Lesson video

Hi, y'all.

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.

The title of today's lesson is Multiplying and dividing where two or more values are negative.

This is within the unit, arithmetic procedures with integers and decimals.

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

So previously we've looked at multiplying and dividing with negatives, but only when one of the integers was negative.

So we're going to take that learning and we're going to extend it in today's lesson to looking at what happens if two or more of the numbers are negative.

Here are some words that you should be familiar with that we will be using in today's lesson.

Absolute value.

Remember, the absolute value of a number is its distance from zero.

So for example, 5 negative 5 both have an absolute zero of 5 because they were both 5 away from zero.

And product.

The product is a result of two or more numbers multiplied together.

Today's lesson is going to be split into three separate learning cycles.

The first one is going to be looking at multiplying when two values are negative, then we'll look at dividing and then we'll extend that in learning cycle 3 to multiplying and dividing and we'll introduce more numbers.

We're gonna start with multiplying when two values are negative.

Let's review multiplication using the two colour counters.

So you'll be used to see in these two colour counters.

Remember the red side is negative and the yellow side is positive.

Now, very simple calculation here.

And you might be thinking, why are we doing this with counters? But as I often say, it's useful sometimes to start with something easy and then we can build upon that.

Multiplying two positive vintages Looks like this.

We've got three lots of 5 positive counters.

And we can clearly see that that is three groups of positive 5, which is 15.

Or we could think of it the other way round, 5 multiplied by 3.

What does that look like as a diagram? We've got here five lots of 3, which is 15.

So five groups of positive 3 is also 15.

Now let's have a think about 3 multiplied by negative 5.

We've got three groups of negative 5.

Three groups of negative 5 is negative 15.

So we can count the count this up.

Negative 5 multiplied by 3.

Now we know that it will have the same, the same product, don't we? Because we know that multiplication, we can write either way around.

How does it look with two colour counters though? How you read the calculation can help.

And we can read this calculation as the negative of five groups of 3.

So the negative of five groups of 3 because we can't have negative 5 groups, so we're going to say as the negative of five groups of 3.

And that's how we write it.

So you make five groups of 3.

And to make it the negative of this, we flip all of the counters over so they all become red.

And we can see now that the answer to that calculation is negative 15.

This is probably a new concept to you.

Don't worry if you're a little confused right now.

We're going to go through this very, very slowly and build up.

Five lots of negative 3, here's our five lots of negative 3.

That's negative 15.

What about negative 3 multiplied by 5? Again, we know it's got the same product because we've just switched them around, but we need to think about what it looks like with the two colour counts.

Let's reread this calculation.

Let's reread it as the negative of three groups of 5.

The negative of three groups of 5.

We're gonna write that as the negative of 3 multiplied by 5.

We make three groups of 5 and then because we want the negative of it, remember, we're gonna flip all of those counters over and the yellow counters all become red.

This shows us now that we're definitely happy that the answer to negative 3 multiply by 5 is negative 15.

So to make them negative for something, we flip the counters over.

That's really important.

We can use the same idea if we're multiplying two negatives together.

Let's have a look and see what that looks like.

Negative 3 multiplied by negative 5.

We could read this of as, sorry, the negative of three groups of negative 5.

That's how we'd write it.

We're gonna make three groups of negative 5 because we've said we can read that as the negative of three groups of negative 5.

Here's three groups of negative 5.

What do we do to make the negative of it? Yeah, that's right.

We flip the counts over.

So we flip them all over and they become yellow.

So the answer to negative 3 multiplied by negative 5 is 15.

Let's just run through that again.

We're going to read it as the negative of three groups of negative 5.

So we drew three groups of negative 5, but because we wanted the negative of it, we flipped those counts over.

Or negative 5 multiplied by negative 3.

We could read this as the negative of five groups of negative 3.

Looks like that, and we make five groups of negative 3.

But we wanted the negative of that.

So what do we do? Yeah, that's right.

We flip the counters over and we can see it gives the same.

The most important step here is recognising that if we want to find the negative of something, we flip our counters over.

So here are the products we've just looked at.

3 multiplied by 5 is 15, 3 multiplied by negative 5 is negative 15, Negative 3 multiplied by 5 is negative 15, and negative 3 multiplied by negative 5 is 15.

