Lesson video

Hi there.

My name's Ms. Lambell.

You've made such a super fantastic choice, decided to join me today and do some maths.

Come on, let's get started.

Welcome to today's lesson.

The title of today's lesson is, "Securing Understanding Of Rounding," and that is within our unit, estimation and rounding.

By the end of today's lesson, you'll be able to recognise when values have been rounded and you'll also be able to state sensible suggestions of what that unrounded value may have been.

Some key words that we'll be using in today's lesson are, conjecture and degree of accuracy.

A conjecture is a mathematical statement that is thought to be true but has not yet been proved.

A degree of accuracy shows how precise a number or a measurement is.

So for example, we could measure something to the nearest centimetre, we could give a value rounded to the nearest 10, we could round a value to one decimal place.

These are some examples of different degrees of accuracy.

Today's lesson, I've decided to split into two separate learning cycles.

In that first learning cycle, we're going to concentrate on the unrounded numbers, and in the second one we will concentrate purely on the degrees of accuracy.

Let's get started on that first ending cycle, unrounded numbers.

Here we go.

Jun and Sofia are talking about these facts.

Let's take a look at the facts they're looking at.

The first one, the attendance at the 1996 World Cup was 97,000.

The government spent 172 billion on grants in 2021 to 2022.

Steps taken to get to the moon, 400 million.

And the population of the UK when I researched this lesson was 67.

33 million.

Jun says, "Do you think these are exact values?" So he's asking Sofia, "Do you think those were the exact values given in each of those statements, in each of those facts?" Sofia says, "It seems very unlikely." Do you think they are the exact figures? Like Sofia says, it does seem very unlikely that they are the exact actual figures in each of those situations.

In certain situations, it is not necessary to know the exact value unless we're going to be working within a given degree of accuracy.

Attendance at the World Cup in 1966 was 97,000.

Jun says we did not need to know the exact value to get a sense of the number of people that were at the match.

That's right, isn't it? It wouldn't matter if it was 97,001, 97,010.

The 97,000 gives us a really good idea of how big the attendance was.

Sofia's think a little bit further than that.

So she's saying that she thinks that the number is probably, and remember here it's just a conjecture.

It's probably given to the nearest thousand.

So it's something that she thinks is true, but it's not yet been proven.

Do you agree with Sofia? Do you also think that the attendance at this football match has been given to the nearest thousand? What did you think? It certainly looks as though it has been rounded to the nearest thousand because our integer ends in three zeros.

We'll stick with the same figure, 97,000.

Jun now says, "There must have been more than 97,000 people at the match." We've decided that we think that this has been round to the nearest thousand, but Jun is saying that must mean if we've rounded it to the nearest thousand that there were more than 97,000 people at the match.

Sofia says, "Jun, remember when we drew out number lines, we went to the multiple above and below the value." Sofia's remembering a previous lesson that she did where she was looking at rounding numbers and using a number line and she wrote the multiple below and the multiple above, and we could actually round values up or down.

Here on our number line, we've got 97,000.

The other side would be 96,000 and 98,000.

There are some values that start with 96,000 and something that could potentially round to 97,000.

We could make a conjecture about the exact attendance and this would just be a conjecture because it would be something that we think could be true, but we've not yet proven.

For example, we could have an attendance of 96,782, that would round to 97,000 or we could have a attendance of 97,005, that also to the nearest thousand rounds to 97,000.

We could even go so far to saying we've got an attendance of 97,499.

That would still round to 97,000.

We can make a conjecture about what those values might be, but we would have to look up the information to find out the exact attendance at the football match.

Let's take a look at the other three statements.

The government spent 172 billion on grants in 2021 to 2022.

Steps taken to get to the moon, 400 million.

Population of the UK, 67.

33 million.

What do you think each of the above has most likely been rounded to? Pause the video, have a think about those three statements and what you think they may have been rounded to and come back and we'll see how you've got on with that.

Super.

What did you think? I think it looks like this has been given to the nearest billion.

The government grant was probably not exactly 172 billion, so it's been given to the nearest billion.

The next one has been given to the nearest hundred million.

The third one, now I think that one's a little bit tricky because it's got that word million written on the end.

It makes you think that maybe it's been measured to the nearest million, but I'm not sure that that's true.

Let's take a look at what that's going to look like in a place value grid.

So we are looking at 67.

33 million.

So our decimal point is going to go after our millions column.

Now we can put in our 67.

33.

Remembering we need the placeholder zeroes in the other columns.

