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Secondary Magazine - Issue 115: It Stands to Reason

Created on 07 October 2014 by ncetm_administrator
Updated on 28 October 2014 by ncetm_administrator


Secondary Magazine Issue 115'Le Penseur' by Juanedc (adapted), some rights reserved

It Stands to Reason

There’s the old adage: a lost traveller asks for directions from a local resident, only to be told “well, to get there I wouldn’t start from here”. Teaching fractions in Years 7 and 8 can sometimes feel the same – but your pupils are where they are even if that’s not where you want them to be. If so, it’s well worth taking time to understand what they do and don’t understand about fractions, and what they can and can’t explain, and in this issue we share some fun ideas how you could do so.

Have a look at this spider diagram. You may like to add some things to a copy of the diagram yourself (click the diagram below to download as a PDF). See how many other links you can make:

spider diagram

Now try this with your pupils. How many of these links do they make? Within the diagram you can see different meanings of fraction notation as both names of numbers and as operators. Fractions can be interpreted as:

  • part of a whole unit
  • comparisons between part of a set and the whole set
  • a point between two whole numbers
  • result of a division operation
  • comparing the sizes of two
    • sets of objects
    • measurements

It is the necessity of understanding that a fraction can be used in these different ways, and the need to work out which way that a particular fraction is being used in a given context, that makes fractions a hard concept for many pupils.

This Always, sometimes, never activity helps pupils by consolidating their ideas about what fractions are:

always, sometimes, never activity

The Fraction Fascination task from NRICH encourages pupils to reason with the sizes of different fractions within a unit whole. The second activity is a good challenge.

NRICH: Fraction Fascination

Another activity that promotes reasoning skills is Oh Harry, where pupils are encouraged to link fractions to measurements of quantities. You might want to adapt the context to something a bit more “grown up”!

NRICH 'Oh, Harry' activity

A key use of fractions is to make comparisons. Could all your pupils look at this diagram and describe the relationship between the two lengths?

diagram showing two different lengths to describe

(Rich suggestions would be "The red length is two thirds of the green length" and "The green length is three halves of the red length").

And in this diagram?

diagram showing two different lengths to describe

In this video clip, part of the microsite The teacher as researcher/teaching as researching, one teacher writes and talks about her work with Cuisenaire rods inspired by the writings of Gattegno and Goutard. She uses fractions to compare the lengths of the different colours. Pupils can be given rods and asked to suggest relationships between the rods, or given some relationships and asked to draw the rods that represent the relationships – the challenge is to do so with as few rods as possible.

graph using fractions to compare lengths of favourite colours

This idea of comparison can be explored further by asking pupils what fractions can they see in this diagram:

diagram with shaded areas for description

Your pupils will probably identify 25 and 35 but note if they say "25 of the whole" or just "25". Does anyone say something of the form "x is 23 of y" or "x is 32 of y"? Can such a statement be explained clearly?

Activities such as this one make good starters or mid-lesson revitalisers, and will help you ensure that your pupils are all starting in the right place for the journey ahead of them. There are further resources here:

  • some ideas about fractions themselves and issues related to teaching fractions can be found in the NRDC booklet, Fractions, by Rachel McLeod and Barbara Newmarch
  • this NRICH article, Understanding Fractions, which includes links to other activities
  • this NCETM Departmental Workshop, Fractions.

Image credit
Page header by Juanedc (adapted), some rights reserved



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