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- [Clicking] I've had this for so long now, I keep trying.

Oh, no.

I keep trying and I suppose that's the important thing, but each time I do, it leaves me frustrated.

I wouldn't say angry, not quite that level of emotion, but frustrated for sure.

And I'm about to start teaching a maths lesson.

So I'm thinking it's probably best that I put this down [Thump] so that I can focus maybe with a few deep breaths first, so that I can focus on this maths lesson.

Do you need to step away from any distractions, so that you're able to focus and give me your undivided attention for the next 20 minutes? If you do, press pause.

I'm going to use this time to take those deep breaths.

And I'll see you back here, when you're ready to start.

In this lesson, we will be comparing and ordering fractions and decimals.

We'll start off with a comparing fractions activity before we look at comparing fractions and decimals, then take a look at fractions between fractions.

Really important step before I leave you with your independent task.

Things that you're going to need: Pen or pencil, some paper, pad or book from school and a ruler.

Press pause, get yourself sorted with these items, come back, and we start.

Here we go.

Getting us started, which symbol is missing between each pair of fractions? Greater or less than or equal to? Is the statement an equality? Are the two fractions equal or is the statement an inequality? Is there an imbalance between the two fractions? Are they not equivalent? Press pause, work out the missing symbols, and think about how you know, then come back and check.

Hold up your paper, let me see which symbols you picked.

Okay and oh, I like I can see on some of these some written explanations for how you know, as well.

Let's work through these symbols, shall we? So for the first one we've got 1/2 is less 4/8, is less than 5/8.

I say 4/8 because that's what I used to explain it.

5/8 is greater than 4/8, 4/8 is equal to 1/2.

So 1/2 is less than 5/8.

Next 12/32 is equal to 3/8.

There's a relationship.

12 divided by four is three 32 divided by four is eight.

So those fractions are equivalent.

81/100 is greater than 4/5, 4/5, 8/10 80/100.

Oh, it was close.

But 81/100 is 1/100 more than 4/5.

So it's great to find a one.

8/20 is less than 45/100, 8/20.

20 to 100 multiplied by five, eight fives.

40/100 is less than 45/100.

So important there to use those relationships and connections to tenths and hundredths, at times, to help give those explanations.

How did my explanations compare to your own? Similar? Different? Did we have the same solutions perhaps in differences in our explanations? Next, how many decimals could we place on this line? Take a look at it.

I'm representing it in three different ways, but the number represented is one.

The space between the start and the end of the line, the number stick, the bead string, is one.

How many decimals could we represent? On the top line, how many might you say? Ah, so one decimal at each division? So, divided into 10 equal parts so we could represent 10 decimals.

How about then on the bead string or the number line in the middle? The whole has been divided into 100 equal parts.

Ah, we could represent 100 different decimals.

I wonder what each equal part represents then, if we've got this space between zero and one? If it's divided into 10 equal parts, each equal part's worth 1/10.

What about if it's divided into 100 equal parts? Each equal part is worth 100, 100/100, one, 10/10, one.

Okay.

I've got some fractions I'd like you to complete.

Can you tell me what each of these fractions is? So start on the left.

Good, 2/10.

The space between zero and one is divided into 10 equal parts.

That's the second of those 10 equal parts, 2/10.

Next nice, half.

How many tenths? 5/10, halfway along the number stick between zero and one.

The last one? 9/10.

Good.

Okay, how about on the bottom? We're now thinking about the space divided into 100 equal parts.

What would this one be? Then, this one.

So you can tell me more in a moment.

And then, this one.

So I've just shown you where on the number line each of them would be.

So what would the first one be? Good, 10/100.

Or what? How many tenths? 1/10 equal to 10/100, good.

The next one on the bottom, in the middle.

Good 60/100.

How many tenths? 6/10.

And the last one? Good, 80/100 or 8/10.

Let's have a look at this now.

What's changed? Let me go back and forwards.

What's changed? Yes.

Now, the number line is representing the space between zero and 0.

1, zero and 1/10.

Now, let me just help us to understand what that means.

If we look at this one again, the space between zero and one, where would O.

1 be? There, 1/10.

So all we've done here is increase the number of equal parts, the space between zero and 0.

1, a bit divided into.

So instead of 10 parts here, notice that again.

10 parts.

Now we've got how many? 100 equal parts between zero and 0.

1.

So what does each of those parts represent? 0.

