Mathematics from zero
What is a fraction
A pizza is cut into 4 equal slices. You take one slice. You did not take a whole number of pizzas — you took part of one. To write down exactly how much, whole numbers are not enough. You need a fraction.
After this lesson you can say what a fraction is, name its two numbers and what each one means, explain why the parts must be equal, and read a fraction whose top number is as big as — or bigger than — its bottom number.
A fraction names a part of a whole. When something is split into equal parts and
you have some of those parts, a fraction records exactly how much. It is written as
two numbers, one above the other, with a line between: the slice of pizza above is
one quarter, written 1/4.
The bottom number is the denominator; the top is the numerator. The denominator
— the bottom number — says how many equal parts the whole was cut into. The
numerator — the top number — says how many of those parts you have. In 1/4 the
denominator 4 means the pizza is in 4 parts, and the numerator 1 means you took 1
of them.
The parts must be equal. A fraction only makes sense when the whole is split into
parts of the same size. If a pizza is cut into one huge piece and three tiny ones,
taking “one of four pieces” is not 1/4 — the pieces are not equal, so no single
fraction describes a piece. Equal parts is the rule that makes the denominator
meaningful.
The numerator can equal or pass the denominator. If you take all 4 slices, you
have 4/4 — four of four parts, which is one whole. If you have a second pizza and
take one more slice, you have 5/4: five quarter-slices, more than one whole. A
fraction is not always less than one; it is simply this many of that-sized parts.
Read the fraction 3/8.
The denominator is 8: the whole — say a chocolate bar — is split into 8 equal
pieces.
The numerator is 3: you have 3 of those pieces.
So 3/8 means “three of eight equal parts” — three eighths of the bar. Because 3 is
less than 8, this is less than one whole bar. If the numerator were 8 you would have
8/8, the whole bar; if it were 11 you would have 11/8, more than one bar.
Why this works
Why insist the parts be equal? Because the denominator is a promise about size. 1/4
tells you the piece is one of four equal shares — so it is a known amount. If the
parts could be any size, 1/4 would tell you nothing about how much you have. The
equal-parts rule is what gives a fraction a definite value.
Common mistake
A common mistake is reading the top number as the total. In 3/8 the 3 is not how
many parts the whole has — that is the bottom number, 8. Top is how many you have;
bottom is how many equal parts the whole was cut into. Always read the denominator
first: it sets the size of one part.
A cake is cut into 8 equal slices. Type the denominator of the fraction for one slice.
You have 3 of those 8 slices. Type the numerator.
In the fraction 2/7, type the denominator.
A bar is in 5 equal parts and all 5 are shaded. How many whole bars is that? Type the number.
A pie is cut into 6 parts and none are eaten. Type the numerator of the eaten fraction.
In the fraction 3/8, what does the bottom number 8 tell you?
A fraction names a part of a whole, written as a top number over a bottom number. The bottom number is the denominator: how many equal parts the whole was cut into. The top number is the numerator: how many of those parts you have. The parts must be equal, or the denominator means nothing. The numerator can be smaller than, equal to, or larger than the denominator — equal means one whole, larger means more than one.