Lesson video

Hello, how are you today? Welcome to today's math lesson.

My name is Dr.

Shorrock and I'm very much looking forward to guiding you through the learning today.

Today's lesson is from our wee unit, understand place value within numbers with up to eight digits.

This lesson is called problem solving using knowledge of the composition of powers of 10.

As we move through the learning today, we will deepen our understanding on strategies that we can use to solve word problems. We will look at the importance of using representation such as the bar model or a number line to support us.

Now, sometimes new learning can be a little bit tricky, but don't worry, I am here to guide you and I know if we work really hard together, then we can be successful.

Let's get started then, shall we? How can we solve problems using knowledge of the composition of powers of 10? This is the key word that we will use in our learning today.

You've probably heard this word before, but it's always useful to practise.

My turn, million.

Your turn.

Fantastic.

1,000,000 is composed of 1,000 thousands and is written as a one followed by six zeros.

10 1,000,000s make 10,000,000 or a 10 with six zeros.

Let's start our learning today thinking about how we can represent word problems. And we have Laura and Jacob to guide us today.

Did you know? Ooh, I wonder what this fun fact is going to be.

It is about 1,000,000.

How else could we say that? Oh, that's right.

One million.

Did we know it is about 1,000,000 metres from Oxford, in England, to Rome, in Italy? Wow.

And we've got a plane and the plane is about 1/4 of the way through its flight from Oxford to Rome.

Can you visualise that? What do you see? I can see a dot for Oxford and Rome, and I can see a plane that's 1/4 of its way through.

What might the question be here, do you think? Any ideas? How many metres of the journey are left? Okay, so we've got the parts of the journey that it's been and then how many metres are left, and that's the bit we're being asked about, isn't it? How many metres are left? What should we do first, do you think? Well, just like I've done in my head, we visualised it, but then we need to represent this as a bar model.

We know it's about 1,000,000 metres from Oxford to Rome, and this is known and it's going to be our whole.

That's the whole journey.

So that can be our whole for our bar model.

We know the plane is 1/4 of the way through the flight.

Well, what does that mean? That's right.

It means the whole must have been divided into four equal parts.

Each part is 1/4 of the whole.

We can use our bar model to help us.

We can see that the first quarter is how far the plane has flown and the remaining three quarters are how far the plane has left to fly.

Is there a different way we could've represented that problem? That's right, Jacob.

We could have represented this as a number line.

We know the whole, 1,000,000, has been divided into four equal parts and each part is 1/4 of the whole.

And we know 1/4 is how far the plane has flown and 3/4 is how far the plane has left to fly.

Once we've represented the problems, we can then form an equation and solve it, which we will look at doing later, but for now it was really important that we could represent that problem.

Take a look at the bar model and the number line.

Does that make sense to you? Can you see the part of the bar model and the number line of where the plane has already flown.

and you can see what we have got left to find? Let's look at a different problem then.

A charity is looking to raise 1,000,000 pounds.

Can you visualise that? What do you see? I've thought of it as like a bit of a thermometer and the charity needs to raise 1,000,000 so that's at the top of my thermometer.

The diagram shows how much that they've raised so far.

What might the question be, do you think? That's right.

How much more money do the charity need to raise to reach their target? What should we do first, do you think? We visualised it, haven't we? That's right.

We always then need to represent information as a bar model.

We know 1,000,000 pounds is our whole and it's known.

That's the total amount they want to raise, isn't it? And we know the whole is divided into five equal parts.

Each part is 1/5 of the whole, and we can see three parts are equivalent to the amount of money raised so far, and two parts show what remains to be raised.

Is there a different way that we could've represented the problem? We've done it as a bar model.

Anything else? That's right.

Thank you, Jacob.

We could also represent this as a number line.

We know the whole has been divided into five equal parts and each part is 1/5 of the whole.

We can see 3/5 is the amount of money that's already been raised and 2/5 remains to be raised.

Once we've represented the problem as a bar model and/or a number line, we can form an equation from these to solve it.

Let's check your understanding on this.

Could you tell me which bar model is an accurate representation of this problem? The mass of a polar bear can reach around 1,000,000 grammes.

Wow, that's a lot of grammes, isn't it? That's a heavy polar bear.

The mass of a lion is 1/5 times this mass.

So look at the bar models a, b, c, and d, and see which one you think is an accurate representation of that problem? Pause the video while you do that, maybe find someone to talk to about each of these bar models, and when you're ready to go through the answers, press play.

(Shorrock gulps) How did you get on? Did you say, well, it must be c.

