Lesson video

Hello there.

My name is Mr. Tilston.

I'm a teacher.

If I've met you before, it's really nice to see you again, and if I haven't met you, it's nice to meet you.

Today's lesson is all about mixed numbers, and I know that you already know something about mixed numbers, so let's see if we can build on that today.

We're going to be looking at adding.

If you're ready to begin, let's begin.

The outcome of today's lesson is this: I can add mixed numbers, crossing the whole.

We've got some key words.

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

Your turn.

And my turn.

Improper fraction.

Your turn.

You've met these concepts before, I'm sure, but can you remember what they mean? Can you explain them? If not, let me give you a little reminder.

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

So for example, here's one and a half.

That's how we say it.

That's how we write it.

That's a mixed number.

Could you think of a different example of a mixed number? And an improper fraction is a fraction where the numerator, that's the top number, is greater than or equal to the denominator, that's the bottom number.

So for example, five thirds or nine eighths, they're improper fractions.

Can you think of a different improper fraction? Today's lesson is split into two cycles.

The first will be adding fractions to mixed numbers, and the second adding mixed numbers to mixed numbers.

So if you are ready, let's begin by adding fractions to mixed numbers.

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

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

Andeep picks up a handful of white number rods.

Now, you might have some coloured rods in front of you.

Now if you have, you could use them.

He compares them to a whole yellow rods.

If you've got white rods and yellow rods, you could model this along with Andeep.

Here we go.

So this is what you could create.

This is what he's created.

What do you notice? What can you see? What could you say? What's this? The whole has been divided into five equal parts.

So can you see that? So the yellow represents one whole and it's divided into five equal parts, and each part is therefore one-fifth of the whole.

Okay, so what could we say he's got then? I have seven of these parts.

He says, I have seven one-fifths.

We say seven-fifths, and we write it just like that.

So that's how to write seven-fifths.

That's an improper fraction.

The numerator is greater than the denominator.

Izzy gives Andeep three more, one-fifth rods.

Can you see? So what's he got now? How could you describe that? To find out the total value of the rods Andeep has now, says Izzy, We can add.

Yes, we can.

What would happen if you added those two amounts together? Well, we've got this, seven-fifths, our improper fraction plus three-fifths, our proper fraction.

What does that give us? Seven-fifths, add three-fifths is equal to two, says Izzy.

Is she right? What do you think? Do you agree? Well, Andeep doesn't.

He says, I respectfully challenge you, and I think that's a really nice way to say that you disagree with somebody.

He says we are adding two fractions, so the sum must also be a fraction.

What do you think about that? Izzy says, but what do you notice about the addends? What do you notice? We've got seven-fifths and three-fifths.

What do you notice about them? One is a proper fraction and the other is an improper fraction.

Can you remember which one's which? The proper fraction is three-fifths, and the improper fraction is seven-fifths.

So the addends are already different types of fractions.

Well, that's true.

Let's put the number rods together.

Now, what could you say we've got? Then, says Izzy, we can represent this as a bar model to support us to solve this.

Well, let's do that.

So this is seven-fifths plus three-fifths.

Improper fractions are added in the same way as proper fractions.

If the denominator is the same, we add the numerator.

So seven one-fifths plus three one-fifths is equal to 10 one-fifths or 10-fifths.

Two groups of five-fifths can be made from 10-fifths.

So what could we say now? What do you notice? Five-fifths is the same as one whole.

How many one wholes have we got? We've got two.

So seven-fifths plus three-fifths is equal to 10-fifths, but that's also equal to two.

Although writing 10-fifths is mathematically correct, it is usually written in its whole number form.

So it's more conventional to say two than 10-fifths.

Let's explore this with the rods Andeep has.

10 is a multiple of five, so it must be equivalent to a whole number.

There are two full groups of five-fifths in 10-fifths.

Once again, five-fifths, 10-fifths, two.

Andeep's rods have a value of two.

Here's one and here's two.

So that five-fifths could be expressed as one, and the other five-fifths also expressed as one.

Well, let's have a little check.

Let's see how much of that you've understood.

Find the value of these rods.

