Choose the word which best fills the gap: When we ________ double brackets every term in the first bracket must be multiplied by every term in the second bracket.

Expand and simplify (x+2)(x+5)

Expand and simplify (x+2)(x-5)

Expand and simplify 2(x+3)-2(x+5)

A rectangle has side lengths (x+4) and (x-3). Match the expression that would represent it's area.

Yasmin expanded (a + x + 3)(x + 2). Below are her workings out. Which expression matches the area of the rectangle she has created?

Expanding Double Brackets Quiz

﻿Hello! My name is Mrs Buckmire. Today I will be teaching about difference of two squares. Now make sure you have a pen and paper and if you want to get a little more practical and it would be good if you have some paper that you are allowed to cut up permission to use scissors and just any straight edge, but you don't need that for this lesson, Okay. Please pause the video whenever I ask you too, but also whenever you need to, okay. Learn your own pace, and it can be useful to rewind the video sometimes if you are struggling to understand something, sometimes just hearing it a second time can be really helpful. Okay lets start. So, expand and simplify these things. So, X plus two times X take away two, X subtract five times X plus five and X plus two-point-five. What do you notice? Okay, pause the video now. Okay how did you do? So, expand it and I'll go for the first one, and I reckon you have probably done them. So, we have X plus two. So, there is my X and then my plus two, and then I have X and I am taking away two. So, lets just take away two here. So, take away two. So, here this one. This whole thing is X squared. This one will be two X. Here we are going to have subtract two X and here we are going to have negative two times positive two, good it is negative four. So, altogether I have X squared plus two X, take away two X, take away four. Excellent, so you end up with just X squared You're collecting like terms plus two x, take away two x equals zero. So, X squared take away four. Well I think you got that. What about this one? You get X squared, take way 25 and this one, Hmm X squared take away 6.25. What do you notice? The answers to all of them have those middle terms. We add them and subtract them so, we end up with zero. So, we end up with zero coefficient of X. So, all of them don't have X in there final answers. Anything else? Interesting, they all have one X squared and also they all have a square number. What else? They all have subtraction as well. There are many links here there must be some thing going on, because even in the question that is all one bracket has plus and one bracket has subtraction, and we know actually in multiplication is commutative so, all of it doesn't matter, even though in the second one it goes negative and positive and it is the same as having a positive and negative as in the operating side of bracket. Alright, lets look at this BLANK Okay so, a square with side length B is cut from a square of side length A. How can you rearrange the three pieces left to form a rectangle? What would the area of this rectangle be? Now this is where we can use are BLANK. So, if you got scissors, paper you can cut and pen and a ruler then use those, make sure you got those ready. Lets think about this, a square with with side length B is cut from a square side length A. So, I got my square here and so I am going to label it, this length is going to be my A and this length is also then yeah still A so, square. Okay remember if you don't have square paper you can always kind of fold the rectangle in half there and then cut the excess. So, the beauty about a square is it's diagonal is a line of symmetry so that's how actually with a rectangle we can get a square. Okay, now there's our square side length A and square side length B is complement so, watch I am going to do that trick. So, just get square side length B. This is where a ruler would come in handy, I actually don't have a ruler, I know terrible math teacher. So, I am going to actually just use this pen this side edge, that's why I said we can be creative we don't need rulers really. That's for measuring here so, we're not measuring, we're putting in so, here you go. See how I made my square and that of length B. B. Okay, what I'm actually going to do is cut this out so, it's cut from this square. Actually what could be useful, let's continue that line so, we can see where we cut to make it easier. So, I'm just going to do dashes here so I can see that it looks like the image on the right hand side of your screen. Bump, Bump. Okay so I'm cutting. Okay so I'm going to cut B out and I'll try to do my best cutting skills here. Let's see, there, this is my square B being cut out. There, okay. We're interested in what is the area of this piece that is left, okay. And can you arrange the three pieces which are rectangles so I would cut here, and we can cut here. And we now got three rectangles, and we're thinking okay, can we arrange these three pieces to form a rectangle what would the area of this rectangle be? Okay? Pause the video and have a go. Now you don't need to do the practical you can just thinkin about the answer, but do pause it and have a go. In three, two, one. Okay, you've had a think to yourself? Let's see Now all this is actually already a rectangle so I'm going to leave this here I think actually I can just put this at the end like this and it should be the same, ah look how good my cutting is it's the same size. Hmmm, why? Let's think. So this was here, and it was the whole of "a" this side was So what is the distance from here to here? Well the whole thing was "a", now I've taken off this part and this length is... good it's the same as this one. So this length was "b", so this length is "a" takeaway "b" Now, actually this length here, the whole thing is "a" and actually we're cutting out "b" aren't we? So this actually is also, this length is also "a" takeaway "b" so can you see that? The whole thing is "a" and then we're cutting off "b" So that length up to here is "a" takeaway "b" So actually this length matched up with this length so when I turn it, yes it makes a rectangle So, what would the area of the rectangle be? Hmmm... Area of a rectangle, how do we work that out? Good! It's length times width. So let's say "a" subtract "b" is the width What is the length? Excellent, it's "a" plus "b" So the length of the whole rectangle is, and let's actually just write it below, is "a" plus "b" and this width or height or whatever you want to call it is "a" takeaway "b" times "a" plus "b" Okay, let's do this one that's... Okay, I'm back and I want to just formalise that So this first one was "a" squared takeaway "b" squared We had our whole square, which was "a" squared That's the area of it and we're taking where that "b" is from and I cut out that corner, didn't I? So that's the representation of it algebraically and now, we end up with A plus B times A takeaway B and here's a little image of that so remember that side ended up being A takeaway B and this side was still A all the way and then we were plussing B here so I end up with my square... well.. yeah square, and then rectangle and rectangle. Okay? But look back at that patchwork if it helps you visualise it as well, but we know that actually these two are.. these two are equal to each other and what's most important is the name of it is the difference of two squares Whenever you hear difference of two squares, oh its A squared takeaway B squared, which equals A plus B times A takeaway B So why's it called difference of two squares? 