027. Multiplying Fractions, Mixed Numerals and Whole Numbers

Learning Intentions

• To understand that multiplying fractions is easier if you first cancel common factors from numerators and denominators in each fraction
• To know that a whole number can be written as a fraction with a denominator of
• multiply fractions, mixed numerals and/or whole numbers, giving an answer in simplest form

Pre-requisite Summary

• Understand that a fraction can represent division and part of a whole
• Know the meaning of numerator and denominator
• Be able to Solve factors of whole numbers
• Understand equivalent fractions and simplest form
• Know that a whole number such as can be written as
• Be able to convert mixed numerals to improper fractions
• Be able to Simplify fractions by dividing numerator and denominator by a common factor
• Recall multiplication facts for small whole numbers

Worked Examples

Worked Example 1

a) Write each whole number as a fraction with denominator :

b) Explain why this does not change the value of the number.

Worked Example 2

Multiply and simplify:

a)

b)

c) State which common factors can be cancelled first.

Worked Example 3

Multiply by first cancelling common factors:

a)

b)

Worked Example 4

Multiply a fraction and a whole number:

a)

b)

c) Write the whole number as a fraction first.

Worked Example 5

Convert mixed numerals to improper fractions, then multiply:

a)

b)

Worked Example 6

Multiply mixed numerals and give the answer in simplest form:

a)

b)

Problems

Problem 1

a) Write each whole number as a fraction with denominator :

b) Explain why this does not change the value of the number.

Problem 2

Multiply and simplify:

a)

b)

c) State which common factors can be cancelled first.

Problem 3

Multiply by first cancelling common factors:

a)

b)

Problem 4

Multiply a fraction and a whole number:

a)

b)

c) Write the whole number as a fraction first.

Problem 5

Convert mixed numerals to improper fractions, then multiply:

a)

b)

Problem 6

Multiply mixed numerals and give the answer in simplest form:

a)

b)

Exercises

Understanding and Fluency

Exercise 1.

Write each whole number as a fraction with denominator :

a)

b)

c)

Exercise 2.

Multiply and simplify:

a)

b)

c)

Exercise 3.

Multiply by cancelling common factors first:

a)

b)

c)

Exercise 4.

Multiply a fraction and a whole number:

a)

b)

c)

Exercise 5.

Convert mixed numerals to improper fractions, then multiply:

a)

b)

c)

Exercise 6.

Multiply mixed numerals and simplify:

a)

b)

c)

Exercise 7.

Mixed practice:

a)

b)

c)

Exercise 8.

Mixed practice:

a)

b)

c)

Reasoning

Exercise 9.

Explain why writing as is valid.

Exercise 10.

A student says but does not simplify first. Explain why cancelling first can make the calculation easier.

Exercise 11.

A student cancels the in with the in the same fraction and gets . Explain the mistake.

Exercise 12.

Explain why mixed numerals are usually converted to improper fractions before multiplying.

Problem-solving

Exercise 13.

A recipe needs cup of milk for one batch. How much milk is needed for batches?

Exercise 14.

A rope that is m long is cut into pieces each of length of a metre for a craft display. Find the product .

Exercise 15.

A garden bed is m wide and m long. Find the product of these measures.

Exercise 16.

A student completes of a worksheet, then completes of that amount again for revision. Find .

Exercise 17.

A machine makes trays each hour for hours. Find the total number of trays made.

Exercise 18.

A ribbon of length m is used to make decorations, each Use of a metre in a design calculation. Find the product .

Potential Misunderstandings

• Students may try to add numerators and denominators instead of multiplying
• Students may not Recognise that a whole number can be written as a fraction with denominator
• Students may cancel numbers that are being added rather than factors being multiplied
• Students may cancel within a single numerator and denominator incorrectly instead of across factors in a product
• Students may forget to convert mixed numerals to improper fractions before multiplying
• Students may multiply correctly but leave the answer unsimplified
• Students may simplify using numbers that are not common factors
• Students may forget that cancelling common factors does not change the value of the product

Next: 028. Dividing Fractions Using Reciprocals