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Mathagony Aunt

Q. I am a student teacher and have always struggled with fractions, can you explain how the "turn upside down and multiply" rule for dividing fractions works?

A. Before teaching division make sure pupils have a thorough understanding of equivalent fractions, in particular that when you multiply or divide the bottom line (denominator) you must do the same to the top line (the numerator) so that the value of the fraction doesn't change.

So 3Z4 can be written as: - that is, 3Z4 is equivalent to 6Z8.

Let's consider what happens for Rewrite this as a fraction, replacing the divide sign by a line.

Then multiply both top and bottom by the reciprocal of the second fraction (turn it upside down and multiply top and bottom).

As you can see this has the effect of creating a division by 1 so looks as if you have just turned the second fraction upside down and changed the division sign to multiplication.

There is another way of looking at what we are asking when we divide a fraction by a fraction, and that is again by looking at equivalent fractions but this time partially using the rule of addition of fractions, first make the denominators the same.

So 3Z4 V 2Z3 becomes: Which we can think of as asking the question: "How many eight-twelfths are there in nine-twelfths?"

or 9 V 8 = 11Z8 as before.

Q. We had an argument in the staffroom about the number of units of alcohol in two small bottles of lager. How do you work out the number of units of alcohol in a drink?

A. I recently ran a quiz based on alcohol at the Specialist Schools Trust's maths and computing conference in London.

I had to do some research beforehand because I, too, didn't know how to calculate the number of units of alcohol in a drink.

First, you multiply the volume of drink by the percentage of alcohol and divide by 1,000. So two 300ml bottles of lager at 5 per cent alcohol by volume is 600 x 5 divided by 1,000, which is three units of alcohol. That means that a binge for women is just four bottles of lager, as a binge is six units of alcohol for women and eight for men.

A bottle of wine (700ml at 12 per cent alcohol by volume) therefore contains 8.4 units. The alcoholic content of a drink can be written in different ways: alcohol % vol; Alc % vol; % vol; ABV %.

There are 8g of ethanol in one unit of alcohol. The system of units was introduced 20 years ago and how to calculate it is something we should help pupils understand, particularly as alcohol content can vary in a single type of drink - for instance, lager can vary between 3.5% ABV and 9% ABV.

Research shows that about one third of pupils in secondary schools are binge drinking on average three times a month.

I am creating a school tour, "Public Maths: the Alcohol Challenge", using quizdom voting units so that pupils can immediately access information about their own drinking habits and understand the drinks industry. Alcohol is a part of our culture and for the majority is integral to socialisation, so it forms part of the citizenship curriculum and we can study it through maths.

An interesting website is at: l I thought you might like my poem below. I wonder what pupils would create using the same idea.

Reflections of Mathematics

Matters of Universe,

Abstractly accessing


Heaven sent.

Enduring Math Agony and Ecstasy.





Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.

Email your questions to Mathagony Aunt at Or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX

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