Lesson video

Hi-ya, my name's Ms. Lambell.

I'm really pleased that you've decided to join me today to do some maths.

I'm really looking forward to working alongside you.

Welcome to today's lesson.

The title of today's lesson is, "Checking and securing, finding fractions of an amount." This is within the unit, "Understanding multiplicative relationships with fractions and ratios." By the end of today's lesson, you'll be able to find a fraction of any given amount.

A keyword that will be using in today's lesson is reciprocal.

It's worth just recapping the definition of the word reciprocal.

The reciprocal is the multiplicative inverse of any non-zero number.

Any non-zero number multiplied by its reciprocal will be equal to one.

I've divided today's lesson into three separate learning cycles.

The first one will be looking at finding fractions of amounts, and concentrate on using a bar model to do this.

Then we'll move on to looking at fractions of amounts as a multiplication, and then in the final learning cycle, we will look at fractions that are greater than one.

Let's get started on that first one.

Here we go.

What fraction of each bar is shaded in each of the following diagrams? I'll put them on the screen one at a time, give you a moment to think about your answer, and then I'll reveal the answer.

Here we go.

Here's the first one.

It's 2/5.

The bar has been split into five equal parts, so we know we're talking about fifths, and two of them are shaded.

So 2/5 of the bar is shaded.

Here's the next one.

And that one's 4/9.

So again, the bar split into nine equal parts, and four of which are shaded.

And the next one.

That's 1/3.

Next.

Give you a bit longer on this one, there's a bit more counting to do.

11/12, now I'm wondering there, did you count that there were 12 altogether, and then count 11 purple ones, or did you just subtract one from 12? I wonder.

Next one.

And this one is 3/4, and the final one.

And that one is 10/15.

So we know how to identify what fraction of a bar is shaded.

We can use that now to help move our learning on.

What value does the whole bar represent? So this time I've shaded the entire bar, what value does that represent? It represents one, that means that each part is worth 1/5.

It's just like the first one that we looked at, that was split into five equal parts and I'd shaded two.

So we know that each part is 1/5.

Aisha says, "Could we use this bar model to help us find 2/5 of an amount?" Well, there's 2/5.

We know that from the previous slide, you did that.

Lucas is saying, "Yes you can.

We can change the value that the whole bar represents." So for example, the whole bar may represent 20.

Let's find out then what is 2/5 of 20? Here's my bar, and it represents 20, the amount we're finding a fraction of.

We share the 20 into five equal parts, because we want to find fifths.

There's my five equal parts, 20 divided by five is four.

Each part is worth four.

So Lucas says each part's worth four.

Yep, that's right, Lucas.

Well done.

Aisha says, "Yes, and as we want to find 2/5, we need to multiply four by two to give us two parts." So there's our two parts, there's our 2/5, we've got two lots of four, two multiplied by four, so it's eight.

Lucas says, "That means that 2/5 of 20 is eight," and that's right, isn't it? Calculate 5/8 of 26 pounds 40.

So Lucas wonders whether we could use it to answer this question.

So can we use that bar to answer this question? And Aisha says "Yes, the whole bar represents 26 pounds 40." So in the previous example it represented 20, exactly the same bar that this time is going to represent 26 pounds 40.

We need to divide that by eight.

Just have a think for me, why do we need to divide it by eight? Yeah, that's right because we are fining 5/8.

We need to split our bar into eight equal parts.

Each part is worth 3 pounds 30.

We want 5/8, so we need to find five lots of 3 pounds 30, or 3 pounds 30 multiplied by five, which is 16 pounds 50.

What is 2/3 of 2 3/4? Aisha says, "And this type of question, could we use the bar?" And Lucas says, "I don't see why not." Here's our bar representing 2/3.

The whole this time is 2 3/4, so we need to do 2 3/4 divided by three.

"How do we do this?" Lucas says.

And Aisha says, "Lucas, you remember, convert the mixed number into an improper fraction." "Oh yes," Lucas says, "and multiply by the reciprocal of the divisors." So we are going to change it into an improper fraction, which is 11 over four, and we multiply it by the reciprocal of three.

Remember I said that was gonna be important word today.

The reciprocal of three is 1/3.

And then remember when we're multiplying fractions, we multiply across the numerators, and we multiply across the denominators.

So that's 11/12.

We want 2/3, so we want two lots of 11/12, which is 22 over 12, which cancels down to 11 over six.

