Lesson video

Hello everyone, it's Mr. Millar here.

In this lesson, we're going to be looking at manipulating equations and inequalities.

So first of all, I hope that you're all doing well.

And it's the final lesson of this unit, so if you've been watching all the videos so far, then really well done.

So let's start off by having a look at the "Try this" task.

We've got all statements below What's the same? What's different? So I won't give you any further instruction.

Pause the video now for two or three minutes to find a couple of similarities, and a couple of differences between these four statements.

Great, so well done for doing that, and there's lots of things that you could have said.

So for example, you could have noticed we've got a four X and a couple of them, we see 23 in some of them, so lots of similarities and also some differences as well.

So the top two are inequalities, of course, and the bottom two are equality's so equal signs, but there's some other things I want to point out, which are important.

So if we look at the bottom two equalities, if we take a two X plus seven equals 11.

5, and then we decide to multiply both sides by two, have a think, what would you get if you multiplied both sides by two? Well, you would get four X plus 14 equals 23.

And you should notice that that is exactly the same as the final statement, except for the fact that the 23 and the four X plus 14 have swap sides but they're still the same thing.

And actually you can do something similar for the top two statements.

So if you look at the first statement and multiply both sides by four, what would you get if you multiply both sides by four? Well, you would get four X on the left is less than or equal to 23 minus 14.

And if you do something similar to the second statement, let's say you added four X, first of all.

So we would get 23 is greater than or equal to 4 plus four X, and then you decided to subtract 14 from both sides.

You would get 23 minus 14 is greater than or equal to four X, which again is the same thing as you get in the first segment, except it's the other way round.

So yeah, lots of similarities and differences here.

But the one thing that's important, which I did a couple of times is that if I do the same thing to both sides, so for example, multiplied by two, multiplied by four, add four X, I'm doing the same thing to both sides, it maintains, it keeps at the same inequality or equality.

And it's this that we're going to be looking at in today's lesson.

Let's get ready for the connect slide when you are ready.

Okay, so for the connect task, you need to say whether the following statements are always, sometimes or never true.

If we know that two X plus three Y is greater than 15, and we're going to be using this idea of maintaining inequality, if we do the same operations of both sides.

So if I just give you an example, let's say we had two X plus three Y plus one is greater than 16.

Then what we do is we compare it to the original inequality and we say, "okay, "what I've done here is I have just added one to both sides.

"So if I just do the same thing to both sides, my inequality will stay true." So this would be always true.

So what I want you to do in each of these three cases is I want you to start off with the original inequality.

So I'll just write them down for you in all three cases.

Okay so you're going to compare the original inequality to the new inequality and therefore decide if the following are sometimes, always, or never true.

Pause the video for two or three minutes and figure these ones out.

Okay, so let's go through them.

The first one, well, you should have noticed that the second equation, the second inequality, excuse me, is the same as the first one, except that both sides have been multiplied by two.

Therefore this is going to be always true because our inequality is maintained.

Okay, what about the next one? Well, you should have noticed that there was a plus three in the original and a minus three in the new, so if you had subtracted three Y from both sides, you would have got two X is greater than 15 minus three Y.

And so just because this new inequality looks a little bit different to the original one doesn't necessarily mean that it's not always true.

So have a think if two X is greater than 15 minus three Y will two X be greater than 14 minus three Y? Well, the answer of course, is that it is going to be always true.

And that's because if two X is greater than 15 minus three Y it's it's certainly going to be greater than two X, is certainly going to be greater than 14 minus three Y.

Because that is going to be smaller than 15 minus three Y.

So that is also always true.

And the final one is actually going to be sometimes true.

And you would have noticed that there's actually nothing we could do here to, so, so there's nothing that we could do here to make the original inequality the same as the new one.

And it's sometimes true because if, for example, Y is a positive number, so let's say that Y is equal to one, then it is going to be true because, you know, if we have six Y then it's going to be greater than three Y, so it's going to be true.

But if Y was negative, negative one, say for example, then it's going to be not necessarily true.

And that is because a plus six Y would be a negative number.

Let's move on now to the independent task when you're ready.

Okay, so here is the independent task, a couple of questions for you do based on what we've done so far.

So pause the video now, five to six minutes to have a go at these questions.

Okay, great.

