Hone

Lessons · Teaching · multiplying and dividing fractions

Multiplying is easy and dividing is multiplying upside down

To multiply fractions, multiply the numerators and multiply the denominators. To divide, turn the second fraction upside down and multiply. No common denominator is needed for either, which is the opposite of addition and catches people out.

Hone is a place to practise a career, one idea a day. This is one of its lessons, written out in full and free to read without an account.

In beta. This lesson was written for Hone and has not yet been checked by a state certification officer. Practice material, not professional advice. What that means.

What it is for

'Of' means multiply, and half of a third is the picture that proves it: cut a square into thirds, take one, halve it, and the piece you hold is one sixth of the square. A teacher who can draw that does not need the class to trust a rule.

How to think about it

Multiply straight across, then simplify. Better, cancel any common factor between a top and a bottom before multiplying, because the numbers stay small. To divide, write the division, flip the second fraction, change the sign to multiply, and proceed as before.

Worked example

1/2 x 1/3 = 1/6, and the picture is half of a third of a square
Multiplying by a number below one makes the result smaller.
2/3 x 9/10: cancel the 3 into the 9 and the 2 into the 10 first, giving 1/1 x 3/5 = 3/5
Cancelling first keeps the arithmetic to single digits.
3/4 divided by 1/2 = 3/4 x 2/1 = 6/4 = 3/2
How many halves are in three quarters? One and a half of them.
Dividing by a fraction below one makes the answer bigger
Which is why the grouping picture of division is the one to keep.

Your turn

Name what is done to the second fraction before a division becomes a multiplication.

Turn the second fraction  and multiply

The trap

Finding a common denominator before multiplying. It is not wrong, it is wasted work, and it is a sign the learner is applying the addition rule without knowing which rule they are in.

Next in Teaching

Every Teaching lesson on one page

Practise multiplying and dividing fractions on HoneA question on it now, a real problem where there is one, and it is remembered for review. Free, no email needed.