Fractions · Grade 5

What do the numerator and the denominator of a fraction mean?

The denominator, the number underneath, says how many equal parts the whole was divided into. The numerator, on top, says how many of those parts we took. Three quarters means the whole was cut into four equal parts and we took three of them — and the word equal is the condition without which there is no fraction at all.

Learning objectives

Two numbers, two questions

Every fraction has two numbers, and they answer different questions. The denominator below answers how many parts the whole was cut into. The numerator above answers how many of those parts we took.

So in three over four, the four says the whole was cut into four, and the three says we took three of them. Both numbers talk about the same parts: one counts them in the whole, the other counts them in our hands.

One consequence looks backwards at first: the larger the denominator, the smaller each part. A fifth is smaller than a third, because we cut the same whole into more pieces.

The word equal is the condition

A cake cut into four unequal pieces is not divided into quarters. A fraction assumes every part is the same size, and without that the number written down says nothing at all.

This is why drawing helps. A rectangle cut into equal parts with some of them shaded shows the fraction, and it shows immediately when the division is not equal.

A fraction is also a place on the line

You can think of a fraction as part of a cake, and you can think of it as a point between two whole numbers. Both pictures are true, and the second one is the more useful later on.

To place a fraction on the line, cut the gap between zero and one into as many parts as the denominator says, and count along as many as the numerator says. Three quarters sits at the third mark out of four.

Worked examples

  1. A rectangle is cut into 8 equal parts and 5 are shaded. What fraction is shaded?

    1. The number of parts the whole was cut into is the denominator: 8
    2. The number shaded is the numerator: 5
    3. Write the numerator above the denominator

    Answer: Five eighths

  2. Which is larger: a third or a fifth?

    1. A third cuts the whole into three parts; a fifth into five
    2. More parts means each part is smaller
    3. So each third is larger than each fifth

    Answer: A third, even though the number underneath is smaller

  3. A pizza is cut into six pieces, two of them much larger than the rest. Is one piece a sixth?

    1. A fraction requires division into exactly equal parts
    2. Here the parts are not equal

    Answer: No. You can say one of six pieces, but not a sixth

Common mistakes

Thinking a bigger denominator means a bigger fraction
The denominator counts how many parts the whole was cut into, so a bigger one gives smaller parts. A tenth is smaller than a half, even though ten is bigger than two.
Reading the fraction as two separate numbers
Three over four is not three and four but a single quantity, a little under one whole. Reading it as two numbers is what later leads to adding denominators.
Ignoring the requirement that the parts be equal
Without an equal division the number written is not a fraction. That is not fussiness — it is what makes comparing two fractions possible at all.

What to remember

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