Lesson video

Hi, my name's Ms. Lambell.

You've made an excellent choice deciding to stop by and do some math for me today.

I'm really pleased.

Let's get started.

Welcome to today's lesson.

The title of today's lesson is "Checking Understanding of Multiplication With Fractions." And that's within our unit, arithmetic procedures with fractions.

By the end of this lesson, you'll be able to use mathematical structures that underpin multiplication of fractions.

Some keywords that will be used in to write today's lesson are numerator and denominator.

You'll be super familiar with these, but just a quick recap.

The numerator is the expression in a fraction which is above the fraction line.

So in this case here, 5, and the denominator is the expression that is below the fraction line.

So in this case, that's the three.

Today's lesson, we're going to split into two separate learning cycles.

In the first one, we're going to explore the area model.

You'll be familiar with the area model when you were looking at integers, a multiplication of integers.

So we're going to see how that relates to fractions.

Now we're working with fractions.

And in the second learning cycle, we'll use what we've discovered in the first learning cycle to make sure that we can confidently multiply fractions together.

Let's get going with that first learning cycle.

Each of these large squares has an area of 1 metre square.

What is the area of each section? I'm going to put each of the diagrams up onto the screen separately, and then when I'm done, I'll ask you to pause the video and decide what area of each you think is shaded.

Here are the diagrams. There they are, all eight of them.

I'd now like you to decide what is the area of each shaded section.

Pause the video, and when you've got your eight answers, you can come back.

Okay, we're gonna go through each of those individually now.

Here's our first one.

The shape is split into two equal parts.

One equal part is shaded, so half of the square is shaded.

If we look at that as an area model, remember, I told you that each of the squares was 1 metre square.

So that means that each of its dimensions are 1 metre.

Here, we have half of the square, so that's half a metre.

So the area is a half.

Let's look at the second one.

This time, the shape has been split into four equal parts.

One equal part is shaded, which means a quarter of the square is shaded.

But let's take a look at the area.

This time, we have a half and a half, and we know to find the area, we multiply those two together.

So we now know then that a half multiplied by half must be a quarter because we know that a quarter of the shape is shaded.

So a half multiplied by half is a quarter.

Onto the next one, how many equal parts do we have here and how many are shaded? That's right, we've got six equal parts, and one of them is shaded.

That means the fraction of our shape that is shaded is 1/6.

Let's think about the area.

The area is a half, this time multiplied by a third.

I have a third of the shape because I'd split it into three equal parts vertically.

So we now know then that 1/3 multiplied by 1/2 or 1/2 multiplied by 1/3 must be 1/6.

And the next one, I'm gonna ask you to write down for me what you think each of the steps is going to say this time.

This is split into eight equal parts.

One equal part is shaded.

We therefore know that 1/8 of the square is shaded.

Let's have a look at that area of that purple rectangle.

Here, we've got a half, and this time, this is a quarter because I've split it into four equal parts horizontally.

That means that a quarter multiplied by half must be equal to 1/8.

And the next one, this one, we've split into six equal parts, and two equal parts are shaded.

If we look here, we can see that 2/6 of the square is shaded.

The area of the purple rectangle is a half multiplied by 2/3.

So we can see the width of that rectangle is 2/3 of the width of the square.

So 1/2 multiplied by 2/3 is 2/6.

But we know that 2/6 is actually 1/3 in its simplest form.

So this is also 1/3 of the shaded square.

And now this one.

This time, the square has been split into 12 equal parts.

Six equal parts are shaded.

6/12 of the square is therefore shaded.

The height of our purple rectangle is what fraction of the whole square, and that's 3/4.

And the width is 2/3.

This must mean that 3/4 multiplied by 2/3 is 6/12.

But what do we know that 6/12 is in its simplest form? Yes, you're right, 1/2.

So we now know that that is equal to a half.

Now, let's take a look at this one.

So again, I'm gonna ask you to write down what you think this is showing us before I go through it.

So pause the video and then come back when you've got an answer.

Great work.

So this time, we're split into 15 equal parts.

Eight of them are shaded, which means that's 8/15ths of the square.

The area of the purple rectangle is four-fifths multiplied by 2/3.

