Lesson video

Hello, my name is Mrs Buckmire and today I'll be teaching you about negative scale factors.

So first, make sure you have a pen and paper.

Remember, pause the video when I ask you to, cause it means I want you to have a think and I want you to have a go at a question, but also pause whenever you need to.

So if you need a bit more time or something, just pause it.

And also you can scroll back and rewind as well.

So, it's often helpful to hear something again, if you didn't quite understand it the first time.

Okay, let's begin.

So before you try this, I want you to use two of the five number cards to fill the spaces.

Okay? So, 3 times something take away 5 times something and just put the numbers in.

And I want to know what is the greatest number you can produce and what is the least number you can produce, okay? So you might want to try just throwing in numbers and seeing what you work out.

You might want to think, oh, be a bit more strategic about which ones you put in to find the greatest or least.

Pause the video and have a go.

Okay.

So what is the greatest number? So if you were a bit more strategic, you might think, hmm.

If I want it to be as great as possible, well, all of these numbers are negative, the ones that the, the number cards.

So three times the negative, well, I want the absolute value to be as small as possible so that it's close to my zero, so that it's not too negative.

So actually the one with the smallest absolute value is -2.

So I'd put -2 in here.

And then take away, now again, all of them are negative and I know from my order of operations that I'm going to end up doing this first.

So 5 times, so 5 lots of a negative.

And so it's going to be negative.

So I want it to be as big as possible because I'm going to be taking it away.

So the absolute value, so when I say big, I want the absolute value to be as big as possible.

So the one with the biggest absolute value is -6, wait a second.

Do you know what absolute value is? What is it? Yes, absolute value is the distance from zero.

So now we can see, this is the greatest number.

So it's 24, because 3 times -2 is -6 subtract 5 times -6 is -30.

So when we're subtracting negative, we can add the additive inverse.

So it's going to be -6 plus 30, which equals to 24.

So what did you get for the least? 8 was the least one.

And again, we can see we've used one with the smallest absolute value and the one with the biggest absolute value.

So well done if you had a bit of strategy behind it, but well done if you got it anyway.

Good job.

Okay.

So if you've learnt about some multiplication with scaling, with me before you might've seen this, but otherwise I want you to make sure you understood the model that I was using.

So it's the idea that here this 4 has actually come from this 2.

So it's been stretched, and it's been stretched 2 times as big to get to 4.

So we can see that actually, 2 times 2 equals 4.

So if I make this a little bit smaller, we can see decimals are fine as well.

And that number, that's the scale factor.

So scale factor is how much we've stretched it by, by how much amounts, like here at 1.

1 is stretched by one lot and 0.

1 lot more times as big.

So what do you think when it equals one? Yeah, it hasn't been stretched.

It was just 2.

And then what about when then our scale factor becomes less than one? What do you think happens? Yeah.

So although we say stretch, it can still be stretched and becoming smaller.

So it actually becomes shorter as we get into, kind of our parts of a whole, into our fractions and decimals.

So at zero as expected, well it's just zero, 0 lots of 2 is 0.

What do you think would happen when it becomes negative? Excellent.

It's going to be stretched in the opposite direction.

So there's our decimals where it's less than the size.

And then when it gets to -1, it has been stretched in the negative direction, it's one whole.

So the absolute value is equal here.

So the absolute value of -2 is 2, absolute value of 2 is 2.

And actually these two are additive inverses.

So 2 plus -2 equals 0.

And then what about then, what we started with was 2, so if we go to -2, we can see it's the same as a scale factor of 2, but actually it's in the opposite direction.

So it's -4, it's been stretched in the opposite direction twice.

So there we had where it was 2, stretch in that direction.

And then there we have when it is -2.

Okay.

Let's just cement that learning a bit more from that GeoGebra work.

So, each of these arrows can be connected to 12 using a scale factor.

So, for the example, I'm going to have a look at this arrow here and as we can see, it's going in the opposite direction.

So what do you think the scale factor is? Yes, it's going to be a negative, that's right.

So, how much? So we can see the absolute value is one and a half times the original value.

So, the scale factor is going to be -1.

5.

And so that's -1.

5 times 12, and that's actually equal to a value of -18.

So have a go at A, B, C, D, E and F.

Do pause and just have a think.

Even if you just say it out loud, what do you think the scale factor would be? And even better, if you can actually write out a similar calculation to what I've done.

This is on the worksheet as well so if you can't quite see on the screen very well, look at the worksheet, because then you can zoom in.

Okay.

So the example I gave, so for A, it's going to be 2 times 12, so it's actually, it's twice as big so it's been stretched 2.

