Lesson video

Hi there, my name is Mr. Tilston.

It's great to see you today.

I'm a teacher and whilst I teach all of the different subjects, my favourite one is definitely maths.

So you can imagine how delighted I am and how excited I am to be here with you today to teach you this lesson, which is all about mixed numbers.

I think you've heard of mixed numbers before.

I'm quite confident about that.

So let's explore that even further.

If you are ready, I'm ready, let's begin.

The outcome of today's lesson is I can solve problems involving mixed numbers and our keywords, quite simply, mixed numbers.

So my turn, mixed numbers, your turn, and let's have a little reminder about what mixed numbers are.

I'm sure you know though, a mixed number is a whole number and a fraction combined into one mixed number.

So for example, here, look, you've got one and 1/2, and that's a mixed number.

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

The first will be representing word problems and the second, solving those word problems. So if you're ready, let's begin by thinking about representing word problems. 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.

Sofia walks 3/4 of a kilometre to her friend's house and then another one kilometre to school.

Now, we're not going to solve this problem, but we are going to represent it.

Can you think about some different ways that we could represent it? Can you visualise that, asks Sofia.

Well, here we go, it could be represented like this.

So we've got that 3/4 of a kilometre that she walks to her friend's house, and then there's another one kilometre to school.

Now, Sofia's saying, what might the question be? So let's see if you can use your prediction skills.

Can you think what the question might possibly be with this information? What don't we know? What might we want to find out? And the question is this, how far does Sofia walk in total? Did you guess that? Well done if you did.

She says, let's start by representing this as a part-part-whole model.

That's a great way to represent things.

She says, can you identify the parts and or the whole? So let's look again, what do we know and what don't we know? She walks 3/4 of a kilometre.

Is that part or a different part or the whole? And then another one kilometre to school.

Is that a part, a different part or a whole? That 3/4 of a kilometre and that one kilometre are parts.

So can you identify any key vocabulary that might help us to decide what kind of problem this is and what sort of operations we're going to use? And then another.

That's a little clue, isn't it? What operation could that be if there's another? and then the question is, how far does Sofia walk in total? That's another clue.

We need to find the total.

This means we must have the parts and indeed we do.

So we're going to represent it as a part-part-whole model.

I'm sure you've seen lots of these before.

I'm sure you've used some of these very recently.

Let's use one here.

So we've got two parts.

What do you notice about those parts? Here we've got a whole number.

So a whole part.

It's not a fractional part, it's not a mixed number.

It's a whole number part.

And here we've got a fractional part.

So what happens when we've got a whole number part and a fractional part? We are going to have a whole, or in this case, the total, when we combine those together, it's going to give us a mixed number.

At the minute we're not interested in what that is.

We're just representing the problem.

We need to find the unknown whole.

The part-part-whole model can then be used to form an equation for us to use to calculate the answer.

So I won't give it away yet, but what operations might be involved in that equation? Now let's look at a different problem, shall we? Jacob needs to drink two and 1/2 litres of water during a training session.

Two and 1/2.

What do you notice about that number? He only has half a litre in his water bottle.

Half a litre.

What do you notice about that number? Can you see how this problem is slightly different to the previous problem? We've got different knowns and different unknowns.

Jacob says, can you visualise that? Can you picture in your head what that might look like? So here we go, so he needs to drink two and 1/2 litres of water, but he only has half a litre in his water bottle.

So that might look a little something like that.

So we know the fractional part.

What might the question be? See if you can use those prediction skills.

See if you can think what the question might be.

And the question is this.

How much more water does he need to put into his bottle? Did you predict that? Well done if you did, Jacob says, let's start by representing this as you guessed it, a part-part-whole model.

Can you identify the parts and or the whole? So let's have a look at those numbers and see if you can decide, is it a part, a part or a whole? So he needs to drink two and 1/2 litres of water.

What do you think that is? A part, a part or a whole? He only has half a litre in his water bottle.

