Lesson video

Hi, there.

My name is Mr. Tilstone.

It's my great delight to be here with you today for what will be the final lesson of the negative numbers unit.

So here's where we get the chance to tie all those skills together.

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

The outcome for today's lesson is, "I can solve problems involving negative numbers in different contexts." And our keywords today.

So my turn, your turn.

My turn, difference.

Your turn.

And my turn coordinates.

Your turn.

I'm hoping that you might have encountered those words quite recently, but let's have a little recap.

It won't hurt.

So the result of subtracting one number from another is the difference and differences are always positive.

Coordinates are a set of values that show an exact position.

They can be shown on a grid as pictured.

So here, you've got your x-axis, your y-axis, and then your origin.

The zero part where the axis meet.

So our lesson outline is split into two cycles today.

The first will be temperature problems and the second will be problems on a coordinates grid.

But let's start with temperature problems. So in this lesson, you'll meet, and you might have met her before, Laura.

So Laura's here today to give us a little bit of a helping hand.

Okay, temperature problems. So the temperature inside the cabin is 20 degrees Celsius.

The temperature outside the cabin, much colder, <v ->10 degrees Celsius.

</v> So we've got quite a difference there.

One positive temperature inside and one negative temperature outside.

So it might be a winter's day, it might be a very cold country, but it's a cold temperature.

What is the difference between the inside and the outside temperatures? The difference.

Okay, well let's start by drawing a number line containing all the known information that will help us to establish the unknown information.

So here's our horizontal number line.

And we know that minus 10 or -10 is the temperature outside the cabin.

So we can put that on the number line on the left-hand side.

And we know that the temperature inside the cabin is 20 degrees Celsius, so we can put that on the right-hand side.

Then we're going to bridge two and from zero to calculate the two distances from zero and then add them together to calculate the difference.

Let's do that.

So here's our zero.

Somewhere in between those two.

So a little closer to the -10, but don't worry too much about that.

So we're going to, first of all, do our first jump off.

First distance from -10 to zero is 10.

<v ->10 is 10 away from zero</v> and then 20 is 20 away from zero.

So we've now got two differences.

All we've got to do, add them together.

And the arithmetic there fairly straightforward, I think.

That's 10 add 20.

So it's calculate the difference.

Add the value of the distances from zero, 10 add 20 equals 30.

So there's a 30 degree difference between the outside and the inside temperatures, or if you like, between the inside and the outside temperatures.

So the temperature inside the cabin is 20 degrees Celsius.

The temperature outside the cabin is negative 10 degrees Celsius.

What's the difference? Well, Laura says, "I can do this quickly even without a number line." Oh, well that would save time, wouldn't it? If she didn't have to do a number line.

"I can visualise, ah, the two distances from zero." She can picture it.

"The difference is the total distance between the temperatures." So she's got -10 is 10 degrees away from zero, 20 is 20 degrees away from zero, 10 or 20 equals 30.

No need for a number line.

This contain the same information there.

Let us do a check.

Let's see if you've got that skill.

And you might want to draw a number line.

You might be able to do it without a number line by visualising like Laura just did.

The temperature inside the cabin this time is 18 degrees Celsius.

So still positive.

The temperature outside the cabin is -15 degrees Celsius.

It's still negative but different.

What is the difference between the inside and the outside temperatures? Can you visualise the two distances from zero to solve the problem? Okay, well give that a go.

Pause the video, we'll see you shortly.

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

Well, the inside temperature, that's 18, which is 18 away from zero and the outside temperature -15, that's 15 away from zero.

Add those two together and you've got 33 degrees Celsius.

Well done if you've got that.

You're on track in today's lesson.

You might have looked at the other way around though.

Instead of adding 18 add 15, you might have done 15 add 18.

Doesn't matter which way you start when it comes to differences.

You could go positive to negative or negative to positive.

It does not matter.

Here we go, if you did choose to use a number line, that's what it would look like on a number line.

So the difference between the temperatures, whatever way you choose to do it, is 33 degrees Celsius.

Okay, the temperature was -15 degrees Celsius at night and then it increased to six degrees Celsius in the day.

Increased.

Do you know what that means? It went up, so it increased.

What was the change in temperature? Okay, so I think just like before, we've got two values established.

We've got our negative value, we've got our positive value.

All we've got to do now is calculate the difference between them.

