In my new job at Twinkl I get the time to dig into a lot
more of what people are saying in the world of maths education than I have in
the past. An absolutely lovely result came up today in my google alert for
maths education that, despite 20 years teaching mathematics, I had never come
across – or forgotten if I have.
It might be well known to others, I don't know, but I was
flabbergasted that there was something so simple about fractions that I wasn't
already aware of, and I felt compelled to share it.
Take two equivalent fractions, say 
and 
. If you subtract the
numerators and denominators, the resulting fraction is also equivalent to these
two fractions. In this case 
.
The result is equally valid when neither
fraction is in its simplest form, for example and 
: 
.
The proof of this is actually relatively
straightforward. Given two equivalent fractions 
, then we have 
However, if 
. Substituting 
gives 
.
I think this is a lovely proof to offer
GCSE or A-Level pupils, although you would probably have to provide a hint for
the 
step.
It also suggested to me a nice (if
inefficient) route to determining whether two fractions are equivalent - particularly
if one of the fractions has a numerator and denominator that is relatively large.
Simply subtract the smaller numerator/denominator from the larger (or multiples
thereof), and if the result is equivalent to the smaller fraction, then so is
the original fraction.
For example, given the fractions 
and 
, one might do the
following:
At this point we can see that both and 
both simplify to 
, and so as these are
equivalent, then is also equivalent.
Now, of course, in the age of calculators
that can simplify these fractions immediately, this is not something we would
potentially encourage pupils to carry out as a process but, I think, with the
right pupil at the right time, showing something like this could be a nice way
to opening them up to a wider view of mathematics than that which is covered by
efficient calculation alone.
Credit to ‘Math with Bruna’ whose YouTube video was
included in my google alert which brought this lovely result to my attention.