Lesson video

Hello, I'm Mr. Coward, and welcome to today's lesson on forming quadratic equations, part one.

For today's lesson, all you'll need is a pen and paper, or something to write on and with.

If you could please take a moment to clear away any distractions, and turn off any notifications, that would be brilliant.

And if you can, please try and find a quiet space to work where you won't be disturbed.

Okay, when you're ready, let's begin.

Okay, so time for the "Try this" task.

I'd like you to pause the video and have a go.

Pause the video in three, two, one.

Okay, welcome back.

Now, let me talk through this task.

So, here, I think of a number.

Now, I don't know what that number is, but then I add seven, then I multiply by five, and then subtract one, and my answer is 54.

Now, you could've done this in lots of different ways.

One of the ways that I like to do it is I like to form an equation.

So I like to say that my first number that I didn't know is called X.

And what do I do to X first? Well, I add seven.

And then, I multiply it by five, so that means the whole thing gets multiplied by five, so we can use a bracket to represent that.

I then subtract one, and then my answer is 54.

And you can do inverse operations to work backwards to find the number.

So inverse operations, what did I do last, I subtracted one, so I add one.

Then, what did I do second to last, I multiplied by five, so I divide by five.

So I get X plus one, I don't need the brackets anymore.

Equals 11, then I subtract seven, 'cause that's the opposite of add seven, and I get my answer, X equals four.

So my original number is four, and let's check.

Four plus seven, 11.

11 times five, 55, 55 subtract one, 54.

Okay, so, that was just using my instructions here.

So, my number was four.

Second one, I think of a positive number, I square that number.

Oh.

So we start off with X, and then we square it, then we subtract one, and we get 48.

Now, working backwards.

Oh-- Come on, X, there we are.

X squared equals 49, and some of you, who know your seven times table and your square numbers, would've been able to see that it was seven.

Now.

We're going to pause there, because it's just seven, because it's just a positive integer.

But if it wasn't positive, it could've actually been negative seven, and that's something we'll look at in future lessons, and so consider that a little teaser.

Okay, so today's lesson is all on quadratic equations.

So what exactly is a quadratic equation? Well, an equation that can be rearranged such that the highest power of the variable is a square.

So what does that mean? Well, it's an equation, which means it has an equals sign.

And either side of that equals sign are two expressions.

And the highest power is a square, so the highest power of one of the variables is two.

Now.

Here are some examples.

This, the highest power is two, it's an equation, two things are equal to each other.

Two things are equal to each other.

The highest power is two.

It's a quadratic equation.

This is a quadratic equation, now this one has two variables in, and that slightly changes things.

I'm pretty sure that only one of the variables can be squared.

I don't think that both variables can be squared.

But you don't need to worry about that, that's just me being a bit pedantic on my definitions there.

Because I think once you have another squared, that changes it.

Okay, over here, we've got one variable, the highest power is a square, it's an equation.

Over here.

One variable, the highest power is a square, so this one, we're allowed both of them to be squared 'cause they have the same variable.

But yeah, that's just an interesting case that, you know, I don't want to.

You to think that we could have, for instance, X squared plus Y squared equals nine, because then that would no longer be a quadratic.

Okay, so some of these are non-examples.

So this one.

Why isn't this a quadratic equation? The highest power is three.

Why isn't this a quadratic equation? Well, it doesn't have an equals sign.

Well, why is this not a quadratic equation? So this one's an expression, by the way.

This one is not a quadratic equation, it's actually an exponential equation, okay, it's a number to the power of something, okay, the variable is in the power.

So that's an exponential equation.

And here, this is just a linear equation.

There's no power at all.

So that's not quadratic, the highest power is not two, the highest power is one, it's a little secret one that we don't actually write.

Okay, so what I want you to do now is decide which are quadratic equations, which are linear, and which are neither.

So, pause the video and have a go.

Pause in three, two, one.

Okay, welcome back.

Now, here, well, this one, this one is linear.

Okay, the highest power is a secret one, so that is a linear equation.

This one is neither, it's got a power of seven, which is much bigger than two.

This one, this one's a quadratic, equals sign, highest power of two.

This one.

This one is a quadratic.

This one is not an equation.

This one, highest power is three, this is actually called a cubic when the highest power is three.

It's not a quadratic, it is a cubic.

This one, yeah, this one's fine.

This one, nope.

Do you remember what that one's called? An exponential equation.

And this one, nope, this is-- Well, you could rearrange it, and then it becomes linear.

So, this is actually a linear equation, but it's kind of in disguise.

Yeah, yeah, that's an interesting one, actually.

Because, if you times both sides by X, you get 4X equals 11, so it is linear.

So I would say, yes, this is a linear.

Yeah, okay, that is a linear equation.

It's just a hidden one.

Okay, so we're going to be using, for today, we're going to be using some algebra tiles, and I don't know if you've seen algebra tiles before, so, let me just quickly go over them.

This has got a length of X, and that has got a length of one, okay? So that is one, and that length is X.

So the area of that is one times X, X.

That is a length of one, that is a length of one, so the area of that is one.

That is a length of X, and if I put those next to each other you'd see, and that is a length of X, so that is an area of X squared.

So when we're dealing with algebra tiles, these numbers or letters that are on them represent the area.

