# Fraction stories: Multiple Representations

This task has been designed for late primary, early secondary.  I can imagine using this as an assessment task to establish the extent of pupils understanding of different meaning of fraction.  Perhaps controversially, this is what I would expect pupils (at least here in Scotland) to be able to do on arrival at secondary.  This is regularly not the case.

I have been running CPD on Fractions this year and often talk about the multiple interpretations of fractions.  This task might be a useful tool in teacher learning, and might help to focus teachers on some of the various meanings of fractions which need to be developed.

The first column (left of page) is focused on part-whole representations.  Working top-down, we have a pre-segmented rectangle, which is the basic case.  Next is one where pupils have to split up the rectangle themselves and shade.  This is slightly harder, as the pupil is generating the representation.  We want to look out for equal parts here!

The next one is a fraction of a set of discrete objects, followed by a set of discrete objects which has a number of items greater than the fraction denominator.  This is an important conceptual realisation for pupils.

Into the middle column pupils are asked to write in words, as language is an important representation as any other.  The number line placement focuses on the measurement meaning of fraction.  How does the fraction measure up against one whole?

The next two questions are essentially the same, but pupils need to appreciate this.  They are focused on the operation meaning of fractions.

The final column begins with the quotient interpretation, followed by a task on the ratio meaning.  There is a quite deliberate set of questions here.  The first two are about part-whole companions, while the second are about part-part comparisons. This is an important contrast.

The second from last is an opportunity for pupils to create 3 equivalent fractions.  This is perhaps the most pivotal idea for progression with fractions into secondary.

The final question asks for pupils to describe two fractions which add to give our fraction.  These additive relationships should be in place, well before formal teaching at Third Level.  This is not about finding common denominators etc.  Instead it is about pupils’ fraction sense.