Lesson video

Hi there, my name's Ms. Lambell.

You've made such a super fantastic choice deciding 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 Checking and Securing Understanding of Rounding and Truncation, and this is within the unit, Rounding, Estimation and Bounds.

Now, let's take a look at some keywords that we will be looking at in today's lesson, and these are rounding, degree of accuracy, significant figure, and truncation.

Rounding means to change a number into another number that is approximately the same value, but it's easier to work with.

So for example, 561 to the nearest 10 is 560.

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

For example, here, something may be measured to the nearest centimetre, something may be given to the nearest 10, or it may have been given to the nearest one significant figure.

Significant figures.

Remember, these are the digits in a number that contribute to the accuracy of the number, and the first significant figure is the first non-zero digit.

That's really important, isn't it? Truncation is when we simplify a number by cutting off one or more of the digits and replacing them with zeros if this is necessary to preserve the place value.

For example, 3,678 truncated to one significant figure is 3,000, and 0.

3678 truncated to two decimal places is 0.

36.

Today's lesson is split into three learning cycles.

In the first one, we will look at rounding to a given place value.

In the second one, we will concentrate on significant figures, and in the third one, we will look at truncation.

Let's get started that first one, so rounding to any given place value.

An elephant's mass is estimated to be 5,000 kilogrammes.

Aisha says the elephant could be 5,433 kilogrammes.

Alex says the elephant could be 4,856 kilogrammes.

Can both Aisha and Alex be correct? They can both be correct.

In order for them both to be correct, what degree of accuracy has the 5,000 kilogrammes been given to? What do you think? What did you come up with? They're both correct if the mass has been rounded to the nearest 1,000 kilogrammes.

Let's just check that.

Make sure that both of their possible masses do round to 5,000 kilogrammes when we round them to the nearest 1,000 kilogrammes.

Let's start with Aisha.

Here's Aisha's number, and I'm using my place value grid to help me to round this to the nearest 1,000.

I identify the thousands column and the digit in the thousands column.

Remember then we need to consider the number immediately to the right, and that's the number in the hundreds column here, which is a four.

Halfway in this column would show a five.

We have a four in this column, which means that our number is closer to the lower multiple of 1,000.

Therefore, we round down to the multiple of 1,000 that is below 5,433.

Aisha is correct if the estimated weight has been given to the nearest 1,000 kilogrammes.

Now, let's check Alex's.

Again, I'm using my place value grid.

I'm identifying the number in the thousands column 'cause we're checking what Alex's mass is rounded to the nearest 1,000 kilogrammes.

The digit in this column is the four.

Let's look at the digit immediately to the right.

Halfway, remember, would be a five.

Eight is greater than five, so we know that it's closer to the higher multiple, and therefore, we round up to the multiple of a 1,000 that is above 4,856.

Alex is correct.

The estimated weight has been given to the nearest 1,000 kilogrammes.

An elephant's mass is estimated to be 5,000 kilogrammes.

We know that.

And here, we've got Aisha's possible suggested mass.

Would as Aisha's mass be correct if the estimated mass was given to the nearest 100 kilogrammes? Let's take a look.

Here's my place value grid.

It's a slightly simplified version.

I identify the 100 because I'm rounding to the nearest 100 kilogrammes, and the four is the digit in this column.

I look to the right, so I'm gonna look at the three that is in the tens column.

It's less than five, therefore, we round down to the multiple of 100 that is below 5,433, which is 5,400.

So no, 5,433 kilogrammes actually rounds to 5,400 kilogrammes when it's rounded to the nearest 100 kilogrammes.

Really important then, when we're rounding, we identify the column that we are rounding to just as we did.

You don't necessarily need to use the place value grid.

You may be able to do it without, but it is absolutely fine to continue to use that place value grid, remember? We look at the digit to the right of this column.

Remember, that's immediately to the right.

If it is less than five, we round down, and if it is five or greater, we round up.

We've got Alex and Aisha again, and they've been given a number to round to the nearest 10.

I want you to think about who you agree with and why.

Aisha's number is 469 and Alex's number is 464.

Aisha says my one's digit is more than five, so I round up to 470.

