Lesson video

Hi, welcome to today's lesson.

My name's Mr Peters and in this lesson today we're gonna be thinking about how we can find the size of the whole when we know what the size of a unit fraction is.

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

So, by the end of this lesson today you should be able to say that I can find the whole when the size of a unit fraction is known.

In this lesson today we'Ve got two keywords which we're gonna be referring to throughout.

I'll have a go at saying them first and then you can repeat them after me.

The first word is unit fraction, your turn.

And the second word is denominator, your turn.

So, let's think about how we can describe these then.

Well, a unit fraction is a fraction that has a numerator of one.

And the denominator is the number at the bottom of the fraction.

It tells us how many equal parts the whole has been divided into.

In this lesson today we've got two learning cycles.

The first one is identifying the whole and the second cycle is finding the whole in different contexts.

If you're ready, let's get started with the first cycle.

In this lesson today we've got four students with us.

We've got Sofia, Sam, Lucas, and Alex and as always, they'll be sharing their thinking and any questions that they have along the way to help us with our learning.

So, the pupils at Oak Academy have been talking about some of their favourite computer games that they get to play.

Lucas enjoys playing a game called Dragon's Breath where he has to battle and defeat the opposing army who arrive on the back of dragons.

Sofia likes to play iDance.

In her game you have to keep the robot up to speed with the beat of the music, otherwise you end up losing power.

Let's see what Lucas was talking about with regards to his game.

He said that last night he had one fifth of his lives left before he completed the level.

That sounds like you were nearly defeated, Lucas.

You managed it just in time.

You had one fifth of your lives remaining, didn't you? So, yeah, a good question, Sofia.

How many lives did you start off with then, Lucas? Well, we know that if this heart represents one fifth of the total lives that Lucas started with, then we know that this one heart represents one of five equal parts.

Therefore, each one of the parts has a value of one heart and therefore, all together, we would have five hearts to make the whole, wouldn't we? So, Lucas would've started off with five lives before he started that level.

Ah, well done for completing that level, Lucas.

I wonder how tricky the next one was.

Now thinking about what Sofia was talking about with her game.

She said that she only had one bar of energy left for her robot before she managed to complete the round.

There we go, we can see that here.

You're right, that was close, Lucas, wasn't it? And so how many green bars did you start off with then, Sofia? Well, we know here that one green bar is one tenth of the whole.

So this is the unit fraction of the whole, isn't it? Look and there we've got 10 bars there, haven't we? We know that each one of these bars must be equal, so there are 10 equal parts all together and therefore Sofia will have started with 10 green bars before she started the level.

Well done, Sofia, for getting through that.

Let's look at how we can record this in a table.

We've got the games recorded on the left-hand side and then we can show what one part of the whole amount of lives that each game gives you here.

Lucas had one part of his whole lives remaining, didn't he? And that represented one fifth of the whole.

We know that as a result of that, the denominator tells us how many equal parts the whole has been divided into, therefore the whole must've had five parts to start us off with, and therefore we can now draw what the whole would've looked like.

It would've looked like five hearts.

For Sofia's game, we know that Sofia's section here represented one tenth of the whole.

Therefore looking at the denominator again, the denominator tells us that we have 10 equal parts.

So therefore we would've had 10 equal parts to represent the whole and again, we can draw that here.

Yep, that's a great spot, Lucas, isn't it? We were talking about that with the denominator there, weren't we? We noticed that the denominator tells us the number of equal parts that the whole has been divided into.

So, if we know the size of the unit fraction, then it can help us to identify how many parts make up the whole, can't it? Okay, time for you to check your understanding now.

What I'd like you to do here is have a go at completing the table for these two games here.

Okay, welcome back, let's see how you got on.

Okay, the first game was called Decade and one part of the lives that you were given in that game was a piece of lightning, and we can see that that piece of lightning represented one third of the whole.

So, we know that the whole would be made up of three equal parts and therefore the whole amount of lives that you'd be given in that game would look like this.

For the second game, it's called Long Pass, and your lives here are marked in stars.

