Lesson video

Hello, my name is Mr. Peters.

Thank you for choosing to learn with me today.

In this lesson, we're gonna be thinking about decimal numbers of hundredths and how we can compare and order these.

If you're ready to get started, let's get going.

By end of this lesson today, hopefully you should be able to say that I could compare and order decimal numbers with hundredths.

Throughout this lesson, we've got a number of keywords we're gonna be thinking about.

I'm gonna have a go at saying them first and then you can repeat them afterwards.

Are you ready? My turn, compare.

Your turn.

My turn, place value.

Your turn.

My turn, order.

Your turn.

My turn, ascending.

Your turn.

My turn, descending.

Your turn.

Great.

Let's find out what these words mean in a little bit more detail.

So when we compare something, we're thinking about looking for the similarities and differences within that item or that value.

The place value of a digit within a number is dependent on where it is placed within the number.

When we think about ordering numbers, what we mean is we can use a set rule to place these numbers in a certain way.

For example, you could go from the smallest to the largest, and that is exactly what we mean by ascending order.

In an ascending order, we go from the smallest to the largest number, and the opposite of that would be a descending order.

In a descending order, we'd place our numbers from the largest to the smallest.

Look out for these words today and use them where you can 'cause these will really support us with our learning as we go.

Okay, so in this lesson today, we've broken this lesson down into three cycles.

The first cycle, we'll think about comparing numbers with the same whole number.

The second cycle, we'll think about comparing numbers that look similar.

And the third cycle, we'll think about comparing and ordering a range of decimal numbers.

Let's get going.

In this lesson, you'll meet both Izzy and Alex who will help us along the way sharing their thinking to help us with our understanding of the mathematics.

So in this lesson today, Izzy and Alex are starting off by representing numbers.

Izzy is saying that this block here is going to represent one whole.

Alex is saying that this block here will represent one 10th.

And then this little block here, this block will represent one hundredth.

We're going to need this understanding throughout this lesson.

So have a look here.

Izzy has her number and Alex has his number.

Do you agree with Alex? Alex is saying that, "I think I have a bigger number because I have more blocks altogether." Take a moment for yourself to have a little bit of a think.

Okay, let's explore this in a little bit more detail.

So in order to identify who has the larger number, we need to be able to compare numbers.

We can see here that on the left-hand side, we've got Izzy's number, and on the right-hand side, we've got Alex's number.

And in the middle, we've got a circle where we're going to use what we call an inequality symbol to help us compare these numbers.

Inequalities look like this.

This symbol here represents less than, and we can see that because on the left-hand side, we have one block, and on the right-hand side, we have three blocks.

This symbol here represents greater than.

And again, we can see that because on the left-hand side this time, we've got three blocks.

And on the right-hand side, we've got one block.

So there are more blocks to the left than there are on the right.

And then finally, this symbol here represents equal to, and we know we've seen this symbol before, but the reason why it represents equal to is because there are two blocks on the left and two blocks on the right and they're the same amount and that's why the bars are level.

Let's have a look at these numbers in a little bit more detail again.

Izzy's number has one one, four tenths, and three hundredths.

We can write this as 1.

43.

Alex's number has two ones, it has three tenths, and four hundredths.

And we can write this as 2.

34.

Now, Alex said before, didn't he, that he had a bigger number because he had more blocks? Did you agree with him? So you may think that he was half right 'cause Alex is right.

He does have the larger number, but it's not because he has more blocks.

The reason he does have the larger number comes down to the number of ones that he has in this example.

So when we start thinking about comparing numbers, we can start doing this by looking at the whole numbers, first of all.

Izzy has one one and Alex has two ones.

So we can say that one whole is less than two wholes and therefore, Alex has the larger number because he has two wholes.

We can also explore how this looks on the number line.

Here is Izzy's number, 1.

43, and here is Alex's number, 2.

34.

Again, Alex's number is larger because it has more ones and it's also further to the right-hand side on our number line.

So we know that the larger the number, the further to the right it would be on our number line.

So this time, we're gonna place these numbers within our place value chart.

Have a look carefully.

What do you notice? That's right.

Whilst we've got the place value headings in place, we haven't actually put the decimal point in place to separate the whole numbers from the fractional parts.

