Lesson video

Hi there, my name is Mr. Tilstone.

I hope you're having a lovely day, and I hope that you're ready for some maths.

I certainly am.

If you are, will you help me to count down from five? Are you ready? 5, 4, 3, 2, 1.

Let's begin.

The outcome of today's lesson is this.

I can recognise and explain the equivalence of one-half and two-quarters.

And we've got some keywords.

If I say them, will you say them back? Are you ready? My turn.

One-half.

Your turn.

My turn.

Two-quarters.

Your turn.

And finally, my turn.

Equivalent.

Your turn.

Those words are going to come up a lot today.

Our lesson is split into two parts or two cycles.

The first one will be recognise two-quarters, and the second will be equivalence.

So for now, let's focus on recognising two-quarters.

What do you know about quarters? Hmm.

In this lesson, you're going to meet Alex and Sam.

Have you met them before? They're here today to give us a helping hand with the maths.

Alex and Sam are exploring quarters with this delicious yummy-looking cake.

Alex says, "I can divide this cake into quarters." Could you do that, do you think? How would you divide it into quarters? Something like that? Sam says, "Each part is one-quarter." And why is it one-quarter? What makes it one-quarter? Well, the whole has been divided into four equal parts.

And here's one of them.

"I have one-quarter of the cake," says Alex.

"What if I give you another slice?" says Sam.

What will he have then? He won't have one-quarter anymore, will he? What will he have? "Now I have two-quarters of the cake." "What if I give you another slice?" says Sam, It's very generous of her, isn't it? What would Alex have now? He wouldn't have one-quarter, he wouldn't have two-quarters.

What would he have? "Now I have three-quarters of the cake." "What if I give you another slice?" Well, that is kind, isn't it? But he wouldn't have one-quarter, he wouldn't have two-quarters, he wouldn't have three-quarters, he would have four-quarters of the cake.

"Now you have the whole cake.

Four-quarters is equal to one whole." Alex and Sam continue to explore quarters.

They're not going to use cakes this time.

I'll bet Alex has had quite enough cake for today.

They're going to use a square.

And he says, "I can divide this into quarters." How would you do that? Is there more than one way? This is the way that he's chosen.

Was that the way that you were thinking or were you thinking of a different way? "I can shade one-quarter, "says Sam.

Which quarter would you shade? It could be this one.

It could be one of the other quarters too, but she's chosen this one.

"I can shade two-quarters," says, Alex.

What would you do now? Maybe you chose that quarter.

It could have been one of the other two.

Now, Sam says, "I can shade three-quarters." How about that? That's one way to shade three-quarters.

Alex and Sam continue to explore quarters.

"Now the whole shape is shaded," says Alex.

"That means that four-quarters are shaded." So what does that tell us about four-quarters? Four-quarters is equal to or equivalent to one whole.

Alex and Sam continue to explore quarters.

They're quarters experts.

Are you? Alex says, "I can divide these cakes into quarters." What would you do? How would you divide those cakes into quarters? Hmm? Would you do this, one for you, one for you, one for you, one for you, and then there's some more cake, so one for you, one for you, one for you, one for you, and there's still some more cake, so one for you, one for you, one for you, and one for you, and all the cakes have gone? "I can circle one-quarter," says Sam.

Which one-quarter would you circle? This is the one that Sam's chosen.

It could have been one of the other three.

"I can circle two-quarters," says Alex.

Which two-quarters could you circle? Maybe you chose those two, maybe you chose a different two, but that's two-quarters.

And then Sam says, "I can circle three-quarters." Which three-quarters would you circle? Maybe it's these three-quarters or maybe different ones.

What about now? What have we got now? "Now all of the cakes are circled." "That means that four-quarters are circled." So what can we say about four-quarters? It's the same as a whole.

Let's have a little check for understanding.

Match the images to the correct fraction.

So we've got a few different images here and we've got one-quarter, two-quarters, three-quarters, and four-quarters.

Which is which? Pause the video and explore.

How did you get on with that? Let's have a look.

Well, this is one-quarter.

So our whole was split into four equal parts and that was of them.

And this is two-quarters.

Our whole was split into four equal parts and that was two of them.

This is three-quarters.

Our whole was split into four equal parts and that was three of them shaded.

And here's four-quarters.

Our whole was split into four equal parts and we've circled all of them.

Alex and Sam use numerals to write each fraction this time.

Can you remember how to do that? This is how to write one-quarter.

1/4 of the cakes are circled, 2-4 of the cakes are circled, 3/4 of the cakes are circled, and 4/4 of the cakes are circled.

So remember that four means quarters.

This is how to write 1/4.

So remember that fraction bar means has been divided into, the four tells us how many equal parts, and the one tells us how many of those parts we've got.

