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.

Hiya, welcome to today's lesson.

The title of today's lesson is "Subtraction of Positive and Negative Integers" and it's within the Unit Arithmetic Procedures of Decimals and Integers.

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

Some words that you'll be familiar with which we will be using throughout today's lesson are additive inverse and absolute value.

So just a quick reminder of what they are.

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

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

So for example, 5 and -5 are both 5 away from 0, and so they both have an absolute value of 5.

So we'll be using these words throughout today's lesson, so that's why it's worth having a quick reminder of them.

We're going to divide today's lesson into 3 separate learning cycles, the first of which is rewriting a subtraction as an equivalent addition.

Now, at the moment, that might sound really, really strange.

Writing a subtraction as an equivalent addition is actually fairly straightforward.

We'll look at that first and we'll go through it step by step and you'll be fine.

The second learning cycle, we are going to look at subtraction of positive and negative integers.

So by now, you should be confident with adding positive and negative integers.

We're going to extend that today to looking at subtraction of those.

And then in the third learning cycle, we are going to deepen our understanding of subtraction of integers.

Here's just a quick recap of what we mean by an additive inverse.

Remember the additive inverse means that the sum of the two values is 0.

So the additive inverse of 80 is -80, additive inverse of -12 is 12, the additive inverse of -2/5 is 2/5, and the additive inverse of m is -m.

All of them are additive inverse, remember, because they sum to zero, they make a zero pair.

We're going to start by looking at this calculation and we are going to use the double-sided counters, the positive and negative counters.

On the left, we are going to look at how you may have seen these calculations before.

And on the right-hand side, we are going to look at what we mean by rewriting a subtraction as an equivalent addition 'cause I don't know about you, but I prefer adding to subtracting.

Like I said, you may have seen something similar to what is on the left-hand side.

5 subtract 3.

We start with 5 negative counters and we subtract 3 of those.

We take 3 of them away, and we are left with 2.

What would that calculation look like if we wrote it as an equivalent addition? Well, let's take a look.

We would actually add the additive inverse of +3, which is -3.

So the calculation on the left, 5 subtract 3, is equivalent to 5 add -3.

You might be thinking at the moment, you're not overly convinced with that.

So let's just have a look at what that looks like with the counters.

So we are starting with 5 positive counters and we are adding in 3 negative counters.

We can see here then, we've managed to form 3 zero pairs at the end leaving us with just 2.

The two things are equivalent.

Let's have a look at another example.

This time 3 subtract 5.

We start with 3 counters and we need to subtract 5 of them, but we don't have 5.

So, previously, you may have looked at adding 0 pairs until you did have enough counters to subtract the 5 positive counters.

So I'm going to add a 0 pair, and I'm gonna continue that until I have 5 positive counters which I can then remove.

But we are going to actually consider every subtraction as an equivalent addition.

What does that mean for this question? It means we're going to do 3 add, we're changing the subtraction but we have to add the additive inverse.

The additive inverse of 5 is -5, so we've rewritten that subtraction as an equivalent addition.

With the counters, we start with 3 positive counters and we add 5 negative counters.

Here, we've got our 3 zero pairs and so we are left with -2.

Notice, we end up with exactly the same answer.

Let's look at another example.

5 add 3.

Now, you might be thinking to yourself here, why is she making us do 5 add 3? Just want to show you what it would look like.

So that would be 5 positive counters add 3 positive counts.

What would that look like as an additive inverse? It would look the same, okay? We are not subtracting, so therefore we don't need to change that calculation.

We're only altering the calculation if it's a subtraction.

We now look at -5 subtract 3.

So this might be how you are familiar with doing it from before.

Start with 5 negative counters and I'm subtracting 3, but I don't have 3 positive counters to subtract.

So I add my 0 pairs so that I do have 3 positive counters to subtract, and if I subtract those, I end up with -8.

What does it look like as an additive inverse? <v ->5, and we're going to add the inverse of 3,</v> and the inverse of 3 is -3.

If we look at the counters for this one, we can see we start with 5 negative counters and we add 3 negative counters.

We're ending up with exactly the same.

So writing a subtraction as an equivalent addition allows us to be able to answer many questions in a much easier way.

Like I just said, additive inverses allow us to write any subtraction as an addition of additive inverses.

4 subtract 9.

So this is 4 subtract, and remember that's +9, it has no symbol attached to it, so therefore it's +9.

The inverse of +9 is -9.

So we would rewrite this as 4 add -9.

4 add -9.

Let's look at the next one.

This time I'm starting with -4, and I'm subtracting 9 again, that's a +9.

So it's going to be -4 add the additive inverse of negative, oh sorry, the additive inverse of 9, which is -9.

What about if we switch numbers around? <v ->9 subtract 4.

