Lesson video

Hello there, I'm Mr. Tilstone.

I'm really delighted to be working with you today on a lesson about negative numbers.

So if you are ready to begin, let's go.

The outcome of today's lesson is, I can explain how the value of a number relates to its position from zero.

Our keywords today, we'll do this in a my turn, your turn style, are my turn, scale, your turn, my turn horizontal, your turn.

Let's see what those words mean.

You may or may not have encountered them before.

So a scale is a number line with equal divisions or for equal values.

You find scales on rulers, scales for mass and capacity.

On a thermometer the scale is the number line showing the temperature, and we're going to focus on that today.

A horizontal line goes in this orientation, side to side like the horizon, that's how we remember it.

Our lesson today is split into two cycles.

The first being the value of numbers on a horizontal number line, and the second applying knowledge of number lines.

So let's begin with the value of numbers on a horizontal number line.

Are you ready? So in this lesson today, you're going to meet Jun and Lucas.

You may have seen these guys before.

They're gonna be giving us a little bit of a helping hand.

Have a look at these thermometers and think about the scales on these thermometers.

Are the scales horizontal or vertical? So do they go like the horizon, so they're horizontal or are they vertical? Let's have a think.

They're vertical.

Temperatures above zero are positive, you may have encountered that knowledge before, and temperatures below zero are negative.

Have a look at these ones.

Something changed there, didn't it? What's the same and what's different? Let's have a think about that.

What's the same and what's different compared to the thermometers that you saw before? Hmm.

Well first of all, this time the scales are in fact horizontal.

They go like the horizon, they're horizontal.

The temperatures though have not changed that stayed the same.

They're the same temperatures as before.

And this time though, the temperatures to the right of zero on the scale are positive.

So before the ones above zero were positive.

This time the ones to the right of zero are positive.

And temperatures to the left of zero this time are negative.

And before it was temperatures below zero are negative, so a small change there.

But whether horizontal like the bottom example or vertical like the one on the left, temperatures greater than zero are positive.

So however we look at it, temperatures greater than zero are positive, and temperatures less than zero are negative.

And that's true with both of those thermometers despite the differences in orientation.

So we're gonna chant those very important generalisations with Jun and Lucas.

So I'll read it first.

So negative numbers are less than zero.

Can you read that with me please? Negative numbers are less than zero.

Now just you.

And positive numbers are greater than zero.

Let's say that together, ready? Positive numbers are greater than zero.

Now just you.

Fantastic.

Very important information.

So let's have a look this time at a number line.

It's a scale.

So it's got a positive side and a negative side.

So the positive side goes 0, 1, 2, 3, 4, and 5.

So 1, 2, 3, 4, and 5 are positive numbers, zero is not positive, it's not negative either, they're right of zero and they're positive numbers.

The left hand side, it goes negative one, that's a first number to the left of zero, so negative one, negative two, negative three, negative four, negative five.

And you could hear as I was saying those, those are negative numbers.

So to the left of zero, negative numbers, and we're going to really focus today on that horizontal scale.

And remember, zero is neither positive nor negative, it's a special case.

It's the only one that is.

So which side of zero would these numbers be positioned on the number line? So remember to the left of zero negative to the right of zero positive.

So we've got negative 12.

There's a little clue in how I said that, isn't there? 14, seven, negative seven, negative three and five.

So we're not too concerned at the minute about exactly where they would be, just whether they're on the left or right of zero.

So negative numbers are less than zero, positive numbers are greater than zero.

Thanks for the reminders.

Here we go.

So to the left of zero, we've got negative seven, negative 12 and negative three.

They are negative numbers, they go to the left of zero, and to the right of zero, we've got five, seven, and 14.

Have you understood that so far? Do you think you're doing okay? Shall we have a little bit of a check to see? Where would 13 be positioned on this number line? Would it be A, to the left of zero, or B, to the right of zero? Talk to your partner, pause the video, make a decision.

