Ordering fractions on a number line II

Ordering fractions on a number line II

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about ordering fractions on a number line.

Practical task:

  1. Cut out the fractions below.
  2. Order the fractions from the smallest to largest.
  3. Place the fractions in their correct positions on the number line.
  4. When you have finished, glue the fractions into the boxes.

Fraction number line

\(2 \over 3\) \(1 \over 2\) \(3 \over 4\) \(1 \over 3\) \(1 \over 10\) \(4 \over 5\) \(6 \over 10\) \(1 \over 5\)
Task administration: 
  • This task is completed with pencil and paper only.
  • This resource looks at developing understanding of ordering proper fractions, this includes both: unit fractions (1/10 , 1/5 , 1/3 , 1/2 etc.) and non-unit fractions (6/10 , 2/3 , 3/4 , 4/5 etc.).
  • Students could be asked to explain how they know that one fraction is larger/smaller than another when they are placing them.
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: 

1/10 , 1/5 , 1/3 , 1/2 , 6/10, 2/3 , 3/4 , 4/5 .

Based on a trial set of 30 Y7/8 student.

NOTE: Students should be able to order all eight fractions to indicate understanding of ordering fractions.  Various incorrect strategies can be used to order the fractions almost correctly, e.g., students who state that 4/5 is greater than 3/4 may still have a misconception about fraction size.  Two important ideas in this resource are:

  • where they place 6/10 – as the largest fraction, or just after 1/2 ; and
  • how they justify their choice.
Teaching and learning: 
Ordering (or comparing) non-unit fractions requires an understanding that fractions are more than unit parts, they represent a part-whole relationship between the numerator and the denominator.  If students are only introduced to unit fractions without some reference to non-unit fractions they may form some strategy or system of understanding which is inadequate for all fractions, e.g., ordering by denominator first, and using the numerator as a "tie-breaker" for the fractions with the same denominator. Students may then need to learn a new strategy for ordering or comparing all fractions.
Diagnostic and formative information: 

All students placed x/10 (x being either 1 or 6) as the smallest fraction.  This suggests an awareness that tenths were the smallest unit fractions.

Common error Likely misconception
6/10 , 1/10 , 4/5 , 1/5 , 3/4 , 2/3 , 1/3 , 1/2 or 1/10 , 6/10 , 1/5 , 4/5 , 3/4 , 1/3 , 2/3 , 1/2 Students apply the rule that the larger the denominator the smaller the fraction and then order the numerators (either applying the same rule as the denominators or ordering them as whole numbers) when the denominators are equal.  They see the fraction as some kind of whole number relationship – not a relative relationship.
1/10 , 1/5 , 1/3 , 1/2 , 2/3 , 3/4 , 4/5 , 6/10(i.e., all the unit fractions are in order) Students have a misconception about the relationship between the denominator and numerator.  They try to relate the two numbers by:

  • ordering the numerators first and then ordering the denominators if the numerators are equal;
  • multiplying the numerator and denominator (4×5 > 3×4); or
  • adding numerator and denominator (4+5 > 3+4).

By using any of these three strategies students can order 7 of the 8 fractions correctly, but not 6/10 .  It is therefore important to check how students work out the order of the fractions.
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 see the Fractional Thinking concept map: part-whole fractions.

If required, students could go back to partitioning and explore constructing the parts (unit fractions), combine 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 , … even exploring 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 or 3/4 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 ask for non unit fractions with the same denominator,

e.g.,or 1/5 , 3/5 , 6/5 ; 1/4 , 3/4 , 5/4 ; 1/2 , 2/2 , 3/2 .