Let's do that now.

There we go.

So now we can see Izzy's numbers on the top row, 1.

43 and Alex's numbers on the bottom row, 2.

34.

So to compare two numbers, we need to start comparing looking at the column with the largest place value.

In this case, the largest place value is the ones.

And as Alex is saying, he knows that 1.

43 is smaller than 2.

34 because 2.

34 has two wholes or two ones within it, and 1.

34 only has one whole or one one within it.

Therefore, 2.

34 is the larger number.

So we place in here the less than symbol because the smaller number has come first.

1.

43 is less than 2.

34.

Right, let's have a look here.

This time, we've got three numbers to compare.

Let's have a think about how we can place these within our place value chart.

What'd you notice about the numbers? One of them actually has two digits before the decimal point, doesn't it? And two of them have two digits after decimal point, whereas only one of them has one digit after the decimal point.

Let's see how this affects our thinking here.

Well, we know that when we need to compare numbers, we start with the largest place value.

What is the largest place value here? Well, in this example, the tens is the largest place value.

That's because one number has two tens and the other two numbers don't have any tens.

So we can already identify what the largest number is.

The largest number is going to be 23.

47 because it has two tens and the other numbers don't have any tens.

Now let's think about the other two numbers.

We've looked at the largest place value.

So now we need to look at the next largest place value, which in this case would be the ones.

So let's look at the ones.

We know we don't need to worry about the top number anymore because we've ordered that one.

So let's look at the bottom two.

Well, the middle number has seven ones, and the bottom number has two ones.

So which number is the largest? That's right.

7.

68 is greater than 2.

3.

So we can place 7.

68 in the middle.

And we know that the smallest number would be 2.

3, so we can place that down on the left-hand side.

There we go.

Those are our numbers now ordered correctly.

These numbers have been ordered in ascending order.

That means going up from the smallest to the largest.

Okay, time for you to check your understanding now.

Which one of these representations shows the largest number? Is it A, B, or C? Have a little go.

Well done.

It was B, wasn't it? And why was B the largest number? That's right.

B has three ones, doesn't it? A only has one one and C doesn't have any ones.

So B would be the largest because the largest place value in these numbers is the ones, and B has three of them.

Well done if you got that.

Have a look at these ones then.

Which number here would be greater than 2.

47? Take a moment to think.

That's right.

It's 3.

12.

And how did you know? Yep, that's correct again.

2.

47 has two ones and 3.

12 has three ones.

And there is no larger place value before it.

There are no tens.

So in this case, the largest place value is the ones and A has three ones.

And the original number, 2.

47, has two ones.

So A is greater than.

Okay onto your first task for today then.

I've given you some empty boxes here and I've got three numbers.

One on the left-hand side, which will be the largest number.

One on the right-hand side, that would be the smallest number.

And one on the middle, which would be between those two numbers.

Using the digits nought to nine only once, what I'd like you to do is place them in the boxes to make the largest number you can, which would go on the left, the smallest number you can, which would go on the right, and the number between the largest and the smallest.

Once you've tackled that, I'd like you to go through this set here and compare each of these values either side of the circle.

Within the circles, you can use your inequalities to compare them.

So is it greater than, less than, or equal to? Good luck and I'll see you again shortly.

Okay, let's have a look and see how we got on.

Here's an example that I came up with to solve this.

My large number was 9.

51, my middle number was 2.

36, and my smallest number was 0.

48.

And as you can see, I've got a solution here where I've only used each digit between zero to nine only once.

My thinking was I'm going to start comparing the ones, so I'm gonna place the largest number I have, which is a nine, in the largest number in the ones column.

And I was gonna place the smallest number I have, the zero, in the smallest number in the ones column.

That made it easy to identify what could be placed within the middle.

I wonder if you found a similar solution or a different solution.

I wonder how many different solutions you did manage to find.

Did you find more than anyone else? Well done if you did.

Okay, let's compare these then.

0.

81 is less than 1.

32, 1.

50 is greater than 0.

50, 1.

26 is less than 2.

06, 23.

01 is equal to 23.

01, and finally, 23.

10 is greater than 22.

10.

Well done if you managed to get all of those.