What is your thought, what did you say? In fact, they are to the right of zero.

13 will be to the right of zero, because it's a positive number.

Another check then, where would negative 50 be positioned on the number line, to the left of zero, or to the right of zero? Have a chat, pause the video.

Well, negative 50 is a negative number, that means it is to the left of zero.

Okay, so we've got a number line here, it's got positive numbers on, it's got negative numbers on, and it's got zero on.

It's going up in ones.

Let's have a look at these two values.

We've got three and negative three.

Is there anything that you notice about those numbers? What about four and negative four? Hmm.

I'm noticing something about that pair of numbers, something they seem to have in common.

For both positive and negative numbers, the larger the value of the number, the further it is from zero.

So four is further from zero than three, so it's got a larger value.

Four is four away from zero and three is three away from zero.

So four is further from zero.

Four is greater than three.

How about this? Negative four is four away from zero, but negative three is three away from zero.

So negative four this time is further from zero.

So negative four is less than negative three.

Negative 10 would appear to the left of negative seven.

It's further away from zero, it's less than negative seven.

Here we go.

So here we've got two, a positive two, and here we've got negative three.

Let's use the stem sentences to describe the positions of negative three and two.

So, mm is a mm away from zero.

Mm is mm away from zero.

Mm is closer to zero.

Mm is further from zero.

Let's see if we can fill that in.

So we've got negative three is three away from zero.

Two is two away from zero.

Two is closer to zero.

Negative three is further from zero.

We can see that quite clearly when it's represented on the number line.

So let's use those stem sentences to describe the positions of negative three and four.

So same again, just as before.

Negative three is three away from zero.

Four is four away from zero.

Negative three is closer to zero.

Four is further from zero.

A positive and a negative number can be the same distance away from zero but in opposite directions.

Let's have a look at that on the number line.

So look, we've got negative three and three there they're same distance away but in opposite directions.

And then we've got another example here look.

Four and negative four are the same distance away from zero, but in opposite directions, the distance is the same.

Let's have a check for understanding.

You're going to complete the stem sentences.

So mm is a mm away from zero.

Mm is mm away from zero.

Mm is closer to zero.

Mm is further from zero.

Work with a partner if you can, pause the video, and we'll give you some feedback shortly.

Did you manage to successfully fill in those stem sentences? Let's see what the answers are.

So we've got negative four is four away from zero, and five is five away from zero.

You might have done those the opposite way round.

That's fine too.

Negative four though is closer to zero and five is further from zero.

Well done if you got that.

I think you're probably ready now for an independent task.

So task one is, sort the following numbers into their correct columns in the table.

So we've got negative four, seven, 25, negative 10, negative 40, 12, and negative seven.

So you've got that number line look, got zero in the middle, it's got a left hand side, it's got a right hand side.

So what numbers will go where? See if you can sort them out.

Task two, sort the following numbers into their correct positions on the number line.

So we've got 10, negative three, negative 10 and three.

And you can see they've already been marked out, you've just got to decide which one goes where.

Task three, which whole numbers can be found between negative 11 and negative seven? So we've got more than one answer for that one.

Task four complete the stem sentence, mm is mm away from zero, mm is away from zero, mm is closer to zero, mm is further from zero.

And finally task five for each number line, circle the number that is furthest from zero.

So good luck with all of that.

Pause the video and I'll see you again shortly for some answers.

See you soon.

Welcome back.

Let's see how you got on.

So for task one, sort the following numbers into the table.

So to the left of zero were all the negative numbers.

So negative four, negative 10, negative 40, and negative seven.

You could have those in any order by the way.

And then to the right of zero, that's all the positive numbers, so that's 7, 25 and 12 again in any order.

Well done if you got those.

Number two, the numbers go in these positions on the number lines.

So from left to right it goes negative 10, negative three, three and 10.

And the whole numbers that can be found in between negative 11 and negative seven are as follows, we've got negative 10, negative nine and negative eight.

So well done.

