Identifying the number of pieces not the fraction difference
Students who write their answer as the number of pieces could be asked what fraction the pieces represent.  Students may not be able to work with fractions as numbers and may need to represent the fraction using sliced rectangle shapes to work out what the difference to a half and the whole are, e.g., for

a)  get students to show the whole (10 parts), half (5 parts), the fraction we are looking at (³/₁₀) and then how many parts (2) to make half, and what fraction each is (tenths).

Writing incorrect fractions
Most of the above common errors involving writing incorrect fractions could also 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).

Students who wrote equivalent fractions for half or ⁹/₂₀ (question d).
Students who answered with an equivalent fraction of half (or ⁹/₂₀) can be asked how they know these fractions are "closer to half than ⁹/₂₀ , but not half".  They will need to explore the idea that any equivalent fraction has the same value (and ratio) as any other equivalent fraction.

Writing a fraction that is the same distance to a half, but greater
Students who identified a fraction that is the same distance to half, but greater can be asked to show how it is closer than ⁹/₂₀ on a number line.  To make it easier to compare they can use common denominators for the fractions they are comparing. For example students could try to show how ¹¹/₂₀ is closer to half than ⁹/₂₀ .  Ask them to quantify the difference (like in question c), compare that difference, and talk about what makes something closer. They can make conjectures and share their ideas to develop a general rule.

Incorrect fraction notation
Students who constructed a fraction with a non-whole number numerator can be asked if they have another equivalent fraction for their solution. They simply may not be aware that with fraction notation both numerator and denominator should be whole numbers (and denominator ≠ 0).

Using decimals
Some students might prefer to compare fractions by first converting them into decimals.  They will often rely on a calculator to do this.  It is important to challenge them to look at how using information about the denominator and numerator could be used, and therefore how the problem can be solved using fractions.

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.

Greater or less than half?
For students who correctly identified the magnitude in question c), encourage them to identify whether the difference was more or less than half, and give in their simplest form.

Extending the idea of quantifying closeness
As an extension students could also use a number line to investigate a number of fractions between two fractions (e.g., ⁹/₂₀ and ¹/₂) and identify the closest fraction to half.  They could also develop a conjecture and discuss a general rule that explains how to recognise which fraction is closest.

Insufficient explanations
These are explanations that are very close but require details or clarification. Most of the students who gave these sort of explanations correctly identified the three fractions greater than half, but did not fully explain their reasoning.
Students who answered with an insufficient explanation could share their explanation (or explanations from this resource) in a group to identify details and clarify ideas that are needed in their explanations.  Students could develop criteria for an explanation and do some peer assessments.

They could also have a "sufficient" explanation modelled and made explicit to them. A summary of what makes an explanation sufficient is:
  a. specific details;
  b. appropriate use of examples;
  c. justification;
  d. clarity to another reader.
 

Examples of students' explanations
*These explanations could be cut out and used for class or small group discussion.

Insufficient (meaning that they were almost sufficient)*

• Imagine a pizza in your head and then divide into the number on the bottom of a fraction and then count the number of pieces – the number on the top.
• Because anything lower than half the number is higher than half.
• Because anything lower than half the number is higher than half.
• Because I just divide the bottom number by half.
• If the top number is under double the top number it is over half.

Incorrect explanations (based on whole number or unit fraction only misconceptions)
Some incorrect students' explanations for the problem were:

• The smaller the denominator the bigger the fraction. (rule works only for unit fractions)
• None of then are larger than half, they are all lower than half. And the smallest fractions have the largest numbers on the bottom and the smallest on top. (still using the above explanation to determine fraction size)
• Because the numerator counts to the denominator, e.g., ⁵/₇ is ¹/₂ because the 7 is 2 more than 5, and the 5 is just 1 so it makes ¹/₂ . (developed a system to try understand the part-whole relationship between the denominator and numerator)
• Because if you add the top and the bottom number it will give you the answer. (developed a system to try understand the part-whole relationship between the denominator and numerator – add the two together).