Lesson video

Hi there! My name is Mr. Tilstone.

I hope you're having a good day today.

Let's see if we can make it even better by having a successful maths lesson.

Today is all about mixed numbers.

If you're ready, I'm ready.

Let's begin.

The outcome of today's lesson is this.

"I can compare and order mixed numbers when the whole is the same".

And our keywords, my turn, mixed number.

Your turn.

I'm sure you're very familiar with what mixed numbers are by now, but let's have a check anyway because it's very important.

A mixed number is a whole number and a fraction combined.

For example, this is one and a half.

That's a mixed number.

Can you think of any other mixed numbers? Our lesson is split into two cycles.

The first will be making connections to prior learning, and the second comparing mixed numbers that have the same whole.

So let's start by thinking about some learning from the past about fractions and comparing fractions.

In this lesson, you're going to meet Aisha and Andeep.

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

Andeep and Aisha are trying to compare some numbers, and here they are.

What kind of numbers are these? These are mixed numbers.

They've got a whole number part, and a fractional part.

We've got one and three-eighths, and we've got one and five-eighths.

What do you notice about these fractions? What's the same and what's different? They're rather similar.

The whole number part is the same.

They've both got one as their whole number.

The denominator's also the same.

Did you notice that? They are both split into eighths.

The numerator's different.

One's got three and one's got five.

Andeep say, "Both of these are mixed numbers with the same whole number part." So how could you compare them? We can't compare them using the whole number part.

Aisha says, "Let's start by looking at a simpler case." So let's think not about mixed numbers, but about fractions that have got the same denominator.

So Andeep and Aisha revisit their learning on proper fractions.

Three-eighths, she says, is the smallest, because it's only three lots of one-eighth.

Five=eighths is the greatest, because it's five lots of one-eighth.

So five is greater than three.

We can represent each fraction as a model to prove it.

So three-eighths is represented by that top bar model, and you can see that's got three of the eight parts, and five-eighths, that's the bottom bar model.

That's got five of the eight parts.

Three-eighths takes up less of the whole, so it's the smaller fraction.

The whole is the same size each time, and there are an equal number of parts, but fewer of those parts are shaded.

So that's why three-eighths is smaller.

You can see that, can't you? We can use an inequality symbol to compare.

We can say three-eighths is less than five-eighths.

We can also represent this on a number line to prove it.

Here we go.

You can see three-eighths, and that's closer to zero than five-eighths is.

Five-eighths is further from zero.

Three-eighths is the smallest because it's nearer zero, and five-eighths is the greatest because it's closer to one.

So that's a different way to look at it.

We can use a generalisation to help us compare these proper fractions.

And you might remember this generalisation from prior learning.

Let's have a look.

I'm going to say this, and then we'll say it together, and then just you say it.

When we compare fractions with the same denominator, the smaller the numerator, the smaller the fraction.

Can you remember that? Let's say that together.

Are you ready? Let's go.

When we compare fractions with the same denominator, the smaller the numerator, the smaller the fraction.

Can just you say it now? Off you go.

Three is smaller than five.

So three-eighths is smaller than five-eighths.

And we can use that inequality symbol.

Okay, let's do a quick check.

Let's see if you've got that.

Compare these proper fractions.

They've got the same denominator.

So four-fifths and two-fifths.

And then can you fill in the blank? Four-fifths is "mm" than two-fifths.

So we want an inequality symbol and a word, please.

Pause the video.

Let's have a look.

So we can say four-fifths is greater than two-fifths, and we can show that with words or the inequality symbol.

If the denominator is the same, which it is in this case, we compare the numerators.

The greater the numerator, the greater the fraction.

Both of these fractions have the same denominator, five, so we compare the numerators.

Four is greater than two, so four-fifths must be the greater fraction.

Well, let's see if you got that.

Let's do some practise.

Number one, shade the fractions and then compare them.

Number two, compare these numbers using those inequality symbols.

Number three, starting with the smallest, order these sets of numbers.

And you might have noticed that in all of these questions, the denominator is the same.

So really focus on that numerator, please.

Good luck with that.

Pause the video, and I'll give you some answers very shortly.

Welcome back.

How did you get on? Are you feeling confident? So number one, shade the fractions and compare them.

