Lesson video

Welcome to our second lesson in this fractions unit.

Today we will be multiplying proper fractions.

All you'll need is a pencil and a piece of paper today, pause the video and grab your things if you haven't done so already.

So today we're going to be multiplying pairs of proper fractions.

You'll start with a quiz to test your knowledge from the previous unit, then we'll look at the procedure for multiplying fractions, then simplifying fractions before you go onto some independent learning, and then our final quiz.

So let's start with your initial knowledge quiz, pause the video now and complete the quiz and click restart once you're finished.

Great work.

So, now we're looking at patterns when comparing the product with its calculation.

So when you look at the product here, 1/15, what do you notice about its relationship with the two fractions in the calculation? And the same for the second one.

Pause the video and make some notes.

So you should have noticed that the numerator of the answer is always found by the multiplication of both numerators.

So here you have one multiplied by one gives you one, and the denominator of the answer is always found by the multiplication of both denominators.

Three multiplied by five is equal to 15.

Let's check it over here.

So numerator times numerator, one times one is one, denominator times denominator, four times five is 20.

So the actual procedure for multiplying fractions is extremely straightforward.

What happens when we're multiplying a fraction by an integer? So by a whole number, this integer, it doesn't have a denominator.

So what do we do here? So we can express the five as a fraction with a denominator like this.

So five as a fraction is 5/1, but you might be asking yourself, how does 5/1 represent five? Well, this is what the fraction actually tells us.

Five is the number of equal parts being counted.

And one, the denominator, is the number of equal parts that the hold has been split into.

So 5/1 just means five wholes or five.

And if you think about what we already know about fractions, 5/1 can be expressed as five divided by one and we know that five divided by one is five.

So when we have an integer multiplied by a fraction, we can express the integer as a fraction over one.

So now we've done this representation, look at the patterns that you see now in the calculations and think about whether you can suggest an efficient method for multiplying fractions.

Pause the video while you have a think.

So our efficient method is simply numerator multiplied by numerator and denominator multiplied by denominator, five times two is 10.

One times three is three and so on.

You can see that pattern across all of them, and where there is an integer, so this one was originally five times 2/3, where there is an integer, a strategy that we can use is to express the integer as a fraction over one, and then do our straightforward multiplication.

So now I'd like you to use this strategy to find the productions below, pause the video while you do your working.

So for your first one, you could have expressed this integer four as a fraction over one, so 4/1.

And then you were multiplying the two numerators, four times on those four.

And then your two denominators, one times three is three.

So four multiplied by 1/3 is 4/3 or a third of four is 4/3.

The same here.

You could expressed this integer as a fraction over one, two times six is 12, seven times one is seven.

Here we had it straightforward, two times three is six, seven times four is 28, so 6/28, and you may be looking at these thinking, these are not in a simplified form, or these could be expressed as mixed numbers.

At the moment, I want to just leave them as they are.

So then the next one, numerator times numerator, two times four is eight and three times five is 15.

Then again, we can express this as a fraction over one, or you may have actually noticed that all you could do is multiply the integer by the numerator, three times three is nine, and then your denominator will stay the same as the original fraction.

And let's just check that, three times three is nine and eight times one is eight.

So it's up to you which strategy you use.

Then the final one, two times one is two and one times two is two, great work.

Now we're going to look at expressing our fractions in their simplest form.

And we looked at this in our previous fraction unit, but it's really good for us to just have a little bit of revision of how to do this So, I've got 2/3 times 3/4 is equal to 6/12, and I want to express it in its simplest form.

So what I need to think of is what are common factors of six and 12 so that I can efficiently divide both six and 12 to simplify the fraction.

Now remember, you could draw yourself a factor bug of six and then one of 12 as well.

So we start with the antenna, with one and six and then two and three.

And then for 12, we've got one and 12, two and six and three and four.

So we're looking for our highest common factor.

And I can see that that is six.

So I know that whatever I do to the numerator, I have to do exactly the same to the denominator to keep the relationship the same.

Six divided by six is one and 12 divided by six is two.

So in its simplest form, 6/12 is equal to 1/2.

Now let's have a look at expressing the answer as a mixed number.

So three times a half is three halves or that's a half of three is three halves, and we're going to convert it to a mixed number.

So first of all, we need to ask ourself, how many lots of two halves are there in three halves? There is one with 1/2 left over.

So again, I can think of it as how many times does two go into three.

That was one time with one leftover.

So as a mixed number, that one is one and a half.

So now I've put the answers up from our earlier questions and I'd like you to whiz through now these answers and see whether they can either be simplified or converted to mixed numbers, pause your video while you do this.

So the first one, 4/3, is an improper fraction.

