Lesson video

Hiya, my name's Ms. Lambell.

Thank you so much for popping along today to do some maths.

I hope you enjoy it.

Welcome to today's lesson.

Today's lesson's title is "Checking and securing understanding of negative numbers in a context." This is within our unit "Arithmetic procedures with integers and decimals." By the end of this lesson, you'll be able to interpret negative numbers in a context and also think about negative numbers without a context.

We'll be using the word integer throughout today's lesson.

Remember, this is a positive or negative whole number, or zero.

So for example, -2, 0, 153.

Those are all examples of integers.

But today, mainly, we will be focusing on negative integers.

We're going to divide today's lesson into two separate learning cycles.

The first of those is reviewing negative numbers in context, and then we'll look more closely at using those negative numbers.

So let's get started on our first learning cycle, which is reviewing negative numbers in a context.

And remember, we did say we were also going to think about negative numbers without a context.

I've got four different scenarios.

I'm going to read each one of them in turn.

And then what I'd like you to do when I've read them all is to pause the video and have a think about what you think the answers to those might be.

So, the first one, Alex places a dot on a number line.

It is five away from zero.

Where might Alex's dot be? Aisha places a dot on a number line.

It is seven away from three.

Where might Aisha's dot be? Lucas places a dot on a number line.

It is 100 away from one.

Where might Lucas's dot be? And the final one, Jun places a dot on a number line.

It is one away from one quarter.

Where might Jun's dot be? So what I'd like you to do is pause the video, read all of those back through, and decide where you think the dots are going to be.

When you've had a go at that, come back and we'll take a look at those problems together.

Now, in each situation, there are actually two possible answers.

I'm wondering if you came up with two possible answers for each of those situations.

You may have.

If you didn't, don't worry 'cause we're gonna take a look at those together in a moment.

The other thing we're going to look at is how far apart the answers were.

So maybe just pause a moment and have a look.

How far apart, if you did end up with two answers on each number line, how far apart were each of them? Like I said, we're going to look at each of those in turn now together.

Alex places a dot on a number line.

It is five away from zero.

Where might Alex's dot be? I'm wondering if you ended up with your dot in the same place as Alex.

Let's take a look at a number line to start with.

Here's my number line.

My number line goes from -10 all the way up to 10.

It's five away from zero.

So five away from zero.

So I've got an arrow there from zero, five to the left.

Okay, and we can see that the dot ends up at -5.

What about if I go five away from zero in the other direction? We've got a dot at five.

Did you end up with both of those dots? Maybe you ended up with the positive one, not the negative one, or maybe you did end up with both.

Well done if you ended up with both.

That's superb.

So Alex's dot could have been either -5 or 5.

Let's take a look at what Aisha was looking at.

So Aisha was placing dots on the number line, but she was going to put it seven away from three, seven away from three.

So let's find three on our number line, and we'll go seven to the left.

Now, if I move seven places to the left of three, I end up at -4.

But she could have moved seven places to the right of three, and we can see that that meant the dot would have been at 10.

Aisha's dot, then, could have been either at -4 or 10.

Lucas also places a dot on a number line, or maybe two dots.

It's 100 away from 1.

So getting a little bit more challenging now because the numbers are getting bigger.

Let's see how we got on.

Now, I've decided not to draw a number line going all the way from 1 to 100 because it would be a bit cramped on my page, so I've just decided to represent this in a slightly different way.

There's 1 because we are working out what is 100 away from 1.

So 100 to the left.

What would be 100 to the left of 1? Now, if I was going to do that, I would think, right, well I'm gonna get to zero first.

So if I get to zero, I've moved one place to the left.

And then I need to move another 99 places to the left, meaning I'm going to end up at -99.

Well done if you got that.

Easier going to the right, isn't it, because we're with positive numbers, which we're much more confident with using.

If I'm at 1 and I move 100 places to the right, I'm gonna end up at 101.

Lucas's dot could have been at -99 or 101.

Jun had the most challenging question.

Well, I think it's the most challenging because we're now looking at fractions.

It's one away from one quarter.

Where might the dot be? There's a quarter, and I need to go one away.

Now, I'm going to do the one to the left first 'cause that's more challenging.

Let's get that done out of the way.

