Lesson video

Hi, my name's Ms. Lambell.

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

I'm really pleased.

Let's get started.

Welcome to today's lesson.

The title of today's lesson is Deepening Understanding with Multiplication of Fractions, and that's within our unit Arithmetic Procedures with Fractions.

By the end of this lesson, you'll be able to generalise and fluently use strategies to multiply mixed numbers.

Some key words that we'll be using in today's lesson are product, proper fraction, improper fraction, and mixed number.

A quick recap of what those mean to help us write this lesson.

A product is a result of multiplying two or more numbers together.

A proper fraction is a fraction where the numerator is smaller than the denominator.

An improper fraction is a fraction where the numerator is equal to or greater than the denominator, and a mixed number is an improper fraction that has been written into its integer and fractional parts where the fractional part needs to be a proper fraction.

I've split today's lesson into two separate learning cycles, the first of which we'll be looking at multiplying by a mixed number, and in the second, we'll look at comparing different strategies for doing this.

Let's get going on that first one.

Here, we have some missing numbers.

What I'm going to do is ask you to pause the video and have a go at working out what numbers you think need to go in each of the boxes.

You can pause the video now and come back when you've got your answers.

Good luck.

Okay, how did you get on with that? Let's take a look and see if you've ended up with what I had.

In the first box it was four.

We can see there that we had four lots of 12, 12 add 12, add 12, add 12 is four lots of 12.

The next one, we can see here we've got three lots of 12, so we must have multiplied by three and then we also have a six and what fraction of 12 is six and that's a half.

So altogether we've multiplied 12, by three and a half.

Just take a look at the next one.

Working out what goes in the box this time.

We are looking at 12 multiplied by three and a quarter.

We can see that we have our three lots of 12, three multiplied by 12, so we need to work out what goes in the box.

We need to know what is a quarter of 12.

Well, a quarter of 12 when I find quarters, I quite like to half and half again, half the 12 is six and half the six is three.

So the missing number here is three, and then the final one, what goes in the boxes here, maybe a little bit more challenging this one, but we can see we've got three lots of 12 signified by the three.

And then we've got a third.

We need to work out what a third of 12 is.

Remember when we're finding a third of something, we are splitting it into three equal parts, so we need to take our 12 and divide it by three to give us four.

We'll look at one of these again later on and see how that links to what we are going to be doing in today's lesson.

Want us to just quickly go back to when we were looking at multiplying together integers.

Andeep remembers that when we calculated something like this before, we used the area model.

So when we were multiplying integers together, we used the area model to help us to be able to do this.

Sofia is a little bit rusty and she says, can you help please? Can you remind me what that looked like? So at the moment, Sofia is not quite sure what Andeep means by the area model.

I'm sure she's just been a little bit forgetful, but let's have a look and we'll see and remind ourselves just in case you are a little bit rusty, like Sofia, what the area model is for multiplication and it looks like this.

So remember, this is a scaled down version.

This is not drawn to scale.

It's representing the area, but it isn't actually a scale drawing, which is absolutely fine because we need to just work out what the area of that is.

So I've written my 13 and I've partitioned it into 10 and three and 24 has been partitioned into 20 and four.

Now we're going to use this area model to help us to find the answer.

In the first box, we're going to multiply together 10 and 20, which is 200.

In the top right box, we're gonna multiply together 10 and four, which is 40.

In the bottom left box, we're going to multiply three and 20, which is 60, and in the bottom right box we're gonna multiply together three and four, which is 12.

So remember to find what goes in the box you just look at what's to the left and what's to the right and you find the product of those two things.

We then need to add all of those up.

That's our four separate areas.

So we add those up.

That gives us an answer of 312.

So hopefully now Sofia has remembered what the area model is, and if you were a little rusty, that's given you a little reminder too.

Sofia says, oh yeah, she remembers now.

So she does know.

She'd just forgotten.

Maybe she was having one of those mornings.

Let's take a look, like I said, we would take a look at one of those questions we looked at earlier to see how that links to what we are going to be doing today.

So we're going to take a look at this question and here we can see that we have 12, add 12, add 12, add four, and that is a representation of 12 multiplied by three and one third.

So here we have our three multiplied by 12, three lots of 12 and that could have been calculated, remember by partition in the 12 into 10 and two, three multiplied by 10, add three multiplied by two, but I expect most of you don't need to partition that number.

