Lesson video

Hello there.

My name is Mr. Tilstone.

I'm a teacher, and I love maths.

So it's a real pleasure to be here with you today to teach this math lesson, which is all about fractions.

Hopefully you're getting really confident at adding and subtracting fractions.

Today we're going to do some problem solving based on that skill.

So if you are ready to begin, let's go.

The outcome of today's lesson is this, I can solve problems involving addition and subtraction of mixed numbers.

And we've got some key words.

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

Your turn.

And my turn, improper fraction.

Your turn.

What do those terms mean? Let's have a reminder.

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

So for example, this is one and a half.

That's a mixed number.

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

For example, five-thirds and nine-eighths are both improper fractions.

Today's lesson is split into two parts, or two cycles.

The first will be interpreting data, and the second will be problem solving.

So let's begin by thinking about interpreting data.

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.

Izzy records how much water she drinks each week for four weeks.

Well, first of all, Izzy, good for you.

Drinking water is one of the best things you can do.

She presents her results in a table.

Another good idea.

Here we go.

So we've got a column for the week, and a column for the amount of water drunk.

What do you notice? Think about those fractions that you can see.

Izzy says, "I notice the units we are working with are litres and tenths." What questions could we ask about the data? We've got some mixed numbers here, what could we ask? Well, we could ask, how much water does Izzy drink in total over the two weeks where she drinks the most.

What do we need to do first if we are asking that? So let's see where she did particularly well, and drank the most water.

It was these two weeks.

So week two and week four.

Let's represent this as a bar model to help us form an equation to solve.

So we've got nine and eight-tenths, nine and three-tenths, two mixed numbers.

We have two parts, and the whole is the unknown total.

So have a look at that bar model.

We've got the part.

We've got the part, we just haven't got the whole yet.

If we combine those parts, we'll get the whole.

We can use the bar model to form an equation.

Can you see an equation forming there? What operation could we use here.

To find the whole, we add the known parts, and we've got two known parts.

Nine and eight-tenths and nine and three-tenths.

So that's the equation.

Nine and eight-tenths, plus nine and three-tenths is equal to something.

Can you remember how to add mixed numbers? Well, we could use part-part whole models like this, and then first we could add the whole number parts.

So let's do that.

That's nine plus nine, that equals 18.

And then we can add the fractional parts, that's eight-tenths plus three-tenths, and that's 11-tenths.

When we recombine them, this is what we get.

18 and 11-tenths.

Something's not conventional about that, that's not quite right.

What do you notice? We need to do a final conversion because the fractional part of the mixed number is an improper fraction.

So it's not really a mixed number.

So let's do that.

18 plus 11-tenths equals 18 and 11-tenths.

But we could say this, we could partition that 11-tenths into 10-tenths and one-tenth.

And what could we say about that 10-tenths? One full group of 10-tenths can be made from the 11-tenths, just like so, and then there would be one more tenth.

Now that looks a bit easier, doesn't it? We can add those together.

18 plus one plus one-tenth, equals 19 and one-tenth.

This is 19 and one-tenth.

It's always good practise to check calculations, so let's do that.

Have we got another method? Let's check this calculation using a number line instead of partitioning.

Okay, let's do that.

So here's nine and eight-tenths, one of our add-ins.

First we add the whole number part, so let's do that.

So if we add a whole number on, that will take us to 18 and eight-tenths.

Then we add the fractional part, partitioning as needed.

So we're going to bridge through the whole number, so that's going to be two-tenths, taking us to 19, add one-tenth, taking us to 19 and one-tenth.

So we didn't add three-tenths, we added two-tenths and one-tenth.

So that is equal to 19 and one-tenth.

The number line method gives us the same answer, so we're likely to be correct.

I wonder which method you preferred.

Over the two weeks, I drank 19 and one-tenth litres of water.

Good going, Izzy.

That's very healthy.

So let's do a little check.

How much water does Izzy drink over the two weeks where she drinks the least? So we looked at the most before, what about the least now? Have a look.

Take your time.

Work with a partner if you can.

Off you go, pause the video.

So, did you first identify the two weeks where she drank the least amount of water? It was these two.

And now we're going to combine them.

So eight and seven-tenths, plus eight and four-tenths.

