Lesson video

Hiya, my name's Miss Lambell.

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

I hope you enjoy it.

Welcome to today's lesson.

The title of today's lesson is Addition of Positive and Negative Integers.

And it's within the unit Arithmetic Procedures with Integers and Decimals.

By the end of this lesson, you'll be able to use the mathematical structures that underpin addition to add positive and negative integers.

There are some key words that you probably won't be familiar with that we will be using throughout today's lesson.

We're just gonna go through those briefly now, but don't worry if you don't understand any of them.

We will look at them, like I said, in more detail as we progress through the lesson.

The first one is additive inverse.

The additive inverse of a number is a number that, when added to the original number, gives a sum of zero.

The next one is zero pair, and it uses the previous keywords.

A number and its additive inverse make a zero pair.

Like I said, don't worry if you don't understand that now.

You will do by the end of this lesson I'm sure.

And the final one is absolute value.

The absolute value of a number is its distance from zero.

So for example, five and negative five are both five away from zero and so both have an absolute value of five.

Please don't worry about any of those.

We'll look at them in more detail as the lesson goes on.

Today's lesson we will be splitting into three separate learning cycles, representing integer using counters, addition of positive and negative integers and deepening understanding of addition of integers.

Let's start off with our first one then, which is representing integers using counters.

What do you notice about the following images? Do you notice anything at all about them? Pause the video, have a little think, and then when you're ready, come back.

You may have said something like these.

They're symmetrical.

The dot's on the same numbers in each, but one of them is negative.

So if we look at the first one, the dots are on negative three and three.

The second one, the dots are on negative five and five, and so on.

So remember when we talked through those key words, we said the absolute value of a number is its distance from zero.

Negative three is three from zero.

Three is also three from zero.

This means that both negative three and three both have an absolute value of three.

Negative three and three are additive inverses of each other.

This means that they sum to zero.

So we start at negative three and we add three.

We end up at zero.

That's what an additive inverse is.

Well, we're ready for a quick check for understanding already, and it's a true or false.

The absolute value of 200 and negative 200 is 200.

Do you think that is true or false? And as always, I don't just want you to guess true or false.

I want you to be able to back your answer up confidently with one of the justifications below.

A, because one is negative and one is positive, or B, they are both the same distance from zero.

Pause the video now, and then you can come back when you're ready.

Super.

Hopefully you recognise that this statement was actually true.

If we looked at them on a number line, we would have to have a really big number line, but we could visualise it.

They are both the same distance from zero.

So remember, it's not just the fact that one is positive and one is negative.

It's actually that they both are the same distance from zero.

One is 200 to the left of zero and one is 200 to the right.

Additive inverses.

This is probably a new concept to you, but don't worry, it's actually quite easy once you get the hang of it.

A hundred, 100.

The additive inverse of 100 is negative 100 because the sum of those two things is zero.

They make a zero pair.

When we sum them, they're zero.

Negative eight.

The additive inverse of negative eight is eight because when we find the sum of negative eight and eight, it's zero.

Negative a half.

The additive inverse of negative a half is a half.

The sum of those two is zero.

They make a zero pair.

A pair of numbers that make zero.

We can even apply it to algebra.

A, the additive inverse of A is negative A.

When we sum those two, they make zero.

They are a zero pair.

Which of these are correct additive inverses? So read them through and decide which are correct additive inverses.

Pause the video, come back when you're ready.

There is more than one.

Great work.

Let's have go and look and see whether you've got these right.

I'm sure you have.

So A, yes, it is.

Negative 82 and 82 make a zero pair.

They've sum to zero.

H and negative H.

Yes, they are also additive inverses.

They sum to zero.

Negative a quarter and three quarters, no.

If we were to sum those, we'd end up with a half.

We need it to add up to zero if it is going to be an additive inverse.

The final one, 0.

9 and negative 0.

9, yes, they are additive inverses.

So like I said, it was a new concept but hopefully now with just those few quick examples, you understand what we mean by an additive inverse.

Two values that when summed make zero.

They make a zero pair.

