Lesson video

Hi, 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, "Multiplication and Division where only one value is negative" And this is within the unit, "Arithmetic procedures with integers and decimals".

So up until now we've looked at negative numbers, we've looked at them in context and we've looked at adding and subtracting with negative numbers.

So we're going to move our learning on now to look at what happens when we start multiplying and dividing with negative numbers, and we're gonna build that up nice and slowly.

And today we're just going to be looking at what happens when one of the integers we are multiplying or dividing by is a negative.

Two key words that we will use in today's lesson are dividend and divisor.

Just a reminder of what those are then.

The dividend is what we are dividing and the divisors is what we are dividing by.

So for example, in the calculation, six divided by three, six is the dividend and three is the divisor.

You'll be familiar with these words, you may see them in today's lesson, which is why I've brought your attention to them.

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

The first learning cycle, we are going to just look and concentrate on multiplication, and we're only going to be looking when one value is negative.

So first learning cycle, we just concentrate on multiplying.

When we're happy with multiplying, we'll be then ready to move on to the second learning cycle, which is when we're going to be looking at division, but we're going to be concentrating on just looking at when the dividend is negative.

So remember the dividend is what we are dividing, not what we are dividing by.

Let's get started then on that first learning cycle, multiplying when one value is negative.

Jun and Laura each have 24 negative counters.

So remember we've talked previously about our counters for negative numbers, and negative ones remember, are the red ones, the red side if you've got double-sided counters.

Jun sets his counters out like this.

That's what Jun's done.

Let's just check, has he used all 24? He has.

We could set that out also, if we didn't have counters, as a bar.

So we've got here negative six in each.

These two diagrams show us four groups of negative six, and we know that that's negative 24 'cause that's the number of counters that Jun had.

We can also say he's got four lots of negative six, which is negative 24.

Lots of in maths we use a multiplication symbol.

Four multiplied by negative six is negative 24.

So the mathematical calculation that represents these two diagrams is four multiplied by negative six and the answer is negative 24.

Laura sets hers out like this.

So Laura set hers out and in a slightly different way.

What would that look like as a bar? How could we describe Laura's diagrams? If we use the same structure as we did for Jun's diagrams, we can do the same for Laura's.

Laura has six groups of negative four.

We could also read that as six lots of negative four and we know that that can be written as six multiplied by negative four.

So they both started with 24 counters, but they've set them out in slightly different ways, but both of them still have the negative 24 counts.

Here's another representation of a different number.

What number is this a representation of? Yes, that's right, it's a representation of negative 12.

Jun says, "I can see three groups of negative four" And Laura says, "I can see four groups of negative three".

Who is right? Actually, they're both right.

Let's take a look and see how they can both be right.

So Jun says, "I can see three groups of negative four".

So zoom's looked horizontally.

So if we look horizontally, we can see Jun's quite right, that we've got three rows, so three groups and there's negative four in each group.

If we look vertically, we can see why Laura says there's four groups of negative three.

So vertically we can see we've got four columns of negative three.

So four groups of negative three.

They've still both got negative 12 counts.

What I'd like you to do now is to have a go at this question.

I'd like you please to write down two number sentences that describe this array.

So you're gonna pause the video and when you've got your two sentences you can come back and check.

Great work.

These are the two sentences that you should have.

Eight groups of negative four.

So if we look vertically, we can see that there are eight columns, and in each of the columns there are negative four counters, so eight groups of negative four, or we could see four groups of negative eight.

So looking horizontally we can see we have four rows and there are negative eight in each row.

The number of counters doesn't change, so the total of those is going to be the same for both.

Doesn't matter which way round we described it.

So like I just said, it doesn't matter which way you interpret the array, you're going to end up with the same thing.

This array shows us four multiplied by negative five.

So we're looking at the rows, four rows, negative five in each, and if we were to count up all of the counters, there would be 20 of them, and they're negative, so that's negative 20.

Four multiplied by negative five is negative 20.

What about if we look vertically? So if we look vertically, we can see we have five columns, and in each column there are negative four counters.

Five lots of negative four.

Five multiplied by negative four is also negative 20.

You are ready now to have another go at check for understanding.

In this check, what I'd like you to do is to identify all of the alternative ways of writing six multiplied by negative three.

So of those four options, which one of those are just an alternative way of writing six multiplied by negative three? Pause the video when you're ready, come back.

Great work.

Let's have a look at our answers then.

So you should have identified A, C and D.

All three of those are just an alternative way of writing six multiplied by negative three.

I'm sure you've got all of those right and well done.

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

I'd like you to calculate the answers to the following questions.

And remember, no calculators.

