Ordering fractions on a number line
- 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.
^{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.
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. |
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} .