Lesson video

Hi there, my name is Mr. Tilstone.

I'm here today to teach you a lesson all about negative numbers and I'm really excited about that too.

So if you are ready, let's begin.

The outcome of today's lesson is, "I can identify and place negative numbers on a number line." And keywords we've just got the one today.

That is my turn, your turn.

So my turn interval, your turn, and let's find out what interval means, shall we? An interval is the space on a scale.

So we've got a scale here.

It is usually shown using a small mark or line and it can be marked or unmarked, and we've got examples of both here.

So negative six you can see is on a marked interval and there's an unmarked interval in between zero and two.

That's got no numbers attached to it.

So that's called an unmarked interval.

We've got three cycles in today's lesson, "Counting to work out on marked intervals, positioning whole numbers and positioning fractions and decimals." We're going to start with counting to work out unmarked intervals.

Are you ready? In this lesson, you're going to meet these two characters.

They may be familiar to you already.

We've got Sam and we've got Laura to give us a helping hand.

Okay, so here we've got Sam and we've got a number line and the number line has got a zero on and it's got negative values to the left of zero, and it's got positive values to the right of zero.

Sam is counting using a negative number line, which goes up in ones and has all marked intervals.

She goes negative five, negative four, shall we carry on? Negative three, negative two, negative, zero, one, two, three, four, five.

She's going left to right.

The numbers are increasing in value as they go.

This time she's going to go from right to left.

She's going five, four, shall we carry on, ready? So nine, five, four, three, two, one, zero, negative one, negative two, negative three, negative four, negative five.

I wonder if you are able to do that without even looking at the screen.

Well done if you could do that.

So the numbers this time are decreasing in value.

They're going down.

So what is the same and what is different about this number line compared to the last one? So just have a little look at it.

I can see it's got some things in common with the last number line and it's got some things that are different.

Let's have a little think.

Well, it says Sam, "Just like the last number line, this one goes backwards and forwards in ones." it does, but the difference is that, this number line has some unmarked intervals.

Can you see? Some of those intervals don't have numbers attached, so it's got unmarked intervals, but we're still counting in ones just like before.

So if we were to mark those intervals, that's what it would look like.

But we're not using those numbers, we're just using the even numbers.

So the odd numbers this time are not marked on this number line.

We could still count in ones.

So if we wished we could go negative 10, negative nine, negative eight, negative seven, and so on and so on.

So let's do that.

Let's practise counting backwards and forwards in ones, including the unmarked values.

Are you ready? Okay, ready? Negative 10, negative nine, negative eight.

Do this with me.

Negative seven, negative six, negative five, negative four, negative three, negative two, negative one, zero one, two, three, four, five, six, seven, eight, nine, 10.

And once again, well done if you were able to do that without even looking at the screen, because that must mean you've got a number line in your head.

This time, let's only count the marked intervals.

Are you ready? Count with me.

Negative 10, negative eight, negative six, negative four, negative two, zero two, four, six, eight, 10.

So number lines do not always count in multiples of one.

The multiples can be all sorts of numbers.

So let's do a little check for understanding.

Let's see if you've got that learning so far.

So I'd like to know what are the values of A and B? So work with a partner, see if you can agree what is A worth, what is B worth? Pause the video.

Let's see what you came up with.

Let's see if you've got the right answer.

So A is negative nine.

So even though that is an unmarked interval, we can figure out that's negative nine and B is also unmarked, but we can figure it out.

It's in between negative four and negative two.

So therefore it's got to be a negative three.

Let's have a look at another number line.

Okay, this one's different again.

So think about the last number line that you saw.

Think about this one.

What's the same, what's different? I can see a couple of things.

Well, just like before, the number line shows negative and positive multiples of two.

So it's counting in twos just like before.

This time however, there are no unmarked intervals, so everything is marked.

Whole number values exist between the intervals.

So there even though that's not got an interval there, there is a value there.

So what do you think it might be? So it's halfway between negative eight and negative six, negative seven.

Okay, let's do a check.

What whole number is this arrow pointing to? So it's not pointing to a marked interval, it's not pointing to an unmarked interval.

It's pointing in between two marked intervals, but it is worth something.

What is it worth? Have a think, and pause the video.

Well, let's see.

It's in between negative four and negative two.

You could say it's greater than negative four, as well and less than negative two.

And the answer is negative three.

