Understanding fractions

National Curriculum: this is in the Year 3 programme of study, part of key stage 2 (Years 3 to 6). Schools have to teach the key stage 2 programme by the end of Year 6, and can introduce content earlier or later within it.

Children in Year 3 are 7 or 8, and turn 8 during the school year.

What it is

A fraction names a number of equal parts of a whole. In 3/5, the bottom number says the whole was split into 5 equal parts, and the top number says 3 of them are being counted. The parts have to be equal: three pieces out of five cut any way at all is not 3/5. Schools call the bottom number the denominator and the top one the numerator, though a child can hold the idea long before the words.

1/5 is a unit fraction, one of the parts. 3/5 is a non-unit fraction, several of the same part. Reading 3/5 as three fifths, where fifth names the size of one part, is what lets a child see that 5/5 is one whole and that 5/4 is a little more than one.

The bottom number works the opposite way round to whole numbers. The more parts a whole is cut into, the smaller each part is, so 1/8 is smaller than 1/4 even though 8 is more than 4. A cake cut for 8 people gives smaller slices than the same cake cut for 4, and that is the fact the notation is recording.

When children learn it

Naming and writing fractions such as 1/5 and 3/5 is a Year 3 requirement. It follows halves, thirds and quarters of shapes and of amounts in Year 2. Year 3 also compares fractions that share a bottom number, counts along a number line in fractions past 1, shows equivalent fractions, and adds fractions that share a bottom number. Tenths and hundredths written as decimals come in Year 4, and adding fractions with different bottom numbers in Year 5.

What it looks like when it goes wrong

These patterns come up again and again in children's answers on this skill. Each one points at a specific missing idea, not carelessness.

The shaded parts counted as a whole number. A bar in 5 equal parts with 3 shaded, named as 3. They counted what was shaded and stopped. The whole those parts came out of has not become part of the answer, so there is nothing yet for the bottom number to say.
The parts that are left named instead. The same bar named as 2/5: they counted the parts that are not shaded. Both numbers are in the right places and the notation is understood, so ask which parts the question is about before treating the fraction idea as missing.
The fraction upside down. 3 shaded out of 5 written as 5/3. The two numbers are being placed by which is bigger, or by which was counted first, rather than by what each position means. Reading it aloud catches it: five thirds is more than one whole, and there is one bar on the page.
A bigger bottom number read as a bigger fraction. Asked which is more, 1/4 or 1/8, they answer 1/8, because 8 is more than 4. The bottom number is being read as whole-number size rather than as how many parts the whole was cut into. What to do about it is on comparing fractions, the Year 3 skill this one leads to, where it is the error most of the wrong answers come from.

How to help at home

Cut something real into equal parts and ask two questions in this order: how many equal parts is the whole cut into, and how many are we counting. The first answer goes underneath, the second goes on top.
Name the part, not only the number: one part of five is a fifth, and three of them are three fifths. Saying it that way keeps the size of a part in view, which is what the bottom number is about.
Settle the bigger-bottom-number question by sharing something: cut a cake for 4 people, then the same cake for 8, and ask which slice is bigger.
Ask for the same fraction of two different wholes, such as half a pizza and half a page. A fraction is a number of equal parts of whatever the whole is, not a picture that has been memorised.