So one star represents one fifth of the whole, that means it would be five equal parts which would be representing the whole all together and therefore we could draw that just like this.

Well done if you managed that.

Let's see what Sam and Alex were talking about with the games that they like to play.

Sam likes to play a game called Newspaper Kid.

In this game you have to deliver the newspapers as fast as you can without bumping into any of the obstacles along the way.

Alex's game is called Planet Drive.

In his game you have to drive the rocket in and out of the planets and the asteroids without being hit.

So, let's see how Sam got on with her game then.

She said, during her game of Newspaper Kid, that she slipped in a puddle at first, then she had to run away from a rather possessive dog, and she also fell off her bike at one point which means she was left with just one life in her game.

In Newspaper Kid, lives are represented by half a circle, so one semicircle represents a life.

Well, let's think about how many lives she had remaining then.

Sam had one quarter of her life remaining which is the unit fraction.

And therefore, the whole must have been four equal parts all together.

And there we go, we can now see the amount of lives that Sam would've had at the beginning of the game of Newspaper Kid.

Right, and let's have a think about Alex's game then.

He said he lost five segments of his whole life before he managed to complete the level and he only had one segment left as well.

In his game, each life is represented by a triangle.

So we can see here that this is the one triangle that he had left remaining.

So, take a moment to have a think for yourself, what did he start off with then? That's right, Alex said he lost five segments and had one segment remaining, so that's six segments all together.

Therefore, this one triangle represents one sixth of the whole.

If this represents one sixth of the whole, that means the whole must need six segments all together, and we can see that here below with our hexagon.

There's the one life remaining of the six which Alex had throughout the game.

Okay, so let's take a moment again now to represent this in our table.

For Newspaper Kid, we have one semicircle remaining.

That was one quarter of the whole, wasn't it? If one semicircle represents one quarter of the whole, then we know that there were four equal parts because the denominator is four.

So we need four equal parts, and therefore the whole would look like this.

And for Planet Drive we know that one triangle represented one sixth of the whole, the denominator again is six, so we need six equal parts and, as a result of that, our whole would look like this.

Yep, and that's right, Lucas, the denominator is definitely helping us to identify how many parts the whole has been divided into and therefore telling us how many parts the whole is needed if we know the size of the unit fraction.

Well done you two.

Okay, and another quick check then.

Take a moment to have a think.

Can you identify the number of equal parts the whole would look like and draw the whole each time as well? Okay, so for The Climb 2K24, each part was one square, wasn't it? And that was one sixth of the whole.

Therefore the whole would need six parts, wouldn't it? And the whole could look like this.

And for the game Shadows, well one part was represented by one of these segments, this was one of eight equal parts, so we need eight equal parts for the whole and therefore the whole could look like this.

Well done if you managed to get those.

Okay, and onto our task for today then first of all.

What I'd like you to do here is complete the table for each of these games here below.

And once you've done that, maybe you'd like to come up with your own example of a game which you could add to the bottom of this table as well.

And then once you've done that, have a look here.

This shape here represents one quarter of the whole.

I wonder what the whole could look like.

And how many different wholes you can come up with.

Good luck with those two tasks and I'll see you back here shortly.

Okay, welcome back, let's go through these then.

The first one is Farm Hero 2.

One part of the life is a carrot and that represents one third of the whole, that means we need three equal parts and that would be three carrots to represent the whole.

For Witches and Wizards, we don't actually know what the part looks like, but we do know what the whole looks like.

Here, the whole is represented by four stars, so one quarter of that whole is what we're looking for, that would be one star, and therefore we can also say that the number of parts needed for the whole would be four as well.

For the next game, Dive, we actually have two blue circles.

This time we don't know what fraction of the whole this is, do we? But we do know that the whole has five equal parts.

If that's the case, then if these two circles represent one of those five equal parts, then that would be one fifth of the whole.

So if two circles is one fifth of the whole, then we would need 10 circles to represent five fifths of the whole because each one fifth is represented by two circles.

Hm, for Origins, that sounds like a really interesting game, I wonder what that's about.

