Lesson video

Hello.

How are you today? I trust you are feeling well and you are ready for this maths lesson.

My name is Dr.

Shorrock, and I look forward to learning with you today.

Today's lesson is from our unit, Order, compare, and calculate with numbers with up to eight digits.

This lesson is called Estimate and identify numbers on number lines.

As we move through the learning today, we are going to deepen our understanding of how we can determine how number lines are composed and then use this to place and estimate where numbers lie on a number line.

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

Shall we get started then? How do we estimate and identify numbers on number lines? These are the key words from our learning today.

We have estimate and midpoint.

It's always good to practise saying words aloud.

So let's have a go at this together.

My turn, estimate, your turn.

Nice.

My turn, midpoint, your turn.

Fantastic.

Now, when we estimate, we find a value that is close enough to the right answer.

So it's not the exact right answer.

It's just close enough, but usually we give it some thought or some calculation.

And the midpoint, well, that's just the middle of something.

In this case, we're going to be looking at the middle of a number line.

It's the point that's halfway along.

So let's get started, shall we, today? we're going to start by thinking about how we place and identify numbers on a number line.

And in today's lesson, we have Aisha and Lucas to help us.

Let's look at this number line.

What do you notice? Where does it start? Where does it finish? That's right, it starts at 12,000 and finishes at 13,000.

Now, Aisha and Lucas are discussing where to place 12,500 on this number line.

Where would you place it and can you prove you are correct? Lucas says, "Well, let's just have a guess.

I think it goes here." What do you think? Do you agree with Lucas? Aisha doesn't.

She's respectfully challenging him, 'cause we shouldn't really guess, should we? Instead of guessing, we can use our number sense superpower so that we can be accurate.

And what do you notice about the numbers? Is there something that you notice that might help us? That's right, there is a difference of 1,000 between the first and last numbers.

We've got 12,000.

If we added 1,000, we would get 13,000.

Can we use that to help us? Because what do we know about 1,000? Well, we know half of 1,000 is 500, don't we? So we can use this fact to place 12,500 on the number line.

Because if half of 1,000 is 500, then halfway between 12,000 and 13,000 must be 12,500.

So 12,500 would be the midpoint of this part of the line and we used our number sense superpower there, didn't we, to position 12,500? But is there another strategy that we could have used? Yes.

Can you think of another strategy? Can you see that the whole line segment has been divided into 10 equal parts? So if we divide that 1,000 divided by 10, we get 100.

So each part must be worth 100.

So the number line is increasing in multiples of 100.

Each step is worth 100.

The digit five in 12,500 is worth 500.

So 12,500 must be positioned at the end of the fifth part.

One part, two parts, three parts, four parts, five parts.

Let's check your understanding with this so far.

Could you look at this number line? Which letter shows the correct placing of 23,700? Is it A, B, or C? Pause the video while you have a think about it.

When you're ready for the answer, press play.

How did you get on? Did you say, "It must be B"? But why is it B? Well, the number line is increasing in multiples of 100 because the difference between 23,000 and 24,000 is 1,000, and there are 10 equal parts.

The letter B is at the end of the seventh part.

The value of the digit seven is 700, so it must be the end of the seventh part.

Now look at this different number line.

What do you notice this time? Is there something that's the same, something that's different to before? You might have noticed that this number line starts at 10,000 and finishes at 20,000.

So this time, there's a difference of 10,000, isn't there? But there are still 10 equal parts, and Lucas and Aisha are discussing where to place 12,500 on this number line.

So the number they are placing is the same as before, but this number line is different.

Where would you place 12,500? Can you prove you are correct? Lucas says, "Well, let's not guess this time." We're gonna start by determining the multiples that the number line is going up in.

So the difference is 10,000 between 10,000 and 20,000.

So this is the whole and there are 10 equal parts.

So we can divide to find the value of each part.

10,000 divided by 10 is 1,000.

So the number line is going up in multiples of 1,000.

And we know that 12,000 is less than 12,500, which is less than 13,000.

So 12,500 must be in between 12,000 and 13,000.

And we know 12,500 is halfway between 12,000 and 13,000, so we can position 12,500 on our number line.

Let's check your understanding of that.