What's that? Is that part a part or a whole, and what don't we know? So two under half litres in this case is the whole, that's all of the water that he will need.

And the half a litre is the part, the part that he's already got.

Can you identify any key words that might help us to decide what kind of problem this is? What kind of equation it will be? He only has half a litre, so that suggests he hasn't got all of it.

How much more water does he need to put into his bottle? We need to find the how much more.

And this is a comparison.

We must have the whole and one of the parts, and indeed we do.

So remember the whole is the two and 1/2 litres, our mixed number and one of the parts is half a litre, our fractional part.

So let's represent that as a part-part-whole model.

So here we go.

So it looks slightly different as you can see.

So we know the whole, we know the total.

That's two and 1/2.

And we know one of the two parts, we know the half the fractional part.

Here's a whole, here's a fractional part, and here's the unknown part.

So this time we're not looking to find the whole, we are looking to find a part.

We need to find the value of the unknown part, says Jacob, the part-part-whole model can then be used to form an equation for us to use to calculate the answer.

So we are not going to think about that yet.

We'll return to that later.

What could the equation be? We'll return to that.

Let's have a little check, complete the part, part whole model to represent this problem.

Jacob has taken four and 1/2 metres of string to school, four and 1/2, to use in his project after he has used some, he has half a metre left.

How many metres did he use? So you've got your part-part-whole model.

So make a quick sketch of that.

Can you fill that in with the information that you know? Pause video and have a go.

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

So four and 1/2 metres and half a metre are two known bits of information.

Are they parts or wholes, he's used some, so that's a little clue about what kind of problem it's going to be, what kind of equation it will be.

And that word left is also another clue.

How many metres did he use? So four and 1/2 is our whole, it's our mix number.

That's where that goes in the part-part-whole model.

And then there's the half a metre that he's used and we don't know the other part and we're going to return to that.

So well done if you added those numbers to those parts of the part-part-whole model, it's time for some practise.

Now remember, you're not solving these problems, not yet.

Hold your horses, that will come later on.

You're just representing them using part-part-whole models.

So 1A, Sofia is making cakes for the school fair.

The recipe needs four kilogrammes of white sugar, four kilogrammes and 2/5 of a kilogramme of brown sugar.

How much sugar does Sofia need in total? So you're going to draw that part-part-whole model and decide where to put those pieces of information, those numbers.

And I can see there we've got a whole number part and we've got the fractional part.

B.

Jacob has a three and 7/10 metres length of ribbon.

He uses three metres of it.

How much ribbon does he have left over? What will that look like as a part-part-whole model? Sofia reads for four and 3/4 of an hour on Saturday.

On Sunday she reads for 3/4 of an hour less.

How much time does she spend reading on Sunday? So we've got some different kinds of problems there.

But for all of them you can use that part-part-whole model.

Think very carefully about where to write the numbers.

Number two, look at this part-part-whole model, then write two examples of the word problem that this model could be representing.

In your first problem, make the unknown part different.

So that's one kind of problem.

And in your second problem, make the unknown part the subtrahend.

So in other words, the number that you are subtracting from the total, good luck with that, pause the video.

If you can work with somebody else, I always recommend that, you can help each other out and bounce ideas off each other.

Pause the video and good luck.

Welcome back, how did you get on with that? Let's have a look, let's give you some feedback.

So here were two known bits of information.

The four kilogrammes, what's that? That's a whole number and 2/5 of a kilogramme.

What's that, that's a fractional part.

So we don't know the mixed number.

We don't know the total and that's what the question's asking.

But that and tells us that we're going be combining them together.

And this is what your part-part-whole model will look like.

Now you may have written your whole number in the bottom right and your fractional part in the bottom left, that's fine too.

And B, Jacob has a three and 7/10 metres length of ribbon.

He uses three metres of it.

How much ribbon does he have left over? So that's a different kind of problem, isn't it? This is a difference problem.

So three and 7/10 is our mixed number.