And if you've got that, you're on track with today's lesson so far, well done.

Okay, let's have a look at this number line.

This is a different kind of number line, isn't it? So what's the same and what's different about this number line compared to the last one that you looked at or the last ones that you looked at? Okay, have a little think.

What's the same, what's different? Well this time it's a thermometer.

So it is a number line, but it's a specific kind of number line.

It's a thermometer.

Just like the last number line, this one contains both positive and negative numbers.

So that's something that's got in common.

Some differences though, aren't there? There are a few differences.

It is vertical this time, so it's not horizontal, it's vertical.

It also has lots of unmarked intervals, lots of them, only the multiples of 10 are marked.

So we can see some unmarked intervals with no numbers next to them, but we can see some marked ones and the marked ones are worth multiples of 10.

So let's focus on this little bit.

There are five intervals between zero and 10.

You count them if you like.

One, two, three, four, five, five intervals.

And there are five twos in 10.

So the intervals are going up in twos.

Let's use counting to check.

So it goes, zero, two four, six, eight, 10.

That works, that works.

That's what the intervals are worth two.

So the unmarked intervals are worth two.

Now I know the value of the intervals.

I can count forwards and backwards on the number line from difference starting points.

So let's say we started here at negative 10, we could count forwards from there.

So count along with Sam as she counts forwards from negative 10 to 10.

Are you ready? We'll do it together.

So we've got negative 10, negative eight, negative six, negative four, negative two, zero, two, four, six, eight and 10.

So we were able to count those numbers even though they're not all marked on the number line.

This time count along with Sam, as she counts backwards from 10 to negative 10.

So it's going to be the same numbers, but in the reverse order, are you ready? 10, eight, six, four, two, zero, ready? Negative two, negative four, negative six, negative eight and negative 10.

Let's have a check.

So which whole number is this arrow pointing to? So let's have a look.

It's in between negative 10 and negative 20.

It is on an unmarked interval.

What's it pointing to? Pause the video.

It is pointing to negative 12.

So well done, if you've got that, you're on track.

"A stem sentence can be used to determine the value of the unmarked intervals." So there are um intervals between zero and um.

There are um, ums and um, so the intervals are going up in ums. And this has been colour coded to help you.

So if you notice this is a purple line, there's two purple lines.

So they contain the same number.

There's two green lines, so they contain the same number and there's two black lines, so they contain the same number.

So that's just a little help.

Let's have a look, let's have a think then.

So let's start with there are um intervals between zero and um.

There are four intervals between zero and 20.

One, two, three, four, I could see that.

So there are four fives in 20, so I'm using my times-table's knowledge there.

So the intervals are going up in fives and we can use counting as a check.

five, 10, 15, 20, fantastic.

Now that we've established that the unmarked intervals are worth five, we can label those unmarked intervals including the negative ones.

So let's label the negative numbers too.

So we've got negative five, negative 10, negative 15.

Let's have a check.

"Complete the stem sentence and use counting as a check.

Label the unmarked intervals." So that stem sentence again, there are um intervals between zero and um, there are um, ums and um.

So the intervals are going up in ums. Pause the video and we'll have some feedback very shortly.

Did you manage to figure that one out? Let's have a look.

There are five intervals between zero and 100, and there are five twenties in 100.

So the intervals are going up in twenties and you could use counting as a check.

So let's start with zero, zero, 20, 40, 60, 80, 100.

That works, that fits.

We can also label the negative numbers too.

Negative 20, negative 40 and negative 60.

Are you ready for some independent practise? I think you are.

So task A1 is to use that counting strategy that we just used to determine the value of the intervals and then label those unmarked intervals.

Same thing for task two.

This time we've got zero and 50 labelled.

See if you can work out what the intervals are worth based on that knowledge.

And for task three, label the intervals again.

So this time you've got the fact, we've got a negative five, a 10, and a 25.

We haven't got zero this time.

So you're going to have to do a little bit of thinking about this one.

I think you can use that same strategy as before them.

And finally, for this cycle question for label the intervals.

So this time we've got -100, positive 100 and zero In between them.

Can you figure out what the other ones are worth? And then B, "Would -160 be one of the unmarked intervals if the number line were extended? Explain your answer." Okay, very best of a luck with that.

Pause the video and I'll see you soon for some feedback.

Welcome back, let's have a look then.

So for task A1, the intervals were as follows.