We also looked at 5 multiplied by 3 is 15, negative 5 multiplied by 3 is negative 15, 5 multiplied by negative 3 is negative 15, and negative 5 multiplied by negative 3 is 15.

Those are the calculations we've just looked at and we've solved those problems using the double-sided counts.

Andeep says, "I've noticed that if both numbers are positive or both numbers are negative, the answer is positive.

And if there is one of each, the answer is negative." Do you agree with Andeep? It certainly does look like he's right, doesn't it? Let's check Andeep's idea with a different pair of numbers.

So we're gonna use 2 and 4.

2 multiply by 4.

I've got two groups of 4 or four groups of 2.

The answer to that, I think we'll all agree is 8.

Let's look at the next one.

Two groups of negative 4 or negative 4 groups of 2.

So we we want the negative of four groups of 2.

So remember we flipped those counts over, so that's negative 8.

So just thinking back to what Andeep said, if we had a positive and a negative, the answer's negative.

Well that's definitely worked, hasn't it? Let's have a look at this one.

So we've got the negative of 2 lots of four.

There's our two lots of 4, but we remember the negative of it, so we're gonna flip it over or four lots of negative 2.

And we can see that's negative 8.

So again, we have a negative number multiplied by positive number.

Certainly looking like hand deep is right because the answer is negative.

Let's look at the final calculation in this set.

Negative 2 multiplied by negative 4.

So we are going to have the negative of two groups of negative 4.

Here's our two groups of negative 4, but we want the negative of that.

So what are we gonna do? Yeah, we're gonna flip the counts over, or we could read that as four, the negative of four groups of negative 2.

There's my four groups of negative 2, but remember we wanted the negative of that, so we're going to flip those counters over.

So Andeep said if the signs were the same then it was gonna be a positive answer and it is, isn't it? We had a negative 2 and a negative 4 and our answer was positive 8.

So Andeep was right.

And we're gonna generalise now.

This is for the product of two integers.

If both integers are positive or both integers are negative, the product is the absolute value of the product.

So for example, 3 multiplied by 4 is 12 and negative 3 multiplied by negative 4 is 12.

If there's one positive integer and one negative integer, the product is the negative of the absolute value.

So for example, negative 3 multiplied by 4 is negative 12 and 3 multiplied by negative 4 is negative 12.

So in general, if both of our numbers are positive, both negative, it's the absolute value.

If we have one of each, then it's the negative of the absolute value.

Now you are ready to have a go at some questions.

So before you go onto a big task, I'd just like you to do this check for understanding so that we can be confident that you are gonna be successful at the task.

So you're gonna pause the video and you're gonna match each question on the left to the correct product on the right.

Good luck and come back when you're ready.

Great work.

Let's have a look then.

So we should have negative 5 multiplied by 4, matched with negative 20, negative 7 multiplied by negative 4 with 28, negative 6 multiplied by 4 with negative 24, 7 multiplied by negative 4 with negative 28, negative 5 multiplied by negative 4 with 20 and negative 4 multiplied by negative 6 with 24.

So remember, if your symbols are the same, it's going to be absolute value.

If the symbols are different, it's going to be the negative of the absolute value.

Well done with those.

Before I set you going on some independent tasks, we just need to make sure that we're all happy with how to do and fill in a number pyramid.

This number pyramid is a product number pyramid.

It means that the product of the two bricks below go in the box above.

So in this box we would find the product of 2 and negative 3.

The product of 2 and negative 3 is negative 6.

So in this box it would be the product of the two bricks below, which is negative 1 and 3, the product of negative 1 and 3 is negative 3.

We would then continue until we get to the top.

So that's 3 because the product of the two boxes below, negative 3 and negative 1 is 3, and we've got negative 18, negative 9, and finishing at the top with 162.

You are going to be doing some of those in the independent task, which is why I wanted to just make sure we all understood how to fill those in.

You are now ready to have a go at this task.

So you've got your number pyramids and remember the each brick is the product of the two bricks below it.

If you managed to get all of those done, you could have a go at creating your own number pyramid with a base of four bricks, which has a positive number at the top.

Good luck with that.

Come back when you're ready.

Well done.

Here are your answers.

I'm not going to read them out because you might get a bit confused as to which brick I'm talking about.