1, it's been divided into 100 equal parts.

Each equal part is worth one thousandth.

Whereas in the top number line, when it's divided into 10 equal parts, 0.

1 divided into 10 equal parts, each equal part is worth 1/100.

When it's divided into a hundred equal parts, 1000th.

Okay, this is a lot to get our heads around this isn't it? So we're thinking of 0.

1 as 100/1000 on that bottom number line.

Whereas on the top, we're thinking of it as 10/100, 0.

1.

So in thinking about the number line, the number stick, bead string divided into 100 or 10 equal parts.

And the whole space, it's only worth 1/10, but we can think about 1/10 divided into 100 equal parts, each part equal to 1/1000.

Or we can think about 1/10 divided into 10 equal parts, each equal part worth 1/100.

Good.

So tell me about these three fractions.

What are the missing numerators? First one? Good, 2/100.

Next? 5/100 and then 7/100, all the way up to 10/100, 0.

1.

How about it on the bottom? So looking again, this time they're worth thousandths.

What are the missing numerators from the left? Good, 10/1000, 1/100.

Next? 60/1000 or how many hundredths? 6/100.

Last, 80/1000.

Well done.

Or how many hundredths? 8/100.

Really good work there, that's challenging thinking.

We've got so, so small when we're thinking about zero to 1/10, and then we've made it even smaller by dividing that into a 100 equal parts.

I think, though, you're set to have a quick go at this task.

Place these fractions on the number line.

Use your connections to tenths, to hundredths, to work out their location.

Come back when you're ready to check.

How did you get on? Can I have a look at anything you've drawn? Maybe you've got a list of the fractions and decimals in order.

Maybe you've got your own number line.

Let me see.

Looking really good.

Shall we compare? Now, I'm just going to draw some arrows for you so you can check these off.

I'm going to start with just 0.

15, nice and easy, 15/100.

The number line here, zero to one, divided into a 100 equal parts.

Each equal part is worth 1/100, so 15/100 there.

The next one I'm going to look at is 0.

64, 64/100.

It's past halfway, it's past halfway.

64/100, there.

And then I'm going to look at 0.

46, 46/100, just under halfway.

46/100.

Okay, the next three, not quite as easy because they're not in decimal form already.

So I could think about 3/5.

I could then think, "Ah, how many, tenths? 6/10." 6/10 along the number line there.

Equally, that's 60/100.

Next one, 9/20.

That's where it is.

How did I find it? Well, I could think about that space between zero and one divided into 20 equal parts.

It's those smaller divisions on the top of the number line and that's the ninth of them.

Also, I could think about well, if that's 4/10 and 5/10, that's equal to 8/20 and 10/20.

9/20 is halfway between the two.

What's left? Ah, 3/25.

I left that one to last for a reason, 3/25.

Okay, so here I was thinking relationship to 100.

3/25, this is where it is and I though about that as 12/100.

3/25, 25 four times, three four times, 12/100 along that number line.

Okay, as we were looking through that activity, I gave an example of fractions halfway between.

Let's just go back.

It was the 9/20.

9/20 is halfway between eight and 10/20, fractions between fractions.

But of course, 8/20 and 10/20 originally was 4/10 and 5/10.

So notice how I approached finding the halfway point between them, some equivalent fractions work.

We're going to have a go at that.

Let's bring you back.

Finding some fractions halfway between zero and one quarter, which fraction is halfway between? Use the rectangles to help you.

Absolutely, 1/8.

How about between one quarter and two quarters? Which fraction would be there, halfway between? Super, 3/8.

And that would continue.

We could fill in the other eighths between two quarters and three quarters, three quarters, and one.

Here we would have them then.

Now, another way to think about it and it's not correct.

The space between one quarter and two quarters, We could have said 1.

5 quarters or the space between three quarters and one, three and a half quarters, but that's not accurate.

That's not how we record fractions, which is why equivalent fractions are so much more helpful and useful here.

Now, sometimes our work with equivalent fractions takes us to very, very small fractions based on very, very large denominators.

So for example, one quarter, two quarters, three quarters we can think of as a number of hundreds, connections to their decimal form.

What are the missing numerators here? Good, 25/100, 0.

25.

How do we say 50/100 as a decimal? 0.

5.

And what's three quarters as a decimal? 0.

75.

Now to help us think about eighths as decimals, we need to increase the number of equal parts, increase the size of our denominator, making each of those equal parts smaller of course, and work with thousandths.