So we now know that four-fifths multiplied by 2/3 must be equal to 8/15ths.

Now, we'll take a look at this one.

The shape is split now into 20 equal parts, and 12 of them are shaded.

That means that 12/20ths of the share, sorry, that means that 12/20ths of the square is shaded.

Here, we've got one metre.

We've shaded in the whole width of the original square, and 3/5.

This means that 3/5 multiplied by 1, well, we know that multiplying by 1 doesn't change the value of something, so that's 3/5.

If I take away those vertical lines, we can see why it's 3/5.

We can see that those two things were equivalent.

I'd like you now, and you've already done a little bit of practise this 'cause I asked you to do some of those yourself before I went through them.

I'd like you to have a go at this question.

You can now pause the video and when you've got your answer, come back and I'll be waiting to check in with you.

Good luck.

Let's see how you got on.

The shape is split into 30 equal parts, did you get that? Yep, great.

10 equal parts are shaded, and you got that too.

That means that 10/30 of the square is shaded.

And you got that? Great.

Let's take a look now at what that look, what the area is of the rectangle.

2/5 multiplied by 5/6.

2/5 multiplied by 5/6 is equal to, sorry, is equal to 10/30.

What does 10/30 simplify to? That's 1/3.

So we know the area of this is 1/3.

If you didn't quite get to 1/3, if you only got to 10/30, that's not wrong.

But remember, we do like to give our answer in its simplest form if we can.

Now, you're ready to have a go at this task.

Each of the large squares have an area of 1 metre squared, so they're exactly the same as the ones that we've just done together.

For each square, I would like you to do the following things, please.

I'd like you to label the dimensions of the shaded rectangle, so the dimensions of the purple rectangle.

I'd like you to write down the calculation represented by the area of the shaded rectangle, and then I'd like you to work out the area of the square that is shaded.

It's just what we've been doing on the previous slides, but with different rectangles shaded in.

So you're gonna be super good at this.

I'll wait here for you.

You can pause the video now and then come back when you're ready.

Great work.

Now, let's check those answers.

There's a lot of information here on this slide, so I'm not going to try and read it all out.

What I'm gonna say is pause the video, check your answers, and then when you are ready, come back, and we can get going on that second learning cycle.

You can pause the video now.

Did you get 'em all right? You did amazing, well done.

And then we can move on to that next learning cycle.

Multiplication of fractions.

Okay, what would the diagram to represent this calculation look like? We've got 5/6 multiplied by 3/4.

Just have a think.

Maybe sketch it out.

Doesn't need to be totally accurate.

Just try and make your parts look as equal as you can, but it won't matter if they're not perfect.

Here's my square.

I'm splitting it into 6 equal parts horizontally to represent the 6 of the first fraction, and 4 equal parts vertically to represent the quarters of the second fraction.

I'm multiplying 5/6, so that my arrow here is representing my 5/6 and 3/4.

I can now shade in that rectangle.

Now, we know to find the arrow of a rectangle, we multiply the length by the width.

So 5/6 multiplied by 3/4 is 15 over 24.

So Sofia says it looks like you just multiply the numerator together and the denominators together.

Do you agree with Sofia? You do? Yeah, let's have a look.

5 multiplied by 3 is 15, and 6 multiplied by 4 is 24.

So Sofia, you've definitely stumbled across something really, really useful there.

Does it always work, I wonder? It does.

When we multiply two proper fractions, you multiply the numerators to get the numerator of the product and multiply the denominators to get the denominator.

So for example, 3/5 multiplied by 3/4, we multiply the numerator together, 3 multiplied by 3, and we multiply the denominators together, 5 multiplied by 4.

And this gives us 9/20.

In general then, we can say that A over B multiplied by C over D is equivalent to A multiplied by C over B multiplied by D.

Here's a diagram to represent that.

We can see the green rectangle, the area, that would be A multiplied by C, that gives us numerator.

And that is a fraction of the entire shape, and the area of the entire shape is B multiplied by D.

We can see why that generalisation works.

Sofia is interested to know if you had more than two fractions, if you would still just do the same? What do you think? Let's take a look at this one.