And then we'll convert that back into a mixed number, which is 1 5/6.

Aisha she wants to know, "What question does this bar model represent?" Lucas says, "It is split into nine equal parts." Let's just check.

Yeah, that's right, Lucas.

And two are shaded.

Yeah, right again.

And so it could be what is 2/9 of 6,300 grammes, certainly looks to me like that's what this bar model is representing.

What did you think? You agree? Brilliant, well done.

Now you are gonna have a go.

Which of the following bars is not set up correctly? Which one is not set up correctly? I'd like you to pause the video, decide which one is not set up correctly, and then when you're ready you can come back.

Good luck.

What did you decide? Hopefully you decided it was C.

The bar in C shows 4/8, not 3/8, so the only mistake there was to shade in one extra part of the bar model.

What is 3/5 of 350 pounds? There's my bar model that represents 3/5.

The whole this time is 350 pounds, and we want to work out the shaded part.

So we take our 350 and divide it by five, because we're working out five equal parts, which is 70.

So it's going to be 70 multiplied by three, which is 210.

Notice, this time I didn't write the values in the each of the boxes, so you can skip that step if you want to.

But if you prefer, you can actually write 70 pounds in each of the boxes.

You are ready to have a go at one on your own now.

Here you go.

Pause the video, when you've got your answer, come on back.

What did you get as your answer? Let's check.

This time the whole is representing 400.

Divide that by five.

That's 80.

We want four parts, so we want four lots of 80.

Here I've written it as 80 multiplied by four, remember that multiplication can be done either way around, and that's 320.

Now you can have a go at these questions.

I'd like you to pause the video, give these a go, and then you can come back when you're ready.

Good luck.

And the next lot.

Great work.

Let's check those answers.

Pause the video, check through your answers, and then when you're done, you can come on back.

Good.

Now we can move on to our next learning cycle, which is fractions of amounts as multiplication.

Aisha and Lucas are answering the following questions.

1/7 multiplied by 14, and 1/7 of 14.

What is the same and what is different about those two calculations? Pause the video, have a think.

Come up with some ideas, and then come back when you're ready.

Here are some examples of things that you might have written.

Both have one as a numerator.

Both have seven as a denominator.

They both use the integer 14.

One has a multiplication symbol, and the other says, of.

So the first three were things that were the same, and the final one was something that was different.

Here are Aisha's and Lucas' strategies for working out the answer to this question.

1/7 multiplied by 14, is one multiplied by 14 over seven, which is 14 over seven.

We know the fraction line represents division.

14 divided by seven is two.

And here is the other method.

1/7 of 14.

So using that bar model that we just looked at in that first learning cycle, the whole bar represents 14, and we're splitting it into seven equal parts.

What is this same and what is different about those strategies? Pause the video, come up with some ideas, and then come back when you're ready.

Let's have a look at some examples of things that you may have thought of.

Both divide 14 by seven, both have the same answer of two.

Only the second one used a bar model.

So again, the first two are two things that are the same and then the final one was something that was different.

Aisha and Lucas are answering the following questions, 3/5 of 20, and 20 multiplied by 3/5.

What is this same and what is different about these calculations? Have a think.

Now some examples of things that you may have said.

They both have three as the numerator.

They both have five as the denominator.

They both use the integer 20.

One has a multiplication symbol, the other says, of.

In the second one, the fraction is second in the calculation.

Did you come up with all of those? Well done if you did.

Here are their strategies for answering this question.

So Aisha and Lucas are showing us their strategies.

So we've got one here using the bar model, and then we've got multiplication as a fraction.

What is the same and what is different about these strategies? Some examples, both divide by five, both multiply by three, both get the same answer of 12, the second one has fewer steps.

Aisha is now saying, "Does this mean that multiply and of are the same thing?" And Lucas says, "Because we are using the same numerators, the same denominators, the same amount, and only multiplication and division, they are equivalent." Aisha says, "Yes, and multiplication and division have equal priority." And you should remember that when we looked at priority of operations.

We can see here actually they're the same thing.

But both multiplying by three, both of them are dividing by five.

Remember, a fraction is just an alternative way of showing a division.

That's why the two things are equivalent.

In maths then the word of and the process of multiplying are the same thing.

3/5 of 600 is exactly the same as 3/5 multiplied by 600.

