Close to a half

Close to a half

Pencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about identifying how close some fractions are to a half.

Question

a)  Select all the fractions below that are larger than \(1 \over 2\).

    • \(4 \over 9\)

    • \(6 \over 10\)

    • \(2 \over 5\)

    • \(4 \over 6\)

    • \(3 \over 8\)

Question 1Change answer

b) Explain how you can tell when a fraction is larger than \(1 \over 2\).

Question 1Change answer

c)  How far away from \(1 \over 2\) is each of these fractions? 
     Write your answer as a fraction.
     i)  \(3 \over 8\) 
     ii) \(4 \over 6\) 
Task administration: 
This task can be completed with pencil and paper or online (with SOME auto marking).
Levels:
3, 4
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
fractions, ordering fractions, key competencies
Description of task: 
Students identify fractions that are greater than a half, explain their reasoning, and identify how far from half some fractions are.
Curriculum Links: 
This resource can help to identify students' ability to apply strategies to solve subtraction problems involving fractions.
Key competencies
This resource involves explaining why fractions are greater or less than a half. This relates to the Key Competency: Using language, symbols and text.
For more information see http://nzcurriculum.tki.org.nz/Key-competencies
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Additive thinking, sets 6-7
Additive thinking, sets 7-8
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  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.:

  • the top number is more than half than the bottom number.
  • other sufficient explanations/diagrams justifying their answer.

Note: Some students also identified how much larger than half the "larger fractions" were.

 difficult
c)

i)
ii)

1/8
1/6
Some students identified whether the difference was more or less than half.

difficult
very difficult
Based on a representative sample of 145 students.
Teaching and learning: 
This resource is about being able to work out whether fractions are less than or greater than a half, and by how much (closeness).  Being able to compare a fraction to a half is an indication that students are developing an understanding of fraction size that moves students' understanding towards fractions as numbers.  Students can tell how close a fraction is to a half when they realise that the numerator must be exactly half of the denominator for the fraction to equal a half.

Prior knowledge
An understanding of fractional notation and about "half-ness". 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.

Diagnostic and formative information: 

Over two-thirds of students correctly identified at least one fraction greater than half without any incorrect fractions identified.

  Common error Likely misconception
b)   Part-whole 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
Next steps: 
Part-whole misconception about fractions
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 part-whole understanding of fractions before trying to compare fractions.  Ultimately the relationship between top and bottom numbers is a division relationship.  Students could begin by

  1. partitioning and identifying the parts (unit fractions),
  2. combining these parts to make non-unit fractions that are between 0 and 1 (also called proper fractions), and
  3. naming these new fractions (part-whole fractions).

Asking students what the whole shape is, and then what part they are finding, can support them to develop a more part-whole understanding about fractions. The idea of equal-sized parts and the correct number of partitions is an important piece of knowledge to support part-whole 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 non-unit fractions involves combining similar parts, and naming these new fractions (part-whole 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.

For more information about comparing fractions, and fractional thinking, see the Fractional thinking concept map: equivalence.
 
Numeracy resources
Book 7: Teaching Fractions, Decimals and Percentages, 2006
  • Trains (Early/Advanced additive)
  • Wafers (Advanced counting/Early additive).