Which arrow shows the correct placing of 43,500? Pause the video, maybe talk to somebody about this, and compare your thoughts.

When you're ready to hear the answer, press play.

How did you get on? Did you work out that the difference between 40,000 and 50,000 is 10,000 and there are 10 equal parts? So each part must be worth 1,000.

So the number line is increasing in multiples of 1,000.

And if we count up in 1,000, we can mark 43,000 and 44,000 on our number line and 43,500 is halfway between 43,000 and 44,000.

So then we can place 43,500 on our number line.

Let's look at this different number line.

What do you notice? Where does it start? Where does it finish? How many intervals are there? Well, Aisha and Lucas are trying to identify what the number represented by the letter A could be.

Any thoughts? What do you think? Can you prove you are correct? Well, Lucas is saying the letter A represents 1,300,000 because it's after the third part.

So he's saying it's after the third part, so each part must be worth 100,000, mustn't it? Do you agree with Lucas? Oh, Aisha doesn't, does she? I wonder why she's challenging him.

What do you think? Well, Aisha is saying that she knows halfway between 1 million and 2 million is 1,500,000.

So 1,300,000, well, it must be before halfway, mustn't it? Because it's smaller than 1,500,000.

And the letter A is actually after, isn't it? The letter A cannot be 1,300,000.

So let's start by determining the multiples that the number line is going up in.

Good idea.

The difference between 1 million and 2 million is 1 million.

So that's our whole, and there are five equal parts.

So we can divide to find the value of each part.

Well, we know 10 divided by five is two.

So 10 100,000 divided by five must be 200,000.

So each part is worth 200,000.

The number line's going up in 200,000.

So we've got 1 million, 1,200,000, 1,400,000, 1,600,000, 1,800,000, and 2 million.

So we can see that the letter A represented 1,600,000.

Lucas was wrong, but I think he's learned a lot, hasn't he? It's okay to make mistakes 'cause Lucas' brain has now grown 'cause he now understands how to work out what the letter A could be.

He has learned to determine the multiples that the number line is increasing in first and not just guess.

1,300,000 is actually positioned halfway in between 1,200,000 and 1,400,000.

Let's check your understanding.

Could you look at this number line, be thinking? What do you notice? Lucas is saying that the number represented by the letter A is 2,300,000, but he's incorrect.

Can you tell me what the correct number should be? Pause the video while you have a think.

When you're ready to hear the answer, press play.

How did you get on? Did you say, well, Lucas just thought that it must be 2,300,000 because the letter A is at that third end of that third interval? But you can't do that, can you? You need to work out what the difference is, and the difference between 2 million and 3 million is 1 million.

And then we need to divide by the number of parts.

There are four.

So that means this number line is increasing in multiples of 250,000.

So we've got 2 million, 2,250,000, 2,500,000, 2,750,000.

So the correct number that A is representing is 2,750,000.

It's your turn to practise now.

For question one, could you label the missing numbers on the number line? So remember, don't rush into this.

Stop and think.

What do you notice? What's the difference between the starting number, and the end number, and how many intervals are there? for Question two, could you work out the difference between A and B for each number line? So you're going to have to work out the value of A, the value of B, and then find the difference.

How do we find the difference? Can you remember? That's right, we need to subtract the smaller number from the larger number.

Pause the video while you have a go at both those questions.

And when you are ready to go through the answer, press play.

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

So for question one, you had to label the missing numbers on the number lines.

If we look at the first number line, there are four parts, and the difference between the starting number 4 million and the end number 5 million is 1 million.

We need to divide by four.

So each part is worth 250,000.

The number line is increasing in 250,000.

So we must have 4,250,000, 4,500,000, and 4,750,000.

For our next number line, we've got five parts, and the difference between the first number 300,000 and the last number 400,000 is 100,000.

We divide it by five, which is 20,000.

So this number line is increasing in steps of 20,000.

320,000, 340,000, 360,000, 380,000.

For question two, you had to work out the difference between A and B for each number line.

So here we can see the difference between our first number and our last number is 20,000, and there are four equal parts.

We need to divide by four.

So each part must be worth 5,000.

That means A must be 285,000 and B must be 295,000.

Then we can find the difference by subtracting which is 10,000.