So we can establish that there are five intervals between zero and 10.

So therefore the intervals must be worth two.

So it can go two, four, six, eight and 10 and then extending it the other way, negative two, negative four, negative six.

For number two, this time there were five interval between zero and 50, and there are five tens between zero and 50.

So they're going up in tens and down in tens.

So 10, 20, 30, 40, 50.

And then the other way, negative 10, negative 20, negative 30.

And the intervals here this time are going in fives.

So from left to right, negative 15, negative 10, negative five, zero, five, 10, 15, 20, and 25.

4A, "Label the intervals." So that's going in 25.

So from left to right, it is -100, -75, -50, -25, zero, and 25, 50, 75, and 100.

And 4B, "Would -160 be one of the unmarked intervals if that number line were extended?" Well, no, it is not a multiple of 25, so it wouldn't appear as one of those unmarked intervals.

It would appear in between two of them though.

So that number would exist on that number line, somewhere between -150 and -175.

Are you ready for cycle two? That's positioning whole numbers.

God, let's go.

Okay, this number line is missing information, including the zero.

Is it possible to determine the value of the intervals using counting? Yes, you can choose any two marked intervals and work out the difference.

So any two of those would do.

To me it makes sense to choose two of the positive ones.

I think that'll be slightly easier.

Let's go for 20 and 60, but we're going to use the same strategy.

So there are um ums in um.

So the intervals are going up in ums. So I'm thinking about the difference between 20 and 60 and that's 40.

So there are um ums in 40.

What do we think? So we've got a one, two, three, we've got four intervals this time.

So there are four ums in 40 getting closer, aren't we? What do you think? There are four tens in 40.

So the intervals are going up in tens.

Another strategy in this particular example is, to look at the fact that two of the values have an equal distance from zero and you've got a midpoint in between them.

So that midpoint must be zero, zero must appear in the middle of the two numbers and then it becomes easier to notice that the intervals are worth 10.

So if you've got zero there, look, you can see it goes zero, 10, 20.

So it becomes quite easy to work out what they're worth.

So a couple of things to notice about that number line.

Let's do a check.

So what's the value of each of those letters? So you've got A, B, C, and D.

They are all appearing on unmarked intervals, but what are they worth? So you're going to need to establish first of all, what the unmarked intervals are and then work out what those specific ones are, okay? Pause the video and we'll see you shortly.

How do you get on with that one? Let's have a look then.

So A is -70.

So the number line is going in tens.

So it goes -80, -70, and then B, that's another couple of intervals away.

So that's negative 50, C is zero.

Now I actually started with C on that one.

Because I noticed that it had -20 and 20 and I noticed that was a midpoint between them.

That's how I helped to determine that we're going up and down in tens.

So C is zero and D is 80 'cause it's halfway between 60 and 100.

Okay, over to you for some independent work.

So number one, you're gonna place zero, one, negative one and negative two on that number line.

We've got negative four and four as the marked intervals and we've got lots of unmarked intervals.

Task two, what numbers are the arrows point to here? So again, you're going to need to establish what those intervals are worth.

The information we've got here is that zero is there and 10 is there.

So count how many intervals there are between those as your starting point.

For number three, now this time we've got a vertical number line.

You're going to position zero, 25, -25, 50 and -75 on that number line.

So good luck, I'll see you soon for some answers.

Welcome back, how did you get on with that? So for number one then it went as follows.

Negative two was just there, negative one there and then zero.

And then one next to it.

For number two, the arrows are pointing to, as follows from left to right, negative six, negative two, negative one and eight.

And then for number three then, the numbers are as follows.

So from top to bottom it goes, 100, 50, 25, zero, negative 25, negative 75 in those exact positions.

The final cycle then.

So positioning fractions and decimals.

So Sam and Laura are conjecturing about where to place half on a number line that shows positive and negative values.

They can't decide.

So they're having a nice healthy debate and that's what good mathematicians do.

Sam says, "I think it is to the left of zero because it's not a whole number." Okay, it's not a whole number, I agree with it.

So is it to the left of zero.

And Laura says you are right, but it is not, but it is greater than zero.

So therefore, it must be between zero and one." So have a thing.

Do you agree with Sam or do you agree with Laura? Laura says, "Half can be written as 0.

5.

So it's got five tenths more than zero." This number line has unmarked intervals in between the multiples of one.

