Lesson video

Hello there.

My name is Mr. Tilstone.

I hope you are having a nice day.

I certainly am.

Let's see if we can make it even better.

Today we're going to be doing a lesson all about mixed numbers.

Now you might be thinking to yourself, "I know a bit about mixed numbers already." Fantastic.

Let's see if we can take that knowledge even further.

If you are ready, I'm ready.

Let's begin this lesson.

The outcome of today's lesson is: I can compose and decompose mixed numbers to solve addition and subtraction calculations.

Our keywords, we've just got the one today, my turn, mixed number, your turn.

And what does mixed number mean? I think you might know, but it's worth having a little bit of a recap.

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

So for example, one and a half, that's a mixed number, and that's how we say it, one and a half, but that's how we write it, with the number 1 and the fraction 1/2, no hand.

Our lesson today is split into two parts or two cycles.

The first cycle will be addition equations involving mixed numbers, and the second: subtraction equations involving mixed numbers.

So if you are ready, let's start by thinking about addition equations involving mixed numbers.

And in this lesson you're going to meet Jacob and Sofia.

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

Jacob has been playing tennis for one hour.

Have you ever played tennis? Sofia has been playing tennis for half an hour, so not quite as long.

What might the question be, do you think, here? So we've got two different times, one hour, half an hour.

What do you think is going to be asked here? Now remember the cycle title is "addition equations involving mixed numbers" so what could it be? And this is the question: How long have they been playing for altogether? Did you guess that? Well done if you did.

And Sofia says, "Let's represent this as a part-part-whole model." Well that's a good idea.

Have you done this before? You just might have done this very recently.

We have two parts and we need to find the whole.

So can you picture how we might set that out with a part-part-whole model? What about like this? We don't know the whole.

That's the top part of the part-part-whole model.

We do know the two parts though.

So what do those numbers actually refer to then? So the one is the one hour that Jacob spent playing tennis and the half is the half an hour that Sofia spent playing tennis, and we're trying to find out how long they spent altogether playing tennis.

The part-part-whole model can be used to form an equation to solve.

So here you can see one plus you can see half.

So we can write an addition equation.

One plus half equals what? What do you think the answer is to that? One and a half? That's how we say it.

And that's how we write it.

That's a mixed number that you can see there.

Quantities made up of a whole number part and a fractional part can be expressed as mixed numbers.

One half added to one is equal to one and a half.

The children have been playing tennis for one and a half hours altogether.

A snail travels five metres in an hour, new problem.

After a few more minutes, the snail has travelled a further two-tenths of a metre.

Hmm.

What do you think the question is going to be this time? What do you think? Think about what question we had last time.

Think about the cycle title.

See if you can make a prediction.

And the question is this: How many metres has the snail travelled in total? Did you guess that? Well done.

Sofia says, "Let's represent this as a part-part-whole model." Yes, let's do that.

That worked really well last time.

So once again, can you picture that before we do it? Before we show you what might that look like? Think about the information that we've got so far.

How might we put that into a part-part-whole model? What bits are known, what bits are unknown? She says, "We have two parts and need to find the whole." So what is each part? Can you see? It looks like this.

So what was the five? The five is the five metres that the snail travelled initially.

And then what's the two-tenths? The two-tenths is the two-tenths of a metre that the snail then travelled in a few minutes.

So the part-part-whole model can be used to form an equation to solve.

Can you think about what it might be this time? What could it be? How could we use that addition sign like we did last time? How can we make an equation? What would it say? It would be five plus two-tenths equals something.

We don't know the something yet.

That's what we're trying to find out.

So that's what the part-part-whole model looks like and that's what the equation looks like.

Quantities made up of a whole number part and a fractional part can be expressed as mixed numbers.

And I'm sure you are getting really good at this now, that is your mixed number.

We say five and two-tenths, but that is how we write it.

That's a mixed number.

Two-tenths added to five is equal to five and two-tenths.

The snail has travelled five and two-tenths of a metre.

Let's have a quick check for understanding.

