Lesson video

Hello, there, my name is Mr. Tilstone.

I'm a teacher.

I teach all of the different subjects, but the one that I enjoy the most is definitely math.

One of my favourite parts of math is percentages.

So it's a real treat and a real pleasure to be here with you today to teach you this lesson, which is all about percentages.

If you're ready to begin, let's begin.

The outcome of today's lesson is this.

I can solve problems where a percentage part is known but the whole is unknown.

Our keywords, we've just got the one.

My turn, percentage, your turn.

What does percentage mean? A percentage is a proportion of a whole.

Our lesson is split into two parts, two cycles.

The first will be finding a missing whole and the second missing whole problems with two steps.

Let's begin by finding a missing whole.

In this lesson, you're going to meet Aisha and Lucas.

Have you met them before? They're here today to give us a helping hand with the math.

Aisha loves reading.

She's read, get this, 625 books in the library.

Oh my goodness.

Well done, Aisha.

That's fantastic.

"I've read," she says, "25% of the books in the school library." Hmm.

So we know how many books she's read and we know what percentage that is, but we don't know yet how many books there are in total in the library.

"I wonder how many books are in the library altogether?" Wonders Lucas.

So yes, what's 100% of the books? How many? If you know the percentage parts and size of the part, you can calculate the whole.

Bar models are brilliant for this sort of thing.

We can see them math very clearly when we use a bar model.

So she says, "I've read 25% of the books in the library." And you can see that bar model has been split into four equal parts and the middle part.

So we could say that each one is worth 25%, and we know that 25% is 625.

That's the part that Aisha has read.

We're trying to find out how many books there are in the library altogether.

How can we use that? What would you do now with that information? Look at the bar model.

Well, we could think of three other 25% sections too.

Each of them will be worth 625 bucks and then they can use multiplication.

We could add them all together.

That's not very efficient, but multiplying is.

So 625 multiplied by 4 will give us the total number of books in the library.

To multiply by four, here's a good tip.

You can double and double again.

I bet you knew that.

So 625 doubled is 1,250, and double that is 2,500.

That's how many books there are in the library altogether.

And she's read 25% of them.

That's very impressive.

Aisha has read 20% of her book.

She says, "I've read 45 pages of my book so far." Hmm.

45 pages of Aisha's book is equal to 20% of the whole book.

So what don't we know? We dunno how many pages there are in the whole book but we could work it out.

This is what we know.

So we're using a bar model again.

This time it's showing 20%.

So it's split into five equal parts for the middle part and each one's worth 20%.

So 20% is 45.

What would you do next? What's your next step to find the total? That's the part that Aisha has read.

And then we can use 20%, four more times.

Each of them worth 45 pages.

So what can we do now? Again, we can use multiplication, 45 multiplied by 5.

To multiply by five, here's a good way to do that.

We can multiply by 10 and halve the answer.

Have you ever done that before? Let's do that.

So 45 multiplied by 10.

That's nice and easy, isn't it? 450.

And then half of 450 is 225.

That's a really good way to multiply by five.

No need to use a written method.

There are 225 pages in Aisha's book altogether.

That bar model really was helpful, wasn't it? Let's use it again for this one.

Lucas starts watching an episode of "Rocket Girl".

He says, "I've watched seven minutes so far, which is equal to 10% of the episode." And you may notice our bar model is split into 10% sections.

So seven minutes of the episode is equal to 10% of the whole episode.

Now what don't we know here? We don't know the total running time of the episode, but we do know 10%.

What would you do now? What's your next step? That's the part that Lucas has watched.

100% which is the whole is equal to 10 lots of 10%.

So what could you do now? You could multiply 7 by 10.

7 multiply by 10 is equal to 70.

So the episode is 70 minutes long.

Once again, I think that bar model really helped to bring that math alive.

Let's do a little check.

Aisha starts watching an episode.

How long is it? Let's have a look.

She says, "I've watched 28 minutes so far, which is equal to 50% of the episode." What could you do here? Is there a useful bar model that you could draw to represent that math? And Lucas says, "Could you use a bar model just like we've done before to help you work out the answer?" How could you use a bar model? What could you draw? Pause a video and off you go.

