dividing fraction views

the intention of these resources is to present three ways of making some sense of how to divide by a fraction, linking a diagram to what happens numerically (/algebraically)

firstly, what happens when a whole number is divided by a fraction?

secondly, Fawn Nguyen's lovely approach uses rectangles to change a division of fractions into one with common denominators
this method was advocated by Caleb Gattegno

I have altered Fawn's text slightly, thinking that you could present this to students
(Fawn has much to say, in general ways, about how to utilise rectangles to make sense of problems)

when you have common denominators to both fractions you could multiply the dividend and divisor by this common denominator but (with language being used) 'an' somethings divided by 'bm' somethings should be the result required

algebraically:

thirdly, Mr Novak's approach (from moveitmaththesource) uses the idea that when you are presented with a division sum you can adapt it by multiplying both parts of a division (dividend and divisor) by the same amount - in this case the reciprocal of the divisor - without changing the sum

just as when you divide e.g. 7 by 3 the remainder is expressed in terms of the divisor