Lesson video

Hello there.

My name is Mr. Tilstone.

I feel really honoured and excited to be working with you today on a lesson about negative numbers.

So if you are ready, let's begin.

The outcome for today's lesson is I can use knowledge of positive and negative numbers to calculate intervals.

We've got two keywords today.

So my turn, your turn.

My turn, bridging, your turn.

My turn, partitioning, your turn.

Let's find out what those words mean.

You might have heard of them before.

So bridging is a strategy which uses addition or subtraction to cross a number boundary.

You might remember doing this in key stage one.

An example would be turning eight plus seven into eight plus two plus five, thus bridging through 10.

And we're going to be doing something very similar to that with negative numbers.

Breaking a number up into two or more parts is called partitioning.

And in the example above seven has been partitioned into two and five, has been broken into two parts.

Our lesson today has three cycles: differences between positive and negative numbers, adding and subtracting positive to negative, and adding and subtracting within negative numbers.

But let's start with cycle one, which is differences between positive and negative numbers.

Are you ready? In this lesson today, you're going to meet Izzy and Lucas, who you might have seen before.

They're going to be helping us today.

Okay, let's have a look.

We've got a thermometer here, a vertical thermometer, and they are typically vertical like this one.

The marks can be multiples of different numbers and you might have seen thermometers going up in tens, fives, twos, all sorts.

This particular one uses multiples of one.

The numbers can appear on the left or on the right or like this one, both.

So there's all sorts of different ways a thermometer can be set out.

However, they can also appear in horizontal form, and in this lesson the focus will be on horizontal thermometers.

So numbers greater than zero are positive.

Numbers less than zero are negative.

And zero itself is neither positive nor negative.

So the temperature was four degrees celsius in the day and then it dropped to -3 degrees Celsius at night.

What was the change in temperature? Hmm.

Visualise picture in your head, a thermometer showing the first temperature and then another one showing the second.

Can you do that? Can you imagine that? It might look something like this.

So it's four degrees Celsius in the day, we can see that on the top thermometer, a -3 degrees Celsius at night.

So the first one is to the right of zero and the second one to the left of zero.

Zero is an essential number to consider when calculating the difference between positive and negative numbers, it's what we're going to use as a bridge.

Bridging through zero from one value to the other will help to establish the difference.

Start by establishing the difference from zero of the positive value, so that's the four.

Then establish the distance from zero of the negative value, so that's -3.

So the distance between 4 and 0 is 4 and the distance between -3 and 0 is 3.

And all we have to do is combine the differences.

So 4 + 3 = 7, and you could do it the other way round two, 3 + 4 = 7.

The difference between those temperatures is seven degrees Celsius.

This method fits the story because it's showing a reduction in temperature.

The temperature is reducing.

It's going down.

However, the difference is the same whether we start from the positive number and work backwards or start with a negative number and work forwards.

It's quick and efficient to draw a single number line to show the change, including only the numbers you need to think about.

So before we did two different number lines to represent the two different temperatures when all the numbers in between them, but we don't need to do that.

We can be more efficient.

Let's have a look.

So let's start by drawing the horizontal number line.

And then on the right hand side of that, we're going to include one of the pieces of information that we know, which is four degrees, the positive temperature.

We're gonna put that there.

Then on the other side of the number line, we can put our negative temperature, that's -3.

Now, somewhere in between those, and don't worry too much about the positioning, is going to be a zero.

So we can draw a bridge between four and zero and the difference between those is four.

And we can draw another bridge from 0 to -3, and that's three.

So we've got two different distances from zero.

Combine those together and we've got 4 + 3 = 7.

You could do 3 + 4 = 7.

But the change in temperature is seven degrees, and we've done that by bridging through zero.

How about this one? The temperature was -5 at night and then it increased to six degrees in the day.

So once again, we've got a negative temperature and a positive temperature.

And the question is not what's the difference, but what's the increase in temperature? Again, let's do a number line, a horizontal number line.

