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 Solve 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 and are improper fractions or mixed numerals.

Worked Example 2

Convert each mixed numeral to an improper fraction:

a)

b)

Worked Example 3

Convert each mixed numeral to an improper fraction:

a)

b)

Worked Example 4

Convert each improper fraction to a mixed numeral:

a)

b)

Worked Example 5

Convert each improper fraction to a mixed numeral:

a)

b)

Worked Example 6

Convert in both directions:

a) to an improper fraction

b) 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 and are improper fractions or mixed numerals.

Problem 2

Convert each mixed numeral to an improper fraction:

a)

b)

Problem 3

Convert each mixed numeral to an improper fraction:

a)

b)

Problem 4

Convert each improper fraction to a mixed numeral:

a)

b)

Problem 5

Convert each improper fraction to a mixed numeral:

a)

b)

Problem 6

Convert in both directions:

a) to an improper fraction

b) to a mixed numeral

c) Explain how multiplication and division are used in each conversion.

Exercises

Understanding and Fluency

Exercise 1.

State whether each is an improper fraction or a mixed numeral:

a)

b)

c)

Exercise 2.

State whether each is an improper fraction or a mixed numeral:

a)

b)

c)

Exercise 3.

Convert each mixed numeral to an improper fraction:

a)

b)

c)

Exercise 4.

Convert each mixed numeral to an improper fraction:

a)

b)

c)

Exercise 5.

Convert each improper fraction to a mixed numeral:

a)

b)

c)

Exercise 6.

Convert each improper fraction to a mixed numeral:

a)

b)

c)

Exercise 7.

Convert in either direction as needed:

a)

b)

c)

Exercise 8.

Convert in either direction as needed:

a)

b)

c)

Reasoning

Exercise 9.

Explain why is called an improper fraction.

Exercise 10.

A student says . Explain the mistake.

Exercise 11.

Explain why converting to a mixed numeral involves division.

Exercise 12.

A student converts to . Explain why this is incorrect.

Problem-solving

Exercise 13.

A ribbon measures metres. Write this length as an improper fraction.

Exercise 14.

A jug contains litres of water. Write this amount as a mixed numeral.

Exercise 15.

A recipe uses cups of flour. Write this as an improper fraction.

Exercise 16.

A plank is metres long. Write this length as a mixed numeral.

Exercise 17.

A class walked km. Express this as an improper fraction.

Exercise 18.

A container holds 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