'cause it's the difference between two squares, so we have A squared, so a squared number takeaway B squared. Difference is takeaway. So that way and it works out. So this is what you need to write down. So this helps. Copy this down in your notes and maybe say out loud "A squared takeaway B squared equals A plus B times A takeaway B" or A plus B times A takeaway B equals A squared takeaway B squared. Okay this is like something that if you can really remember this it could be helpful sometimes when you're trying to um expand and factorise and also work out certain calculations, it can make things easier. Okay, lets explore this a bit more. Right, first let's check you understand this, okay? So can you expand this double bracket? Pause the video and have a go. Three, two, one. Okay, did you use this rule? If you didn't have another quick go. Okay. So we had, I'm going to go through it Here is X plus three times X takeaway three That equals A squared A is X so X squared takeaway B squared, B is three, so three squared. What is three squared? Good, it's three times three Which is? Excellent. Nine. So the answer is B Well, if you weren't fooled by the other ones So remember, the area is subtracted, so it needs to be the difference of two squares, and you must square that second number. Okay? Okay, I would love you guys to have some more practise in independent tasks so this is kind of practise of all your expanding of so far, so Binh has completed some questions Correct and explain her mistakes, so she might got some right, but she might got some wrong. She's expanded and simplified the following expressions so they're there, her working out is there I want you to correct it and write out what the correct answer is, and then explain what do you think she's done wrong. Okay? Just try and get into the mind of Binh and what do you think she did? All right. Pause the video and have a go. Okay, how did you do? How did she do? She get them all right or wrong? Let's see The first one, well this was good X squared plus five X plus four X plus 20, what? That doesn't equal 30 X squared, ahh! What do you think she did? It looks like she just added them all together, why can't she do that? Girl you can only collect like terms so it should be X squared plus nine X plus 20. All right the next one, okay X squared, good, plus four X, takeaway five X, good, wait a second... plus 20? Negative five times positive four is negative 20, aghh! So here it looks like she multiplied negative five and four incorrectly so maybe she thought it was positive and misread it, or maybe she just didn't write, um, just didn't know negative times a positive makes an um is a negative? Okay there's a mistake there so that's should be the correct answer now. What about the next one? So X squared subtract four X plus five X, yes takeaway 20, yes, all looks good. What about here? Ahhhh, plus five X takeaway four X is plus X. It's a positive X here. So you have a negative four, plus five, we're getting a positive zero, and a positive one so there she collected like terms incorrectly. So maybe she, I don't know She made a mistake there, didn't she? X plus four? Where you got X squared plus 16 Do you know what she's done? I reckon she just squared them without actually thinking this means X plus four times X plus four, even if that does not look like a four. Let's do that better. Ah, here. Oops. Okay, does that look like a four to you? Hopefully? Okay so when we expand that what do we get? We get X squared plus four X plus four X plus 16 Did you do that? Yeah? So what was your final answer? Good, X squared plus eight X plus 16, ah, look about four eight X, don't know what that could represent so, here it's really important to remember that expanding double brackets out, X plus four bracket squared means that actually you need to write out the brackets always. Don't try to just do it from your..just do it automatically, actually write out the brackets every time it's just, it's really useful when you're first starting. And this one, I recognise this. Did you recognise it? It's difference of two squares, so it should be, I don't know what they've done. It should be X squared takeaway 16 plus four squared and what more could they have done? Maybe like when they have the minus four? I don't know what they've done to be honest. It's difference of two squares. Maybe you came up with something you thought Binh had did. I can't work it out. Well done, if you had a good go at those, so Binh didn't do so well, but maybe you did. Right, on to our Explore. Xavier says I can work out 45 squared takeaway 15 squared by calculating 60 times 30. Hmmm Is he correct? How could you calculate 35 squared takeaway 15 squared? 19 squared takeaway nine squared? 38 squared takeaway 12 squared? Can you write down what the calculation would be? And explain why this works. And then I would love you to generate your own calculations that can be found using a similar trick Hmmm. I think in a little matter of a lesson you can be really good at this, okay? So you pause the video and have a go at it and we'll go through them, but have a go for yourself first. Maybe try to come up with the best, the biggest calculation you can do using this similar trick one that you think no one else would of thought of. If you're really confident. All right, pause in three, two, one. Okay, so was he correct? Did you work it out? Or did you kind of reason he is correct? It is true Tick, tick, how could you calculate this? What's the key thing? What's this whole page about? What's our whole lesson about really? Difference of two squares, yes! So, you could do 35 plus 15 times 35 subtract 15 which would be 50 times 20. Is that not so much easier than 35 squared takeaway 15 squared? You might be able to do 50 times 20 in your head lets see the other one, so this one became 19 plus 9 times 19 takeaway nine So nine takeaway nine is easy it's times ten 19 plus nine, I get 28 You could do that in your head, couldn't you? This could be a trick you could impress some people with. Let's see, 38 plus 12 times 38 takeaway 12 So that equals 50 times, take away 12 and we're going to get 26, so do you agree? Yes? Oh that one might be hard to do in your head, but actually it definitely makes it easier, doesn't it? So, when we say explain why it works, well it's all about the difference of two squares, and you can go back to our diagram or you could, maybe like show it algebraically so just to say oh, its the same as this, and generate your own or begin to imagine how great and imaginative you have all been where you create your own calculations so do check they work, maybe even have a calculator to check them or you can do some multiplications and make sure, oh is this definitely correct? Really, really well done everyone. Thank you so much for your hard work, you should be really proud of yourself. What is the difference of two squares? Tell me? Excellent! You try and remember that. Try and say it out loud again and stick that into your memory. I would love you to do the quiz that the ideal time for you to show off what you've learned and I hope to see you in another lesson maybe. Have a lovely day! Bye!