So the best thing to do is to pause the video, mark your answers and come back when you are ready.

Lovely.

Let's move on then.

So we are now comfortable with multiplying when two values are negative.

We're going to extend that now to looking at what happens when we are dividing and two values are negative.

Let's take a look.

We're going to start by looking at 15 divided by negative 5.

And the way we are going to do this is firstly, we're going to start by looking at how many groups of 5 we can make from 15.

So we're going to do 15 divided by 5.

15 divided by 5.

You can make three groups of 5.

15 divided by 5 is 3.

We want the negative of this, the negative of five groups.

So we flip the counters over because actually, we weren't dividing by 5, we were divided by negative 5.

15 divided by negative 5 is negative 3.

What calculations does this array show? Have a think about that.

And here's a bar model that represents the same thing.

Hopefully you said that this array represents two multiplied by 4 is 8 because we've got two rows of four counters.

Here in the bar model, we can see that we've got two groups of 4.

It also represents 4 multiplied by 2 because we've got four columns with 2 in each column, or in the bar model, We've got four parts with 2 in each part.

8 divided by 2 equals 4.

We know that from the facts that we've got above.

It's asking us how many groups of 2 can we make? How many groups of two? We can see that we make four groups of 2.

And in the bar model we can also see that we can make four groups of 2.

8 divided by 4 is 2.

So this is asking us, how many groups of 4 can we make.

Groups of 4, we can see that there are two of them.

And in the bar model again, we can see that there are two groups of 4.

What calculations does this array show? This array shows 2 multiplied by negative 4 because we've got two rows of negative 4 counters and that's negative 8.

And in the bar model we can see here that we've got two groups of negative 4.

It also shows 4 multiplied by negative 2 because we've got our four columns and those negative 2 counters in each column.

In the bar model, we can see that if we split 8 into four parts, there's negative 2 in each part.

That must mean then, that negative 8 divided by negative 2 is 4 because we know that we can rearrange the multiplication into a division.

Negative 8 divided by negative 2 is 4.

How many groups of negative 2 can you make? One, two, three, four.

One, two, three, four.

Now let's take a look at how many groups of negative 4 we can make.

One, two, one, two.

So negative 8 divided by negative 2 is 4 and negative 8 divided by negative 4 is 2.

We are grouping.

Here are the calculations we've just been looking at.

Just take a moment, pause a moment to have a look at those calculations.

See if you notice anything.

I wonder if you spotted the same as Andeep.

Andeep has noticed that just like when you multiply 2 negative numbers together, you get a positive answer.

It's the same when dividing.

So we're going to generalise just like we did for multiplying, but this time we are looking at the quotient.

If both integers are positive or both are negative, the result of the division is the absolute value of the quotient.

The quotient remember is the answer to the division.

12 divided by 4 is 3 and negative 12 divided by negative 4 is also 3.

If there is one positive and one negative, then the result of the division is the negative of the absolute value.

Negative 12 divided by 4 is negative 3 and 12 divided by negative 4 is negative 3.

So remember, just the same rules that apply with multiplying as they do with dividing.

Now I'd like you to pause a video and have a go at matching up the question on the left hand side to the correct quotient.

When you're ready, come back.

Great work.

Let's have a look at the answers.

25 divided by negative 5 is negative 5, negative 27 divided by negative 3 is 9, negative 16 divided by 4 is negative 4, negative 24 divided by negative 6 is 4, 30, sorry, negative 30 divided by negative 6 is 5, and 45 divided by negative 5 is negative 9.

You're now ready to have a go at this task.

This is a times table grid.

Your job is to find the missing numbers.

Remember in a times table grid, to find any of the boxes within the grid you find the product of the row and the column.

Good luck.

Come back when you're ready.

Well done.

Now we can check those answers.

There are a lot there.

I'm not going to read all of those out.

So pause the video, check your answers, and then when you are ready, come back and we'll move on to our final learning cycle for today's lesson.

Like I said, we're now onto that final learning cycle for today's lesson.

Done fantastically well up till now.

Let's keep going.

Here, we've got Aisha and we've got Alex.

And we've got a calculation, but there are some ink splats on it.

We don't know what any other numbers are.

