Lesson video

Hello, my name is Mrs. Buckmire, and today's lesson's about further division.

So make sure you have a pen and paper.

Sometimes I'll ask you to pause the video, to have a go at something, so please do pause it, but also remember to pause it whenever you need to, you can also rewind the video.

Sometimes it helps to hear something again.

Okay, let's begin.

Okay, so try this, complete the calculation.

So you want different calculations there.

The final one, I'd love to see your creativity here.

So can you think of two numbers that you think no one else in the world would have written down at this stage? What do you think? Maybe you can use some negatives, maybe some decimals, maybe some fractions mixed fractions.

That's just a chance for you to show off.

Okay.

Pause the video and have a go.

Okay, so here are my answers.

So six times negative four was negative 24.

Nine times, it should be negative two.

Negative five times negative six equals 30.

10 times negative nine, and negative 12 times by negative seven, makes seven times eight to 12 equals 84.

So what did you get for this one? One example I put was negative two times negative 22.

Well done.

If you've got something different, that was also correct as well.

Did you use fractions? Fantastic.

Check your work really carefully.

And I hope you enjoyed doing that.

Okay.

So we can actually derive the following fact family using the product.

Negative two times negative four equals eight.

So one from this, if I know negative two times four equals eight, then I also know that negative four times negative two equals eight as well, yeah? Commutativity.

Do you remember that word? What else do I know? Involvement division.

Yes.

I know that actually, eight divided by negative two equals negative four.

And remember that is just another way of doing it over in a fraction form.

This is equal to eight divided by negative two.

That's just another way of writing it.

And so if I know this, what else do I know? Another division fact? Yes.

That eight divided by negative four equals the negative two.

So what I want you to do is write similar calculations, for each of the following products.

So negative two times three equals negative six, and negative two times negative six equals to 12.

Now make your fact family involving these negative numbers.

If it helps, you can draw a diagram.

So if you've seen previously, you might draw like a stretch diagram or the addition diagram and whatever you need to do to help you out, but have a go at writing similar calculations.

Pause the video and do have a go.

Okay, so negative two times three equals negative six.

Well, that means I've just written it out first as it was, and then again, I'm using their commutative law.

So just swapping the numbers round.

So three times negative two equals negative six.

It's all about division.

So first one, negative six divided by negative two equals three, and then we can have negative six divided by three equals negative two.

So for the next one you could have got these.

Pause the video and check your work carefully.

Okay.

So now you're ready for the independent tasks.

There are three questions here.

The last one, again, I just want a bit of creativity there for question three, and this is all just revision of things that you have done before, but just trying to relate the facts more, derive the information from a given fact.

Okay.

Pause and have a go.

Okay.

So the first ones, I'm expecting you to be quite confident with these.

So I think because we've practised them before, I can just show you the answer.

So negative 15, negative 15, negative two, negative two.

And again, here.

So pause the video and check those.

If you're not sure you can rewind to the previous task and have a listen in, to those relationships.

Okay.

So the question two, now, the answers you should have got.

Negative two, negative four, negative eight, negative six, negative 12, negative four.

So what I thought is more interesting.

Can you see relationship between A, B and C? And can you see a relationship between D, E and F? Hmm.

What do you think is going on here? Yeah, so what's happened is each time it's always negative 24 being divided, but here is divided by 12, then divided by six, then divided by three.

So actually to get from 12 to six, we need to divide by two.

So then to get from negative two to negative four, what do we have to do? Good.

We times by two, I wonder if that relationship holds true here.

So negative, so positive six to three, we have to divide by two.

Negative four to negative eight, we have to times by two, it's quite interesting.

And the same happens below.

So negative eight and negative four, we divide by two, negative six and negative 12, we times by two.

Negative four to negative two, we divide by two.

Negative 12 to 24, we times by two.

Okay.

How many different ways can you place integers in the the two spaces to make the equality true? How many different ones did you get? Ah, there're so many, I'm just going to give you some examples.

So now if you did positives, maybe you could have a positive on the first one.

So you could have done 60 divided by three equals 20, 60 divided by two equals, whoops.

30, that should be 60 divided by 10, equals six.

So what about if you try to use negatives? Yeah.

So 60 divided by negative three equals negative 20, 60 divided by negative two equals negative 30.

And you see how I'm actually just using the answers I already made? Okay, so what about that next one? Negative six divided.

It could have been divided.

So I'm kind of going to use the same information.

So divided by three equals to negative 20, you could have got negative 60 divided by negative three equals a positive 20.

Negative 60 divided by 20 equals to negative three.

Negative 60 divided by negative three equals two.

Oh, I've done that one, I've done that one.

Oh Mrs. Buckmier, what are you doing? And peeing yourself.

Let's change that one to, what one do I want it to be? Yeah, it was actually negative 20.

