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 Securing Understanding of Multiplication with Fractions.

By the end of this lesson, you'll be able to multiply unit, non-unit, and improper fractions.

Some key words that we'll be using during today's lesson, so it's worth having a quick recap as their meaning and those are product, commutative, and prime factors.

I've split our lesson today into 2 separate learning cycles.

In the first 1, we're going to look at comparing products, and in the second 1, we'll look at multiplying 2 or more fractions.

Let's get going with that first learning cycle.

Here we've got Lucas and Sam.

Now, Lucas has a piece of material which is 6 metres long and 1 metre wide, so 6 metres long and 1 metre wide.

What is the area of Lucas' material? How do we find the area of a rectangle? Yeah, we multiply the length by the width.

The area of Lucas' pieces material is 6 metres squared.

Lucas says, "I need to cut my material into 2 long strips." Dotted line there showing where Lucas is going to cut its material.

What calculation will represent the area of Lucas' new strip? What calculation? We know to calculate the area of a rectangle, we multiply the length and the width.

Here, we would do 1/2 multiplied by 6.

We can see if Lucas cuts his piece of material horizontally, each part would have a width of 1/2 a metre.

Sam says, "But 1/2 of 6 is three and when you multiply a number it should get bigger." We'll take a look at what Sam has said.

So Sam says, "When you multiply a number, it should get bigger." Will the product of 2 numbers always be greater than either of those numbers? What did you think? Can you think of any pairs of numbers when that is not the case? So could you give me a non-example? I wonder what you said.

Let's take a look at the following.

1/2 multiplied by 4, that means 4 or lots of 1/2, doesn't it? I'm going to look at that with a representation and I've decided to use circles.

There's 1/2 of a circle and we want 4 lots of that.

We can clearly see that we've created 2 complete circles, 2 holes.

What about if we did 1/2 multiplied by 2? 1/2 multiplied by 2 is 2 lots of 1/2.

There's my diagram and we can see that we've created 1 hole.

1/2 multiplied by 1, that means I want 1/2, so that's just 1/2.

Notice what I'm doing as I go down through these calculations, I'm dividing by 2.

My next calculation is going to be what do you think? It's going to be 1/2 multiplied by 1/2.

Now, what does that look like as a diagram? There's 1/2.

I want 1/2 of that.

There's my 1/2 of that, the dark purple section.

Now, I can see of my whole circle, I've shaded in 1/4.

So this is 1/4.

Remember we divided by 2.

My next calculation is going to be 1/2 multiplied by a quarter.

So I want to know what 1/2 of 1/4 is.

So 1/4 is shaded, 1/2 of that is the darker purple section.

And I can see now of the entire circle, 1/8 is shaded.

When did the products start being less than 1/2? When we started multiplying by a number less than 1.

We noticed there we were bigger than 1/2, bigger than 1/2.

Then when we got to 1, it was exactly 1/2 'cause we know multiplying by 1 doesn't change the value of something.

But then when we started multiplying by a fraction less than 1, we ended up with a product that was smaller.

The product of two numbers where one number is between zero and one, we'll give a product less than the larger number.

Let's have a look at some examples.

So here we've got 0.

04, that is between zero and one, multiplied by 5, your answer is going to be less than 5.

5 multiplied by 1/8.

Here we're multiplying 5 by a number that is between zero and one, therefore our answer is going to be smaller.

The product of two numbers where both are between zero and one will give a product less than either of the numbers.

An example here, 0.

04 multiplied by 0.

5, both of those are between zero and one, therefore the product of those is going to be less than both 0.

04 and 0.

5.

And now a fraction example, 1/3 multiplied by 4/5.

Both of those fractions are between zero and one.

The product of those then has to be less than both 1/3 and 4/5.

We're going to now sort these into the following groups.

I just want you to look at the calculation.

I don't want you to work out an exact answer.

I just want you for each one to decide whether you know the answer is going to be less than 10 or whether it's going to be greater than 10.

Remember it's going to be less than 10 if we're multiplying by something between zero and one.

It's going to be greater than 10 if we're multiplying something by something above one.

Let's take a look at the first one.

Where do you think this is going to go? 10 multiplied by 7/10.

Remember I don't want you to work it out, I just want you to use the knowledge that we've gained during the first part of this lesson.

That's going to be less than because we're multiplied by 7/10 and 7/10 is between zero and one.

Next one, 6 and 1/7 multiplied by 10.

That's going to go in greater than.

We're multiplying 10 by something greater than one.

So therefore the product will be greater than 10.

