A half, a third, a quarter · Lập trình với Martin

Half a pizza, a third of a chocolate

Split a pizza into two equal pieces. One piece is a half. Into three equal pieces? A piece is a third. Into four? A quarter.

These parts are called fractions — numbers for pieces of a whole. They are written as two numbers one above the other with a line: a half is ½, a third ⅓, a quarter ¼. The number under the line says into how many equal parts the whole was cut. The number above the line says how many parts you took.

Two things are sacred with fractions:

First: the parts must be equal. Breaking chocolate into a big piece and a small piece and saying “you have a half” — that is cheating, not a half.

Second: the more parts, the smaller the piece. A third of a pizza is smaller than a half, even though 3 is more than 2. Share among more people and each gets less — you know that from life.

And how do you work out a half of 12 sweets? By dividing — which you already know.

Tereza has a bar of chocolate with 12 pieces and shares with Jonáš. “Half and half,” they both say. Tereza counts: 12 ÷ 2 = 6, six pieces each. Then Sofie comes. “So into thirds:” 12 ÷ 3 = 4, four pieces each. And when Filip joins too, it comes out quarters: 12 ÷ 4 = 3 pieces for each. Jonáš grumbles that his portion keeps falling: 6, 4, 3… Tereza shrugs: “More mouths, smaller piece. That is not greed, that is fractions.”

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Read a fraction from the bottom: the number under the line = into how many parts I cut, the number above the line = how many parts I take. ¼ = cut into 4, take 1.

How to work out a part of a whole

A half, a third or a quarter of a number = dividing.

Method for “a half of 12”:

A half means split into 2 equal parts.
I divide: 12 ÷ 2 = 6.
A half of 12 is 6.

Same for a third of 12: 12 ÷ 3 = 4. A quarter of 12: 12 ÷ 4 = 3.

The rule: the denominator (the number under the line) is the divisor. A half → divide by two, a third → by three, a quarter → by four.

Check by putting together: the parts must make the whole again. A half of 12 is 6 → two halves: 6 + 6 = 12. It fits. A third is 4 → three thirds: 4 + 4 + 4 = 12. It fits. If the parts do not make the whole, the part is calculated wrongly.

More parts at once: three quarters (¾) of 12? Work out one quarter (12 ÷ 4 = 3) and take it three times: 3 × 3 = 9. First divide, then multiply.

When it does not come out smoothly: a third of 10? 10 ÷ 3 = 3, remainder 1 — ten sweets you cannot share fairly into thirds, one is left. That is a correct answer too. With a pizza the leftover piece could be cut, with sweets you draw lots.

A half and a quarter of 20

There are 20 children in the class. A half go on a trip, a quarter have a club. How many children is that?

A half of 20: I divide by two: 20 ÷ 2 = 10 children.

A quarter of 20: I divide by four: 20 ÷ 4 = 5 children.

Check by putting together: two halves: 10 + 10 = 20. Four quarters: 5 + 5 + 5 + 5 = 20. Both fit.

Notice: a quarter (5) is exactly half of a half (10). Split in half and split in half again = a quarter.

A third of an hour

How many minutes is a third of an hour?

Step 1: an hour = 60 minutes (the time lesson).

Step 2: a third → I divide by three: 60 ÷ 3 = 20 minutes.

Check: 20 + 20 + 20 = 60. It fits.

And a quarter of an hour? 60 ÷ 4 = 15 minutes — exactly that “quarter of an hour” we talked about with time. The words quarter and half with clocks are fractions in disguise.

Three quarters of 8

Sofie has read three quarters of a book that has 8 chapters. How many chapters has she already xong?

Step 1: one quarter: 8 ÷ 4 = 2 chapters.

Step 2: three quarters: 3 × 2 = 6 chapters.

Check: one quarter is left, so 2 chapters. 6 + 2 = 8. It fits.

Recipe for a part of a whole: divided by the bottom number, times the top one.

⚠️

A fraction holds only when the parts are the same size. And watch a tempting mistake: a third is not more than a half, even though 3 > 2. The number under the line says among how many people you share — more people, smaller portion.

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.

Work out halves, thirds and quarters of a whole. The bottom number of the fraction = the divisor.

Practise the questions

On paper

Draw and cut — fractions are born from real sharing.

Draw three equal rectangle chocolates. Split the first with lines into halves, the second into thirds, the third into quarters. Colour one part each time and label it with a fraction.
Work out: a half of 18, a third of 21, a quarter of 24, a half of 50.
Three quarters of 20 (first a quarter, then three times). Write both steps.
Cut a strip of paper into quarters with no measuring. (Hint: fold in half twice.) Explain why it works.

Now you 💪

You can read ¼: cut into 4 equal parts, take 1.
You work out a third of 21 by dividing: 21 ÷ 3 = 7.
You know why a third is smaller than a half, and you can explain it with portions.

✅

Done when: You work out a half, a third and a quarter of any whole and you check the result by putting the parts together.

What to take from this lesson

A fraction describes a part of a whole: under the line the number of parts, above the line how many you take.
A part of a number = dividing: a half of 12 is 12 ÷ 2 = 6.
The parts must be the same size, otherwise it is not a fraction, but cheating.
The more parts, the smaller the piece: a third < a half.
Check by putting together: the parts must make the whole.