We can now see that the final non-zero digit is in the 10000s column.

Therefore it seems as though this has been rounded to the nearest 10,000.

Obviously we don't know exactly, but that would be what we would reasonably consider what is most likely.

Jun is thinking of a number.

He uses rounding to give clues.

After each clue Sofia writes down her guess of the number.

I want you to think about, will her answer change each time? Actually, what I'd really like you to do is to have a go at this task with Sofia.

I'm going to read the clue, then I'm gonna give you a moment to write down a number that you think works and then we'll take a look at what Sofia has written.

Here's the first clue.

It is 5,000 to the nearest thousand.

Write down a number for me that is 5,000 to the nearest thousand.

Sofia guesses 4,800.

Wonder if you guessed the same thing.

It would be a bit freaky if you wrote exactly the same thing I think, but you could have.

Let's go on to clue two.

It is 4,500 to the nearest hundred.

I'd like you to write down a number.

Has your number changed or were you able to use the same number as you had written previously? Sofia is going to have to change her guess.

Let's take a look at clue three.

It is 4,530 to the nearest 10.

Do you have to change your original guess? If you do, change it now, and let's take a look at what Sofia's written.

Sofia, her previous guesses will not work for this one.

She's now changed her guess to 4,528.

And the final clue.

It is 4,528 to the nearest whole number.

Can you write down a number that is 4,528 to the nearest whole number? Now actually Sofia's guess works here.

Her previous guess, but she could, if we'd known that actually this was not an integer value, then we could, we'd know that we needed some values outta decimal point.

And Sofia here has made a guess of 4528.

4.

Did you manage to pick one number that was correct for all of those four clues? I'd be very surprised if you did, but if you did, well done.

Jun's homework has been damaged by his dog.

His answers are still there, but some of the numbers are missing from the questions.

What could the missing numbers be? 192 rounded to the nearest something, that's the bit that the dog's eaten, naughty dog, is 190.

What do you think is the missing part of Jun's homework? Let's take a look.

Well, we generally round to the nearest 10, 100, 1000, et cetera.

Here we don't need to think about decimal places because we are looking at a whole number, 192.

So out of those 190 is only a multiple of 10, therefore it must be 192 that's been rounded to the nearest 10.

That was the only thing that could have been missing.

Let's look at the next one.

43 to the nearest something is zero.

It cannot be 10 because 43 would round to 40.

We know 43 rounds to 40 from our previous learning.

It could however be 100 or any power of 10 greater than 100.

It could be 100, 1000, 1 million.

It could be lots of things, but it's probably most likely to have been 100.

And the next one.

Nine, and then there's a bit there the dog's ripped off, rounded to the nearest thousand is 1000.

What are your thoughts here? Let's have a look on a number line.

We've got 1000.

What's the multiple of a thousand that is lower than that? It's going to be zero.

Halfway is 500 and we know that it's got to round to a thousand.

It's got to be between 501,000.

But we know it starts with a nine.

So for example, it could have been 931.

It could be any three digit number starting with nine as these are all to the right of 500, which is the halfway point.

So we know that anything to the right of 500 would round to 1000.

We already know that the number started with a nine.

So it could be any two digits after that nine and that would round to 1000.

What about this one? 295 rounded to the nearest something is 300.

I'll give you a moment to think about that one.

300 is a multiple of 100.

So therefore 295 rounded to the nearest hundred is 300.

It's always worth doing a double check.

Now we've revealed the value, or sorry, here the word is a hundred, just check is 295 rounded to the nearest hundred 300 and it is.

And this one.

We've got a bit of ripped paper.

And then the number eight rounded to the nearest hundred is 200.

Again, gonna give you a moment to have a think about what you think the dog might have chewed away.

Let's take a look at this on our number line, what 200 and we've got 100.

150 is in the middle.

If we're rounding to 200, we know it has to be between 150 and 200.

So it could be 168.

Is 168 the only value that we could have put there? What other numbers could it have been? It could have been any of these.

Now I'm not going to read them all out, but notice some of those values are over 200.

We know that anything between 200 and 250 but not including 250 would also round to 200.

Sofia wants to go back and take a look at this question again.

So it was 295 rounded to the nearest and we decided that the bit that had been chewed was the word hundred.

So she's saying now, "You said that the missing word was hundred, could it have also been 10?" What do you think? Do you agree with Sofia? Let's take a look.

Here's my number 295 and I've decided this time to use my place value grid.