Okay.

So, I think of a positive integer, and I'm just going to say that's X, and I'm going to represent it with my algebra tile.

I multiply it by four.

4X.

So let's write this down as we go.

So you've got X, we multiply it by four to get 4X, and then I add three.

So I've got 4X plus three.

And my answer is 27.

So the total area of this and this is equal to 27, so I say the area of 4X plus three is equal to 27.

Okay, so now I've formed an equation.

Is that linear, or is it quadratic? It's linear.

Okay, I think of a positive integer.

I square it.

So, when I'm squaring it, do X times X, which gives me my area of X squared.

I then add on three, and my answer is 103.

So here, I've got X times X, which is X squared, and then I add on three, and that total area, this shape and this shape, is equal to 103.

Okay, and I don't want you to see if you can work out what it is for now, I just want you to get a sense of what is the difference between a linear and what is the difference between a quadratic.

So here.

What we do is we square a number, and when we square a number, that gives us the highest power of two, which means it turns it into a quadratic.

So, squaring an unknown, or a variable, turns something into a quadratic.

This one, we didn't do any squaring, so it didn't become a quadratic, whereas this one, we did do some squaring, so it became a quadratic.

Okay, let's do a slightly trickier one now.

So, I think of a positive integer, let's call it X.

I add on three.

And then I multiply by four.

So I multiply everything by four.

So I multiply that by four, and I multiply that by four.

And you can see, with my algebra tiles, that it's 4X plus twelve, that's my total area.

And my total area is equal to 52.

So that is equal to 52.

Okay? Now--oh, sorry, it's overlapping a little bit, but it doesn't matter too much.

I think of a positive integer, so let's say X, I add three this time, so I've got X plus three.

And then, I square it, so I do X plus three times X plus three.

Like that.

So that gives me that area there.

So, we can write this in two ways, we can write this like this, X plus three squared.

Equals 25, okay, 'cause the total area is 25, or you could expand that, and have X squared plus 6X plus nine equals 25.

Just expanding that bracket, so X plus three.

X plus three.

And if you expand that, you'd get that, okay? So there's two different ways that we could write this area.

Okay, this one is linear, this one is a quadratic.

Okay, I think of a positive integer, let's call it X, I add on three.

I then add one more than my original number.

Well, my original number is X.

And one more than X is X plus one, so I need to add on that.

Like this, add on that, okay? There's my original number, plus one more.

So I get 2X plus four, and that has got a total of 28.

Okay, so the total area of that is 28, because my answer is 28.

I think of a positive integer.

X.

I add on three, so I get X plus three.

I then multiply by one more than my original number.

So that would be X plus three-- I'm going to write it down here-- multiplied by X plus one.

So.

What I'm doing here is I'm multiplying two unknowns, two variables, and that gives me a quadratic.

So when we multiply two unknowns, we get a quadratic.

So that is equal to 120, or we could write it like this, 120 equals, expanding these brackets, X squared plus 4X plus three.

And that's quite easy to see from your diagram.

Okay? So, this one.

We added my original number, and that kept it linear.

But here, we multiplied by my original, and that made it non-linear, because we multiplied two of the same variable together, which turned it into a square.

And that's really important, that idea that, where I have the square in it, and we're multiplying it by itself in some way, and that moves it from being linear to being a quadratic.

Okay.

Compare the two boxes.

What is the same, what is different? So maybe we need to pause the video and have a read.

Okay, over here, I square it, then I multiply by five.

Over here, I multiply it by five, then I square it.

How's that going to change our expression? So let's think, we start off with X, and then we square it.

And then I multiply by five, which we can write as 5X squared.

Over here, I multiply it by five.

So that's 5X, then I square it, which means I square the five as well.

So I square both the things.

Five squared, 25, X squared, X squared.

So here I've got 25X squared, which equals 27, okay? And you can write it like that, or you can write it like that, and I slightly prefer to write it like that.

It's just a bit neater and it's a bit tidier.

Okay, so I just wanted to draw your attention to those differences there.

And it's all to do with order of operations.

Okay.

So, last one, I think of a positive integer.

I square it.

Then I add five, and then I divide by six, so we can write everything being divided by six with this big long line there, okay? And we've got that as equal to 42.

Okay, so, X squared plus five divided by 6 equals 42, so that's like a fraction line there.

Okay, so, what I want you to do now is I want you to have a go at the independent task.

So, pause the video to complete your task, resume once you've finished.

Okay, and here are my answers.

So mark your work.

Okay, so now it's time for the explore task.

Make up three quadratic equation "I think of a number" problems, make one easy, one medium, and one hard.

So I want you to be creative and try and make up the problems that we've been doing in this lesson, but I want you to make them quadratic.

So, pause the video and have a go.

Pause in three, two, one.

Okay, welcome back.

Now, I can't possibly go through all the different answers for this.

There's so many.

But, yeah, and it's unfortunate that I can't go through them, 'cause I'd really like to see them.

But, as long as you had a squared in, or you're multiplying it by itself at some point, that should give you a quadratic.

Okay, so if you'd like to, please ask your parent or carer to share your work on Twitter, tag it @OakNational and #LearnwithOak.

Okay, so that is all for today's lesson, thank you very much, and I will see you next time.