So we've got six quarters plus two quarters.

How can you express that? Is it more than one way? Pause the video.

So six quarters, or six one quarters, plus two quarters, or two one quarters is equal to eight quarters, but eight is a multiple of four.

So what does that tell us? It must be equivalent to a whole number.

There are two full groups of four quarters in eight quarters.

So eight quarters is equal to two.

Andeep represents three and three-fifths using rods.

That's a mixed number.

Izzy gives him another three one-fifths.

What's the total value of Andeep's rods now? So this is a slightly different question, isn't it? This time we're adding a fraction to a mixed number.

Three and three-fifths plus three-fifths is equal to what? What do you think? Let's represent this in a part to part whole model.

Good idea.

Here we go.

So we've got three and three-fifths, that's one part, and three-fifths, that's the other part.

We're gonna combine those two parts to make the whole, what do we get? Well, let's partition the mixed number into its parts.

It's got two different parts, hasn't it.

It's got the three, the whole number part, and the three-fifths, the fractional part.

Now we can add the whole parts.

Well, there's only one whole part, so that's still just three.

Then we add the fractional parts.

So what fractional parts are left? We've got three-fifths and another three-fifths.

Three.

one-fifths, add three one-fifths is six one-fifths, or six-fifths.

Something's not right about that though.

My rods have a total value of three and six-fifths, says Andeep.

I respectfully challenge you, she says.

Andeep has not properly converted the sum into a mixed number.

He's got one more step to go.

In a mixed number, the numerator must always be smaller than the denominator.

How many groups of five-fifths are there in six-fifths? How many whole groups? There is one group of five-fifths and then one more-fifth.

So what can we do with that one group of five-fifths? There's a five-fifths.

What can we say about that? It's a whole number.

So it's an extra whole.

And then when we add those together, nice and straightforward now, isn't it? Three plus one is equal to four, and then the extra one-fifth is equal to four and one-fifth, and that's the answer.

Andeep says, My rods have a value of four and one-fifth.

Can you see that? There we go.

That five-fifths can become a whole.

And then we've still got that extra fractional part, the one-fifth.

Five and seven-fifths is not a mixed number.

Can you convert it to one? Pause the video.

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

So we've got five and seven-fifths.

We've got a problem.

It's not a mixed number at the minute.

It looks like one, but it isn't, because the numerator, the fractional part, is bigger than the denominator.

So we need to do something.

How many groups of five-fifths are there in seven-fifths? Well, you can get one whole group and then two more-fifths left over.

So let's express it that way.

Let's express that seven-fifths is two different parts, a five-fifths and a two-fifths.

Then what can we do with a five-fifths? That's one whole.

Now we can easily add them together.

That equals six and two-fifths.

Very well done, if you've got that, you are on track and you're ready for the next part of the learning.

And the next part of the learning is the practise.

So number one, express these numbers in a more conventional way.

So they're not correctly written at the moment.

Can you sort that out? And number two, find the sum, the total of these numbers.

And Izzy says, Remember to write the answers in the conventional format.

So not like before, for example, where we had the mixed number, that really wasn't a mixed number because the numerator, the fractional part, was greater than the denominator.

Okay, very best of luck with that.

Pause the video.

Welcome back.

How did you get on with that first set of questions? Let me give you some answers so you can check.

So number one, three and six-fifths, that's not how we would write it.

That's a mixed number.

It looks like a mixed number, but it's not because the numerator of the fractional part is greater than the denominator.

We've gotta sort that out.

That's four and one-fifths.

That's six-fifths, can be thought of as five-fifths or an extra whole plus one-fifth.

So that's four and one-fifth.

Two and zero quarters.

Well, that's still two, that's got nothing extra.

That's just got the whole number and nothing else.

So two.

Three and four quarters.

Well, those four quarters would make an extra whole, an extra one.

So we can say it's four.

And then three and nine quarters.

Well, if we think about that nine quarters, how many fours go into nine? Two.

So it's got two extra whole parts and then one fractional part, one quarter left.

So two and one quarter in addition to the three, which makes five and one quarter.

Very well done if you've got that.