This time, we're not using the word difference, we're using the word change, but it means the same thing.

So we know the nighttime and daytime temperatures.

The problem is still to calculate the difference between the temperatures.

The temperature rose from -15 degrees to zero and then on up to six degrees.

So you can see, we've got that jump, that bridge.

The change in temperature is the sum of these.

So here we go.

Just like before, we've got two distances to zero.

From the negative to zero and from zero to the positive.

15 is the difference between -15 and zero and six is a difference between zero and six.

Others two differences together and we've got 21.

So the change in temperature was 21 degrees Celsius.

Let's do a check.

The temperature was six degrees Celsius in the day and then it dropped a -14 degrees Celsius at night.

What was the change in temperature? Can you visualise a change on a number line? Because if you don't have to use a number line, all the better, that means you've saved some time.

Okay, pause the video and give that a go.

Okay, let's have an answer to that then.

So here we go.

So the difference between -14 and zero is 14 and the difference between zero and six is six.

Six add 14 or 14 add six is 20.

So it doesn't matter which way around you have your add-ins, it's going to give us a difference of 20.

The change in temperature was 20 degrees Celsius.

Very well done if you got that.

You're doing well.

Okay, the temperature inside the cabin is 20 degrees Celsius.

Nice and warm.

The temperature outside the cabin is 30 degrees lower.

Hmm.

Now this is a different kind of problem, isn't it? Have you spotted the difference already? So what is the outside temperature? is the question.

Now, before we knew the inside temperature, we knew the outside temperature.

This time we do know the inside temperature.

We don't know the outside temperature.

We do know the difference though, so we're going to use that.

The difference this time is known before it was unknown.

The temperature outside the cabin is not known.

So can you visualise this information on a number line? Where is the missing information? So in my head at the minute, I've got a number line and I've got some information on that number line.

I can picture it and it looks a little something like this.

So we can partition the difference of 30 degrees and bridge through zero.

So we went from 20 degrees to zero, which is a difference of 20 degrees.

And of that 30 degrees we've still got 10 degrees left.

So instead of counting back 30, we counted back 20 and then 10 we took two steps, but two nice easy steps.

So the temperature outside the cabin is -10 degrees Celsius.

And once again, that technique is called partitioning.

So we've done two things there.

We've partitioned and we've bridged through zero.

The temperature inside the cabin is 20 degrees.

The temperature outside the cabin is 30 degrees lower.

What is the outside temperature? Let's see what Laura's got say about this same problem.

She says, like before, "I can do this quickly, even without a number line." She's very fluent, isn't she, Laura? I just partitioned the 30 into 20 and 10 because I knew the inside temperature was 20 degrees Celsius.

No need for a number line for Laura.

So the outside temperature had to be -10 degrees Celsius.

Brilliant.

Let's do a check.

The temperature outside the cabin is -8 degrees.

The temperature inside the cabin is 25 degrees higher.

So we're going the other way this time.

We're going from negative to positive.

Before, it was from positive to negative.

So what is the inside temperature? Visualise the information on a number line.

Where is the missing information? Is it a jump or a value on the number line? So have a good think about that.

Approach it in the way you need to approach it.

If you can do it without a number line, that is what I would recommend.

So pause the video and give that a go.

Let's have a look then.

So outside the cabin, -8, inside the cabin, 25 degrees higher than that.

That's not to say, 25 degrees inside the cabin, it isn't.

It's 25 degrees higher than -8.

So we're going to calculate that.

So we know there's a difference between those temperatures and we're going to bridge through zero to get there.

We're gonna bridge that 25.

So what could we partition the 25 into? Hmm.

Let's have a think.

We could do that.

So we could go from -8 to zero and then zero to 17 by partitioning the 25 into eight and 17.

So the temperature must be 17 degrees Celsius.

Very well done if you got that.

The temperature outside the cabin in the daytime is -3 degrees Celsius and it drops to -20 degrees Celsius overnight.

What's the difference in the temperature? Have you noticed something about that problem? Something different to the other ones.

I have.

Before, we were dealing with a positive and a negative value.

This time, both values are negative, but we are finding out the difference.

The higher temperature is known, the lower temperature value is also known, but the difference between the negative values is unknown.

So we're calculating a difference again, this time between two negative values And we could do a number line.

So we've drawn a quick number line.

