Lesson video

Hello, my name is Mrs. Buckmire and today our lesson title is Factorising Quadratics 1 so I'll start to teach you how we can factorise quadratics.

Don't worry if you're not sure what that means.

We'll come on to that later, but make sure for now you have a pen and paper and you're ready to learn.

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

If you need more time, you go through it, press that pause button and just resume it when you're ready, okay? And sometimes it is helpful to rewind just to listen to something again, just especially when I'm explaining certain mathematical technical things, maybe so just rewind the video, listen to it one more time and sometimes that can help you to understand it.

Okay, let's start.

So your Try This is to expand the following brackets.

You've got three questions there.

What's most interesting is you thinking about what's the same or what's different about each one.

So as you're working them out maybe start to think about that, okay.

Pause the video and have a go.

Okay, so the first one, so you should have got this.

So remember, I prefer if you've collected like terms and simplify it.

For the next one and this one.

Pause it if you need to check over.

If you made any mistakes, just have another quick go at it, maybe.

So what I was most interested in is what is the same and what is different? What did you say? Good, all of them are the same.

They all have x squared in it, very true.

Excellent, the number even is 12.

It doesn't have to be the last number.

I could have written these in any order.

I like to, I think, mathematicians, we like to do the highest power first in general, but it's fine if you do it in different order so the number is always 12 What's different? Good, each of them have a different coefficient of x.

I wonder why.

Okay, so here we've got at the top, we've got our questions you've already done.

At the bottom, oh even at the bottom, we've got the questions we've already done at the top are those that have been expanded and here are arrays to show that.

So in this array, we have this side, length is x and this side length is one, two, three, four, five, six.

So represents this side length of x plus six and here we have x and this side length is two so plus two, and we can see the area of this whole thing is x squared.

Plus now these are all like, you can imagine a wall length one and height x.

So they're all x, x, x.

And we have all together eight lots of x and then here, well this is length one and this is length one.

So each of these are one, so we have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12.

That's a bad 11 there.

So there's 12 ones there so you might have, maybe, if you're in school, you might have seen algebra tiles, maybe some of you.

So these are kind of could be seen as that as well.

So with their sets and lengths and dimensions.

So all together in here, the area is x squared, plus eight x plus 12 and these get factorised into x plus 6 as a factor so there's one length of the rectangle and x plus two is another factor.

So there are two factors of x squared, plus eight x plus 12.

So goin' that way, it's called factorising.

And you guys already know if we're going upwards, what's it called? X and it's called expanding.

So expanding is the reverse of factorising.

And factorising is the reverse of expanding.

So for this one, so this area is x squared.

These are all our x bars, really, and then here's our ones.

How many ones do we have? Good, 12 again.

And we can see that this 12 is here so that we can always know that this corner is going to be whatever number this is.

So here, 12 so this was 12.

This whole area was 12 and same with this.

This whole rectangle here was 12.

And it's easy to see that this part is x squared because that's our square and here as well.

Okay, let's look at this a bit closer.

So x squared plus 5 x plus six so I'm going to show you how I can factorise it using an array.

Okay, so first I know this area and this area.

What is it? Good, that was x and x, that area, this one is my x squared.

And this one, the bottom rectangle, purple, Yes, it's six.

Okay, so now I don't know this and I don't know this, but I know that when I multiply them together, because they're the same length as this and this, I'm going to get to six so I'm kind of make a table and in it I can put in the factors of six, so the factor pairs.

So what more times to give it to equal six? Well, one times six and three times two.

So now I know it's either going to be x plus one times x plus six or x plus three times x plus two.

You can pause it and maybe expand each one to see which one gets there.

Okay, so when I expand this one and we're going to do the x squared plus six x plus x plus six, so we got x squared plus seven x plus six and what about this one? Good, I got x squared plus three x plus two x plus six.

So x squared plus five x plus six.

So it must be this one.

So this is the factorisation.

So it should be x plus three and x plus two.

So main thing to remember is from the style and we have the a and b must multiply to get this so we need to do our pairs of factors and there might be more, there might be more than two, and then we can actually write them out and expand them to see actually, which rectangle is it.

Which array is it? Pause the video and make good notes so you understand it.

Okay, let's do this one together.

So you've drawn a rectangle.

Okay, so I've this here and this here.

What are those areas? Good, so side length of the green one at the top is x and x so that area was x squared.

And this one? Good, it is nine.

What next? Good, so we have a and b? So we're going to make our table.

You fill out the table, pause it.

We're doing ahead.

Good, so the table factor pairs that multiply to give nine, yes, so one and nine and three and three.

Any others? No, okay, so we have this and this.

Do pause it and expand those and decide which one's right.

Good, so I get, make sure you've paused it, you've had a go.

You should have your answer now.

Okay.

I get for this one, I got x squared plus 10 x plus nine.

And this one I went x squared plus six x plus nine.

Maybe you're noticing something.

Maybe you're noticing something about once your multiplying the answers.

Now there's x plus three times x plus three.