And number two, find the sum of these numbers.

We're going add these together.

So 12 ninths plus six ninths, that's 18 ninths.

But what you notice, 18 is a multiple of nine.

We can count in nines to get to 18,, nine, 18, twice.

So we can say that is two, two wholes.

Five eighths plus 19 eighths, well, that's 24 eighths.

But once again, 24 is a multiple of eight, correct? So 8, 16, 24.

So that's three.

Two quarters plus two and three quarters.

Okay, well if we add them together, we get two and five quarters.

But that's not correct, is it? Two and five quarters is not mathematically correct? So that five quarters we could think of as four quarters and one quarter or one and one quarter.

So that equals not two and five quarters, but three and one quarter.

Three and five-sevenths plus three-sevenths, well that equals three and eight-sevenths, but eight is greater than seven, the numerator is greater than the denominator.

So we can't have that, but we could think of it as seven and one-seventh.

So that one and one-seventh added to the three is equal to four and one-seventh.

Three and two-fifths plus nine-fifths.

Well, that's three and 11-fifths, but we can't say three and 11-fifths.

So let's turn the 11-fifths into a mixed number and add it to the whole number.

So five, 10, that's two whole numbers, and then one-fifth left over.

So that's two and-fifth.

And add that to the three and you've got five and one-fifth.

And then seven-sixth plus four and one-sixth.

Or we could swap those addends over because addition is commutative, and we could say four and one-sixth plus seven-sixth.

That gives us four and eight-sixths.

But that's not correct.

That's not mathematically correct.

We need to do something.

We need to think about that eight-sixth.

Well, it's got one whole within it.

It's got six-sixth and another two-sixth.

So that's one whole and two-sixth.

So four plus one and two-sixth is equal to five and two-sixths.

Well done if you've got those.

You're doing really, really well and you are ready for the next part of the lesson, which is adding mixed numbers.

So let's do that.

Andeep and Izzy represent two numbers using rods.

Here they are.

What numbers are they? What can you see? We've got our denominator established.

What is it? It's fifths.

So what we've got, I can see two mixed numbers there.

Can you? What numbers are being represented? Well, we've got three and two-fifths, three wholes and two-fifths left over.

And we've got two and four-fifths.

We've got a whole number part that's two and a fractional part that's four-fifths.

What's the total value of the rods? So if we added those together, what have we got? How much have we got? We could put the whole parts together, then the fractional parts together, says Izzy.

And I think that's a great idea.

Let's do that.

So here's the whole parts together and the fractional parts together.

Five one-fifths are equivalent to one whole.

So let's convert, shall we? Instead of five-fifths, let's have another whole.

So that's how many whole numbers we've got and that's a fractional part that we've got left over.

In addition, the total value of the rods is therefore six and one-fifth.

Let's represent Izzy's method using part-part-whole models to partition the mixed numbers.

Good idea.

So this is three and two-fifths, which is equal to three, the whole number part, and two-fifths, the fractional part.

And we've got two and four-fifths.

That's equal to two, the whole number part, and four-fifths, the fractional part.

Now can you remember what we did first? What did we add together first? The whole number parts.

So let's do that.

That's three plus two, that's five.

Then we combine the fractional parts.

So that's two-fifths plus four-fifths, which is six-fifths.

We've got a whole number and an improper fraction.

That won't turn into a mixed number unless we do something.

We need to sort it out.

What do you notice about the sum of five and six-fifths? Five and six-fifths is not a mixed number because the denominator is greater than the numerator.

So from that fractional part, the six-fifths, we could create a whole number and a fractional part.

In a mixed number, the numerator must always be smaller than the denominator.

We need to do a final conversion to express this sum as a mixed number.

How many groups of five-fifths are there in six-fifths? How many wholes can you create from that? Well, there's one group of five-fifths and one more-fifth.

So we could partition that six-fifths as such.

What do we do with that five-fifths? We can turn it into another whole.

Now that's much easier, isn't it? I can see what we need to do now.

Add those together and we've got sixth and one-fifth.

Now that is a mixed number.

Let's do a check.