It can't be a because we have the mass of a lion's 1/5 times this mass, so the whole needs to be divided into five equal parts, so it can't be a.

b? Well, we know the mass of the polar bear must be our whole because it's the larger amount and in b, 1,000,000 is one of the parts, so it can't be b.

And c? Well, c is correct 'cause the whole is 1,000,000 and we know we need to divide it into five equal parts.

And for d, well, we've got the whole is 200,000 and that's actually the value of one of the parts, so that would be incorrect.

It's your turn to practise now.

For question one, could you represent these problems as bar models? A famous pop star, Taylor Rush, charges 1,000,000 pounds to play at a music festival.

Another pop star, Ariana Petite, charges 1/10 times this amount.

How much more does Taylor Rush charge? Question b, Laura thinks of a number.

1/5 of her number is 200,000.

What number is Laura thinking of? And part c, every four months, The London Eye has about 1,000,000 visitors.

How many visitors is this per month? For question two, could you look at this scale and represent finding the difference between the numbers represented by a and b as a bar model? Now pause the video while you have a go at these.

There is no need to solve these questions.

The really important thing is that you can represent them and that's what I'm looking to see, how you have represented these on bar models.

Pause the video and when you are ready to go through the answers, press play.

(Shorrock gulps) How did you get on? You were asked to represent these problems as a bar model.

So for a, we had Taylor Rush charging 1,000,000 pounds.

That's the known amount and it must be the whole because it's the larger amount.

And we need to find 1/10, so there must be 10 equal parts.

One part is equivalent to how much Ariana Petite charges.

The other nine parts represent how much more Taylor Rush charges.

For part b, Laura is thinking of a number, the whole is unknown.

We don't know what number she's thinking of, do we? But we do know that we have 1/5, so there must be five equal parts, and each part is worth 200,000.

For part c, every four months, The London Eye has about 1,000,000 visitors, so we know this must be the whole and it is known, and we need to divide it by four because we're thinking about there's four months and we want to know for each month, so there must be four equal parts.

1,000,000 is the whole and there are four equal parts, and one part would be equivalent to the number of visitors per month.

For question two, you are asked to look at this scale and represent finding the difference between a and b as a bar model.

We know to find the difference, we need to subtract the part from the whole.

The whole is the larger number b, and the known part is the smaller number a.

So my bar model looks like this, where the whole is b and one of the parts is a, and the difference would be the unknown part.

How did you get on with representing all of those in a bar model? Well done.

Fantastic learning.

You're trying really, really hard and that's what's really important.

We're going to move on now and think about, well, we've now represented these problems. How can we take a bar model or a number line and solve the problems? So let's revisit this problem.

It's about 1,000,000 metres from Oxford to Rome.

The plane was about 1/4 of the way through its flight, and the question was how many metres of the journey are left? We had represented this as a bar model and as a number line, and now we can use those to form equations to solve.

The whole has been divided into four equal parts.

So we can take our whole 1,000,000 and divide it by four.

But it's a large number, isn't it? Jacob, you're right.

How do we solve equations like this when the numbers are so large? Ah, Laura knows we can use unitizing unknown facts to solve this.

We know 1000 divided by 4 is equal to 250.

So 1,000 thousands or 1,000,000 divided by 4 must be equal to 250,000.

So 1,000,000 divided by 4 is equal to 250,000.

The plane therefore has flown 250,000 metres, but we were asked to see how far it has left to fly.

So we need to subtract the known part, 250,000 from the whole 1,000,000.

1,000,000 subtract 250,000.

Well, I can partition the 250,000 to help me.

1,000,000 subtract 200,000 is 800,000, subtract 50,000 is 750,000.

So the plane still has 750,000 metres of its journey left.

Let's revisit this problem.

(Shorrock gulps) A charity is looking to raise 1,000,000 pounds and the diagram shows how much they have raised so far.

And we needed to work out how much more money the charity need to raise to reach their target.

We represented this in a bar model and on a number line, and we can now take those to form equations to help us solve the problem.

We've got 1,000,000 is our whole and it's been divided into five equal parts.

Again, yeah, it's such a large number, Jacob.

How do we solve this? That's right Laura, we can use unitizing again, can't we? We know 1000 divided by 5 is equal to 200.

So 1000 thousands or 1,000,000 divided by 5 is equal to 200,000.

So the value of one part is 200,000.

We need to work out the value of three parts, 600,000, so the charity have raised the equivalent of 600,000 pounds, but we needed to work out how much remains to be raised to reach their 1,000,000 pound target.

So we need to subtract the amount they have raised from the whole.

1,000,000 subtract 600,000 is equal to 400,000.

So the charity still needs to raise 400,000 pounds to reach their target.