</v> <v ->9, I'm gonna change that subtraction</v> to add in the additive inverse of 4, which is -4.

So I end up with -9 add -4.

What does it look like if I subtract a negative? So I start with -4.

I want to change my subtraction to an addition and I'm gonna add the additive inverse of -9, which is 9.

So I end up with -4 add +9.

We may also see that written as -4 add 9.

We're going to see whether you can do some of those for yourself.

But before we do that, because I think this can be quite confusing, it is probably the first time you've seen it, is we are going to do the ones on the left-hand side together first.

We are going to write each of the subtractions as an addition of additive inverses.

So the first one we start with 5 and we're subtracting 8.

And remember that's +8.

So we're going to end up with 5 add, the additive inverse of +8 is -8.

Next one, we start with 8, and we're going to add the additive inverse of 5, which is -5.

The next one we're starting with -5 and we're adding the additive inverse of 8, which is -8.

And the final one, we're starting with -8.

This time we're adding the additive inverse of -5, which is +5.

And like I said previously, you don't have to write both of those.

We can just have -8 add 5.

Just gonna pause a moment and let you just look through those.

Like I said, this is probably a new concept to you.

I know when I was introduced to it, it took me a little while to get my head around it.

But just remember all we are doing is changing the subtraction to an add and then we're adding the additive inverse.

And you were super good at additive inverses earlier and in previous lessons.

Okay, I think we're ready now for you to have a go at some of those on your own.

Here are the questions I'd like you to do.

So I'd like you to change each of the subtractions to an addition of its additive inverses.

Pause the video.

Good luck.

Do your best.

Come back when you're ready.

Well done.

Hopefully, you didn't find that's a daunting once you'd got your head round it.

So the first one you should have -2 add -3, the second one, 2 add -3, the third one, 3 add -2, and the 4th one, -2 add +3 or just -2 add 3.

I'm really hoping that you've got all of those right, but don't worry, we'll continue to practise that skill as we move forward.

For task A, what I'd like you to do is to match each of the subtractions on the left-hand side to its equivalent addition on the right-hand side.

So you're going to pause the video now, you're going to match up each one.

Notice, I've tried to make this challenging for you, so I just used 10s and 7s so you can't guess from the numbers.

Have a go at this.

Good luck.

Come back when you're ready.

Welcome back.

Let's have a look and see how you got on with that.

Hopefully, once you've got your head around it, you didn't find it too challenging.

So the first one, -10 subtract 7 matches with -10 add -7.

The second one, 10 subtract -7 matches with 10 add 7.

Remember there you may have written 10 add positive 7, which is fine.

10 subtract 7 matches with 10 add -7.

<v ->7 subtract 10 matches with -7 add -10.

</v> <v ->10 subtract -7 matches with -10 add 7,</v> or again you might have +7 in brackets at the end.

7 subtract 10 is 7 add -10.

And -7 subtract -10 matches with -7 add 10.

And again you may have +10 at the end, which is fine.

Well done on those.

Hopefully, you've got all those right? I'm sure you did.

We're gonna move on now.

So we now know how to write a subtraction as an equivalent addition.

So we're going to use that and we're actually going to take it one step further and we're gonna calculate the answers.

But remember you are really, really good already at adding positive and negative integers and as we've just created an addition from subtractions, we're just going to be doing the same as we've done previously.

So let's get started on it.

We're gonna do fantastically well.

Let's go.

We need to remember this.

So we've looked at this previously.

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

So if they're both positive, they're both negative, they have the same signs, we find the sum of their absolute values, just remembering that the negative ones will give us a negative value as an answer.

If one integer is positive and the other is negative, then we have to find the difference of their absolute values.

Remembering, if the negative number is the greater, we know our answer is going to be negative.

And if the positive number is the greater, we know our answer is going to be positive.

That shouldn't be new to you.

That's something that you will have looked at previously before starting on this lesson.

It was just to recap.

We'll now look at some calculations and how we can apply what we already know, what we've just learned in the first learning cycle where we've learned how to change a subtraction to an addition, and then we're going to find our answers.

So we've already rewritten, we've already got really good knowledge of what's gonna happen when we add integers and so therefore we should be able to put the two together now.

4 subtract 9, we're gonna rewrite that as 4 add -9.

Remember, the additive inverse of +9 is -9.

The answer to that, so all we need to do now is we need to look, we have got positive and negative numbers, therefore our answer is going to be the difference of the two.

The difference of 9 and 4 is 5.

The negative integer is bigger, so therefore our answer is going to be negative.

The answer to this is -5.

Let's move on and look at another example.

And like I said, it was the difference of the absolute values.

Next one, -4 subtract 9.

We'll rewrite that as an addition.

So -4 add -9.

This time, we've only had negative counters, so we are going to find the sum of the absolute values, which is 13, and we've only got negative counters.

