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Hi everyone, thank you for joining me for a math lesson.

My name is Miss Jeremy.

Today, we are looking at ordering and comparing six digit numbers.

So, find yourself a nice quiet space for your learning.

And once you're ready, press play to begin your lesson.

Let's start by looking at our lesson agenda for today.

We're going to begin with a warmup about number lines with five digit numbers before looking at number line placement.

We'll then be calculating missing intervals on number lines before our independent task and quiz, at the end of the lesson.

The resources you're going to need for today are a pencil and some paper, and a nice quiet space for your learning.

So pause the video now, find your resources and press play once you're ready to begin.

Let's start by looking at our warm up for today.

So as you can see, the question says, which five digit numbers are being pointed to.

And we've got two arrows on our number line pointing to the number a and the number b, and we need to work out what those numbers are.

So, as you can see, it's quite handy because some of the numbers have been already marked out on our number line.

Our large intervals have already been determined for us.

And we can see they're going up in 10000s.

We've got 10,000, 20,000, 30,000 and so on and so forth.

So, what we need to work out before we can work out what these arrows are pointing to is what those smaller intervals are and what the distance is between those smaller intervals is.

So, I'd like you to have a look.

You can see that there are 10 small intervals for every large interval, and each large interval is equal to 10,000.

What do you think those smaller intervals are equal to? I'm going to give you five seconds to have a go at working out what the small intervals are.

Okay, have you got it? So, in order to find the smaller intervals, we need to do our larger interval jumps, which are 10,000 divided by the number of small intervals, which is 10.

So, each of those small intervals is worth 1,000.

So, we know that the large intervals are 10,000, the smaller intervals are 1,000.

Now, we can have a go at working out what those arrows are pointing to.

Well, I can see that arrow a, is directly halfway between 20,000 and 30,000.

And you can see that the halfway line just stretches slightly lower than the other smaller intervals.

What number is half way between 20,000 and 30,000, three seconds.

The number is 25,000, so the number for arrow a, is 25,000.

Let's think about b, now, it's not halfway, it's over halfway between 50,000 and 60,000.

So, if it were halfway, it would be 55,000, but it's gone 2,000 over that.

Have you worked out what it is yet, three seconds.

Arrow b is pointing to 57,000.

And you can see that because halfway would be 55,000, 55,000 plus 2,000 is 57,000.

So, let's look at another number line.

Let's try and identify the missing numbers on this number line, this time, our jumps are not going up in 10000s for our larger intervals.

We're going to have to look at the larger, and smaller intervals again, because they've changed for this number line.

First of all, let me look at my larger intervals.

My larger intervals are marked out by these longer lines here, and they've very kindly being filled in for us.

And you can see that the jumps are going up in 100000s.

So I can see it goes 100,000 to 200,000, 300,000.

Let me see if I can work out the smaller intervals now.

Well, if one large interval is equal to 100,000 and there are 10 small intervals in every large interval I can do 100,000 divided by 10 to work out what a small interval is worth.

And I know that a small interval must be, three seconds to work it out.

A small interval, in this case, is equal to 10,000.

So, I've done the first step of my success criteria, I've calculated my intervals.

So, now, looking at arrow a, well, I'm going to look at the number that comes before arrow a, and the number that comes after arrow a.

In this case, arrow a is directly halfway between 100,000 and 200,000.

What number is directly between 100,000 and 200,000? Well, I know that it must be 150,000, and I can even count up in 10000s, if I want to, to double check.

So, starting at 100,000, 100,000, 110,000, 120,000, 130,000, 140,000, 150,000.

Arrow a, is pointing to the number 150,000.

Now, let's look at b, b isn't halfway between any two of the larger intervals, but we can still work it out.

So, I can see that it's between 300,000 and 400,000.

Let's count upwards from 300,000 using our small intervals of 10,000, are you ready? 300,000, 310,000, 320,000, 300 and.

? 30,000, so arrow b, is pointing to 330,000.

So, we looked at the number before and after, and we identified the values of a and b.

That's a way that you can use the larger intervals, if they're marked out on your number line, to determine what the smaller intervals are, and then to use that information to help you work out what numbers are being pointed to.

So now, is a chance for you to complete one of your own.

Have a look at this number line.

We've got the large intervals pointed out, but we don't know what the small intervals are yet.

You're going to have to calculate that.

And then, I'd like you to use that information, looking at the number before and after the arrow, to identify the values that are being pointed to by arrows a and arrows b, pause the video to complete your task and resume it once you're ready to see whether your answers match mine.

