Lesson video

Hello there and welcome to today's lesson.

My name is Dr.

Rowlandson and I'll be guiding you through it.

Let's get started.

Welcome to today's lesson from the unit of 2D and 3D shape with compound shapes.

This lesson is called problem solving with 2D shapes, and by the end of today's lesson, we'll be able to use our understanding of 2D shapes to solve problems. Here are some previous keywords that will be useful during today's lesson.

If you want to remind yourselves what any of these words mean, pause video while you do so and press Play when you're ready to continue.

The lesson contains two learning cycles.

In the first learning cycle, we're going to focus on solving abstract problems and then the second learning cycle, we're going to look at some more contextual problems. Let's start off with abstract problems involving 2D shapes.

Here we have a compound shape.

The compound shape is constructed by joining two congruent rectangles.

The total area of the shape is 48 metres squared and what we need to do is find the perimeter.

We're going to work through this together shortly, but before we do, you may want to pause the video, while you think about what you might do to solve this problem or how you might start this problem and then press play when you're ready to continue.

Okay, I wonder what we thought.

Well Andeep is going to talk us through this problem.

He says, "With geometry problems, "it can sometimes be helpful "to draw additional lines on the diagram." So, Andeep draws a line here where it shows the two separate congruent rectangles.

He then says, "The rectangles are congruent, "so the areas are equal "and the total area is 48 metres squared." That means to find the area of each rectangle, we can do 48 divided by two to get 24.

So, the rectangles have an area of 24 metres squared each and he then says, "With geometry problems, "it can sometimes be helpful to cover up, "parts of the diagram while you focus on other parts." So, for example, we could cover over the right hand rectangle like this and then just focus on the left hand rectangle.

Andeep says, "I could work out the length of this rectangle now," because we can see we have a rectangle with an area of 24 metres squared and a width of four metres squared, so we could use that information to find that missing length by doing 24 divided by four to get six.

With geometry problems, it can also be helpful to write extra information on the diagram such as missing lengths.

We have a few miss and lengths left on this compound shape, but we could write some of them on straight away.

For example, we could write this four metres and six metres on the right hand rectangle, because we know it's congruent to the left hand rectangle.

Then we have two more lengths still to find.

There's the top length which is horizontal.

That is a sum of the other two horizontal lengths.

That would be four plus six is 10, so that top length is 10 metres.

And then we have one last vertical length here.

This vertical length is the difference, between the other two vertical length that we can see, so that'll be six subtract four, which is two, so it must be two metres.

Andeep then says, "I now have all the information I need to find a perimeter." The perimeter is a distance around the outside of this shape, so that'll be 10 plus four plus six, plus two plus four plus six, which gives 32 and in this case it's metres.

A full set of calculations for Andeep solution can now be seen on the screen.

It required five separate calculations, but could Andeep have done this in a slightly different way? How might you have done it differently? Pause the video while you think about this and press Play when you're ready to continue.

Well, Laura has a different idea.

<v ->Laura calculated the same answer 32 metres,</v> but she did it in a different way.

She said, once I found these two limbs are the six metres and the 10 metres, I transform the shape into one large rectangle like this.

She says, I then found the perimeter of this rectangle by doing two lots of six plus 10, which gives 32 metres.

Why does Laura's method give the same answer as Andeep's method? Pause a video while you think about that and then press Play when you're ready to continue.

Well, let's start off with why the length of the right hand edge is six metres.

Here's the sum of the two metres and the four metres that we labelled on earlier and then the length of the base is 10 metres, because it's the sum of the four metres and six metres.

So, the perimeters of the compound shape and its surrounding rectangle are equal.

I show on the other hand, got a different answer.

She got perimeter of 40 metres.

Let's see if can figure out why her answer was different.

She says, I found the perimeter of one rectangle first, so the perimeter of this left hand rectangle is two lots of six plus four, which gives 20 and it's metres in this case.

She then says, I then doubled my answer, because there are two rectangles.

So, two lots of 20 gives 40 metres.

Why does Aisha's method then not produce a correct answer? Pause the video while you think about what went wrong here and then impress Play when your ready to continue.

The issue here is that not all of the perimeter of the rectangle on the left contributes, towards the perimeter of the compound shape in this problem.

What we can see when we look at this rectangle is that there is part of that vertical length on the right, which does not contribute to the perimeter of the compound shape.

That four metres we can see there that's not around the edge of the compound shape that's going through it.

And what we can see with the rectangle on the right is not all of this contributes to the perimeter of the compound shape either.