For the second number line, we had a difference of 1 million and there are five equal parts.

So we're dividing by five, which is 200,000.

So we're counting up in steps of 200,000.

A must be 1,400,000 and B must be 1,800,000.

We're finding the difference by subtracting which is 400,000.

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

Fantastic learning so far.

I'm really impressed with how hard you are trying, and it's really important in math to try our best.

We're gonna move on now and have a look at how we estimate numbers on a number line.

So looking at this number line, what do you notice? Where does it start? Where does it end? Is there something different about this number line? Aisha and Lucas are estimating the position of 65,000 on this number line.

Now why do you think they are estimating? That's right, they're going to have to have a guess that's close enough to where it really should be, but there's no intervals marked, aren't there? So it's going to be an estimate.

How would you do this? Where would you estimate the position of 65,000 to be? And can you prove you are correct? Lucas says that he's noticed this number line is unmarked, so we're gonna use our number sense.

And we're gonna start by determining that midpoint, the middle of this line segment.

Halfway between 50 and 100 is 75.

We can use unitizing now.

So halfway between 50,000 and 100,000 is 75,000.

So we can start by marking that midpoint.

It will help us to be more accurate when we estimate.

65,000, well, it must be before the midpoint, mustn't it? Because it's smaller than 75,000.

But where do you think it would be in that part of the line? Well, it's going to be near 75,000 than 50,000, isn't it? So we can estimate the position of 65,000.

Lucas has used his number sense to estimate the position as accurately as he can.

Let's check your understanding with that.

Look at this number line.

Could you estimate the position of 85,000? Think about what you notice.

where does it start? Where does it finish? What might that midpoint be? Pause the video while you have a think.

And when you are ready to go through the answers, press play.

How did you get on? Did you work at that midpoint to be 75,000 and mark that on? And then did you notice something about the number we've been asked to estimate? That's right, it's 85,000 isn't it? So it must be further along than the midpoint.

But do you think 85,000 is near 75,000 or near 100,000? That's right, it's near a 75,000, isn't it? So we can estimate with some degree of accuracy where the position of 85,000 is.

How did you get on with that? Well done.

Let's look at this number line.

What do you notice this time? Where does it start? Where does it finish? Is there anything marked? Aisha and Lucas want to estimate the number that the arrow is pointing to.

What would you estimate as a number the arrow pointing to? Can you prove you're correct? What might help you? Aisha is saying that we can see the arrow is past the midpoint of the line, and the midpoint must be 850,000 because there is a difference of 100,000 between 800,000 and 900,000.

And half of 100,000 is 50,000, so that midpoint must be 850,000.

So can we use that to help us? I think we can, can't we? So the arrow, the number the arrow is pointing to must be greater than 850,000, but smaller than 900,000.

It's about halfway, isn't it, between the midpoint 850,000 and 900,000.

And we know that halfway between 50,000 and 100,000 is 75,000.

So halfway between 850,000 and 900,000 must be 875,000.

So using our number sense superpower, we can estimate that the arrow is pointing to 875,000.

Lucas says he could estimate that the arrow is pointed to 875,122.

Well he could, couldn't he? But when we estimate, we usually use a number that is a multiple of 10 'cause that's gonna be close enough.

We don't have to be precise because we can't be precise.

We are estimating.

Let's check your understanding with this.

Could you look at this number line and estimate which number is represented by the arrow? So remember to think about where does the number line start, where does it finish, and what might that midpoint be, and where is the arrow in relation to that midpoint? Pause the video while you have a think about this.

When you are ready to hear the answer, press play.

How did you get on? Did you identify that that midpoint must be 650,000? And that can help us then because we can see the arrow is positioned before the midpoint.

So the value must be smaller than the 650,000.

It's actually about halfway between 600,000 and 650,000.

So we can say the value of the arrow is about 625,000.

How did you get on with that? Well done.

Look at this number line then.

What do you notice now? Where does it start? Where does it finish? Aisha and Lucas Estimate where they think 5,192,012 would be on this number line.

Can you estimate where you think that number would be and prove you're correct? Well, Lucas has noticed that 5,192,012 is not a multiple of 10.

So to position this number, we can use Unitizing and focus on the number of thousands.

We're going to ignore those smaller value digits.

