GCSE: I don't understand how to rationalise denominators

First, I would make sure the pupil understands why rationalising fractions is necessary. Then I would acquaint them with the relevant vocabulary so they can express their thought process better and understand instruction more easily.

A square root or 'surd' cannot go on the bottom of a fraction. In order to divide we need a quantifiable number to split our numerator by.

Then, I would assess their ability for mentally converting surds and non-surds, squaring essentially, and spend a little time developing the skill or recommend some online mental maths testing resources if the skill needed addressing.

I would like you to convert the following mixed numbers into surds please: 3V2 , 2V5 , 4V3 , 5V10

A: V18 , V20 , V48 , V250

I would like you to simplify the following surds please: V36 , V45 , V44 , V132

A: 6 , 3V5 , 2V11 , 2V3V11 or 2V33

Then I would explain why we multiple the surd through the fraction.

As a surd is a square root, we can convert it to a rational number by squaring it, ie V2 x V2 = 2. Fractions hold the same value if we multiply the top and the bottom by the same number. For example 2/4 is the same as 1/2, we've multiplied the numerator and the denominator of 1/2 by 2. 3/4 = 12/16 because we multiplied top and bottom by the same number. Therefore, if we want to get rid of a surd on the bottom of a fraction, we can multiply the whole fraction by the surd, without changing the value of the fraction.

I would give worked examples such as

4/V5: 4 x V5 = 4V5 , V5 x V5 = 5 , therefore 4/V5 = 4V5/5

I would then check the pupil understood and answer questions/go over things if they needed it. Then I'd give them example questions to try.