That width of four metres on the left, does not contribute to the perimeter of the compound shape, because it goes through the middle of the compound shape, rather than being at its edge.

But what we can see in Aisha's calculation is that she's included that length of four metres twice within it, once for the rectangle on the left and also for the rectangle on the right as well, but it shouldn't be included in the perimeter at all.

But once Aisha recognises this issue, she can fix her answer by subtracting eight.

That is two lots of the four metres that shouldn't be counted in the perimeter.

If she does 40 subtract eight, she gets 32.

And that's the same answer as what Laura and Andeep got.

So, let's check what we've learned.

Here we have a compound shape that is constructed by joining two congruent rectangles.

The total area of the shape is 120 metres squared.

What is the area of each rectangle? Pause video while write it down and press Play when you're ready for an answer.

The answer is 120 divided by two to give 60 metres squared.

So, now we know that what is the value of X.

Pause video, why you work it out and press Play when you're ready for the answer.

The answer is 60 divided by 10, which is six.

So, now we have the long length for this compound shape labelled Y.

What is the value of Y.

Pause the video while you work it out and press Play when you're ready for an answer.

The answer is the sum of 10 and six, which is 16.

So, now we have all but one length labelled with a number.

We have the short length which is still unknown, which is Z.

What is the value of Z? Pause video while you work it out and press Play to continue.

The value of Z is, the difference between 10 and six, which is four.

So, now you have all that information, you can calculate the perimeter.

Pause while you work it out and press Play to continue.

The answer is six plus 10, plus four plus six plus 10 plus 16, which is 52 and in this case it's metres.

What you've done there is you've taken a complex problem and you've broken it down to lots of small questions and by answering each small question, we've solved the overall problem.

Well done.

So, let's now look at a different abstract problem.

Here we have a diagram that contains a circle inscribed inside a square.

And what we need to do is find the total area of the shaded regions.

Now, we're going to work through this together shortly, but before we do, perhaps pause the video, while you think about what you might do or how you might get started with this question.

Pause the video while you do that and press Play when you're ready to continue.

Well, Jun's going to talk us through this problem.

Let's hear what he says.

Jun says, "When a problem requires multiple steps, "it can sometimes be helpful to start by writing a plan." So, here's what Jun plans to do.

First he's going to find the area of the square and then he's going to find the area of the circle and finally to find the area of the shaded regions, he will do the area of the square, subtract the area of the circle and what this plan does is it takes our big problem and breaks it down into three smaller problems. And Jun says, "This can help me to focus on one piece of information "at a time while also keeping the final goal in mind." So, let's start working these things out.

The area of the square is eight squared, which is 64.

When it comes to the circle, the diameter of the circle from one side to the other is the same as the length of the square.

So, that's eight metres.

That means the radius of the circle must be four metres, therefore the area of the circle is Pi times four squared, which is 16 Pi.

We could write it as a decimal, but we need to be careful not to round any numbers, until we have our final answer.

Otherwise, our final answer will not be quite accurate.

Then finally, step three to find the area of the shade of regions, will do the area of the square.

Subtract the area of the circle, so that's 64, subtract 16 Pi, and that will give us 13.

7 metres squared and because this is our final answer, that number is rounded and it's rounded to three significant figures.

So, let's check what we've learned.

Here, you have a square, calculate its area.

Pause the video while you do it and press Play when you're ready for an answer.

The answer is 10 squared, which is 100 metres squared.

Here we have a circle.

What is the radius of this circle? Is it A, five metres? B, 10 metres or C, 20 metres? Pause while you choose and press Play when you're ready for an answer.

The answer is five metres.

The diameter is 10 metres, so this must be five metres.

Now you know the radius.

Calculate the area of the circle and give your answer in terms of Pi.

Pause, why do that and press play to continue? The answer is Pi times five squared, which is 25 Pi metres squared.

So, let's put all these pieces together now to solve a more complex problem.

The diagram contains a circle inscribed inside a square.

The area of the square is a hundred metres squared.

The area of the circle is 25 Pi metres squared.

And what you need to do is calculate the total area of the shaded regions, giving your answer to one decimal place.

Pause the video while you do that and press Play when you are ready for an answer.

Here's our answer.

We do 100, subtract 25 Pi to get 21.

5 metres squared to one decimal place.

Okay, it's over to you now for task A, this task contains two questions and here is question one.

You are given two compound shapes and some information about them and you need to use that information to find the perimeter of each compound shape.