We can add by partitioning.

So we add the whole numbers, then the fractional parts.

The whole numbers added together give us 16, and the fractional parts gives 11-tenths.

Well, that's an improper fraction, so that's not correct yet.

We need to do something.

We need to do a conversion.

A final conversion is needed.

So we can turn that 11-tenths into one and one-tenth making altogether 17 and one-tenth.

So Izzy drinks 17 and one-tenth litres of water over those two weeks.

Good going, Izzy.

Very healthy indeed.

And well dome to you if you got that.

Another question that we could ask is, how much more water does Izzy drink week in four than week one? Well, let's have a look.

Here's week four, nine and three-tenths.

Here's week one, eight and seven-tenths.

Andeep says, "The question has the words how much in it, so we need to add." Do you agree with that? "I respectfully challenge you," says Izzy.

That's a nice way to say it.

"Let's represent this as a bar model." Good idea, Izzy.

Bar models really help us to picture the maths.

We want to know how much more, this is a comparison.

So we have the whole and a part.

We need to find the unknown part.

So this is a whole, nine and three-tenths and this is a part, eight and seven-tenths.

We are looking to find the difference between them.

We can use the bar model to form an equation.

So can you see what we're doing this time? It's not actually adding, is it? Andeep wasn't correct there.

We're looking to find the difference between those two numbers.

So how can we do that? To find an unknown part, we subtract the known part from the whole.

So this is nine and three-tenths, subtract eight and seven-tenths is equal to something.

Can you remember how to subtract mixed numbers? What strategies have you got? Have a look at the fractional parts of the minuend and the subtrahend, and what do you notice? Well, we can't use partitioning.

It's a good strategy to use when we can use it, it's very efficient, but in this case we can't use it.

It won't be efficient.

Do you know why? Because the fractional part of the minuend, that's three-tenths, is smaller than that of the subtrahend, that's seven-tenths.

We're trying to do three-tenths, subtract seven-tenths.

We can't do that.

What do you notice about the numbers? The numbers are close together, so we could use a number line and find the difference.

That would be a much more efficient way to do it.

That would work.

So let's have a look.

So here's a number line, a quickly sketched number line with eight and seven-tenths on one side, and nine and three-tenths on the other.

And if we add three-tenths onto the eight and seven-tenths, it takes us to the next whole number, that's nine.

Then if we add another three-tenths to go from the whole number, that takes us to nine and three-tenths.

So we've got two lots of three-tenths, add them together and we've got six-tenths.

Now I'm sure you'll agree, that method was much better and much more efficient than trying to partition in this case.

It is always good practise to check calculations.

So if you can do another method, do it.

If you've got time to do that, that's what we recommend.

So let's check this calculation by converting to improper fractions.

That's another way to do it.

Okay, can you remember how to do that? We are to multiply the whole number by the denominator and add the numerator.

So nine multiplied by 10, plus three, is equal to 93.

So we can say nine and three-tenths is 93-tenths.

And eight multiplied by 10, plus seven, is equal to 87, so we can say that's 87-tenths.

And that's our calculation.

93-tenth subtract 87-tenths.

The denominators are the same, so we can subtract the numerators.

We can leave the denominators as they are.

That's 93 subtract 87, and it's easier probably just to count on from 87 to get to 93, and that's six-tenths.

We've got a difference of six, those numerators.

So six-tenths.

What do you notice about the answer? It's a proper fraction, so we don't need to do a final conversion.

We've got six-tenths.

Converting to improper fractions gives the same answer as before.

We are likely to be correct, and Izzy drink six-tenths of a litre more in week four.

Let's do a little check.

How much more water does Izzy drink in week two than week one? Pause the video.

Well, let's see.

In week two, she drank nine and eight-tenths of a litre.

And in week one, eight and seven-tenths.

So we're looking to find the difference between those.

We need to use subtraction.

The whole number part and the fractional part are greater in the minuend, so eight-tenths is greater than seven-tenths, so we can use partitioning, and that's good news because that's a very efficient method.

So that gives us one and one-tenth.

Izzy drinks one and one-tenth litres more in week two than week one.

It's time for some practise.

Number one, look at the data presented in this table about how many hours Izzy spends playing the guitar, and answer the questions that follow.