Let's see how much you've understood.

True or false? Seven plus two-fifths equals 72-fifths.

Is that true or is that false? And can you explain why? Pause the video, have a good think about that, and see if you can come up with a really clear explanation.

What do you think? Was that true or false? That was false.

That's not what happens when you add seven and two-fifths together.

Quantities made up of a whole number part and a fractional part can be expressed as mixed numbers.

So it should actually be seven and two-fifths.

Seven is your whole number part and two-fifths is your fractional part.

It's time for some practise.

I think you are ready for this.

Solve these equations.

Now you might notice with some of them that the whole number comes first and the fraction part second, which is the case with the first one.

But you might notice with some of them that the fractional part comes first and the whole number second, which is the case with the second one.

So be careful, have a good think about what that means and have a think about what the answer will be.

And number two, solve the following problems. Start by representing the information in a part-part-whole model, just like the ones you've seen.

And use this to form an equation to solve.

So these are the problems. A: During halftime in her football match, Sofia eats one-quarter of an orange.

Her teammates eat five oranges between them.

How many oranges are eaten in total? So there, I could see a part that was a whole number part and I could see a part that was a fractional part.

So think what we need to do there.

In his garden, Jacob is growing a sunflower.

Yesterday it was two metres tall.

Today it has grown another three-tenths of a metre.

How many metres tall is his sunflower now? And once again, did you spot a whole number part there and a fractional part? What do we do with them? Number three, solve these equations.

Now have a look at these.

You might notice that these have got more than two addends.

So think about what you would need to do, and some have got the information that's missing in different parts of the equation.

Have a good think.

And as always, if you can work with somebody else and share ideas, that would be amazing.

Okay, pause the video, good luck with that, and I'll see you shortly for some feedback.

Welcome back.

How did you get on with that? How did you find that? Do you think you're getting used to this now? Let's have a look.

Let's check.

So number one, solve these equations.

Four plus two-sevenths equals four and two-sevenths.

I said the word "and" but you don't write "and", so you write it just like you can see it there.

That's your mixed number.

And then six-ninths plus five equals five and six-ninths.

So well done if you knew that even though you could see the fractional part first in that equation, we write the whole number part first.

And then the next one, 20 plus eight-tenths equals 20 and eight-tenths.

So this time we had a two digit whole number, but this same principle applies.

And then we had one-15th plus seven equals seven and one-15th.

So once again, the fractional part came first in the equation, but it doesn't come first when you write it as a mixed number.

And then 14 plus two-thirds equals 14 and two-thirds, written just like that.

And 21 plus four-fifths equals 21 and four-fifths, written just like that.

And number two, solve the problems and use the part-part-whole models.

So during halftime in a football match, Sofia eats one quarter of an orange.

Her teammates eat five oranges between them.

How many oranges are eaten in total? So this is how we could possibly represent the part-part-whole model.

And you might have put your addends in a different order, but as long as your two parts were one-quarter and five, that's great.

And we don't know what the answer is yet.

You might also, by the way, have done your part-part-whole model in a different orientation and that's fine too.

So here we can see five plus one-quarter equals five and one-quarter.

And that's how we write it.

So once again, the fractional part came first.

That's the first piece of information that we knew, but that's not how we write it.

We write the whole number part first.

Five and a quarter oranges are eaten in all.

And in his garden Jacob is growing a sunflower.

Yesterday it was two metres tall.

Today it has grown another three-tenths of a metre.

How many metres tall is his sunflower now? So you could see two parts there, the two metres and the three-tenths of a metre.

There were your two parts.

Combine them together, add them together.

And this is what we get.

The equation is two plus three-tenths equals two and three-tenths.

And that is how we say it.

And that is how we write it.

The sunflower is now two and three-tenths metres tall.

And so we had three addends there, so three parts, but two of them can go together to make a whole number part.

So that's the three plus five.

So that's how we could treat that.

That's a whole number.

That's eight and two-tenths.

Five plus four-fifths plus 14.

