Lesson video

Hello! My name is Mrs. Buckmire, and today I'll be teaching about quadratic contexts.

Okay? So make sure you have a pen and paper, and if you have a pencil instead that's fine of course, it could be useful to have pencil though, because we might do a bit of plotting.

And remember, when do I want you to pause the video? Good, whenever I ask you to, but also, whenever you need to.

Just learn at your own pace.

And remember, rewind the video if you need to hear something again, that can be helpful.

Okay, lets begin.

So, for our try this, I want you to compare these two situations, and tell me what's the same and what's different.

So I think of an integer, I multiply it by the next consecutive integer, or a rectangle has length one centimetre longer than its width, I find the area of the rectangle.

Okay? So look at those, and tell me what's the same, what's different, now Yasmin says: "I'm going to specialise by creating some examples for each one." So maybe you could follow Yasmin's lead.

Pause the video and have a go in 3.

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Okay, so what do we think? So, what's the same and what's different? So what's the same? Hmm.

they both have an unknown, what do we call an unknown in math normally? Good, a variable.

So both have a variable, that's actually something which can change, we don't really know what it would be.

What else? Yeah, they're worded problems, true, what's the same? What else is different, sorry? So this one is talking about like, a number and multiplying it by a consecutive, wait.

What's consecutive? Yeah, one more.

Oh, one more! The one on the right, the rectangle one, is also about one more, isn't it? So it says one centimetre longer, and this one, by the next consecutive integer, so it's one more.

And an integer is a? Whole number.

Okay, on the left since it says it must be an integer, it must be a whole number, but the rectangle one does it have to be an integer? Not necessarily, so that's something that's different.

Well done you came up with some more examples.

Okay, so, if I were going to evaluate this, maybe you already specialised like Yasmin said.

So maybe you did something like, thought of an integer, say the number 5, and then you multiply it by the next consecutive integer, six, to get thirty.

Or maybe you thought "Oh, rectangle has its length one centimetre longer than it's width," so let's say the width was 5 centimetres, and the length is 1 centimetre longer, 6 centimetres, so the area is 5 times 6 which is 30 centimetres squared.

Phew, there we go.

There's another thing that could be the same, because if you use the same numbers, you end up with the same answer.

But this one, actually, has units based on the area.

It's the area of a rectangle, while this one actually is just a number, there's a difference.

So if you specialised, you might have come up with similar examples like this.

So how can we go from this to actual algebraic expressions? Okay, so here, we've started with an integer, I'm going to say x, so x can be any number, or any integer, so any whole number, and then you times it by, so how can I write the consecutive integer? Good, it's one more than x, so x plus 1.

Okay, and in math, do we like to write times? Very confusing, having this x as a times, so let's just have the very efficient not put an extra operation that we don't need, because actually we know that x brackets x plus 1 is x times x plus 1.

So this would hold truth for all of these problems, so actually I could just find any value of x that is an integer, and then I could do the integer times the integer plus one, and I'd be thinking of this integer, I'd be solving this problem, and thinking about this problem.

Okay, we're at this one, let's get rid of.

Actually let's do it again here, let's keep it here.

Okay, so I have x centimetres is here, that's going to be my width, and x plus 1 centimetres is what the length is.

So the area is going to be x times x plus 1, so again exactly the same, and we know that we can even expand that as it get x squared plus x, to work it out as well.

That's an equivalent expression.

So, pause the video if you'd like to write this down.

Okay, so for our independent task, what I want you to do is match up the situations on the left hand side with some of the expressions.

And there might be more than one answer for some, there might only be one answer, okay? So the first question is, Annabel is x years old, her friend Rosie is 2 years older than Annabel.

I find the product of their ages.

Which expressions match this situation? And then, a rectangular garden is 2 metres shorter than it is wide.

I find the area of the field, which expressions match the situation? So pause the video and have a go.

In 3.

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Okay, so Annabel is x years old, now the point of this is when we're working with algebraic expression, its best to actually like, define what your variable is, so here they defined that Annabel is x years old, and that's really useful.

So, her friend Rosie is 2 years older than Annabel, so as an expression, how old is Rosie? Fantastic.

So Rosie is x plus 2.

So if I find the product, what does product mean? Good, it means timesing.

So here I'm going to do x times x plus 2, and we like to be efficient, so lets get rid of that times and write it like that.

