Lesson video

Hello, my name is Mrs. Buckmire and today I'll be teaching you about expanding double brackets.

First, make sure you have a pen and paper.

Please the pause the video whenever I ask you to, but also whenever you need to.

It's all about your learning, go at your own pace.

If you need more time, pause the video and take your time.

And also remember it can super useful to rewind the video and hear certain things again.

Maybe I said something too fast or you weren't sure about something.

If you listen to it again, sometimes that can be really beneficial.

Okay, let's begin.

So for our try this, we have Ben here, who says 53x19= x and there's the calculation.

And it goes for this calculation work here.

So pause the video and decide what's the same and what's different about their answer.

They get the same answer, maybe even check if it's correct? And then say, oh, what's the same, what's different.

Okay, pause and have a go.

In three, two, one, and pause.

Okay, so what did you think? So I already said they both got the same answer.

That's something that's the same.

They both also partitioned 53 into 50 and three.

And Ben has partitioned 19 into 10 and nine.

But Xavier's written it by 20 subtract one.

Xavier's has addition and subtraction in that second line of working out.

Ben only has addition.

Anything else? Okay, let's see, let's look a bit closer.

So here we can see Ben might have worked this out using an array.

So here they've let this-- oh that is such a bad straight line.

They've let this side, you know the one I'm talkin' about equal to 50.

And in this part three, so the whole thing's 53.

This part 10, this part nine, so the whole things 19.

So here would be 50 times 10, which is our 500.

And three times 10 is here, which is 30.

Which is the area, this area is 450, and this area is 27.

And they added to get 1,007.

So Xavier's array, hmm.

You know what that would look like? It'd look like this.

So here's our 50 and our plus three.

So how are we going to do the 19? We'll say that whole rule then from there to there is 20 and then we're going to subtract one.

I might even just actually just add a little negative sign there to remind us that we're subtracting one.

That look like a negative sign to you? Okay, so therefore, the area originally of the whole rectangle, this whole rectangle, was 50 times 20, which was 1,000.

And the whole rectangle here was three times 20 which was 60, but then we're going to subtract 50 times one which is, let's do it in black so you can see it, which is 50, I subtract three times long, which is three.

So that's where we get these two subtractions from.

So these diagrams are quite different.

But we can still use area and use an array to work this out.

Okay, I'm going to show you another example and I want you to complete it.

So it's all about using that distributed property.

So here.

So work that out, maybe pause it and just calculate what the answer would be.

And now can you actually draw the array, what would that look like? Pause and have a go.

Okay.

Okay, so what about if we did it using this method? Can you complete that? So do the workin' out, the next line, what would they look like? And also, can you draw the array? Sending the answer in three, two, one.

And here you go.

Okay, so here you can see what 20 plus three, and 20 subtract one, so check your answers carefully.

Make sure you understand that.

Because now we're going to go to algebra.

Okay, so there was the answer.

And now we're going to compare that to.

So, we have our x plus three, and we can draw that like that that's fine.

Now for x take away one, so that sides going to be x, so that's fine.

And then we're going to subtract one from it.

So actually our one is going to appear here, and that's the subtract one, okay? So now we're trying to find the area we would have x squared here.

This one would be three x.

This would be the negative of x.

And this would be subtract three.

So when I'm writing it out, I have x squared plus three x, take away x, take away three.

And then what do we do? Yeah, we collect like terms. So, x squared, that's now you have x squares and plus three x, take away x, is plus two x.

And then we take away three, so there is our final answer.

Okay, so pause the video and copy those for your notes to hep you.

Okay, so this your chance for you to check your understanding, so have a go at this one.

Expand and simplify this double bracket.

It could help to do a diagram.

Okay, pause and have a go.

Okay, so we can say now it's not drawn to scale, that's more than fine, okay.

So you can say this is x and this'll be our plus three.

This can be our x, this whole side.

And this can be our negative two.

So here we have x squared, plus three x, negative two x, negative two times positive three, what's that? It is negative six.

So therefore we have x squared, plus three x, take away two x, take away six, collect like terms. So x squared is as x squared.

What's three x take away two x? One loss of x, so it's definitely not D.

And then we got take away six.

So B is our answer.

Well done if you got that.

Okay, so this is your independent task.

Now this is not a long task because the explore task I think might take you a bit longer.

So all I want you to do is match the equivalent expressions by expanding the left hand side.

And maybe they didn't all match, check them really, really carefully.

But pause the video.

It might help if you use an array to help you out.

Remember it don't have to be done to scale.

But have a go in three, two, one.