Now did you notice we had a whole number and then a fraction and then a whole number? But just like before, we can combine those whole numbers together.

14 plus five, not too difficult there is it? That's 19.

So that's 19 and four-fifths.

And then eight-tenths plus three plus 20 equals 23 and eight-tenths.

And in this example, the fractional part came first in the equation, but as always, the whole number part comes first when we write it.

And then the fractional part.

And then one third plus seven plus two, we can combine the seven and two to make nine.

So that gives us nine and one-third.

And then 19 plus one-fifth plus 11, well, we can combine those whole numbers to make 30 and then add the fractional part on.

So 30 and one-fifth.

So the whole number part always comes first in the mixed number, always.

And then 17 plus nine-tenths plus 17.

That was a bit tricky wasn't it? The arithmetic was a bit harder there.

That was a double.

So double 17 equals 34.

So that gives us 34 and nine-tenths and very well done if you got that.

How are you doing so far then? I think you're doing really well.

I hope you're feeling nice and confident and ready for the next part of the challenge.

And that is subtraction equations involving mixed numbers.

We've done addition.

Now let's look at subtraction.

Jacob has a piece of string that is three and one-tenth of a metre in length.

See if you can picture that.

He cuts his string into two pieces.

One piece is three metres long.

Okay, what's the difference between this kind of problem and the ones that we were looking at before? Hmm? Think about what we know and what we don't know.

What's known and what's unknown? Well, this time we know the whole.

That's three and one-tenth of a metre.

This time we know one of the parts.

We don't know the other part, not yet.

So see if you can picture all of that.

So this is a piece of string, three and one-tenths of a metre, and then a part of it's been cut off.

So it's cut it into two different parts.

And one of the parts is three metres long.

What could the question be? What do you think? Can you make a prediction? What am I going to ask you now? And this is the question: What is the length of the other piece of string? So we've cut off that three metre part from the three metre and one-tenths of a metre.

What have we got left? And then Jacob says, "Let's represent this as a part-part-whole model." Why not? That's been working really well for us so far.

So let's continue doing it.

We have the whole and we have a part.

We need to find the unknown part.

Right before I show you, see if you can picture it.

It's going to look slightly different this time.

We can set the part-part-whole model up in exactly the same way as before.

But remember, this time there's a different known piece of information.

We know the whole, we know one part, we don't know the other part.

How might that look? You might want to have a go at sketching that now.

But it might look like this.

So the part-part-whole model can be used to form an equation to solve.

So here we've got three and one-tenth, and then we've got one part of it, that's three.

We don't know the other part.

And this is the equation.

Three and one-tenth subtract three equals something.

Hmm? So we need to find out what's left.

Mixed numbers are quantities that are made up of a whole number part and a fractional part.

So we've seen that lots and lots of times now in lots of different ways with lots of different numbers.

So let's have a think.

You know there's a whole number part.

What's that? That's the three.

And there's a fractional part to that mixed number.

What's that? That's the one-tenth.

But we've taken the whole number part away.

So what's left? Three and one-tenth is made up of three and one-tenth.

So that means when we take away the three, the other piece of string is one-tenth of a metre.

Let's look at a different example.

So Sofia has two and two-fifths of a litre of water.

So let's have a look at that.

Can you see that? That's two and two-fifths.

She accidentally spills two-fifths of a litre.

Hmm? What's different this time to the last question? What's the same? Well, just like before, we know the whole number part.

And just like before, we know one part, which is, this time, the fraction.

So we know the fractional part and we're subtracting that.

So how much water does she have left? "Let's represent this as a part-part-whole model," says Sofia.

Yes, absolutely.

It's working so well.

We have the whole and a part.

We need to find the unknown part.

The part-part-whole model can be used to form an equation to solve.

So here we go.

So two and two-fifths.

Take away that two-fifths.

So the two-fifths was what's spilled out.

And that's what we are left with.

So we don't know what that is yet, but we can work it out.

Can you work it out already? To find an unknown part, we need to subtract the known part from the whole.

