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Halves, thirds and quarters

Numbers · level 2, lesson 6 of 7

A question to start

Two brothers share a sandwich. The older one cuts it "in half", but his cut is not in the middle. He takes the big piece and says: "I'll have the bigger half!"

Is there such a thing as a bigger half? People say it all the time. But a mathematician would say the older brother did not cut the sandwich in half at all. Why not?

And here is a second puzzle. Would you rather have one third of a pizza or one quarter of the same pizza? Four is bigger than three... so is a quarter bigger? In this lesson you will find out what halves, thirds and quarters really are, and the answers to both puzzles will be easy.

The idea

Equal parts

So far, every number in this topic has counted whole things: 3 apples, 12 cookies. But often we want part of a thing: some of a sandwich, a piece of a pizza, part of a ribbon. Numbers for parts of a whole are called fractions.

The most important word in this lesson is equal. To cut something into halves, thirds or quarters, every piece must be the same size.

  • Halves: the whole is cut into 2 equal parts. Each part is one half.
  • Thirds: the whole is cut into 3 equal parts. Each part is one third.
  • Quarters (also called fourths): the whole is cut into 4 equal parts. Each part is one quarter.
halveshalvesNOT halves

The circle and the square are cut into halves: the two parts are exactly the same size. The rectangle is cut into two parts, but they are not equal, so they are not halves.

That answers the sandwich puzzle. If the two pieces are different sizes, they are not halves. A "bigger half" is impossible, because halves are equal by their very meaning.

Why must the parts be equal? Because a fraction is a promise about how much. If "a half" could be any piece, then "half a pizza" might be a tiny sliver or nearly the whole thing, and the words would tell you nothing. Making the parts equal is what makes "a half" mean one exact amount.

How to check that parts are equal

  • Fold it. Fold a paper square along the diagonal. If the two triangles fit exactly on top of each other, they are halves.
  • Turn it. Cut a square from corner to corner. Turn one triangle around: it fits exactly on top of the other.
  • Count it. If a chocolate bar has 12 little squares, each half must have 6 of them.

Equal does not have to mean the same shape

Here are three ways to cut a square into quarters.

a quartera quartera quarter

The shaded parts are different shapes: a small square, a thin strip, a triangle. But each one is a quarter of the same square, because in each picture the square is cut into 4 equal parts. So all three shaded parts are the same amount of square, even though they look different. (If you cut them out, you could cut and rearrange any one of them to cover any other exactly.)

Writing fractions

We write one half as 1/2, or stacked up like this: \(\frac{1}{2}\). One third is 1/3, and one quarter is 1/4.

  • The bottom number says how many equal parts the whole is cut into.
  • The top number says how many of those parts we are talking about.

So 1/4 means "the whole is cut into 4 equal parts, and we have 1 of them". The bottom number is called the denominator (it names the size of the parts). The top number is the numerator (it counts them).

We can also have more than one part. If a pizza is cut into 4 equal slices and you eat 3, you ate 3/4 (three quarters). If a cake is cut into 3 equal pieces and two are gone, 2/3 (two thirds) are gone.

Fraction strips: which is bigger?

Here is one whole strip, and the same strip cut into halves, thirds and quarters.

1 1 whole 1/2 1/2 halves 1/3 1/3 1/3 thirds 1/4 1/4 1/4 1/4 quarters

Look at the shaded pieces. 1/2 is bigger than 1/3, and 1/3 is bigger than 1/4. The more pieces you cut the whole into, the smaller each piece is.

That answers the pizza puzzle. A third of a pizza is more than a quarter. Think of sharing: if a pizza is shared among 3 people, each gets more than if the same pizza is shared among 4 people. More people, smaller shares. The 4 in 1/4 is bigger, but it is the number of people sharing, not the size of the piece.

The strips show more:

  • 2 halves make 1 whole. 3 thirds make 1 whole. 4 quarters make 1 whole.
  • 2 quarters are the same length as 1 half. So 2/4 is the same amount as 1/2.

