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Fractions as a double count

Tasks such as these are usual in classrooms.

Colour two thirds of this shape.

A circle divided into three equal parts radiating from the centre.

What fraction of this shape is shaded?

A square divided into four equal parts by its diagonals. The top and bottom quarters are shaded.


Students learn to connect a part-whole area diagram with the written fraction by thinking, "The denominator tells how many parts, and the numerator tells how many are shaded."

While this is a useful basic strategy, it is problematic for several reasons.

  • The written fraction is seen as a pair of whole numbers, each recording the count of something different: a 'double count'.
  • Many students believe this double count to be the actual definition of a fraction.
  • A fraction is not comprehended as a single number, having a value like other numbers.
  • The double count cannot be generalised to all situations, such as "Share two pancakes equally among four people."
  • Students can lose sight of the whole and interpret a part-whole situation as a ratio, especially when working with fractions of collections.

Wrong number

The double-count perception of fractions can be detected in student responses to basic tasks.

Expanding the view

To avoid the limitations of the double-count perception of fractions, students need to expand their view of fractions beyond the basic part-whole model.

Fraction fiddle

The digital learning object illustrates the relationship between the part-whole model of a fraction and the value of the fraction, by representing its position on a number line.

Curriculum links

Year 3: Model and represent unit fractions including 1/2, 1/4, 1/3, 1/5 and their multiples to a complete whole

Year 2: Recognise and interpret common uses of halves, quarters and eighths of shapes and collections

Year 4: Investigate equivalent fractions used in contexts