Common Misunderstandings - Level 4 Partitioning

There is little doubt that a considerable proportion of Year 5 and 6 students experience difficulty with fractions, decimals and percent. A major factor contributing to this is that many students misinterpret the meaning of the denominator. Also, while students may exhibit an intuitive understanding of proportionality in terms of the out of idea, this is limited to familiar contexts and proper fractions (eg, 3 quarters of a pizza or the fraction of red smarties in a packet of smarties). Few students at this level see fractions as numbers which can be arrived at by partitive division (eg, 3 pizzas shared among 4) and ‘live’ uniquely on the number line.

This could be due to/associated with:

  • viewing the denominator in the same way as the numerator (ie, as a count or ‘how many’ number, rather than an indication of ‘how much’);
  • a limited exposure to practical experiences that show what happens as the number of parts are increased and how fractional parts are named;
  • a groups of only idea for multiplication and division; and
  • little or no access to strategies that support the construction of appropriate fraction representations.

By the end of Level 4 students need to be able to work meaningfully with a wider range of numbers. In particular, they need to have established a meaningful basis for thinking about rational numbers in whatever form they appear (eg, proper fractions, mixed fractions, decimal fractions, and percentages). This requires the recognition that equal parts are required; that the number of parts is related to the name of the part (ie, fifths for 5 parts, sixteenths for 16 parts); that as the number of parts increases, each part becomes smaller; and that fraction representations are created by partitioning discrete or continuous quantities into equal parts (see Partitioning (pdf - 199.51kb) paper). Understanding the relationship between fractions and partitive division is essential for fraction renaming (equivalent fractions). In particular, students need to recognise how the region idea for multiplication is related to fraction diagrams, for example, thirds (3 parts) by quarters (4 parts) produces twelfths (12 parts), and how increasing/decreasing the number of parts can be understood in terms of factors, for example, recognising that 3 parts (thirds) increased by a factor of 4 (as a result of halving and halving again) produces 12 parts (twelfths).

Key indicators of the extent to which students have developed an understanding of fractions and decimals is the extent to which they can construct their own fraction models and diagrams, and name, record, compare, order, sequence, and rename, common and decimal fractions.