You know your 12 times table and you'd know that three multiplied by 12 is 36.

We then have got here the third multiplied by 12 because we were multiplying by three and one third.

So we've multiplied by three and now we find third of 12, which is four.

So remember as we went through that earlier, reminded you that finding a third of something means splitting it into three equal parts and 12 divided by three is four.

What's happened here then is we'd actually partitioned the three and a third into three and separately one third.

So we were already partitioning when we looked at those questions earlier.

Here we have a question two and 5/12 multiplied by 28.

Andeep says, do you think we could use the same method for this? So thinking about that area model, Sofia says, I don't see why not.

Let's have a look and see what it would look like.

So remember, we partition our numbers.

So here we would partition the integer part and the fractional part.

That's where our partition would go.

28 would be partitioned into 20 and eight.

Now we're going to find out what goes into each of our areas.

Remember, it's not to scale.

Two multiplied by 20 is 40; two multiplied by eight is 16.

Remember when we're multiplying a fraction by an integer, it's really important to remember that that integer has a denominator of one.

Our numerators will be five and 20.

Product of those is 100, and then our denominators would be 12 and one and a product of those is 12.

So this is 100 over 12.

The same thing then for the last box, we are going to do 5/12 multiplied by eight.

Five multiplied by eight is 40 and the denominator is gonna stay as 12 because we've multiplied it by one.

We now need to add all of those four separate elements together.

So 40 add 16, add a hundred over 12, add 40 over 12.

Notice I've not bothered to simplify those at this stage.

I'm going to leave my simplification until the end.

Let's deal with the integer part first.

So 40 add 16 is 56 and we know that 100 over 12, add 40 over 12 is 140 over 12.

Now I'm going to simplify 140 over 12 to 35/3 and then I've changed that into an a mixed number, which you are super good at now, and then I can add those together.

Yes, the area model is going to work.

It doesn't matter that we are finding areas that have fractional parts.

Calculate this product using the area model.

Let's just do this one again together.

Partition the one and seven eighths into one and seven eighths, so into is integer and fractional parts and 34 into 30 and four.

One multiplied by 30 is 30; one multiplied by four is four.

Remember here we're multiplying seven by 30 to give us 210.

It's gonna stay as eight because we're multiplying by one, so that doesn't change.

And then the final box, we're gonna do seven multiplied by four is 28 and again that denominator is going to stay as eight.

Then we can find the sum of those four parts, so we get 34, add 238 over eight.

Then simplify your fraction and then finish the calculation.

Here you may decide to change 238 over eight into a mixed number first and then simplify.

The order in which you do those two steps doesn't matter.

I would like you now to have a go at this.

I'd like you to spot the mistake.

Now, when I do spot the mistake questions, what I tend to do is not even look at the answer.

I do the question myself and then I see how my method or answer differs from the one that I've been given.

Pause the video now.

Come back when you've spotted that mistake and don't forget, I want you to spot and correct the mistake, so when you come back you'll have the correct answer.

You can pause the video now.

Good luck.

Great work.

Let's take a look and see where that error was.

The error was here and the reason I've put this in is because actually this is a really common error.

What this person has done is they have multiplied both the numerator and the denominator by 10.

So that means they've actually multiplied by one and so they've not changed the value of the two thirds.

They've not multiplied it by 10, it should have been 20 over three and I'm sure you've got that.

Now that's going to affect our answers over on the right hand side.

So 20 over 30 should have been 20 over three.

Now we will need to change two thirds, so that here is going to change into six and two thirds and then hopefully you spotted there was another error here.

This should have been two and two thirds.

Eight thirds is actually two and two thirds, not two and one third.

That will have affected the addition of three to 28.

The three should actually be nine and one third, and then the final answer, what is it? It's 37 and one third.

I wonder if you got that right.

I'm sure you did.

Let's just recap them.

A really common mistake that people make is to multiply, when multiplying by an integer is to multiply the numerator and the denominator by that integer, but if you do that, you are actually multiplying by one and not the integer you're trying to multiply by.

You are now ready to have a go at task.

Here, I'd like you please to calculate the answers and I'd like you to give all of your answers in their simplest form.