How did you get on? Let's have a look.

Well, you could draw a bar model that looks a little something like this.

Because it's 50%, we could split the bar into two parts.

So 50% is 28 minutes.

That means another 50% is also 28 minutes.

So we can double.

Double 28 is equal to 56.

So the episode is 56 minutes long.

Well done if you've got that.

You're on track and you are ready for the next part of the learning.

Lucas thinks of a number.

Hmm.

"I'm thinking of a number.

20% of my number is 31." He says, "What is my number?" Hmm, I wonder if a bar model could be used here.

"I'm going to work out what number Lucas is thinking.

I'll start with a bar model." Good idea, Aisha.

Showing the part that Lucas is thinking of.

Well, here's the bar model, it's 20%.

So we can split that 100% into five parts like that.

20% is 31.

What do you do now? What's your next step? Because we need to work out what the number was.

The original number was that he's thinking of.

Well, 100% which is the whole is equal to 5 lots of 20%.

So we can multiply the 31 by 5.

31 multiplied by 10 is equal to 310 and then half that is 155.

Lucas's number is 155.

Is that right, Lucas? Yes, it is, 155.

Once again, don't you think that bar model was so helpful there? I do.

Well, let's do a little check.

What number is Lucas thinking of? He says, "I'm thinking of a number.

25% of my number is 35." Okay, 25%.

Think how you could split your bar model up.

It's 25%.

What is my number? And Aisha says, "You could use a bar model to help you work out the answer." I would recommend it.

Pause the video.

How did you get on? Did you use a bar model? I think it's helpful.

So this is a bar model showing 25%.

So 100% split into four equal parts, each is 25%, 25% is 35.

Now what? Well, we can think of four lots of 35 and that will give us.

Well, if you double 35, that's 70 and double that again, that's 140.

So that's the answer.

Lucas's number is 140.

And well done if you've got that.

Brilliant work.

It's time for some practise.

Find the missing numbers using the bar models to help.

So A, Lucas has read 25% of his book and he's read 53 pages.

How long is his book? B, Sofia has read 20% of her book and she's read 65 pages.

Her book is how many pages long? And C, Jacob has read 50% of his book.

He's read 275 pages.

How long is his book? Those bar models will really help there.

Number two, find the missing numbers.

Complete bar models to help you.

A, I'm thinking of a number.

20% of my number is 90.

What is my number? How could you split that bar model to show 20%? And B, I'm thinking of a number.

50% of my number is 606.

What is my number? How could you split the bar model this time to show 50%? What's Aisha's number? C, I'm thinking of a number.

10% of my number is 460.

What's my number? How could you split the bar model to show 10%? And D, I'm thinking of a number.

25% of my number is 1,750.

What's my number? How could you split the bar model to show 25%? How many equal parts will you need to show? Right here, pause the video and away you go.

Welcome back.

How did you get on? Let's have a look.

So 1A, Lucas has read 25% of his book and that is equal to 53 pages.

So therefore, his book is 212 pages long.

Let's see how we got that.

The whole is equal to 53 multiplied by 4.

And the bar model helped us to scaffold that thinking.

53 doubled is 106 and 106 doubled is 212.

And for B, Sofia has read 20% of her book.

She's read 65 pages and her book's 325 pages long.

Here's how we get that.

The whole is equal to 65 multiplied by 5, and we can get there by multiplying by 10 and halving.

So there's no need, in that case, to use a written method.

And see, Jacob has read 50% of his book, which is 275 pages.

Double that, that's 550 pages.

And number two, here are the answers.

So for A, 100% is equal to 5 lots of 20% and one lot of 20% is 90.

So that's 90 multiplied by 5.

You might have multiplied by 10 1/2 to get 450.

And for B, 100% is equal to 2 lots of 50% or one lot of 50% is 606, double it, that's 1,212.

In B, the whole is equal to double 606.

And for C, 100% is equal to 10 lots of 10%, one lot of 10% is 460.

