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The idea of the fixed unit whole

Irrespective of whether the units a student is dealing with are units of length or area or discrete units, when 3/4 is a number rather than 3 equal parts out of 4 equal parts of something, a dimensionless abstract unit has been formed. Statements such as 3/4 > 1/2 are meaningful because they refer to numbers that are located with respect to the number one (a fixed unit whole). When students have not recognised the need for an abstract unit whole, the representations that they make to compare fractions can be inappropriate.

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Comparing 1/5 and 1/6

Developing an appreciation of an abstract unit whole used when we treat fractions as numbers requires a careful introduction of the fraction notation a/b. The relational number a/b relies on the (multiplicative) comparison of like units to create a unit-less quantity.