Alex says my one's digit is less than five, so I round down to 450.

Who do you agree with and why? Aisha is correct.

We need to be careful when we use this phrase round down.

And we do use it.

We use it a lot.

But we need to be really careful as we do not change the digit, it stays the same.

We take a look at Alex's number.

We're rounding to the nearest 10.

So I've identified the digit in the tens column.

I look at the number to the right of that, it's a four, so we do round down, but we round down to the multiple of 10 that is below 464.

This means that the digit in the tens column doesn't change.

Alex's number actually rounds to 460.

Well done if you spotted Alex's mistake.

This time, Aisha and Alex have been given the same number and that's 452.

And they are rounding this to the nearest 100.

Again, I'm going to reveal what they've got to say, and I'd like you to think about who you agree with, and remember, I want to know that why.

That why is so, so important.

Aisha says it ends in a two, so I need to round down to 400.

And Alex says the tens digit is five, so we need to round up to 500.

So Aisha thinks we're going to round it down to 400, and Alex thinks we're going to round it up to 500.

Who do you agree with, and more importantly, why do you agree with them? Let's take a look.

Alex, this time, is correct.

When considering rounding up or down, we must look at the digit in the next column to the right of what we are rounding to.

It's the column immediately to the right.

We can't skip a column.

A horse's mass is estimated to be 540 kilogrammes.

Aisha says the exact weight of the horse is 540.

3 kilogrammes.

That's the exact weight of the horse.

What degree of accuracy could 540 been given to? It could've been given to either the nearest 10 or the nearest integer.

Now, we're going to take a look at rounding to decimal places.

We're going to start by rounding 2.

9305 to three decimal places.

And I'm going to go back to using my place value grid because I find it extremely helpful.

Here's my place value grid.

Remember, we need to make sure that the digits all appear in the correct column.

So the two is in the ones column, and then the rest of the digits follow after the decimal point.

We need to here identify the thousandths column because that is the column that is the third decimal place.

After the decimal point, the third column is the thousandths column.

So we identify this and the digit in it.

Just as we did when we were looking at integers, when we were rounding to nearest 10, 100, or 1,000, we look at the digit to the right.

We consider the digit that is in the 10 thousandths column.

That's the column to the right, and that's a five.

Halfway would show a five in this column.

This is exactly halfway.

What do we do when it's exactly halfway? Do we round up or do we round down? That's right, we round up.

Always round up if it's exactly halfway.

We round up to the higher value.

2.

9305 is equal to 2.

931 to three decimal places.

Now, we'll do one more together, and then I'd like you to have a go at one independently.

We're gonna round 148.

72 to one decimal place.

I've got my slightly simplified version of the place value grid.

Remember, you don't have to use the place value grid if you're confident without.

Let's place our number into the grid, and this time, we are looking for one decimal place.

We're looking for the first digit after the decimal place, and that, this time, is the tenths column, which is the seven.

Remember, we need to look at the digit immediately to the right.

It's less than five, so therefore, we round down.

Remember that when we round down, the digit before the one we've highlighted does not change.

We round to the lower value.

148.

72 to one decimal place is 148.

7.

Now it's your turn.

Round 856.

46 to one decimal place.

Pause the video.

When you've got your answer, come back and we'll check it.

I know it'll be right.

Remember, if you need to draw up place value grid, that's absolutely fine.

Okay, good luck with this, and I'll see you in a moment.

How did you get on? I know you've got the right answer.

Let's place our number into our place value grid.

One decimal place, that's the tenths column.

We look at the digit immediately to the right.

That's a six.

What do we do when it's a six? That's right, yep, we round up.

So the digit is greater than five, so we round up to the higher value.

856.

46 to one decimal place is 856.

5.

Of course you got that right.

Now, a quick check before I set you going on our first independent task for today's lesson.

Which of the following have been written correctly to two decimal places? Pause the video.

Make your decision.

If I were you, what I would do is I would imagine that the answers aren't there, I'd work the answers out myself, and then I'd compare, and then I would be able to see which ones are wrong.

So pause the video, and then when you're ready, come back and we'll check those answers for you.

Super work.