So we can multiply that by 10.

The whole is equal to 460 multiplied by 10, and that's 4,600.

And for D, 100% is equal to 4 lots of 25%.

One lot of 25% is 1,750.

You might have been able to double and double again.

You might have needed a written method.

But however you did it, the answer is 7,000.

That's Aisha's number.

Hey, you're doing really, really well.

Let's move on to the next part of the lesson.

That's missing whole problems with two steps.

Sometimes you need to work backwards first.

Lucas says, "I'm thinking of a number.

75% of my number is 180.

What's my number?" Now have a think about that.

Can you see where and why and how that is more difficult than the questions we've looked at already? "This is tricky," says Aisha, "because 75 is not a factor of 100.

We can't count in 75s, so I can't just multiply the part.

I'll use a bar model to represent this." Good idea.

Always helpful.

So here's a bar model.

So we know the 75% part, that's 180 and there's 25% left.

I need to work out that 25% of Lucas's number.

How can we do that? Well, 75% divided by 3 is equal to 25%.

So we can divide that 180 by 3 to work out 25% of Lucas's number while 18 divided by 3 is equal to 6.

So 180 divided by 3 is equal to 60.

So each of those parts is worth 60.

And so is the remaining part.

So we know that 25% of Lucas's number is 60, and now we can work out the whole of the number.

We can multiply by four.

And again, we can double and double again.

60 multiplied by 4 is equal to 240, and that was Lucas's number.

So that took one more step, but we got there.

"That's right." He says, "75% of 240 is equal to 180." Lucas thinks of another number.

Let's see if we can guess it.

"I'm thinking of a number.

60% of my number is 1,800.

What's my number?" Now again, we've got that same problem before and that 60 is not a factor 100.

So we're going to need to think of it slightly differently.

100% is equal to 5 lots of 20%.

Hmm, that could be helpful.

60% is equal to three lots of 20%.

So we could split a bar into 20% section.

So five sections.

I can represent this using a bar model.

So that's 60% that we know, that's 1,800, and two 20% that we don't know.

Now we can divide that 1,800 by three to work out 20%.

So let's do that.

Let's not think of it as 60%.

Let's think of it as three 20%.

So that gives us 1,800.

So each of those is worth 600 and each of these is also worth 600.

Each 20% section is worth 600.

And now we've got 600 multiplied by 5.

You can multiply by 10 and halve it and that gives us 3,000.

Lucas's number is 3,000.

Well done, Aisha.

60% of 3,000 is equal to 1,800.

Once again, that was tricky, but it only took one more step.

So let's do a little check.

Let's see if you've tuned into this.

What number is Lucas thinking of? He says, "I'm thinking of a number.

40% of my number is 160.

What is my number?" Hmm.

And Aisha says, "I'll help you." Thanks Aisha.

"By using a bar model to represent this." So she's done half of the job for you.

So can you complete the rest? Pause the video.

Well, let's see.

100% is equal to 5 lots of 20%.

So that's how we could think of that 40% as two 20%s.

20% of Lucas's number is equal to 80.

To work out Lucas's number, multiply 80 by 5.

And we've got a good way to do that, haven't we? Multiply by 10, halve it, and that's 400.

80 multiplied by 5 is equal to 400.

You could also use your times tables knowledge there, couldn't you? 8 multiply by 5 is equal to 40.

And that's a related fact.

Make it 10 times the size.

Aisha is watching a film.

She says, "I've watched 12 minutes, which is equal to 15% of the film." How long is the film? Hmm.

"I can divide 15% by 3," says Lucas, "to work out 5%." That could be helpful because 5 is a factor of 100.

So 12 divided by 3 is equal to 4.

So 5% of the film is four minutes.

Then we can work out 10%.

That's eight minutes.

Now we're almost there, aren't we? One last step.

Can then multiply 10% by 10 to work out 100% or the whole and that's hopefully nice and easy for you.

8 multiplied by 10 is equal to 80.

So the film is 80 minutes long.

