Lesson video

Hello, my name is Mrs Buckmire and today I'll be teaching you about multiplication as scaling, okay? So, make sure you have a pen and paper and remember, pause the video when I ask you to, but also pause whenever you need more time.

So when you're checking answers or you want to have a go at something before I say it, just pause the video and, of course, rewind as well to hear something again as sometimes it helps.

Okay, let's see what the "Try this" is today.

So, for the try this, you're going to write a calculation for each number line.

So here's an example.

Now, we see that the longer arrow is 72 and now we can split and we can assume that each of these are equal.

So, I can say 24 plus 24 plus 24 equals 72 and I've found out 24 by doing 72 divided by three because there are three equal parts or I can say that three lots of 24 equals 72 so three times 24 equals 72.

So, you can use this to help you and for these number lines, can you write a calculation for each one.

Even better if you can write two calculations for each one, one being a sum and one being a product.

Pause the video and have a go.

Okay, so, for A, I would have got six times 12 equals 72 or 12 plus 12 plus 12, six times, to equal 72.

What about B? B relates to my example, so my example, I had 24 lots of three equals 72 and this one you'd get three lots of negative 24 equals negative 72, so it's like my example, actually, it's in the opposite direction so it's negative, so negative 24 plus negative 24 plus negative 24 equals to negative 72.

And for C? Excellent, two lots of negative 36 or negative 36 plus negative 36 equals to negative 72, so mark your answers carefully.

Okay, firing on from that, we already kind of see a multiplication as scaling, but I just wanted to make it a bit clearer.

So, here we have one lot of two, so one times two and you can see if I increase this I have two lots of two which is four and we've got repeated addition here so it's two plus two.

If I do three, it's three lots of two.

We can see how see how this arrow is being stretched out so it's like if it's a rubber band, it's being stretched.

So now we have two plus two plus two.

And her four lots of two.

So let's have a look at two lots of two to get four.

Now, what would happen if this two was actually negative? What can you predict will happen? Good, so actually it's still four long, but it's negative four now, so it's in the opposite direction, so it's been stretched in the opposite direction when it is negative.

So, if we put this up to three, before it was three lots of two and it was repeated of twos, but now it's repeated of negative twos, so it's negative six.

It's been stretched in the opposite direction.

Okay, so you just saw me using GeoGebra which is something you can look up online, so GeoGebra to see how, actually, multiplication can be used to stretch.

Now, Anthony has six lots of negative two and he's saying it's equal to two lots of negative six.

And this is the diagram he's drawn, so can you see how, if negative two is stretched from this point to this point then if it's stretched three lots you get to negative six, but if actually it's stretched six lots it's the same as negative six being stretched twice.

So negative two has been stretched six times and negative six has been stretched twice, so we call two and six the scale factors and, actually, I'm just going to write that one for you as well, 'cause that is an important piece of information.

So here we can call these the scale factors.

Okay, let's see another example from Bim.

So, three lots of negative two, so stretching negative two three times, so three is a scale factor and two lots of negative three so two is the scale factor.

This is how it looks and they're equal here, they end up being the same length.

So why is that? What's the same and what's different on either side? Yeah, so it's the same digits, but one's got a positive and one's got a negative, that's interesting.

Why don't you have a go.

So, here we have negative three stretched up to this point and we have negative four stretched up to here.

Write down a statement using products that is true.

Okay, so you would have got four times negative three equals to three times negative four maybe.

Maybe you swapped them around so maybe instead of writing four times negative three, you wrote negative three times four equals to negative four times three.

Do you remember what that's called? Do you know what that's called? That's called commutativity.

So if I was saying it in words, I might say negative three scaled by a factor of four is negative twelve.

So, here, four is the factor.

Or I might say negative four scaled by a factor of three is negative twelve.

So, here, three is the factor for this one.

So, this is actually an identity of a times negative b is exactly equal to always b times negative a and that's actually a rule that's always true.

Okay, for your "Independent task" kind of thinking about that scale factor, and the fact that multiplication can be thought of as this stretch, I want you to have a go at this.

Now I'll just quickly go through what I want you to do, so for A, we can see that from here to here is negative eight, but it's been stretched two times so the scale factor is going to be two.

And then if I was writing it out like this, well I'd write two times negative eight equals to negative 16.

Okay, you have a go at those and then for question two, I want you to identify the equal pairs of calculations.

For some you might not need to work it out, but for others it probably is helpful.

Okay, so going through these.

So for A, we're going to get two times negative eight equals negative 16 and the scale factor, let's write it on, so the scale factor equals to two.

The next one, we're going to get four times negative eight, so what's the scale factor? Good, the scale factor equals to four.

For this one, the scale factor is three.

