Shifting wholes

Examples of the misconception of using 'different wholes' can be seen in the diagrams that students use when comparing the size of two fractions.

In the following example, the student has correctly identified the larger fraction, but has not provided a logical reason.

Hand drawn student work stating 1/3 is bigger than 1/6. 1/6 represented by a rectangle divided into six equal boxes, one coloured; 1/3 represented by a circle divided into three equal parts, with one coloured.

Different shaped wholes.

The drawn response of two different shaped wholes suggests the student does not understand the need for the wholes to be the same, and is possibly just recalling familiar representations for these fractions.

In the following example, the student has drawn a rectangle to illustrate the first fraction, then simply extended that whole to represent the second fraction.

Hand drawn student work responding to ‘Draw fractions 1/4, 1/3 and 1/5 in order’. The fractions are represented correctly by rectangles divided into equal boxes with one box shaded. The rectangles are different sizes.

Different length wholes.

By contrast, this example demonstrates an awareness of the assumption that the wholes are the same. This understanding allows the reasoning about the size of the parts to develop.

Hand drawn student work to determine whether 1/3 or 1/4 is larger. Two circles the same size are side by side; the left divided into four equal parts with one part shaded, the right divided into three equal parts with one part shaded. Correct written explanation.

Same sized wholes.

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 4: Investigate equivalent fractions used in contexts

Year 5: Compare and order common unit fractions and locate and represent them on a number line