This one hasn't got any positive values on, so we don't need anything to the right of zero just to the left.

So we've plotted in where -3 might roughly go and we know that it's been a drop to -20 degrees.

So we can plot both of those onto our number line.

What we've gotta calculate is a difference between those values.

What's the difference between -20 degrees Celsius and -3 degrees Celsius? Hmm.

The difference between three and 20 is 17 and it follows then that the difference between -3 and -20 is also 17 degrees Celsius.

The difference between two values is always expressed as a positive number.

So it's not -17, the difference is 17 'cause -3 is 17 degrees higher than -20 and -20 is 17 degrees lower than -3.

So it's always a positive difference.

Let's have a check.

The temperature outside the cabin in the daytime is -7 degrees Celsius and it drops to -22 degrees Celsius overnight.

So once again, two negative temperatures.

What is the difference in temperature? Try and visualise that number line.

Okay, pause the video, have a go.

Let's check your answers then.

So here's the number line.

Here's our values plotted onto it.

So we know that it was -7 in the daytime, -22 overnight.

Gotta work out the difference between those.

Well, I started by thinking about the difference between 7, 22 and I know you had to add 15 onto seven to get to 22.

So that's the difference.

And likewise, you can do 22, take away 15 to get seven.

So that's a difference.

So therefore, the difference between -22 and -7 is the same.

It's 15 degrees.

The difference between two negative values is the same as the difference between the same positive values.

So I'm using that to help me.

So well done if you realise that too and well done if you got the answer.

The difference, remember, is always expressed as a positive number.

So the temperature outside the cabin in the daytime is -3 and it drops by 17 degrees Celsius overnight.

Slight difference.

Can you spot the difference between this and the last one? What's the nighttime temperature? So this time, we know the daytime temperature, we don't know the nighttime temperature this time.

We do know the difference.

They're the things that we know.

So the difference of 17 degrees Celsius is in between two negative values.

So we're looking for a negative final answer.

So visualise that number line again to help you.

That's what I'm doing right now.

I've got that number line in my head.

It's got a zero on there, it's got a -3 and then I'm gonna do a jump from -3.

That's a jump of 17.

Where will that take me? Okay, and it looks just something like this.

So there's a -3, a difference of 17.

We don't know the nighttime temperature, we've got to calculate it.

The new temperature is another 17 degrees Celsius further away from zero.

So the new temperature is -20 degrees Celsius.

So in the same way that 17 more than three is 20, 17 degrees away from -3 is -20.

The nighttime temperature, -20.

Okay, little check.

We've got some answer descriptions here and we've got some questions.

You've gotta match them up.

Okay, so let's have a look.

Let's have a read.

So the temperature in France is 15 degrees Celsius.

The temperature in Iceland is -17 degrees Celsius.

How much warmer is France than Iceland, right? Is that a positive temperature? Is it gonna give us that a negative temperature or is it a difference or change in temperature? Let's have a look at number two.

The temperature in France is 18 degrees Celsius.

This is 32 degrees higher than the temperature in Iceland.

What's the temperature in Iceland? So once again, what do you might start up to? Positive temperature, negative temperature or a difference or change in temperature.

And the last one, the temperature in Iceland at midnight is -21 degrees Celsius, by midday, the temperature rose by 24 degrees Celsius.

What was the temperature at midday? So for the last time then, is that positive temperature, negative temperature or a difference or change in temperature? See if you can match them up.

Pause the video.

Let's see what you match to what then.

So the first one then is a difference or change in temperature.

The second one is a negative temperature and the third one is a positive temperature.

So well done if you match those correctly.

I think it's time for some practise.

Do you feel ready? Because I think you are ready.

So let's have a look at what you've got to do.

So task A, number one, the temperature inside an aeroplane, these are all about aeroplanes , it's 23 degrees Celsius.

The air temperature outside is -21 degrees Celsius.

Calculate the difference between the temperature.

So you often get this, so you want the inside of the plane to be nice and comfy and warm, but often when you're high up in the air, the outside temperature is very, very cold and a negative temperature.

So you've gotta calculate the difference, first of all.

Number two, the air temperature outside the plane is -27 degrees Celsius and it's 45 degrees Celsius warmer inside.

So what is the temperature inside the plane? And there's a 35 degrees Celsius difference between the inside and outside temperature of an aeroplane.

The inside temperature is 22 degrees Celsius.