Okay, are any of those there? Yes, we have right there.

So that's correct.

We could also expand that so we'd get x times x, good, x squared, plus x times 2, 2x.

Is that there? Oh, it's right next to it.

Okay, anything else? No, it doesn't look like any others are equivalent.

Well done if you got those answers.

Okay, a rectangular garden, now whenever I see any kind of shape stuff I often like to actually draw it.

So here's my garden.

It's 2 metres shorter than it's wide.

So here, it's useful if you decide what your x is, so I'm going to say like, oh, this it it.

This is how wide it is, rectangular is 2 metres shorter than it is wide, so this is x subtract two.

So two metres shorter than it is wide.

This is how short or long it is, and that's how wide it is.

This is the width.

So now, if I'm trying to find the area I'm going to do x times x subtract 2.

Is that equivalent to any? Yup, there is.

So this one.

Any others? Yeah, x squared subtract 2x, as well as the expansion of it.

Any more? No.

Okay, but what if I had done it the other way around? So what about if I said that actually I'm going to have the length being x, so instead I'm going to have the length being x, and the width, and it says the garden is 2 metres shorter than it is wide, so this is the length, the length is x, shorter than it is wide, so the width then, is going to be x plus 2.

Because that means x is 2 metres shorter than x plus 2, wouldn't it? So if I say, let the length be x, and that means the width is x plus 2.

Can you see that? I'll just give you some time to process that.

Okay, so now then what would my area be? Good, my area would be x times x plus 2, so that would make which one, any of them correct? It would make this one correct, let's do it in a different colour, green.

So this is where x equals the length.

Any of the others? Ah, this one as well! So, what I want you to learn from this is what you define your variable as is really really important because it can come up with different expressions based on what it is.

Okay? So like, let's say the width was four.

So let's say the width was four metres.

The rectangular garden is 2 metres shorter than it is wide.

That means that the length of it is two metres, because two metres is two metres shorter than four metres, okay? So that means here, the width is going to be 4 metres, this is 2 metres, 4 times 2 is 8 metres squared, so it works.

And here, the length is the two metres, so this is 2 metres, 2 metres plus 2 is 4 metres, 2 times 4 is 8 metres, so as we see, it does work in both cases, okay? So it just all depends on what you let your variable be, okay? That's why you define your variable, that's why you should say what it is.

Okay, let's see what the explore task has for us.

Okay, so what you're going to do, is you can see here, a rectangle, and a square is cut out of it, in the corner there.

So it's a 9 by 4 rectangle with a square cut out.

So form an expression to describe the area of the resulting shape.

So once that square has been removed, what is the area of the shape left over? Are there different ways to find this expression? Graph it, and what is the maximum and minimum area? Okay, so quite a few different parts to this, so I do you want to spend some time having a go, if you need some support, come back here and I will help you out.

Pause in 3.

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Okay.

So just a bit of help, so form an expression, so I just mentioned it, but let me just say, so it's kind of like, we have our whole rectangle that is 9 centimetres by 4 centimetres, and then we're going to take out the square.

Now this square, we don't know what the lengths are, so we're going to write it as x centimetres by x centimetres.

Okay? So you need to find this area, and you're going to subtract this area from it.

So that's one way, you could also, now, technically, what we have left is this shape.

What's this shape called? Good, you might know it as a compound shape, you might know it as a rectilinear shape, you might even count the sides and say it's a hexagon, okay? So, often, let me see command champs, everyone's like "Oh, I must divide it into rectangles," and that's what you can do.

So, let me rub out my badly drawn shape because you don't need to see it, as long as you know that this bit's been taken away.

So that's that method.

But here, you can actually split it up maybe.

So you can split it like this, you can even split it in a different way, okay? But then trying to work out what the area of each rectangle is, so what's the area of this rectangle, what's the area of this rectangle, that's going to give you different expressions, okay? And then when you have this expression, this expression relates to the area, hopefully, then you can plot it.

So I'm going to have a equals whatever the expression is, you can have y equals and then make a table of values and you can plot it.

And then to find the maximum and minimum area, what I would do is I would actually think about the situation.

Okay? I am going to go through it, but just think, "Oh! Where would the maximum be, what's the smallest square I can take out, what's the minimum, what's the biggest square I can take out?" Okay, hopefully that's enough information.

Just have a go, don't worry if you can't get it all.