Pause the video while you work through this and press Play when you are ready for question two.

Here is question two now.

You have two diagrams which contain a circle inscribed insider square and what you need to do is calculate the area of the shaded regions in each, using the additional information that has given to you.

Pause the video while you do that and press Play when you are ready for answers.

Okay, let's see how we got on with that then.

For part A, here are your two congruent rectangles and your perimeter for the compound shape is 28 metres.

For part B, here are your three congruent rectangles and the perimeter for the compound shape is 36 metres.

And then question two, for part A, you're given the length of the square.

If you use that to find the area of the shaded region, you should get 30.

9 metres squared.

And then part B where you're given the perimeter of the square.

If you use that information, you should finally get an answer of 1.

9 metres squared.

Great work so far.

Now, let's apply what we've learned to solve some contextual problems. Here we have Izzy.

Izzy is calculating the cost to decorate a room.

The diagram shows the shape and the measurements of that room.

She wants a new carpet and to paint the walls and ceiling the same colour.

So, she finds out some details about these things.

The paint is sold in 2.

5 litre tins and each paint tin costs 22 pounds.

One litre of paint covers 10 square metres and then the carpet costs 30 pounds per square metre.

How could Izzy calculate the total cost to decorate the room? Perhaps pause the video, while you think about how you could take this big problem and break it down into lots of small problems and perhaps write a plan.

Pause video while you do that and press play when you're ready to continue.

Well, let's hear from Izzy.

She says, before I start, I'll write a plan for my solution.

And here's a plan.

You gotta start by finding the area of the floor and then she'll work out the cost of the carpet.

She'll also work out the area of the ceiling and the area of the walls and then the cost of the paint and finally the total cost for it all together.

So, let's work through that plan now.

Izzy says I'll start by finding the area of the floor.

There are multiple different ways that I could do this.

Here's one way.

We could split the room into two rectangles, A and B.

The area of rectangle A is 2.

4, multiply by three, which is 7.

2 metres squared.

The area for rectangle B is 1.

5 multiplied by 1.

8, which is 2.

7 metres squared.

So, that means the total area of the floor is 7.

2 plus 2.

7, which is 9.

9 metres squared.

Izzy says, "Now I know the area of the floor, "I can calculate the cost of the carpet." So, the area of the floor is 9.

9 metres squared and the cost of the carpet is 30 pounds per metre squared.

So, that means we can do 9.

9, multiplied by 30 to get 297 pounds.

Izzy then says, "I'll now start working out "how much surface will be painted." So, we know that the air of the floor is 9.

9 metres squared.

Let's think about what the air of the ceiling would be.

She says the air of the ceiling is equal to the area of the floor, so that means the area of the ceiling is 9.

9 metres squared.

Let's now find the total area of the walls.

Each wall is a rectangle with height 2.

4 metres.

We could visualise this problem by drawing a rough sketch of what a wall might look like, coming off each of those edges of the room.

For example, this wall would have a base of 1.

8 metres and a height of 2.

4 metres.

So, that means its area would be 4.

32 metres squared.

This wall will have a base of 3.

9 metres and a height of 2.

4.

We multiply those, we get 9.

36.

This one we'll have a base of three metres and a height of 2.

4, so that'll be 7.

2.

This wall would be 2.

4 by 2.

4, which is 5.

76.

This wall would be 1.

2 times 2.

4 to get 2.

88 and this wall would be 1.

5 times 2.

4 to get 3.

6.

So, we can see the area of each separate wall, that room to get a total area, we could add them all together to get 33.

12 metres squared.

Now, that method required quite a lot of different calculations, but Izzy says the same answer could have been obtained by multiplying the perimeter of the room by the height of the walls.

The perimeter of the room is the sum of all its edges, which is 13.

8 metres.

And then if we multiply that by 2.

4, we get once again 33.

12 metres squared.

Izzy reminds us that she's using the same paint to cover the walls and the ceiling.

So, the air of the walls is 33.

12 metres squared.

The area of the ceiling is 9.

9 metres squared.

So, the total area to be painted is the sum of those two numbers, which is 43.

02 metres squared.

So, now we need to work out how many tins of paint Izzy will need to cover all that surface.

And here's a reminder of those details from earlier.

One litre of paint covers 10 square metres.

Paint is sold in 2.

5 litre tins and each paint tin costs 22 pounds.

And a total area to be painted is 43.

02 metres squared.

