rearrangement times/share with numbers

‘doing the same to both sides’ and ‘balance’ models can help to emphasise the equality of two sides of an equation/formula but seem to me to be not easily assimilated ideas when compared to using inverses, in visual ways, when transforming a statement as a rearrangement

it may well be important to link the ideas of both models

in the case of multiplication/division, by starting from rearrangements of numerical statements  transformations can be appreciated as equivalent statements involving those particular numbers
(given this statement, we can also say this...) 

how these statements are deduced in terms of multiplying being the 'opposite' or inverse of division (and vice versa) can hopefully be appreciated - appealing to what happens visually

it seems helpful to focus on the 'shape' of a resulting transformation, compared with the initial statement e.g. top statement: 'the 3 is upstairs and it is multiply by' to: lower statement, 'the 3 is downstairs and it is divide by' 

a powerpoint is here

visuals of rearrangement