Okay, let's move on to the second phase of our lesson now.

Comparing numbers that looks similar to each other.

Here, Alex and Izzy have got some numbers again.

Have a look at the numbers.

Alex is asking, "What do you notice about the numbers this time?" That's right.

We're using place value counters this time, aren't we instead of dienes.

But also both of these numbers have the same amount of ones this time, don't they? Alex has got three ones.

And Izzy, as she's just pointed out, she also has three ones as well.

Hmm.

How are we gonna compare these two numbers now? Well, let's line them up in our place value columns again.

As always, we'd start with the largest place value when comparing these numbers.

However, as you can see, here the ones column, they're the same, aren't they? So we can't compare them at the moment.

That means we're going to need to go to the next largest place value column.

In that case would be the tenths, wouldn't it? So both our numbers have three ones, but with regards to tenths, well, actually Alex's number has four tenths and Izzy's number has one tenth.

So can we compare these numbers now? That's right.

Izzy's pointed out that four tenths is greater than one tenth.

So therefore, 3.

41 would be the largest number.

We can use our inequalities to express that.

3.

41 is greater than 3.

14.

Have another look.

What do you notice about the numbers this time? Yeah, a good spot.

We still have to say a number of ones, don't we? But also we have the same number of tenths this time.

Hmm, we're going to need to think about this in a little bit more detail again, aren't we? Let's represent this on our place value charts again.

Starting with the largest place value, we know that the ones were the same, so we can't compare them here, can we? As Alex has said, when the ones are the same, we then need to think about comparing the tenths, the next largest place value.

When we get to the tenths, however, they're both the same.

So again, this time, if we cannot compare the temps, that means we need to go to the next largest place value, in this case, the hundredths.

Let's move along to the hundredths.

What can we say now? Well, we know that one hundredth is less than three hundredths, so we know that 3.

43 would be the larger number because it has more hundreds than 3.

41.

And there we go.

We've expressed it with our inequality.

3.

41 is less than 3.

43.

Here's another example.

Have a look for yourself first of all.

Okay, well, we're gonna actually compare this in a slightly different way this time.

We're starting to look at the ones again, aren't we? But we've realised that they're both the same.

This time, however, we're gonna think about the numbers after decimal points as hundredths.

We've got two decimal places to the right-hand side of the decimal point.

And we know that both of these represent hundredths.

The top number represents 68 hundredths and the bottom number represents 67 hundredths.

So instead of going to the tenths and then to the hundredths, we could break these tenths down into hundredths to start off with and read them as hundredths.

So instead of saying six tenths and eight hundredths, we could just call it 68 hundredths.

And the same for the number underneath.

Instead of saying six tenths and seven hundredths, we could just say 67 hundredths.

We know that 68 hundredths is greater than 67 hundredths.

So the larger number here would be 7.

68 because it has 68 hundredths and the other number has 67 hundredths.

Okay, time for you to check your understanding again.

Tick the numbers that are smaller than 5.

62.

Well done.

It's both A and D, isn't it? Let's have a look at why.

Well, both of the numbers have five ones, so they're gonna be similar numbers, aren't they? However, option A has 26 hundredths and option D has 60 hundredths, and we know that 26 hundredths and 60 hundredths are less than 62 hundredths, what our original number has.

Therefore, these are both smaller than 5.

62.

Maybe you could take a moment to write down a couple of numbers for yourself which you know would be less than 5.

6, however, have five ones.

Okay, another check.

True or false: 8.

74 is greater than 8.

47.

Take a moment to have a think.

That's right.

It's true, isn't it? Have a look at the justifications now.

Which one helps you justify your answer? And that's right, A, isn't it? We know it's A because 8.

74 has 74 hundredths and 8.

47 has 47 hundredths and 74 hundredths is greater than 47 hundredths.

Okay, onto task two now.

Here is the sequence of expressions that I would like you to compare.

Starting at the top and moving down one at a time.

Each time you move down one though, I want you to think about the previous one and ask yourself what did you notice? What was the same, what was different each time? Maybe that will help you.

Good luck and I'll see you in a minute.

Okay, welcome back.

Let's tackle these then.

So the first one, 0.

74 and 0.

75.

Well, we know that 0.

