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Hello, my name is Mrs. Hopper and I'm excited to be working with you in this lesson from the unit on comparing fractions using equivalents and decimals.

I love fractions and I hope you do too.

And I hope during this lesson you'll be able to see and understand a little bit more about them and how they link to decimals.

So if you're ready, let's make a start.

So in this lesson we're going to be looking at identifying and describing patterns in the number system.

We've got two key words, equal parts and divide.

I'll take my turn and then it'll be your turn to say them.

So my turn, equal parts, your turn.

My turn, divide, your turn.

Words I'm sure you are familiar with, but let's just check the definitions.

They're gonna be useful to us in our lesson.

So when we divide, we split an object, number or quantity into equal parts.

Each equal part will have the same value as the others in the whole.

And numbers are equivalent when they have exactly the same value.

So let's make a start.

We've got two parts to our lesson today.

In the first part, we're going to be dividing into 2 and 4 equal parts.

And in the second part into 5 and 10 equal parts.

And we've got Lucas and Sam helping us today.

So what is 1,000 divided into two equal parts? I'm not sure this is going to be terribly taxing for you, but let's just have a think because we're looking remember at patterns.

So let's start where we know and then maybe we can extend the pattern to help us work out things that maybe we're not so familiar with.

So what is 1,000 divided into two equal parts? Lucas says "The whole is divided into two equal parts.

Each part is one-half of the whole." So we still thinking about that fraction link.

And 1,000 divided into two equal parts gives us 500 in each part.

And on the number line, one jump of 500, two jumps of 500.

So that middle unmarked line was 500, and we can say that 1/2 of 1,000 is equal to 500.

And 1/2 of, we can represent with multiplication.

1/2 X 1,000 = 500.

And Lucas reminding us, "We read this as one-half of 1,000 is equal to 500." So what is 1,000 divided into four equal parts? Lucas say, "The whole is divided into four equal parts.

Each part is one-quarter of the whole." So we're thinking about a quarter of 1,000 this time and we know that that's 250.

We also know that half of 500 is 250, and so we can mark our jumps of 250 along the number line.

So 250, 500, 750, and 1,000.

One-fourth of 1,000 is equal 250.

And We read this as one-quarter of 1,000.

One-quarter multiplied by is the same as one-quarter of 1,000 and it's equal to 250.

Time to check your understanding.

We've been thinking about 1,000 up to now.

But now we're thinking about 100.

So what is 100 divided into two equal parts? Can you complete the bar model, the steps on the number line, the multiplication, and the stem sentence? Remembering that the whole is divided into hmm equal parts.

Each part is hmm of the whole.

So a few gaps to fill in there.

Pause the video, have a go, and then we'll look at them together.

How did you get on? I'm sure you know what the answers are going to be, but it's pulling it all together and thinking about how we're thinking about our numbers and the relationships between them this time.

So the whole is divided into two equal parts, and each part is one-half of the whole.

So our two missing parts on the bar model were 50 and our jumps on the number line were 50 are unmarked part label 50.

What about the multiplication? 1/2 X 100 = 50.

And we read this as one-half of 100 is equal to 50.

When we're thinking about fractions and certainly fractions of a whole number, we often use that language of one-half of, but we represent that with a multiplication.

Time for another check.

What is 100 divided into four equal parts? Same gaps for you to fill.

So pause the video, have a go, and we'll come back for some feedback.

How did you get on this time? So the whole has been divided into four equal parts.

Each part is one-quarter of the whole.

One-quarter of 100 is 25, there are four 25 in 100, and we can mark those jumps, 25, 50, 75, 100 jumps of 25 on our number line.

What about that multiplication this time? Remember Lucas said, "Four equal parts." So we're thinking a quarter.

So 1/4 X 100 = 25 but we read that as one-quarter of 100 is equal to 25.

So what about 10 divided into two equal parts? Can you see a pattern here? We've done 1,000 and then 100 and now 10.

Well, I'm sure you all know what 10 divided into two equal parts is.

We're thinking again as Lucas says about each part being a half.

We know that a half of 10 is 5, so our jumps will be 5.

A 1/2 X 10 = 5, and we read this as one-half of 10 is equal to five.