2 4/5 of 8,600 grammes is equal to 2 4/5 multiplied by 8,600 grammes.

1/4 multiplied by half is the same as 1/4 of a half.

Knowing that fractions of amounts can be written as multiplications, we can multiply fractions efficiently.

4/9 of 81.

We know that this is equivalent to 4/9 multiplied by 81, which is four over nine, multiplied by 81 over one.

Remember, any integer can be written as a fraction with a denominator of one.

We multiply across the numerators, and we multiply across the denominators.

You don't need to write out all of those steps, but this is just to show you the process that we are going through.

Now we can look for a common factor between the numerator and denominator, so that we can make sure that we are calculating this efficiently.

We could rewrite 81 as nine multiplied by nine, so we've got four multiplied by nine, multiplied by nine over nine multiplied by one.

We can then simplify this.

So we know that nine over nine is one, so we multiply that by what's left, which is four multiplied by nine over one, and one multiplied by 36, is 36.

Again, you don't need to write out all of these steps every time.

Sort the following into the most efficient method.

So we're looking at whether we're gonna do the multiplication first, or whether we're going to do the division first.

Let's look at the first one.

3/8 of 64.

That's going to go in the divide first, because 64 is a multiple of eight.

So that division is nice and quick and easy for us to do.

Let's take a look at the next one.

5/12 of 57.

This is going to be multiply first.

57 is not a multiple of 12, so it's not going to benefit us by doing that division first.

3/4 of 48, where's that one going? Well done.

That's going in divide first, because 48 is a multiple of four.

Next one.

2/9 of 99.

Yeah, that's right.

It's going in the divide first, because 99 is a multiple of nine.

4/15 to 50.

Where's that one going? Good, that's multiply first, because 50 is not a multiple of 15.

3/7 of 58, where's that one? That goes in multiply first.

58 is not a multiple of seven.

9/11 of 55.

Well done.

Good.

Yeah, that's gonna in the divide first, because 55 is a multiple of 11.

Which of the following are equivalent calculations for 7/8 of 48? Pause the video, look through each of those, and then you can come back when you're ready.

Good luck with that.

Let's take a look.

A was a correct equivalent.

B was a correct equivalent.

C was a correct equivalent.

And D was also a correct equivalent.

They're all equivalent.

Did you work that out for yourself? Well done.

Are some ways easier to calculate mentally than others though? Think about that, which one is the most efficient way of working that out? I dunno about you, but I would do it using C, 48 divided by eight is six.

Six multiplied by seven is 42.

I can do that very quickly in my head.

Now you're ready to have a go at the next task.

You're going to find the odd one out in each row of the table.

It has a different answer to the others, so you're going to work out all of the answers.

One of them will be different.

You then take the letter that is alongside that calculation and then if you can, you're gonna have a go at rearranging those letters into a word.

As always, it really doesn't matter if you don't manage to rearrange letters into a word.

What's most important is that you are able to answer the questions.

Good luck with this.

You can pause the video now.

I'll be here when you get back.

Let's check those answers now then.

On the top one, they were all 18, apart from the third one that was 16.

The second row, they were all 24, except the final one, which was 26.

The third row, they were all 27, except for the second one that was 25.

The fourth row, they were all 48, except for the third one that was 49.

The fifth row, they were all 3/4 except, for the second one that was 3/8.

And the final row, they were all 27 over 32, except for the first one, which was 29 over 32.

And if you took all of those letters and you rearranged them, you've got the word, proper.

Now let's move on to that final learning cycle.

We're looking now at fractions greater than one.

What would a bar module look like for calculating 1 4/5 of 30? Let's take a look.

Here's our one, and we know that that is equal to 30.

The whole is 30.

Remember it's what we are finding, it's the amount we're finding a fraction of.

Here's a second whole, but this time we want 4/5 of that.

So there's our 1 4/5.

We know that the whole represents 30.

We're going to take the 30 and divide it into five equal parts, and then we're going to multiply that by four.

30 divided by five, and six multiply by four is 24.

We are trying to find the entire shaded part of the diagram.

So we know that the first bar was representing 30, the second one we've just worked out is 24.

So the answer, the whole entire part that's shaded is equal to 54.

And Lucas says, "How can a fraction of an amount be greater?" Aisha says, "We are finding more than one whole.