The next one's 0.

5 times 12.

The next one has a scale factor of 0.

25.

So we discussed in the video, how scale factors can be less than one.

And that's when actually it's smaller than our original number.

So these ones here, now it's going the opposite direction.

So it's going to be a negative scale factor.

So here it's -2, the next one's -0.

5 and a -0.

25.

So what was the relationship between ABC and DEF? Can you spot anything? Notice anything? Excellent.

So D is the additive inverse of A, if you do 24 plus -24, you get 0.

Remember, additive inverse is when they sum to zero and the same with B being the additive inverse of E and F being the additive inverse of C.

Well done if you spotted that.

So a negative scale factor can be interpreted as a stretch in the opposite direction.

That's something you might want to write down.

Okay.

So onto the independent tasks, there's another similar task where I want you to think about the representation and then I want you to identify equal pairs of the calculations shown.

Okay? So do pause the video and have a go.

Okay.

So let's go through this quickly.

So, I also want you to state the scale factor.

So the first one is 4 times 8.

So it has a scale factor, is 4.

The next one would be 2 times 8.

So the scale factor is 2.

The next one is a half times 8, so it's half, being stretched half as much.

So equals a half, is the scale factor.

What about for D, E and F? Did you get -32 for the sum, 16, is that an answer and -4? So therefore the scale factors would be equals -4, the scale factor is -2 and the scale factor equals negative a half.

Okay.

For next one, identify equal pairs of calculations.

Now you could work them out.

So here they are worked out, but actually you could see that -12 times 6.

Well, I know by my, by actually the stretches that actually A is going to be the same as C.

Cause actually you can see it's just the order that has changed, cause actually, when we stretch it, if we stretch 6 by -12, or if we stretch 6, or if we stretch -12 by 6, we're going to end up with the same answer.

You might also notice B and F are the same.

So you can A and C, you can have B and F, you could have D and E, and you could have then G and H.

Well done if you've got those.

Okay.

So for your explore task, now, this is a bit of a challenge.

So N and M are both positive integers and P is a negative integer.

Remember integer, what is it? Yes, whole numbers.

Okay.

So how many solutions can you find to the following? N times P equals -24.

P times N times M equals -8.

And what happens to your answers above if N, M and P are allowed to be non-integers? Okay.

So how many different solutions? Well, you want to think of integers where N is positive and P is negative and they multiply to give -24.

I think you can work those out.

We've done some work on factors before I reckon, I think you can think of those.

And same here, where P is negative, N and M are positive.

So how many different solutions can you come up with and try and come up with all of them, because they're integers, trying to work with all the solutions and then for the last bit, what happens if they are not integers? Just write a sentence for that.

Okay.

Pause the video and have a go.

Okay.

So let's go through it.

So for N times P equals -24.

Well, I found four sets of solutions.

I've shown them there.

Now my question to you, what's the relationship between the top lines and the bottom lines? Yeah.

So here, the absolute values of N and P have swapped around.

So here, let's use a different colour.

The absolute value of N is 1 and absolute value of P is 24 and then the absolute value of N becomes 24 and absolute value of P becomes 1.

So actually the absolute values swap, but the negative sign remains on P because P is a negative integer.

Nice.

What about B, how many solutions did you get? Six, eight? There were 10 sets of solutions! Now, I'm going to show you one way that I kind of figured it out.

So I did, I kept P the same.

So, every time I had P being -1, so I got these different solutions when it was -1.

And then I kind of just swapped around my N and my M.

So, you can see here 8 times 1, the same as 1 times 8.

Hmm, what's that word called? When in multiplication, you can, the order doesn't matter.

Yes.

Commutative.

So we call it commutative.

We can do that.

So I did that and I did the same thing with 4 times 2 and then actually swapped them around for N and M.

Now you can do that setting P to be -2, setting P to be -8, have a little go and see if you can get to those 10 sets of solutions.

So I've given you four.

If you didn't get to 10, see if you can find the others.

And then that final bit, what happens to your answer above if N, M and P are allowed to be non-integers.

Well, then there'd be loads of solutions.

In fact, an infinite number of solutions.

Well done if you got that.

Right, really, really well done today everyone, if you would like to share your work, I would love to see it.

You can ask your parent or carer to share your work on Instagram, Facebook or Twitter, tag it @OakNational, #LearnwithOak.

I've really enjoyed today's lesson, I hope you have too.

Please do the exit quiz, it is such a valuable way for you to learn and check your understanding really.

Read the feedback for any questions you got wrong, and hopefully that will help you keep progressing and keep getting even better at maths.

Have a lovely day.

Bye!.