Nothing new there I think.

But remember we're looking at patterns so we're working our way through these different wholes.

A 1,000, 100 and 10 now.

So what do you think is gonna be next? Ah, 10 divided into four equal parts.

Now what's gonna happen for the first time when we get to 10 divided into four equal parts? Yeah, we're still thinking four equal parts, a quarter of the whole.

But this time we need to use a decimal to represent it.

So a quarter of 10 if we've have 10 divided.

So if we have 10 divided into four equal parts, each part is 2.

5, 2.

5, 5, 7.

5, 10.

1/4 X 10 = 2.

And we read this as one-quarter of 10 is equal to 2.

Oh, so we've done 1,000 as a whole, 100 as a whole, 10 as a whole.

Now one is our whole, what's 1 divided into two equal parts? Again, we're thinking half and it's a half.

If we've got one whole, half of it is a half and we represent that as a decimal as 0.

And there are jumps on the number line, 1/2 X 1 = 0.

5, and we read it as on-half of 1 is equal to 0.

I bet you can predict what's coming next.

Yes, one divide into four equal parts.

Have you spotted that we had that 250, 25, 2.

5 last time.

So this time, we're thinking one-quarter of the whole and we might remember what one-quarter is a decimal 0.

25, half of 0.

5, a quarter of one, 0.

25, 0.

5, 0.

75 and one.

1/4 X 1 = 0.

25, and we read that as one-quarter of 1 is equal to 0.

25.

So let's put all of those together.

The thinking about the halves this time 1,000, 100, 10 and 1, what do you notice? And Lucas says, "As the whole is divided by 10, the part of the whole is also divided by 10." So 1,000 divided by 10 is equal to 100, 500 divided by 10 is equal to 50.

And then at the other end, 10 divided by 10 is equal to 1, 5 divided by 10 is equal to 0.

25.

So as our whole is scaled down by a factor of 10, our part of the whole is also scaled down by a factor of 10.

And if you've been working on equivalent fractions, that might ring some bells with you as well.

So when the whole is one-tenth times the size, the part is also one-tenth times the size.

Time to check your understanding.

Have a look this time we've moved from our halves into quarters.

Can you spot some of those relationships when we're thinking about one-quarter? So pause the video, have a think about what's the same and what's different and we'll get back to discuss our answers.

How did you get on? What did you spot? Did you spot those 25s moving their way around the place value chart, I wonder.

And Lucas says again, "As the whole is divided by 10, the part of the whole is also divided by 10." Let's look at 100s and 10s.

A 100 divided by 10 is equal to 10, 25 divided by 10 is equal to 2.

The quarters are 25 tens, 25 ones, 25 tenths and 25 hundredths.

Oh, that's interesting depending on where we read to.

So a quarter of 1,000 was 25 tens, a quarter of 100 is 25 ones, a quarter of 10s is 25 tenths, and a quarter of one was 25 hundredths.

That's an interesting way of thinking about it, isn't it? You might want to explore that with a place value chart or maybe like a 10-year chart.

Time for you to do some practise now.

Can you complete the blanks in the table and for part two, what patterns can you see in your answers? So complete the blanks and then think about the patterns you can see in your answers.

Pause the video, have a go, and we'll come back together for some feedback.

How did you get on? So let's think about that language of multiplying as the of.

So a half of a thousand is 500, half of a hundred is 50, half of 10 is five, half of one is 0.

And then with our quarters, a quarter of a thousand is 250, a quarter of a hundred is 25, a quarter of 10 is 2.

5, and a quarter of one is 0.

25.

That hopefully was kind of the easy bit.

Then I asked you to have a think about the patterns that you saw in your answers.

How could you describe what was happening? So this is some of the things that we talked about.

As you look across the table, the wholes and the parts divide by 10 each time, but the digits stay the same.

So let's have a look at that top row.

We had 500 divided by 10 is 50, divided by 10 is equal to five, divided by 10 is equal to 0.

Does that work for our quarters as well? So a quarter of a thousand was 250, divide that by 10 to get 25, divide that by 10 to get 2.

5, divide that by 10 to get 0.

25.

And that match with our 1,000 becoming 100, becoming 10, becoming one dividing by 10 each time.