Let's start with one of the pieces of information we know.

We know it was -5 at night.

We know that in the day it was six degrees.

We know that somewhere in between those is the zero.

We know that's a five degree difference between -5 and 0.

We know there's a six degree difference between zero and six degrees.

All we've gotta do is combine those together.

5 + 6 = 11.

There was an 11 degree Celsius increase in temperature.

That's the bridging strategy.

This can also be described as an 11 degree Celsius difference in temperature.

Let's do a little check to see if you've got that.

So here's the problem.

The temperature was -3 degrees Celsius at night and then it increased to 9 degrees Celsius in the day.

So we've got a negative value, we've got a positive value.

What was the change in temperature? Draw a number line to show your working.

So don't forget to put that zero on.

Don't forget to use a bridging method.

Pause the video and have a go.

Okay, how did you get on with that one? Do you think you've got to grips with this strategy? Let's have a look.

So here's what your number line might have looked like.

And then when we add those two differences or distances to zero, 3 + 9 = 12.

There was a 12 degree increase or difference in the temperature.

Really well done if you got that.

Let's change the context.

So this one's not going to be about temperatures.

Izzy was on the -2nd floor of a hotel.

She took the lift to the fifth floor.

How many floors did she go up? Now, in this case, we're going to use a vertical number line or at least initially to represent this problem.

'Cause that's more like what a lift would look like, it goes up and down.

So we've got some information on there that we've established.

We know the negative value, that's -2.

We know the positive value, that is 5.

And zero is going to be somewhere between those.

So the jump from -2 to 0 is 2, and the jump from 0 to 5 is 5.

Combine those two distances together and we've got 7.

So she went up seven floors.

Now, that same calculation can be performed on a horizontal number line.

And it would look something like this.

2 + 5 = 7.

So in both cases, we've got the negative number on one side of the number line, the positive number on the other side of the number line, zero in between, and then we are considering the jumps, to and from zero.

Now what does each of the numbers in this calculation represent? So we've got 2 + 5 = 7.

So that same context from before, the same story about the lifts.

Let's have a think.

The two represents the difference between the floor where Izzy started and the ground floor, the five represents a difference between the ground floor and the floor where Izzy ends up, and the seven represents a total number of floors that Izzy goes up.

So that's what each of those numbers means.

So Izzy was on the fifth floor of a building.

She took the lift to the -2nd floor.

How many floors did she go down? Now is that the same problem or different? Hmm, it is a similar one.

And it's now become a positive to negative story, starting with a positive and working down to the negative.

However, the answer will still show the difference between the floors.

So it's going to be exactly the same as before.

So whether starting with a positive or the negative, make two jumps, the first going to zero and the second going from zero, then just combine them together.

Let's do a check.

Gonna change the context a little bit here.

So a polar bear was on an iceberg at four metres above sea level.

It then jumped into the sea diving down to -5 metres or five metres below sea level.

What was the total distance that the polar bear went? So consider your positive value, consider your negative value, and use a jumping or bridging strategy to find the difference between them.

Pause the video.

Let's have a look at an answer to that and a possible way that you might have gone about it.

So we've got a horizontal number line here.

We've got the -5 on the one side and the four on the other side and zero in between them.

And we're going to make two jumps.

So the 4 to 0 and the 0 to -5.

And then what we're going to do is combine those differences together.

So 4 + 5 = 9.

So the polar bear went down nine metres.

Time for some practise.

Task one, a penguin was on an iceberg at four metres relative to sea level and then it jumped into the sea diving down to -3 metres relative to sea level.

What was the total distance down that the penguin went? Now, I've given you a little bit of help here by already drawing the number line for you.

So you've gotta fill in the gaps.

Number two, Izzy was on the third floor of a building.

She took the lift to the negative fifth floor.

How many floors did she go down? Draw a number line to work out the change, could be horizontal, vertical.