But Aisha says, "I don't need to be able to see the numbers to know if the answer is positive or negative." Hmm, what are your thoughts on that? Do you think it's possible to know that? Well, Alex wants to know how? I wonder if you can answer that question.

Let's see what Aisha says and see if you agree.

She says, "I count the number of negative integers.

If there is an even number, the answer will be positive.

If there is an odd number, the answer will be negative." Hmm.

What are your thoughts on that? Let's check Aisha's idea.

Let's see whether she's right.

So here I've got a negative number multiplied by a positive number.

And we know from what we looked at in learning cycle one, this means our answer is going to be negative.

It's got one negative number and the product is negative.

Let's look at the next one.

Here, we have two negative numbers.

So according to Aisha, it's got an even number of negative numbers so the answer is going to be positive.

Let's see whether that's right.

So a negative multiplied by a positive is a negative and then I'm multiplying that negative by another negative.

And remember Andeep spotted, if the symbols were the same, if it was two negative numbers, the answer is going to be the absolute value.

Here we're not working out a value, we're just deciding whether it's going to be positive or negative.

So yep, even number, two negative numbers, the answer was positive.

So it's looking like Aisha's right? Let's now have a look at this one.

So this one has three negative numbers and Aisha says if it's an odd number of negative numbers, then the answer's going to be negative.

So let's take a look at this one in more detail.

That's negative and then we're gonna multiply it by the rest.

And then negative and then negative, when we multiply those, we get a positive.

And now we're gonna multiply that by a negative, which is negative.

So Aisha is right.

If you have an even number of negative numbers, the answer is going to be positive and if you have an odd number, the answer is going to be negative.

So true or false, the answer to the following is negative.

As always, not just a true or false, please.

I want that justification.

Justification that shows me you really understand why your answer is true or why your answer is false.

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

Well done.

Let's have a look.

Did you say false? Well done if you said false and you should have also said B.

There is an even number of negative numbers.

We've got negative 3, negative 4, negative 2, negative 7.

So that's four negative numbers.

That's 4 is an even number, and so therefore it is going to be a positive number, not a negative.

Here they are again, Alex and Aisha.

Let's see what they're chatting about now.

Is it the same for division? Alex wants to know.

Aisha says, "Well, multiplication and division are inverses of each other, so yes." In this check for understanding, I'd like you to decide, I don't want you to work out the answers, all I want you to do is to decide which of them will give positive answers.

As always, pause the video and when you are ready, come back.

Well done.

Let's have a look and see which ones will give positive answers.

A, we've got two negatives.

Two is even.

D, we've got one, two, three, four negatives and 4 is even.

B had one negative that's an odd, and C had three negatives that's also odd.

So it was A and D.

Super work if you've got both of those.

Now Alex has done his homework.

And what I'd like you to do, please, is I'd like you to mark his work.

So you are going to be the teacher here.

You are going to check his work and you are going to mark it right or wrong.

But as any good teacher would, they wouldn't just put a cross next to it.

They would also correct the mistakes to help Alex with his learning.

Pause the video now and have a go at these questions.

Well done.

Let's have a look and see whether you agree with this.

Number one was not correct.

The answer should have been 24.

2 was correct.

3 was incorrect, it should have been been negative 1.

4, 5, and 6 were all correct.

7 was incorrect, it was 9.

And 8 was incorrect.

It should have been negative 8.

Don't worry too much if you didn't get 7 and 8, right? I put those in for challenge for those of you that need that.

I forgot to mention before I set you off on this task, that question nine was to make up some of your own questions with incorrect answers and to think about mistakes that people might make.

I wonder if you did that.

Well done if you did.

I'm super impressed.

Now we are ready to do a summary of the learning that we've done in today's lesson.

You've done fantastically well and there's been some really quite difficult concepts there with the idea of these negatives and flipping counters.

But hopefully, now we've been through it and also we've been through those generalisations, you feel happy and confident that you can multiply and divide with any number of negative values.

The product of two negative integers will be positive.

The product of an even number of negative integers will be positive.

The product of an odd number of negative integers will be negative.

And multiplication and division are inverses of each other, so we can just apply the same rules.

So the same is true for both multiplication and division.

Like I said, today's lesson I think, has been quite challenging.

You've done fantastically well.

I'm really pleased that you've stuck with me right to the end, and I hope to see you again in another video.

Thank you.

Bye.