What about going up and down the column? What did you notice? Yeah, as you look down the table, the parts that represent a quarter are a half of the value of the parts that represent a half and that makes sense, doesn't it? If you imagine something divided into two equal parts and then divided into four equal parts, you would see that that quarter is a half of the half.

So our values would half as we went down the table and compared halves with quarters.

So let's pick half of 10, half of 10 is five, a quarter of 10 is 2.

5, and we also know that half of five is 2.

So I hope you spotted those patterns as you worked as well.

Okay, for the second part we're going to think about dividing into five and 10 equal parts.

So what is 1,000 divided into ten equal parts? So Sam says, "The whole is divided into ten equal parts.

Each part is one-tenth of the whole." So we are finding one-tenth dividing by 10.

So one-tenth of 1,000 is equal to 100.

We've got steps of 100 going along our number line as well, all the way up to 1,000.

1/10 X 1,000 = 100.

And remember how we were reading that? We read that as one-tenth of 1,000 is equal to 100.

What about 1,000 divided into five equal parts? So we had 10 equal parts and each part was 100.

I wonder if you can use that to help you work out what 1,000 divided into five equal parts is going to be.

That's right, it's 200, isn't it? We've only got half the number of parts, so each one must be twice the size.

The whole is divided into five equal parts.

Each part is one-fifth of the whole.

And we can now count along our number line, which was in 100s, and we can count along in 200s, and we can see that two of those 100 steps have combined to make one step of 200.

So we've got five steps of 200 instead of 10 steps of 100, 1/5 X 1,000 = 200 and we read it as one-fifth of 1,000 is equal to 200.

Ah, can you see what we're doing here? We started with 1,000, now we have 100.

What is 100 divided into 10 equal parts? Well, we really should know this, shouldn't we? So each part is one-tenth of the whole and each part, of course, is worth 10.

And we can count up in tens all the way to a hundred and mark our number line.

1/10 of 100 or 1/10 X 100 = 10, and I beat Sam too at this time.

We read this as one-tenth of 100 is equal to 10.

And you can see there, we've got our tenths, but this time we want five equal parts.

So each part must be one-fifth of the whole and we can see that we're kind of combining two of those tenths to equal one-fifth.

So 100 divided into five equal parts is equal to 20.

And there we can see our 20s counting up to 100.

1/5 X 100 = 20, one-fifth of 100 is equal to 20.

Over to you to check your understanding.

What is 1 divided into ten equal parts? Can you complete the bar model, the number line, and those stem sentences, and the equation to help us.

Pause the video, have a go, and we'll get back together for some feedback.

How did you get on? What knowledge did you use? Did you use your knowledge of decimals and fractions or did you use the patterns that we've been spotting thinking about 1,000, 100, 10 and now 1, whichever way you did it, you'd have realised that 1 divided into ten equal parts gives parts of 0.

1, and we can count along the number line up to one with those.

The whole is divided into ten equal parts, and each part is one-tenth of the whole.

And we can represent that with the equation.

1/10 X 1 = 0.

1, and we read that as one-tenth of 1 is equal to 0.

I wonder if you can guess what's coming next.

Over to you again, another check.

What about 1 divided into five equal parts? We've left you the number line there of tenths.

Can you use that to help you work out what 1 divided into five equal parts is going to be? And then complete the gaps in the equation, and the stem sentences.

Pause the video, have a go, and we'll come back for some feedback.

How did you get on? Did you realise that it was going to be 0.

2? We're combining two of those tenths to be one-fifth, so I'll jump along the number line is in 0.

2, whole divided into five equal parts.

Each part is one-fifth of the whole.

1/5 X 1 = 0.

2, and we read that as one-fifth of 1 is equal to 0.

Okay, so let's have a look at these altogether.

What have we left out? What are the gaps that need to be filled in? Can we fill those in? And then what do we notice? So we've got 1,000 divided into 10 steps of 100 and 5 steps of 200.

So what about 100? 10 steps of 10 or five steps of 20? Well done.

What about the 10? We've got five steps of two, 10 steps of one, of course we have, and we haven't filled anything into one, but you've just been working on that.

So 10 steps of 0.

1 and five steps of 0.

