Larger fractions III

Larger fractions III

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about showing or explaining which of two fractions is larger.
a) Show or explain which is the larger fraction out of \(1 \over 3\) and \(2 \over 5\)
 
 
 
 
 
 
 
 
b) Show or explain which is the larger fraction out of \(4 \over 5\) and \(6 \over 7\)
 
 
 
 
 
 
 
 
c) Show or explain which is the larger fraction out of \(7 \over 6\) and \(6 \over 5\)
 
 
 
 
 
 
 
 
Task administration: 
This task is completed with pencil and paper only.
Level:
4
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
fractions, ordering fractions
Description of task: 
Students compare two fractions, find the larger fraction and show or explain their reasoning.
Answers/responses: 
    Y8 (11/07)
a) 2/5 and explanation involving any 1 of the following:

  • 1/3 is equivalent to 2/6 , and 2/5 would be larger because fifths are larger than sixths;
  • a common denominator to compare the fractions, e.g., 5/15 and 6/15 );
  • an accurate diagram of both fractions clearly showing larger and smaller pieces; or
  • other sufficient explanations/diagrams justifying their answer.
 difficult
b) 6/7 and explanation involving any 1 of the following:

  • 4/5 is 1/5 away from 1, 6/7 is 1/7 away from 1 and 1/5 is greater than 1/7 , so  6/7 is closer to 1 and therefore larger [difference to a whole];
  • a common denominator to compare the fractions, e.g., 28/35 and 30/35 ;
  • an accurate diagram of both fractions clearly showing larger and smaller pieces, and more shaded for 6/7 ; or
  • other sufficient explanations/diagrams justifying their answer.
 difficult
c) 6/5 and explanation involving any 1 of the following:

  • 6/5 is greater than a whole by 1/5 and 7/6 is greater than a whole by 1/6 .  As 1/5 is greater than 1/6 , then 6/5 is greater;
  • a common denominator to compare the fractions, e.g., 35/30 and 36/30 ;
  • an accurate diagram of both fractions clearly showing larger and smaller pieces, and more shaded for 6/5 ; or
  • other sufficient explanations/diagrams justifying their answer.
 difficult

Based on a representative sample of 128 students.

NOTES:

  1. Accept the use of cross multiplying to find common multiples to show which fraction is larger, but ask if students can explain why this works or show it in another way as well.  Cross multiplying to find common factors could be a learned procedure and not necessarily indicate understanding about fraction size, or the part-whole nature of fractions.
  2. The questions can be solved using different strategies, including equivalent fractions and partitioning knowledge, difference to a whole (for both question b and c).  Encourage students to look at the range of strategies used by the group/class and discuss which are more efficient or accurate, and identify which best suits the fractions being compared.
Teaching and learning: 

This resource is about showing understanding about fraction size.  Accordingly, the student's explanation or representation is paramount.  Simply identifying the larger of two fractions is no guarantee of understanding or even knowledge of the fraction relationship.

Diagnostic and formative information: 

Most students either found a common denominator to compare the fractions or drew a diagram to compare the fractions.  Of the students who drew a diagram to show their reasoning, over half did not draw with sufficient accuracy to show that one fraction was larger than another.  This insufficient diagram was a very common error.  A small number of students attempted to use whole number argument (stating that larger numbers meant a larger fraction), while others argued that smaller denominators meant larger fractions.

  Common error Likely misconception
a)
b)
c)		 2/5 > 1/3  [2 & 5 are bigger than 1 & 3]
6/7 > 4/5  [6 & 7 are bigger than 4 & 5]
7/6 > 6/5  [7 & 6 are bigger than 6 &5]		 Whole number misconception
Applying rules for whole numbers, and not recognising that a fraction is a rational number (a/b).  Students do not understand the relationship between the numerator and denominator in a fraction.
a)
b)
c)		 1/3
4/5
6/5
…because the "lower the denominator the bigger the pieces".		 Part-whole misconception for non-unit fractions
Students do not understand that to compare non-unit fractions they must consider the relationship between the top and the bottom number.
a)-
c)		 Insufficient diagram showing only the largest fraction Inaccurate diagrams
Students do not understand that to compare fractions the diagrams must be accurate and comparable, i.e., they represent the same referent whole.
a)
b)		 State that both fractions are the same/equal
2/5 and 1/3
4/5 and 6/7 .		 Part-whole misconception
Students know they are one "piece" from 1 but are not taking into account the size of the pieces.
Next steps: 

Whole number misconception
Students who think that "the larger the numbers the larger the fraction" are likely to be applying these rules based on their experiences with whole numbers and do not recognise that fractions are rational numbers and therefore a different kind of number.  This misconception can involve totalling (adding or multiplying) all the numbers in each fraction to compare with another.  Students with this misconception (and those who found pieces but did not take into account the size) could start looking at unit fractions and working with diagrams to explore how the fractional notation works for unit fractions, and then use diagrams to illustrate non-unit fractions.  They should then support their diagrams with an explanation of how they know their answer is correct.
NOTE: Using rectangle diagrams when comparing fractions can ensure a similar sized common referent whole. For example, comparing 4/5 and 3/7 :

One rectangle has been partitioned into 5 parts and the other into 7 parts.  The fractions have been shaded and the diagrams can be compared.  Students can further explore other non-unit fractions using these two denominators, e.g., 3/5 and 3/7 , or 3/5 and 2/7 (both these examples have the fractions on opposite sides of a half).  Finish this exercise by removing the images and encouraging students to solve similar comparison problems by visualising the fractions in their head or explaining about the size of the parts and the number of them involved in the problem.

Part-whole misconception for non-unit fractions
Students who respond that "the lower the number on the bottom the larger the fraction/pieces" may be reciting something they have memorised without understanding. Students could develop misconceptions that will lead to problems when they move on to looking at or comparing non-unit fractions. This rule is not transferable to the non-unit fraction comparisons. Check that students can offer another explanation or representation to justify their answer, and encourage them to show their explanation using a diagram (see above).

Inaccurate/insufficient diagrams
For students who drew an inaccurate diagram to compare the two fractions, encourage them to show how they made their diagram.  Ask how they know that each part is a unit fraction – if each part is the same unit fraction how could they be of different size?  Discuss with them what needs to be in place when comparing two diagrams.
For example, students should know that 1/2 is bigger than 1/4 .  Show them a diagram where 1/4 looks bigger and talk about the comparison.  Encourage them to use rectangles as representations of the fractions rather than circles.

Students should begin to recognise that the shapes have to be the same size to compare fractions.
Number lines can also be used. Students could partition the number line for both unit fractions and build up the fractions being compared.  Show on the number line, work out which is larger, and discuss how they know their answer is correct.
Students who answered with an insufficient explanation, or simply repeated that one was larger without further elaboration, need to have a "sufficient" explanation modelled and made explicit to them.  A summary of what makes an explanation sufficient is:

  • specific details;
  • appropriate use of examples;
  • justification;
  • clarity to another reader.

Further teacher and peer modelling of a sufficient explanation should also help supplement their understanding of what is needed in an explanation. Have students share sufficient explanations and discuss as a group or class what features makes them "sufficient".  Click on the link to the English resource writing an explanation for further information.  If necessary once an explanation is given address any areas of concern (as described above).

For students who drew an accurate diagram for comparing, encourage them to explain and show on their diagram, how they know their answer is correct.  Students can also be encouraged to work towards an explanation without a diagram (see above).

Click on the link for further information about comparing fractions (Fractional thinking concept map).