Next one, 10 multiplied by 2 and 4/5.

Greater.

We're multiplying by 2 and 4/5, which is greater than 1, therefore the product has to be greater than 10.

And the next one, 8/9 multiplied by 10.

Yep, great.

You spotted that that's gonna go in the less than 10 because we are multiplying 10 by something that's between zero and one.

8/9 is smaller than 1, but greater than zero.

And the next one, 20/19 multiplied by 10.

That goes in greater than 10.

Now, I tried to catch you out there, but I'm sure I didn't.

20/19 is actually greater than 1, isn't it? It's 1 on 1/19th.

So we need to look carefully at the fraction that's being multiplied by and not just assume that because we're multiplying by a fraction that the answer is going to be smaller.

We need to look at that fraction very carefully and we can see here it's an improper fraction, therefore it must be greater than 1.

Now this one.

Hopefully a fairly easy one for you.

We know that 1/9 is greater than zero, less than one, so it's gonna go in less than 10.

And one final one, 10 multiplied by 3/2.

And that's gonna go in greater than 10.

So hopefully if I did trick you with the 20/19, I didn't with this final one.

3/2 is actually 1 and 1/2.

So we are multiplied by something greater than 1, therefore the product will be greater than 10.

Well done with those.

Now, we can do this quick check for understanding.

So like you've just done with that previous task, you're just going to decide whether it is going to be greater than or less than, whether the product of the two things is going to be greater than or less than the integer in the calculation.

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

Great work.

Let's have a look then.

You should have 25 multiplied by 6/5 is greater than 25.

4/9 multiplied by 10 is less than 10.

12 multiplied by 1 and 1/10 is greater than 12.

And 12/11 multiplied by 111 is greater than because 12/11 is an improper fraction so therefore greater than 1.

Now, I'd like you to consider what is the same and what is different about these two calculations? So I'm gonna pause a moment and let you have a think, what's the same and what is different? We're gonna have a look at these with some bar models.

So let's look at the first one.

There's my four, my four units and I want to multiply by 1/3.

So I've split each of my units into three parts to represent the thirds.

I want four lots of third.

Remember multiplication, sometimes it's useful to think of it as lots of.

I want 4 lots of third.

So I've got 1/3, 2/3, 3/3 and 4/3.

That's 4 lots of 1/3.

That bar model is representing the value of 4 multiplied by 1/3.

Here's my 4 again, but this time it's asking me to find 1/3 of 4.

So I need to split my 4 into three equal parts and I've shown you that here with the purple sections.

Now, I'm going to move those purple sections down onto my bar model that I want to represent 1/3 multiplied by 4.

Hopefully, now you can see that, actually, those 2 things are equivalent.

This shows us that it doesn't matter which way rank we write the product, the result is the same thing.

4 multiplied by 1/3, it's equivalent to 1/3 multiplied by 4.

And this diagram shows us that clearly.

Lucas and Sam have been given this fact from their teacher.

8/15 multiplied by 465/500 equals 248/500.

Lucas says, "I know that 4/15 multiplied by 465/500 is equal to 124/500." And Sam says, "How did you do that so quickly?" He did do it super quickly.

Wonder how? He says, "we know that 4/15 is 1/2 of 8/15." Yeah, that's right, isn't it? 4/15 is 1/2 of 8/15.

So Sam says, "It must be 1/2 of 248/500.

So you were right." Well done, Lucas.

What a superb spot there.

You didn't need to work out the actual answer.

You could use that information that you've been given from your teacher.

Well done.

I'd like you to write down as many different products as you can using this fact.

Like Lucas did on the previous slide, I want you to use this fact to write down as many different products as you can and their answers, but I don't want you to go through the whole process of multiplying them together.

I want you to use that little strategy that Lucas used on the last slide.

Pause the video, now come back.

I'd like you to get at least five.

Impress me with more if you want to.

Good luck.

Great work, well done.

How many did you come up with? And these are just some examples.

I'm not gonna read them out because you may have different ones.

So I'm just gonna pause and let you check and see whether you've got any of those that I've got.

Remember, if you've got one that isn't on your screen, what you can do is you could get your calculator out, you could do the multiplication and you could check your answer.

Remember though, if you use the fraction button on your calculator, it will give you the simplest form, so you might just need to find the products of the numerator and the denominators separately.

This is an interesting one.

How do you think I found that one? I wonder if you came up with that one as well, but if you didn't, just have a think for me.

How did I find 20/25? I found the sum of 134/1250 and 536/1250.

Why do you think I found the sum of those? Yeah, that's right because they were the products before 25 and 16/25.