5,192,012 has 5,192 thousands, and 5,192 thousand is nearer 5,200 than it is 5,100.

So we can use that information to place or to estimate where this number might lie on the number line.

It will be nearer 5,200,000.

And look at this number line now.

What do you notice? Aisha and Lucas estimate the number they think the arrow is pointing to.

What do you think it might be pointing to? What's your estimate and can you prove you are correct? Aisha is saying that we can see that the number line increases in multiples of 1 million.

We're counting up in millions, aren't we? From zero to 10 million.

But the arrow is positioned before the first multiple of 1 million, but it's closer to 1 million than it is the midpoint, isn't it? So Aisha is estimating that the arrow is pointing to the number 900,000 because 900,000 is nearer to 1 million than it is the midpoint or to zero.

What about this arrow? Can you estimate the number that this arrow is pointing to? What do you notice? What's it got to be more than? What's it got to be less than? Is it more than or less than the halfway point? Well, the arrow is positioned between the 5 million and 6 million multiples, but it's closer to 6 million than the midpoint.

So Aisha is estimating that the arrow is pointing to the number 5,800,000.

Let's check your understanding with that.

Could you estimate the number that this arrow is pointing to? Take care to notice the multiples of 1 million that the arrow is between, and think about that midpoint of that part of the line.

Pause the video while you have a think about it.

And when you are ready to hear the answer, press play.

How did you get on? Did you notice that the arrow is positioned between 3 million and 4 million? So that number that the arrow represents must be greater than 3 million, but smaller than 4 million.

It's closer to 3 million, isn't it, than the midpoint.

So we need to have a number that's close to 3 million.

Aisha is estimating that the arrow is pointing to 3,100,000.

Remember, you could say something like 3,119,022, but you don't need to make it hard on yourself.

Lucas is saying that your estimate that you've just done might be slightly different, but it should be greater than 3 million and smaller than 3,500,000 and nearer 3 million.

It's your turn to practise now.

For question one, could you look at these number lines and estimate where 280,000 would be positioned on each number line? Take care to notice what's the same or what's different about these number lines.

For question two, could you draw arrows to show roughly where each of these numbers is located on the given number line? For question three, could you estimate the values of A, B, C, and D? Pause the video whilst you have a go at those three questions.

And when you are ready to go through the answers, press play.

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

For question one, you had to estimate where 280,000 would be positioned on each of these.

For the first number line, 280,000 would be near to 300,000.

For the second question, 280,000 would be fairly near the midpoint, which would be 275,000.

And for the last question, this number line started at zero and went to 1 million.

Well, 280,000, that's going to be less than the midpoint of 500,000 and roughly near halfway between zero and that midpoint.

For question two, you had to draw arrows to show roughly where each of these numbers is located on this number line.

8,000,055.

Well, that's just gonna be past the 8 million, isn't it? 100,000 will be fairly near to zero.

1,150,000, that would be in between 1 million and 2 million and nearer 1 million than the midpoint.

6,500,000 would be halfway between 6 million and 7 million, and 4,833,212 will be close to 5 million.

Then you had to estimate the values of A, B, C, and D.

The value of A is roughly 3,090,000.

You may have got something slightly different, but would need to be between 3 million and 3,000,100 and nearer to 3,000,100.

for part B, you can see that the value of B is roughly 3,390,000.

It would have to be greater than the halfway, so 3,350,000, and nearer to 3,400,000.

For C, the value is roughly.

It's roughly in the middle, isn't it? 3,450,000.

The value of D must be more than 3,900,000, but less than that midpoint of 3,950,000.

And so we've gone for the value of D is roughly 3,920,000.

Remember, you may have got answers that were slightly different to mine.

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

Fantastic learning today.

You really deepened your understanding on how you can estimate and identify numbers on number lines.

We know we need to understand the scale of a number line, so that's the multiples that it goes up in, and that helps us estimate and identify the numbers on it.

We know to determine the multiples that the number line increases in.

We can take the value of the whole and divide it by the number of parts.

And we know that locating the midpoint on a number line or between multiples helps us to estimate.

You should be proud of how hard you have worked today.

I'm proud of you.

I've had great fun learning with you, and I look forward to learning with you again soon.