Ordering fractions on a number line

Ordering fractions on a number line

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This task is about placing fractions on a number line in order from smallest to largest.

Question Change answer

Drag the fractions to the spaces on the number line below.
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Task administration: 
  • This resource looks at developing understanding of ordering proper fractions (both unit fractions and non-unit fractions).
  • Students could be asked to explain how they know that one fraction is larger/smaller than another when they are placing them.
  • Some students may be able to identify the order of the fractions without cutting out and comparing the fractions before ordering.
Level:
4
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Keywords: 
fractions, ordering numbers, number lines
Description of task: 
In this practical task students order proper fractions on a number line between 0 and 1.
Answers/responses: 

2/9 , 2/8 , 2/5 , 1/2 , 3/5 , 2/3 , 7/10 , 4/5

Based on a trial set of 20 Y7/8 students.

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

  • where they place  2/9 , 1/2 , and 4/5 and the fractions around them; and
  • how they justify their choice.
Teaching and learning: 
This resource is a more difficult parallel resource to Ordering fractions on a number line II.  It requires students to place fractions that are less common and requires more of an understanding about the relationship between the numerator and the denominator.   The fractions have been selected to minimise the opportunity for students correctly ordering the fractions based on a misconception, e.g., students who state that 4/5 is greater than 3/4 may still have a misconception about fraction size (i.e., that 4 and 5 are bigger than 3 and 4, therefore 4/5 is bigger than 3/4).
Diagnostic and formative information: 

Students who identify 7/10 as the largest are likely to be using their understanding of whole number rather than of rational number (fractions as rational numbers). A significant number of students correctly identified the location of 1/2 .

Common error Likely misconception
2/9 , 2/8 , 2/5 , 1/2 , 3/5 , 2/3 , 4/5 , 7/10 Students order most of the fractions correctly but fall back on a whole number understanding comparing 4/5 and 7/10 (i.e., 7 is greater than 4).
7/10 , 2/9 , 2/8 , 4/5 , 3/5 , 2/3 , 1/2(denominator then numerator)7/10 , 4/5 , 3/5 , 2/9 , 2/8 , 2/5 , 2/3 , 1/2 Students have a misconception about the relationship between the denominator and numerator.  They try to relate the two numbers by ordering the numerators and then the denominators (or vice versa) either in descending/ascending order.
Next steps: 
Students who have either of the misconceptions identified above need to develop a part-whole understanding of fractions before trying to devise a system to order or compare fractions.  A part-whole understanding of fractions is about recognising that a fraction is a part of a whole, and that it is the relationship between the top and bottom number that describes and quantifies the fraction, i.e.,  it is about the relationship between the numbers, not their absolute size.  Students should be aware that the bottom number represents how many parts make up the whole, and the top number represents how many of these parts are of interest.  For further information about part-whole fractions (and associated resources) see the Fractional Thinking concept map: part-whole 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.  Encourage students to explore a range of many new fractions such as 3/13 , 11/27 , etc, or even include some top heavy (or improper) fractions and discuss how large these fractions are.  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.  Appropriate drawing of fractions can promote understanding and help with comparing the size of fractions.

For students who misplace one fraction such as 2/3 , 3/5 , 4/5 or 7/10 , ask them to explain how they know that the misplaced fraction is larger or smaller and to show this using a diagram – they can even begin by partitioning a rectangle, showing each fraction, and then comparing.  For a resource comparing fractions see Larger fractions (level 3).

For students who could place all eight fractions correctly, draw another number line that has room for these fractions and explore where they think a top heavy fraction such as 6/5 or 5/4 might go.  Scaffold the students to construct these top heavy fractions by starting with the unit fraction of each and asking for non unit fractions with the same denominator,e.g., 1/5 , 3/5 , 6/5 ;  1/4 , 3/4 , 5/4 ; or   1/2 , 2/2 , 3/2 .