So that's how to write 1/4.

1/4 of the cakes are circled.

Now 2/4 of the cakes are circled.

We're going to circle 3/4 in a moment.

How would we write that? Just like this, 3/4 of the cakes are circled.

And finally, 4/4 of the cakes are circled.

There are 12 cakes in total.

You can count them, but there are.

Alex and Sam can write equations to show the amount that has been circled.

So 1/4 of the cakes are circled here and our total was 12, so we can write 1/4 of 12 equals 3.

How could you write that? What would you write for the equation this time? 2/4 of 12 equals 6.

What about now, now that 3/4 of the cakes are circled? Remember the total is 12.

That's our whole.

3/4 of 12 equals 9.

What about 4/4? What equation would we write now? The total is still 12.

We've got all 12 cakes.

4/4 of 12 equals 12.

Let's have a little check.

Look carefully at these eight sweets.

What is missing from each equation? So we've got 1/4 of eight equals something.

Something of eight equals four.

3/4 of eight equals something.

And 4/4 of something equals eight.

And the answer to all of those can be seen within that image.

So look very carefully.

Pause the video and give it a go.

Let's see, so 1/4 of eight equals two and you can see that on one of the hands.

2/4 of eight equals four.

So if you're looking for four sweets, you can see that on two hands, so 2/4 of eight equals four.

And then 3/4 of eight, that would be how many sweets there are in three hands equals six.

And then finally, 4/4 of something equals eight.

4/4 of eight equals eight.

And you can see that in the image with all of the hands.

It's time for some practise.

I think you're ready.

You're doing really well.

Find different ways to divide these shapes into quarters.

Can you shade 2/4 of each shape? So each shape appears twice.

Can you think of a different way each time to divide that shape into quarters and to shade 2/4? There are so many ways you could do this.

Number two, write the fraction that each image represents.

Now, look very carefully.

And then a little extra challenge from Sam.

"Could you write an equation to match the fraction of the amount in B, D, or F?" Hmm.

Let's see how good you are at writing those equations.

Okay, pause the video.

Good luck with that.

And I'll see you soon for some feedback.

Welcome back.

How are you getting on? Are you feeling confident about quarters? Let's have a look.

There are many, many possibilities for this first one.

This is one way to divide that shape into quarters, one of many ways.

And we could shade those two.

We could have shaded a different two, but that's showing 2/4.

Here's a different way to divide that shape into quarters, still using rectangles.

You didn't have to do that.

There's lots of different ways to do it.

What about triangles? Did you get that one? And here are 2/4.

Again, it could have been a different two.

This shape has been divided into quarters, again using rectangles.

And here are 2/4, and this shape has been divided into triangles, but still quarters.

I can still see four equal size parts there.

And we just need shade.

Any two of those.

What about these two? Okay, the next one was a little bit trickier, but what about this? I can see four equal size parts, so it is showing quarters.

Now, we pick any two of them and shade them in.

These two will do.

What about a different way to do it? How about this? Did you get something like that, I wonder? And again, we pick any two of those four equal size parts and we shade them in.

What about those circles? Let's divide those into quarters.

That's one way.

Let's pick two.

Those two.

And here's another way, and again, we can just shade two of those in.

What about those two? There were different ways to do it.

Here are just some.

And number two, write the fraction that each image represents.

Well, A, I can see four equal size parts and that's one of them, so that's our fraction using numerals.

And B, I can see four equals size parts, each one's a cake and that's three of them, so that's 3/4.

For C, I can see four equal size parts and we've shaded two of them, so that's 2/4.

For D, I could see four equal size parts and we've shaded one of them, that's 1/4.

That was a bit trickier.

And for E, I could see four equal size parts and all four of them shaded, so that's 4/4.

And for F, I could see four equal size groups and we shaded two of them, so that's 2/4.

Could you write an equation to match the fraction of the amount in B, D, or F? Well, let's have a look at B.

How many cakes have we got? We've got four.

So 3/4 of four equals three.

Let's look at D.

How many circles have we got? We've got eight.

So the equations are going to be 1/4 of eight, 1/4 of eight equals two.

And then finally for F, we've got 16 pennies and 2/4 of them are shaded, so it's 2/4 of 16, and you can see that is eight.

You're doing very, very well.

You're ready for the next cycle and that is equivalent, so let's go.

Alex and Sam have another look at their work on quarters.

"The whole has been divided into four equal parts.

Two of those parts have been shaded, so 2/4 of each shape is shaded." Would you agree? Let's have a look at them all.

Yes, 2/4 of each shape is shaded.

Sam says, "I agree.

I've noticed something else about these fractions too." Hmm? Have you noticed something else? That's what good mathematicians do.