So at the moment, we've only multiplied two fractions together.

So we're gonna multiply 1/3 by 2/5 using the method that we now know works.

So 1/3 multiplied by 2/5 is 1 multiplied by 2 over 3 multiplied by 4.

And we're gonna still got to multiply that by a quarter.

1 multiplied by 2 is 2, 3 multiplied by 5 is 15.

So we end up with 2 over 15 multiplied by 3/4, and then we can multiply those together, and we get 6 over 60.

But Sofia's interested to know that could she just have multiplied all three of the numerators first and then all three of the denominators? Could she have done that? She could.

So actually, we can have any number of fractions.

If we're multiplying them together, we're just always going to multiply all of the numerators together and all of the denominators together to give us those new numerator and denominators.

Let's do this one together, and then you'll be ready to have a go at the 1 on the right hand side by yourself.

Calculate the answer to 5/6 multiplied by 2/3, multiplied by 3/4.

And we're going to give our answer in its simplest form.

We know, to calculate this, that we are going to multiply the numerators together and the denominators together.

So we end up with 5 multiplied by 5 multiplied by 3 over 6, multiplied by 3 multiplied by 4, which gives us 30 over 72.

And we're great at simplifying 'cause we've done it lots in the last sequence of lessons.

That ends up as 5 over 12.

Now, it's your turn.

I'd like you to have a go at this question.

And remember, I'd like your answer in its simplest form.

Please, no calculators.

You can pause video now.

I'll be here when you come back.

Great work, let's check that answer.

You should have ended up with 2/15, and that comes from multiplying together 2, 1 and 2, which gives us 4, and multiplying together 5, 2 and 3, which gave us 30.

And then we simplify that to 2/15.

You may have noticed that some products need simplifying.

When we were looking at adding and subtracting the fractions, because we were using the lowest common multiple, most of the time, we didn't need to simplify our answers.

We may have had to simplify some when we were using mixed numbers.

Here, we've had to simplify some of our answers at the end.

So for example here, 5/6 multiplied by 3/5 is 15 over 30, 15 coming from the product of 5 and 3, and the 30 coming from the product of 6 and 5.

And we know that 15 over 30 is equal to a half.

But you can simplify before multiplying to be more efficient.

Let's take a look and see what that looks like.

We could have noticed that there are some common factors, and that would give us the opportunity to simplify first.

So for example here, 6.

Actually, we could write that as 2 multiplied by 3 or 3 multiplied by 2.

Here, now, we can see that we've got 1 multiplied by half.

Now, why have we got 1 multiplied by half? 5 multiplied by 3 is 15.

5 multiplied by 3 is 15.

Remember, any fraction where the numerator is the same as the denominator is equivalent to 1.

So we can see here now why the answer to that is a half.

So we've got those common factors of 5 and 3, and we can simplify first.

Andeep tried to simplify this calculation before multiplying.

So Andeep has been listening in, and he thinks he quite liked this idea of simplifying first.

Let's take a look and see what he's done.

So 4/5 multiplied by 2/3 equals 4 multiplied by 2, over 5 multiplied by 3.

And he's spotted that 4 actually can be written as 2 multiplied by 2.

And now he says he can cancel the 2s, giving him an answer of 2/15.

What mistake has he made? Well, 4/5 multiplied by 2/3 is 8/15ths.

We know that, don't we? Because we know we multiply the numerators and we multiply the denominators.

But he's got 2/15.

So we know he's definitely made a mistake somewhere.

And this is the mistake he's made.

He's tried to take out common factors, but he's only taken out common factors of the numerator.

We can simplify if factors are common to both the numerator and the denominator, because remember then, what we're effectively doing is dividing by 1.

And it's okay to divide by 1 because it doesn't change the value.

Which of these could you simplify before multiplying? Pause the video and come back to me when you've got your answer.

Let's check those.

So you should have had A, C, and D.

We can clearly see with A, without going any further, we've got a 5 as enumerator and a denominator.

But we could also find some other common factors because 8 could be written as 2 multiplied by 4.

Here, with C, we could write 6 as 2 multiplied by 3, and then we'd have that common factor of 2 as a numerator and denominator.