Calculate the outside temperature.

And for each of those, think about what do you know? What else do you know and what don't you know.

What you're trying to find out.

And number four, it is 50 degrees Celsius warmer inside the plane than outside.

What could the inside and outside temperatures be? And the outside temperature here will always be negative.

Have a think about that many, many possibilities here.

See how many you can find.

Pause the video.

Good luck.

And I will see you very shortly for some answers.

How do we get on? Let's have a look.

So number one then, the difference between the temperatures.

If we add together our two distances from zero, we've got 21 and 23, that gives us 44.

So it's 44 degrees Celsius.

Well done if you've got that.

Number two, the air temperature outside the plane is -27 degrees Celsius and it's 45 degrees Celsius warmer inside.

So we know the difference in this case, when we add 45 by maybe partitioning and bridging through zero, so the temperature inside the plane is 18 degrees Celsius.

Number three, there's a 35 degree difference between the inside and outside temperature.

The inside temperature's 22 degrees.

Calculate the outside temperature, so we dunno the outside.

We know the other two bits of information.

And once again, we've used partitioning.

So we've partitioned that 35 into 22 and 13.

So the temperature outside the plane is -13 degrees Celsius.

Number four, this is just one of many possibilities.

We've got the inside temperature is 22 and the outside -28.

Your two distances from zero have to equal 50 in this case, and 28 plus 22 equals 50.

Time to move on to the final cycle and that is problems on a coordinates grid.

So hopefully, you've had some quite recent experience of using coordinate grids that have got four quadrants because that's what we're going to use today.

But let's start with a two quadrant one.

What are the coordinates for the other vertices of this rectangle? So can you see a rectangle on this two quadrant coordinate grid? We've got one of the coordinates listed, so that's negative 3, 4.

So it's -3 on the x-axis, 4 on the y-axis.

So negative 3, 4.

What about the other ones? Well, on this side, the top right-hand vertex is 1, 4 because it's 1 on the x-axis, 4 on the Y.

And then our bottom right-hand corner is 1, 1.

1 on the x-axis, 1 on the y-axis.

And finally, we've got -3, 1.

So -3 on the x-axis, 1 on the y-axis.

And I find that stem sentence very helpful.

So what are the coordinates for this point inside the rectangle then? So we've got something inside the rectangle.

We've got our vertices established for the rectangle.

What about that one? What do you think the coordinates are here? So let's see.

We've got -2 on the x-axis and we've got 3 on the y-axis.

So that's -2, 3.

What other coordinates would describe points that lie inside this rectangle? What about this one? Remember, it's, mm on the x-axis, mm on the y-axis.

Well here, we've got -1 on the x-axis and 2 on the y-axis.

So that's -1, 2.

Or what about this one? It's a tricky one.

It involves zero.

They're always a bit trickier.

So think, mm on the x-axis, mm on the y-axis.

It's not really gone anywhere on the x-axis, has it? So zero.

So 0 on the x-axis, but 3 on the y-axis.

Okay, what coordinates describe points that would lie inside this trapezium? Hmm, so we've got a different shape this time.

Coordinates that would lie inside it.

So we've got our first quadrant, our positive, positive one.

So both values have gotta be positive here.

So, mm on the x-axis, mm on the y-axis.

That's 1, 2.

Here, we've got our second quadrant.

So we've got, mm on the x-axis, mm on the y-axis, <v ->2 on the x-axis, 1 on the y-axis.

</v> That's -2, 1.

Here in our third quadrant, we've got, mm on the x-axis, mm on the y-axis, <v ->3 on the x-axis, -2 on the y-axis.

</v> And in our fourth quadrant, our final quadrant, we've got, mm on the x-axis, mm on the y-axis.

So that's 1 on the x-axis, -1 on the y-axis.

And here we are.

That's how we express those coordinates.

There is one point in each quadrant.

Notice how the positive and negative values change depending on which quadrant the point is in.

Let's have a check.

Which of these points would lie inside this rectangle? So we've got a rectangle there.

Look, and it spans all four of the quadrants.

Which ones would be inside? Hopefully, talk to your partner, if you've got a partner next to you.

So we've got 1, 1, -3, 4 -2, -2 and 1, -2.

So pause the video and decide which ones would lie inside that rectangle.

Did you manage to come to an agreement with your partner? Let's have a look.