And we know that the sum of those is 20/25.

So I could sum those 2 separate products.

You are now gonna have a go at some of these.

Given that 8/35 multiplied by 175/600 is equal to 140/2100, which of the following are true? You can now pause the video, check back with me when you've got your answers.

Good luck.

Now, let's check those answers.

A was true.

1/2 of 8/35 is 4/35.

So we have 140/2100 to give 70/2100.

And D was true because 2/35 is a quarter of 8/35.

So we found a quarter of 140/2100, which is 35/2100.

Well done If you identified that those two were true.

Is it possible to tell which of the following has the greater product without calculating an answer? So I want us to get really good at what I like to call number sense, just having a sense about numbers and being able to answer questions like this without actually working out an answer.

So I'm just gonna give you a moment to think about that.

If we draw an area model for each of these, they would both have one part shaded.

Which product would the hole be divided into smaller parts? 1/6 multiplied by 1/7.

Because it has more parts, and so remember if something has more parts, those parts will be smaller.

We therefore know that 1/5 multiplied by 1/6 is greater than 1/6 multiplied by 1/7.

Is it possible to tell which of the following has the greater product without calculating the answer here? What do you notice about the numerator of each product? They're the same.

What do you notice about the denominators of each product? Again, they're the same, aren't they? So, therefore, those must be equal.

And this one, is it possible to tell which of the following has the greater product without calculating the answer? Both calculations use 2/3.

The first calculation multiplies by 2/5, which is less than one.

and we know from our previous learning, but if we multiply by a number between zero and 1, the product will be smaller.

So here the product is going to be smaller than 2/3.

And in the second calculation, we are multiplied by 5/2, which is greater than one, so therefore the product is going to be greater than 2/3.

So we know that 2/3 multiplied by 2/5 is less than 2 1/3 multiplied by 5/2.

Now, you're gonna have a go at some questions independently.

You're going to use this fact to answer the following questions.

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

Good luck.

Well done.

And then question number two, without calculating, decide which is the greater product in each pair.

So remember here we're not calculated, we are just using our number sense to decide which one is the bigger product.

And then I'd like you to write those six products in order from the smallest to the largest.

Good luck with that.

Pause the video now.

Well done.

Now, let's check those answers.

Here we have the answers to question number one.

A, 170/5049; B, 680/5049; C, 85/5049; D, 850/5049; E, 255/5049; and F, 935/5049.

How did you get on with those? You got them all right, well done.

Let's have a look at these then.

So the first pair were equal.

The second pair, the one on the left was greater.

And the third pair, the one on the right was greater.

And then we write those products in order from smallest to largest.

You should have 1/10 multiplied by 1/9, then 1/8 multiplied by 1/10, and then 3/5 multiplied by 4/19 and that's actually equal to 4/5 multiplied by 3/19, then 4/7 multiplied by 3/8 and then 3/8 multiplied by 7/4.

Well done on those.

We can now move on to our next learning cycle.

We're gonna be multiplying two or more fractions together.

So earlier, we looked at these two calculations, 4 multiplied by 1/3, which is 1 and 1/3 and 1/3 multiplied by 4.

And remember those diagrams I showed you, we could see that those two things were equal and they were both 1 and 1/3.

What was the product as an improper fraction? The product, remember, is the result of multiplying two or more numbers together.

So the product here is 1 and 1/3.

And as an improper fraction, that is 4/3.

Therefore we know, sorry, that 4 multiplied by 1/3 is 4/3.

Hmm.

How does this link to our generalisation that a/b multiplied by c/d is equal to a multiplied by c/b multiplied by d? Just have a think about that moment.

Any integer, remember, can be written as a fraction with the denominator of 1.

We could rewrite 4 as 4/1 multiplied by 1/3.

By doing this, we can clearly see that we are multiplying the numerator and we are multiplying the denominators to give us 4/3, which was 1 and 1/3 It's really important that we remember that any integer could actually be written as a fraction with a denominator of one.

That's really important to remember and often something that we forget.

Let's calculate 15 multiplied by 1/4 multiplied by 2/5, or we know that 15 could be written as 15/1 and we're gonna multiply that by 1/4, multiply by 2/5.

So when we're multiplying fractions, we multiply the numerator and the denominators.

This will give us 30/20.

I'm gonna simplify that to 3/2 and then I'm gonna convert that because it's an improper fraction into a mixed number, giving me 1 and 1/2.

Lucas and Sam are calculating this, 12 multiplied by 2/3 multiplied by 5/6.

