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Different wholes

There are two main fraction scenarios: specific and general.

  • A fraction that refers to the partitioning of a specific whole describes a particular quantity.

    For example, cutting a mini-pizza in half produces a different amount of food than cutting a family-size pizza in half. Although we know \(\frac{1}{2}\) = \(\frac{1}{2}\) in a proportional sense, the actual quantities represented by a half are not equal when the 'wholes' in the particular context are not equal.

  • A fraction where the actual size of the whole is not relevant is used in a general and abstract sense.

    No specific context is required and it is assumed that the 'whole' for all the fractions is of the same size.
    For example, logic is sufficient to determine that one-half is larger than one-quarter without needing to know 'half of what?' and 'quarter of what?'

Confusion and misconceptions arise when students do not distinguish between the specific and the general scenarios for using fractions.

Shifting wholes

Students' lack of awareness of the importance of equal wholes becomes apparent when they are asked to provide specific examples to justify a general comparison of fractions.

Fixing wholes

Students need direct experiences in making comparisons between fractions in situations where the wholes are the same, and in situations where the wholes are different.

Cut and find

Comparing and discussing concrete models of fractions can help students realise the significance of paying attention to whether the wholes are the same size or different sizes.