Lesson video

Hello there, my name is Mr. Tilston.

I'm a primary school teacher and I really love maths.

So it's my great pleasure and great delight to spend this math lesson with you, and we're going to be talking about perimeter.

If you're ready, I'm ready.

Let's begin.

The outcome of today's lesson is I know that unknown side lengths can be calculated from the perimeter and known side lengths.

Our keywords, we've just got the one, my turn perimeter, your turn.

Hopefully you know what this word means by now, hopefully you've been exploring it quite a lot recently.

But let's have a reminder, the distance around a two dimensional shape is called the perimeter.

Here's a rectangle, and that is its perimeter.

Now you might have explored that recently using counting, adding, doubling, that's perimeter.

Our lesson is split into two cycles, two parts, the first will be finding missing lengths in rectangles and the second problem solving with rectangles.

So if you're ready, let's start by finding missing lengths in rectangles.

In this lesson, you're going to meet Laura and Lucas.

Have you met them before? They're here today to give us a helping hand with our maths and very good they are too.

With the exception of squares, which is a special kind of rectangle, all rectangles have two long sides and two short sides.

When the four lengths are added together, they give the perimeter of the rectangle so your might have explored adding to find the perimeter before with rectangles.

So we've got a bar model here, and this is showing the long sides, are two long sides and the two short sides of a rectangle.

Here we go.

So two long sides and two short sides.

It might not be that exact same size, but we would have two long sides and two short sides.

Might be a smaller rectangle in a different orientation, and again.

What is known and what is unknown about this rectangle? So let's have a look.

What do you know? What can you work out quite easily and what do you need to spend a bit more time working out? Let's have a look.

Well, one of the things that we know because we've been told is the perimeter of the rectangle, it's 24 units.

What else do we know? We know the length of each of the long side.

It's only given one, but we know them both because they're opposite each other and they're in a rectangle.

So they must both be eight, eight units.

So that was quite easy to figure out.

Now what we don't know is the length of the short sides, but we have got enough information to calculate it.

How could we do that? Have you got any ideas? But we do know the total of the two long sides, we know eight at eight, that's a double.

So the long sides total 16 units.

So that's something that we know.

What about the short sides? What must they total? Well, the difference between 16 units, that's the long sides and 24 units, that's the perimeter, will be the sum of the short sides.

So therefore 16 plus something equals 24.

16 plus what equals 24? Maybe you could use bridging to work that one out.

Or you might even think of it as a subtraction, 24, take away 16 equals something.

Either way the answer is eight.

So in total, in total, the short sides total eight units.

That still hasn't answered what one of them is.

What could we do now? What's the length of one short side? There's two short sides.

So halving eight will give us the length of one of them.

Half of eight hopefully is quite easy for you to do.

Half of eight is four.

So the short sides have got a length of four units.

What's different about this example here? Have a look at this.

What do we know this time? And what don't we know this time? Well, we know the perimeter again, it's different though, it's 20 units.

This time though, both of the short sides are known, one's given, but we know the other one 'cause it's the same.

So the short sides total six, three plus three equals six.

And we can use that same strategy to work out the missing side length.

It's just that this time it's the long sides.

So the short sides total six units.

So six plus something equals 20, you could do it that way or 20 takeaway six equals something.

Both of those will give the answer, see which one you like.

I like counting on, I'd probably use that first one.

That's 14 either way.

So that means that the long sides total 14, both of them are 14 altogether, so how can we find out the length of one long side? We half it.

Half of 14 is what? So half of 14.

Seven, because double seven is 14.

With use our doubles, our known doubles.

So that is the length of the long side.

So it takes a few steps, but we can work out the missing side length when we know the perimeter and one of the side lengths.

That's all we need.

Let's have a check, what is the length of each missing side? So you know the perimeter is 22 units and you know one of the sides, the short side is five.

Pause the video and give that a go.

Let's see, what strategy did you use there? Well, there's something I know straight away and that's the short side is five because they're opposite each other.

What do the short sides total? 10 units.

What can we do now? We've got two numbers there that we can work with, 22 the perimeter and 10 the total of the short sides.

Well, we need to find out the difference between those numbers using either counting on or subtraction.

By the way, we get 12.

So 10 plus 12 equals 22 or 22, takeaway 10 equals 12.

So the long sides total 12, almost there, one last step.

We need to halve it.

Half of 12 is six.

So the long sides are each six units in length.

Well, then if you got that, you are on track.

A rectangle has a perimeter of 40 centimetres and a long side of 12 centimetres.

So we know the perimeter again and we know the length of one long side.

What's the length of each short side? Now this time you might notice you haven't got the rectangle, you've got to visualise it, picture it and use the information that you've got.

All right, pause the video and give that a go.

Let's see.

Well, if one long side is 12 centimetres, the other long side is 12 centimetres.

The long sides total, that's double 12, hopefully quite straightforward, that's 24 centimetres.

Then 24 plus something equals 40 or 40 takeaway 24 equals something.

Either way you were going to get the same number, so choose your method.

16.

So the short sides total 16 centimetres altogether.

One last little step, what do we do with that number 16? We halve it.

Half of 16, you should double.