They notice things.

Hmm? "What have you noticed, Sam?" asks Alex.

She says this, "One-half of each shape is shaded." Oh, did you notice that too? Alex says, "Doesn't that mean that the whole has been divided into two equal parts?" These aren't, are they? They're divided into four equal parts.

Alex is remembering something about his work on halves there.

He's right.

So Sam says, "Can you imagine some of the lines have disappeared, like this?" So you can picture this with your mind.

Oh yeah, that lines disappeared.

Now that is halves.

Look at that.

Now the same shapes with the same amount shaded are showing halves.

"Oh yes, I see it now.

1/2 of each shape is shaded," says Alex.

"If I bring the lines back again, one-half of each shape is still shaded." So are you ready? Can you picture those lines back? Let's see.

So at the moment, they're showing halves and now they're showing two-quarters.

Hmm.

So what can we say about one-half and two-quarters? Does that mean that two-quarters and one-half are the same amount? What do you think? Does that mean that? "Yes, yes, Alex, well done.

Two-quarters equals one-half.

There's a special name for this." Hmm? Can you remember? It's a little bit like equals.

Alex has remembered.

He says, "I think I remember.

Are they equivalent?" It was one of our keywords today.

And yes, Alex, you are quite right.

One-half and two-quarters are indeed equivalent.

Two-quarters and one-half have equivalence.

That means that they are equal or equivalent.

They are fractions that describe the same amount.

You can show their equivalence using two strips of paper.

Hopefully you've got some paper in front of you and you can explore this too.

Let's have a look.

So we've got two equal sized strips of paper.

If we fold both strips in half, like so, and then fold one of the strips in half again, like so, and then unfold them both, like so, we've got two equal size strips, but they're divided into different equal amounts.

One strip is divided in half and the other in quarters.

Shade or cut off one-half.

Shade or cut off two-quarters.

There we go, so we've cut them off this time.

And what can you say? What can you see? Two-quarters and one-half are exactly the same size because they are equivalent.

That's a really important learning point.

I'd like to say that again, but I'd like you to say it with me.

I'll read it really slowly.

Are you ready? Let's go.

Two-quarters and one-half are exactly the same size because they are equivalent.

Hmm.

Very good.

Let's do that one more time.

I'm going to zip my lips and you're going to say it.

You ready? Zip, go.

Let's do a little check for understanding.

Use two identical strips of paper to show that one-half and two-quarters are equivalent.

Okay, have fun with that.

Pause the video.

Did you manage to prove our generalisation that one-half and two-quarters are equivalent? Hopefully you did.

Hopefully you took some paper.

Two different size bits of paper.

Didn't have to be as thin as these, could be a different size.

And you showed that two-quarters and one-half are equivalent just like that.

One strip of paper has been divided into quarters because there are four equal parts, and one has been divided in half because there are two equal parts.

Two-quarters of one strip and one-half of the other strip are shaded.

And you can see they're exactly the same amount.

And if you thought of that as something else, like let's say a chocolate bar, I would be just as happy to have two-quarters of that chocolate bar as half of that chocolate bar because it's the same, it's the same amount.

They're the same size, which shows that these fractions are equivalent, our keyword.

Alex and Sam want to explore the equivalence of one-half and two-quarters.

"I will find one-half of 12 apples." Here we go, so 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

What have we got? What's one-half of 12 apples? One-half of 12 equals six.

And Sam says, "I will find two-quarters of 12 apples." What do you think the answer's going to be? Hmm, and let's find out.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

And here's two-quarters.

What do you notice? Two-quarters of 12 equals six.

They're the same.

One-half of 12 equals six.

Two-quarters of 12 equals six.

They're equivalent.

Both the fractions of 12 are equal to six.

That is because one-half and two-quarters are, as we've already proved, equivalent.

Alex and Sam want to explore the equivalence of one-half and two-quarters.

And this time Alex says, "I will find one-half of 20 using a bar model." So not a particular object, just 20.

So he's made his bar model, split into two equal parts, and he's going to find one-half of 20.

How could he do it? What would you do? He's going to use marks like this.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20.

And that's one-half.

So one-half of 20 equals 10.

I like how he set his marks out by the way into those fives, like dice fives.

And Sam says, "Because they're equivalent, I think that two-quarters of 20 will also be 10." What do you think? Do you agree with Sam? Because one-half of 20 equals 10, does that mean that two-quarters of 20 equals 10? Shall we find out? "I will prove my idea by finding two-quarters of 20 using a bar model." What bar model will she draw? How will she split it up? Four equal parts just like this, and she's going to use the same system using marks.

Let's go.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20.

And she's finding two-quarters and that's two-quarters.

What do you notice? Two-quarters of 20 equals 10.