How can you write that? Good, more efficiently, we can just write x plus three squared.

Maths, we always want to be efficient, okay? So x plus three brackets squared 'cause they're times by itself.

Right, well if you got that, let's do one by yourself now completely.

Okay, so do this one by yourself.

Okay, was one of your answers there? There's a good odd one of your answers were there.

Okay, so we had this area, which is x by x and the area of the purple is eight 'cause eight ones.

Do we have a and b? So let's actually do a little table.

Let me do it myself so I have a and b.

So you could have one times eight.

You could have two times four.

So I have x plus eight times x plus one, and I have x plus two times x plus four and when I multiply them out, this one gets to x squared plus nine x plus eight and this one gets to x squared plus six x plus eight.

Yeah, so your answer, was it there? Yeah, C.

Well done if you've got C.

Okay, I'm hoping you did factoring, you didn't just expand the others, this is to practise factorisation, okay, so make sure for the next one you're having a go at factorising.

Have you got that? Okay, Zaki spilled some smoothie all over his words.

He can no longer read the coefficient of x but he is a fantastic student.

so he's going to write out all the possible factorisations and the questions so he knows that yet one of them will be the answer.

So I want you to do that for Zaki.

Question two is to factorise so there's questions a to f and I want you to factorise all of them so you actually can't do the little expanding trick 'cause there's no answers there.

You need to factorise.

Okay, so factorise and then check them by expanding on which one it is.

Okay, pause the video in three, two, one.

Okay, so all the possible factorisation.

So is where we have this diagram.

So a and b needs to equal to 30.

So therefore I'd write out all my factors and pairs of 30.

And there are a lot, there is one and 30, two and 15, three and 10, five and six.

I think that's all of them so if it was x plus 30 x plus one, you end up with this.

X plus 15 x plus two, x plus 10 x plus three, x plus six x plus five.

So pause the video and check you've got those answers.

Okay, so the independent tasks feedback question 2, so it was to factorise.

X squared plus eight x plus 15 became x plus five x plus three.

Here's the next one.

Okay, so did you notice anything about those questions? So b and c, you might not.

If you did it with arrays, you might not have had to do a new array 'cause you already had one in the answers and also for a and d as well so I thought I'd be helpful.

E and f, they both end in a hundred, but what is special about f? There's something special about it that hopefully you recognise It was x plus 10 x take away 10 is difference of two squares so hopefully you learned that if you were with me in a past lesson, you might've learnt, so x squared take away b squared equals to a plus b times a take away b so here a was x and b was 10.

Really well done if you spotted that and one if you got those correct.

Okay, so for our Explore.

So this just pure explore with positives and negatives and different numbers.

So what different expansions can be made by picking two blue cards and two green cards, 'kay, so just have a play with them and then try and see if you can arrange the cards so that the brackets expand to give the following quadratics.

Okay, so have a go one to eight.

Now there's some negatives in there we haven't really done but maybe you can have a little play and see if you can find the factorisations of them.

I think all of you guys can have a go at this.

Just play around with it.

Even if there's ones you can't do, just put in some of these numbers.

They're all using those cards above and see what you get.

And then tell me, what do you notice? It's all about pattern spotting with mathematicians.

Okay, how'd you do? Got lots and lots, did you do lots of expanding, you're very good at expanding now? Maybe you did do some using factorisations? Okay, so this one x plus four x plus two.

This was x take away four x take away two, x plus eight x take away one, x take away eight x plus one, x plus four x take away two, x take away four x plus two, x plus eight x plus one, x take away eight x take away one.

You check yours really carefully.

If you need to pause and look really carefully and check those answers.

So what did you notice? Yeah, so there were some negatives.

So it's interesting how, let me use this.

So here it was positive, positive.

Everything's positive here.

This is positive, negative, there's positive, negative, positive, negative, positive, negative.

I wonder if that's always the case, I'm not sure.

Well, actually this has a negative and positive, but actually it's negative, negative so yeah, it's not always the case.

Negative, negative has a negative and a positive.

Negative, negative has a negative and a positive.

Negative, positive has two negatives here.

Interesting, so maybe I wonder how you get those answers.

Okay, we had some other similar answers, four and two, four and two, eight and one, eight and one, I'm kind of talking about the absolute value so actually the digits four and negative two, but this one's negative four and two so although yes, the absolute values are equivalent, they're actually, this one's negative and this one's positive, this one's positive and this one's negative.

Even they both have negative eight.

this one has plus two and this one has negative two.

There are so many things to notice, 'kay, you couldn't have noticed them all.

Well, I can't say them all out loud, but really well done if you had a good go at it, Thank you so much for all your hard work today.

I hope you've enjoyed this lesson and you understand how to factorise quadratics a bit better.

Please do the x equates.

It's an ideal opportunity to actually even learn more 'cause when you actually test yourself and you see what you do know, then anything you don't know, you can look at the feedback and learn some more, so ideal opportunity.

Have a lovely, lovely day and hopefully I'll see you in another lesson, bye.