Sofia is saying, "Could it not have been rounded to the nearest 10?" Let's round 295 to the nearest 10, we identify the digit in the tens column, we look to the column immediately to the right of this and it's five.

Remember if it's exactly halfway, which will mean there is a five in the next column, then we round up to the larger value.

So, Sofia is quite right.

It could also have been rounded to the nearest 10.

The reason we probably went through a hundred is because we looked at the size of the bit that had been cut out by the dog.

But Sofia, well done spotting, that actually some of these questions have more than one answer.

Now we can have a quick check for understanding.

Let's have a look at what you're gonna do.

So, what number am I? So it's very similar to what we did earlier where I asked you to write down some answers.

And also Sofia joined in.

You're going to get five clues.

I'm going to reveal each clue one at a time.

And after each clue, I'd like you to write down a number.

When you get the next clue, you are fine to choose a different number if your previous number no longer works.

Let's get going.

Clue number one.

When rounded to the nearest thousand, I am 2000.

You got something written down? Great.

Now you may need to change your answer, which is absolutely fine.

Clue two.

When rounded to the nearest hundred, I am 2000.

Did you need to change your number or could you stick with the previous number? As long as you get the right answer, it doesn't matter how many times you need to choose to change your answer.

Clue three.

When rounded to the nearest 10, I am 1,980.

Make sure you've got something written down.

Clue four, I am a multiple of five.

Think about what we know about the ones column for any multiple of five and make sure that your number satisfies this.

And finally, clue number five.

I am not any of the rounded values.

This means that I am not 2000 or 1,980.

Now there should only be one answer that you could possibly have written down.

And it doesn't matter whether I'm talking to just you or you and your friends or you and your neighbour in the classroom.

Every single one of you will have written down the same number if you've managed to follow my clue successfully.

And that number is 1,975.

Did you get that? Now you can have a go at your first independent task.

So there are a few different questions and they're on different slides.

So I'm going to read the clues to you and then I'm gonna ask you to pause the video and then come back when you're ready.

Sofia's thinking of some numbers.

She uses rounding to give two clues.

After each clue, she guesses her number and writes it down.

Does your guess change each time? So you're going to write down a number after clue one, try not to look at clue two and then you are going to put down a different answer or the same answer if you can when you get to clue two.

Pause the video, have a go at this one and then come back when you're ready.

And part B, two new clues, here you go.

Pause the video and then come back when you're ready.

And part C.

Ooh, didn't change a lot there, I have changed something.

And question number two.

So Jun's dog again has been extremely naughty and has again managed to chew up Jun's homework.

Your job is to work out what's been chewed off of Jun's homework.

Pause the video.

Come back when you've got those answers and I'll be here waiting for you.

Good luck.

That's great.

And question number three, are these statements always, sometimes or never true? A number rounded to the nearest 10 will always have two digits.

A number rounded to the nearest 10 will end in one zero.

A number rounded to the nearest 10 will be less than five away from the original number.

Give some examples to show why.

Decide if it is always true, sometimes true or never true and also give me those examples.

Pause the video now, come back when you're ready.

That's superb work, well done.

Let's check those answers.

Here they are just examples.

Okay, so I've got 282 and 292.

Again, yes, they did change the next one, again, they're just examples.

You may have different things.

And for this one, Sofia's did change and for the final one, Sofia managed to guess the correct answer straight away.

You'll probably have different values to those.

Question number two then.

For A is 100, B, 1000.

C, an example could have been 1,785.

D, you could have 1000 or you can have 100 or you could have 10 and E, 397.

And all of those statements are sometimes true and if you want to have a look at my examples there, you can pause the video and read those and un-pause when you are ready.

You will have things that are very similar but not exactly the same probably.

Now we can move on to the second learning cycle in today's lesson.

And that is degrees of accuracy.

As we mentioned in the first learning cycle, the degree of accuracy we choose is often dependent on the context.

Here we had our different statements and I've slightly changed one of those and that's the steps taken to get to the moon.

I've actually now written it to its nearest million rather than its nearest a hundred million.

Here, the context of the question helped us to decide what would be an appropriate degree of accuracy.

Steps taken to the moon.

Well that's a lot, loads and loads and loads so it'd be really silly to give that to the nearest 10 or a hundred.

So here we've given it to the nearest million.

The attendance at a world cup, if we gave it to the nearest million would be zero.

So that wouldn't be very sensible either.

So you can see that the context often helps us to decide what would be a suitable degree of accuracy.

Over the next sequence of slides, I'd like us to think about what would be an appropriate degree of accuracy.