So the questions get a little bit trickier as they go further down, but you've got all of the skills and all of the knowledge to be able to be successful at these.

Just remember, some of those methods that you've used in previous lessons for the multiplication.

Good luck with this.

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

And question number two.

So question number two, I'll challenge you a little bit here.

Write down as many products as you can with the following answers.

So for example, if I was going to be very unimaginative, for A, I could write down one multiplied by negative 60.

I'd like you to write down as many products as you can for each of those answers.

Challenge yourself to come up with at least four in each, but I'm sure you can impress me by coming up with a lot more than four for each.

Pause the video now, have a go.

Get loads and loads of products and I'll see you when you've got your answers and come back.

Well done.

There's a third question in this task you'll be pleased to hear.

And it's this one.

So I've given you here eight separate calculations, and what you're going to do, without a calculator remember, is you are going to calculate the answers to those questions.

So you're gonna work out the answers to questions A through to H.

When you've got your answers, you need to find them in the grid, and it will give you a letter.

If you can't find your answer in the grid, it just means you've made a little mistake.

So just check through your answer carefully to see if you can see where you've gone wrong.

Once you've answered all eight, you should have eight letters, and those eight letters you're gonna try and rearrange to make a word.

So you're gonna rearrange the letters to make a word.

And then if you've got time, you could have a go at writing your own word using the code.

You might choose to write your name, your favourite sport, whatever you like.

So pause the video now, give these questions a go.

Good luck and hopefully you'll end up with the same word that I have.

Well done.

Now, did you manage to come up with your own word that you wrote in code? Brilliant.

Also, have you found the word that was made from the rearranged letters? Let's see, let's go through the answers.

We'll go back to one first.

One A is negative 15.

B, negative 32.

C, negative 72.

D, negative 60.

E, negative 75.

F, negative one.

Sorry, negative 1,130, and G, negative 585.

Well done if you've got all of those right, particularly the last two, which were a little bit more challenging.

Question two.

So here are some examples of products that you may have written, and you can see there at the end of the A is very similar to that one example I gave you.

Hopefully you've been a little bit more imaginative than that and that you've come up with, like I said, at least four for each.

I won't read out those answers.

Your answer may not be on the screen.

That doesn't mean it's wrong.

What it might be worth doing is just checking your answer with a calculator.

So pause the video now.

You can check the screen to see if any of your answers are there, but if they're not there, don't worry remember, just check your answer using the calculator and make sure it gives you the product that you were expecting to.

Now let's have a look at three.

Let's see whether you managed to unscramble those letters and get that word.

The answers first though.

So A was negative 180.

B, negative 126.

C, negative 528.

D, negative 165.

E, negative 162.

F negative 135.

G, negative 180, and H, negative 336.

Well done if you've got all of those.

That would've given us the letters once we've rearranged them, it would've spelt the word, "Negative".

Superb work if you managed to unscramble those and get that word, "Negative" Well done.

And even better if you manage to write your own word using the code.

What a shame I can't see what you've done.

So, we've looked now then at multiplication, and we're happy with multiplication.

We know what we're doing, we're thinking as groups of, and then we can just complete the multiplication.

Let's have a think now then at what happens when we are dividing.

And remember I said we're only going to be looking at when the dividend is negative.

In our future further learning, we will look at what happens when the divisor is negative.

But for today we're just going to concentrate on that dividend being negative.

Here I have 24 negative counters and I need to divide by four, so I need to share them into four groups.

Here are my four groups.

So I'm going to put my counters carefully.

Remember when we're sharing, we need to share them into equal groups, which is why I'm doing them one group in turn.

I've shared my 24 negative counters equally between those groups.

We could also represent this calculation with a bar model.

Let's see what that would look like.

Here's my bar.

So my total that I've got is negative 24, and I'm sharing it into four groups.

So there are my four groups.

So I'm taking my 24 and I'm sharing it into four groups.

So really I could think of that as 24 shared into four groups, which is six, but I was sharing negative 24, and we can see from my loops at the bottom of the page that actually there are, yes, there are six counters in each loop and there are negative counters.

So we should put negative six in each of the bars.

Negative 24 divided by four then is negative six.

That's the answer to that calculation.

I think you'll agree that that's, that makes sense, doesn't it? If I've got 24 negative counters and I share them into four groups, I'd have six negative counters in each group.

We know that negative 24 divided by four is negative six, and we're confident of that because we've just shown that on the previous page with the sharing into the groups and also that bar model.

So Jun now says, "Does that mean that negative 24 divided by six is negative four?" Have a think about that for a minute.

What do you think? I think Laura's got something to say back to Jun.