So therefore our answer has to be negative, doesn't it? Let's look at another one.

<v ->4 add 9.

Do we need to rewrite that?</v> No, I hear you shout, of course, we don't.

It's an addition.

We're only rewriting subtractions.

So -4 add 9, we just need to look.

We've got positive and negative, so it's going to be the difference of the absolute values.

And this time, the positive number is the greater, our answer is going to be positive.

So +5.

Now, we're gonna look at -9 subtract 4.

Do we need to rewrite? Yes, we do 'cause it's a subtraction.

Remember, that's what we are doing.

We are rewriting subtractions.

Reason we didn't write the previous one because it was an addition.

So I'm gonna rewrite it as -9 add -4.

This time I only have negative counters, so I'm finding the sum of those values.

And if I've only got negative counters, therefore my answer has to be negative.

And one final example, -4 subtract -9.

Let's rewrite it as an addition first.

Remember, addition of additive inverses.

<v ->4 add +9.

</v> And it's okay to drop that positive in that bracket.

This time, we've got positive and negative numbers so we know we are going to be finding the difference of the two absolute values, the difference of 9 and 4 is 5.

Now, all we need to do is consider which was the greater, the negative integer or the positive, and it was the positive.

So therefore our answer has to be positive.

It might be worth pausing the video now and just looking at all of those different things and making sure that you can make sense of it.

Obviously, we will continue to look at this together and there'll be some independent practise but there's quite a lot of information there.

So if you feel you need time, that's okay.

Just pause the video, and then when you are ready, you can come back and we'll move on.

So just as we did before, I thought it might be useful to actually do some of these questions together first.

So we're gonna do those on the left-hand side and then you are gonna have a go independently at the questions on the right-hand side.

So remember we are rewriting, sorry, the subtractions as an equivalent addition.

That's what we did in the first learning cycle.

So you're expert at that now.

The only extra bit we're going to do now is we're going to actually work out the answer, but we are going to use the knowledge that we already have from our previous understanding and learning to make that step nice and easy.

First one, 7 subtract 13.

We're gonna rewrite that as an equivalent addition.

So 7 add -13 because the additive inverse of +13 is -13, remember.

Here we've got positive and negative, so we need to find a difference.

The difference of 13 and 7 is 6.

The negative number is greater, so our answer is going to be -6.

Let's look at the next one.

-13 subtract 7.

Rewrite as an equivalent edition.

So -13 add -7.

This time, we've only got counters that are the same, or if you're not thinking about counters anymore, we only have negative integer.

So we find the sum of the absolute values, the sum of 13 and 7 is 20.

But if I've only got negative integers, my answer has to be -20.

Moving on again, -7 subtract -13.

Rewrite the subtraction as an equivalent addition, <v ->7 add 13.

</v> Here we have positive and negative.

So we are finding the difference.

The difference between 13 and 7 is 6.

This time the positive number is greater.

So our answer is going to be +6.

Remember, you don't have to put the positive there.

You also don't need to put the brackets.

I think you are more than ready now to have a go at some of these independently.

So I'm going to put them on the screen.

Once they're on the screen, pause the video, have a real good go at them, remember no calculators, and then come back when you are ready to check your answers.

Great work.

How did you get on with those? Really well, I'm sure.

So the first one, our answer is -15 add 10, that's it.

Rewritten as a equivalent addition, which gives us a answer of -5.

The next one would be written as 10 add -15, which also gives us an answer of -5.

And the final one we would rewrite is -15 add -10, which gives us an answer of -25.

Well done if you've got all of those right.

If you didn't, just pause the video and check and see if you can understand where you went wrong.

I'm sure if you did make a mistake, it was just a little slip.

We should now be ready to have a go at some of these independently.

So a slightly different task here.

You've got a maze, you've got a start in the top left corner, and you've got a finish in the bottom right-hand corner.

The way the maze works is, you work out the answer at the bottom of the box.

So you're going to start by calculating what 5 subtract -1 is.

Remember, you are going to rewrite that subtraction as an addition, an equivalent addition using those additive inverses.

You are then going to work out the answer.

The answer to that is either going to be 4 or 6 'cause those are the only two boxes that I can move into.

You need to work out is it 4 or is it 6? No guessing.

Once you've worked it out, you can draw an arrow or you can highlight your way through the maze.

It's entirely up to you how you indicate how you're moving through.

You then work out the answer to the question at the bottom of that box.

And then you find that.

You're only allowed to move vertically, so up and down, or horizontally, left or right.

What I'd like you to do now is pause the video and try and find your way carefully through the maze.

Good luck.

Come back when you're ready.

Absolutely superb work.

Let's check that you didn't get stuck in the maze.

Let's have a look.

So we should have got gone to 6 and then down to 1, across to -5, and then across to -9, across to -4, and then we went down to 3 across to -2, down to -6.