If I move one away from a quarter, let's think about it again, I would go to zero, that's taken a quarter.

How much of my one is left? That's 3/4.

I'm actually going to end up -3/4.

And again, slightly easier moving into the positive numbers to the right.

If I'm one away from 1/4, then my other dot would have been 1 1/4.

The other thing, then, we're thinking about is how far away were the two dots? So how far away? And maybe you noticed that in each situation, the answers were double the distance away.

So if we look at this one, it said it was one away from a quarter, so we move one to the right and one to the left.

So therefore, it must have been two, the distance between the two dots must have been two.

In the previous one, we moved 100 to the right and 100 to the left, meaning the distance between the two dots was 200.

All of the distances between the two dots was double the distance we were working with.

Now I want you to have a think about will there always be a positive and a negative answer? Have you managed to think of a situation when there won't be a negative answer? Well done if you have because there are lots of them.

Here's my example.

I had three away from 10.

If I move three to the left on a number line from 10, I'm gonna end up at seven.

If I move three to the right, I end up at 13.

Neither of those are negative.

So there are lots and lots or an infinite number of situations where we wouldn't end up with a positive and a negative answer.

Maybe you could start thinking about would there be a situation where I would end up with two negative answers? Now, we did say that we were going to look at things which have a context, and often those things are in everyday life.

Negative numbers are actually all around us.

Can you think of any places where you may see negative numbers in your everyday life? I'm sure you came up with some really, really great ideas.

I've just got three here.

These are not the only three.

I'm sure you came up with some much better ones.

But these are some quite common ones.

So firstly, temperature.

So temperature, I think we all know that sometimes it can get pretty chilly and our temperatures go below zero.

Our temperatures are negative temperatures.

Also, buildings.

Tall buildings and lifts.

You may get into a lift in a car park which is below ground, and you might see that you were on level -3 because you are three below zero, the ground being zero.

And you may have seen it in sea level.

So this might cross over into geography.

Sea level would be zero, so anything above sea level would be positive and anything below zero would be negative, so anything below the top of the sea would be negative.

But like I said, I'm sure you came up with some much more interesting examples of where you might have seen negative numbers in everyday life.

Andeep is thinking about -5.

<v ->5 is between -1 and -7.

</v> Do you agree with Andeep? And I've given you there a number line that may help you, but you might be able to do this without the number line.

Do you agree with Andeep? Let's take a look at what it looks like on the number line.

<v ->5 is between -1 and -7.

</v> So Andeep is correct.

Here's -1, here's -7, and here's -5.

So we can clearly see that Andeep was right.

<v ->5 is between -1 and -7.

</v> Well done, Andeep.

He's now saying -2 is halfway between 3 and -9.

So this time, he's not just saying that it's between those two.

He's actually specifying that it's exactly halfway between the two.

Do you agree with Andeep this time? This time, he's not correct.

Let's take a look at why he's not correct.

So we're starting, we're looking at 3 and -9.

Here's my dot at 3, and here's my dot at -9.

He's saying -2 is halfway, so let's put an arrow at -2.

Now, I think we can see just by looking that that's not halfway.

It is between 3 and -9, but it's not halfway.

I think we can see by eye that it's closer to 3 than it is to -9.

Let's just double check that, though.

If we look, we can see that 3 is 5 away from -2, and -9 is 7 away from -2.

In order for it to be halfway between, those two values would need to be the same, wouldn't they? I now want you to pause the video and actually work out where Andeep should have put the dot if he wanted it exactly halfway between 3 and -9.

Pause the video.

When you've got an idea, come back and see me.

Right, I'm wondering whether you used a number line for that or whether you did it without a number line.

Either is fine, but well done if you've done it without a number line.

Here's three, and here's nine.

So if we want to work out halfway, first thing we need to do is know the distance between those two points.

And the difference between 3 and -9, and we could count on the number line if we needed to, is 12.

If I want halfway, well, half of 12 is 6, so halfway must be 6 from -9 and 6 from 3.

So let's have a look.

6 from -9 is -3.

So -3 is halfway between 3 and -9.

I don't need to check what's 6 below 3 'cause it should give me the same answer, but you might want to double check that you end up in exactly the same place.