Three metres is our whole number part and he uses some of that.

How much ribbon does he have left over? So three and 7/10 is our mixed number, our total.

And then we've got one known part, that's the three metres that he's used.

So what we don't know is the difference, and that's what that might look like.

And again, you might have had that whole number and the unknown part swapped over.

You might have had the whole number in the bottom right, that's fine too.

And see, Sofia reads for four and 3/4 of an hour on Saturday.

On Sunday she reads for 3/4 of an hour less.

How much time does she spend reading on Sunday? So we've got four and 3/4 of an hour, that's known and that is our mixed number.

That is going to be our total and 3/4 of an hour, that's a fractional part.

And this time it says less.

So that tells us something about the structure of this problem and what kind of operations we'll be using later.

How much time does she spend reading on Sunday? Let's see how we can represent that.

It might look a little something like this.

So we've got four and 3/4 and we've got a known part, the 3/4 of an hour.

So we are looking to find the difference between those.

So that's the unknown part.

And again, your 3/4 might have gone in the bottom left of this part-part-whole model.

That's fine.

And number two, wide range of possibilities for this one.

Here's just one possible example.

How about this? Sofia had collected nine and 1/4 litres of rainwater in a tank in her garden and used one quarter of a litre to water some plants.

How much rainwater did Sofia have left? And here the unknown part is a difference.

So well, and if you wrote a problem involving the difference and B, Jacob had nine and 1/4 metres of wool, he used some and then had 1/4 of a metre left, how much wool did he use? So in this case, that unknown part is a subtrahend.

We are subtracting it from that mixed number total.

It is time to move on to the next cycle.

And that is solving word problems. So we're going to revisit those problems and this time we're going to solve them.

So let's revisit this first problem.

Remember this, Sofia walks 3/4 of a kilometre to her friend's house and then another one kilometre to school.

How far does Sofia walk in total? So we made a part-part-whole model and this was it.

Now what do we do with it, how do we find that unknown? We need to find the unknown whole.

What kind of number is it going to be? To find the whole, we're going to add together the known parts, so we know the whole number.

We know the fractional part.

That's 3/4 plus one.

Now as with any addition, it doesn't matter which order we put the add-ins in.

So we could also think of it as one plus 3/4.

And that will help us to form our mixed number.

Quantities made up of a whole part and a fractional part can be written as mixed numbers and you've had lots of practise doing that, I'm quite sure.

So this is the answer to that, one and 3/4.

So remember it's not 3/4 and one.

We have to write in that order with the whole number first and the fractional part second, one and 3/4.

And although I'm saying one and 3/4, you don't write and, you just write it exactly as you can see it there.

Sofia walks a total of one and 3/4 of a kilometre.

Let's revisit a different problem.

Remember this one, Jacob needs to drink two and 1/2 litres of water during a training session.

He only has half a litre in his water bottle.

How much more water does he need to put in his bottle? And we made this part-part-whole model, if you remember.

So here we go.

We've got the total, we've got the mixed number this time, we've got the fractional part.

So what haven't we got? We haven't got the whole number part.

How can we find it? It's not going to be adding this time is it? Sofia says we need to find the unknown part and this time it's a subtraction.

So we've got two and 1/2 subtract half equals something.

So what whole number are we going to be left with if we do that? Mixed numbers are composed of a whole part and a fractional part as you can see, two and 1/2.

And when we take away that half from that two and 1/2, we're left with two.

So that is the missing hole number, the missing part.

Jacob needs to put another two litres of water into his bottle.

Let's do a little check.

Use the part-part-whole model to form an equation and then solve it.

Jacob has taken four and 1/2 metres of string to score to use in his project after he has used some, he has half a metre left.

How many metres did he use? So that's our part-part-whole model.

Can you form an equation? Can you solve it? Think carefully about the operation that you're going to need.

Pause the video.

Let's see, this time it was a subtraction.

We've got the mixed number part, the four and 1/2.

So four and 1/2.