Okay.

So have a look at their different relationship, pause and check your work carefully.

Okay.

So for your explore, what I want you to do is consider each of the following statements and decipher each, if it's always, sometimes, or never true.

Here n represent any number, so, n divided by two greater than two, zero, sorry.

Is it always, sometimes, or never? True, n divided by negative two is less than zero and n divided by negative two equals n times negative half.

So if you feel super confident, you go for it, even if you're not confident, maybe just have a little go, thinking oh maybe I could try this.

Maybe I could do this.

I'll give you some support in a second.

Okay so, pause port.

If you've done this with me before, you'll know that I like to just try things out.

And as mathematicians, when we don't understand something, sometime we just, just throw some numbers at it.

So maybe choose a number, probably one divisible by two.

'Cause that might make it a bit easier.

So like, I might choose a number, let's say eight and I might try it out.

I also might try out the number zero, 'cause that's always interesting, and more often interesting.

And I might try out negative eight.

'Cause therefore I've got a positive, I've got a negative and I've got zero.

And it's saying for always to be true, it would work for all of them and maybe I have to think a bit more carefully about the structure of the question.

If it's never true, maybe it won't work for them more.

Maybe I can think, oh, why is that? And then if it's sometimes true, maybe it'd work for one of the numbers, but not the other numbers.

So you can think through that.

So have a go, substitute in it in, and seeing what kind of answers you get.

Okay, how did you do? So I did say we could try the numbers eight، zero، negative eight.

Maybe you just thought about it.

But I'm going to show you one method that you could kind of use to work this out, one strategy even.

So if I'm subbing eight, so eight divided by two equals to four.

Is four less than zero? No.

So four is not less than zero, so it does not work.

So is it never true? Hmm.

Let's investigate a little bit more.

So what about negative eight? So negative eight divided by two.

Well, I know that two times negative four equals negative eight.

So this equals negative four.

So negative four, is it less than zero? Yes it is.

So it is true.

So sometimes it's true and sometimes it's not true.

Hmm.

Maybe you can write an inequality about when it is true, and when it isn't true.

So it is true when this was negative eight.

So when else is it true? It's true for negative six, it's true for negative two.

Is it true for negative three? Yes it is, 'cause negative three divided by two is still a negative number, even if it's a decimal or fraction.

So it's true for all our negatives it seems, so whenever N is less than zero.

So it's a negative number, it seems to be true.

And that makes sense, we think about our diagram as well.

We're always stretching the negative one divided by two.

It doesn't matter how much we stretch it, It's still going to be negative, when we're dividing it by a positive two.

Maybe you got that.

Okay, so what about this other one? Let's try it.

Now, if you haven't had a go yet, feel free to pause it.

I'm going to go straight in, so eight divided by negative two.

When I know negative two times negative four equals eight.

So this equals negative four, which is less than zero.

So it works.

Is it always true then? Hmm.

What about the negative eight? Negative eight divided by negative two.

Now negative two times four, stretch four, would be negative eight.

So that equals four.

Ah, four is not greater than, not less than zero.

It's greater than zero.

So it does not work.

So again, it's only sometimes true.

Hmm.

Have a go at rounding up equality? When is it true? Did you think about some more examples? Yeah, so it's true for six, five, four, nine, 100.

So it's true whenever n is greater than zero.

So if n is greater than zero, then n divided by negative two is less than zero.

And I was just to say, this means this arrow often used by mathematicians, it means it implies that.

So you can mean, so Maths, we like to be super efficient and just use that to communicate those words.

And more about when N is less than zero.

Yeah.

So it's true for the top bottom line, but it's not for the bottom line.

So this one is true, this one is not true, same here.

Pause if you want to check your work.

Okay and the final one.

So, n divided by negative two equals the n times negative a half.

Hmm.

So n divided by negative two, equals two.

Let's just try that out.

See if I had like eight divided by negative two.

Well, I know that negative four times negative two equals eight.

That would be negative four.

Is that the same as negative four times negative a half? Sorry, I said that wrong.

Is it the same? What was that n again? N was eight.

So is it the same as eight? I'm replacing each of my n's with eight, is the same as eight times negative a half? Well, yeah, 'cause if eight times negative a half.

Well, if we have eight and then we're stretching it in the negative direction instead, so opposite direction, and it's only a half, that's absolutely valid so it's going to be negative four.

So it is the same.

So it works for eight.

Hmm.

Will it always work? Yes, it will always work.

So n divided by negative two equals the n times negative a half, always.

Really well done if you've got that right.

Excellent, excellent work today, everyone.

If you had to get the tries this and the explore, you should be really, really proud of yourself, okay? I would love it for you to do the exit quiz.

It's a really helpful way for you to check your understanding, and I will see you next lesson, hopefully.

Bye.