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

Select the word that best fills in the gap: Sometimes ___________ can be used to model situations.

"I think of two numbers with a difference of 3 and multiply them together." Which expression bests represents the statement?

A square is cut out of the rectangle. Give an expression for the area.

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

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

Quadratic Contexts

﻿Hello, my name is Mrs Buckmire. Today I'll be teaching you about maximum and minimum area. So first make sure you have a pen and paper and remember pause video when I ask you to. And also whenever you need to, it's sometimes useful to rewind the videos when just listening to certain bits again, and sometimes it can help you understand it better. Right let's begin. Okay so for the, try this, the width and height of a rectangle sum to 10 centimetres, what is the greatest possible area? How do you know? Here's some examples of possible rectangles? There's one, there's another, maybe find the area of these, see if you can make a rectangle where the side lengths do sum to 10 cent where the are is even bigger. Pause the video and have a go. Okay. How did you do, did you work on the areas of these? What was the first one? Good, eight times two is 16 centimetres squared. Remember new units. What about the second one? You maybe had to use a calculator, but fine if you didn't Maybe you can work it out without a calculator. Excellent. 23.31 centimetre squared. Did you try maybe six times four, 24 centimetre square. That's our biggest yet. Was there anything bigger? Well done. We have five centimetres and five centimetres to make 25 centimetres squared. That was the biggest one. That's quite interesting. So really what's happening with these examples? This one side's getting shorter and this side is kind of getting longer, isn't it. Shorter. Then it gets longer like similar with these and actually five ends up in the middle of six and four. Interesting. Okay. So the greatest possible area was 25 centimetres squared. And so how do you know, like, yes, we noticed about how one side gets shorter, one side gets longer, kind of seemed like an average between them at the end with a six and four. But we can actually use a graph to work this out. So I could plot by completing a table of values. So this table, what do you notice? All of my X values are what? Yes they're all positive. They all even as well, but that doesn't matter. You could actually do it where we have zero, one, two, three. I just couldn't be bothered to do every single value. But the main thing is actually X always has to be positive. Why have I done that? Good. So X is one side length. Hm. Wait a second. If I didn't know it was 25. What would the other side length be? If it's not necessarily square, it might be a rectangle. So it would be X plus this side equals 10. So this side would be 10 subtract X. Do you agree? Okay, so now I could say the area equals the X times 10, subtract X, and then now I've got actually an equation that I can substitute these values in and find out the different values. Okay. So then wouldn't that just work it out, would I actually need to have a graph? So the beauty of a graph is even like the kind of, so you can see how I can actually write in the odd numbers here, but about all these other numbers in between like 0.5, 2.5, 4.36 6.78, then no, I'm not going to plug into my calculator, every single value, but I want to go off on a curve. I can see those values and I can actually see, Oh, if X was this, this would be the area for an infinite number. If I have a smooth curve. Okay, let's explore this further. Okay. So the graph was Y equals X brackets, 10 subtract X. What type of graph is that? Excellent, it's a quadratic. What type of quadratic? Is it going to be a happy face or sad face? It's a negative quadratic. We have a negative for the X squared and it's going to be a sad face therefore, to why me and my sad face is it's going to look like, and we have to zoom out because it's so big. It's going to look like that. Interesting. How can we use this, this graph. All of these points are plotted. They all represent coordinates where we have the X value. So we have one of the sides and we have the second value is the areas. The Y value equals the area. You can see how there's an infinite number of points. Now, why am I not bothered by these points? Good. So here, our X is negative. We know the X is the length, so it has to be positive. Really. I'm just concerned with this side of the graph, put here. So you can see it clearly this side of the graph. Okay. So when is the area? The maximums, the area is the y value Where is y as big as possible? At the very top. So here, so at 25 and that's what we expected. We said didn't we, we said when X equals five, y is 25. So when is it the smallest? At zero. So actually, yeah, if x is zero. Then really we don't have a square at all. So we could just say that like X being X has to be greater than zero, but it could be like really, really close to there couldn't it be. So that's what X is zero is when we're going to have our minimums, those points were actually quite important, but this shows and proves that when x equals five, 25 is the biggest area. And that's the maximum point. Okay. So for your independent tasks, I've given you a diagram, and it shows two, nine centimetre by five centimetre, rectangles that overlap to make the shape of a square. Can you describe the total area as a quadratic expression, Then I want you to sketch this expression as a graph, and then tell me what happens when X is equal to five centimetres. What happens when X is greater than five centimetres? And what is the maximum area and minimum area? Okay, there's quite a few different paths, I've tried to break it down, just try your best and have a go with it. I will go through the answers afterwards. So first just have a go yourself, pause and three, two, one. Okay. So firstly describing the total area now. We can find the area of this whole rectangle by doing nine times five, which equals 45 centimetre squared. That's fine. And that's the same actually as this whole rectangle, isn't it, which is also nine times five, but what's the problem. Good. We've counted this part twice. Cause it was, once when it was part of this rectangle, and once as part of this rectangle, so we need to subtract this area. What's the area? It's a square. Length x ...

