What is 1/2? It is a half, 0.5 and one divided by two

What is a fraction? It looks like (because it is) a division; it is left undone to avoid producing decimal answers and historically fractions are a way of avoiding doing division. The number on the top is called the *numerator* (it numbers the count; the number on the bottom is called the denominator (it gives the name of the fraction). Think about ‘two thirds’, 2/3, numerator over denominator. a/b is also a fraction, but an algebraic one. In Chinese, the translation is ‘3 under 2’.

A *reciprocal* is 1/n, one over a number, a “-th”; examples are half, third, ninth, sixty-fourth, millionth.

Our problems generally lie with previous teaching; apparently many of us are confused by *any* form of division. Perhaps even *all* forms of division. Talking to students shows that for many the problem started with the idiot idea of *dividing **into* rather than *dividing **by*; keep only the idea of ‘__dividing by__’: Lose the other one.

Equivalent Fractions cause an awful lot of trouble for people in lower sets. Which does not mean that everyone in a high set (stupid label, since usually a lower numbered label) understands equivalence.

Here is a collection of equivalents to a half, where you are expected to see these numbers as being fractions:

2 3 4 5 6 7 There is a common factor between the top

4 6 8 10 12 14 and the bottom numbers

Second, the idea of a *factor* (fundamental to years 7-9) seems to be a problem. If you know your tables backwards and forwards, you won’t have a problem. 3x4=12 says twelve has factors of 3 and 4 (and 1 & 12, 2 & 6). People who know their tables have little difficulty with simplifying these fractions:

2 3 4 8 6 9 2 3 4 8 16 18

12 12 12 12 12 24 24 24 24 24 24 24

but might have more difficulty with

42 36 42 28 56 96 21 13 14 38 16 18

72 112 112 84 112 240 98 78 98 95 144 144

Example 3/5 = 6/10 = 12/20.. Multiply the top by the same as the bottom. Cancelling is the reverse: 36/100 = 18/50 = 9/25… keep going until both numbers have no common factor. Factors are an important part of Yr8 work; important in the sense that later visits assume you already ‘get’ the topic. Turn that around: if you’re a teacher you really must make sure everyone understands factors; if you’re a student you really want to be happy you understand what they are. If you’re not getting thoirugh to a teacher (after all, they do ‘get’ it, so they may not see what is at all hard about the idea), ask one or mnore of your friends - best of all, someone who gets it now, but took a while.

Calculator method: Put any of the last dozen fractions into your calculator. Push the equals button. It will do any possible cancellation. Test that 4/8 becomes ½. Try some of the problems above. Many calculators also change top-heavy [improper] fractions into mixed fractions (so 15/9 skips 5/3 and becomes 1 ⅔). You might discover how to suppress this, so the fractions stay in the improper form (which, generally, I am happy to see as a final answer).

Remaining problems: You need to understand equivalent fractions before you will succeed with fractional arithmetic. You will often be given fractions on occasions where no calculator is available. A rounded decimal is not the same as a fraction: 0.33 is not a third. Yes, it is close, but not worth a cigar, as the saying goes.

Some calculators express fractions well and must be nudged into a differnt mode to show decimals instead. Some default to decimal and must be nudged in a similar way to handle fractions. As a user you must be able to recognise the two states and switch between them easily. Ideally, you can ALSO switch between decimal and fractional forms without needing to ask your calculator (any electronic assistant) for help. I am certain that, whatever form future electronics take, until they are part of your internal body, you will be required to do such conversions both with and without aids. Iin other words, there will be tests held with and without calculators.

DJS