I'd recommend the horizontal ones.

Question three, a penguin was in the sea at -5 metres relative to sea level.

It jumped onto an iceberg at four metres relative to sea level.

What was the total distance up that the penguin went? So you could use a number line to help solve that.

Number four, there's a 10 degree difference between the night temperature and the day temperature.

The night temperature's negative, the day temperature's positive.

So there's a 10 degree difference between them.

What could the temperatures have been? Give at least three possibilities.

You might find yourself on a bit of a roll with that and want to do even more, but at least three please.

Okay, pause the video, good luck, and I'll see you soon for some feedback.

How did you get done with that? Let's have a look.

So number one, the missing values were -3 on one side, four on the other side, then that make makes the jumps 3 and 4.

When you add 3 and 4 together, you get 7.

Number two, draw number line to work at the change.

It might have looked like this, yours might have been vertical as well, but this is a horizontal one.

So we've got -5 on one side and then a jump of five to get to zero and then three on the other side and a jump of three to get there.

Combine those together and you've got eight.

A penguin was in the sea at -5 metres relative to sea level.

It jumped onto an iceberg at four metres relative to sea level.

What was the total distance up that the penguin went? You might have done a number line to work that one out.

You might not have needed to use a number line by this point.

You might have pictured that number line in your head.

But either way, the answer is nine metres.

So well done if you've got that.

Okay, so that 10 degree difference, so we're looking for any three or more of the pairs below.

And you might have been very smart and put something involving fractions or decimals as well.

They're acceptable too, but the whole number of pairs are as follow.

So you could have -9 and 1 <v ->8 and 2, -7 and 3,</v> <v ->6 and 4, -5 and 5,</v> <v ->4 and 6, -3 and 7,</v> <v ->2 and 8, -1 and 9.

</v> So well done if you've got three of those.

I wonder if you had a system like that.

Cycle two, so we're going to be adding and subtracting going from positive to negative.

So back to thinking about temperature.

The temperature currently shows <v ->4 degrees Celsius.

</v> The temperature rises, so it goes up, by seven degrees Celsius.

What is the new temperature? Hmm.

So what's different about this question compared to the previous questions? Have a think about that.

This time, the difference between the temperatures has already been established.

We're not working out the difference.

We know the difference.

The difference is seven degrees.

So the focus on the question is what the new temperature will be.

So let's look on a number line.

So this is a horizontal number line.

So it's currently four, a -4 degrees, which is here, and it rises, it goes up, we know the difference is seven degrees.

So it involves an increase in temperature, so the jump will be to the right.

The new temperature can be calculated by bridging to and then from zero.

So we're still using bridging but in a slightly different way.

So here we go, so we're bridging to 0 from -4.

So what's that? What kind of jump have we made there? We made a jump of four.

And then we're going to partition that seven into four and three.

So three is what remains of the seven.

So the new temperature is going to be three degrees.

So the seven was partitioned into four and three.

It was broken up into two parts.

So we use that distance to zero as the first part and then the distance from zero as a second part.

The temperature currently shows <v ->6 degrees Celsius.

</v> The temperature rises, so it goes up, it increases, by nine degrees Celsius.

So that's the difference already been established.

What's a new temperature? So we're going to use that partitioning strategy once again.

Going to jump to zero and then from zero, partitioning that nine degrees.

Okay, so once again, we can draw a number line, and just the values that we know and need.

So in our number line this time, just like before, we're going to use a horizontal line, and then on the left hand side of it are negative value, which is -6.

Now the other value that we know is the nine degrees and we're going to partition that nine degrees.

So we're going to jump to zero and then from zero.

So think about the two numbers that we're going to need.

Okay, so if we jump in to zero, that's six degrees.

And then what's left of the nine degrees? Three degrees.

So that's going to be our second jump.

So the new temperature is three degrees.

The nine was partitioned into six and three to bridge through zero.

Let's do a check.

