The length and width of a rectangle add to 4cm. Stacey thinks only 2 different rectangles are possible. Do you agree?

A rectangle's length is 2cm greater than its width. Which expression gives the area?

A triangle's height is 4 times greater than its base. Which expression gives the area?

The length and width of a rectangle add to 14 cm. Which expression gives the area?

The length and width of a rectangle add to 14 cm. What is the largest possible area?

A triangle has a base and height that sum to 20cm. What is the upper bound of the triangle's area?

Maximum and Minimum Area

﻿Hello everyone! My name is Mrs. Buckmire, and today's lesson title is Distributive Property. I'm hoping you've had this before, but no worries if not. I will introduce it, and hopefully by the end of the lesson you will understand what I'm talking about. Now first, you need to make sure you have something to write with and something to write on. Please pause the video whenever you need to, and also when I ask you to. And, remember, rewinding the video sometimes does help. Just remember, just work at your own pace. That's the beauty of these videos. Okay, let's begin. So your first try this, how do you calculate 23 times 42? Now, what I am most interested in is why does your method work? So think about, if you've got lots of different methods, maybe think about which one can you actually explain, which one can you kind of justify and reason about why it works? Okay? So do pause and have a go at this. It's not just about getting the answer. Really, I don't really care about the answer. I don't think I know the answer. It's more about actually how did you work this out? Okay? Pause and have a go. In three, two, one, and pause. Okay. So, there are lots of different methods, and some of you guys might use the column method, but I think that's quite tricky to explain. So, the one that I'm going to try and explain is the grid method, or using an array. So, here, we use this array, and this rectangle, and we label each sides. So, my top side is 42, and my left hand side, my vertical side, is 23. And then we know well here, the area here, would be 800, whatever units we're using. This area, this small rectangle here, would be 40. The area of this one would be 120, and the area of this would be six. So all together I could add it up, so I get 920, 960, 966. So while I didn't give out that answer, but did you use this method, as in can you explain your method, okay. So here, what we've really done, is we split it up, like this. And, so, we've kind of partitioned it into 20 plus three and 40 plus two. And then we've made sure that everything in this bracket is multiplied by everything in this bracket. So we can see 20 is multiplied by 40 and two, and three is multiplied by 40 and two. Okay. So hold that in your brain, and let's look at a different example. Okay. So we've got algebra now, but remember we had 20 plus three, we could do it, okay. So we're going to use arrays again. So the way we're going to do it is we're going to let this side length be x, and this part be five. Here will be x and here is two. So you can see how this one makes a square, so this is x squared. What's the area of this one? Hmm.. well it's x times two, but you can just write that as 2x. Down here, five times x, excellent, 5x, and five times two, good, 10. So now we can add these together. So we'd get x squared, plus 2x, plus 5x, plus 10. Now what? What else could we do? We had a collect in like terms, so which terms are alike? X squared is just x squared, there are no other squared values. Now there's a 2x, so two lots of x, plus five lots of x, so there's seven lots of x. And then 10 so that's just our ones, so plus 10. So this is our final answer. So we call that expanding double brackets. So expanding double brackets we're multiplying everything in one bracket by everything in the other bracket. And this array, I think really helps us to make sure we do that correctly. Okay. So pause the video and take notes now if you need to. You might want to add on--oh remember to simplify because that's something that people forget. If you want me to write simplify. I could just write you a little note here. Remember to simplify. Simp-li-fy. At the end. Okay, let's see. This is your opportunity to check your work and have a go yourself. So pause the video and see if you can do what I just did. Okay, so that's why I told you to make notes cause now you can refer to your notes. Maybe rewind back and do make proper notes. Let's see how you do. Pause in three, two, one. Okay. Was one of your answers there? That's a good start. Okay. So what we're going to do is we're going to let this be x, and this be, let's go for, three, and this one be x, and this one be two. Hmm... do you think the order of that really matters? No. It doesn't matter. Okay. So this area, x times x, is x squared. X times three is 3x. X times two is 2x, and this area is going to be six. So I have x squared plus 3x plus 2x plus six. Okay. All of these though look like they've been simplified. So which one would be the simplified version of this? So x squared, there is no other x squared, so definitely start with x squared, which they all have. And then 2x plus 3x, three lots of x plus two lots of x is five lots of x. So this is the only one with five lots of x, so it looks like it's D. Let's just check, and then plus six. Good. So this is correct. So this one is wrong. Well, for a start, there is no x's here anyway and we definitely had x's in our rectangle, so B is definitely wrong. This one, incorrect coefficient. And this one, incorrect coefficient, and incorrect last number as well. Okay. Now the order these are written in do not matter, so if you, when you worked it out, got x squared plus six plus 5x, that's fine, okay. In addition, it's commutative which means the order doesn't matter for that operator, okay. You are ready for your independent task. So what I want you to do is match the equivalent expressions. So the left hand side to the right hand side, maybe they don't all match. Don't just assume they match, okay. Work them all out and check them really carefully. It might be helpful to draw the arrays because that can help check that you're correct, okay. Pause in three, two, one. Okay, so how did you do? Let's go through the first one. So, here I'm going to have x and 24. Now you notice here it's not really drawn to scale. That doesn't matter. You can kind of under scale and think about this must be bigger, but I don't think it matters. So x times x is x squared, plus 24x. Now I actually often do write a little plus here. I'm surprised I haven't yet. But that can be helpful. Positive x here. What's wrong? What did I do wrong? It's 24x, 24 times x is 24x. You should always double check you see. Even I am prone to making mistakes. So plus 24 times plus one, or positive 24, sorry, times positive one is positive 24. So here we have x squared plus 25x, add those together, plus 24, ah, this one. Okay, good. Starting well. So here, let's go x plus six. X plus four. So we have x squared, 6x, 4x, 24. So x squared plus 10x plus 24. Yay! All right, x squared plus 12 plus--times x plus two. What did you get? It