So 1, 1 would be inside.

There it is.

<v ->3, 4 would not be inside.

There it is.

</v> <v ->2, -2 would be inside.

There it is.

</v> And then 1, -2 would be inside.

There it is.

So be very careful when you're reading and plotting coordinates to use the negative symbol correctly.

How could you change -3, 4 so that it is within the second quadrant section of the rectangle? Hmm, little challenge for you there.

Could be -3, 1.

Okay, what pairs of coordinates would describe points that would lie on the top green side of this rectangle? So you can see a rectangle.

Can you see the green line? What pairs of coordinates would describe points on that line? What would they have in common? Hmm.

So imagine the line extending beyond 5 and -5.

What other pairs of coordinates would lie on the line? And what do you notice about those coordinates? So here we go.

What have they got in common, first of all? Have they got the x-axis in common? No they haven't.

Have they got the y-axis in common? Yes, they have.

They're both 4 on the y-axis.

So there we go.

So anything on that line would be 4 on the y-axis.

We'll have different values for the x-axis, but 4 for the y-axis.

So what's the same and what's different about the points on this coordinate grid? Let's have a look.

Hmm.

Again, they've got something in common, haven't they? And something different.

An axis in common and an axis that's different.

What's the axis they've got in common? Let's have a look.

They've all got the same X coordinate but different Y coordinates.

So what is that X coordinate? How far across are all of these? -3.

So anything in line with those has got to be -3 something.

Imagine if the x-axis stretch beyond 5 and -5, which other points will be on the same line as -3, 2, -3, -3, and -3, -5.

Did you notice each of those starts with -3? So other things on the.

Other items on the line would also be -3 something.

Time for some final practise.

So prediction, nine of the coordinates for the points you plot in this task will have at least one negative value.

Hmm.

Okay.

So plot one point in each quadrant and join them to create a quadrilateral.

A quadrilateral.

Think how many sides that's going to have.

Plot 2 more points in each quadrant which fit inside your quadrilateral.

So two more inside that shape that you created.

And was that prediction correct? That nine of the coordinates will have at least one negative value.

Why or why not? That's one to think about, isn't it? Number two, which of these points is the odd one out and why? We've got 3, -3, -3, -3, 4, -3 and -3, -5.

Which one is the odd one out and why? B, plot them on a coordinate grid to help you explain.

And you've got a coordinate grid there.

And C, right? Two more points that will be in the group and two more that will be odd ones out.

So are you ready for this? Pause the video for one final time and give it a go and I'll see you shortly.

Welcome back for the last time.

How did you get on? That was a bit more challenging, wasn't it? Hopefully, you found some success there.

Let's have a look.

So for 1A, lots of possible shapes you might have plotted here.

But the point being there were quad laterals had to be four-sided shapes.

Plot two more points in each quadrant that fit inside your quadrilateral.

Again, lots of possibilities there too, but they've got to be inside that shape.

And was the prediction correct? Why or why not? You will have plotted 12 points using 12 sets of coordinates.

Three of them will be in the first quadrant with no negative values.

All the other points will have at least one negative value unless they contain a zero.

B2A, which of these is the odd one out? The odd one out is -3, -5.

And it's the odd one out because it doesn't have a Y coordinate of -3, but the others do.

So it wouldn't be on the green line.

So B, plot them in a coordinate grid.

Have a look at the example there on the screen.

And for C, two more points that will be in the group and two that will be odd ones out.

Well, there are lots of possible answers.

But all of the coordinates in the group will have a Y coordinate of -3 and the odd ones out will not.

And we've come to the end of our lesson.

Our lesson today has been solving problems involving positive and negative numbers in a range of contexts.

So when solving problems involving temperature, consider what is known and what is unknown.

Think carefully about the language being used and what operation is required.

So we saw in the first part of the lesson, we often had two bits of knowledge that we knew and one that we didn't know that we had to work out.

Coordinate grids can display both positive and negative values.

Patterns in coordinates exist.

Horizontal lines have the same Y coordinate.

Vertical lines can have the same X coordinate.

It's been a real joy working with you today and it's been a joy working with you if you've been with me for the rest of the lessons in this unit too.

Hopefully, now you've got lots of new knowledge and new skills all about negative numbers and hopefully, I'll see you again soon for some different learning about something else.

But until then, take care and goodbye.