Multiplying anything by a number greater than one will give a greater product." And Aisha's right, isn't she? If we multiply it by anything that is greater than one, we always make the number greater.

1 4/5 of 30.

Here's our method for answering this question using multiplication.

Let's just check through each step to make sure that we're happy with each of them.

We've changed 1 4/5 into an improper fraction, which is 9/5.

We are multiplying by 30.

We are then going to multiply the numerator, which are nine and 30, and we're going to multiply the denominators.

Now here you'll notice, I haven't bothered to put the one as a denominator for 30, 'cause I know multiplying anything by one doesn't change the value.

So I've just left that as five.

Nine multiply by 30 is 270 over five, and that is 54.

Second example 2 2/3 of 18.

Change the mixed number into an improper fraction.

2 2/3.

You say 8/3.

Multiplied by 18.

Here notice that I've done it slightly differently.

Here I've noticed that 18 is a multiple of three, so therefore I can do my simplification before I do that multiplication.

18 over three is six, so I end up with eight multiplied by six is 48.

We could also have done that on the top example.

So it's always worth checking to see whether you can simplify before doing that multiplication, just to save you a little bit of time.

You'll get the right answer either way, so it doesn't matter.

Aisha and Lucas are calculating 3 3/5 of 45.

Here's Aisha's method.

Give you a moment, just have a look through what Aisha's done.

And here's Lucas' method.

Again, I'll give you a moment to just look through what Lucas has done.

What is the same and what is different about the two methods.

Here are some examples of things that you might have said.

They both get the same answer, but Aisha has partitioned the 3 3/5.

So we look at Aisha's method, she's done three multiplied by 45, and 3/5 multiplied by 45.

Which method do you prefer? Now I'm all about efficiency, so I definitely prefer Lucas' method.

Looks a lot shorter to me.

But does it really matter? No, we get the same answer either way.

So it's about choosing the method that's best for you, the one that you feel most comfortable with.

But try if you can to always use that most efficient method.

When might as Aisha's method be more useful though? So we've just said that Lucas' is the most efficient, okay, but when might as Aisha's method be more useful? Can you think of any examples? Aisha's method might be more useful when we are dealing with large mixed numbers.

So for example, 11 5/8 multiplied by 32.

Because when we convert 11 5/8 into an improper fraction, we end up with quite a large numerator, that we might prefer not to be multiplying.

Let's give this one a go.

What is 11 5/8 of 32? Oh, it's the example I just said.

Let's have a look at how we would use Aisha's method to work this one out.

11 5/8 multiplied by 32.

We're gonna partition the 11 5/8 into its integer part, and its fractional part.

So we are gonna multiply each of those separately by 32.

11 multiplied by 32, is 352.

Add five multiplied by four.

Now where's that four come from? Can you see why it's four? That's right, because 32 divided by eight is four.

We've done the simplification first, so that we don't have to do five multiplied by 32.

That's 352 add 20, which is 372.

Your turn now.

What is 12 4/5 multiplied by 25.

Pause the video.

Good luck with it.

Remember no calculator, come back when you've got your answer.

Good luck.

Well done, Partition the 12 4/5 into the integer part and fractional part.

This gives us 12 multiplied by 25, which is 300.

And then notice here, 25 divided by five is five, so that's why it's four multiplied by five.

So we get 300, add 20, which is 320.

Did you get that right? Well done.

Here you go then, here's some for you to have a go at now.

Pause the video.

Good luck.

Come back when you've got your answers.

I'll be here ready and waiting to see how well you've done.

You can pause the video now.

Well done for coming back, now let's check those answers.

A, 324.

B, 84.

C, 799.

D, 463 1/3.

And E, 531.

Five outta five you get.

Of course you did.

Now we're ready to summarise the learning that we've done in today's lesson.

You've done fantastically well.

We started off by using a bar model for finding a fraction of an amount and there's an example there.

We then went on to looking at, and we discovered that multiplication and of are equivalent in maths.

So sometimes it might be easier to think of a question as an of, rather than a multiplication.

And sometimes it might be easier to think of it as a multiplication rather than of.

The two are equivalent.

And then finally, when finding a fraction of an amount, the order in which you multiply and divide does not matter, because they have equal priority.

So it's worth just checking and seeing which is the most efficient way of doing it.

Thank you so much for joining me today.

I really enjoyed working with you, and I look forward to seeing you again sometime.

Thank you.

Bye.