I'll show you another example first and we'll leave that on and then you do a different one.

So let me show you one and four ninths plus four and seven ninths.

See if you can remember each step before I do it.

Add the whole number parts, what's that? That's five.

Then what do we do? Add the fractional parts.

What's that? That's 11-ninths.

Is that okay? No, it's not.

We need to do a final conversion.

So one more full group of nine-ninths can be made from 11-ninths and then two leftover.

So that's one and two ninths, making altogether, six and two-ninths.

Okay, see if you can do the same with one and six-eighths plus three and five-eighths.

Add the whole number parts, add the fractional parts and do a final conversion.

Good luck, and off you go.

Were you successful? Let's have a look.

Well, we have the whole number parts and that's four.

We have the fractional parts, that's 11-eighths.

Then we do a final conversion if needed, and it is one more full group of eight-eighths can be made from 11-eighths and then three more eights.

So that's one and three eighths.

So that's four plus one and three eighths.

That's five and three eighths.

Very well done, if you've got that, you are on track and you are ready for the next part of the learning.

Let's revisit this problem.

How else could we find the total value of the rods? We could use a number line, says Izzy.

Well that's a great idea because you won't always have number rods with you.

So a number line would be really helpful if you could use one of those.

So she says, let's start with a greater addend.

So three and two-fifths plus two and four-fifths.

It doesn't really matter which order we add them in, but it's more conventional to start with a greater addend.

And that's three and two-fifths.

So we'll put that at the start of our number line first.

We add the whole number part of the second addend.

So let's do that.

We're going to add two.

So three and two-fifths plus two.

Where would that take us? That would take us to five and two-fifths.

Now what do you think we're going to do next? We're going to add the fractional part.

The trouble is this is going to cross one, but we can partition this part in different ways, but we want to be efficient.

If we add three-fifths, this will take us to the next whole number.

So we're bridging the next whole number.

So instead of four-fifths, we're going to treat it as three-fifths, which is takes us to six, and then the extra one-fifth, which takes us to six and one-fifth.

So that's six and one-fifths.

Well that was pretty efficient, wasn't it? That was even quicker than using number odds.

If we use a number line efficiently, we do not have to do a final conversion.

I like it.

We could also start with the other addend.

No problem.

So two and four-fifths.

Now what did we do first on the number line last time? What did we add first? Can you remember? We added the whole number part.

So two and four-fifths plus three takes us to five and four-fifths.

Now what did we do next? We added the fractional part, didn't we? But what did we do to that fraction? What did we have to do? We had to partition it so that we could bridge.

So instead of adding another two-fifths on, what could we do? We could treat it as one-fifth to take us to the next whole number and another one-fifth.

And that takes us to six and one-fifths.

We've got the same number as before.

Addition is commutative.

The sum remains the same.

Okay, let's do a check.

Use a number line to add these two numbers together.

We've got three and four-sixths, plus two and five-sixths.

Where would that take us? What's the total? Pause the video.

It didn't really matter if you started with three and four-sixths or two and five-sixths.

But let's start with three and four-sixths.

Now a larger addend, so three and four-sixths.

And then we add the two, the whole number part, that takes us to five and four-sixths.

Now we're going to add the fractional part, but we're going to need to partition so that we can bridge.

So that's two-sixths will take us to six.

And then three-sixths.

So that five-sixths has been partitioned into two-sixths and three-sixths.

And then six plus three-sixths.

That's nice and easy.

I think that's six and three-sixths.

And that's the answer.

Well, that if you've got that, you may well have used your number line differently.

So for example, you may have started with two and five-sixths, but as long as you've got six and three-sixths, that's the correct answer and you are ready for the next part of the learning.

The next part of the learning and the final part of the learning, in fact, is some practise.

Number one, I showed you a few methods there using rods, a number line, or partitioning, whichever way you like best solve these calculations.

10 and two-fifths plus four and four-fifths, two and six-sevenths plus four and one-seventh, and three and nine-tenths plus two and four-tenths.

Now if you've used one and you really like that method, it might be worth checking that you are correct by using a different strategy.