A fraction of a group

Fractions also work for a group of things. What is half of 8 cookies?

half: 4half: 4

Half means 2 equal parts. So split the 8 cookies into 2 equal groups: share them between 2 people, as in lesson 5. Each group has 4. So half of 8 is 4. Finding a fraction of a group is dividing:

  • half of 8 is 8 ÷ 2 = 4;
  • a third of 12 is 12 ÷ 3 = 4;
  • a quarter of 12 is 12 ÷ 4 = 3.

And for more than one part, take more than one group: three quarters of 12 is 3 groups of 3, which is 9.

Fractions on the number line

Fractions are numbers, so they have places on the number line. They live between the whole numbers. Look at the stretch from 0 to 1.

01 1/2 01 1/3 01 1/4
  • Cut the stretch from 0 to 1 into 2 equal steps. The first step ends at 1/2, exactly halfway.
  • Cut it into 3 equal steps. The first step ends at 1/3.
  • Cut it into 4 equal steps. The first step ends at 1/4.

A fraction with 1 on top, like 1/2, 1/3 or 1/4, is called a unit fraction: it is one single part. On the number line, you can see again that 1/4 is closer to 0 than 1/3, and 1/3 is closer to 0 than 1/2. So 1/4 < 1/3 < 1/2.

Worked examples

1. Is it halves? A rectangle is cut into two pieces. One piece is 5 squares and the other is 3 squares. Is it cut into halves?

No. The pieces must be equal. Halves of 8 squares would have 4 squares each.

2. Naming the fraction. A ribbon is cut into 3 equal pieces. Ana takes 1 piece. What fraction of the ribbon does she have?

3 equal pieces, 1 taken: 1/3, one third.

3. A fraction of a group. What is a quarter of 20 stickers?

Quarter means 4 equal parts, so share 20 into 4 groups: 20 ÷ 4 = 5. Check: 4 groups of 5 is 20. ✓

4. Comparing. Which is more: half of a chocolate bar, or a third of the same bar?

Half. Cutting into 2 equal pieces makes bigger pieces than cutting into 3.

5. Two thirds of a group. What is two thirds of 15 marbles?

One third of 15 is 15 ÷ 3 = 5. Two thirds is two of those groups: 5 + 5 = 10.

Try it

  1. A pizza is cut into 4 equal slices. What is each slice called? How many slices make the whole pizza?
  2. Draw a rectangle and cut it into thirds. How can you check the parts are equal?
  3. What is half of 14? A third of 9? A quarter of 16?
  4. Put in order from smallest to biggest: 1/3, 1/2, 1/4.
  5. Tim says "I cut the cake into quarters" but he cut it into 4 pieces of different sizes. What would you tell him?
  6. A day on Mars, a sol, is a little longer than an Earth day. If a rover sleeps for half a sol and drives for a quarter of a sol, is it sleeping or driving for longer?
  7. What is three quarters of 8?
  8. Is 2/4 of a pizza more, less or the same as 1/2 of the same pizza? Use the fraction strips to explain.
Answers
  1. Each slice is one quarter (1/4). 4 slices (4 quarters) make the whole.
  2. Draw lines so that 3 parts are equal: for example, two lines going down, cutting the rectangle into 3 strips of equal width. Check by measuring each strip's width with a ruler, or by folding the paper in three.
  3. Half of 14: 14 ÷ 2 = 7. A third of 9: 9 ÷ 3 = 3. A quarter of 16: 16 ÷ 4 = 4.
  4. 1/4, 1/3, 1/2. More pieces means smaller pieces.
  5. They are not quarters. Quarters must be 4 equal pieces. Unequal pieces are just "4 pieces".
  6. Sleeping. A half is bigger than a quarter (in fact, a half is two quarters, so it sleeps twice as long as it drives).
  7. A quarter of 8 is 8 ÷ 4 = 2. Three quarters is 3 groups of 2: 6.
  8. The same. On the fraction strips, two quarter pieces put end to end are exactly as long as one half piece.

Common mistake

"4 is bigger than 3, so a quarter is bigger than a third." The bottom number counts how many pieces the whole is cut into. More pieces means smaller pieces. So 1/4 is smaller than 1/3.

Unequal parts. Cutting something into 4 pieces does not make quarters unless the pieces are equal. Always check by folding, turning or counting.

Where it leads