Let's see what Laura's got to say.

Laura says, "Yes, it does." Wonder, did you say yes, it does as well? Well done.

If negative 24 divided by four is negative six, then negative 24 divided by six is negative four.

Because if you think about it, we're just sharing into more groups, so we're sharing into six groups rather than four, so there'll be four in each group.

And remember, we were sharing negative counters, that's why our answer is going to be negative four.

Here we have a representation of negative 12, and we are going to divide that by three.

How many groups are we going to share those 12 counters into? That's right, three.

We're gonna share them into three groups.

Here's my three groups, and again, I'm just going to equally share those counters into the groups.

How many counters are in each group? That's right, four.

And they're negative counters.

Remember, we can also share this as a bar model.

I've got negative 12 and I'm splitting it into three groups and there are negative four in each group.

So the answer to negative 12 divided by three is negative four.

What other fact do we now know from this? So thinking back to what Jun and Laura said on the previous slide, what other fact do we now know? That's right, we know that negative 12 divided by four is negative three.

Superb, well done.

Another check for understanding now.

So this one, what I'd like you to do is find me which of the following is the odd one out? So basically, it has a different answer to all of the others.

So you're gonna work out the answers to A, B, C, and D and one of them will have a different answer to the other three, and that's gonna be the answer to this question.

Pause the video and then when you are ready, come back.

Well done.

Let's have a look then and see if you agree with me as to which one was the odd one out.

So A, the answer was negative four.

B, the answer was negative four.

C, the answer was negative five, and D, the answer was negative four.

So C was the odd one out.

All of the answers were negative four except for C, and that one was negative five.

Well done, I'm sure you got that right.

We're now ready to move on to task B.

We're going to be answering these questions, so you're gonna pause the video, have a go at these questions and then when you're ready, come back, and then we can move on to the next question.

Good luck with these.

Pause the video, thank you.

Well done.

Now we're going to move on to question two.

Very similar to the question two that I gave you in the first learning cycle, which is really going to show me that you've understood what we are doing today, and that is I'd like you to write down as many division questions as you can with the following answers.

So for example, for A, I may write down negative 16 divided by four.

So challenge yourself to try and come up with as many different ones as you can.

When you've got as many as you think you can get, I'd like you to come back.

Pause the video now.

Great work.

Now we're gonna move on to question three.

Question three.

Answer the questions, shade in the answers in the grid to reveal something.

So you've got your questions on the right hand side.

You're gonna work out the answer to each of those and then you need to find that answer in the grid.

It may appear more than once, so look through the grid really carefully and find each answer and shade it in.

When you shade it in like it says, it should reveal something.

Good luck with this.

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

Brilliant, that was quick.

Well done.

Has your picture revealed something? Let's see.

We'll go back to question number one and then we'll see whether you've got the same reveal as I have, the same picture.

Question number one then, here are our answers.

A is negative four.

B, negative eight.

C, negative eight.

D, negative eight.

I must have had something about negative eight when I wrote these questions.

E, negative seven.

F, negative 85.

G, negative 65, and H, negative 172.

How did you get on with those? Hope you did really well.

As in task A, these answers here are just some examples of ones that are correct.

Pause the video now and use your calculator to check whether your answers are right, and then when you've done that you can come back and we'll have a look and see whether you've got the same picture as me.

Now we're ready to have a look then and see what was revealed when we shaded in all of those boxes.

Hopefully you can see shaded in here, so we should have shaded in, so our answers, let's go through those first.

Negative six for A, B negative nine, C, negative seven, D, negative 12, E, negative 11, F, negative two, G, negative eight, H, negative 20, I, negative four, and J, negative 15.

And then when you shaded all of those in, hopefully you've got the same as me, which was a division symbol, because let's face it, that's what we were doing, weren't we? We were dividing.

Well done.

Now we're ready to summarise our learning from today's lesson.

So firstly, negative counters can be used to help us with multiplication of positive and negative integer.

So remember, we looked at groups of negative counters, so two lots of negative three.

Here I've got two rows of negative three and then we can see that that is negative six.

You might not need to use the counters, but they can be useful.

Negative counters and bar models can be used to help with division of a negative integer by a positive integer.

So remember we were just looking at negative dividend.

We'll move on to looking at negative divisors a little bit later on through this unit.

So negative 12 divided by three.

Remember that means sharing negative 12 into three groups, and we end up with negative four in each group.

Or we could do that as a bar model.

We take our bar of negative 12, we split it into three equal parts.

Each of the parts then would have negative four.

Well done for today, you've done fantastically well.

I really enjoyed working with you and I look forward to seeing you later on through this unit.