We've only been given the number of equal parts and what the whole looks like.

Well, if the whole looks like that, and that's divided into eight equal parts, then we need to find out what the size of one of those equal parts looks like, which would look like this, and therefore the value of that part would be one eighth, wouldn't it? And then finally, Lucas came up with one.

He came up with one called Takemon and in this game you have lives which are represented by triangles.

This one triangle represents one tenth of the whole, that means it would need 10 equal parts and therefore the whole could look like this as well.

Well done if you managed to get all of those and come up with one for yourself.

Okay, and then finally, finding different ways of representing the whole here.

This shape here represents one quarter of the whole, that means the whole needs four equal parts all together.

So you can see here that I've got three different wholes, each of which can be said to have been divided into quarters, so they've got four equal parts.

Well done if you managed to come up with those or some of your own as well.

Okay, onto cycle two now then, finding the whole in different contexts.

So, some of the pupils now have been drawing lines here and they've covered up certain sections of their lines.

Sam has drawn a line and she says that the line that you can see represents one fifth of the whole of her line.

Alex has also drawn a line and he says that the section that you can see represents one quarter of the whole.

Hm, so who do you think has the longest line? Sam is saying that her unit fraction has a denominator of five, so that means that her line must be divided into five equal parts.

There we go, you can see each one of those parts now within this line.

One of those parts represents one fifth and so all together, the whole line would represent five fifths.

So there we go, that would be the length of Sam's line.

Alex on the other hand has a unit fraction which has been divided into four equal parts.

So, the length of Alex's line would look like this.

Hm, what did you notice there then? Well that's right, Sam, because the lengths of the unit fractions were exactly the same, we can say that your line was the longest because it had more parts.

That's a really, really important point to make, Sam.

In particular, the size of the unit fraction being the same for both of the lines.

Well done if you managed to identify that for yourself.

Here's another example.

Who has the most amount of sweets? Lucas is saying that the portion you can see represents one third of the total amount of sweets that he has and Sofia's saying that the portion she has represents one sixth of the total amount of sweets that she has.

Who do you think had the most sweets to start off with? Well let's have a think about Lucas's then.

The portion that Lucas has is representative of one third of the whole.

Therefore, two sweets represents one third of the whole, this would mean he needs three equal parts, doesn't he? And if one of those parts has two sweets, then three parts would be six sweets.

There we go, so Lucas had six sweets all together, what about Sofia? Well again, this time Sofia has two sweets or chocolates in her portion here, doesn't she? But this portion represents one sixth of the whole.

Therefore, if one sixth represents two sweets, then we would need six sixths to make the whole and each one of those groups would have two sweets in it.

So we can see we've now got six equal parts and each part has two sweets in it.

So all together we've got 12 sweets for Sofia.

So Sofia had the most number of sweets, and again, what's really important to note here is that the unit fraction had the same amount of sweets in it.

One third of Lucas's amount had two sweets and one sixth of Sofia's amount of sweets had two sweets as well.

Here's one more example, let's have a look here.

Who has the largest map this time? Lucas is saying that the section of the map you can see represents one quarter of the whole and Sofia's saying that the section of the map that you can see of hers represents one ninth of the whole.

Hm, what do you notice? That's right, the unit fractions are both the same size again, aren't they? So, what would the wholes look like? Well the unit fraction is one quarter for Lucas's, the denominator of that is four, therefore the whole needs four equal parts and then we can now see how we've represented that here.

And let's look at Sofia's.

That's right, Sofia's unit fraction is actually one ninth, isn't it? That means her map needs nine equal parts all together.

And there we go, we can now see the size of the wholes in comparison to one another and we can say whose map was the largest.

Sofia's map was the largest because the unit fractions were the same in size, weren't they? But her map needed more parts all together, so hers was the largest.

Well done, you two, for sharing your thinking.

Okay, it's time for you to check your understanding again now then.

This is one fifth of the whole, so the whole is five sweets.

True or false? Take a moment to have a think.

