023. Improper Fractions and Mixed Numerals

Learning Intentions

• To know the meaning of the terms improper fractions and mixed numerals
• convert from mixed numerals to an improper fraction
• convert from an improper fraction to a mixed numeral

Pre-requisite Summary

• Understand that the denominator tells the number of equal parts in one whole
• Understand that the numerator tells how many of those equal parts are being counted
• Be able to identify whole numbers and fractions separately
• Know that several equal fractional parts can make one or more wholes
• Be able to use multiplication and division with whole numbers
• Understand that division can be used to find how many wholes and parts are contained in a fraction

Worked Examples

Worked Example 1

a) Define an improper fraction.
b) Define a mixed numeral.
c) State whether ${\displaystyle \frac{7}{4}}$ and $1{\displaystyle \frac{3}{4}}$ are improper fractions or mixed numerals.

Worked Example 2

Convert each mixed numeral to an improper fraction:
a) $2{\displaystyle \frac{1}{3}}$
b) $4{\displaystyle \frac{2}{5}}$

Worked Example 3

Convert each mixed numeral to an improper fraction:
a) $3{\displaystyle \frac{3}{4}}$
b) $5{\displaystyle \frac{7}{8}}$

Worked Example 4

Convert each improper fraction to a mixed numeral:
a) ${\displaystyle \frac{9}{2}}$
b) ${\displaystyle \frac{11}{3}}$

Worked Example 5

Convert each improper fraction to a mixed numeral:
a) ${\displaystyle \frac{14}{5}}$
b) ${\displaystyle \frac{23}{6}}$

Worked Example 6

Convert in both directions:
a) $6{\displaystyle \frac{2}{7}}$ to an improper fraction
b) ${\displaystyle \frac{31}{4}}$ to a mixed numeral
c) Explain how multiplication and division are used in each conversion.

Problems

Problem 1

a) Define an improper fraction.
b) Define a mixed numeral.
c) State whether ${\displaystyle \frac{13}{6}}$ and $2{\displaystyle \frac{1}{6}}$ are improper fractions or mixed numerals.

Problem 2

Convert each mixed numeral to an improper fraction:
a) $3{\displaystyle \frac{1}{2}}$
b) $5{\displaystyle \frac{3}{4}}$

Problem 3

Convert each mixed numeral to an improper fraction:
a) $4{\displaystyle \frac{2}{3}}$
b) $7{\displaystyle \frac{5}{6}}$

Problem 4

Convert each improper fraction to a mixed numeral:
a) ${\displaystyle \frac{7}{2}}$
b) ${\displaystyle \frac{13}{4}}$

Problem 5

Convert each improper fraction to a mixed numeral:
a) ${\displaystyle \frac{17}{5}}$
b) ${\displaystyle \frac{29}{7}}$

Problem 6

Convert in both directions:
a) $2{\displaystyle \frac{4}{9}}$ to an improper fraction
b) ${\displaystyle \frac{26}{3}}$ to a mixed numeral
c) Explain how multiplication and division are used in each conversion.

Exercises

Understanding and Fluency

1. State whether each is an improper fraction or a mixed numeral:
   a) ${\displaystyle \frac{5}{3}}$
   b) $1{\displaystyle \frac{2}{5}}$
   c) ${\displaystyle \frac{12}{7}}$

2. State whether each is an improper fraction or a mixed numeral:
   a) $3{\displaystyle \frac{1}{8}}$
   b) ${\displaystyle \frac{17}{4}}$
   c) $6{\displaystyle \frac{5}{6}}$

3. Convert each mixed numeral to an improper fraction:
   a) $1{\displaystyle \frac{1}{2}}$
   b) $2{\displaystyle \frac{3}{4}}$
   c) $3{\displaystyle \frac{2}{5}}$

4. Convert each mixed numeral to an improper fraction:
   a) $4{\displaystyle \frac{1}{3}}$
   b) $5{\displaystyle \frac{5}{6}}$
   c) $7{\displaystyle \frac{3}{8}}$

5. Convert each improper fraction to a mixed numeral:
   a) ${\displaystyle \frac{8}{3}}$
   b) ${\displaystyle \frac{10}{4}}$
   c) ${\displaystyle \frac{13}{5}}$

6. Convert each improper fraction to a mixed numeral:
   a) ${\displaystyle \frac{15}{2}}$
   b) ${\displaystyle \frac{19}{6}}$
   c) ${\displaystyle \frac{22}{7}}$

7. Convert in either direction as needed:
   a) $2{\displaystyle \frac{2}{3}}$
   b) ${\displaystyle \frac{18}{5}}$
   c) $4{\displaystyle \frac{3}{7}}$

8. Convert in either direction as needed:
   a) ${\displaystyle \frac{25}{4}}$
   b) $6{\displaystyle \frac{1}{5}}$
   c) ${\displaystyle \frac{32}{9}}$

Reasoning

9. Explain why ${\displaystyle \frac{9}{4}}$ is called an improper fraction.

10. A student says $3{\displaystyle \frac{1}{2}}={\displaystyle \frac{4}{2}}$. Explain the mistake.

11. Explain why converting ${\displaystyle \frac{17}{5}}$ to a mixed numeral involves division.

12. A student converts $2{\displaystyle \frac{3}{4}}$ to ${\displaystyle \frac{5}{4}}$. Explain why this is incorrect.

Problem-solving

13. A ribbon measures $3{\displaystyle \frac{1}{2}}$ metres. Write this length as an improper fraction.

14. A jug contains ${\displaystyle \frac{11}{4}}$ litres of water. Write this amount as a mixed numeral.

15. A recipe uses $2{\displaystyle \frac{2}{3}}$ cups of flour. Write this as an improper fraction.

16. A plank is ${\displaystyle \frac{23}{5}}$ metres long. Write this length as a mixed numeral.

17. A class walked $4{\displaystyle \frac{3}{8}}$ km. Express this as an improper fraction.

18. A container holds ${\displaystyle \frac{29}{6}}$ kg of rice. Express this as a mixed numeral.

Potential Misunderstandings

• Students may think an improper fraction is “wrong” because the numerator is greater than the denominator
• Students may confuse the whole number part and the fractional part in a mixed numeral
• Students may add the whole number and numerator directly when converting to an improper fraction
• Students may forget to multiply the whole number by the denominator before adding the numerator
• Students may change the denominator when converting between forms
• Students may write the remainder from division as the new denominator instead of keeping the original denominator
• Students may not recognise that an improper fraction and its mixed numeral form represent the same quantity
• Students may simplify or alter the fraction before completing the conversion in a way that changes the intended process

Next: 024. Ordering Fractions