Let's have a look and see what Sofia's got to say.

She's saying that we're going to write down the number of pupils at my school.

This is a sort of situation where actually the exact number of pupils could be considered appropriate, but also we could rank the number of pupils to the nearest 10 and that would also be appropriate.

Let's take a look, another situation.

What's Sofia asking us to look at now? The population of her village.

What are your thoughts here? This would probably be sensible to the round to the nearest hundred.

It would give us an idea of the size of the village that Sofia lives in, whether it's a small village or a much larger village.

The number of people visiting a music festival.

What would it be here? Here it would probably be sensible to give this to the nearest thousand just like we did with the football attendance at the World Cup.

It gives us a really good idea of how big the crowd is without giving the exact value.

The same thing here.

It would give us an idea at the size of the festival.

It often also depends on the units that we are using to measure.

All of those values we've just looked at were values that could only be given as integers.

But when we're measuring we are looking at data that can be given as decimals.

Sofia's height, what do you think the smallest unit of length we could reasonably measure Sofia's height to? What do you think that would be? The smallest unit of length we could reasonably measure her height to be.

And I think that would be the nearest centimetre.

Therefore what would be an appropriate degree of accuracy if we measured her height in centimetres? So if we measure in centimetres, then we would write her height to the nearest centimetre.

So for example, she could be 135 centimetres in height.

But what about if we decided to actually measure her height in metres? What degree of accuracy would be sensible here? Would it still be the nearest integer? No, actually because we know that we can measure to the nearest centimetre and the centimetre would be in the second decimal place.

If I was to stick with a height of 135 centimetres, we would write this as 1.

35 metres.

So we would write it correct to two decimal places.

What about the distance from Sofia's house to school? What would be the smallest unit of length we can reasonably measure to in this situation? I think you'll agree it's probably metre.

We wouldn't want to be getting our little ruler out and measuring it to the nearest centimetre.

Using metres then what would be an appropriate degree of accuracy? What do you think? Just as the height it would be the nearest integer.

The nearest metre.

But what if I changed the unit I was measuring in to kilometres? Now what would be a sensible degree of accuracy.

Here we're still measuring to the nearest metre and that would be in the third decimal place because there are 1000 metres in a kilometre.

What about the width of your little fingernail? What's the smallest unit of length we can reasonably measure to in that situation? Have a look at your little fingernail, get your ruler out.

What would be the smallest unit of length we could reasonably measure to? And that would be millimetres.

Using millimetres, what would be the appropriate degree of accuracy? And that would be to the nearest millimetre.

To the nearest integer.

Using centimetres, what would it be? And that would be the nearest millimetre, which if we're writing in centimetres, would be to one decimal place because there are 10 millimetres in a centimetre.

What about how heavy Sofia's apple is? So she's got an apple in her lunchbox.

How heavy is her apple? Again, what is the smallest unit of mass we can reasonably measure to? And that would be the nearest gramme.

Using grammes, what would be an appropriate degree of accuracy? And as before, that's going to be to the nearest gramme integer.

If I change my unit to kilogrammes, what's my appropriate degree of accuracy now? And it's three decimal places.

So just as it was when we looked at metres and kilometres, it was three decimal places because there are 1000 metres in a kilometre, there are also 1000 grammes in a kilogramme.

So here we could give it to three decimal places.

For this check what I'd like you to do is decide which of the following would be an appropriate degree of accuracy if we were going to record the population of London.

Pause the video, have a real good think about this and then when you're ready, come back and we'll check those answers for you.

What did you decide? It could be the nearest million.

There are more than a million people are living in London, but we may also decide to give it a little bit more accurately and give it to the nearest 100 000.

Now we've got Laura, Sam, and Jacob.

We're going to look at some other context when an appropriate degree accuracy is important.

Laura, Sam and Jacob decide to buy their teacher a gift to say thank you.

Laura says, "The gift cost 10 pounds." Sam says, "We have to pay the same amount." Yep, Sam agree with you to make it fair, everyone must pay the same amount and Jacob says, "I'll work it out.

I need to do 10 divided by three." That's right, isn't it? Total cost of the gift is 10 pounds and there are three of them.

10 divided by three.

Yeah, well done Jacob.

So Laura's asking Jacob now he said he was gonna work it out.

"What does that work out as each, Jacob? "Calculator gave me an answer of 3.

33." And remember we could have a recurring dot over there but just Jacob's decided he's gonna put 3, 3, 3, 3 a given number of times.

So Sam says, 'Well that means we are gonna pay 3 pounds 33 each then." If they each pay 3 pounds 33.