Okay, now getting a bit trickier.

So D, what did you put as the scale factor? Good it's been stretched the same amount as in not really stretched at all, not changed at all, so it's just one, so the scale factor, we can say, equals to one.

What about E? Good, a stretch can make it smaller as well, so actually the scale factor here is going to be one half.

So, let's write it as a half times negative eight equals to negative four is what it would be.

And the final one, the scale factors equals a quarter, so a quarter times negative eight and it equals to negative two.

Okay, so for question two, maybe you worked them all out.

Here are the answers if you did, you can see them.

So, which ones were equal? So, you wouldn't have to work them out, you might have seen that, actually, a is equal to g, because of that rule that we discovered.

So, actually a and g are equal.

You might have seen that b and f are equal.

Which one's equal to c? Yeah, that one you had to work out, so it was c and e.

And then we had d and h being the same.

Well done if you got those right.

Okay, so now is our "Explore" task and this is a tricky one, but I believe you can do this, okay? So, I want you to generate examples of n where: three times n is less than two times n, and where three times n is greater than two times n take away one, so your n in this one might be different, and, for c, where two times n take away one is less than three times n which is less than two times n.

Okay? So there's a lot of thinking, a lot of, what I want you to do, I just want you to have a go, so often, as mathematicians, when we come across something that we're not sure about we just plug in some numbers and have a go so you might plug in some positives, some negatives, maybe zero.

Just put in some numbers and have a good go at this, okay? Even if you only find one example that's fine, maybe you find two or three that work out or maybe, as a challenge, you could find a rule where, actually, whatever happens, this is true and then actually it works out as an example.

Have a go.

Okay, you've had a go? You need some support? Right, why don't you try, so I'll give you some values of n to try.

So, I did say it, but why don't you try n as negative one? n as zero, n as one, n as two, so let's just try it as two first, maybe it's the easiest.

So, if we do three times two, we get to six and we do two times two we get to four so is six less than four? No.

So that does not work.

Okay, let's try with one.

So, when n equals one, three times one equals three, two times one equals two.

Ugh, no! It doesn't work.

Okay, let's try negative one, so three times negative one equals negative three, so it's been stretched by a scale factor of three in the negative direction, well, negative one was already in the negative direction so stretched three times so it's negative three, and two times negative one gets to negative two so is negative two bigger than negative three? Yes it is.

Good, 'cause it's further on the right on our number line, so actually this works when n equals negative one it works so maybe try some other ones, maybe try, like, negative two.

And then similarly put it into these different ones as well and see which ones work.

Good luck with it.

Okay, so for this first one, generate examples of n where three times n is less than two times n.

Now, I showed in the support that n, when n equals negative one, it holds true and, actually, for any n less than zero, for any negative n, it will actually be true.

So really, really well done if you got that.

Okay, so what about b? What did you get? When did it work? Did it work for zero? Yes.

When n equals zero we get three times zero being greater, because that equals zero, is greater than two times zero take away one.

Two times zero is zero, take away one, just leaves it as negative one.

Zero is greater than negative one so it works for n equals zero and, actually, it'll work for any n being greater than negative one.

Now, massive, massive credits if you figured those bits out, if you figured out inequalities and that is going above and beyond and really, really well done.

So, finally, for two, for c, did you work out anything? This one was hard.

Yeah, you had to use fractions and not even just an old fractions, negative fractions.

So, for example, if you used n equals negative a half, you'll get two times negative a half take away one is less than three times negative a half which is less than two times negative a half.

Right, let's check that out.

So, if I have negative a half, so it's like that.

So, if I have two lots of it take away one, well, two times negative a half becomes, yeah, negative one and negative one take away one is negative two.

So negative two is that answer.

And then three times negative a half, so I'm at negative one here, considering this to be zero, so take away another negative a half makes me at negative one and one half.

So, negative one and one half, is that bigger than negative two? Yes it is.

So, because it's closer to zero, it's closer to the right, so therefore that's true, so negative one and a half or negative three over two, depending on what you did, and then finally two times negative a half, what's that? We already worked that out, didn't we? That was negative one, which is bigger, so it holds true for n equals negative a half.

That one was the hardest one.

That was a bit of a challenge, so really, really fantastic work if you got that.

Okay, well done everyone! The main thing that I want you to learn from this lesson was that multiplication can be seen as scaling it.

So we have our scale factor, which was how many lots of the original we have and remember it can be decimals as well, okay? And, you know, you guys have had a go and understood about commutativity so that's when the order doesn't matter and even that other identity which you might have written down, maybe you remember it.

Can you? Yeah, a times negative b equals to b times negative a? Fantastic, have a go at the exit quiz and see what you can do.

Bye.