Okay, the answer's false, isn't it? Use one of these justifications to help you reason why.

That's right, it's B, isn't it? The denominator shows five equal parts, doesn't it? But each one of those parts has actually two sweets in it, so we need 10 sweets all together to represent the whole.

Just because the denominator was five, doesn't mean we have five sweets, does it? We need to find out the size or the value of the group in the examples, don't we? Before we can decide how many are needed to make up the whole.

Okay, and here's another check for understanding now then.

Each parallelogram represents a unit fraction of the whole.

Which shape has the largest whole? Take a moment to have a think.

That's right, it's C, isn't it? A is one part of two equal parts, B is one part of three equal parts, and C is one part of four equal parts.

So, because the unit fractions are all the same in their size, we can say that the fraction with the largest denominator will be the one with the largest whole all together.

Okay, and onto our final task for today now then.

What I'd like you to do is draw each whole and explain which is the largest each time.

You've got this for A, B, C, D, E, and F.

Good luck with those and I'll see you back here shortly.

Okay, welcome back, let's see how you got on.

So again, we can see that the unit fractions are the same size here, aren't they? They're the same length.

We've got one sixth and one third here, but the length of the bars are the same.

Therefore the top number line is one of six equal parts and the second one is one of three equal parts.

So we can say that the top line was obviously longer.

Have a look at the second example.

Hm, well this time one sixth represents two bars and one third represents two bars here.

So which one's gonna be the largest this time? Well, if one sixth represents two bars, then we need another five lots of two bars to make the whole, don't we? Because we need six lots of two bars to make the whole, so this is what the whole would look like for the top bar, and for the second bar, two bars is equal to one third, so we need another two sets of one third, don't we, which would be another four bars all together, to find out the size of this number line.

So we can say again that actually the top number line was the largest one here.

Yep, and once again you're right there, Lucas, with your explanation.

Because each part is the same size and they are unit fractions of the whole, then we can say that the fraction with the largest denominator would have the largest whole, or in this case, number line.

Here's another example.

We've got one third and one quarter here.

Well, if this circle represents one third of the whole, that means it needs three equal parts and if the other one represents one quarter of the whole, it needs four equal parts.

So in this case we can see that the purple circles have the largest whole, don't they? Let's have a look at the examples underneath this as well.

Well, they're both unit fractions again, the parts have the same amount again, don't they? They both have three parts in each of them, however, each one of those parts is a different unit fraction, one of them is one third and one of them is one quarter again.

So the first one would need three parts that are all equal and the second one would need four parts that are all equal.

So we could represent it like this.

Once again we can see that the whole of the purple circles is larger than the whole of the green circles.

Yeah, and a good spot there, Lucas, you noticed for D, didn't you, that each unit fraction has three circles for each part, didn't it? But because they were the same size still, you were easily able to compare them, weren't you? Well done.

And the last one as well here then.

If this square represents one quarter of the whole and the purple square represents one ninth of the whole, therefore we can see quite easily here that one quarter of the whole could look like this, therefore the whole of the green squares would look like this and the whole of the purple squares would look like this, it would need nine equal parts compared the green square's four equal parts.

The area of the purple squares is obviously larger.

Okay, and for the last one then, what do you notice this time? Well, we've got the same unit fraction representing each of the parts, haven't we, here.

But one of the parts is one piece and the other part is two pieces, but are they the same size? They are, aren't they? Actually, if this green square represents one sixth of the whole, then it would look like this, and actually, if those two purple parts represented one sixth of the whole, then it would look like this as well.

So we can say that actually these were the same because not only were the unit fractions the same, but the size of each of the unit fractions was the same as well.

Well done if you managed to get that too.

Okay, and that's the end of our learning for today then.

Let's think about how we can summarise what we've learned today.

We know that if you know the size of a unit fraction, you can work out the size of the whole.

The denominator of the unit fraction.

The denominator of the unit fraction tells us how many equal parts the whole has been divided into.

And you can compare the size of the wholes easily when the unit fractions are the same size.

Thanks for joining me again today, take care and I'll see you again soon.