Let's have a look then.

Let's check those answers.

A was wrong.

B was correct.

C was incorrect and D was correct.

Did you manage to identify those two correct, the two correct ones and the two wrong ones? Fantastic, well done.

So you're now ready to have a go at this task.

What you need to do is you need to round each of the following to the degree of accuracy that I've given to you in brackets.

A is 0.

058, and we were gonna round that to two decimal places.

If it says Hth, that's hundred thousands.

It says M, that's millions, okay? You're then going to find each of those numbers in the number search.

Pause the video, and then when you've got your answers, we'll come back and we'll check those for you.

Great work.

I'd like you to pause the video again, and you're gonna check your answers.

So the answers are there on the left hand side, and that's where you will find them in the number search grid.

Pause the video now and then come back.

We'll be ready to do learning cycle two.

Great work.

Did you manage to identify them all in the grid? Super.

Let's move on now then to our second learning cycle.

We're going to look now at rounding to significant figures.

We're going to round 547 to two significant figures.

Aisha says this is the same as rounding to the nearest 10.

Do you agree or disagree with Aisha? Explain why you agree or disagree.

What did you decide? Alex says, I agree with you, Aisha, as the second significant figure is in the tens column.

547 is 550 when rounded to two significant figures.

And that's the same as rounding to the nearest 10.

Aisha looked and she noticed that the second significant figure was in the tens column, so therefore, it's the same as rounding to the nearest 10.

Well done, Aisha.

They're now looking at rounding 0.

02568 to two significant figures.

Alex says this will be the same as rounding to the nearest hundredth, as the second significant is the two.

Do you agree or disagree with Alex? And remember, I want an explanation as to why you agree or disagree.

And what should you decide? Did you agree with Alex or disagree? Aisha says, remember, Alex, the zero in the 10th column is not significant.

The second significant figure is the five.

Remember, we don't start counting significant figures until we see a non-zero digit.

The two is the first significant figure, and the five is the second significant figure.

0.

02568 is 0.

026 when rounded to two significant figures.

Really important we don't start counting until that first non-zero digit.

Alex says this is the same as rounding to three decimal places then.

And Aisha says, yes, Alex, the second significant figure is in the thousandths column, which is in the third decimal place.

So we can see how these things link together.

When we round to significant figures, we need to remember, like I've just said, that the first significant figure is the first non-zero digit in a number.

That's really, really important.

So for example, in 0.

00483, the four is the first significant figure.

In this number, it would be the eight, the first number that isn't zero.

In this one, the one.

In this one, the two.

In this one, the three.

In which of the following is four the first significant figure? Pause the video, make your decision, and then come back when you're ready.

Great work.

Let's check those answers.

A, no, four is not the first significant figure.

The first significant figure is actually two.

B is correct.

The first non-zero digit is four.

C is correct.

The first digit that we hit is a four.

And D is incorrect.

The first significant figure in this number is six.

What is 0.

002036 rounded to three significant figures? Let's pop that in our place value grid.

Which digit is the first significant figure? That's right, it's the two, the first non-zero digit.

Well done.

In which column is the third significant figure? Okay, well done.

It's the one in the hundred thousandths column.

Hundred thousandths column.

If two is the first significant figure, the zero after that is the second significant figure, so the three is the third significant figure.

We look at the digit to the right of this.

It's greater than five, so we're going to round the number up.

0.

002036 is 0.

00204 when rounded to three significant figures.

Let's do another one together and then you can have a go at one independently.

What is 0.

0805 rounded to two significant figures? My place value grid.

Identify the digit that is the second significant figure.

The eight is the first, so it's in the thousandths column.

It's that zero.

Let's look at the number to the right.

It's a five.

Remember, five or above, we round up to the higher value, so 0.

0805 is 0.

081 to two significant figures.

Time for your go.

301.

48 rounded to three significant figures please.

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

Great work on that.

Now, let's check your answer, which I know, of course, is going to be right.

Let's place our number into our place value grid.

Three significant figures.

Identify the third significant figure.

The first one is the three that's in the hundredths column, so the third is in the ones column.

That's a one.

Look at the digit to the right.