Okay, how did you do? So let's do the first one with an array.

So, x plus 24.

So this is x, this'll be plus 24.

This can be x, this is going to be negative one.

So we have x squared, 24x, pull it to 24x, negative x, negative 24, so x squared, 24x, take away x is 23x.

There we go, those two match up.

Let's do x plus eight.

So we can have x here, we can have plus eight here.

We have x here, we have take away three.

So, x squared, 8x, negative three x, and negative 24.

So remember plus eight times negative three is negative 24, do they match up? Hmm, what one is it going to be? Eight x take away three x is by it, well the next 24, so it goes all the way up here.

Let's see.

The last two you should have got.

X minus six times x plus four equals x squared minus two x minus 24.

And x take away 12 times x plus two equals x squared take away 10x, take away 24.

So it did all match, hah, mo mistakes, that's good.

Well done if you got those.

Okay, so this is your explore.

I told you it's meaty, chunky toss.

So, I want you to fill this grid out.

Now, what you can see is-- and I'll just try and show you over here, is a increases in this direction.

And this is a, so it's x plus one.

So in our next one, we can see if x plus two, the next one, x plus three.

As we go left then, x is going to, a even, a is going to decrease, so this will just become x times x minus two.

And this'll be x take away one times x take away two.

You see? Okay, so as we go down, now wait.

If you understand this, you've seen this before, you go through it, I do want you to tell me what do you note about the highlighted cells.

Be careful what your focus is.

I want you to see if you can create similar examples to these highlighted, these are the highlighted cells.

So if you're confident, you go ahead and pause the video.

Ignore me now, I'm just going to give you a few more examples for those that need it.

And remember, always use the array.

You can use the array when you're expanding, if you need to.

Okay, so you can pause it.

Okay, if you're still with me, a few more examples.

So when we're going down, b is increasing.

This is our b value, this value here.

So it goes along x take away two, it's increasing by one.

So it's going take away one here.

And going up is going to become smaller.

X take away two become x take away three.

Make sense? So here going up, what do you think is going to be? What expression is here? Good, it's x times x take away three.

And what about down here? B is increasing from this one.

Or you can just do the a increase.

So remember there's two ways, two directions you can play with here.

I'm going to do it as-- oh, it didn't come up.

Here you go.

Let me write it in.

It is x plus two, x take away one.

'Cause b is increasing, it says increases by one.

Okay, so everyone have a go.

Or you can at least expand the ones that are there, and see what you get.

Okay, here are all the answers, okay.

It's a bit overwhelming so please pause the video and check your answers carefully.

Okay, so the question was, what pattern can you see in the highlighted squares? Well, rectangles, really.

But they are called cells.

Let's have a little look.

So, we have x squared, we have x squared take away one.

We have x squared take away four.

We have x squared take away nine.

Hmm.

All these numbers added on to x squared are negative square numbers, aren't they? And here we have x times x, x plus one times x take away one Hmm, x plus two times x take away two.

X plus three times x take away three.

Did some of you guys continue it? Did you do x plus four, x take away four? Maybe x plus five, x take away five.

What did you notice? Yeah, x squared take away 16, x squared take away 25.

Hmm, they're all, well there's a special name for it.

Maybe you've heard of it? It's difference of two squares.

We will look at that in another lesson.

Anything else you notice? So maybe you noticed that when we had two negatives, which is all of this collum, par this one.

This last number was positive, that's interesting.

And when we have positive and negative in this collum, and in this collum, the last number is negative other than these ones where there's actually no number by itself, we're always multiplying everything by x.

And maybe you noticed that here we had add two on, add zero on, then take away two, take away four, take away six, kind of going this way.

We seem to be subtracting by two each time.

And going this way, we seem to be subtracting by three.

There are so many patterns.

Like, I could not possibly list them all.

But hopefully you've had a good time just discovering, and noticing different things.

And maybe even helped, maybe didn't actually have to expand.

You could just say, oh this is a pattern I noticed, I can complete it like that.

Fantastic.

Math is all about being efficient.

If you can take those kind of smart mathematical sense shortcuts, that's fine.

Thank you so much for your hard work today.

I've really enjoyed teaching you.

Hopefully you've enjoyed the lesson.

Hopefully you've learned something, which is the most important thing.

Whatever you've learned, can you write it down.

Just one thing, one thing that you will remember from today's lesson.

And then can you please do the exit quiz.

That's a super helpful way for you to assess your understanding.

Also, just practise and show off what you know, really.

Have a wonderful day and hopefully I'll see you in another lesson.

Bye.