024. Ordering Fractions

Learning Intentions

• To know the meaning of the terms ascending and descending
• compare two fractions and decide which one is larger
• order a list of fractions in ascending or descending order

Pre-requisite Summary

• Understand that fractions represent parts of a whole
• Know the meaning of numerator and denominator
• Be able to identify equivalent fractions
• Be able to simplify fractions where helpful
• Understand that fractions can be compared by using common denominators or by reasoning about size
• Know that ascending means smallest to largest
• Know that descending means largest to smallest

Worked Examples

Worked Example 1

a) Define ascending.
b) Define descending.
c) Write ${\displaystyle \frac{1}{4}},{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}}$ in ascending order.

Worked Example 2

Compare the fractions and decide which is larger:
a) ${\displaystyle \frac{2}{5}}$ and ${\displaystyle \frac{3}{5}}$
b) ${\displaystyle \frac{7}{8}}$ and ${\displaystyle \frac{5}{8}}$
c) ${\displaystyle \frac{4}{9}}$ and ${\displaystyle \frac{2}{9}}$

Worked Example 3

Compare the fractions and decide which is larger:
a) ${\displaystyle \frac{1}{3}}$ and ${\displaystyle \frac{1}{5}}$
b) ${\displaystyle \frac{3}{4}}$ and ${\displaystyle \frac{3}{8}}$
c) ${\displaystyle \frac{5}{6}}$ and ${\displaystyle \frac{5}{12}}$

Worked Example 4

Compare by using equivalent fractions or common denominators:
a) ${\displaystyle \frac{2}{3}}$ and ${\displaystyle \frac{3}{4}}$
b) ${\displaystyle \frac{5}{6}}$ and ${\displaystyle \frac{7}{9}}$

Worked Example 5

Write the fractions in ascending order:
a) ${\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{8}},{\displaystyle \frac{5}{8}}$
b) ${\displaystyle \frac{2}{3}},{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}}$

Worked Example 6

Write the fractions in descending order:
a) ${\displaystyle \frac{5}{6}},{\displaystyle \frac{2}{9}},{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{3}}$
b) ${\displaystyle \frac{7}{10}},{\displaystyle \frac{4}{5}},{\displaystyle \frac{3}{10}}$

Problems

Problem 1

a) Define ascending.
b) Define descending.
c) Write ${\displaystyle \frac{1}{3}},{\displaystyle \frac{2}{3}},{\displaystyle \frac{1}{6}}$ in ascending order.

Problem 2

Compare the fractions and decide which is larger:
a) ${\displaystyle \frac{1}{7}}$ and ${\displaystyle \frac{4}{7}}$
b) ${\displaystyle \frac{5}{9}}$ and ${\displaystyle \frac{2}{9}}$
c) ${\displaystyle \frac{6}{11}}$ and ${\displaystyle \frac{3}{11}}$

Problem 3

Compare the fractions and decide which is larger:
a) ${\displaystyle \frac{1}{4}}$ and ${\displaystyle \frac{1}{6}}$
b) ${\displaystyle \frac{5}{7}}$ and ${\displaystyle \frac{5}{9}}$
c) ${\displaystyle \frac{2}{3}}$ and ${\displaystyle \frac{2}{5}}$

Problem 4

Compare by using equivalent fractions or common denominators:
a) ${\displaystyle \frac{3}{5}}$ and ${\displaystyle \frac{2}{3}}$
b) ${\displaystyle \frac{7}{8}}$ and ${\displaystyle \frac{5}{6}}$

Problem 5

Write the fractions in ascending order:
a) ${\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{2}}$
b) ${\displaystyle \frac{5}{6}},{\displaystyle \frac{2}{3}},{\displaystyle \frac{1}{3}}$

Problem 6

Write the fractions in descending order:
a) ${\displaystyle \frac{1}{2}},{\displaystyle \frac{7}{8}},{\displaystyle \frac{3}{8}},{\displaystyle \frac{5}{8}}$
b) ${\displaystyle \frac{4}{5}},{\displaystyle \frac{1}{10}},{\displaystyle \frac{3}{5}}$

Exercises

Understanding and Fluency

1. State whether each list is in ascending or descending order:
   a) ${\displaystyle \frac{1}{5}},{\displaystyle \frac{2}{5}},{\displaystyle \frac{4}{5}}$
   b) ${\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{4}}$
   c) ${\displaystyle \frac{2}{3}},{\displaystyle \frac{5}{6}},{\displaystyle \frac{11}{12}}$

2. Compare the fractions and choose the larger:
   a) ${\displaystyle \frac{2}{7}}$ or ${\displaystyle \frac{5}{7}}$
   b) ${\displaystyle \frac{3}{8}}$ or ${\displaystyle \frac{1}{8}}$
   c) ${\displaystyle \frac{7}{10}}$ or ${\displaystyle \frac{9}{10}}$