It just took an extra step, didn't it? Bradley is taking part in a race.

Bradley has cycled 2.

4 kilometres, which is 4% of the total length.

How long is the race? Hmm.

What could be a useful, helpful thing to do here? We know 4%.

What could we do? What could we do next? I can divide 4% by 4 to work out 1%.

That's a good idea because then we can just multiply that by 100.

I can divide by 4 by halving and halving again or 2.

4 halve is 1.

2 and 1.

2 halve is 0.

6.

I can then multiply 1% by 100 to work out 100% or the whole.

So let's do that.

So we've got 0.

6 multiplied by 100.

So move those digits two places to the left and we've got 60 kilometres.

Bradley's race is 60 kilometres long.

Lucas is making a cake.

He says, "I use 150 grammes of raisins.

This is 30% of the raisins in the bag.

What was the mass of the whole bag?" And Aisha says, "Use the table to help you work out the answer." Pause the video.

How did you do? Well, we could work out 10%, couldn't we? From that 30%, we could divide that by 3.

That would give us 50 grammes.

And now we can use that to work out 100%.

We can multiply it by 10.

50 grammes multiplied by 10 is equal to 500 grammes, and that's the answer.

Well done if you've got.

You are on track and you are ready for the next part of the learning.

And the next part of the learning is the final part of the learning and that is practise.

Number one, find the missing numbers, complete bar models to help you.

So A, I'm thinking of a number.

30% of my number is 240.

What is my number? And B, I'm thinking of a number.

60% of my number is 2,100.

What's my number? So think what you can do to those bar models to represent this.

See, I'm thinking of a number.

75% of my number is 2,700.

What's my number? And D, I'm thinking of a number.

40% of my number is 192.

What is my number? The arithmetic's quite challenging on that one I think.

And two, find the missing numbers.

A, Helen is taking part in a race.

She's travelled 4.

5 kilometres, which is 3% of the whole distance.

So how long is the race? And Sofia has a very large bar of chocolate.

She's eaten 45 grammes of it, which is 15% of the whole bar.

What's the mass of the whole bar? If your teacher is okay with you working in partners, I always recommend it.

And then you can share ideas and strategies and help each other out if things go a bit wrong.

Pause the video and away you go.

How did you get all? Would you like some answers? Let's have a look.

Here are the answers.

So for A, Jacob's number is 800.

30% divided by 3 is equal to 10%.

So 800 is equal to 10 lots of 80.

That's how we get 800.

And for B, 60% divided by 3 is equal to 20%.

3,500 is equal to 5 lots of 700.

And you might have used multiplying by 10 and halving to get that.

And for C, Jacob's number is 3,600.

75% divided by 3 is equal to 25%.

And 3,600 is equal to 4 lots of 900.

And for D, 40% divided by 4 is equal to 10% and 480 is equal to 10 lots of 48.

And number two, here are the answers.

So A, 3% divided by 3 is equal to 1%.

Then we can use that.

We can multiply that by 100, and that's 150 kilometres.

And B, 15% divided by 3 is equal to 5%.

We can use that 5%.

5% double is equal to 10% and now we've got that 10%, 30 grammes multiplied by 10 is equal to 300 grammes.

That was the bar of chocolate.

That was a mass.

We've come to the end of the lesson.

It's been quite challenging, but I hope you've enjoyed that level of challenge.

I certainly have.

Today, we've been explaining how to solve problems where the percentage part and size is known but the whole is unknown.

If I know about a part, I can work out the whole from that part.

If I know what 10% is, I can multiply by 10 to calculate 100%.

If the part is a factor of 100, I can multiply the part by the factor to calculate the whole.

And I think the secret of your success there has been bar models.

Bar models are a brilliant way of illustrating the math, unlocking the math, and helping you to see what operations are needed and what your next steps are.

Well done on your accomplishments and your achievements today.

I think you deserve a pat on the back.

Go for it.

I'd love to spend another math lesson with you at some point in the near future.

But until then, have a great day.

Be the best version of you that you can possibly be.

You can't ask for more than that.

Take care and goodbye.