should of been this one. And so that means hopefully this one matches. Let's actually work it out. So, x squared, and then it is plus 3x plus 8x plus 24. Is that what you got? So it does work out. Lovely. Well done if you got those correct. All right. You are ready to explore. You notice your independent task is a bit shorter because this is a mammoth task, okay. But, I'm hoping it's really really helpful. So what you can see is you've got a grid. Let me just show you this. So, as a increases in this direction, so all of the quadratics are in the form of x plus a, x plus b, okay. And b increases in this direction downwards. It will all make sense once I show you, okay. So, there's the one we're starting with. Now what you'll see is to the right of it, can you see that our a has increased? So, in this direction, a is increasing, a increases, that's my a value. So a has gone from one to two, and there's the answer. It's going to increase again. Look at that. It's now x plus two times--so it was x plus two, x plus three, and now it's x plus three, x plus three. So again, a has increased by one, okay? So what do you think it's going to be here? A is going to decrease by one, x is going--yeah a is going to decrease by one. So it's going to be x times x plus three. Hmm... what about here? It's going to decrease again. So it was zero here. X plus zero, so now it's going to decrease, so it's x takeaway one, x plus three, okay? So in a similar fashion, going up and down, here b increases. So this b value has gone up by one, and then you need to work out what the quadratic would be when it is expanded, okay? Here, it increases by one again, you need to work out the expansion. So when it goes up, what's it going to be? Good, x plus one times x plus two. And up again? Yes, x plus one times x plus one, okay? So here are the ones to help you. Let's give you a few more. So here x, x plus two. X plus two, x plus two. X, x plus four. X plus two, x plus four. And what I want you to do is expand them. Now what you might spot is a pattern, so you don't actually have to expand, what how many are there? Two, four, five times. You don't actually have to do expansion for 25 takeaway four lots, okay? You can actually start to spot patterns, and I don't mind you just filling it in based on the pattern, but do try and practise expanding and look at a few. So once you've got the whole grid filled in, even the ones here that I haven't put the questions in, you should know what the questions are based on the information. Then, I want you to tell me what patterns can you spot? Okay, mathematicians, we love a good pattern, so you try and spot the pattern, then maybe when you're working out you're already using the patterns anyway. And then think, okay, could I generalise these patterns? Okay. So you have a little go at that. Don't worry about the generalisation if you struggle with that aspect. It's more of an extension. But, I definitely want all of you to complete the grid and spot some patterns, okay? Right, pause the video in three, two, one. Whoa. Okay. Here are all the answers. I would recommend pausing and checking it, so I would say yeah, do pause it and make sure that you've got these correct, and then I'll talk about the patterns. Okay. All correct? A couple of little mistakes? All right. If you spotted some patterns, maybe when you were looking at the patterns, you were like oh, actually I made a mistake there, and actually I did that. I made a pattern mistake somewhere and the reason I spotted it is because I didn't follow the pattern, okay. So it could be helpful. So, what do we notice? So, you might notice how, so, for example, here, all the numbers, they're all negative, and here they're actually positive. Here it goes negative one, negative two, negative three, negative four, negative five. And here they're all zero, there are no numbers here. Here it goes one, two, three, four five. Here two, four, six, eight, 10. What's that? They have a two times table, yes? And this three, six, nine, 12, 15, the three times tables. Interesting. Anything else? Maybe you noticed things about the coefficient of x. What does that mean? Coefficient? Good. It's the number in front of x. So here, the coefficient actually increases. So, one, two, three, four, five, and here actually. It goes from zero, one, two, three, four. Two, three, four, five six. So in every case actually, it seems like it's only increasing by one. Whereas the numbers were increasing by one, then two here, and then were increasing by three on each step here. That's interesting. Do you notice anything else? Oh, okay. Maybe you notice something about how the numbers add together compared to the coefficient. So one plus one equals two. One plus two equals three. Is that always true for all of these questions? Hmm... you could pause it and check that. There's no coefficient in front of x squared other than one. So, all of them just have one in front of them. They don't have any 2x squared or anything like that, and that kinds of makes sense because we just have x and x, and you always have that square with x times x don't we? Well done if you spotted anything else. There's so many things I can't cover everything. So if you wrote extra things and you think ah, that's legit, good on you. Excellent work today! Thank you so much for your hard work. I really hope you enjoyed the lesson and understand a bit more on how to expand double brackets. What I would like you to do is write down one thing, one thing that you think oh, I must remember this. If someone asks me oh, yeah, you watched that video, what did you learn? This is what I can say, okay. So one thing you think oh, this is something I need to remember. Then, can you please do the quiz. I think it's a super super helpful way for you to assess your understanding, and also for you to learn a bit more, so you can kind of see when you make a mistake, you can learn from it because I've given you some feedback. If you need to come back to the video and watch it again or watch certain bits again, that's more than fine as well. Have a lovely, lovely day and hopefully I'll see you in another lesson. Bye!

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.

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

Expand and simplify (x+11)(x+2)

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

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

Expand and simplify (x+1)²

Distributive Property Quiz

In this lesson, we will learn more about applying the distributive property of multiplication with variables. We will use area models to visualise the process.

Maths

Y9 Maths

Key Stage 3

Year 9

Distributive property

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

Expanding double brackets

Difference of two squares

Factorising quadratics (Part 1)

Factorising quadratics (Part 2)

Sketching quadratics

Interesting quadratic patterns

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

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?

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

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:Distributive property

Core Content

Quadratic expressions (9.9):