What a fraction really is
Maths · Key Stage 2 · The School
Bottom counts the cut, top counts the take
In 3/4, the bottom (denominator) says the whole was cut into 4 EQUAL parts; the top (numerator) says we took 3 of them. Equal parts is the law — 3 unequal slices of pizza are not quarters, whatever the menu says.
Fractions live on the number line
A fraction is a number, not just a picture of pizza. 1/2 sits exactly halfway between 0 and 1. 5/4 is allowed — it sits past 1, at 1 and a quarter. Placing fractions on a line is the single best cure for fraction fear.
Equivalent fractions are the same point
1/2, 2/4, and 4/8 are the same number wearing different clothes: multiply top and bottom by the same amount and the value does not move. That is why they land on the same spot on the line.
Comparing two fractions you cannot picture
Which is bigger, 3/5 or 5/8? Neither is a familiar one, and comparing the digits gets you nowhere. Three routes, all reliable. Compare each to a landmark: half of 5 is 2.5 and we have 3, so 3/5 beats a half; half of 8 is 4 and we have 5, so 5/8 beats a half too. That narrows it without settling it. So rename both with the same bottom — fortieths take both — giving 3/5 = 24/40 and 5/8 = 25/40. So 5/8 is bigger, by a single fortieth, which is why it felt close. Or place both on a number line from 0 to 1 and look. The line is the slowest route and the one that never lies.