Here's Lucas' workings.

I'm gonna pause a moment and let you have a look through and see what Lucas has done.

And here's Sam's workings.

So again, I'm just gonna pause a moment to give you a chance to look through and see what Sam decided to do.

Now, you've looked at both of those methods.

I'd like you to consider for me what is the same and what is different.

Their methods are the same until we get to 120/18.

Then Lucas simplified and then converted to an improper fraction to a.

Sorry, converted the improper fraction to a mixed number and Sam changed it to a mixed number and then simplified.

So actually, it doesn't matter which way round you decide to do it.

You can simplify first or you can simplify after you've converted to a mixed number.

We're gonna do this one together now and then you'll be ready to have a go at one independently.

Remembering that 6 is 6/1.

So I've written my numerators are 3, 6, and 2, so I'm going to find the product of those and then my denominators are 5, 1 and 3, so I'm going to find the product of those.

This gives me 36/15.

I've decided to simplify before converting my improper fraction into a mixed number.

So I've got 12/5 and then I've converted that into a mixed number, which is 2 and 2/5.

You are now gonna have a go at this one, so pause the video and then come back when you're ready.

Let's check that answer.

1, 3 and 4 are the numerator and 5, 4 and 1 are the denominators.

Did I need to put the 1? No, I didn't need to, but I quite like to because then I'd know I've got the correct number of numerator in the correct number of denominator, so I know that I've not made any errors.

Now, we can calculate that.

That's 12/20 and simplified is 3/5.

That's a proper fraction, so no need for converting.

Now, Sam and Lucas are calculating this, 6/11 multiplied by 1/6.

Here's Lucas' method and here's Sam's method again.

What is this same and what is different about those two methods? Let's take a look.

They have both got an answer of 11.

What? Sorry, 1/11.

Lucas multiplies and then simplifies and Sam uses factor pairs to simplify first.

So if we look at Sam's method, he's written the numerator and the denominators, but then he's reordered the denominator to say 6 multiplied by 11 rather than 11 multiplied by 6.

And the reason Sam's done that is because he knows that 6/6 is 1, so therefore we can cancel out the 6/6 because multiplying by 1 doesn't change the value of something, remember, which gives us 1/11.

Why might finding factors of numerator and denominators be useful here? I've written 15 as 3 multiplied by 5 and 77 as 7 multiplied by 11, 91, I've written as 7 multiplied by 13.

And 225, I've written as 3 squared multiplied by 5 squared.

I've now rewritten that, taken it out of exponent form.

And I have also collected my common numerator and denominators to the beginning of my fraction, just so I can clearly see where I've got a common numerator and denominator.

And I can see here 3/3, that's one.

5/5, that's one.

7/7, that's one.

I now need to just look at what's left.

So the answer is going to be 11/3 multiplied by 5, multiplied by 13, which is 11/195.

When we've got small numerators and denominators, there might not be any benefit to doing what we've just done here by finding those common factors, but certainly when we get to much larger enumerators and denominators, we probably would because it's gonna take us a while to just work out 15 multiplied by 77 and then 91 multiply by 225.

And we can see here that we don't actually need to do that because we can use our common factors before we do the calculation.

Lucas and Sam, they're listening into what we've been doing and they're thinking about factor pairs.

So they've decided actually, yeah, I've got some quite big numbers here.

I'm gonna try and use those factor pairs to help me with this calculation.

Let's have a look and see what they've done.

Lucas says, "I'm going to use 8 for 16 and 40." So looked at 16 and 40 and said, "Right, I'm going to use the factor of 8 because it is common to both." And Sam decides that they're going to use 4 for 16 and 40.

Whose method do you think is going to be the most efficient? Well, let's take a look.

Lucas' method, Lucas decided that he was going to take a factor of 8, so he was going to write 16 as 8 multiplied by 2 and 40 as 8 multiplied by 5.

Then he's rearranged his numerator and denominator with the common factors at the beginning.

That just makes it a little bit clearer to see what's common.

You don't have to do that step.

And then we can see that we are left with 2 and then 5 multiplied by 11 on the denominator.

Sam's method, so Sam decided that was going to use the common factor of 4, so rewritten 16 as 4 multiplied by 4 and rewritten 40 as 4 multiplied by 10.

Done the same thing as Lucas.

So collected those common ones together at the beginning of our enumerators, denominators then cancelled out the 3/3 and the 4/4 because that's just the same as multiplied by 1, which gave 4/110, which was 2/52, sorry, 2/55.

Whose method was most efficient? Lucas' method was more efficient.