What do we do then? So we found the actual distance between -9 and 3 in whatever method we like, so maybe by counting on the number line, or maybe I would start at -9, so I need to take 9 to get to 0, and then another 3 to get to 3, which is 12 in total.

I want halfway, so I half 12.

And then I can do either way.

I can either go 6 up from the dot on the left or 6 left from the dot on the right.

Quick check for understanding.

Which of the following are true? <v ->8 is between -9 and -10.

</v> B, -7 is less than -5.

C, -9 is greater than -10.

D, -4 is between -1 and -8.

Please pause the video, decide which of the following are true, there may be more than one, and when you're ready, come back.

Good luck.

Great work.

Let's have a look and see if you were right.

I'm sure you were.

So the correct ones, we wanted, we were looking for ones that were true.

<v ->7 is less than -5.

That's correct.

</v> It's to the left on the number line.

<v ->9 is greater than -10.

</v> <v ->9 is closer to 0,</v> so therefore it must be greater.

And then D, -4 is between -1 and -8.

Well done if you identified that B, C, and D were all in fact correct answers.

Now we're going to look at temperature.

So temperature is something we're really familiar with with negative numbers, and so we can use that to help us to really understand our negative numbers.

Here we have a table that shows the temperature in January of eight different cities.

Maybe you could have a think about if you know where those cities are.

Do you know where Alice Springs is? Do you know where Quebec is? Maybe, maybe not.

If you don't know, you might like to go away and look it up.

So, which city is the warmest? Hopefully you've identified 29 is the highest of those numbers, so Alice Springs is the warmest of those eight cities.

Next question, which city is the coldest? Which one of those were you most likely to be shivering in? Little bit harder, isn't it? But think about which is furthest away from zero on the thermometer, and that is -8, Moscow.

Moscow is the coldest.

It is furthest away from zero on our thermometer.

Now, what is the difference in temperature between Moscow and New York? I've highlighted the two temperatures in the table so that you can see clearly which two temperatures you are trying to find the difference between.

Have a go at finding the difference between -8 degrees and 2 degrees.

Now, I've mentioned this previously, the way I like to do this is to think about it on my number line or even draw my number line out, and then think about I like to start with the smallest number here.

That's -8.

First thing I'm going to do is I'm going to work out how far I need to move to get to zero.

To get to zero, I move eight places.

Think about it as a temperature.

The temperature needs to rise by eight degrees to get to zero degrees.

But New York wasn't zero degrees, it was two degrees above that, so we need to move another two degrees.

In total, I have moved 10 degrees.

The difference between the temperatures in Moscow and New York is 10 degrees.

Now a check for you.

So thinking about what we just did, and like I said, draw a number line if you need to, you may visualise that number line in your head or you may have a totally different way of doing it.

Any way is fine as long as we get the right answer.

What I'd like you to do, what is the difference between the temperatures in Los Angeles and Quebec? Pause the video, and when you're ready with your difference, come back and check in and see if you're right.

Great work.

Let's have a look and see if you're right.

So we wanted the difference between the temperatures in Los Angeles and Quebec, and the answer was 20.

Los Angeles was 13, Quebec was -7.

So using that method, -7 to 0 is 7 plus another 13 degrees above that, is 20 degrees in total.

Well done if you got that right.

Now we're over to you, some independent tasks for you to have a go at.

I think here I've got four questions for us to have a go at.

So what I'd like you to do is to pause the video and have a go at question number one and question number two, bearing in mind there is more than one possible answer for question number two.

When you're ready, you can come back and see how you got on.

And question number three, question number three.

This time, we're working out what number, so I don't just want to know a number between them, I want to know, please, what number is exactly halfway between those following pairs of numbers.

Remember again, you can use a number line to help you.

If you need to watch back the example that we did on this a little while ago, then obviously you could rewind the video, have a look, and then come back to these questions.

But now pause the video, come back when you're ready.

And finally, onto question number four.

So, I've created here some questions based on that table of temperatures that we had for those eight cities.

So again, you're gonna have a go at those questions, and then when you're done, you can come back.

So pause the video, and I'll see you in a moment.

Good luck.

Wow, I certainly challenged you there, didn't I, with four questions.

Let's see how you got on with those.

Let's start with question number one.