It's less than five, so we're gonna round down to the lower number, so our answer is 301.

What was that you said? You got it right? Of course you did.

Now, before we move on, I want us to double check we've really secured our understanding of significant figures.

Here, you are going to decide which of the following are correctly rounded to the given number of significant figures, and I've given those to you in the brackets at the end of each number.

So pause the video, and as always, I'll be here waiting when you come back, and we'll check those answers.

Super work.

Well done.

Let's go.

What did you decide for A? A is incorrect.

0.

27 rounded to one significant figure should be 0.

3.

B is correct.

C is incorrect.

It should be 330.

And D was correct.

An elephant's mass is estimated to be 5,000 kilogrammes, so we are back now to our elephant's mass as being estimated to 5,000 kilogrammes.

Aisha says this must have been rounded to one significant figure.

And Alex says, actually Aisha, I'm not sure that is true.

Can you think of a mass that rounds to 5,000 kilogrammes, which has not been rounded to one significant figure? Example.

Remember, these are just examples.

You may have something different.

4,983 rounds to 5,000 kilogrammes when it's given to two significant figures.

5,004 kilogrammes rounds to 5,000 when it's given to three significant figures.

So just because it has only one non-zero digit, we cannot assume that it's been given to one significant figure.

Often, when we're looking at these sorts of questions, we're asked in part B to round something to a given number of significant figures.

And we're asked to calculate something on our calculator.

We'll do this one together.

We're going to give our answer as a decimal, and we need to write down all of the digits on our calculator display.

We're going to type it into our calculator, and this is what you should get.

So if you need to pause the video, go away and find your calculator, of course you can, but I know you've always got it to hand because you're super organised.

Your calculator display should look exactly the same as mine.

So our answer that we round our answer down is 1.

160525376.

We need to give our answer in part A correct to two significant figures.

The second significant figure is in the tenths column.

The first one is in the ones column, and the second is in the tenths column, and that's the first decimal place.

So we're just effectively rounding this to one decimal place.

Our answer is 1.

2.

Now, your turn.

You're going to spot and correct my mistakes please.

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

How did you get on? Brilliant.

Well done.

Let's have a look then.

What is wrong with the first calculation? There's my calculation and what is wrong with it? Well, we can see here we've got 3.

6 to the power of five multiplied by 15.

But if we look at the calculation, it's 3.

6 to the power of five and then multiply that by 15.

The multiply by 15 is part of the exponent on my calculator display and it shouldn't be.

You must remember to press the right arrow after entering in an exponent.

What's wrong with their answer? Their error with the answer is they've missed the negative symbol and they've also missed off the power of 10 part of the answer.

This is what the answer should've been, and so therefore, your answer, correct to three significant figures, should have been negative 2,000.

Well done if you got that right.

Now Task B.

First question, question number one is super easy, so you'll be back really, really quickly.

Circle the first significant figure in each of the following.

Pause the video, and then when you're ready, come back, and we'll look at question number two, which will be a bit more challenging, I promise.

Okay, question number two, pause the video, come back when you're ready.

Just realised that that was question two and three, and now, question four.

Pause the video.

Great work on those, and there is a calculator question, just a final one is question number five.

Wow, there's quite a lot to do there wasn't there? Let's have a look at the answers.

Pause the video, check your answers, and then come back when you're ready.

Answers to questions two and three will have something similar to this.

Question number four, again, pause the video and check your answers.

And finally, question number five, pause and check.

Super work on those.

Now, we can move on to our final learning cycle for today's lesson where we're going to look at truncating.

Aisha has truncated the answer given on her calculator display to two significant figures, and she says that her answer is 1,300.

Aisha's answer was an integer.

Okay, so we know her answer was an integer.

What is the lowest value Aisha's answer could have been? Alex says the lowest value would've been 1,250.

Do you agree with Alex? Alex is not correct.

He would've been correct if Aisha had rounded the answer to two significant figures, but remember, she was truncating her answer.

Here's Aisha's answer.

When truncating, we do not change any digits before the truncation.

This means that the only digits that can change are the zeros.

We were considering the lowest integer Aisha's answer could've been.