Good. So take away X squared. So we have 45 plus 45 subtract X squared. So it's 90 take away X squared. That's going to be the total area. Okay. Can you sketch the expression as a graph? Now you might have, actually done a table of values and plotted it. Okay. When you do it, I know they all look a bit like this. You can check using this. The highest value is going to be 90 and then is going to go through here at around like 9.5. So just under 9.48, I'll put it on 9.48. It's like 9.486. Shall we round it better. Let's find the two decimal places. So go through around 9.49. So just look at your graph. Is it between like, is it around 9.5 and minus 9.49. So it's going to look like this. So as you would have plotted it, it was key. What's wrong with my drawing right now. Yeah. But I'm going to have any negative Xs, all of this part doesn't matter. So really just this positive bit. And even then we can't have negative area. So this bit doesn't matter. Okay. So just check and see if your plot, Oh, your sketch looks a bit like this. Okay. So what happens when X is equal to five centimetres? With X equals five centimetres, then it's going to be going five along here. So it's going to end up around there and then it's going to be going five centimetres in both of them. So it will be like here. So if I drew it on, it would look a bit like there, Because this length is five centimetres. This, that means this one is five centimetres, five centimetres. This whole thing is nine centimetres. It's just, this part is four centimetres. This bit is five centimetres. This is four centimetres. Interesting. So what happens to the area then when X is equal to five centimetres, does it get bigger or smaller? Well, even actually, if we use our equation, which was a equals 90 takeaway X squared, we get 90 takeaway 25, which equals two 65 centimetres squared. So what happened at X is greater than five. Well then we can't actually create a square. Can we? So actually five is the maximum cause five is the longest that this length is. So if it was six centimetres, if we go past this, it wouldn't really be a square. So actually that is not a possible situation. So what is the maximum area you might want to look at the graph again? I can sketch the graph again for you. Yes. When we did the graph, it's like this, What was the biggest number up here? That's the maximum area, gives a maximum area was at 90. So that's actually when X equals what is it? Zero. So when X equals zero, the maximum area equals 90 centimetres squared. I'm about the minimum area. So the minimum areas, is actually what we found out when X equals to five centimetres, because that's the biggest X can be, the biggest X can be is five centimetres. So the biggest, the area of the square can be is 25. The minimum areas, actually 65 centimetres squared, even though in the graph, there is some values of zero because of the rectangle length bounding it, we actually can't get higher than four. So actually five sorry, than five. So actually five is the biggest it can be. So 90 take away 25 is 55 centimetres squared. Really really well done if you've got that right. Okay. So for the explore, going to explore more of these upper and lower bounds, you have to think about the situation. This is all about context. That's why we're practising it so much. So do the area of these shapes have an upper or lower bound. What are they? So rectangle that has a width three centimetres greater than it's height, a triangle where the base and height, sum to 12 centimetres and a parallelogram where the height is three times its width. Pause and have a little go. There will be support. If you get stuck, you can come back and have a look. Okay. So I worked out, as I've just done a little sketch first. I think actually drawing is super, super helpful. So here now remember what we let up there will be. It's like one of the most important things. So I might say where the width is. So let's say the height is X, then width is just three centimetres greater, then what will that be? So the height equals X, we have our height here. So what would the width be? It's as an expression. Good X plus three. That's the width.

