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## Multiplying Fractions: How to Do Them & How to Teach Them Easily

Fractions is one of those subject areas that takes time and hard work to grasp. Sometimes quite counter- intuitive and sometimes taught via methods to be blindly followed, fractions can worry both students and teachers.

But worry not! They do make sense!

They are real and understanding them is NOT impossible.

This is our blog on __multiplying fractions__ and we really hope that we cover all aspects and approaches.

## Background Knowledge

This is easily overlooked. But before tackling multiplying fractions, students __must__ be comfortable with:

- Multiplying numbers together. (If not, get them practising with Times Tables with Emile)
- What a fraction is and what it represents.

Students __should__ know:

- What a numerator is.
- What a denominator is.

It is perfectly possible to multiply fractions without knowing these labels, but students may struggle to follow your directions if they are unsure.

Students may find it useful if they also understand basic algebra so that you can explain the steps using variables.

## The Multiplying Fractions Method

To multiply any two fractions, we can follow the steps below.

- Multiply the numerators.
- Multiply the denominators
- Re-arrange the setup.

What does this look like?

So where the problem is: :

- Multiply the numerators 2 x 4 = 8
- Multiply the denominators 3 x 5 = 15
- Rearrange:

Easy right?

But this is a learning by rote approach. No understanding…you just times the top number and times the bottom numbers.

This is all perfectly acceptable, in my opinion, as a solid starting point.

Stated algebraically:

(This can be a good challenge for those working at greater depth. Express multiplying two fractions together algebraically where the first fraction is ^{a}/_{b} and the second fraction is ^{c}/_{d} To push those further, can they express the multiplication of 4 fractions algebraically. How about ‘n’ fractions?)

## Visually Representing the Multiplication of Fractions

^{2}/

_{3}x

^{1}/

_{2}

The fraction square on the left has 2 blue sections of a square which has 3 equal sections. I hope we can agree this blue area is therefore two thirds of the square.

Similarly, I hope we can agree the second fraction square represents a half of the shape.

If we overlay these coloured areas, we can see they cover two sixths of the fraction square (I’m not concerned yet by the improper fraction).

## Starting with Fractions and Whole Numbers.

An alternative and equally valid place to start investigating multiplying fractions is to look at multiplying fractions by whole numbers.

We can start by taking 4 x ^{2}/_{3}. This can be represented with a range of manipulatives such as Numicon or Lego.

Four lots of two thirds could be presented by:

How many thirds are there in total? How many blue sections? What does each blue section represent?

Students should be able to determine there are 8 blue sections.

If the teacher tells them that each blue section is a third of a whole, will the students be able to figure out there are eight thirds?

Can they express this as a fraction?

So it looks like 4 x ^{2}/_{3} is the same as eight thirds or:

^{3}/

_{4}without drawing squares? Can they represent this algebraically? (N x

^{p}/

_{q}=

^{N x p}/

_{q })

### Extending from whole numbers to fractions.

First confirm that students know that whole numbers can be expressed as a fraction (for example 4 = ^{4}/_{1}).

How would this affect:

Could anyone guess how we could calculate:

Hopefully all this investigation is leading them to:

## The Multiplying Fractions Rhyme

If all else fails, there is the rhyme:

Multiplying fractions: no big problem,

Top times top over bottom times bottom.

And don’t forget to simplify,

Before it’s time to say goodbye

## Equivalent Fractions

Equivalent Fractions should have been covered in Years 3, 4 and 5 of the National Curriculum. Please note that dealing with equivalent fractions while multiplying fractions is NOT part of the Year 6 National Curriculum.

However, I think this is a great opportunity to reinforce the concept.

As an example take ^{2}/_{3} x ^{3}/_{4}

- We can times the top numbers (numerators) together: 2 x 3 = 6
- We can times the bottom numbers (denominators) together: 3 x 4 = 12
- Rearranging gives us:

^{6}/

_{12}is equivalent to

^{1}/

_{2}. So the most “satisfying” answer is:

## Improper Fractions

An improper fraction is one where the top number is bigger than the bottom number i.e. the numerator is bigger than the denominator. For example, ^{5}/_{4} is an improper fraction because 5 is bigger than 4.

When multiplying fractions, it is not uncommon to end up with an improper fraction.

Just see the example in section 5, where we end up with 4 x ^{2}/_{3} = ^{8}/_{3}. Here ^{8}/_{3} is an improper fraction as 8 is bigger than 3.

A better representation may be a whole number and a fraction together – known as a mixed number. Here ^{8}/_{3} = 2^{2}/_{3}

Improper fractions are covered in years 5 and 6 of the national curriculum, but **not **in relation to multiplying fractions.

## Prime Factors

Prime factors can be used to good effect in multiplying fractions and what I really like is that it shows how learning about prime numbers can be useful. If I can show how something is useful then at least some of the students will be engaged! In the past, I have sold this approach as a cheat for doing difficult calculations.

Without a calculator, how do we determine the answer to:

Now the principle works on the same line as this:

As long as we divide the top of the fraction and the bottom of the fraction by the same number, we won’t affect the answer. (We know this from our work with equivalent fractions.)

Here we can remove (or cancel) the “3”s from the top and the bottom.

So now with the horrible calculation:

2240 = 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5

720 = 2 x 2 x 2 x 2 x 3 x 3 x 5

Therefore:

We can remove four of the “2”s and one of the “5”s from both top and bottom:

Similarly:

So looking back at the original calculation:

So just to bring it back to something more manageable:

I believe these sorts of investigations are a brilliant basis of some greater depth activities.

## Practise or Learn By Questions

We’re huge fans of practise with live/immediate feedback.

The more questions encountered with immediate correction, the more the approach is embedded and numeracy confidence improves.

Obviously, we suggest using Fractions with Emile. It assesses students and delivers games and activities at an appropriate level. What’s more it allows students to compete against