Representation of negative numbers using positive and negative counters can be really useful.

Now you may have seen these before and you may not.

If you haven't got your own, you could even just quickly make some out of bits of paper.

And you may be able to, though, use two coloured counters and these counters are red on one side and yellow on the other.

Each counter represents one.

Yellow side up has a positive value of one and I'm going to represent those as a yellow counter with a plus on, and red side up has a value of negative one, and I'm going to use a red counter with a negative on to remind me that that's negative.

What is the total of those counters? The two counters that I have there on the screen.

What is the total of them? Hopefully you notice they're actually additive inverses of each other.

We've got a one and a negative one.

They make a zero pair.

They're additive inverses.

Well done if you've got that right.

Now we're going to look at what counters are representing.

So what is the total of the counters on the screen? Sometimes it's easier if you line them up as zero pairs.

So I'm going to line up all of my positive counters on the left and my negative counters on the right.

I can then clearly see if I have any zero pairs.

I have one zero pair here and another zero pair here.

Those two things make zero.

So I'm just left with the counters that are not in the boxes.

So this represents positive two or just two.

Remember we don't have to put the positive.

What is the total of these counters? Have a think about that.

Again, I'm going to line them up.

My positive counters on the left and my negative on the right.

You may be able to do it without, but here we go.

I've lined up my positive.

Now my negative.

Make my zero pairs.

And I can see that I'm just left, once I've got those zero pairs, with negative two.

Those counters represent negative two.

That's not the only way of representing negative two with counters.

We'll look at that in a moment.

Now your turn.

What integer do these counters represent? So if you need to, you could write them out, as I did, the positive numbers on the left and the negative numbers on the right and find your zero pairs and see what you're left with.

Or you may have an alternative method for doing it.

Whichever way you get there, it doesn't matter, as long as you're right.

Pause the video, come back when you're ready.

Well done.

Let's see if you are right.

So the answer was negative one.

I'm sure you got that right.

What about these counters? This time slightly different question.

What would you need to add to these counters so that it represents three? So this time I want you to decide what counters do you need to add to the diagram so that it represents the number three.

Pause the video, come back when you've got your answer.

You would need to add four yellow counters, four positive counters.

Well done if you've got that right.

Now I mentioned earlier that the representation I showed you for negative two wasn't the only representation for negative two.

So here, we're just going to have a look at some examples and non-examples of representations of negative two.

So on the left there'll be examples, ones that do represent negative two, and on the right non-examples, ones that don't represent negative two.

This represents negative two because we can see we've got one zero pair with two red counters left over.

This one does not.

We've got one zero pair and only one negative counter left over.

This example here, we would have three zero pairs with two negative counters left over.

This one we would have two zero pairs with two positive counters left over.

And then we have two more examples, one of each.

You might want to pause the video and check you understand why the one on the left is an example and the one on the right is a non-example.

We're now ready, or you're now ready, to have a go at a task.

So very much like one of the check for understandings we did.

I'd like you please to add counters to make each of the representations correct.

On the left hand side, I've told you what integer I want to be representing, and then I have a representation of some counters, but unfortunately they're incomplete.

What I'd like you to do is to add counters to make each of them complete.

Pause the video and then come back when you're ready.

Here are our answers.

And again, these are just examples.

I've decided to go for the least number of counters to make it correct, but you may have added positive ones to the first one and then more negative ones.

It doesn't matter.

So my first one, I've decided to add four negative counters, the second one, three positive, the third one, three positive, and then the final one, three negative counters.

Just check that you understand if you've made any errors.

Now we can move on.

We're now ready to add positive and negative integers and we're gonna continue to use the counters to help us.

Here is a calculation and here are the counters that represent that calculation.

So let's have a look at that.

We've got four, we've got four positive counters.

And then we're adding two negative counters.

What's the total of the counters that we can see on the screen? So again, you might decide that you'd quite like to line them up, find your zero pairs, and then see what we've got left.

So four, add negative two, is two.

We can see that we've only got two positive counters left.

Let's look at another example.