And then D, 8 could be written as 2 multiplied by 4 and 10 could be written as 2 multiplied by 5, given a common factor there of 2.

And also, 9 could have been written as 3 multiplied by 3.

So again, there'd be a common factor of 3 in both the numerator and the denominator.

Now, I'd like you to have a go at these questions.

What I want you to do is to concentrate on simplifying first and please make sure you've shown all the steps that you took.

So here, remember, we're talking about efficiency, and we're thinking about simplifying before we do the multiplication.

You can pause the video now.

I'll be here when you come back.

Good luck.

Great, let's check those.

So here, 5 multiplied by 2, over 6 multiplied by 3.

That 6 is 2 multiplied by 3, so I can do that.

Notice then, I've rearranged the order on the tops that I can see that I've got my 2 and a 2, which is really just 1, 2 over 2 is 1.

So that leaves me with 5 over 3 multiplied by 3, so that's 5 over 9.

The next one, 4 multiplied by 5, over 5 multiplied by 8.

Here, I've rewritten the 4 multiplied by 5 as 5 multiplied by 4.

You don't have to do that, but I just quite like to have my common factors at the beginning so I can see them clearly.

We've rewritten 8 as 4 multiplied by 2.

We can now see that we have a numerator and denominator of 5, numerator and denominator of 4.

So therefore, that's effectively multiplying by 1, so it doesn't change the value, so this is gonna give us 1/2.

And the final one, 8 multiplied by 3 over 9 multiplied by 10.

Rewrite the 8 as 2 multiplied by 4, rewrite the 9 as 3 multiplied by 3 and the 10 as 2 multiplied by 5.

And then we can see here what's common.

Notice this time, I've not rearranged them, just to show you that you don't need to.

I can see clearly, I've got a 3 and a 2 in my numerator, a 3 and a 2 in my denominator.

So if I take those factors out, I end up with 4 as my numerator, and 3 and 5 is my denominator, and then I'm gonna find the product of 3 and 5, and that's 15.

Now, we're ready for this task.

You're gonna calculate the answers to the following, and make sure that you simplify all of your answers.

You then need to find them in the table and write down which letter each gives.

And then you can rearrange your letters to make a word.

Remember, you don't have to worry about doing the word part of it if you don't want to.

You could just work out the answer to the questions.

But remember, if you can't find your answer in the box, it either means you've made an error or it means that you've not fully simplified your answer.

So you're gonna pause the video and then you're gonna give these a go.

Remember, it's no calculators.

I want to see all of the steps you've done, and I'd also like you to try and use that simplifying before method if you possibly can.

Good luck with these.

You can pause the video now, and I'll be here when you come back.

Now, let's move on to looking at question 2 in this task.

What's the missing digits in each of the following calculations? You know how to multiply fractions together, so I'm now gonna ask you to find the missing digits in those calculations.

Pause the video, good luck with those, and then come back when you're ready.

Super.

Let's check the answers then.

So A was 1/4, B-1/3, C-2/5, D-3/4, E-5/8, F-1/2, G-1/3, H-7/10, and I-3/5.

And that gave you the letters A, R, U, M, O, N, R, E, and T.

And if you rearrange those letters, you get the word "numerator." Did you manage to get that? Like I said, don't worry if you didn't.

The the most important thing is that you were able to multiply those fractions together and get the correct answers.

Now, let's take a look at the answers to question number 2.

The missing digits - A was 8, B-50, C-3, D-5, E-12, F-9, G-5, and H-7.

You've got all those right, did you? Well done.

Quick recap then to summarise the learning that we've done during this lesson.

When we multiply a fraction by a fraction, you multiply across the numerators and across the denominators.

Multiplying fractions is super easy.

Remember, do try and look and see whether you can simplify before you do that multiplication.

Taking a look at our example, we've got 3/5 multiplied by 3/4.

We multiply the numerators which gives us 9, and we multiply the denominators, which gives us 20.

Could we have simplified before here? No, we couldn't.

Well done with today's learning.

It's been fantastic.

I've really enjoyed working with you.

Bye, and I hope to see you again soon.