The temperature currently shows <v ->8 degrees Celsius</v> and the temperature rises by 12 degrees Celsius.

What's the new temperature? Well, I'm going to tell you it's four degrees Celsius.

And we've got a number line here showing it.

I want you to do this.

So I want you to use the language, partitioned and bridge, to explain the process that's been used to solve this calculation.

So how have we solved that one? Talk to your partner and see if you can come up with a really good explanation.

Pause the video.

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

You might have said something along the lines of this.

The 12 was partitioned into eight and four to bridge through zero.

So that's using our two keywords.

Let's do a check.

The temperature currently shows <v ->2 degrees Celsius.

</v> The temperature rises by 16 degrees Celsius.

Use the strategy of partitioning and bridging to and then from zero to work out the new temperature.

Pause the video and give that a go.

Let's see, so the 16 has been partitioned into two and 14, so therefore the new temperature is 14 degrees Celsius and we don't really need to draw a number line as long as we understand them how to partition the number.

The same strategy can be used when temperatures decrease from positive to negative.

So the temperature currently shows three degrees Celsius.

The temperature drops by seven degrees Celsius.

So it's going down, it's decreasing.

What's the new temperature? This question involves a decrease in temperature.

So the jump will be to the left and we're going to partition that seven degree drop into two numbers.

So first of all, we've jumped from three to zero, we've made that bridge, which is a difference of three, and then let's consider what's left of the seven.

So we've partition the 7 into 3 and 4.

So it's going to take us to -4.

That is the new temperature.

The new temperature is -4 degrees Celsius.

The 7 this time was partitioned into 3 and 4.

The temperature currently shows 3 degrees Celsius and the temperature falls by 9 degrees Celsius.

What is the new temperature? This can be calculated using a number line and just the numbers given in the problem to be efficient.

So we draw our horizontal number line.

This time we know the positive value, so we're going to include that and we're going to jump to zero.

And we're going to think about partitioning that nine degree drop into two numbers.

We've got the first one here though.

The first one's three.

So three and what make nine? 3 and 6 make 9.

So therefore, the new temperature is -6 degrees.

The 9 this time was partitioned into 3 and 6 to bridge through zero.

We're ready for a little check.

Let's see if you've got that.

So the temperature currently shows four degrees Celsius and the temperature falls by seven degrees Celsius.

What is the new temperature? Draw a number line to show you are working.

So remember you know the positive value and then you're gonna partition that seven into two different numbers on a number line.

Okay, pause the video.

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

So we've got our number line here, horizontal number line, we've got our positive value on the right hand side, we've got our jump to zero, that's the difference of four degrees, and then we're partitioning the 7 into 4 and 3.

So therefore, it's -3.

The new temperature is -3 degrees celsius.

If you've got that, you are well on track in today's lesson and congratulations for that.

I think it's time for a practise task.

So number one, A, the temperature rises from -8 degrees Celsius by 11 degrees Celsius.

What is the new temperature? And the number line's been given to you already there.

You've just got to fill in the gaps.

And then B, the temperature falls from seven degrees Celsius by 15 degrees Celsius.

What's a new temperature? You may want to use a number line, but you don't have to.

Number two, a lift in a hotel goes up 10 floors from the floor marked -3.

What floor is it on now? So again, bridge to and from zero.

Use partitioning.

A diver makes a jump of 15 metres from 10 metres above sea level.

What's their new elevation? So they're going down, but can you work out the new elevation? So again, you're going to bridge to zero and from zero.

Time to pause the video and see if you can apply those skills.

Welcome back, how did you get on? So number one, the missing numbers were as follows.

So we've got a jump of 8 from -8 to 0, a jump of 3 from 0 to 3.

And then for B, the temperature falls from 7 degrees by 15 degrees celsius.

What's the new temperature? <v ->8 is the answer to that.

</v> <v ->8 degrees Celsius.

</v> And we can work that out by partitioning.