Lucas did it in one less step.

But why was it more efficient? Because we want to be efficient, don't we? And therefore we need to understand why Lucas' method was more efficient.

What do you think? The reason was because he used the highest common factor of 16 and 40.

That meant that the simplification had happened before, which is what happened with Sam's method.

Lucas and Sam are now looking at this calculation.

Busy bees they are today.

25/52 multiplied by 39/75.

We can see here we've rewritten 25 is 5 multiplied by 5 and 39 is 3 multiplied by 13, 52 is 2 multiplied by 26 and 75 is 25 multiplied by 3.

So that's the method that Lucas has decided to use.

And here's Sam's method.

If we look at Sam's method, we can see that Sam has exactly the same numerator as Lucas in their calculation, but if we look at the denominator, we can see that instead of writing 2 multiplied by 26, Sam has written 2 multiplied by 2, multiplied by 13, which is also equal to 52.

And then a 25 multiplied by 3 is actually written that as 3 multiplied by 5 multiplied by 5.

Whose method do you think is most useful for completing the multiplication? I think it's Sam's method.

I think Sam's method is most useful.

What do you notice about Sam's method? They've written each number using its prime factors.

So 52 as a product of its prime factors is 2 multiplied by 2, multiplied by 13, not 2 multiplied by 26.

Now, we're gonna take a look at why Sam's method is more useful.

Here's Lucas' method.

and I'm gonna rewrite the numerator and denominator with the common factors first.

Then we can see here we've got one and we can see what we're left with.

If we take a look at Sam's method, let's rewrite them.

So the common factors are at the beginning, we can see here we've got common factors of 3, 5, 5 and 13.

Therefore we're only left with nothing on the top, nothing on the top.

That's interesting.

Do I put 0/4? No, you're right.

It's 1, isn't it? Because, really, I could write that and multiply by 1, so it's 1/4.

Now, Lucas would've got to 1/4, but it would've taken a lot, lot longer.

So it's always best if you can rewrite the numbers as a product of their prime factors.

And if you need to revisit that, you could always go back to that video.

You could pause this one, go back to that video and then come back when you reminded yourself how to write something as a product of its prime factors.

Now, let's have a go at this one together.

I'm going to write each of the numerator and denominators as a product of its prime factors and I use the calculator to do that.

If you need to, you could go back to that video and remind yourself how to use the prime factor button on your calculator.

So I've written them as their prime factors.

I've then combined my numerator and my denominators.

Next, I've rewritten my numerator and denominators with all my common factors at the beginning.

You don't have to, I just find it a little bit easier to see which ones I've got that are in both the numerator and the denominator, which are 2, 3, 3, 11 and 13.

And now we can work out the answer.

It's going to be 3/55.

We've got 3 left as the numerator and then 5 multiplied by 11 is 55.

Notice how easy that question became once we'd written each of those numbers as a product of its prime factors.

Now, you are ready to have a go.

Use the prime factorization button on your calculator to write each of those numerator and denominators as a product of its prime factors and then cancel any common factors you can and then see what you're left with.

Good luck.

Come back when you're ready.

You can pause the video now.

Here are those written as product to their prime factors.

Here's my rearrangement.

And we can see then we're just left with 13 on the numerator and 5 and 17 on the denominator given us 13/85.

Now, you're ready for task B.

And I want you to think here about efficiency and I'd like to use the most efficient method to answer these questions.

Good luck with those.

Remember no calculators.

I want to see all steps of your working out, all of your answers in their simplest form.

Good luck.

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

I'll be here waiting.

Well done.

Here are the answers.

A, 24; B, 3 and 1/3; C, 1/5; D, 1/9; E, 11/13; F, 6/121; G, 1/22; and H, 3/10.

How did you get on with those? Well done.

Now, we can summarise the learning that we've done during today's lesson.

And there was a lot there.

We've done a lot.

Well done.

The product of two numbers where one number is between zero and one will give a product less than the larger number.

So for example, 5 multiplied by 1/8 is less than 5.

The product of 2 numbers where both are between zero and one will give a product less than either of the numbers.

So 1/3 multiplied by 4/5 is going to be less than both 1/3 and 4/5.

Any integer can be written as a fraction with a denominator of one.

And this is useful when we are multiplying fractions and factors can be used to make the multiplication of fractions more efficient.

And there's an example there of one that we did in the lesson.

Like I said, this lesson, there's been a lot of outstanding learning happening and I'm really pleased you managed to stick with me to the end.

Thank you for your time and I'll see you again soon, I hope.

Bye.