74 is less than 0.

75, and we could reason that a number of ways.

Firstly, we both know that they've got zero ones.

We know that they've both got seven tenths.

However, 0.

75 has five hundredths and 0.

74 has four hundredths.

Therefore, 0.

75 would be the larger number.

You may have reasoned it to say that 0.

74 is 74 hundredths and 0.

75 is 75 hundredths, and therefore, 75 hundredths is larger.

Okay, let's look at the next example.

What did you notice? Yeah, that's right.

1.

74 is greater than 0.

75.

Ah, the hundredths stayed the same, didn't they? But the ones changed this time.

And actually, the number on the left now has a one, whereas number on the right has zero ones, even though it has more hundreds is not the larger number.

It is now 1.

74, which is the larger number.

Okay, underneath that, have a look.

It's the same again, isn't it? 1.

74 is greater than 0.

76, but what changed? Yep, the number of hundredths on the right-hand side got even bigger than the number of hundredths on the number on the left-hand side, didn't it? But it still doesn't have any extra ones.

So the number on the left-hand side has one one, and the number on the right-hand side has zero ones.

Therefore, 1.

74 is still the largest.

What do you notice this time? Yeah, good spot.

Actually, the number on the right-hand side now has a one, doesn't it? So that means we're gonna have to reverse our inequality symbol to represent 1.

74 being less than 1.

76 'cause we know that the hundredths on the right-hand side are larger than the hundredths on the left-hand side.

And another look.

What's changed this time? Ah, both of the tenths in the columns have increased by one tenth, haven't they? So the number on the left-hand side was 1.

74, but now it's 1.

84, and the number on the right-hand side was 1.

76, but now it's 1.

86.

So whilst the number of tenths have increased, has that changed the way the symbol is? It doesn't because the numbers have both increased by the same amount.

Actually, we know that the number on the right-hand side will still be greater than the number on the left-hand side.

So we can read it as 1.

84 is less than 1.

86.

Onto the last two then.

What do you notice? Well, we've gone from having one whole in both the numbers to now having 10 wholes in both the numbers.

Are the hundredths the same still? Yeah, they are, aren't they? So actually, whilst the number of wholes has changed for both numbers, they're still the same number of wholes.

So the inequality would stay the same way.

10.

84 is less than 10.

86.

And then the final one.

What do you notice this time? Well, yeah, one of them is now a fraction, isn't it, instead of a decimal.

How do we read that fraction? 10 wholes and 84/100.

And we know we can write that as 10.

84.

So now that we know that, what's the difference between the expressions above and the expressions below? Ah, they've just swapped size, haven't they? So actually, the 10.

86 is now on the left-hand side, and the 10.

84 would be on the right-hand side.

Therefore, we would need to swap our inequality around to say that 10.

86 is greater than 10.

84 or 10 and 84/100.

Well done if you managed to get that.

Okay, and onto our last cycle now.

Compare and order a range of decimal numbers.

Alex is saying here that he has fewer counters than Izzy does, therefore, his number must be the smaller number.

What do you think? Izzy is saying that she disagrees because Alex's number has more ones than her number.

Therefore, Alex's number would be the largest number.

How do you feel about that? Let's check this out now in our place value charts.

So here I've got Alex's number in the top row, and underneath that, I've got Izzy's number.

Alex's number is represented of two ones, and Izzy's number is represented by 1.

45.

Izzy's saying, "Remember, we need to start with the largest place value.

In this case, the largest place value here is the ones." Alex has two ones and Izzy's has one one.

Therefore, Alex's number would be the larger number because his has two ones and Izzy's only has one one.

What'd you notice about Alex's number though? That's right.

He doesn't actually have any hundredths or additional tenths, does he, at all? Therefore, whilst his number only has one digit, it does not necessarily mean that his number is smaller.

It's all about the value of that digit.

And in this case, the value of Alex's digit was two ones, and we know that two ones is larger than one one, which is what Izzy had.

Here's another example.

Look carefully this time.

What do you notice? Well, one of the numbers has two digits and one of the numbers has three digits.

But remember, as we've just spoken about, even though one of the numbers has less digits, it doesn't necessarily mean it's a smaller number.