And for number two, the floor is now floor seven.

And for B, it is -5 metres is the new elevation.

Well done if you've got that.

The final cycle, this is adding and subtracting within negative numbers.

So before, we've looked at the difference between positive and negative.

This time we're going to stay in the negative numbers.

The temperature falls three degrees Celsius from -2 degrees Celsius.

What is the new temperature? So here we've got that marked out on the number line though.

So we've got -2 here and it's a fall of three, so it's going to be three degrees less.

So think about which way that's going to go.

Left or right.

So what's different about this question? Well, this time the calculation does not bridge zero.

It doesn't go past zero.

It stays on the left-hand side of zero, in the negative side.

Just like before, the decrease in temperature means a jump to the left on the number line, this time by three places.

Here we go.

So the new temperature is -5 degrees Celsius.

The temperature is -9 degrees Celsius and it rises by 6 degrees Celsius.

What is the new temperature? So we've got our thermometer here, we've got a a number line essentially, and it's got marked intervals that are going up and down in tens and it's got unmarked intervals that are going up and down in ones.

So we can see that a -9 degrees Celsius and it's going to rise by 6.

That means it's not going to bridge zero.

So we don't need a bridging strategy.

The increase in temperature means a jump to the right on the number line, this time by six places.

The increase in temperature is less than the distance to zero.

So six is less than nine essentially.

So no bridging will be involved.

Here's our jump, six places, and our new temperature is -3 degrees Celsius.

So six degrees warmer, the -9 degrees is -3 degrees.

Let's have a check.

The temperature is -8 degrees Celsius.

Calculate the new temperature if it: A, rises by five degrees Celsius, and B, falls by five degrees Celsius.

Have a go at that, pause the video.

Let's have a look.

Okay, so if it rises by five degrees Celsius, the new temperature is -3 degrees Celsius, and if it falls by five degrees Celsius, the new temperature is -13 degrees Celsius.

And you might have noticed all those temperatures stayed in the negatives.

So it didn't involve bridging.

It didn't go past zero.

Very well done if you've got that.

You're on track.

So I think you are ready for some final practise.

So task C, number one, A, the temperature falls from -8 degrees Celsius by nine degrees Celsius.

What's the new temperature? Remember, this is going to stay in the negative numbers.

B, there's a temperature increase from -12 degrees Celsius by seven degrees Celsius.

What's the new temperature? C, the temperature is -12 degrees Celsius.

By how much must it rise to reach -3? And D, the temperature is -5 degrees Celsius.

By how much does it fall to reach -13 degrees Celsius? And number two, A, a whale is swimming below the surface of the water at -50 metres.

So you can see the whale there.

It goes up 25 metres, and then down five metres.

Where is it now? Pause the video.

Good luck and I'll see you soon.

Okay, so how did you get on with that final task? Staying within the negative numbers.

So number one, A, the new temperature's -17 degrees.

B, -5 degrees.

C was nine degrees Celsius.

That's how much it must rise.

And D, eight degrees Celsius is a difference between those two temperatures.

And then the whale swimming below the surface of the water at -50 metres, goes up 25, down five metres.

Where is it now? And that is -30 metres.

And well done if you got that.

We've come to the end of the lesson.

So our lesson today has been using knowledge of positive and negative numbers to calculate intervals.

When calculating the difference between 14 degrees Celsius and -4 degrees Celsius, for example, the strategy of adding the two distances from zero might be used.

So we've got a different zero 14, a difference of four, 14 + 4 = 18.

When calculating the new temperature through a rise of 10 degrees Celsius from -4 degrees Celsius, for example, partitioning that 10 into 4 and 6 to cross the zero, to bridge the zero, is helpful.

This strategy can be used when going from a positive to a negative number.

And then finally, differences between two negative numbers can also be calculated.

It has been a great pleasure working with you today on this negative numbers lesson and hopefully I'll see you again in the near future.

But until then, take care and goodbye.