So now what do you notice about those bar models? Well, yeah, we divide the whole by 10 to calculate one-tenth and we can work that out for any of them.

A 1,000 divided by 10 is equal to 100.

A 100 divided by 10 is equal to 10, 10 divided by 10 is equal to 1, and 1 divided by 10 is equal to 0.

And you might have spotted that the fifths are double the value of the tenths.

You can see that in each of the bar models.

And these are useful ones to remember.

One-tenth is a decimal is 0.

1 and one-fifth is a decimal is 0.

That's a really useful one to remember as you go forward with your maths.

Time to check your understanding.

Here we've got some number lines.

What do you notice about the two number lines? Pause the video, have a go, and we'll come back for some feedback.

So what did you notice? Well, Sam says, "Whatever the value of the whole, one-fifth is always double one-tenth or one-tenth is always half the value of one-fifth." So we can see here that we've got a number line from 0 to 1,000, and our tenths are 100, and our fifths are 200.

So yes, double the value of a 10th is equal to a fifth or half the value of the fifth is equal to a tenth.

And is that true on our number line from zero to one? Yes it is.

A 10th is worth 0.

1 and a fifth is worth 0.

2, double the value of the tenth.

The tenth is half the value of fifth.

I wonder what else you noticed as well.

Time for you to do some practise now.

Can you complete the blanks in the table and then have a look and see what patterns you can see in your answers? So pause the video and we'll come back together later for some feedback.

How did you get on? Well, we've completed the blanks there.

One-fifth of 1,000.

Remember that's how we read that multiplication.

A fifth of 1,000 is equal to 200, a fifth of 100 is equal to 20, a fifth of 10 is equal to 2, and a fifth of 1 is equal to 0.

1/10 X 1,000, which we read as one-tenth of 1,000 is 100, a tenth of 100 is 10, a tenth of 10 is 1, and a tenth of 1 is 0.

So that was kind of the easy bit completing the blanks.

Now what about those patterns? What did you see? So we could see that as we look across the table, the whole and the parts divide by 10 each time, but our digits stay the same.

So if we're thinking about five equal parts in 1,000, 100, 10 and 1, they'll always be a two involved.

So one-fifth of 1,000 was 200 and a fifth of 1 was 0.

And the same for our tenths, there's always going to be a 1 as a digit there, and they're always dividing by 10 each time.

So 1,000 divided by 10 is 100, 200 divided by 10 is 20, 10 divided by 10 is 1, 1 divided by 10 is 0.

We can see that multiply divide by 10 relationship, divide by 10 if we are looking right across the table, multiply by 10 if we are looking left across the table.

What about going up and down the table? Yes, as you look down the table, the parts that represent a tenth are a half the value of the parts that represent a fifth.

So a tenth is a half of a fifth.

So if we're considering the same whole, then a tenth of that whole will always be half the value of the fifth of that whole.

So let's look at 100, a fifth of 100 is equal to 20, and a tenth of 100 is equal to 10.

One-tenth is half of a fifth and 10 is half of 20.

I hope you spotted those relationships and patterns too.

And we've come to the end of our lesson.

So we've been identifying and describing patterns in the number system.

We've looked at it in bar models and on number lines, but you may well have looked at it on place value charts as well.

What have we learned? We've learned that one-half as a decimal is 0.

5 and one-quarter as a decimal is 0.

25.

One-tenth as a decimal is 0.

1 and one-fifth as a decimal is 0.

And when you are dividing 1, 10, 100 and 1,000 into 2, 4, 5, and 10 equal parts, you can see those patterns in the parts.

We could see those multiplying and dividing by 10 patterns as we went from 1 to 10 to 100 to 1,000.

And we could see the doubling in halving patterns when we talked about halves and quarters and about fifths and tenths.

We also learned that you can read 1/5 X 1 = 0.

2 as one-fifth of 1 is equal to 0.

That multiplication is often useful to read as of, especially when we're dealing with finding fractions of something and that of, means the same as multiplying.

We can replace it anytime we look at a multiplication, but it's often the language around fractions is when that of is most important.

Thank you for all your hard work and your pattern spotting today.

And I hope I get to work with you again soon.

Bye-Bye.

I've finished the video