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Fractions

A fraction is one number, not two. Almost every difficulty children have with fractions traces back to reading 3/4 as a three and a four that happen to be stacked — which makes adding the tops and the bottoms look perfectly sensible, and gives 1/2 + 1/2 = 2/4.

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Adding with the same denominator

3/8 + 4/8

  1. Both are eighths, so they can be added straight away.
  2. Three eighths and four eighths is seven eighths.
  3. The denominator does not change — you still have eighths.
  4. The answer is 7/8.

Worth knowing. The 8 stays put. It is the name of the pieces, not a number being added, and saying "eighths" out loud makes that hard to forget.

Adding with different denominators

1/4 + 1/2

  1. Quarters and halves cannot be added as they are — the pieces are different sizes.
  2. Halves can be rewritten as quarters: 1/2 is 2/4.
  3. Now both are quarters: 1/4 + 2/4.
  4. One quarter and two quarters is three quarters. The answer is 3/4.

Worth knowing. Finding a common denominator is the whole job. Once both fractions have the same name, the adding is the easy part.

Where it usually goes wrong

  • Adding the denominators, so 1/2 + 1/2 comes out as 2/4.
  • Leaving an answer unsimplified when the question asked for lowest terms — 6/8 rather than 3/4.
  • Assuming a bigger denominator means a bigger fraction, so 1/8 is put above 1/3.

The Professor says the denominator out loud as a name: not "four" but "quarters". Three quarters plus one quarter is four quarters, and nobody adds the word.

Practise it

All level paths

Questions teachers and parents ask

Why can't you add the denominators?

Because the denominator names the size of the piece, and adding two names makes no sense. Half a pizza and half a pizza is a whole pizza, not two quarters — the pieces did not get smaller because there were more of them.

Does the answer always have to be simplified?

It depends on what was asked. 6/8 and 3/4 are the same number, so both are correct answers to the arithmetic; whether a child is allowed to stop at the first is a teaching decision, and the levels here say which one they want.

Which is bigger, 1/3 or 1/8?

1/3. The bigger the denominator, the more pieces the whole was cut into, and so the smaller each piece is. Cutting a cake into eight gives smaller slices than cutting it into three.

When are fractions taught?

Adding and subtracting with like denominators is grade 4 (4.NF.B.3), unlike denominators grade 5 (5.NF.A.1), multiplying grade 5 (5.NF.B.4), and dividing a fraction by a fraction grade 6 (6.NS.A.1). They are revisited every year in between.

Where this shows up

Standards

  • 3.NF · Number and operations — fractions
  • 4.NF · Number and operations — fractions
  • 5.NF · Number and operations — fractions
  • 6.NS · The number system

Formats