Comparing 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
Comparing two fractions means deciding which one is bigger. Ordering means doing that for several at once. Year 3 asks for it in two cases, and they behave differently enough that a child can be secure in one and stuck in the other.
When two fractions share a bottom number, every part is the same size, so the top numbers decide it: 5/7 is bigger than 2/7, because five parts of that size is more than two of them. When each fraction is a single part, such as 1/3 and 1/5, the bottom numbers decide it the other way about: cutting a whole into more parts makes every part smaller, so 1/3 is bigger than 1/5.
Those two rules sound like opposites and are one fact said twice. The bottom number sets the size of a part and the top number counts how many of those parts there are, which is what the two numbers in a fraction mean. Two bars of the same width, stacked one above the other and cut into different numbers of parts, show both cases at once. The widths have to match, or the comparison is between two different wholes.
When children learn it
Comparing and ordering unit fractions, and fractions with the same denominator, is a Year 3 requirement. A unit fraction is a single part, such as 1/3, and the denominator is the bottom number. It sits with the rest of the Year 3 fraction work: naming and writing fractions of a set of objects, showing equivalent fractions with diagrams, counting in tenths, and adding and subtracting fractions that share a bottom number. Year 5 goes on to compare and order fractions whose bottom numbers are all multiples of the same number, and Year 6 to compare and order any fractions, including fractions bigger than 1. The term a class meets this in varies from school to school.
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 bigger bottom number read as the bigger fraction. Asked which is bigger, 1/3 or 1/5, they answer 1/5. A child answering that way is not being careless. Every number they have met until now was bigger when it looked bigger, and they are applying that to a number which is not counting how much there is: the 5 says the whole was cut into five parts, and five parts of one whole each have to be smaller than three parts of it. Expect it to take time, and expect it back whenever the pairs on the page change.
- A pair with the same bottom number put the wrong way round. Asked which is bigger, 2/7 or 5/7, they answer 2/7. This often turns up soon after the single-part case has been taught, with "more parts means smaller" carried over to a pair where the number of parts is the same in both. Ask what is the same about the two fractions and what is different before any answer is given.
- The two numbers written the wrong way round. Asked which is bigger, 3/5 or 4/5, they write the pair down as 5/3 and 5/4 and compare those. The comparing may be done correctly on the pair they wrote. Which number belongs on top is what is unsettled, and that is what the two numbers in a fraction mean, the skill this one is built on. Naming the fraction shaded on a bar a few times is what to go back to, rather than more comparing.
How to help at home
- Ask two questions before any answer: are the bottom numbers the same, and if they are not, is each fraction a single part? That decides which case it is, and a child who sorts that out first rarely reaches for the wrong rule.
- Compare on bars of the same width. Draw two strips the same length, cut one into thirds and the other into fifths, shade one part of each, and ask which shaded part is bigger.
- Ask for an order rather than a pair sometimes: put 3/8, 7/8 and 1/8 in order, smallest first. Ordering three shows up a rule that was applied by luck.
- Count along in the same fraction out loud: one fifth, two fifths, three fifths, four fifths. Fractions with the same bottom number come in the order you count them, and that is the whole of the rule for that case.