Ooh, Jacob's querying this method, but did we need to work out how much money they had raised? What do you mean, Jacob? Jacob is saying, "Well, if we look at that bar model, we could've just found the value of one part and multiplied it by two." 1,000,000 divided by 5 is equal to 200,000 and multiplied that by two is 400,000.

That's the same answer but a bit more of an efficient way of doing it because there were fewer steps.

Well done.

Thank you for sharing that with us, Jacob.

Let's check your understanding.

Could you use the bar model to support you to form an equation and then solve the problem using unitizing? The mass of a polar bear can reach around 1,000,000 grammes.

The mass of a lion is about 1/5 times this mass.

What is the mass of a lion? Pause the video while you do that.

Maybe compare your answers to somebody else's and when you're ready to go through the answers, press play.

How did you get on? Did you form an equation by taking the whole 1,000,000 and dividing it into five equal parts? So divided by five.

We know that 1/5 of 1000 is 200.

So 1/5 of 1000 thousands is 200,000.

1,000,000 divided by 5 is equal to 200,000.

So the mass of a lion is about 200,000 grammes.

How did you get on with that? Well done.

It's your turn to practise now.

Using your representations from task a, could you form an equation and then solve these problems? As a reminder, question a was a famous pop star, Taylor Rush, charges 1,000,000 pounds to play at a music festival.

Another pop star, Ariana Petite, charges 1/10 times this amount.

How much more does Taylor Rush charge? Part b, Laura thinks of a number.

1/5 of the number is 200,000.

What number is Laura thinking of? And part c, every four months, The London Eye has about 1,000,000 visitors.

How many visitors is this per month on average? Question two, using your representation from Task A, could you find the difference this time between the numbers represented by a and b? Pause the video while you have a go at those questions.

When you're ready to go through the answers, press play.

How did you get on? For question one, you were asked to use your representations that you'd made from the previous task and form equations and solve the problems. The first question involved a famous pop star, Taylor Rush, charging 1,000,000 pounds to play at a music festival.

So that is our whole amount in our bar model.

Another pop star charged 1/10 times this amount and that is why we needed to divide the bar model into 10 equal parts.

And we had to figure out how much more does Taylor Rush charge, so we would need to start off by finding out how much Ariana Petite charges.

1,000,000 has been divided into 10 equal parts.

Each part is worth 100,000.

So Ariana Petite charges 100,000 pounds, but we needed to work out how much more Taylor Rush charges.

So we would need to subtract 100,000 from 1,000,000, which is equal to 900,000.

Taylor Rush charges 900,000 pounds more than Ariana Petite.

For part b, Laura is thinking of a number and 1/5 of the number is 200,000, and we needed to figure out what is the whole.

What is that number Laura is thinking of? We know we have five equal parts so we can multiply 200,000 by 5.

200 is 1/5 of 1000, so 200,000 is 1/5 of 1,000 thousands.

So Laura is thinking of the number 1,000 thousands or 1,000,000.

For part C, every four months, The London Eye has about 1,000,000 visitors.

It's every four months, so that told us that we needed four equal parts so that we could find out how many visitors there were in one month, so we know we need to divide it by four or find 1/4.

1,000,000 divided by 4 is equal to 250,000.

We know that then there about 250,000 visitors per month to The London Eye.

For question two, you were asked to use your representation from Task A to find the difference between the numbers represented by a and b.

We also needed then to determine the values of a and b.

And we can see the whole number line has been divided into five equal parts.

10,000,000 divided into five equal parts.

Well, each part must be worth 2,000,000.

We can then find the difference between a and b.

b would be equal to 8,000,000 and a will be equal to 2,000,000.

8,000,000 subtract 2,000,000 is equal to 6,000,000.

So the difference between the numbers represented by the letters a and b is 6,000,000.

And we can see that because we know each part is worth 2,000,000 and there are three parts in between a and b, so that must be 6,000,000.

How did you get on with those questions? Well done.

Fantastic learning today.

I am really impressed with how hard you have tried and that's what's really important.

I have had great fun as we have deepened our understanding of how we can represent and solve problems using our knowledge of the composition of powers of 10.

We know that having this understanding of the composition of the powers of 10 supports us when we solve problems. And for example, if we know that 1/4 of 1000 is 250, then we know that 1/4 of 1000 thousand or 1,000,000 is 250,000.

And just as with smaller numbers, word problems can be represented and should be represented visually as a bar model to support us to form an equation which we can then solve.

Really well done today, everyone.

Really impressed with the progress that you have made.

I have had such great fun learning with you, and I look forward to learning with you again soon.

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