Let's start comparing them by looking at the largest place value.

Well, they've both got the same number of ones, haven't they? They both have one one.

Therefore, we need to look at the next largest place value, which would be the tenths.

Now we can start to compare them.

The top number has five tenths and the bottom number has four tenths.

So we can, that 1.

5 is greater than 1.

45.

Izzy's also saying we could have read it as hundredths instead.

1.

5 is the same as saying 1.

50 or 50 hundredths.

So we can now read it as one one and 50 hundredths in the top row.

And in the bottom row, we can read it as one one and 45 hundredths.

Therefore, we know because the ones have the same place value, then 50 hundredths is greater than 45 hundredths, and therefore, 1.

5 or 1.

50 is greater than 1.

45.

We know that 1.

5 and 1.

50 are exactly the same thing.

So now that we've become more familiar with comparing decimal numbers with different numbers of digits, we can start to think about ordering these.

Let's have a look at these numbers here and see if we can try ordering them in descending order from the largest to the smallest.

So we've written them all up into our place value chart.

As always, let's start with the largest place value, shall we? In this case, that's the tenths, isn't it? There's only one number that has a tenths digit, and this number is 41.

Therefore, this number would be the largest number.

Now let's have a look at the ones column, the next largest place value.

Hmm.

I can see here that two numbers have a ones digit of one and two numbers have a ones digit of four.

So we'll need to start thinking about the numbers with the fours.

And as Izzy is saying, that means we're gonna have to look at those numbers more carefully and start moving along looking at the tenths column now because we can't compare those two numbers just by looking at the ones.

So here we are.

One of those numbers is 4.

01, and one of those numbers is four or 4.

00.

And we know this is the same thing.

Looking at the tenths, we can see they both have the same number of tenths.

They both have zero tenths.

However, the first number, 4.

01, has one additional hundredth, and the bottom number, four, doesn't have any additional hundredths.

So we can say that 4.

01 is greater than 4.

00 or four because it has one additional hundredth extra.

So the next largest number then would be 4.

01, and the third largest number would be four or 4.

00.

And we know these are the same.

Now let's look at our final two numbers that we've got.

Of course, we're going to need to go back to the ones and check those.

Well, they've both got the same ones again, haven't they? They've both got one one.

So this isn't gonna do.

We're gonna have to go and look at the tenths again.

Let's have a look at the decimals.

Well, do you know, this time I'm gonna compare them using my understanding of hundredths.

The number on the top has 40 hundredths, and the number at the bottom has four hundredths.

Therefore, I know that 40 hundredths is greater than four hundredths.

So the next largest number is 1.

40.

That would mean that the smallest number would be 1.

04.

And here we go.

We can represent this in our order now.

So now we have ordered these numbers in descending order from the largest to the smallest.

Okay, final check for understanding now.

Which numbers are larger than 5.

48? Take a moment to have a think.

That's right, it's both B and C.

How do we know? Well, 5.

48 has five ones, doesn't it? And 5.

49 has five ones, but 5.

48 has 48 hundredths, and 5.

49 has 49 hundredths.

So we know that's larger.

And C, well, C is made up of six ones and 5.

48 is made up of five ones, isn't it, and 48 hundredths.

So we know that six ones is greater than five ones.

So six would also be the larger number.

Well done if you've got that.

And here's another example.

Doing some more comparing here.

Match these numbers to the correct place in the order.

On the left-hand side, we want the largest number, and on the right-hand side, we'd like the smallest number.

And in the middle, would be the number that's not either the largest or the smallest.

Take a moment to have a think.

That's right.

20.

1 would be the largest number because it has two tens.

10.

2 would be the second largest number because it has one 10.

And 1.

02 would be the smallest number because it doesn't have any tens.

It only has one one.

Okay, onto our last few tasks for today then.

First things first, what I'd like you to do for this question is work down each column vertically.

So starting on the left-hand side, and then we're going down to the right-hand side one.

What you're going to need to do again is use your inequality symbols to compare each of the expressions on either side of the circle.

Once you've done that, I'd like you to think about looking at these numbers and placing them in ascending order.

And then finally, a nice little task here.

Alex is thinking of a number between two and three.

The number has three digits and those digits sum to five.