When we subtract that half from it, what is left over? It's going to be a whole number part, it's four, well done if you said four.

Jacob used four metres of string.

It's time for some final practise.

Number one, use the part-part-whole that you drew in task A to support you to form an equation and then solve these problems. Now if you've got the part-part-whole models wrong in task A, don't worry, but make sure you have got the right ones before you do this.

A, Sofia's making cakes for the school fair.

The recipe needs four kilogrammes of white sugar and 2/5 of a kilogramme of brown sugar.

How much sugar does Sofia need in total? You've got your part-part-whole model.

Can you solve it? Can you write an equation and can you solve it? B was Jacob has a three and 7/10 metre length of ribbon.

He uses three metres of it.

How much ribbon does he have left over? Once again, you've got your part-part-whole model.

You need to write an equation and you need to solve that.

And C, Sofia reads for four and 3/4 of an hour on Saturday.

On Sunday she reads for 3/4 of an hour less.

How much time does she spend reading on Sunday? Remember to look for those clues like that word less is a clue.

So once again, you've got your part-part-whole model, you need to write an equation and solve it.

And then for two, remember you wrote two problems for task A, question two, well you're going to write an equation for each of those and you are going to solve them.

So good luck with all of that and I'll see you soon for some feedback.

Welcome back.

How did you get on with that final practise? Confident, let's see.

So Sofia's making cakes for the school fair.

The recipe needs four kilogrammes of white sugar and 2/5 of a kilogramme of brown sugar.

How much sugar does Sofia need in total? So this was an addition.

The in total was a clue there.

So to find the total, we need to add the known part.

So that's four plus 2/5s equals four and 2/5s.

Sofia needs four and 2/5 of a kilogramme of sugar in total.

And you might have even written that in the top part of your part-part-whole model where the question mark is.

And for B, Jacob has a three and 7/10 metres length of ribbon.

And he uses this time three metres of it.

How much ribbon does he have left over? Well this time we need to subtract the known part.

That's three, from the whole.

That's three and 7/10.

So three and 7/10 subtract three equals 7/10s.

Jacob has 7/10 of a metre of ribbon left over.

and C was Sofia reads for four and 3/4 of an hour on Saturday.

On Sunday she reads for 3/4 of an hour less.

How much time does she spend reading on Sunday? So we're looking at the difference.

To find the unknown part, we need to subtract the known part from the whole.

So that's four and 3/4, subtract 3/4 and that equals four.

Sofia spends four hours reading on Sunday.

Well done if you got that, and you were returning then for question two to your two problems that you created earlier on, and in the example that I gave you, Sofia had collected nine and 1/4 litres of rainwater in a tank in her garden and used 1/4 of a litre to water some plants.

How much rainwater did Sofia have left? So here the unknown part is the difference, and this is the equation that's nine and one quarter.

Subtract one quarter equals nine.

Sofia has nine litres of water left.

She's got a whole number left.

And then in this one Jacob had nine and 1/4 metres of wool.

He used some and then had one quarter of a metre left.

How much wool did he use? And here the unknown part is the subtrahend.

So that's nine and 1/4.

Subtract something equals 1/4 and nine and 1/4 subtract nine equals 1/4.

So Jacob uses nine metres of wool.

We've come to the end of the lesson.

Today's lesson has been solving problems involving mixed numbers.

Word problems can involve mixed numbers and you've done lots of examples there of that.

Mixed numbers may be used as part of the question or may form the answer.

And again, you've done both kinds.

Mixed numbers can be used to represent measures and you've done that lots of times here with litres and metres and so on.

When we solve word problems, representing them as a part-part-whole model can support us to form equations that we can then solve.

I find the part-part-whole model really useful.

It helps me to see whether I'm adding or subtracting.

You've done really, really well today and I hope you're proud of yourself and of your achievements and your accomplishments.

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

But until then, take care.

Enjoy the rest of your day, whatever you've got in store and goodbye.