And the known part in this case is two-fifths and the whole is two and two-fifths.

So take away that two-fifths from two and two-fifths and what's left? The fractional part's gone and it just leaves the whole number.

Mixed numbers are quantities that are made up of a whole number part and a fractional part.

We've said that lots of times now, I'm sure you are getting used to that.

It's a really useful generalisation.

Two and two-fifths is made up of two and two-fifths.

So that means when we take away that two-fifths, there is two litres of water left.

How are you doing? Do you think you're keeping up with this? Do you think you are ready for a little check? I think you are.

Let's see.

Use the part-part-whole model to help you to complete the equation.

So look at the part-part-whole model.

What do we know? What don't we know? So we can see 11 and two-sevenths.

That's our mixed number.

We can see 11.

That's a part.

And the question is this: 11 and two-sevenths subtract 11 equals what? Pause the video.

So when we take away that whole number from the mixed number, we're just left with a fractional part.

And that's two-sevenths.

Very well done if you said two-sevenths.

You are on track and you are ready for the next part of the lesson.

And here are those final practise questions.

So number one, solve these equations.

And just like before, just be careful when you're answering them.

Some of the information might be presented differently each time.

It might help for you to draw a part-part-whole model each time, but you don't have to, that's optional.

And number two, look at the calculation that Jacob has done.

He has made a mistake.

Well, we're all human.

It happens sometimes, doesn't it? Let's see if we can help him.

Spot and explain his mistake.

Can you calculate the correct difference? So he said seven and three-quarters subtract three-quarters equals seven and one-quarter.

Hmm? Help him out.

And see if you can explain that to Jacob in as clear and effective way as you possibly can.

Pause the video, good luck with that and I'll see you soon.

Welcome back.

How did you get on with that final cycle? Shall we have some answers? Why not? So number one, solve these equations.

12 and two-sevenths subtract 12.

Well, we've taken away the whole number part of that mixed number, leaving just a fractional part.

And that's two sevenths.

Five and one-eighth subtract one-eighth equals five.

We've taken the fractional part away from the mixed number.

18 and three-20ths subtract three-20ths equals 18.

We've taken the fractional part away from that mixed number, leaving the whole number.

And our next question looks a little bit different, but it's exactly the same principle as before.

So we've got three equals three and one-quarter subtract one-quarter.

And 10 equals 10 and nine-tenths subtract nine-tenths.

And 15 equals 15 and one-sixth subtract one-sixth.

Number two, you might have spotted that Jacob made a mistake.

He did.

If a mixed number is composed of three-quarters as its fractional part and we subtract three-quarters, we cannot be left with one-quarter, 'cause what we've done, really, we've taken away all of the fractional part.

So we should just be left with a whole number.

So here's what it should have been.

So he said seven and three-quarters subtract three-quarters equal seven and one-quarter.

No, Jacob should have partitioned the mixed number into its whole and fractional parts.

And then he would've noticed that the difference must be seven.

So when you take away that fractional part, it just leaves the whole number.

If you drew that as a part-part-whole model, that would help.

So maybe that's my tip for Jacob.

Draw it as a part-part-whole model and he'll soon realise his mistake, I think.

We've come to the end of the lesson.

You've been amazing today and I've really enjoyed spending this time with you.

Today we've been composing and decomposing mixed numbers, so we've been making them up and we've been taking them apart.

Mixed numbers are quantities that are made up of a whole number part and a fractional part.

Addition and subtraction calculations represent how mixed numbers can be composed or decomposed.

And you've done both.

You've written addition equations, you've written subtraction equations.

Knowing how mixed numbers are composed supports us to solve addition and subtraction calculations.

And you've done that with contexts and you've done that without contexts today.

Very well done on your achievements and your accomplishments today.

You've been absolutely fantastic.

I think you should give yourself a pat on the back.

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

But until then, enjoy the rest of your day, whatever it is you've got in store, keep being successful, keep working hard, and I will see you soon, I hope.

Take care and goodbye.