Let's go through them.

So the first one, given that X minus six equals Y, fill in the gaps to make each of these equations hold.

So in each case you would start off with a X minus six equals Y, X minus six equals Y, and X minus six equals Y.

So what's happened in the first one? Well, we can see a, a Y in the original one and a three Y here, and also a minus six and a minus 18, that should give you the clue that we've multiplied both sides by three.

So therefore the new coefficient of X will of course be three.

The next one, well if we had Y and now we had a Y minus three, that's told us that, what have we done to both sides? Well of course we have subtracted by three.

So therefore, Y minus six minus three is going to be, sorry, X minus six, minus three is going to be X minus nine.

And the final one.

Well, this time it's a, it's a bit of an interesting one.

There's actually a number of ways that we could do it.

So let's say for example, that we added two X to both sides.

If we added a two x to both sides, well, yes, we would get three X minus six on the left hand side, I'm going also get Y plus two X on the right hand side.

And the second question, given that X minus six equals Y, which of the following inequalities are always true.

Well, the first one is definitely going to be always true.

And that's because if we add a one to the right hand side, it's going to make it bigger than the left hand side.

So if it was equal originally, now the right hand side would be bigger.

The next one, well, if we think about it and we started off with X minus six equals Y, but now we're only subtracting four from X instead of six.

So therefore we are making a statement on the left hand side bigger than it was before.

So this is actually going to be never true.

And the final one, well, again, this is fairly similar to the third one in the first example, this is always, this is going to be sometimes true.

Not always true, because it depends whether the original X was positive or negative.

If it was positive, then it would be true because we're doubling a positive number, but if x was negative we'd be doubling a negative number.

So we'd be making it even smaller.

So it would be less than Y.

Great, so when you're ready, let's move on now to the explore task.

Okay, so here is the final slide, the explore side slide.

So Anthoni has written a pair of statements that are true at the same time.

How many ways can you complete it using the number cards one to nine? So have a look at these two statements and see if you can spot something, a relationship between them.

It's similar to what we've been doing so far this lesson.

So see if you can spot it, I'll give you a couple of seconds.

Well, you should notice that, of course, that four A is twice two A and minus six B is twice minus three B.

So what does that tell you? Well, what that tells you of course, is that to get from the bottom one to the top one, you would multiply by two or divide by two to get from the top to the bottom.

So therefore thinking about what the numbers and the boxes have to be, what would they have to be? Well, the first box would have to be eight because eight is twice the size of four and the bottom of box would have to be four, for the same reason.

So that is one possibility, but actually it turns out that if we do something a little bit clever, there are more possibilities.

So let me just rub this out for one second.

And what I'm going to do now is I'm going to rearrange both of these equations to put up all of the A's on one side.

So the first one, I keep the four A on this side and I move this minus something to the other side.

So what I'm going to have is I'm going to have eight plus that something minus six B.

And I do the same thing for the bottom one, even feel free to have a think, a pause video, about what I would do here.

Well, this time I want to keep two A by itself.

So I need to move this minus four.

So how I do that? Well, I add four to both sides.

So I'm going to get two A equals four plus something minus three B.

And so, would you agree with that? These two statements are the same as the originals.

I've just moved everything so that the A's on one side, on one side.

Anyway, this is useful because now I can see that this four A is double two A, and this minus six B is double minus three B.

So now I'm looking at my numbers here, eight plus something, and four plus something.

And I know that this eight plus something has to be double four plus something.

So this actually leads to a number of other possible solutions.

For example, I'll give you the first one.

If the first missing number was a two, I have eight plus two equals 10, so I'd have to have four plus one to make five.

And therefore I can see that the first equation is double the one below it.

So this gives rise to a number of other possible answers.

So feel free to pause the video and see if you can get all the other possible answers.

Okay, so well done if you found them and the other possible answers are having in the boxes are four and a two, a six and a three, an eight and a four, which is actually the one that we had already.

And that is it, so there's actually four possible answers here, which may seem surprising when you first have a look at the problem.

Okay, that is it for today.

That is it for this unit.

Really well done for watching all these videos.

I hope you got something out of it.

And in case I don't see you again, or you don't watch any of my other videos, best of luck in your math careers.

Thanks very much for watching, have a great day.

And bye bye.