So here we've drawn the counters to help us with this calculation.

So we've started with three positive counters because the calculation starts with three and we've added five negative counters.

So three positive and five negative.

Let's line them up.

Let's find our zero pairs and let's see what we've got left.

So this time we're left with negative two.

Now what I'd like us to do is to start with the calculation with no counters.

So I'm gonna ask you just to briefly pause the video and draw out the counters that you think represent this calculation.

And if you can, then move on to finding the answer.

When you've had a go at that, come back and we'll take a look and see whether you've got that right.

Well done if you were brave enough to give that a go by yourself already, but we're gonna go through it together now anyway.

So we're starting with five positive counters and we are adding two negative counters.

Hopefully that's what you've drawn.

And then we can line them up.

You may not need to.

Find your zero pairs.

And then we've got the calculation answer is what we're left with, which in this case was three.

If you did have a go, well done and even better if you've got an answer of three.

What about this one? What do you think this would look like in counters? A little bit different to the ones we've done already.

Just maybe pause and have a think what it might look like in terms of counters.

This time we're starting with negative three.

So there's my negative three counters and I'm adding another two negative counters.

That's what my calculation looks like.

Was that what you imagined it would look like? Now, this time I have no positive counters.

I've just got a whole string of negative counters.

So my answer here, I can't form any zero pairs because there are no positive counters.

My answer is negative five.

We're definitely ready now to have a go at this check for understanding.

Which of the representations I've given you, A, B, or C, do you think is the correct representation of the calculation negative four add two? Pause the video, come back when you're ready.

What did you decide? Hopefully you decided it was B.

I started with four negative counters and I added two.

Even better, and I'm sure you did this because you're amazing, you went on to then tell me that the answer to that is negative two.

We're ready now to have a go at a different task.

So at the top I have given you some calculations which you can complete using counters if you need to or you may not need to.

What you are then going to do is you are going to find each of those answers, and they may appear more than once, in the grid.

You need to shade in all of the correct answers in the grid and it will reveal a word.

So work out all of the answers, find them in the grid, shade them in, and hopefully, if you've done it correctly, you'll reveal a word.

Pause the video, good luck with this, hope you're having some fun and then come back when you're ready.

Does yours look like it's a word? Let's hope so, because if it does, we've definitely done it right.

These are the boxes that we should have shaded in.

Not very clear there what the word looks like.

So I've written it, I've shaded those boxes in a bit more to show us a little bit better.

And it was zero.

Well done if yours looks exactly my mine does on the board.

So zero, remember that's really important.

In today's lesson, we've been talking about zero pairs.

We've been talking about additive inverses, which make zero.

So really, really important word.

Let's move on now then to our final learning cycle.

And we're going to start looking now at what happens if we've got more than two numbers.

So, so far we've just looked at if I've got two numbers, I want us to start looking at what about if there's more than two? Because we don't necessarily know that we're only ever gonna need to add positive and negatives and just two of them.

Here we have Sam and Aisha.

Sam says, "I know this represents a negative number without counting." "How can you do that?" Aisha says.

Well, let's see what Sam's got to say.

See whether what he says makes sense.

He says there are more negative counters, so it must be a negative number.

Does that make sense? There are more negatives, so it must.

I think so because once we've made those zero pairs, we'd only have red counters left.

Aisha says, oh, she thinks she's got it.

So she wants to give it a go.

Good on you Aisha.

Here's one that a Aisha's gonna have a go at.

Okay, you might want to decide whether you think that's going to be positive or negative.

Remember, we're not necessarily working out an answer here, we're just trying to decide whether we think that answer is going to be a positive or negative integer.

Let's see what Aisha thinks.

So she says, "There are more yellow counters, so it must represent a positive number." That's right.

And actually Sam went on to say that it represents four.

We could create three zero pairs, we would have four yellow counters left over.

Hmm, not sure I'd want to be drawing out counters or counting out counters when my numbers get much, much bigger.

Let's have a look and see whether Sam and Aisha have an idea of a way we could do this without counters.