I'm just gonna leave you with one important thing before you go and remember that each integer has a denominator of one.

You can pause the video now and have a go at these.

Good luck.

I'll be here when you get back.

Well done.

Thank you for rejoining me.

Let's check your answers.

A 65, B 15 and 3/7 C, 11 and 1/12, D 50 and 2/7, E 45 and 1/3 and F 121 and 1/2.

So I did ask you to put those in their simplest form.

If you forgot to do that, just double check mine that your answers do simplify to the ones that I've given you here on the screen.

Now we're ready to move on to our second learning cycle during this lesson and we are going to be comparing strategies.

It is really important to remember in maths that there's not just one way of answering a question.

There are different strategies, some of which you may prefer; some of which are more efficient.

Let's take a look at what we're looking at in here.

We've got two and three quarters multiplied by one and two thirds.

Andeep says, can we use the area model with two mixed numbers? And Sofia says, yes we can.

So we know we can use 'em with integers.

We know we can use 'em and we've got a mixed number and an integer, so surely we must be able to use it when we've got two mixed numbers.

Let's take a look.

We'll partition each of these mixed numbers into their integer part and their fractional part, so two and three quarters and one and two thirds.

We're then going to find the area of each of the sections.

So two multiplied by one is two; two multiplied by two thirds is four over three.

Three quarters multiplied by one is three quarters and three quarters multiplied by two thirds is six over 12.

We then need to find the sum of those four separate parts.

I need to find a common denominator because I'm adding together fractions and my LCM remove my lowest common multiple of three, four, and 12 is 12, so I've converted each of those so that they are in twelves.

This gives me two and 31 over 12.

I need to convert the 31 over 12 into a mixed number and then finish off.

Now, I've gone through that quite quickly.

You might now need to pause the video and just go through each of those steps and make sure you understand them.

And if you don't, just rewind the video and go back and have a look at my explanation again.

I've just had a thought.

If we can multiply proper fractions by multiplying the numerators together and the denominators together, could we change these into improper fractions and do that? So fear's response is, oh yes, of course we could.

Let's check that we get the same answer.

So Andeep's thinking about the fact that actually when we were looking at multiplying fractions together, we started off looking at the area model and then we moved on and we found that there was a method which meant we just multiplied the numerators together, and multiplied the denominators together.

But remember, we understood why we were doing that.

We had our little visual to help us with that.

Let's take a look.

Here's the method we've just used.

The area model that's at the top.

I'm just gonna grey that out so it doesn't distract us for a moment.

Now we're going to take a look at Andeep's idea and that is to convert each of the mixed numbers into an improper fraction.

Two and three quarters is 11 quarters and one and two thirds is five thirds.

Now we know we multiply our numerators together and denominators together.

This is going to give us 55 over 12.

Then we need to convert that into a mixed number.

We see we get the same answer, just have a think.

Which method do you prefer? Now, personally I prefer the second method because it looks to me like there's a lot less work to do, so it's more efficient.

But remember, it doesn't matter which method do you use just as long as you are confident with using at least one of them.

Sofia has decided that she doesn't need to draw the area model out.

Here are her work-in's for this calculation, two and four-fifths multiplied by three and two sevenths, so she's partitioning, she's using that partitioning method.

She's done two multiplied by three at four-fifths multiplied by two sevenths.

She's got six, add eight over 35, and that is six and eight 35th.

Why will Sofia's method not work? Notice I have highlighted there the word not.

It doesn't work.

Your job is to work out why.

Explain to me why it doesn't work.

Pause the video and have a think about that.

I look forward to hearing what you think when you come back.

What did you decide? The problem here is Sofia has actually not found the product of three and four fifths and the product of two and two sevenths.

Think back to the area model, visualise it in your head.

We had to find the area of four separate parts and here Sofia has only found the area of two separate parts.

Now this is a common mistake.

It's really important if you are going to use the area model and you are not going to draw it out, you need to remember that that area model for two mixed numbers is going to be split into four parts, so you must make sure that you are adding together four parts at the end.

If you think you might make that error, then you may decide that actually converting into improper fractions and multiplying and then simplifying at the end might be a better method for you.

One disadvantage converting mixed numbers to improper fractions is that you can end up with very large numerator and denominators.

Remember that you can write the new numerators and denominators using factors to simplify before multiplying, so it is a disadvantage, but actually we have strategies to deal with that.

