Close to a half
Y6 (06/2009)  
a) 

^{6}/_{10} and ^{4}/_{6} selected.

moderate 
b) 

Explanation involving the top number (numerator) being more than half the bottom number (denominator), e.g.:
Note: Some students also identified how much larger than half the "larger fractions" were. 
difficult 
c) 
i) 
^{1}/_{8} 
difficult 
Prior knowledge
An understanding of fractional notation and about "halfness". If students are not aware of fractions as a relationship between a part and a whole, they may have difficulty identifying how the numerator and denominator are used to work out the size of a fraction, and may try other relationships, such as subtraction, addition, or "the largest" of the numbers.
Over twothirds of students correctly identified at least one fraction greater than half without any incorrect fractions identified.
Common error  Likely misconception  
b) 
Partwhole misconception about fractions Students treat the fraction as two separate numbers and describe some other system of attempting to judge fraction size from these numbers. 

c) i) ii) 
Already a half (or 0) 
Stating that the fraction was already one half Students identify that the fraction is already a half. Students wrote 0 or stated that the fraction is a half or already a half. 
c) i) ii) i) ii) i) ii) 
^{4}/_{8} ^{3}/_{6} ^{5}/_{8} ^{2}/_{6} ^{7}/_{2} ^{6}/_{4} 
Writing incorrect fractions Writes the fraction for a half using the given denominator. Works out the difference to one, not a half, e.g.,^{ 3}/_{8} + ^{5}/_{8} = ^{8}/_{8} (=1) Works out the additive difference of numerators and denominators to ^{10}/_{10} (based on whole number misconception that fraction notation makes to ten), e.g., ^{3}/_{8} + ^{7}/_{2} = ^{10}/_{10} 
Students who treat the fraction as two separate numbers and describe some other system of attempting to judge fraction size from this need to develop a partwhole understanding of fractions before trying to compare fractions. Ultimately the relationship between top and bottom numbers is a division relationship. Students could begin by
 partitioning and identifying the parts (unit fractions),
 combining these parts to make nonunit fractions that are between 0 and 1 (also called proper fractions), and
 naming these new fractions (partwhole fractions).
Asking students what the whole shape is, and then what part they are finding, can support them to develop a more partwhole understanding about fractions. The idea of equalsized parts and the correct number of partitions is an important piece of knowledge to support partwhole representations of fractions before attempting to explain ideas about equivalence. This could also be supported by using a number line to show where fractions fit in between whole numbers.
Stating that the fraction was already a half rather than how far from a half
Students who indicated the above error could be asked whether the fraction was equal to a half. If not, then whether it was more or less and then how far from a half. Getting students to show diagrams to support their reasoning may support their conceptualisation of the problem. However this error could also relate to them only experiencing sequencing to build up fractions. Students may benefit from going back to partitioning and exploring how constructing nonunit fractions involves combining similar parts, and naming these new fractions (partwhole fractions).
Writing incorrect fractions
Most of the above common errors involving writing incorrect fractions could be addressed by getting students to draw the fraction they are considering and use the given denominator to also draw a half. For example, How far away from ^{1}/_{2} is ^{3}/_{4}?
Students could draw ^{3}/_{4} and then ^{2}/_{4} and indicate the difference.
Then quantify the size of the difference.
Students could then look at the denominator and numerator to identify any pattern (that the difference is ^{1}/_{4} (which is ^{3}/_{4} – ^{2}/_{4}).
Using decimals
Some students might prefer to compare fractions by first converting them into decimals. They will often rely on a calculator to use this. It is important to challenge them to look at how using information about the denominator and numerator could be used.
Whole class discussion
Many of the above next steps can be explored in whole class discussion where students share and critique their own and others' strategies to develop a fuller understanding.
Book 7: Teaching Fractions, Decimals and Percentages, 2006
 Trains (Early/Advanced additive)
 Wafers (Advanced counting/Early additive).