Only changing the digits in the box, the blue box, can you make a number that is lower than 1,300? No, the lowest value Aisha could have is 1,300.

What is the greatest integer Aisha's answer could have been? Remember, we can only change the digits in the box.

What's the greatest integer Aisha's answer could have been.

Now, you may already have told me that when I revealed the first question, but there's a hint there if you need it.

We need to replace the zeros in the tens and the ones column with the highest digit, and we know the highest digit is nine.

Aisha's answer was an integer, remember? So Alex says the lowest integer it could be was 1,300, and the highest was 1,399.

Which of the following have been correctly truncated to three significant figures? Pause the video, decide which are correct, and then come back when you're ready.

Let's take a look.

A was incorrect.

This has been rounded to three significant figures.

So we look, we count.

The one is the first significant figure, the four is the second, the seven is the third significant figure.

But we've looked at the number after.

Remember truncation, we just replace any numbers after the third significant figure with a zero.

B was incorrect.

Here, they have truncated to one significant figure.

Remember, the first non-zero digit is the first significant figure.

The first significant figure in this number is the one, the second is the six, and the third is the eight.

That was the mistake in that one.

C is correct, and D is incorrect.

Here, the magnitude of the number has changed.

Remember, that's the size of the number.

The first digit of nine starts in the millions column.

If we look at our first number, we got 9,999,999.

999.

So it started in the millions column, but in the truncation, it's in the a hundred thousands column.

It's really important we remember to replace every digit with a zero.

Now, we're gonna have a look at some problem solving, so some Who am I's, and I really like these.

I really enjoyed making them up for you.

Use the clues to answer this question.

I am a multiple of four.

When truncated to two significant figures, I am 3,100.

The thousand and tens digits are the same.

Aisha says if it is a multiple of four, the last two digits must be a multiple of four.

Alex says, yes, and we know that it's four digits long starting with 31.

Do you agree with Aisha and Alex's statements? Yeah, both statements are true.

I've decided because Alex said it was a four digit number, I've drawn out four boxes to put my digits into.

So using Alex's statement, we can fill in the thousands and the hundreds digits.

So we know, because it's been truncated, that the three and the one must be the same.

What other digits can we fill in now? Yeah, using the third statement, we know that the tens digit must be a three as it is the same as the thousands digit.

Aisha says the last two digits have to be a multiple of four, so the number is 3,132.

32 is a multiple of four.

Is that the only solution to this problem? No, it could also have been 3,136.

Now, you're ready to have a go at your final independent task for today's lesson.

You're gonna complete the table.

Pause the video, and then when you come back, I will reveal question number two for you.

Good luck.

And question number two.

You gotta work out the numbers given by the clues.

Pause the video, good luck with these, and then when you come back, we'll check those answers for you.

Well done.

How did you get on? Did you manage to work out my two mystery numbers? Well done.

Here we go.

Pause the video, check your answers, and then come back when you're ready.

And question two.

A was 135, and B was 2,552.

Well done if you managed to discover my two mystery numbers.

Let's just summarise now then what we've done during today's lesson.

It is important to understand the difference between rounding and truncating.

Rounding means to change a number into another number that is approximately the same value, but is easier to work with.

When we ranked, we considered the digit to the right of the degree of accuracy.

For example, for 567 to the nearest 10, is 570 because we look at the digit after the tens column, and here it is greater than five, so we round up to the next multiple of 10.

Truncation is when we simplify a number by cutting off one or more of the digits and replacing those with zeros if necessary to preserve the place value.

567, so the same number, if it's truncated to two significant figures is 560.

We find our second significant figure, which is the six, and the digits before the truncation stay the same.

The digits after the truncation are replaced with zeros if that's necessary to preserve the place value.

And here, it is to make sure that the five and the six remain in the correct columns.

When dealing with significant figures, the first non-zero digit is the first significant figure.

Remember, don't start counting significant figures until you get to that first non-zero digit.

Fantastic work today.

We've been through absolutely loads and loads of stuff, and I've been super impressed with how you've done.

Hopefully, you'll join me again for another lesson, and I really look forward to seeing you soon.

Take care of yourself.

Goodbye.