So Sam's saying, "If we drew the counters, what would we draw?" So rather than drawing them, trying to visualise what we would draw.

What do you think we'd draw for negative 20 add negative 50.

What do you think? Well, Aisha says we would draw 20 red counters.

Yeah, that's right, isn't it? Negative 20, we know the red counters are negative.

And then another 50 red counters.

Yeah, add negative 50.

So Sam says we'd only have red counters.

Is Sam, right? He is, isn't he? Because we've only got negative numbers.

Yes.

Aisha says "Yes, that's right.

And we would have 70 red counters in total." So the answer to that calculation is negative 70.

I'd only have red counters and there would be 70 of them.

Oh, Sam's really going for it now.

"So can we just add the absolute values and make it negative?" So he's spotted that the sum of 20 and 50 is 70, and they're the absolute values and you just need to make it negative.

Hmm.

What are your thoughts on that? Aisha says, "Maybe." She's not convinced.

Let's have a look at this example then and see whether Sam is right.

So like he says, we're gonna see if it works for this one.

So Aisha says we would draw 20 yellow counters for the positive 20, and 50 red counters.

Ah, Sam says, "Hang on a minute, we could make 20 zero pairs then." Hmm.

"That means we would have 30 red counters left." Meaning the answer to that is negative 30.

So Sam had an idea that we found the absolute values and then we put a negative in front.

But we've shown here that that doesn't work, but it worked on the previous one.

I wonder why.

Let's look at this question in a slightly different way.

Thinking about additive inverses.

So we mentioned them earlier, why did we mention 'em if we weren't gonna use them? We're going to use them now to make sure that we have a real understanding of why the answer to this is negative 30.

So the same calculation as it was on the previous slide.

We're going to split our negative 50 into negative 20 and negative 30.

Just check you agree.

If I have negative 50 counters, I could split them into two piles of negative 20 and negative 30.

Yeah.

Notice then I've created additive inverses here on the left hand side.

The ones in the blue box.

20 add negative 20 is a zero pair.

So they are additive inverses.

So all I've now got left from my 50 is my negative 30.

Now you may find it easier to think of it that way or you may still continue to prefer to think of the counters.

It's entirely up to you.

Each way gets you the right answer and no way is the definite correct way.

Now Aisha's noticed something.

Aisha's noticed that the answer is the difference between the absolute values if you ignore the negative sign.

So she's saying this time they're different.

So is the difference between 20 and 50 30? Yes.

Oh, so that does look like it might be right.

And Sam says, "Oh yes, and we know it will be negative as a negative integer has a greater absolute value." So is that right? Yes.

Negative 50 is greater than 20.

Okay, so there would be more negative counters.

You might like to think of it that way.

We can now start to generalise what is going to happen based on what we are adding.

Here we've got 32 add 45.

I think you'll agree we'd only have positive yellow counters there.

That's fairly straightforward.

We know that that's 77 and we probably wouldn't even bother with the counters.

What happens if I turn all of those counters over and make them red? I'd have negative 32 add negative 45.

This time I've turned all of my counters over so they are all red.

I only had negative counters, and so my answer is negative 77.

Notice the absolute value of the answers are the same as the sum of the absolute values of the addends.

Here we've got addend 32 and 45, the absolute values, and then sum them.

That was when we only had positive counters or only negative counters.

Now let's take a look and see what happens when we've got positive and negative.

Negative 32 add 45.

This time we have more positive counters.

So therefore we know our answer has to be positive.

And we need to find this time the difference between 45 and 32.

So the answer is 13.

What about if we turned all of those counters over and we had 32 yellow and 45 red? This time there are more red counters, more negative counters, and so therefore our answer has to be negative.

But the difference between 45 and 32 hasn't changed.

So our answer is negative 13.

So in general, we can notice that the absolute value of each sum is the difference between the absolute values of the addends.

In general, addition of two integers.

If both integer are positive or both are negative, find the sum of their absolute values.

So if I've got a bigger positive number add a smaller positive number, my answer is going to be an even bigger positive number.