So if for example, you used a number line, why don't you then check using the rods? You should get the same answer.

And number two, Andeep adds together two mixed numbers, and this is what he gets.

He's done one and four-sixths plus four and three-sixths is equal to five and seven-sixths.

It's not correct, is it? Can you spot and explain his mistake, help him out.

And number three, solve this problem.

I walked 10 and one quarter kilometres in one day and nine in three quarter kilometres the next day.

How far did I walk all together? How could you represent that? Good luck with that.

Pause the video.

And away you go.

Welcome back.

How did you find that final round of questions.

Are you feeling confident? Do you think you've got this? Well, let me give you some answers and you can check.

So number one, this is using a number line.

You might have used a different method though.

This is 10 and two-fifths plus four and four-fifths.

So we're going to add the whole number part, that's nice and easy, I think.

That's 14 and two-fifths.

And then the fractional part.

And in this case, we're going to have to partition that fractional part so that we can bridge.

So let's turn it into three-fifths and one-fifth.

So that takes us to 15 and one-fifth.

Well then if you've got that, you might have used a different method, but your sum should be the same.

It should be 15 and one-fifth.

And B, two and six-sevenths plus four and one-seventh.

This is using the number odds.

So combine them together, combine the whole number parts and that's what we've got.

And then combine the fractional parts and it gives us another whole.

So that's six and seven-sevenths, but we wouldn't write that.

That's not conventional is it? We would write seven because that seven seventh creates an extra whole.

You may have used a different method, but your sum should be the same.

It should be seven.

And what about this one, three and nine tenths plus two and four tenths.

What did you do? How did you do it? Well, we add the number parts, the whole number parts, that's five.

Then we add the fractional parts.

That gives us 13-tenths.

That's not correct.

That's not conventional.

We need to do a final conversion.

How many wholes can we get from 13-tenths? We can get one whole and three tenths left over.

So that's one and three-tenths.

Add it to the five is equal to six and three-tenths.

Well done if you got that.

You might have used a different method, but your sum should be the same.

So six and three-tenths.

And then Andeep adds together two mixed numbers and this is what he's got.

Five and seven-sixths, but that's not correct.

Can you spot and explain his mistake? Well, you might have noticed that Andeep has calculated correct, but he's not given his answer in the conventional form for a mixed number.

That's not a mixed number.

A mixed number has a numerator that's smaller than the denominator.

He forgot to make his final conversion.

One more full group of six-sixths can be made from seven-sixths, and there will be one more six.

His answer should in fact be six and one six.

If you said that, you corrected him, very, very well done.

And number three, I walked 10 and one quarter kilometres in one day and nine and three quarter kilometres the next day.

How far did I walk all together? Well, we could write that in the fractional forms like this.

That's 10 in one quarter plus nine and three quarters.

And there's lots of ways you could have done this.

Might be a number line, partitioning, whatever.

First, we add together the whole number parts, that's 19.

Then the fractional parts, that's four quarters, but we wouldn't say 19 and four quarters, would we, 'cause four quarters creates an extra whole.

So we'd do the final conversion.

Well done if you said 20 kilometres.

We've come to the end of the lesson, and my goodness, you've been magnificent today.

Very well done, you.

Why don't you give yourself a well-deserved pat on the back.

Let's recap on what we've been doing today.

We've been adding mixed numbers, crossing the whole.

Improper fractions can be added in the same way as proper fractions.

A mixed number can be partitioned into its parts when adding it to improper or proper fractions.

And a part-part-whole model can be useful for that.

To add two mixed numbers, the whole number parts can be added and the fractional parts can be added.

That's a useful strategy.

If the resulting sum has a fractional part that is greater than one, then this needs a final conversion to express it as a conventional mixed number.

A number line is a useful tool to support addition of mixed numbers, and I dare say, it's perhaps the most efficient way, the quickest way.

So you might want to consider using that when you feel really confident.

And the lesson is over.

I hope you have a wonderful day, whatever you've got in store and that you are the best version of you that you could possibly be.

I hope I get the chance to spend another math lesson with you at some point in the near future.

But until then, take care and goodbye.