Okay, how did you get on? Let's have a look at the answers.

So, I can see straightaway, again just like before, our big intervals, our large intervals are increasing by 100,000 each time.

And that tells me that the small intervals, just like last time, are increasing by 10,000.

Like last time, we've got an arrow that's pointing halfway between two numbers.

So arrow a is equal to 350,000 arrow b, in this case, is just over that halfway mark between 400,000 and 500,000 and, in this case, is equivalent to 470,000.

Give yourself a little tick, if you got that correct on your page.

So, let's move on, let's make it a little bit more challenging.

Have a look at this number line.

What is different about the way that this number line is presented? I'm going to give you five seconds to have a look.

How is this different to the number lines that we've already seen today? So you might have noticed that in this number line we're given the starting point and the ending point of the number line, but we are not told what the large or small intervals are in between that starting and ending point.

So, what we're going to have to do is work that out.

We're going to have to determine what the larger intervals are, then we're going to have to determine what the smaller intervals are, then we'll use that information to help us work out what arrow a is pointing to.

So, the first thing I'm going to do is look at my starting point and my ending point.

I can see this number line starts at 400,000 and ends at 900,000, and I'm going to use a strategy called trial and error to work out what the intervals in between are.

I'm going to have a guess, and I'm going to assume that this number line is going up in 10000s for the larger intervals.

And I'm going to assume that, and I'm going to calculate to see whether that would work.

So, we're imagining it's going up in 10000s, I'm going to count using my 10000s.

So, 400,000, 410,000, 420,000, 430,000, 440,000, 450.

no can't possibly be the case because I wanted to end up at 900,000 and, actually, if I count up in 10000s I end up at 450,000, can't possibly be right.

So, if it's not 10000s, why don't I try 100000s? Let's have a go at that and see if that's any better.

So, this time we're counting up, but we're imagining those larger intervals are spaced 100,000, apart.

Starting at 400,000, count with me.

500,000, 600,000, 700,000, 800,000, 900,000.

Brilliant, yes, we've worked out that our larger intervals are spaced 100,000 apart.

Now, if we know that we can fill in some information.

I can fill this in, I can say that that is equal to 500,000.

That number there is 600,000.

What's the next number? It's 700,000, and the next number's 800,000 and we've already got 900,000 at the end there.

So, we know what those larger intervals are.

And if we know that each large interval is 100,000 and there's 10 small intervals and one large interval, what is a small interval equivalent to, three seconds.

The small interval is spaced.

or our small intervals are spaced 10,000 apart.

So that's going to help me a lot.

Now I can actually have a look at arrow a and work out what arrow a is pointing to.

Well, I can see that here, at this point here is 800,000 and I'm going to count up in my 10000s, those small intervals to work out where arrow a is.

So, count with me, 800,000, 810,000, 820,000.

Arrow a is, 830,000.

Let me talk you through those steps that we completed there.

We looked at the starting point, we looked at the ending point.

We used trial and error to have a go at working out what those larger intervals were.

If it didn't work, we tried something else.

Once we found our larger intervals, we used that information to find our smaller intervals.

And then, we managed to identify what the missing value on our number line was.

So, let's have a go at a slightly more challenging one.

This time we're dealing with six digit numbers on a number line that has a starting point, an ending point, but none of the large, or small intervals filled in.

So, just like last time, you're going to have to identify what the larger intervals are and what the smaller intervals are.

I'd like you to pause the video now to complete your task and resume it once you're finished.

Okay, how did you get on with that activity? So, let's have a look at it together.

So, in this case, as you can see, our number line starts at 600,000 and ends at 700,000.

So you might've seen that the larger intervals on this number line couldn't have been going up in 100000s because the entire number line was only going up in 100000s.

So, from 600,000, all the way to 700,000, that was 100,000 in itself.

So actually your larger intervals, in this case, were equal to 10,000.

And that meant that your smaller intervals were equal to 1,000.

And what you might have seen is that a was directly on one of the larger intervals.

It was equivalent to 620,000.

And you might have seen that b was halfway between, or nearly halfway between two of the different larger intervals.

And you might've seen that it was worth 687,000.

So, if you've got those correct, give yourselves a little tick, just like that.

And we'll have a look at the next challenge.

So, now, have a look at this question.

So far, what we've been looking at is number lines, where we're given the starting point and we're given the end point, and we need to work out one of the numbers somewhere in the middle, this time we flipped it around a little bit.