Here we've got three-fifths and one-fifth.

That's how you could shade those.

You might have shaded different parts, by the way, of those bars, that's just one way to do it.

And we can say three-fifths is greater than one-fifth.

And we've got three-sixths shaded in B, and then four-sixths shaded underneath that.

So we can say three-sixths is smaller than four-sixths.

And then we're using those inequality symbols.

So we can say, five-sixths is greater than three-sixths.

The denominator is the same.

We are comparing the numerators.

Five is greater than three.

Simple as that.

And for B, two-ninths is less than seven-ninths.

Same again.

Same denominators, different numerators.

Two is less than seven.

And for C, 21 100ths is greater than seven 100ths.

For D, 33 99ths is greater than 31 99ths.

Only just, only by a small amount, but it is greater.

And then E, 12 25ths is less than 13 25ths, and again, only by a teeny tiny amount.

And then finally, for F, 101 1000ths, "mm" 100 1000ths, we can say 101 1000ths is greater than 100 1000ths, by a teeny teeny, tiny tiny bit.

And number three, starting with the smallest, order these sets of numbers.

So the denominator is the same.

We need to focus on the numerators.

And one-eighth is less than three-eighths, which is less than seven-eighths.

For B, three-twelfths is less than seven-twelfths, which is less than nine-twelfths.

And for C, 10 50ths is less than 15 50ths, which is less than 20 50ths.

And in all of these examples, it was a numerator that we needed to focus on, because the denominator was the same.

Right, I think we're up to speed now, and I think we're ready to look at comparing mixed numbers that have the same whole.

So let's revisit our mixed numbers from before.

Let's see if we're ready for this now.

So we have one and three-eighths and one and five-eighths.

Because we know that three-eighths is smaller than five-eighths, one and three-eighths must be smaller than one and five-eighths.

Do you agree, or do you disagree? What do you think? Well, let's look at the composition of these mixed numbers, how they're made up.

Here's one and three-eighths.

The two parts of that whole are one, and three-eighths.

Here's one and five-eighths.

The two parts of that mixed number are one, and five-eighths.

So the whole number parts, you might notice, are exactly the same.

So we don't need to focus on that part.

We need to compare the fractional parts, the three-eighths and the five-eighths.

And you've already done lots of questions like that, haven't you? In the previous cycle.

One and three-eighths is composed of one and three one-eighths.

One and five-eighths is composed of one and five one-eighths.

Three is smaller than five, so one and three-eighths must be smaller.

Let's represent that on a number line.

What do you notice? When we count from one integer to the next in eighths, three-eighths always comes before five-eighths.

So one and three-eighths is smaller than one and five-eighths.

We can also represent this using a different model.

Both mixed numbers have one as their whole number part.

So that's one.

And here's the fractional part.

Fewer segments are shaded in for one and three-eighths, so it must be smaller.

You can see, looking at those circles, you can see that clearly the one on the right, the one and five-eighths is greater.

And the one on the left, the one and three-eighths, is smaller.

We can use that inequality symbol to express this.

One and three-eighths is less than one and five-eighths.

What have you noticed about how we compare mixed numbers that have the same whole number parts? If we have the same whole number parts, then it's the size of the fractional parts that determines which number is greater.

And that's the case here.

We do have the same whole number parts.

They're both one.

The fractional parts are different, and that's what is important here, and that's what we're going to compare.

Andeep says, "First we compare the whole number part.

If this is the same," which it is, "then we compare the fractional part." Well, let's do a check.

Look at these numbers.

See what you notice.

Which is the greatest? Pause the video and give that a go.

Well, first of all, did you notice they both had the same whole number part? It was two in both cases.

So we're looking at the fractional part when comparing them.

One was two-fifths, and one was four-fifths.

And the answer is two and four-fifths is greater, because four is greater than two.

The whole number parts are the same, so we need to compare the fractional parts.

Two-fifths is composed of two one-fifths, whereas four-fifths is composed of four one-fifths.

So it must be greater.

Four-fifths would be further along the number line, and would take up more space in a model, proving that it is greater.

So there's lots and lots of different ways that we can prove that.

But two and four-fifths is greater than two and two-fifths.

Right, I think you're ready for some final practise.

Let's see how we get on with this.