Just see how far you can go.

Okay, so I did just show you some methods, so if we're doing that one where it was the whole rectangle take away the square, so we're just taking that part away, so the whole rectangle is 36 36 centimetres squared, we'll take away the smaller square so it's x times x which is x squared centimetres, and that's how it's written.

So I'd write it as, a equals 36 take away x squared.

Now really, the units don't matter, because I'll be substituting and at the end we can like, check if our units are correct.

Okay, and then what about this one? If we do it like I said, like this, so this rectangle here, rectangle 1, what's the length of it? So, this length is going to be 9 subtract x, so it's 9 take away x, and what's the height of it? The height is this height, which is x, so we have that as the area is x brackets 9 subtract x and then the area of this rectangle down here, it's a bit longer, number 2, so number 2, so the height of it is going to be this, which is going to be 4 take away the x.

So the height is 4 subtract x, and the length of it, the length is 9.

So it's going to be 9 times 4 take away x.

So the whole area equals to x brackets 9 subtract x plus 9 brackets 4 take away x.

Phew, okay.

Looks like I'm running out of space, so let me just multiply, expand that up here, so a equals 9 x, what's x times x? Good.

x squared plus 9 times 4? 36, and then positive 9 take away x negative, so take away 9 x, phew okay, so these are going to cancel out, can you see that? So, this cancels with this, whoops.

So therefore, that wasn't a good colour.

A equals 36 take away x squared.

So there are different ways to find it, I much prefer this first one.

It's way nicer, isn't it? But this one is also a valid method as well, and if you did that, and you actually could have done it in different ways as well, so you could have made it into 3 rectangles, you could have made it into 2 different rectangles, there's lots of different ways.

Well done for having a go.

Okay, so a square is cut out.

Graph it, so here is the graph shown.

It is a negative quadratic.

So it's that kind of sad face, and we can see up here it goes through at 36, and here it goes through our x axis at negative 6, and this one is at 6.

So well done if your graph looks like that.

So what is the maximum minimum area? So remember, this is the graph of A equals 36 subtract x square, so that's how it's a negative quadratic, there's a negative in front of the x squared.

Okay, so the maximum, we want this square to be as small as possible, so about just 1x is 0.

So when x equals 0, we get the smallest square, so that means we have the maximum area, 36 take away 0 is 36.

And we see that from the graph don't we? So if we see here, when x equals 0, and we go up, we hit it at 36, which is the highest point on the graph, so the highest area, the maximum, is 36.

Okay, and we can even be fantastic at adding units, okay, what about the minimum area? So, can we look at the graph and see what the minimum is? Ooh, well this is negative, we can't get negative sides anyways, so that can't be right.

What about here? Oh, it just keeps going, it seems to keep going forever.

Hmm, okay.

Let's think about this.

So what is the biggest the square can be? Can it be a 9 by 9? No, because the longest that this side of the square can be is 4.

So really, the longest it can be is 4 by 4.

So let's say about here.

So, the longest it could be is 4 centimetres by 4 centimetres.

So when it is 4 by 4, it will be, the area will be 36 take away 16.

So the minimum area is going to be 20 centimetres squared.

Okay? So just think about that, the largest we could get our square to be was 4 by 4 because we can't get to 9 because actually, we can't, we don't have space to go down to 9 on this side of the paper.

So that was the min area.

Well done if you've got that! So remember, without graphs, sometimes it's super helpful.

Like, we can see all the 36.

But in context, we need to think about when is it valid.

So it's not really valid for all these negative values, x is never going to be negative, and here, it kind of stops at, it stops at 4 because 4 was the longest that x could be, because of this constraint with the side.

So though yes the graph goes lower than the x area, I did the y value on the x value there.

So it stops at 4, so where's 4? 2 4 here.

Which is at 20.

So this, it stops here, where x is 4, because of this 4 value here.

And that's at 20.

Okay? So I just wanted you to have a go, explore in class.

I hope you enjoyed doing that.

I really wanted to say well done with the lesson today.

If you'd like to share your work, I'd love to see it.

Please ask your parent or carer to share your work on twitter tagging @OakNational and #LearnWithOak, it'd be fantastic if you could do the exit quiz, it's an ideal opportunity for you to test out what you know, and just show off what you've learned really.

Have a lovely lovely day and thank you for all your hard work.

You should be proud of yourself.

Bye!.