They add up to make five.

Which number could Alex be thinking of? Good luck, and I'll see you back here shortly.

Okay, here we go.

Let's use our inequalities to compare the following numbers then.

So 0.

4 and 0.

6.

We know that 0.

6 is greater than 0.

4, so we can say 0.

4 is less than 0.

6.

What's changed this time? Well, the number on the left-hand side now has a digit of six, and the number on the right has a digital of four.

However, the value of these has changed, isn't it? The six is now six hundredths, and the four is also four hundredths.

So we can say that 0.

06 is greater than 0.

04.

What do you notice this time? Well, the 0.

04 is swapped back to the left-hand side now, but on the right-hand side, we've got 0.

6.

So the value of the six has changed.

It's no longer six hundredths.

It's now six tenths.

So we can say that 0.

04 is less than 0.

6 because six tenths is greater than four hundredths.

What do you notice this time? Well, the 0.

04 has been made 10 times bigger, and it's become 0.

4.

And the 0.

6 has made 10 times smaller.

It's become 0.

06.

So again, the values of these digits have changed, and 0.

4 is greater than 0.

06 because 0.

4 is four tenths, and 0.

06 is six hundredths.

Eight is equal to 8.

00.

We know that the zeros after the decimal point don't matter if there aren't any additional tenths or hundredths placed.

So we don't even need to write them there, do we? It still represents eight.

Okay, onto the next column, 0.

3 and 0.

25.

We know that 0.

3 is greater than 0.

25 because it has three tenths.

And on the right-hand side, the number only has two tenths.

0.

11 and 0.

09.

Well, we could read that as 11 hundredths and nine hundredths.

So 11 hundredths is greater than nine hundredths.

Underneath that, 2.

02 and 2.

22.

I'm gonna read it again as hundredths.

So they've both got the same ones and they've both got two ones, but the number on the left has two hundredths, and the number on the right has 22 hundredths.

So we can say that 2.

02 is less than 2.

22.

Then we've got five on the left-hand side and 4.

99 on the right-hand side.

Hmm, don't get confused here.

The five represents five ones, and the four on the right-hand side represents four ones.

Therefore, five ones is greater than four ones, even though it has nines in the tenth and the hundredths.

So we can say that five is greater than 4.

99.

And finally, we've got 160, and on the right-hand side, 1.

60.

They've got the same digits, haven't they? However, the value of these digits is remarkably different.

On the left-hand side, we've got 100 6 tens and zero ones, and on the right hand side, we've got one one, six tenths, and zero hundredths.

If we're looking at the largest place value, the number on the left has 100, and the number on the right doesn't have any hundredths.

So we can say that 160 is greater than 1.

60.

Well done if you've got all of those.

Okay, here are these numbers placed in ascending order, going from the smallest to the largest.

So we've got 0.

03 would be the smallest, 0.

1 would be the second smallest, 0.

13 would be the third smallest, 0.

3 would be the fourth smallest, and 0.

31 would be the largest.

And then finally here for Alex's number, there are a number of different numbers that we could have possibly put here to find as a solution.

You could have had 2.

03 where if you sum those digits together, that would make five.

You could have had 2.

12.

Again, summing those digits would've made five.

2.

21, again, they would've sum to five, and finally 2.

30.

So there were four possible solutions for this task that we came up with.

I wonder if there were any others that you could have came up with.

Brilliant.

Well done today.

That's the end of our lesson, and hopefully you're being a lot more confident when thinking about comparing and ordering decimal numbers which include hundredths.

Just to recap some of the key points from our learning today.

To compare two numbers, we need to compare the same place values, don't we? And when doing this, we always start with the largest place value.

If the numbers in that place value are the same, we then move to the next largest place value.

We could compare tenths and hundredths by reading the numbers as hundredths, for example, 7.

68 could be read as seven ones and 68 hundredths.

And then finally, once you've been able to compare numbers, we can start thinking about ordering these numbers to a set rule.

And those rules could be an ascending rule, getting from smaller to larger, or a descending rule, getting from largest to smallest.

Thanks for learning with me today.

I've really enjoyed teaching you this lesson, and hopefully you are feeling a lot more confident.

Take care, and I'll see you again soon.