So question number one, we had -3 is less than -1, B, -11 is less than -8, C, -57 is greater than -58, and then a bit of a challenging one with a decimal, <v ->0.

5 is greater than -0.

6.

</v> And then, like I said, for question number two, these are just some examples of numbers that you may have written to make the statement true.

So for A, -8, -10, -13, B, -178, -210, -313, C, -5.

1, -5.

4, -5.

8, D, -12.

34, -12.

5, -12.

75.

Remember, they are just examples.

Question three, these are not examples.

Remember, I wanted to know here the number that was exactly halfway between those two integers.

A, -8, B, -75, C, -8, and then some slightly more challenging ones, D, -66, and E, -9.

5.

Did you get five out of five? If you did, that is absolutely superb.

Well done.

And then finally, onto our temperatures.

So the answer to A, we wanted to write them in order from coldest to warmest.

So Moscow, Quebec, Tromso, New York, London, Los Angeles, Jaipur, and Alice Springs.

B, which two cities had the greatest difference in temperature? And that was Alice Springs and Moscow.

So the warmest and the coldest, which we'd already identified when we went through some questions.

Which two cities had the least difference in temperature? And here, you could have actually had two different answers.

And finally, part D of question number four, what is the difference in temperature between London and Los Angeles? And that's eight degrees.

And find another pair of cities with the same temperature difference, and that was New York and Tromso.

Great work on those questions.

We're now going to move on to our second learning cycle.

We're using negative numbers in a context.

Here, we have Izzy, and she gets her school lunch from the canteen.

You may do too.

She forgot to ask her dad to top up her account.

I wonder if that's ever happened to you.

She has 1 pound credit on her account.

I'm not sure that's enough to get her something for her lunch.

Her lunch costs 3 pounds.

Does she have enough money? No, she hasn't got enough money.

She's only got a pound credit, but her lunch is gonna cost her 3 pounds.

If she buys her lunch, what will the balance on her account be? So if the school let her purchase the lunch, what will her new balance be? Well, she can pay a pound of the cost.

So she's got to pay 3 pounds.

She can pay a pound of it with the pound credit.

So she will still owe the canteen 2 pounds, and we would write her balance as -2 pounds.

And notice, I've put the -2 pounds in brackets.

Sofia.

Here's our canteen price list.

So we've got lots of things we can buy.

Apple, melon pot, meal deal, sandwich, ice lolly, fruit juice, and water, all at different prices.

Sofia buys an apple, a sandwich, and a bottle of water.

She has 2 pounds on her account, so she's 2 pounds in credit.

What is the balance on her account after she has bought those items? I'd like you to pause the video and give this question a go.

We will go through it together, but it'd be interesting to see whether you could get this question right yourself before we go through it.

So pause the video, good luck, and have a go at this one.

Well done.

Let's have a look then.

So the total cost of the items she's going to buy, the apple, the sandwich, and the bottle of water, the total cost is the total of those three items, which is 3.

25 pounds.

Now, she's got 2 pounds on her account.

So just as previously in the previous learning cycle, I encouraged you to think about going from the smallest number to zero, and then moving on again, here, we're going to start at the biggest number.

So we know that we need to be calculating 2 pounds subtract 3.

25 pounds.

So that's the 2 pounds that she has in her account, and I've got to subtract 3.

25 pounds.

So I'm going to use that method of going to zero first.

How do I get from 2 pounds to 0 pounds? I subtract 2 pounds.

But we actually want to subtract 3.

25 pounds, so I really need to subtract another 1.

25 pound.

So that means that Sofia's new balance is -1.

25 pound.

She owes the canteen 1 pound and 25 pence.

So just take a moment there to have a look at the structure of that diagram that I would use to help me with this question.

Let's take a look at another example.

So we'll just do this one together because in a moment, you'll have an opportunity to try some of these independently.

So we'll just go straight into looking at this one together.

We use the same canteen price list.

Jacob goes to the same school.

Jacob buys a meal deal and an ice lolly.

He has 75 pence on his account, so he's not got much on his account.

What's the balance on his account after he has bought these items or those items? Let's first work out the total cost of what he's buying.

That's the meal deal and the ice lolly.

So a meal deal is 2.

65 pounds.

An ice lolly is 80 pence.