So you could see that that needed to be converted to a mixed number.

And there's one lost of three in and four with one leftover.

So that's equal to one and 1/3, the same in this second one, an improper fraction can be converted to a mixed number.

How many sevens are there in 12, there's one with five leftover, so that's one and 5/7.

6/28, that is a proper fraction, but it can be simplified so that I know that both of these are multiples of two, so I can divide both numerator and denominator by two.

And that will give me 3/14.

for D, 8/15, there no common factors of eight and 15, therefore that's already in its simplest form.

This one is an improper fraction, can be converted to a mixed number.

There's one eight in nine with one leftover, so that's one and 1/8, and then 2/2 is equivalent to two, two wholes.

Okay, so you're going to complete some independent work now, which is just using the procedure that we have learnt today and applying it to some fraction multiplication.

So pause the video while you complete your task and then click restart when you've finished and we'll go through the answers together.

Question one, you were asked to efficiently calculate the product and then express it in its simplest form.

So we know that we're multiplying the numerators and then we're multiplying the denominators.

Two times two is four, three times three is nine.

And that is already in its simplest form because they have no common factors.

B, three times one is three, four times two is eight.

Again, that's in its simplest form.

C, two times three is six and three times five is 15.

I know that six and 15 can both be divided by three.

They have that common factor.

Six divided by three is two and 15 divided by three is five.

And on to D, two times three is six, five times four is 20.

And I know that these numbers both have a common factor of two, so I can divide the numerator and denominator both by two to get 3/10 as the fraction in its simplest form.

For question two, you are multiplying an integer by a fraction and asked to express your answer as a mixed number.

So you could use which strategy you're comfortable with, either multiplying integer by numerator.

And then you know that the denominator remains the same.

So five times two is 10 and that's as thirds.

And then we can convert that to a mixed number, three lots of three go into 10 with one leftover.

Three and 1/3.

You may have used the other strategy where you convert the integer into a fraction over one, and then just use the same, multiply numerators, multiply denominators.

Three times three is nine, four times one is four, and we knew that there are two fours in nine with one remaining, so that's two and 1/4.

Four times three is 12 with the denominator staying the same, 12/5.

And we know that there are two fives in 12 with two remaining.

Sorry about that crazy two.

And then finally again, you could either multiply the integer by the numerator or convert it to a fraction over one, two times seven is 14 and one times five is five.

And we know that there are two fives in 14, with four remaining, so that's two and 4/5.

And the next question you were asked to fill in the missing values in the calculations below.

So we'll work across the top first, three times three is nine so our product is 9/20, And here we're being asked five times what is 20.

So you can think about it like this, five times something equals 20.

You could use your knowledge of times tables to fill that in very quickly, or you can rearrange using your knowledge of the inverse to get this unknown by itself.

20 divided by five is four, five times four is 20, onto the next one.

We're being asked three times something is six.

We know three times two is six, and then something times three is 21.

We know that seven times three is 21.

Now this one we have to balance the two.

So we know that five times four is equal to two times something, so let's look at this here.

Five times four is equal to 20, and if we want this to be equal to 20 as well, then this numerator must be 10.

And then the bottom part, we know that eight times something is equal to 20 times two, 20 times two is 40, so this must be eight times five, which is equal to 40.

And then the same again on the bottom one.

On D, you're balancing these, something times eight is equal to six times four, six times four is 24.

So in order for this to equal to 24, it must be three, three times eight is equal to 24.

And then the denominators, four times nine is equal to something times six, four times nine is 36.

Something times six is 36, so this denominator must be six.

Now on to a word problem.

So there is 3/7 of a cake in a tin and I eat 4/5 of what is remaining, what fraction did I eat? So I'm looking to find 4/5 of 3/7, 4/5 of 3/7.

And we know that that is the same as 4/5 times 3/7.

Or the using the commutative law We know that that's the same as 3/7 times 4/5 and that is equal to 12/35 and that is in its simplest form.

Your final question, the values in the outer triangles multiply together to produce the fraction in the centre triangle.

So you're multiplying three fractions here.

So one times two is two, times three is equal to six.

Two times five is 10, times four is equal to 40.

So you may have put 6/40 in there, or you might have simplified it to 3/20.

And then finally, we're looking at one of the outer triangles.

We have the product here.

So 1/3 times 1/4 times something equals 2/24.

One times one is one, times two is two, three times four is 12, times two is 24.

So you may have written that as 2/2 or converted it to the integer, which was two.

So now it's time to pause the video and complete your final quiz and click restart once you're finished.

Great work today.

In our next lesson, we'll be learning to divide a proper fraction by an integer.

I'll see you then.