Her prediction was right.

And Alex says, "I wonder if you could have just used my bar model.

Sam says, "Yeah, you're right.

That would've saved me some time.

If I know that half equals two-quarters, I don't need to draw another bar model, do I?" "No Sam, you don't.

You could also draw some more lines on my bar model to show two-quarters." Oh, this is a good idea.

So let's draw some more lines.

So at the minute, Alex's bar model is showing halves, let's make it show quarters by drawing extra lines on and nothing else has changed.

The amount hasn't changed, the total hasn't changed.

It's still showing one-half of 20 equals 10 and it's still showing two-quarters of 20 equals 10.

Let's have a check.

Use this bar model to find two-quarters of four.

So what equals-two quarters of four? Pause the video.

Let's see.

1, 2, 3, 4.

And we want two-quarters.

At the minute, that's showing one-half, that's one-half of four.

But what if we did this? Is that what you did? Did you divide your halves into quarters? That means that two equals two-quarters of four.

So we could say two equals one-half of four or two equals two-quarters of four.

I think you're ready for some final practise.

In fact, no, I know you're ready for some because you're doing really well.

1A, explain to a partner how this bar model could help you to find two-quarters of 16.

So hmm equals two-quarters of 16.

B, tick the shapes that have two-quarters shaded.

And number two, Alex and Sam are talking about how to find two-quarters of 32.

Who do you agree with? Alex says, "I would divide 32 counters into four equal parts and count how many counters in two of those parts." And Sam says, "I would divide 32 counters into two equal parts and count how many counters in one of those parts." Hmm.

Who do you agree with? Take your time with that.

Read it again if you like.

Have a good think.

Okay, pause the video.

If you can work with somebody else, I always recommend that and then you can share ideas with each other.

I'll see you soon for some feedback.

Welcome back.

Let's have a look.

So 1A, explain then to a partner how this bar model could help you to find two-quarters of 16.

At the minute, it's showing one-half of 16, which is eight, but how can we use that to find two-quarters? Well, you might have said something a bit like this.

I know that one-half and two-quarters are equivalent.

So if I find one-half of 16, it will be equal to two-quarters of 16.

You might have even drawn some lines on that as well.

Tick the shapes that have two-quarters shaded.

This is one of them, four equal parts, two of them shaded.

Did you get this one? That was quite tricky because it just looks like it's halves, not quarters, but we could turn it into quarters.

So it is showing one-half, but it's also showing two-quarters.

And what about this one that's got four equal parts, two of them shaded? And what about this one? That was another tricky one because again, it looks like halves, which it is.

It's showing one-half, but it's also showing two-quarters.

It's tricky to see two-quarters in this shape, but you can see that one-half is shaded.

Since one-half and two-quarters are equivalent, that means two-quarters must be shaded.

It's just a bit harder to prove or to see.

And then Alex and Sam are talking about how to find two-quarters of 32.

Who do you agree with? Alex says, "I would divide 32 counters into four equal parts and count how many counters in two of those parts." And some says, "I would divide 32 counters into two equal parts and count how many counters in one of those parts." Now, did you spot that actually they were both right? Let's see, that's four equal parts and here's your 32 counters.

And in two of those parts, we've got 16.

And here are two equal parts.

And our 32 counters have been divided up and we've still got 16 in one-half because two-quarters and one-half are equivalent.

So they're both correct.

You can find two-quarters of 32 by dividing it into two equal parts because one-half and two-quarters are equivalent.

They're the same.

We've come to the end of the lesson.

I've had so much fun today exploring this concept and I hope you have too.

Today, we've been recognising the equivalence of one-half and two-quarters, and hopefully now you know that one-half equals two-quarters.

One-half is equivalent to two-quarters.

When a whole is divided in half, there are two equal parts.

When a whole is divided into quarters, there are four equal parts, and you knew that already.

I'm confident about that.

Now, two-quarters and one-half are equivalent because they are fractions which describe the same amount when the wholes are the same size.

That's a really important learning point and we're going to say it again.

I'll say it with you.

Are you ready? Let's go.

Two-quarters and one-half are equivalent.

Now just you say it.

Are you ready? Off you go.

That's what I want you to take away from today's lesson.

And let's have a look at some examples.

So we can see that one-half of 12 equals six, but also two-quarters of 12 equals six.

And one-half of each shape is shaded, or two-quarters of each shape is shaded.

You've been amazing today.

I hope you're proud of your achievements.

Give yourself a nice gentle pat on the back.

You've definitely earned it.

I hope I get the chance to spend another maths lesson with you in the near future.

But in the meantime, have a great day.

Try really hard at the rest of your lessons today if you've got more lessons coming up.

Take care and goodbye.