So let's stick with what we know to begin with.

What would each of those fractions be equal to as a number of thousandths? Call them out.

Absolutely.

You can see the relationships, 25/100, 250/1000.

From there, are we able to work out the number of thousandths that those eights would be equal to? How many for 1/8? 250 is, is 2/8.

125.

And then using that 125/1000, we're able to work out 3/8, 5/8, 7/8.

It's 125, more than 2/8, sorry.

It's 125, more than 4/8, 125/1000, that is.

And there are nice connections, then, to the decimal form.

So you can tell me that decimal of 5/8, 0.

625.

What's the decimal of seven eights? 0.

875.

Tell me 1/8 as a decimal.

0.

125.

And we've got really deep there.

Lots of connections, equivalents, hundredths, tenths, thousandths, and converting fractions like 3/8 into a decimal.

In a moment, you're going to have a go at an activity where you continue to think about equivalents and fractions that are halfway between two other fractions.

I'd like you to prove to me now.

You can pause in a moment.

Show me that three quarters is halfway between 7/10 and 4/5.

I've given you a number line to get you started, but what changes might you make? What equivalents might you bring in to prove that this is correct? Pause.

Come back when you've got some ideas.

Should we have a look at that? Should we take a look? Should we take a look? So hold up your paper, let me see how you went about it.

Okay, so let me share some of my mathematical thinking for this.

Took a look at how it was presented, 7/10, 4/5.

Straightaway, I'm thinking that 4/5 and make that a number of tenths, 8/10.

So now I'm looking at 7/10 and 8/10, fractions between fractions, between 7/10 and 8/10.

We can make another change.

We can think as well as the decimal form, we can be thinking 70/100 and 80/100.

Or which number is halfway between 70/100 and 80/100? 75/100, 0.

75.

We know that that is the decimal form already.

It helps to confirm it.

So we've shown that three quarters is definitely halfway between, sorry, between 7/10 and 4/5.

And we've shown that by making connections to equivalent fractions of tenths, of hundredths, and the decimal form as well.

I'd like you to pause and have a go at activities similar to that last one, but independently.

Come back when you're ready to share.

How did you get on? How was your paper? Let me see how you've represented those number lines or if you've perhaps joined slightly different versions as you've copied from the screen.

Fantastic.

Let's have a look then.

So the fraction between 1/5 and 3/10 is 5/20.

You're thinking 1/5, 2/10, 3/10, 3/10.

Then we're thinking twentieths.

4/20 and 6/20 is 5/20.

Second one.

So the solution is 5/16.

We reach it by finding one quarter, finding one, sorry, 3/8, which is halfway between one quarter and two quarters, 2/8 and 4/8.

But now I'm looking halfway between one quarter and 3/8.

So again, it makes some changes.

I can think of that as 2/8 and 3/8.

Still not helping.

4/16 and 6/16, 5/16 is halfway between the two.

Okay, last one.

Halfway between one quarter and 5/16.

Okay, one quarter.

We can mark that up.

And one quarter of the way along the line, half of a half.

Halfway between zero and a half.

5/16 Halfway, one half, two quarters, 4/8, 8/16.

One quarter, 2/8, 4/16.

So if one quarter is 4/16, 5/16, it's going to be here.

Right.

And we're looking for that to find that division, that marker between the two fractions.

What's that worth? So, okay.

One half, sorry.

One quarter, 2/8, 4/16.

That's helpful.

Oh, halfway between 4/16 and 5/16? So let's continue.

Use of our equivalent fractions understanding, 8/32 and 10/32.

Halfway between the two, 9/32.

Wow, that was a tough session.

I'm so pleased that you have made it all the way through to the end.

Tough, but with those connections to our equivalent fractions, made a lot easier.

Lots of thinking needed still though.

So where does that break coming up for you? I bet.

First, if you'd like to share any of your fantastic learning from this session with Oak National, please ask a parents or carer to share your work on Twitter, tagging @OakNational and #LearnwithOak.

Wow.

You completely took my mind off this.

Well, I say completely.

At least during the lesson.

Now is time for me to either take a well-earned break, well away from this puzzle or to perhaps give it a little bit more time.

What do you guys do next? Please take a break as well if you've got any more lessons lined up for the day.

I look forward to seeing you again soon for some more maths and until then, look after yourselves.

Bye.