Dividing Fractions – Dividing vs Multiplying By The Reciprocal

A fraction is commonly interpreted as the number of parts considered over the number of parts in the whole. So the fraction 1/3 for example, describes 1 of the parts out of 3 parts in the whole.

But fractions also need to be recognized as a representation of division. For 1/3, one part is  being divided by 3 parts. 1/3 means 1÷3.

Dividing Fractions:

Dividing by a Number is the Same as Multiplying by that Number’s Reciprocal

Students need to develop an understanding of how multiplication and division can be manipulated to represent the same expression. They should develop the ability to fluidly move between both representations when problem-solving.

6÷3=2 but it is also true that 6∙1/3=2

Why does this relationship work?

6÷3 is the same thing as 6/3.

6/3=6/1 . 1/3 so 6.1/3

6÷3=6.1/3

We can verify this result using a visual model like a number line. This model shows 6÷3=2.

And we can verify that  6.1/3=2.

Multiplying is the same as Dividing by the Reciprocal

How often have you seen a student reach a point in problem-solving where they need to tackle a step like 9.1/2 and the value represented by the expression becomes very unclear? A solid understanding of the relationships involved will help them recognize that 9.1/2 is the same as 9÷2.

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9.1/2=9/1 . 1/2=9/2 so 9÷2, which is easier to recognize as 4 1/2.

Not Just a Catchy Phrase

The relationship between division and multiplication should be taught with enough support so students can make sense of why it works. The goal is to develop a solid understanding that allows students to think about what is represented, what it means, and how they can manipulate it in order to solve a problem.

Also read: Dividing Whole Numbers By Fractions [Examples]

Frequently Asked Questions on Dividing Fractions