Ordering fractions

Ordering fractions

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This task is about ordering fractions from smallest to largest.

Question Change answer

Drag the fractions below in order from smallest to largest.

  • 1-over-two.png

  • 6-over-five.png

  • 3-over-two.png

  • 3-over-four.png

  • 6-over-ten.png

  • 2-over-three.png

  • 2-over-five.png

  • 1-over-ten.png

Task administration: 
This task can be completed online only and has auto marking displayed to students.
Level:
4
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Keywords: 
fractions, ordering fractions
Description of task: 
Students sort fractions into order from smallest to largest
Answers/responses: 
a)
1/10 , 1/3 , 2/5 , 1/2 , 6/10 , 2/3 , 3/4 , 6/5
NOTE: an error may be a transposition or fraction out of order.
2 marks(all correct)
or
1 mark(1 error)
  Total 2 marks

Extension:

Ask students to explain their reasoning for how they know one fraction is less/more than another.

Diagnostic and formative information: 

A number of students placed most of the fractions in correct order, but made 1-2 errors. Most students knew to have the tenths as the smallest denominator, and most of those one-tenth as the smallest fraction.

Common error Likely misconception
1/2 , 1/3 , 1/10 , 2/3 , 2/5 , 3/4 , 6/5 , 6/10 Whole number (numerator first)Ordering the fractions by the numerator and then the denominator.  Students are applying whole number ordering principle to the top and them the bottom indicating a lack of understanding of the nature of a fraction as a rational number.
6/10 , 1/10 , 6/5 , 2/5 , 3/4 , 2/3 , 1/3, 1/2 Whole number (denominator first)Ordering the fractions by the denominator and then the numerator, indicating a lack of understanding of a fraction as a single rational number. Students have some understanding that a larger denominator means a smaller fraction, but they think this is true for the numerator as well.
1/10 , 6/10 , 2/5 , 6/5 , 3/4 , 1/3 , 2/3, 1/2 Whole number (denominator first)Ordering the fractions by the denominator and then the numerator as above.  Students have some understanding that a larger denominator means a smaller fraction, and that a larger numerator means a larger fraction.
1/2 , 1/3 , 2/3 ( 2/5 , 1/10 , 6/5 , 3/4 ) 6/10 *variable order in brackets Whole number (larger numbers)Ordering fractions by the larger the top and bottom numbers the larger the fraction (these numbers could be added or multiplied together).

Based on a trial set of 27 Y7/8 students.

NOTE: Students should be able to order all 8 fractions to indicate understanding of ordering simple proper fractions.  Two important ideas in this resource are:

  • where they place 2/5 , 6/10 , and 6/5 ; and
  • how they justify their choice.
Next steps: 
After cutting out and placing the fractions, students could be asked to mark down which fractions they started with to work out the order, i.e., did they start with 1/2 or 1/10 or 6/5 ? or with the fractions they knew and then add the others?  By sharing their strategies for ordering, the strategies can be critiqued to find the easiest or most efficient.

Understanding partitioning and the part-whole relationship

Students who have any of the whole number misconceptions identified previously need to develop a part-whole understanding of fractions before trying to devise a system to compare or order fractions.If required, students could go back to partitioning and explore constructing the parts (unit fractions), combining these parts to make non-unit fractions that are between 0 and 1 (also called proper fractions), and naming these new fractions (part-whole fractions).

Using diagrams to compare
After this, encourage students to compare just two fractions (including non-unit fractions) before trying to order a number of unit and non-unit fractions.  For example, students could explain (using materials, diagrams or reasoning) how they know the larger of 2/3 and 1/5 , and 1/3 and 2/5 .  Appropriate drawing of fractions can promote understanding and help with comparing the size of fractions. For a resource comparing fractions see Larger fractions (level 3).

Simple fractions correctly placed all fractions

For students who correctly placed all fractions, ensure that they can also order other odd numbered fractions and top heavy (improper) fractions such as 3/13 , 11/27 , 13/9 , 29/3 , etc.

For further information about Fractions and number lines, lick on the link Fractional thinking: conceptual map.

Numeracy resources:

Book 7: Teaching Fractions, Decimals and Percentages, 2006:Trains (p.19) Early additive/Advanced additive/Early multiplicative.