Lesson video

Hello there.

My name is Mr. Tazzyman, and today we're gonna be learning together.

I'm looking forward to it and I hope you are too.

So if you're ready, we can get started.

Here's the outcome for today's lesson then.

By the end, I want you to be able to say, I can identify when unit fractions cannot be compared.

Here are the keywords that we are gonna be using whilst we discuss the mathematics.

I'm gonna say them and I want you to repeat them back to me.

I'll say my turn, say the word and then your turn and you can say it.

My turn.

Whole.

Your turn.

My turn, part.

Your turn.

My turn.

Denominator.

Your turn.

My turn.

Unit fraction.

Your turn.

Okay then, let's just check what each of these means.

The whole is all of a group or number.

A part is a section of the whole.

And the bar model at the bottom of the screen there shows you that, the top bit is the whole.

That goes from left to right without being divided up.

And then on the bottom row we've got two parts that make up the whole.

A denominator is the bottom number in a fraction.

It shows how many parts a whole has been divided into.

A unit fraction is a fraction where the numerator is one.

Here's the outline for today's lesson on identifying when unit fractions cannot be compared.

The first part, we're gonna think about when unit fractions can't be compared, and in the second part of the lesson we are gonna look at whether or not things are fair.

Let's begin with the first part.

In this lesson, you will meet Lucas and Sam.

They're gonna help us on our journey through these slides by giving us some of their ideas, some of their thoughts, and responding to some of the prompts that you'll see on screen.

Hi Lucas, hi Sam.

Okay, ready to go? Let's do this.

Look at the statement.

Is it always, sometimes or never true? And the statement says one quarter is less than one half.

What do you think? Do you think that's always, sometimes, or never true? Let's see what Sam and Lucas thought.

Sam says, "Always.

One quarter is smaller than one half." Lucas says, "Sometimes.

One quarter can be greater than one half." Lucas draws a representation to prove his point.

"Here is one quarter." You can see there that that whole has been split into four equal parts and one of them is shaded.

And here's one half.

You can see there that that whole has been split into two equal parts and one of those is shaded making one half This one quarter is greater.

What do you think? Do you think that's a fair comparison? "I see now," says Sam.

"It works because the wholes are different." Both those circles are different sizes.

Sam says, "I'll try that in a 3D context too." Here's one quarter and here's one half.

There's two different representations there.

One quarter and one half, but they're different sizes.

What do you think? Again, it works because they're different wholes.

Sam draws some representations.

Here is one quarter, here is one half.

They have the same whole.

Here, one half is greater than one quarter.

So that's helped to prove that it is sometimes true because we've had an example in which it isn't, and an example in which it is.

Sam and Lucas look at just the fraction notation.

You can see it at the bottom there.

We've got the division bar, the denominator and the numerator.

One quarter and one half.

What about now? There aren't any representations.

What do you think? Look at that fraction notation.

How would you compare those two? Lucas says, "When it's just notation, we know the whole is the same." So anytime you see fraction notation, the whole is always the same.

So the greater the denominator, the smaller the fraction.

And they put an inequality symbol in between.

One quarter is less than one half.

It's time to check your understanding of what we learned so far.

What mistake has been made in the visual proof of the statement below? One quarter is less than one half when the whole is the same.

And this person's put always, and then they've drawn two representations.

They do support what they've said, but what mistake have they made? Pause the video, discuss it, have a think and I'll be back in a minute to reveal the mistake.

Welcome back.

Did you manage to spot the mistake? Well, Sam explains, "The whole is not the same.

This means that one half is definitely greater." On the representations, one quarter is less than one half, but in the statement you can see it says when the whole is the same.

So although it's less than one half, it actually doesn't support the statement.

Those two wholes would have to be the same size for it to be a supporting representation.

Here is another statement.

Is it always, sometimes or never true? One quarter is equal to one half.

What do you think? Well, let's see what Sam and Lucas thought.

Sam says, "Never.

One quarter is less than one half." Lucas says, "Sometimes.

One quarter can be equal to one half." Lucas draws representation to prove his point.

"I'll start with identical parts and then make the wholes." That's a really good tip when you are constructing wholes and parts, start with the part to make the whole.

He's got two parts there and then he's putting the rest of the parts to make his wholes.

There's one quarter and one half.

They are equal.

What do you think? "I see now," says Sam.

"It works because the wholes are different again.

The parts are equal size, but the one half is a bigger part of its whole." Sam changes the representations.

"Let's pretend to cut off the tops of the boxes to make lines." Cuts off a little bit.

Cuts off a bit more.

Now we have two lines showing one half and one quarter.

"They are of equal length." "Yes, the parts are equal length, but the one half is a bigger part of its whole." Here are both statements.

Sam and Lucas look at just the fraction notation.

One quarter, one half.

"What about now? There aren't any representations." "What do you think?" "When it's just the notation, we know the whole is the same." So the greater the denominator, the smaller the fraction.

"They are unequal.

One quarter is less than one half." "So unit fractions can't be compared if the whole is different." "I agree.

Unless you want to compare unit fractions of different wholes." Okay, it's time for your first practise task.

For each of the following statements decide whether it is always true, sometimes true or never true.

Draw or create a visual proof to support your thoughts.

A, one fifth is less than one third.

B, one fifth is equal to one third and C, one fifth is greater than one third.