Now, there are multiple different ways you can do this, but one way could be to by working out how many litres of paint you need by doing 43.

02 divided by 10.

That gives 4.

302, so that means we need 4.

302 litres of paint to cover the area that we have to cover.

So, now we know how many litres of paint we need.

We can work out how many tins of paint, would give us that amount.

We could do 4.

302 divided by 2.

5, because each tin contains 2.

5 litres.

That would give an answer of 1.

7208, but you can't buy 1.

7208 tins in a shop.

We need to buy two tins.

So, to get a cost of paint, we can multiply two which is the number of tins by 22, which is a cost for each tin and that'll give 44 pounds.

So, we know the cost of the paint is 44 pounds, we know the cost of the carpet is 297 pounds.

So, now Izzy says, "I can calculate the total cost to decorate the room." We can add together two numbers that we've worked out to get 341 pounds.

Izzy then says, "My original plan can also be used "as a breakdown of the costs." For example, here was our plan and all the numbers are there next to each of those stages, so that if Izzy looks back at this or anyone else looks at it, they can follow where the answer has come from.

And what we can see here is quite a common thing that people get when they buy in the services of, for example, a trades person.

If Izzy hired a decorator to come and decorate her room, that decorator would not only give her a price for how much the whole thing would cost, they would usually give a breakdown of costs as well, so that Izzy can see why the amount is what it is.

So, let's check what we've learned.

Here we have a compound shape, which calculations would find the area of this shape and it may be more than one correct answer.

Pause the video while you choose and press Play when you're ready for answers.

The answers are A, B and D.

Here we have Sofia who wants to buy carpet for a room.

The area of the room is 20.

24 square metres and the carpet she wants costs 25 pounds per square metre.

How much would a carpet cost? Pause the video while you write your answer down and press Play when you ready to see what the answer is.

The answer is 25 multiply by 20.

24, which is 506 pounds.

The perimeter of this room is 20 metres and the walls are three metres high.

So, what is the total area of the walls? Pause the video while you work it out and press Play when you're ready for an answer.

The answer is 20 times three, which is 60 square metres.

Sofia is using the same paint for the walls and ceiling and one litre of paint covers 10 square metres of surface.

The area of the walls is 60 metres squared, the area of the ceiling is 20.

24 metres squared.

How many litres of paint is required? Pause the video while you work this out and press Play when you're ready for an answer.

To get our answer, we would add together 60 and 20.

24 and then divide by 10 to get 8.

024 litres.

Sofia needs 8.

024 litres of paint and paint is sold in 2.

5 litre tins.

Therefore, Sofia needs to buy three tins of paint.

Is that true or is it false and choose a justification from below.

Pause the video while you do that and press Play when you're ready for an answer.

The answer is false.

And when it comes to the reason why, yes, when we divide 8.

024 by 2.

5, we get 3.

2096, which does round down to three.

But if you only bought three tins, she wouldn't have enough.

She wouldn't have 8.

024 litres of paint.

Therefore, she'll need to buy four tins and have a little bit of paint left over.

Okay, so it's over to you now for task B.

This task contains one question and here it is.

Alex is calculating the cost to decorate the room shown in the diagram.

He wants new carpet and to paint the walls and ceiling the same colour.

You're provided with information that will be helpful with this problem.

And what you need to do is use information provided to calculate the total cost of the materials required.

Now, I recommend breaking this problem down into lots of small steps.

Maybe even start by writing a plan that will guide you through the problem.

Pause the video while you work through this and then press Play when you're ready to see the answer.

Okay, let's see how we get on.

The final answer is 744 pounds 80.

Now, if you didn't get that, don't worry about it too much, because here's a breakdown of where that number came from.

What you can do is compare each part of your working with all the numbers you can see on the screen here to see where your solution differed from the answers and then you could rework out that particular part and make any other fixes you need after it to get the correct answer afterwards.

Fantastic work today.

Now, let's summarise what we've learned.

When solving problems, consider whether any elements are familiar from other aspects of mathematics.

And also when solving problems, it's very important to try and keep the goal in mind, all the way through the problem, because it's easy to get distracted by too much information and for that reason, it can sometimes be helpful to write a plan, before you start solving a problem, so it can guide you through it or break the big problem down into lots of small questions which are all each a bit more manageable.

And also don't forget that when you are solving problems with geometry, it can sometimes be beneficial to add information to 'em, such as drawing extra lines or writing measurements on that are not provided to you in the first place, but you've worked out along the way.

Thank you very much today.

Have a great day.