Let's take a look at this one, two and four sevenths multiplied by three and two sevenths.

We're going to write them as improper fractions.

We've decided that that's the method we are going to use, so I've written them as improper fractions and you are super good at that.

We are then going to write each of those using the factor pairs, so 14 is two multiplied by seven, and then the others they're prime numbers and so therefore can't be written as any factor pairs.

We're then going to write, arrange the common factors together and here we can see the only common factors we have are seven.

I've now made my calculation much, much easier because I'm doing two multiplied by 23 over five, so I've multiplied them together and then my final step is to convert them back into a mixed number.

Now all of those steps that we've just been through, you've done in previous lessons, so you should be really confident with each of those steps.

Let's just recap what we did.

We wrote to them as improper fractions.

We then wrote each of the numerator denominators using its factor pairs to see whether there were any common factors we could cancel.

We then cancelled the seven over seven.

Why did we do that? Yeah, because we were multiplying by one, so that doesn't need to be there.

Then we changed it into a mixed number at the end.

This method does have huge advantages.

It's worth remembering that middle step, that idea of finding those factors and cancelling out those common factors before you move on to doing your multiplications.

You can use any of the methods we've looked at.

Obviously some may be more efficient in different situations, so for example, if you're doing three and three quarters multiplied by 24, you might prefer to use the area model and partition that three and three quarters into the integer and fractional part, so into three and then three quarters.

Three and three quarters multiplied by five and five, six.

You might prefer to confer into improper fractions.

Remembering to look then once you've done that, to look for common factors that you can cancel out.

We'll have a go at this one together and then you'll be ready to have a go at the one on the right hand side by yourself.

Here I'm going to use the improper fraction method.

I've converted them into improper fractions.

I'm then going to write each of my numerators and denominators using its factor pairs.

Prime factors is best, remember, because then we can cancel out as many as we possibly can.

Then I'm rewriting so that all of my common factors are at the beginning.

Remember, you don't have to do that step, but I get a bit confused about what's common to both if I don't do that.

I can then see that I've got two over two and two over two, which are going to cancel out to leave me with a numerator of three multiplied by seven and a denominator of five, which is 21 over five, and now I need to convert that back into a mixed number, which is four and one fifth.

Now, you can have a go at this question.

Remember, no calculators, pause the video, come back when you've got an answer and try to be efficient by using the prime factors to cancel down before you do that multiplication.

Okay, pause the video now.

Well done.

Let's check that answer.

Step one, convert to improper fractions.

Step two, write each numerator and denominator as a product of its prime factors or at least writing using factor pairs.

Here, I didn't need to do any rearrangement because I can see my common factors are three and three and they're already at the beginning, so my three will cancel out with my three, leaving with three multiplied by two, multiplied by two multiplied by two as the numerator and five as a denominator.

That gives us 24 over five, and then we'll convert that back into a mixed number.

Did you get that right? You did, well done.

Task B now.

You're going to calculate the following and like I've said there, what I'd like you to do is try and use the most efficient method.

You could use the area model or you could use the improper fraction model.

If you are using that, remember the most efficient method is to then, once you've rewritten them as improper fractions, is to write the numerators and denominators, add as a product to their prime factors, and that's the one tiny little bit I will allow you to use a calculator for.

Good luck with these.

Pause the video and come back when you're ready.

Well done.

We'll check those answers.

Here they are.

A was four and 1/6; B three and 23/24, C, four and 3/8, D four and 4/5, E one and 1/2, F, three and 1/2.

How did you get on with those? Great work.

Let's now summarise the learning that we've done in today's lesson.

We started off by looking at partitioning and the idea that we can use the area model to multiply mixed numbers together, and we can see there that we've got an example of how we do that.

We split each of the mixed numbers into its integer and fractional part.

We then find the area of each of the sections.

Remember, there will be four.

You must make sure you are summing all four of those at the end.

Also, mixed numbers can be multiplied by converting into improper fractions.

Remember, if you're going to use this method, try to be efficient and then try to write each numerator and denominator as a product of its primes, so that you can do that cancelling and simplification before you get into some really big numerators or denominators.

Well done on today's learning.

I've really enjoyed you joining me and I look forward to seeing you again.

Bye.