2) — work out each one on its own: 6 × 3 = 18. The order of the brackets among themselves does not matter, as long as both come before the multiplying.

Same numbers, different result

I compare two write-ups.

Without a bracket: 2 + 3 × 4. Times goes first (why, we will say next time): 3 × 4 = 12, then 2 + 12 = 14.

With a bracket: (2 + 3) × 4.

= 5 × 4

= 20.

Two little curves, a difference of six. That is why brackets must not get lost or get added — they change the meaning of the whole question, like a comma in a sentence.

A bracket at the end of a question

Example: 20 − (4 + 6).

Step 1: bracket: 4 + 6 = 10.

Step 2: I rewrite: 20 − 10.

Step 3: I finish: 10.

As a story: you have 20 Kč and you buy a roll for 4 Kč and a yoghurt for 6 Kč. The bracket is the shopping together (10 Kč), the rest is what you get back: 10 Kč.

Without the bracket, 20 − 4 + 6 would come out 22 — as if the shop paid you for the yoghurt. Nice nonsense.

Two brackets at once

Example: (18 − 10) ÷ (2 × 2).

Step 1: first bracket: 18 − 10 = 8.

Step 2: second bracket: 2 × 2 = 4.

Step 3: I rewrite: 8 ÷ 4.

Step 4: I finish: 2.

Write-up under each other:

(18 − 10) ÷ (2 × 2)

= 8 ÷ 4

= 2

Each line is simpler than the one before. That is the sign of a good method.

⚠️

When you rewrite a question, copy everything you have not worked out yet. A common mistake: from (2 + 3) × 4 a child writes only “5” and forgets the four. Check each line: all the numbers must be there except the ones you have just got rid of.

Practice: now you do it 🎲

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

Bracket first, then the rest. Write middle steps on paper.

Practise the questions

On paper

Write each question on three lines: the task, the rewrite, the result.

Work out with middle steps: (4 + 5) × 2, 30 − (6 + 8), (20 − 5) ÷ 3.
Work out both versions and compare: 10 − 2 × 3 and (10 − 2) × 3. Write by how much the results differ.
Add brackets so that this holds: 2 + 4 × 5 = 30. (Hint: without brackets you get 22.)
Make up a shop story for the question 50 − (12 + 8).

Now you 💪

You can say what a bracket means: work me out first.
You work out (2 + 3) × 4 with a rewrite to the middle step 5 × 4.
You can add brackets into a question so that a given result comes out.

✅

Done when: You always work out a bracket first, you rewrite the question line by line, and you do not lose numbers while doing it.

What to take from this lesson

A bracket says: work me out first. Always, no exception.
Method: work out the inside, rewrite the question, finish the rest.
Write middle steps under each other — each line is the same question, just simpler.
Brackets change the result: 2 + 3 × 4 = 14, but (2 + 3) × 4 = 20.
More brackets? Work out each one on its own, then the rest.