So you can see we're counting in ones but with intervals in between them this time that are unmarked.

So that is where half lives.

It lives halfway between zero and one.

So the number line's counting in halves.

So Laura was right half is actually a positive number.

It's to the right of zero.

It's only just to the right of zero, but it is to the right of zero and therefore it is positive.

So with that in mind, what number do you think this could be? So it's halfway in between zero and negative one.

What could that be? It can't be half again, can it? But it could be negative half and in fact it is, it is negative half.

What about that one? What number could that be? So it's in between halfway in between negative two and negative three.

So on an unmarked interval, halfway between those, what could it be? Well I know that halfway in between two and three is two and a half, so maybe I can use that to help me.

It's negative two and a half 'cause it's on the negative side of the number line.

So this number line is counting in halves.

Just like before fractions and decimals can go in between the intervals.

What's the value of a? So we've got marked intervals here, we've got unmarked intervals, but we've also got values in between the marked and unmarked intervals.

So what could the value of A possibly be? Let's have a look at it.

So A is in between negative two and negative three.

It's not halfway in between though, is it? 'Cause that would be negative two and a half.

So it's somewhere between negative two, negative two and a half, what could that be? Greater than negative two and a half, less than negative two.

Halfway between those values.

What's half of a half? Think about your fraction knowledge.

Half of a half is a quarter, so maybe that could help.

So it is negative two and a quarter.

And we could also say that as a decimal, you may have had some quite recent experience of working with fractions and decimals and you might remember that a quarter is the same as 0.

25.

So negative two and a quarter is the same as negative 2.

25.

So 0.

25 is not a negative number because it is greater than zero, not less than zero.

So although it's less than one, it's not negative.

So there we go, there's 0.

25.

Let's do a little check then.

What's the value of C, do you think? So let's have a look at it.

So in between negative three and negative two, it's closer to negative three.

It's halfway between the marked and unmarked interval there.

So see if you can figure that one out.

Pause the video.

How did you get on? Did you get that one? That was tricky wasn't it? Let's have a look.

That's negative 2.

75, which you might also have said as negative two and three quarters.

Well done if you've got that, you're really on track, you're doing really well.

Over to you then for the final practise tasks.

So for task C, number one, fill in those missing numbers please.

So you can see we've got some marked and unmarked intervals and some of them, some of the gaps are on the unmarked intervals and some are in between the marked and unmarked intervals.

So see if you can figure out what the numbers are worth.

And then finally, the last task for today, "Draw lines to place the following numbers on the number line." So we've got 0.

5, negative half, one quarter, 1.

5, and negative 2.

5.

Where would those values appear on that number line? Pause the video and I'll see you soon.

Welcome back.

Okay, let's see how you got on there.

So for number one, it goes as follows.

So from left to right, negative 2.

5 is halfway in between negative three and negative two.

Negative 1.

5 is halfway in between negative two and negative one.

And then we've got negative 0.

5, halfway in between negative one and that unmarked interval, which would be negative half.

And then the last one, 2.

5.

Now you could say those as fractions as well.

So from left to right, that goes negative two and a half, negative one and a half, negative three quarters and two and a half.

And your last task then, so 0.

5 is here.

That's the same as half and that's halfway in between zero and one, negative half is just here.

So halfway between zero and negative one.

One quarter is there.

So that a quarter is half of half.

So that's halfway between zero and half.

1.

5 is here, right (indistinct) in the middle of one and two and negative 2.

5 over here, exactly halfway between negative two and negative three.

Well done if you got that.

That certainly was challenging.

We've come to the end of the lesson.

Today's lesson has been identifying and placing negative numbers on a number line.

Number line showing positive and negative numbers can come in a variety of forms so they can be horizontal or vertical, and we've seen both kinds today.

They can have unmarked intervals.

So we've seen some that have got unmarked ones where there's no number next to the interval.

The marked intervals can show a wide range of different multiples.

We've looked at things like twos, fives, tens, twenties, all sorts.

The unmarked intervals can be determined by counting those intervals.

So count the marked, count between the marked intervals and that will help you to determine what they're worth.

And the intervals can show fractions and decimals as well as whole numbers.

And then lastly, values can exist in between the marked and unmarked intervals.

You've done really well today and I've really enjoyed exploring negative numbers on a number line with you and I hope to see you again soon.

But until then, take care and goodbye.