So double what makes 16? That's eight.

So the short sides are each eight centimetres.

Very well done if you've got that.

I think you are ready for some practise now, let's see if you can put to practise all that you've been learning.

So number one, the perimeter of this rectangle is 30 centimetres.

What is the length of each short side? Now in this case you can see a long side's been given, 10 centimetres, but the opposite long side's been given, is 10 centimetres as well.

Number two, the perimeter of this rectangle is 38 centimetres, what's the length of each long side? Now only one of the short sides has been given this time, that's the only other piece of information you've got.

Number three, complete the table to give the missing side lengths in these rectangles.

So the perimeter has been given.

We've got one that's 21, one that's 26, one that's 34, one that's 40, and one that's 400.

And then some sides have been given.

It might be that one of the long sides have been given or it might be that one of the short sides have been given.

So you've gotta work out the rest of that information and complete the table.

Number four, the floor of the classroom has a perimeter of 40 metres.

So all together all the way around 40 metres.

The long side of that floor has a length of 12 metres.

So it's a rectangular floor, the long side is 12 metres, what's the length of the short side of the floor? And then Lucas says my rectangle, the one you can see has a perimeter of 30 centimetres and a short side with a length of eight centimetres.

Is that possible? Can you explain why? Good luck with that and I'll see you soon for some feedback.

Welcome back, how did you get on finding those missing side lengths? Let's have a look.

So the perimeter of this rectangle, we know the long sides are each 10th of the total, 20 centimetres.

20 plus something equals 30, hopefully that's pretty straightforward for you, that is 10.

20 plus 10 equals 30.

And one last thing to do, what we're going to do with that number 10, the two short sides total 10? So we half it, half of 10 is five.

So the short sides are each five centimetres in length.

Number two of the perimeter is 38 centimetres, what's the length of each long side? Well, we've only been given one short side, but we know the other one 'cause it's the same.

So the short sides total eight centimetres, they're both four centimetres.

And eight plus something equals 38.

That's 30.

Half of 30 is 15.

That means each long side is 15 centimetres.

And this table here is what it looks like, so if it's got a perimeter of 20 centimetres, that means it's got a long side of eight centimetres, that was given.

So the other long side is eight centimetres, and then the two short sides must be two centimetres.

If the perimeter is 26 centimetres and the long side is nine centimetres, then the other long side is nine centimetres and the short side are each four centimetres.

If the perimeter is 34 centimetres, we know that one of the short sides is five.

So the other short side is five and the long sides are each 12.

And the perimeter are 40 centimetres, if one of the short sides doesn't matter which was eight, that means the other short side is eight and the long sides are 12.

And if the perimeter is 400 centimetres, well, that's got a relationship with a rectangle that's got a perimeter of 40 centimetres.

So we can use some of that information as a starting point to help.

So in this case then the long side is 120 centimetres each and the short side, 80 centimetres each.

And for number four, a rectangle was described but not shown, so you might just sketch your own.

And you might have looked a little something like this, so we know that the long sides are 12 metres, so therefore they total 24 metres.

24 plus something equals 40 or 40 takeaway 24 equals something, that's 16.

Half of 16 is eight.

So therefore the short sides are each eight metres.

Well done if you've got that.

Number five, Lucas says my rectangle has a perimeter of 30 centimetres and a short side with a length of eight centimetres.

Is it possible.

No.

It's not.

The short sides would total 16 centimetres in this case, meaning that the long sides would total 14 centimetres.

That would make them the short sides, wouldn't it? The eight centimetres could be the long side however, and the short sides would then be seven centimetres.

So he might have just got his long sides and his short sides mixed up.

But you're doing really well, let's kick it up a notch, shall we, let's do some problem solving with rectangles.

What could the side lengths be for a rectangle with a perimeter of 20 centimetres? Hmm.

So we haven't been given any of the side lengths this time.

If the perimeter is 20 centimetres, then neither of the length and the width together is 10 centimetres because 10 centimetres plus 10 centimetres equals 20 centimetres.

If I chose six centimetres as the width, then the length must be four centimetres.

That's one possibility, isn't it? Can you think of any others? I know that the opposite sides are equal, so six plus six plus four plus four equals 20 centimetres.

So that's a possibility.

Six centimetres and four centimetres.

Here we go.

So that's a six centimetre side, and a four centimetre side.

So the length and width must equal 10 centimetres, we've established that, and I think there's another possibility too, don't you? I can think says Laura, of other pairs of numbers that sum to 10.

So can I.

How about eight plus two? That's another number bond to 10.

Eight centimetres plus two centimetre, equals 10 centimetres, yep.

The sides could therefore be eight centimetres, eight centimetres, two centimetres, two centimetres, that would equal 20 centimetres for the perimeter.

Here we go.

Eight centimetres and two centimetres.

What about this one, five centimetres plus five centimetres? Hmm.

What's going to happen here? It does equal 10 centimetres, it's a number bond to 10.

So the length and width could both be five centimetres.

What would happen? The sides would be five centimetres, five centimetres, five centimetres and five centimetres making 20 centimetres.

And that is what would happen.

What have we made? Oh, says Laura, that's a square, not a rectangle.