3. Compare the fractions and choose the larger:
   a) ${\displaystyle \frac{1}{2}}$ or ${\displaystyle \frac{1}{3}}$
   b) ${\displaystyle \frac{4}{5}}$ or ${\displaystyle \frac{4}{7}}$
   c) ${\displaystyle \frac{3}{4}}$ or ${\displaystyle \frac{3}{10}}$

4. Compare the fractions using common denominators:
   a) ${\displaystyle \frac{2}{3}}$ and ${\displaystyle \frac{5}{6}}$
   b) ${\displaystyle \frac{3}{4}}$ and ${\displaystyle \frac{2}{5}}$
   c) ${\displaystyle \frac{5}{8}}$ and ${\displaystyle \frac{7}{12}}$

5. Write in ascending order:
   a) ${\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{3}{4}}$
   b) ${\displaystyle \frac{2}{5}},{\displaystyle \frac{4}{5}},{\displaystyle \frac{1}{5}}$
   c) ${\displaystyle \frac{1}{3}},{\displaystyle \frac{5}{6}},{\displaystyle \frac{2}{3}}$

6. Write in descending order:
   a) ${\displaystyle \frac{3}{8}},{\displaystyle \frac{7}{8}},{\displaystyle \frac{1}{8}}$
   b) ${\displaystyle \frac{2}{3}},{\displaystyle \frac{5}{9}},{\displaystyle \frac{8}{9}}$
   c) ${\displaystyle \frac{3}{5}},{\displaystyle \frac{1}{10}},{\displaystyle \frac{9}{10}}$

7. Order the fractions in ascending order:
   a) ${\displaystyle \frac{3}{6}},{\displaystyle \frac{1}{6}},{\displaystyle \frac{5}{6}}$
   b) ${\displaystyle \frac{1}{2}},{\displaystyle \frac{2}{4}},{\displaystyle \frac{3}{4}}$
   c) ${\displaystyle \frac{2}{3}},{\displaystyle \frac{3}{5}},{\displaystyle \frac{4}{15}}$

8. Order the fractions in descending order:
   a) ${\displaystyle \frac{5}{12}},{\displaystyle \frac{11}{12}},{\displaystyle \frac{1}{3}}$
   b) ${\displaystyle \frac{4}{7}},{\displaystyle \frac{2}{7}},{\displaystyle \frac{6}{7}}$
   c) ${\displaystyle \frac{3}{4}},{\displaystyle \frac{2}{3}},{\displaystyle \frac{5}{8}}$

Reasoning

9. Explain why ${\displaystyle \frac{1}{3}}>{\displaystyle \frac{1}{5}}$ even though $5>3$.

10. A student says ${\displaystyle \frac{3}{8}}>{\displaystyle \frac{3}{4}}$ because $8>4$. Explain the mistake.

11. Explain why using a common denominator helps compare ${\displaystyle \frac{2}{3}}$ and ${\displaystyle \frac{3}{5}}$.

12. A student orders ${\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{3}}$ in ascending order because $2<3<4$. Explain why this is incorrect.

Problem-solving

13. Three ribbons have lengths ${\displaystyle \frac{1}{2}}$ m, ${\displaystyle \frac{3}{4}}$ m and ${\displaystyle \frac{1}{4}}$ m. Order them from shortest to longest.

14. A recipe uses ${\displaystyle \frac{2}{3}}$ cup, ${\displaystyle \frac{1}{2}}$ cup and ${\displaystyle \frac{3}{4}}$ cup of different ingredients. Order these amounts from largest to smallest.

15. Four runners completed fractions of a lap: ${\displaystyle \frac{5}{6}},{\displaystyle \frac{2}{3}},{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}}$. Order these in ascending order.

16. A student shaded ${\displaystyle \frac{3}{8}},{\displaystyle \frac{1}{2}}$ and ${\displaystyle \frac{5}{8}}$ of three rectangles. Order the shaded fractions in descending order.

17. A set of water bottles contains ${\displaystyle \frac{1}{3}}$ L, ${\displaystyle \frac{3}{5}}$ L and ${\displaystyle \frac{2}{15}}$ L. Order these amounts from smallest to largest.

18. A teacher lists scores as fractions: ${\displaystyle \frac{7}{10}},{\displaystyle \frac{4}{5}},{\displaystyle \frac{3}{10}},{\displaystyle \frac{9}{10}}$. Order them in descending order.

Potential Misunderstandings

• Students may think ascending means largest to smallest instead of smallest to largest
• Students may think descending means smallest to largest instead of largest to smallest
• Students may compare fractions by looking only at the numerator or only at the denominator
• Students may think a larger denominator always means a larger fraction
• Students may not recognise that fractions with the same numerator can be compared by the size of their parts
• Students may not recognise that fractions with the same denominator can be compared by the number of parts taken
• Students may forget to convert to equivalent fractions when denominators are different
• Students may order fractions by the size of the denominator rather than the actual value of the fraction

Next: 025. Adding Fractions and Mixed Numerals