If I've got a big negative number add a smaller negative number, my answer's going to be an even bigger negative number.

If one integer is positive and the other is negative, we find the difference of the absolute values and then consider whether we expect our answer to be positive or negative.

So if I've got a big positive number, add a smaller negative number, my answer is going to be positive and smaller than my starting number.

So I find the difference.

And if I've got a smaller positive number add a bigger negative number, then my answer is going to be negative.

That is just a generalisation.

Remember, you can always draw the counters or imagine the counters.

That is absolutely fine.

Now we're going to take a look at this calculation.

Looks a little bit more complicated 'cause we've got two more numbers, but actually it's just as easy as what we've been doing before because we are going to sum the positive integers and the negative integers separately first.

So let's start with our positive numbers.

Let's identify them in our calculation.

So we've got three and four.

The sum of three and four is seven.

We're going to do exactly the same now with our negative integers.

So let's identify those from our calculation.

We've got negative two, we've got negative seven and negative six.

We are going to sum those.

We've only got negative counters, so we know that this is going to be the negative of the absolute values of those integers.

So it's going to be negative 15.

The absolute value of the negative integers is greater, so our answer is going to be negative.

So we can already decide that our answer is going to be negative.

There are positive and negative counters, so there are going to be some zero pairs.

If we were to draw this out, we would have some zero pairs.

Therefore we need to find the difference between the two absolute values.

Absolute value of seven is seven.

The absolute value of negative 15 is 15.

The difference of 15 and seven is eight.

So the correct answer is negative eight.

Remember we'd already decided our answer was gonna be negative because our negative integer was bigger.

Quite a lot there to look at.

It might be that you want to pause the video and just make sure that you've understood each step of that.

If you are ready though, we can move on and then we'll have a look at a check for understanding.

So we've got a true or false.

The answer to negative three add five add negative seven add two will be positive.

As always, I want you to decide if it's true or false, but not only that, impress me with your understanding from today's lesson with a correct justification.

Pause the video and come back when you're ready.

What did you come up with? You should have come up with false.

Now which justification? Which justification was it for that? That was false and it was B.

Well done if you've got that right and superb work if you managed to get the correct justification.

Now you're ready to have a go at some of these yourself.

So some of these are bigger numbers, just two of them, to check that you really do understand about whether you're finding the sum of the absolute values or the difference of the absolute values, and also considering whether your answer is going to be positive or negative.

Good luck with this.

No calculators, okay? And when you are ready, you can come back.

Pause the video now.

Good luck.

Here are our answers then.

So number one is 54.

Two, negative 20.

Three, negative 128.

Four is 10.

Five was 80.

Six was negative 43.

Seven was negative 53.

I'm gonna be super impressed if you challenged yourself to do the final question, question number eight, which was make up your own addition question using positive and negative integers so that the sum of all of the answers in the task is zero.

Well the sum of answers one to seven is negative 100.

So you needed to write down a calculation whose sum was equal to a hundred.

So this is just an example.

I've gone from negative 80 plus 180.

You will probably have something different.

As long as it's a sum that's equal to a hundred, well done, you've got that right.

We are now ready to summarise the learning that's happened today.

Now, I don't know about you, but I think this has been a really intense and challenging lesson, but you've done fantastically.

This might be the first time that you've ever seen double-sided counters, the plus and the minus counters, and hopefully you see the value of them in helping us to understand our arithmetic with negative numbers.

So we looked at the fact that integers can be represented with positive and negative counters.

Those counters there show a representation of negative two.

So remember, if you want to, you can line them up, find your zero pairs and see what's left.

Counters can be used to represent addition of positive and negative integers, so we can actually use them to help us with addition.

Zero pairs can be used to simplify addition of positive and negative numbers.

So there's an example there where we started with five positive, we added two negative.

We then created those zero pairs, giving us an answer of three.

And absolute values are useful when adding positive and negative integers.

Well done on today's lesson.

Like I said, I think it's been quite a challenging one, but you've done fantastically well.

I'm really pleased that you decided to join me and I look forward to seeing you again.