Remember, it's often easier to start with the part when drawing.

Pause the video here and I'll be back in a little while with some feedback.

Welcome back.

How did you get on? Did you manage to draw some visual proofs? Well, let's look at each of these questions in turn.

A, you said one fifth is less than one third, and for that you should have selected sometimes.

Here's why.

On the left hand side you can see we've got one fifth is less, whereas on the right hand side we've got one third is less.

For the one on the left, the wholes are the same, but the one on the right, the wholes are different.

Now let's look at B.

One fifth is equal to one third.

That's sometimes true and here's a visual proof.

You can see there that the top bar has been split into five equal parts and one of them is shaded and that one shaded part is the same size as the shaded part on the bottom bar, which has been split into three equal parts, meaning that it's one third.

The wholes are different sized, and that's what makes this sometimes true.

Now let's look at C, one fifth is greater than one third.

Again, sometimes.

If we have different sized wholes then it can be.

On the left hand side, you can see one fifth being less because both of those wholes are the same size, but on the right hand side, if we change the size of the wholes, then we can make it look as if one third is less than one fifth.

Okay, then let's move on to the second part.

Is that fair? Lucas and Sam play a game.

Lucas thinks of a real life prompt and then Sam has to try and represent a fair or unfair version of it.

Lucas says, "Let's have one half of a chocolate bar each." Mm.

"This depends on whether or not it's the same size bar," says Sam.

Sam represents the prompt fairly and unfairly.

So we've got a fair version there on the left and we've got an unfair version on the right.

"In this fair version, we are sharing the same bar," says Sam.

And there's a dotted red line to show how it might be halved.

And the fraction notation is used as a label.

"In this unfair version, someone gets one half of the bigger chocolate bar." "The wholes are different," says Lucas.

And he's talking about the unfair part.

This time Sam thinks of a prompt.

"Let's make teams. One quarter of your group versus one quarter of mine." "I'll use a smiley face to represent group members," says Lucas.

In the fair version, group sizes are equal at 16." So you've got two different groups there, both have 16 in them.

And he's then gone on to split that 16 into quarters, divided it into four equal parts.

So the teams would each have four people, an equal number.

In the unfair version, group sizes are different with 16 and eight.

They're then divided into four equal parts and one of those parts is selected.

"So the teams will be four versus two.

That's unequal." Lucas thinks of another prompt.

"I've read one half of a book and you've only read one quarter.

I've read more." "That depends on the size of the book," says Sam.

"That would be fair to say if we were both reading the same book with the same number of pages.

but it would be very different if you were reading a picture book and I was reading a long novel." Sam thinks of a final prompt.

"I'll have one quarter of the pizza and then you can have one quarter." That sounds fair enough to me.

"If we were being fair, we'd both have one quarter of the whole pizza," says Lucas.

And he splits up the pizza into four equal parts and then he shows that one quarter of that would equal one quarter of that.

"If we were being unfair, you'd start by taking one quarter." There it goes.

"Then I take one quarter of what was left over." So you can see that what was left over is less than one quarter, which Sam would've had.

Lucas and Sam discuss what they found by playing the game.

"These examples were all fair." "That's because the whole was the same." You can see there we've got two teams of 16 in each.

They've each got one quarter of a whole pizza.

The chocolate bar is the same and the book is the same.

"To compare unit fractions, the whole needs to be the same." "Unless you want to compare with different wholes." "We have to think about the context then," says Sam.

Okay, it's time for your second task, task B.

What I want you to do is discuss fair and unfair versions of each of these situations.

Then draw or represent a fair and unfair version.

One, A, let's have one third of a cake each.

B, let's play cricket.

One fifth of my class versus one fifth of yours.

C, I'll have one half of the blueberries and then you can have one half.

For number two, I'd like you to think of your own prompts and discuss them with someone.

Can you draw a representation of a fair and unfair version? Lucas says, "remember to use unit fractions." Pause the video here and have a go at those questions.

Enjoy.

Welcome back.

Let's give you some feedback.

We'll go through each question in turn.

Here's A, let's have one third of a cake each.

Lucas says, "It would be fair if it were the same cake.

It would be unfair if one of the cakes was a lot bigger." Have you drawn something similar? You may well have done.

Okay, let's move on to B.

Let's play cricket.

One fifth of my class versus one fifth of yours.

"It would be fair if both classes were the same size.

It would be unfair if one of the classes was a lot bigger." Let's look at C.

I'll have one half of the blueberries and then you can have one half.

"It would be fair if we both had one half of a set of blueberries.

It would be unfair if you had one half of a set and then I had to make do with one half of what was left." Here's number two, thinking of your own prompts.

This is Sam's.

"I'll take one quarter of the time and then you can have one quarter of what's left." Interesting.

Thinking about time as well.

"Let's each have one third of a box of popcorn," says Lucas.

Okay then, let's summarise today's learning.

Unit fractions are parts of a whole that can be represented with a numerator of one in their fraction notation.

Unit fractions can be compared like quantities or values using inequalities.

When comparing unit fractions, the whole should be the same to make accurate comparisons unless you are hoping to compare with different wholes.

I've really enjoyed learning with you today and I hope you have as well.

I'll see you on another lesson soon.

My name is Mr. Tazziman.

Bye-Bye for now.