There are values that exist between those as well, but they are the whole numbers.

And complete the stem sentence, so here we've got negative two is two away from zero, five is five away from zero, negative two is closer to zero, five is further from zero.

And finally in this cycle, circle the number that's furthest from zero.

So in the first one, four's further from zero, the second one negative four's further from zero.

And in the third one, negative five is further from zero.

Are you ready for cycle two? Let's see.

So this is applying knowledge of number lines.

Which number is further from zero, negative seven or five? You might have a little picture in your mind already and be able to work that out and think about that based on what we did in the first cycle.

Well Jun and Lucas are confident that they can answer this without a number line.

Can you too? Let's see what they said.

I can visualise the number line in my head, says Jun, with zero in the middle, negative seven on the left and five on the right.

So he is picturing exactly what you just explored just then with the number lines.

And Lucas says the jump to five is smaller than the jump to negative seven.

So negative seven is further from zero.

The number tells you how far it is from zero.

Negative seven is seven whole number gaps away from zero says Jun, and Lucas says five is five whole number gaps away from zero.

So negative seven is further from zero.

We didn't need a number live for any of that.

Lucas is using a stem sentence to explore this.

Negative seven he says, is seven away from zero.

Five is five away from zero.

Five is closer to zero, and negative seven is further from zero.

So once again, there was no need for a number line.

How are we doing so far? Are you ready for a little check? So you're going to use that same stem sentence or stem sentences to compare the numbers.

You're gonna work in pairs if you can as well.

So we've got two pairs of numbers to compare, negative six and eight, and then negative seven and five.

So that stem sentence again goes, mm is mm away from zero, mm is mm away from zero, mm is closer to zero, mm is further from zero.

So pause the video, feedback coming.

How did you and your partner get on? Did you manage to agree on the answers here? Well let's have a look.

So negative six is six away from zero, eight is eight away from zero, negative six is closer to zero, eight is further from zero.

And for the second pair, negative seven is seven away from zero, five is five away from zero, five is closer to zero, negative seven is further from zero.

Well done if you've got those.

Okay, this time we're going to compare three and negative three.

Here's what Jun thinks.

I think three and negative three are equal to each other because they are both three away from zero.

Lucas disagrees and as mathematicians, that's fine to do.

He says, I think that three is greater than negative three because it is further right on the number line.

Hmm, who's right I wonder? On this horizontal number line, the numbers are increasing in value the further to the right they are, as you can see.

So the further the right they go, the greater the number.

And that remains true even if the number line is extended past zero.

So as we're going right on the number line, the numbers are increasing in value.

So negative four is worth more than negative five.

Negative one is worth more than negative three, let's say.

So they're increasing in value.

Likewise on this number line numbers decrease in value the further left they are.

So three and negative three are the same distance from zero, that's correct, but three is greater than negative three, and negative three is less than three.

So they're not worth the same.

Okay, let's compare negative three and negative five.

This time Jun says, I think negative three is less than negative five because three is less than five.

Lucas says I'm visualising the number line, negative three is closer to zero.

It is also further to the right.

So negative three is greater than negative five.

I like that he was picturing that number line in his head when he did that.

Okay, what do you think? Who do you think's right? Let's have a look.

So we've got that number line again.

Lucas is correct once again.

I can compare the numbers without using a number line.

I can visualise the number line and use what I know about the positions on the number line.

I can also think about temperatures.

So negative three is warmer than negative five because it's closer to zero.

So he's thinking about how close it is to zero.

Inequality symbols can also be used.

You've probably used these before.

Let's have a look again.

So we've got negative five is less than negative three.

And negative three likewise is greater than negative five.

And you can see that on the number line.

Let's do a check.

True or false? Negative 10 is greater than negative one.

Is that true or false? You might have your own justifications for that.

Let's give you some possibilities.

10 is greater than one.

That means that negative 10 is greater than negative one, or negative one is closer to zero than negative 10.

So negative one is greater than negative 10.

