powerpoint is here
a common approach to all ratio, percentage and similarity questions
(published initially in 1995, MEDIAN paper resources)
Van Hiele proposed using 'proportion matrices' in 1986
this work involves getting from one number to another by multiplying and dividing
unlike other proposals, this work involves two steps (rather than one, the 'rate' or scale factor)
there is a clear need to dissuade the use of additive methods
utilising the hcf as a 'stepping stone'
establishing that + 2 is obviously not wrong but is not correct in the context of work on ratio
once students have appreciated ways to get from one number to another in two steps
using the 'unitary method' as a default option (possibly needing a calculator)
they can move on to proportion problems
but, as with all proportion problems, you can also go another way - across
once the 'boxes' format/diagram has been fairly well established, students might be helped to become aware that all ratio and percentage problems are essentially the same
the questions can all be transformed into a 'boxes' format: that of being given three numbers and finding a fourth - by multiplying and dividing only
examples of this 'translation' to a boxes format/routine:
all percentage questions can be set up in a 'boxes' format
with careful thought about where to place the 100
(the 'starting out' amount)
pie charts can be tackled using 'boxes'
similar shapes questions can be done this way
all trigonometry can be done using 'boxes'
and it might even help to think of the mean in this context