Have they covered the cost of the gift? No.

If we multiply three by 3 pounds 33, we actually end up with 9 pounds 99.

They've not actually got enough money to pay for that 10 pound gift.

Oh dear.

Although when rounding to two decimal places, 3.

3 recurring does round to 3.

33, because we know how to round to decimal places, don't we? But because of the context of the question we actually need to round up so that there is enough money to pay for the gift.

Considering the context of the question can help us decide on appropriate degree of accuracy.

Andeep is making some necklaces.

He has a box of 100 beads.

Ah right, he says, "Each necklace is going to use 15 beads." How many necklaces can Andeep make from one box of beads? What be an appropriate degree of accuracy for this question? It would be the nearest whole number as we're working with beads.

So I have to have put on one bead, two beads, three beads, et cetera.

Going to take our 100 and we're gonna divide it by 15.

So we get 6.

6 recurring, notice this time I've decided to use that recurring dot to show that that six recurs.

And we know that 6.

6 recurring ranks to seven and we know that we have given it the appropriate degree of accuracy is to their nearest whole number because we are making whole necklaces.

Can Andeep make seven necklaces? What do you think? No, he would only be able to make six necklaces.

Our calculation shows he can make almost seven complete necklaces.

If we look we get 6.

6, which means he can make six whole ones and actually two thirds of the final one.

So this is a situation where 6.

6 recurring generally would round to seven to the nearest integer.

Because of the context of the question, we actually need to round down to the nearest integer.

Your turn now.

Year eight from Oak Academy are going on a school trip.

162 pupils and 15 members of staff are going on the trip.

Each coach can carry 52 people.

How many coaches are needed? Jacob's had a go at this question.

He says, "I need to give my answer to the nearest integer.

177 divided by 52 is 3.

4." So we need three coaches as 3.

4 rounds to three.

Do you agree with Jacob? That's the first thing.

And the second thing is, is if you don't, I'd like you to decide what mistake he's made and even better if you could correct that for me.

Pause the video, work out what he's done wrong, correct it for me, and then come straight back.

Jacob's calculation is correct.

So everything he's done is correct.

162 add 15 is 177.

He's divided by the number of seats on each coach, which is 52.

He's got 3.

4.

But actually this is a situation where he needs to round up, 'cause if he doesn't, he won't have enough seats for everybody.

Your turn now.

I want you to pause the video.

You need to match up a statement on the left hand side with what would be the most appropriate degree of accuracy on the right hand side.

Here's your second question again, pause the video and come up when you're ready.

Great.

And question number three.

Answer the following.

Make sure that you carefully consider the appropriate degree of accuracy.

And here I'm going to allow you to use a calculator, but do make sure you write down all of the steps you've put into your calculator.

Good luck and I'll see you back in a moment.

You can pause that video now.

Final questions from this learning cycle and also this lesson.

Again, you may use the calculator, pause the video and then come on back when you're ready.

Super.

Let's check those answers.

So question number one.

Number of sweets in a bag would be the nearest 10.

Population of the UK nearest million.

Number of people at a sporting event would be the nearest thousand.

And the cost of holiday, depending on how big the holiday was, would be the nearest hundred, most probably, but you may decide to give it to the nearest thousand.

Question number two, the length of a pencil in centimetres would be one decimal place.

The length of a piece of ribbon in metres would be two decimal places.

Width of the doorway in metres would be three decimal places.

And the capacity of a mug in millilitres would be the nearest integer.

Well done if you manage to match all of those up.

Three A, is you need 17 boxes and we can see the calculation there.

B, we can see that they would need, they would each get 66 sweets.

C, 37 boxes of cookies and D, they will need 10 tents.

How did you get on with those? Great work.

Now we are ready to summarise the learning that we've done during today's lesson.

Using our method of rounding, we were able to make conjectures about unrounded numbers.

So if you think back to that example we did with the the 1966 World Cup, we knew that the attendance was 97,000 and we made a conjecture about what those values might have been based on the fact we thought it had been rounded to the nearest thousand.

The degree of accuracy often depends on the context and remember in some context, numbers that we would normally round down we have to round up and we looked at the example of the 10 pound gift shared between three people.

And on the flip side of that, there are other contexts where numbers that we would normally round up we have to round down.

In the example we looked at there was Andeep making his necklaces.

You've done fantastically well with today's learning.

I've been really impressed with what I've seen and I look forward to seeing you again really soon.

Thank you.

Goodbye.