Have a good think about that.

Pause the video and see if you can decide, is that true or false? Did you manage to come to an agreement with the people around you? Let's have a look.

That's in fact false.

It's not true to say that negative 10 is greater than negative one.

It's actually less than.

Negative one is closer to zero than negative 10.

So negative 10 is greater than negative one.

Remember the further to the right the numbers go, the greater they are.

Time for some final practise.

For task one, for each pair put a tick in the correct column.

So we've got a table here, and we've got a pair of numbers each time.

So the first one's negative seven and 12.

So you've got to decide, is the positive number further from zero, in this case that's 12, is a negative number further from zero, in this case that's negative seven, or are they the same distance from zero? So make put a tick each time for each pair of numbers please.

And for number two, fill in the missing symbols.

Use these inequality symbols, less than, greater than, or equals to each time.

So for each set of questions, describe what you notice.

The first one has been done for you.

So we've got four is less than five.

So see if you can do that with the remaining numbers.

For B, fill in the missing symbols.

Again, use those inequality and equality symbols, for each set of questions describe what you notice.

And then for C, fill in the missing symbols again using those equality and inequality symbols.

And again, describe what you notice each time.

And the last tasks for today with your partner, you're going to choose two number cards and each time decide which one is furthest from zero.

And you can do that as many times as you like.

Okay, well good luck, and I will see you very soon for some feedback.

And welcome back for the last time.

Let's see how we got on.

So for task number one, with negative seven and 12, the positive number that's 12 is furthest from zero with negative 12 and seven, the negative number was furthest from zero.

With negative five and five, they were equally distanced from zero, so the same distance from zero.

With 10 and negative one, the positive number that's 10, was further from zero, quite a lot further in fact.

And with 10 and negative 10, they've got an equal distance from zero, same distance, and with 10 and negative 100, the negative number is further from zero, quite a lot further in fact.

For number two, it went as follows, so the first one was done for you, so four is less than five, five is less than six, six is less than seven.

And the second set, negative four is greater than negative five.

Negative five is greater than negative six.

And negative six is greater than negative seven.

Now you might have put something along the lines of, when two numbers are negative, the one with the lowest digit is greater because it's closer to zero.

And for the next set we've got 41 is greater than 39, 41 is greater than 40, 41 is equal to 41, and 41 is less than 42.

And then negative 41 is less than negative 39, negative 41 is less than negative 40, negative 41 is equal to negative 41 of course.

And then negative 41 is greater than negative 42.

And you might have put something along the lines of, with two positive numbers the one furthest from zero is the greatest, with two negative numbers, the opposite is true, the one closest to zero is the greatest.

And then for C, we've got negative one is less than one, negative two is less than one, negative three is less than one, negative four is less than one.

And one is greater than negative one, two is greater than negative one, three is greater than negative one, four is greater than negative one.

And what you might have noticed is something like this, negative numbers are always less than positive numbers, or positive numbers are always greater than negative numbers.

And for task three, when you were choosing the cards, all sorts of possibilities here, many, many possibilities.

But let's say you took these two cards, negative five and 11, here's what you'd say.

Negative five is five away from zero, 11 is 11 away from zero, negative five is closer to zero, 11 is further from zero.

We are coming to the end of our lesson.

Let's summarise our learning for today, shall we? So our lesson today has been, explain how the value of a number relates to its position from zero.

For both positive and negative numbers, the larger the value of the number, the further it is from zero.

So for example, eight is further from zero than five.

And likewise negative eight is further from zero than negative five.

Positive and negative numbers can be compared using a number line, but ideally they can also be quickly compared without number lines.

And that saves a lot of time, doesn't it? And when comparing two negative numbers consider which is closer to zero, the closer it is to zero, the greater its value.

And that's the end of today's lesson.

I have thoroughly enjoyed working with you on explaining how the value of a number relates to its position from zero.

And I'd really like it if we got the chance to work together again.

But until then, take care, and good bye.