Have a go at the questions below to show off what you know!

Which expression is equivalent to a² - b² ?

Match the expression equivalent to (x+4)(x-4).

Match the expression equivalent to (x+9)(x-3).

Which calculation would work out 64² - 4² ?

Match the expression equivalent to (x+25)-(x-25).

Match the expression equivalent to (73+x)(73-x).

Difference of Two Squares Quiz

In this lesson, we will learn about the difference of two squares. We will investigate the pattern of results when considering multiplying out brackets of the general form (x+a)(x-a).

Maths

Y9 Maths

Key Stage 3

Year 9

Difference of two squares

Quadratic expressions (9.9)

Algebra

THEME

UNIT

Recognising linear and non-linear graphs

Plotting quadratics (Part 1)

Plotting quadratics (Part 2)

Quadratic contexts

Maximum and minimum area

Distributive property

Expanding double brackets

Factorising quadratics (Part 1)

Factorising quadratics (Part 2)

Sketching quadratics

Interesting quadratic patterns

Which of these tables displaying coordinates of graphs is non-linear?

Complete the tables below and plot the graphs. Select a co-ordinate the graphs have in common.

A square is cut out of the rectangle. What is the maximum area?

Antoni says this is not a quadratic as it does not go through the x axis. Do you agree?

Cala says (x+2)(x+4) is the same as (x+4)(2+x). Is she correct?

Factorise x² + 11x + 24.

Cala says that you cannot factorise x² -9x + 8 because there does not exist a pair of factors for 8 that add to give a negative number. Is she correct?

Select the words that best fill in the gaps in order: The __________ is where the curve crosses the y axis, when _______.

Which graph could be a sketch of -x² + 3x - 2? *Hint* Try out some coordinates to check!

I pick two consecutive integers. I square the bigger one and then subtract 4 of the smaller one. Zaki writes a different algebraic expression to me but is still correct. Which one could he have written?

Lesson overview:Difference of two squares

Core Content

Quadratic expressions (9.9):