boxes, teacher's notes

these are teacher's notes for work on 'boxes'
a 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

with small numbers to start with
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 (probably needing a calculator)
they can move on to proportion problems

you could obviously go down by x 2
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:

sharing in a ratio questions involve an extra two bubbles













all percentage questions can be set up in a 'boxes' format
with careful thought about where to place the 100
(the 'start 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