Students identify the shaded fraction of a number of different shapes.

Answers/responses:

Y6 (11/2005)

a)

^{1}/_{4}

very easy

b)

^{1}/_{2} or ^{4}/_{8}

very easy

c)

^{3}/_{4} or ^{6}/_{8} or ^{12}/_{16}

moderate

d)

^{1}/_{4}

moderate

e)

^{1}/_{8} or ^{2}/_{16}

difficult

Results based on a trial set of 173 Y6 students in November 2005

NOTES:

Accept any equivalent fraction for the answers above (the fraction shown were the selection of answers given by students in the trial).

Accept answer if the students write the name of the fraction instead of the notation.

The first two fractions are evenly partitioned and students can solve by counting the number of shaded pieces and then placing them "over" the total number of pieces.

Diagnostic and formative information:

Common error

Likely misconception

c)
d)
e)

^{8}/_{12} ^{1}/_{3} ^{1}/_{6}

Does not understand that the same fractional parts must be equal-sized. They use only the number of pieces it is broken into to work out the denominator, and the shaded pieces to work out the numerator (not taking into account how large each piece is). [Approximately a third of students showed these misconceptions for each question, C, D, and E]

e)

Similar to the above misconception, but also does not understand that a fraction is a part of the whole. Here the fraction is constructed by putting the number of shaded pieces on the top and the number of unshaded pieces on the bottom. It is a good example of the potential danger of relying on remembering a procedure to construct a fraction rather than understanding what the fraction represents.

Next steps:

Students who constructed the name of the fraction by merely counting up the shaded pieces and put them over the total number of pieces, need to be aware when naming a fraction that each piece needs to be equal-sized (if it is the same fractional piece). For example, when you divide something into quarters you divide it into 4 equal-sized pieces. They should be encouraged to draw lines to create equal-sized partitions to help them work out or explain their strategy. Although some students may feel they are not supposed to draw on lines, encouraging their diagrammatic representation of fraction is important and helps eliminate misunderstandings.

For question e), some students recognised that it was a half of a quarter, this indicates a good understanding of how to construct a fraction and requires some more understanding about the part-whole nature of fractions, i.e., "what is the whole?" and "what is the part?".