The total cost of Jacob's lunch then is 3.

45 pounds.

He has 75 pence on his account.

So we need to take his account balance and reduce it by 3.

45 pounds.

I'm going to use that method again of going to zero and then seeing how much further I need to go to find out his new balance.

So, current balance, 75 pence.

I need to reduce that by the total cost, which is 3.

45 pounds.

So I need to subtract 3.

45 pounds.

I'm gonna go to zero first.

How much do I need to subtract from 75 pence to get to 0 pence or 0 pounds? That's 75 pence.

I actually owe 3.

45 pounds, of which I was able to pay 75 pence.

That means now I still have 2.

70 pounds left to pay, so Jacob's balance is -2.

70 pounds.

So using that idea of going to zero and then working out how much extra below zero you need to go can be really useful.

You may have your own method.

That's absolutely fine, as long as you worked out that Jacob owes 2.

70 pounds.

His balance is -2.

70 pounds.

We're now gonna do a quick check for understanding.

What I'd like you to do is to match each box on the left to the new account balance on the right.

So I've given you the account balance.

That's how much money is in that account currently.

I've given you how much the person spent in the canteen.

And I've given you a set of new balances.

What I'd like you to do is to match the old account balance with the amount spent with what the new account balance would be.

Pause the video.

Good luck with that.

Come back when you're ready.

Well done.

So let's have a look at how those matched up.

So 1 pound and 3.

50 pound spent would leave us a new account balance of -2.

50 pounds.

An account balance of 2 pounds and 2.

80 pounds spent would give us a new account balance of -80 pence.

1.

50 pound balance and 3.

50 pound spent would give us a new balance of -2 pounds.

2.

50 pounds on the balance, the amount spent, 3.

75 pounds, would give us a new balance of 1 pound, -1.

25, sorry.

And the final one would give us a new balance of -1.

35 pound.

Absolutely superb if you've got those right.

Well done.

Now we're ready to have a go at some independent learning.

So for this question, you are going to use the canteen price list.

You are going to work out how much they spent in the canteen using the middle column, bought from the canteen.

You've got the current balance, and then I'd like you to work out the new balance.

So very similar to the check we've just done, but this time, I've not given you the total cost.

I'm asking you to work that out for yourself.

Pause the video.

Have a go at those.

Good luck.

Come back when you're ready.

That was quick.

Well done.

Let's have a look at the next question.

We're now looking at the same eight cities, but this time, I'm asking you to complete the sentences.

So if it's got a degrees C on the end, I'm asking you to complete it with a temperature.

And then the other lines I'm asking you to complete with a city.

So have a go at these questions, and when you're ready, come back.

Good luck.

Well done.

Let's check those answers.

So, one, it was, I'll just read out the new balances, so -1.

45 pound, -1.

95 pound, <v ->35 pence, -25 pence, -15 pence,</v> <v ->2.

25 pounds, and -3.

35 pounds.

</v> And then question two.

A, e.

g.

so again, if it says e.

g.

these are just some examples of correct answers, Tromso is 2 degrees warmer than Moscow and 11 degrees colder than London.

Obviously, if you chose different cities, then your temperatures will be different.

B, I chose 9 degrees warmer than Quebec and 3 degrees colder than London.

C, again, just an example, I chose 1 degree and Moscow and 29 degrees in Jaipur.

And D, I chose 3 degrees New York and 24 degrees Alice Springs.

Like I said, you may have different answers depending on which cities that you decided to choose.

Now we can summarise the learning that we've done in today's lesson.

You've done fantastically well.

So we know that negative numbers are present in everyday life.

So the three examples that I gave you were temperature, stories in a building or floors in a lift, and also sea level.

But like I said, you probably came up with a lot more interesting ones than I did.

We know that negative numbers exist and that they represent values below zero.

So we think about temperature, we were talking about anything that is below zero degrees.

A context can help when dealing with negative numbers.

So often, people find it easier to rather think of which is colder rather than which is smaller.

So that context of temperature, because we're so used to it in everyday life, can help us.

Even if it's not a temperature we're dealing with, okay, it still works in exactly the same way.

You've done really, really well with today's learning, and I'm really just pleased that you decided to join me.

Thank you very much.

Bye, see you again.