Oops.

That's a funny W. A triangle with a base and height sum to 12 centimetres. So if I let the base be X, what will the height be in relation to the base? Good. 12 subtract X. Okay. Our parallelogram where the height is three times its width. Let's say X is the height is three times, so X is the width. So what's the height then . Then I'm going to go for perpendicular. That's a very bad straight line, perpendicular height, X, and it's three X. Okay. So now we have all our variables and our expressions even, and for the lengths. Next for me is to find the area. So how do you find every rectangle? Good. You knew the height times width, length times width, whatever Triangle, base times height divided by two. So remember that, maybe just write a little note, base times height divided by two . And parallelogram? Excellent so base times perpendicular height. So therefore you need to do find expressions for all of them. And then next, what I would do is I would actually plot them for values of X greater than zero. So maybe you want to go from X equals zero all the way up to so all the way up to like X equals 20, maybe, and then plot them and put them on a graph and just try and decide, Oh, what's the highest, what's the maximum area. What is the minimum area? Does it have an upper or lower bound and things like that. Okay. Have a little go and just see how far you can get with that. Pause in three, two, one. Okay. So we already went through, so this one, the area equals X bracket X plus three. This one, we said that this was X and this was 12 takeaway X. So when you write it as an expression it's X times 12, takeaway X, these were a equals or divide it by two. And this one, this was X. And we had this as three X. So we have X times three X, which equals to three X squared in the area. Okay. So now I have our expressions. Let's go on to plotting it. Pause to check your work carefully. Okay. So with the rectangle, so we had the area equals two X and X plus 3. So we can see, is this a quadratic now we're only interested when X is greater than zero. So it's only really this half a part of the graph here. And what can we see? Let me zoom out. Oh my gosh, it just keeps going. There's no maximum point, it keeps going forever. So actually in this case, this rectangle has no maximum area. What would the minimum area be? Good so the minimum area will be just zero. We know that, well, X is one of the lengths and we can't have a negative length. So X equals zero being just zero and three, but then it's not really a shape at all would be actually the minimum. So that's quite an interesting problem there to have, okay. Back on Desmos again, this is a graphic calculator, free to use, and we're doing X brackets, 12 takeaway X, and this is for the triangle and it's all divided by two and see, I've set it up that, Oh, okay. It's a negative quadratic. Is this what you expected? This is what you drew. Okay. So what was the maximum area? Good, so the maximum areas here, which is 18 and when X equals six. So when we had the base as six and the height as six, then the area is 18 because six times six is 36 divided by two is 18. And what would the minimum be? So minimum is when our X equals zero again, because we're not going to go into negatives. Can't have negative area, can't have negative length. So actually when X equals zero, oh and when X equals 12, it would make the other one equal to zero. So either way when we just have zero and we have nothing, there's no area. That's what we can think of as our minimum area. So well done. If you've got a graph that looks like this. Okay. And finally, the parallelograms, the area was three X squared. If you put it three X squared, it is a quadratic. And it would look like this and we can see, so I put both sides. Really. You only want the positive values of X, or you can see, it just keeps increasing forever. It goes up to infinity. So actually there is no maximum, there's no upper bound there. And the lower bound, the lowest is going to be zero. So zero would be the lower bound for that one. Really well done if you got that. Excellent work, everyone today, I think you should be super, super proud of yourself. This can be a tricky problem. When you think about it carefully and be strategic with actually what the context is. I really think you can understand it, do go back and listen to any bits that you weren't really sure of. And then have it go to the exit quiz, this an ideal time to kind of assess what you know, and get some feedback as well. I've written in some comments to help you kind of move on in your learning further as well. Have a lovely, lovely day. And thank you for listening. Bye!

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

In this lesson, we will explore how we can use quadratic graphs to solve maximum and minimum problems.

Maths

Y9 Maths

Key Stage 3

Year 9

Maximum and minimum area

Quadratic expressions (9.9)

Algebra

THEME

UNIT

Recognising linear and non-linear graphs

Plotting quadratics (Part 1)

Plotting quadratics (Part 2)

Quadratic contexts

Distributive property

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.

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?

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:Maximum and minimum area

Core Content

Quadratic expressions (9.9):