And then you might have thought you were gonna finish, but no, you went left to 0 down to -2, and then the finish at -8.

Hopefully, you didn't get lost in that maze and that you are still with me because we are now gonna move into our final learning cycle.

And that final learning cycle is about deepening understanding of subtraction of integer.

We are now really, really confident at rewriting a subtraction as an equivalent addition.

We're going to see what that might look like and deepen our understanding.

So we are going to look at what happens if we've got more than two numbers.

So at the moment, we've just looked at two integers and we want to know what happens when we are going to be looking at more.

Actually, we're going to be doing exactly the same thing.

But what we're going to do once we've rewritten the calculation, sorry, the subtractions as additions, we're going to treat the negative and positive integer separately.

So let's rewrite the calculation first.

I know what, why don't you pause the video and have a go at this yourself before I put the answer up on the board.

Well done.

Now we can see, and I'm sure it is right, that you've rewritten that calculation.

Remember, we're getting rid of all of those subtractions and changing them to equivalent additions.

So it should now read -2 add 3 add 7 add -4 add -6.

Well done if you've got that right.

Now, just as we've done previously, we are going to treat this positive and negative integer separately first.

So let's go through, let's start with positive.

So positive integers in my calculation are the 3 and the 7, there aren't anymore.

So I'm going to find the sum of those, that's nice and easy.

I've made it easy for you.

That's 10.

Now, we're going to sum the negative integers in our calculation.

So we've got -2, -4, and -6.

Remember, we don't need to worry about the next.

We've only got negative counters, so we are going to sum those.

So the sum of the absolute values, remember, 6 add 4 add 2 is 12.

So our answer is going to be -12.

Now, we have the sum of the two separate parts, the positive integers and the negative integers.

We just need to find the sum of those.

And the sum of 10 add -12 is -2.

Now, you are ready to have a go at this quick check for understanding.

What I'd like you to do here is to decide which one of the options is the correctly rewritten calculation for -3 add 7 subtract 4 subtract -8 add -1, which one of those is correctly rewritten? Remember, we want to rewrite all of the subtractions as equivalent additions.

Now, if I was going to tackle this question, I would probably ignore all of the options and I'd rewrite that calculation myself, changing the subtractions to additions and then see which one it matched to.

But you do it however you want.

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

Great work.

Let's hope you got that right, but I'm sure you did.

The correct answer was d.

So if we just look through, we've got -3 add 7 and then we're subtracting 4.

So we wanna change that subtract to add in the additive inverse of 4, which is -4.

We've got then subtract -8.

So we need to change that to add in the additive inverse of -8, which is 8, and then we're adding -1 on the end.

Well done if you've got that right.

We are now on to our final task for today's lesson.

Well done.

You've done fantastically well so far.

And this is quite a difficult concept to understand.

So well done for sticking with it.

You are ready now to have a go at these questions.

Pause the video, good luck with it.

Have a go at all of the questions.

I'm sure you can do them all.

They are slightly harder.

Five and six are a little bit harder, but do have a go.

And then when you are ready, come back, and we'll see how you got on.

Good luck.

Here are our answers.

Number one was -2, number two was 1, three, -12, and four, -6.

So, hopefully, you rewrote those subtractions as additions, dealt with the positive numbers, dealt with the negative numbers, and then you came up with that final answer.

And then for five and six, we were looking for the missing integer.

So to make the calculation correct, what was missing? 5 was -6 and 6 was -23.

Really well done if you've got those last two right.

That shows that you have a really, really firm understanding of what we've been doing in today's lesson.

Don't worry if you didn't, you will get there.

Let's now summarise what we've been doing during today's lesson.

The main and overall idea is that all subtractions can be rewritten as an addition of their additive inverses.

No more subtraction.

We're just going to add.

Our example here is that 5 subtract 3 is actually equivalent to 5 add the additive inverse of 3, which is -3.

We've also looked at when there is a combination of positive and negative integers, find the absolute difference of the values.

If the negative part is bigger, then the answer will be negative.

And that if there are only positive or negative integers, we find the sum of the absolute values.

So examples here we can see, I'm only going to look at the middle column 'cause I'm not subtracting anymore.

I'm just looking at those additions.

We've got different symbols, so we find the difference of 7 and 13.

13 is greater and that's the negative, so it's -6.

The next one, we've only got negatives, so therefore we are gonna find the sum of 7 and 13, which is 20.

We only add negatives so it's -20.

And then the final one, we have got positive and negative.

So, therefore, we find the difference, which is 6, but this time the positive integer was larger.

That's why the answer is +6.

Like I said, quite a challenging lesson, but you've done superbly well to stick with it.

Well done.

Thank you so much for joining me.

I look forward to seeing you again.