5.5

## National STEM Learning Centre

Skip to 0 minutes and 7 secondsPAULA KELLY: The reciprocal of a number is defined as 1 divided by that number. The reciprocal of 2 is 1/2. The reciprocal of 3 is 1/3. The reciprocal of a quarter is 4. In general, the reciprocal of n is one over n. The reciprocal of any number is 1 over that number. Choose a whole number. Write it down. Now write down its reciprocal. Multiply your number by its reciprocal. What do you find? For example, 3 multiplied by 1/3 gives us 1. 2 multiplied by 1/2 is 1. 4 multiplied by [? a ?] [? quarter ?] is 1. For any number, n, the product of that number and its reciprocal is always equal to 1.

Skip to 0 minutes and 59 secondsSo what's the reciprocal of a fraction? What's the reciprocal of 2/3? 2/3 multiplied by 3 over 2 gives us 1. So 1 divided by 2/3 is 3 over 2.

Skip to 1 minute and 16 secondsLet's call the reciprocal of 2/3 R. It's likely R will be a fraction. Let's call it a over b. Then we know that 2/3 multiplied by R is 1. 2/3 multiplied by a over b is 1 over 1. 2/3 multiplied by 3 over 2 is 6 over 6, or a whole 1. So R, the reciprocal of 2/3, is 3 over 2. So does this fit with what we've seen so far? To recap, we've learned that the reciprocal of a number is 1 divided by that number. A number multiplied by its reciprocal equals 1. The reciprocal of the fraction a over b is b over a.

Skip to 2 minutes and 2 secondsIf 3 over 2 is the reciprocal of 2 over 3, then 2 over 3 is the reciprocal of 3 over 2.

# Exploring reciprocals

If you think you know what we mean by ‘the reciprocal of a number’, before watching the video try jotting down the value of the reciprocal of four and the reciprocal of two thirds.

If students learn how to perform algorithms and procedures without understanding how and why the algorithm works, it is likely that they will struggle to apply the rules appropriately in unfamiliar contexts. In order to understand why the rule for dividing fractions works, a clear understanding of what the reciprocal of a whole number and the reciprocal of a fraction is.

It is also useful when calculating the gradients of perpendicular lines.

In summary:

• The reciprocal of a number is 1 divided by that number.
• A number multiplied by its reciprocal equals 1.
• If we define a fraction $\frac{a}{b}$, then the reciprocal of that fraction is $\frac{b}{a}$.

In the next step we’ll discover why reciprocals are commonly seen when dividing one fraction by another.

## Problem worksheet

Complete questions 6 and 7 from this week’s worksheet.