Hmm.

And the challenge was a rectangle with a perimeter of 20 centimetres.

So has Laura succeeded or not? What do you think? Lucas says, "No, no, Laura.

It counts.

A rectangle has pairs of opposite sides that are equal in length just like a rectangle." So a square is a special type of rectangle.

So in fact she has succeeded.

Well done Laura.

What could the side lengths for a rectangle with a perimeter of 20 centimetres be? Well, here's the three examples we've looked at so far.

And it's all a bit random though, isn't it at the minute.

And Lucas thinks, I wonder if we could be a bit more systematic about this so that we can find all of the possibilities rather than randomly picking them.

Hmm.

What could he do? Any ideas? We could record the results in the table he says, yes, good tables are always good.

Let's have a look, where could we start? Lucas says, let's start with the smallest possibility.

Yes, great thinking Lucas, the smallest possibility.

That's good systematic thinking, which is one centimetre.

So the width could be one centimetre.

What would that make the length then? Nine centimetres, they make 10 altogether.

And the calculation here is one plus nine, then doubled, which is 20.

All right, see if you can follow his pattern, what will come next? Lucas says we can keep working up.

So what's next? And we've tried one centimetre, what next? Two centimetres.

Two and eight.

Double makes 20.

What next? He can see some patterns, says Lucas, so can I.

Hopefully you can tell what's gonna come next.

Three and seven, double mix 20.

Can you see what we're doing? So the width has gone one, two, three, then we've got four and six and five and five.

And now we've got all of our possibilities, rather than choosing them randomly, we've been systematic.

And Laura says no need to try six because we've already used that as a side length.

Yes, we have.

So if we used it again, it would just be repetition, it wouldn't be a new one.

That's all the possibilities.

Time for a check.

This time the perimeter of the rectangle is 18 centimetres, and that's been explored with a table.

Can you spot the errors? There's more than one, so I'd be very eagle eye here.

All right.

Pause the video and give that one a go.

Let's see, what is your spot? What about this one then? Two and six.

That should be two and seven.

Two and seven make nine, which doubled is 18, so not six.

A possibility has been missed out, did you spot that? So it went width one, width two, but where it was width three? It was missing.

So that one should have been included, so three plus six, then double makes 18.

And then five and four.

That's correct, but it's not a new one is it? We've already had that one.

So it's not that it is wrong, it just wasn't necessary.

Time for some more practise.

So number one, give all of the possibilities for the side lengths of a rectangle with a perimeter of 24 centimetres.

The first one has been done for you.

Sketch a rectangle for each of your possibilities.

So we've got width could be one, length could then be 11, one plus 11, doubled makes 24.

So what are you going to try next for the width? So that you've got all of the possibilities.

Number two, this time the rectangle has a perimeter of 26 centimetres.

What whole numbers could the length and width be? So a, how many possibilities do you predict there will be? And why.

So think about the information you've already got from before, from question one, and then draw a table to show all of the possibilities, was your prediction correct? Have a good think about that before you make that prediction and pause the video and I'll see you soon for some answers.

Welcome back.

How did you find that? Let's have a look.

You had to do a bit of thinking there, didn't you? That's good.

So these are the possibilities for a rectangle with a perimeter of 24 centimetres.

So the first one was already given, so it could be two in 10.

Double makes 24, three and nine, double makes 24.

Four and eight, double makes 24.

Five and seven, doubled makes 24.

And six and six which will make a square, doubled is 24.

And you sketched rectangles.

Mine have looked a little bit like this.

They are just sketches, so don't worry if you're not too accurate.

Number two, these are the possible whole number widths and lengths for a rectangle with a perimeter of 26 centimetres.

So we've got one and 12, double these, 26, two and 11, three and 10, four and nine, five and eight, six and seven.

Is that what you predicted? Or did you predict something different? It has the same number of possibilities, I'll bet you didn't think that, I bet you thought it might have more.

As the rectangle with a perimeter of 24 centimetres.

This is because 24 centimetres had a possibility where the shape was a square.

So there's six centimetre length and six centimetre width, and this one doesn't.

So it kind of cancels out.

Well, we've come to the end of the lesson and I've really enjoyed doing all that thinking, that mathematical thinking and reasoning there.

Today's lesson has been knowing that unknown side lengths can be calculated from the perimeter, and just one known side length.

It's possible to work out the missing side length in a rectangle if the perimeter and one side length are known, and the example on the right shows this.

So we know the perimeter is 36 centimetres, we know the short side in this case is four centimetres, so the other side, short side is four centimetres and that makes eight centimetres and then eight plus something equals 36.

Well, eight plus 28 equals 36.

So that means a long side total 28, half of that is 14, and that's one short side.

If only the perimeter of a rectangle is known, there are a wide range of possibilities for the side lengths, as long as the pairs of sides total the perimeter and they can be found as we did do systematically, perhaps using a table, but certainly starting with the lowest possible value.

Well done on your achievements today.

Give yourself a pat on the back.

You've been great.

I hope I get the chance to spend another math lesson with you in the near future.

But until then, have a great day.

Whatever it is that you are doing, take care and goodbye.