## Comparing different types of units

Fractions typically arise from two types of activities. The first involves measurement: if you want to represent a quantity by means of a number and the quantity is smaller than the unit of measurement you need a fraction. The second involves division: if the dividend is smaller than the divisor, the result of the division is represented by a fraction (Nunes & Bryant, 2009). For example, when you share 3 pikelets among 4 children, each child receives 3/4.

However, some students may represent two of the pikelets as being cut into two parts and the other pikelet into four parts before distributing the pieces. This type of sharing suggests a two-part interpretation, where the partitioning actions and the results of partitioning are separate entities.

The coordination of the number of people sharing and the number of things being shared may not result in a uniform distribution across the objects being shared. Rather than showing three-quarters of each pikelet, these responses make use of halving and repeated halving to create the desired number of equal pieces. Halves and quarters are the easiest partitions to make and can be progressively distributed in this problem.

Practical sharing contexts need not result in the standard fraction representation of 3/4 as an indicated division. In the right hand image, sharing 3 pikelets among 4 children results in one-quarter and one-half each.

As students can have terms for fractional parts that are linked to inappropriate part-whole fraction images (see the video Counting by quarters) it is important to recognise that fraction words as well as fraction notation may not relate to standard regional models.

### Counting by quarters

In a measurement comparison of lengths (Figure a), the part and the unit whole being measured are separate entities. The fraction is constructed from the relationship between the explicit whole and the part (Watanabe, 2002; Wong & Evans, 2008). When dealing with a partitioned rectangle (Figure b) the part is embedded within the whole and the whole needs to be identified.

(a) Comparison representation for 3/4                    (b) Parts of the whole representation for 3/4

Students typically encounter representations of fractions similar to Figure b, with the parts embedded within the whole. Many students focus on counting parts rather than comparing the relationship between two areas. Counting units are more familiar than units of area and are less challenging. Although it is possible that some students may separate out three of the parts while leaving the whole intact (disembed) it is more likely that they will do this by treating the parts as countable units rather than area units (Kamii & Kysh, 2006).