Number one, complete these equations using those inequality symbols.

Before you do that, you might want to have a quick look at those fractions and see what you notice.

Number two, starting with the smallest, put these numbers in order, and see if you can explain your reasoning, why are they in that particular order? Starting with the smallest.

You might want to cross 'em off as you go along as well, so that you're keeping track.

Number three, Andeep has spilt his drink on his homework.

Oh dear, Andeep.

Accidents happen, I suppose.

So A, what could that missing digit be? And there is more than one answer here.

And B, what could the missing digit not be? Hmm, and again, more than one answer.

So see how you get on with that, pause the video, good luck, and I'll see you shortly.

Welcome back.

How did you get on? Are you feeling confident? Let's give you some answers.

So number one, A, four-tenths is less than seven-tenths.

The denominators are exactly the same.

We are looking at the numerators.

What about one and four-tenths, and one and seven-tenths? Very similar, aren't they? Except this time it's a mixed number.

Both of them have got one as their whole number part.

So we can say one and four-tenths is less than one and seven-tenths.

And what about this one? Three and four-tenths and three and seven-tenths.

Same whole number, same denominator, different numerator.

And that's the bit we compare.

So three and four-tenths is less than three and seven-tenths.

What about B? So seven-ninths and five-ninths.

Seven-ninths is greater than five-ninths.

What about two and seven-ninths, and two and five-ninths? Well, the whole number's the same, isn't it? It's the fractional part we need to look at.

The denominator's the same in the fractional part.

The numerators are not.

Seven is greater than five, so therefore two and seven-ninths is greater than two and five-ninths.

And 100 and seven-ninths is greater than 100 and five-ninths.

Did you notice that the fractional part was the same all the way through there? And for C, two-sixths is less than four-sixths, which is less than five-sixths.

And by that same logic, 99 and two-sixths is less than 99 and four-sixths, which is less than 99 and five-sixths.

And we could have said any number there for the whole number part, couldn't we? The same thing would've been true, 'cause we'd already established the order based on the fractional parts.

Number two, starting with the smallest, put these numbers in order and explain your reasoning.

So what did you do? How did you approach this? Well, looking at this, I can see there's lots that have got a three in for the whole number, and one of them has just got three.

It's not a mixed number.

So now we can look at the rest of the threes.

They're all mixed numbers.

And I can see we've got seven, three, one, and five for our numerators.

One is the smallest there, so that's what comes next.

Three and one-eighth.

So we've still got three mixed numbers that we've got three for the whole number part, and the numerators are now seven, three, and five.

Well, three comes next, doesn't it? So that's what we're going to put next.

And then we've got three and seven-eighths and three and five-eighths.

Which has got the smaller numerator? It's three and five-eighths.

So therefore it must then be three and seven-eighths, and then we've got four.

So well done if you got those in that order.

You might have reasoned that three had no fractional part, so it was the smallest, because it would be first on a number line.

Then you may have noticed that the whole number part and the denominator of the fractional part of the mixed numbers was the same.

So to order the mixed numbers, the numerators had to be considered, which is what we did.

Finally, four had to be the greatest, because four's greater than three.

Simple as that.

And Andeep spilt his drink on his homework.

What could the missing digit be? Well, it could be two, three, or four.

Four and two-sixths, four and three-sixths, and four and four-sixths are all numbers that are greater than four and one-sixth, but less than four and five-sixths.

So what could the missing digit not be? It couldn't be zero, or one, or any digit that is five or greater.

We've come to the end of the lesson.

Today we've been comparing mixed numbers when the numerators of fractional parts are different.

When comparing mixed numbers, if they have the same whole number parts, and we've looked at plenty of examples of that today, haven't we? Then it is the size of the fractional parts that determines which number is greater.

When we compare fractions that have the same denominator as they have today, the greater the numerator, the greater the fraction.

And I think as soon as you crack that, it's all very, very straightforward, isn't it? Knowledge of comparing proper fractions can help when comparing mixed numbers, and you've been using inequality symbols to prove that.

Well done on your accomplishments and your achievements today.

You've been brilliant.

Give yourself a pat on the back.

I hope you have a great day, whatever you've got in store, and I hope that you're just as successful in your other lessons as you were in this maths lesson.

Take care, and goodbye.