This time, I'm giving you the middle point, but I'm not telling you what the starting point is, or the ending point, that's what I'd like you to work out for me, I want you to have a little think about what the starting point, what the ending point might be for this number line.

Then, what the larger intervals are.

And then, what the smaller intervals might be.

I'd like you to pause the video to have a look and complete your task, and then resume it once you're finished.

Okay, how did you get on? Let's have a look at this together.

So, if I know that 250,000 is my halfway point, I need to be thinking about two numbers on either side that would provide 250,000 as a halfway point.

So, in this case, I think that 50,000 is the halfway point between two numbers in the 100000s to six digit numbers.

So I think, over here, I'm going to put 200,000 and then on this side, I've got 300,000.

And that would be accurate in this particular example, because 250,000 would be directly between.

Now, I'll just tell you now that's not the only option.

There are other options for this number line, but in this case, that's the option that I'm going for.

If that's the case, if we starting this number line at 200,000, and ending at 300,000, what is the space between the large intervals and what is the space between the small intervals? Giving you five seconds to work that out.

Okay so, in this case, our larger intervals are going up in 10000s, you can see here, if we count up, you can see how it'd work.

210,000, 220,000, 230,000, 240,000, 250,000.

And if our larger intervals are increasing by 10,000, our smaller intervals will be increasing by 1,000.

So, you can see here, even if we're given the middle point, or a point on our number line, we can still work out the starting point and ending point.

And just to reiterate, there's more than one option for this.

That was my choice there, but you might have decided that, actually, your number line was going up in tens and your smaller intervals were going up in ones.

That's totally a potential as well.

Okay, moving on to your independent task for today.

So, as you can see, there are three number lines on the screen, each of the number lines has given you the number 430,000, somewhere on the number line, but it doesn't tell you what the starting or ending numbers are, that's for you to decide.

What you're going to work out is what you think the small and large intervals are increasing by on each of these number lines.

And then, I'm going to ask you to use that information to identify what the starting point and ending point of the number lines are.

Let me give you an example, we're looking at the first one here.

This one says that 430,000 is placed just in this position here, and I'm going to imagine that the smaller intervals are increasing by 10,000 and the larger intervals are increasing by 100,000.

So, if the small intervals are increasing by 10,000, then this little point would be 420,000, this would be 410,000, this would be 400,000 itself.

And so, if that's 400,000 that would be 300,000 there, that would be 200,000, that would be 100,000.

And my starting number would be 0.

And then, my ending point would be here at 500,000, as you can see, I'm counting up in 100000s for my large interval, that's one option.

Actually, there's lots of different options.

You might imagine that the small intervals are going up in 10, and the larger intervals are going up in 100.

If that were the case, what would your starting point be? What would your ending point be? Have a go at working through these, identifying what you think the small intervals, large intervals, starting points and ending points are.

And once you're finished, come back to have a look at some of the answers together.

So, let's have a look at one of the answers together.

This was one of the examples we saw, and this was one of the examples I gave you earlier on.

But we're going to imagine a different scenario, now.

We're going to imagine that this particular number line is increasing by a different amount.

Let's imagine that we are increasing the small intervals by 1,000 instead.

So, if we are increasing by 1,000, that little point there would be 429,000, that would be 428,000, that would be 427,000.

I'm going to jot it just there so we can see.

And if each of those individual smaller intervals are increasing by 1,000, then, the larger intervals would be increasing by 10,000.

So, we'd have to have here, 417,000.

Remember, it's the 10000s digit that's changing.

This would be 407,000, again, it's that 10000s that's changing.

This would be 397,000, and this one would be 387,000.

My number, right at the very end of my number line, would be 437,000.

So, in this particular example, and there were lots of examples, we looked at one earlier on where we were increasing by 10,000 and 100,000, but in this example, where our small intervals are increasing by 1,000 and our large intervals are increasing by 10,000.

You can see that what I've done is counted backwards to get to my starting point, and forwards to get to my ending point.

Have a look at your examples now, just double check.

Are all your smaller intervals equal, are all your larger intervals equal for each number line? Does it make sense where you've placed your starting point and your ending point? All right, that's it for our lesson today.

Thank you so much for joining me.

If you'd like to, please ask your parent, or carer to share your work on Twitter, tagging @OakNational and #LearnwithOak.

Now, it's time to complete your quiz.

Thank you so much for participating so well with today's lesson.

Do join me again for another math lesson soon, bye-bye.