Does you know that Izzy play guitar? Yeah, she drinks water and plays guitar.

That's Izzy's thing.

So have a good look at that table.

And A, how much longer does Izzy play for on Monday than Wednesday? B, how many hours does Izzy play in total over Wednesday and Thursday, in total? And C, what is the difference, the difference in time between the day that Izzy spends the longest and shortest time playing? Good luck with that.

If you can work with a partner, I always recommend that.

So if your teacher's okay with that, definitely do that, then you can share ideas with each other.

Pause the video and away you go.

Welcome back.

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

Well, how much longer does Izzy play for on Monday than Wednesday? We need to find the difference.

So that's four and one-quarter, subtract three and two-quarters.

Now you may notice here that partitioning isn't really going to work, because the fractional part of the subtrahend is greater than the fractional part of the minuend.

So that won't work.

So another method will be helpful, perhaps a number line to find the difference.

The numbers are close, so let's find the difference counting on.

So three and two-quarters, and four and one-quarter.

Let's count on, two quarters will take us to four, and one quarter will take us from four.

Add those together and we've got three-quarters.

So three quarters of an hour longer.

And B, how many hours does Izzy play in total over Wednesday and Thursday in total? So we're adding together, that's three and two-quarters, and one and three-quarters.

Add them together.

We can add by partitioning.

First add the whole numbers, then add the fractional parts and recombine.

Well the whole number part, straightforward, that's four.

But we've got a little bit of an issue when we add together the fractional part, it gives us five-quarters.

Now at the minute we haven't got a proper mixed number, we need to do one final conversion.

So let's do that.

So let's think of that as instead of five-quarters, one and one-quarter, and then add that to the four, and that gives us five and one-quarter.

Izzy plays for five and one-quarter hours in total over Wednesday and Thursday.

I bet she's getting really good.

And what's the difference in time between the day that Izzy spends the longest and shortest time playing? Okay, well let's find the longest and shortest first.

The longest is Monday, and the shortest is Thursday.

So we need to find the difference.

We can convert to improper fractions, that's one method of doing it.

First multiply the whole number by the denominator, then add the numerator.

So when we do that, that gives us 17-quarters, subtract seven-quarters.

The denominators are the same, so we can just subtract the numerators.

Keep the denominators as they are.

That gives us 10-quarters.

A final conversion is needed, we don't want that as an improper fraction.

Let's turn that back into a mixed number like the other numbers are.

So that is equal to two and two-quarters.

The difference in time that Izzy spends playing is two and two-quarter hours.

You're doing really, really well.

You are ready for the next part of the lesson, and that is problem solving.

Andeep's school day is six and one-quarter hours long.

I wonder if that's shorter than your school day, or the same or longer.

One and three-quarter hours are spent at break.

What might the question be, do you think, can you predict? The question is this, how many hours of learning are there in his school day? Did you predict that? I dunno if you did.

How would you start to answer this? Well, let's start by representing it as a bar model.

We've got some information.

This is the total length of the school day, so it must be the whole.

Where would you put that in the bar model? Just here, the top bar.

Break is a part of the day, so it's our known part.

So that's one and three-quarters.

That's one of the parts, we dunno the other part.

How many hours of learning are there in his school day? The bar model has helped us to see that we're looking to find the difference between those two mixed numbers.

We can use the bar model to form an equation to solve.

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

So let's do that.

Can you form an equation? What will it be? It will be this, six and one-quarter, subtract one and three-quarters, is equal to.

And what do you notice about the numbers that we are subtracting? Have a look at the fractional part of those numbers.

Think about that.

The minuend and the subtrahend.

The unit we are working with is quarters.

Well done if you noticed that.

The most efficient method will be to count back, because the subtrahend is small.

And we can't use partitioning, because the fractional part of the subtrahend is greater than the fractional part of the minuend.

So first we subtract the whole number.

Let's do that.

Subtract that one.

That takes us to five and one-quarter.

What will we do next? Then we subtract the fractional part, partitioning as needed.

So we're going to bridge.

So let's subtract one-quarter, that takes us to five, and then subtract another two-quarters, that takes us to four and two-quarters.

"There are four and two-quarter hours of learning in my school day," says Andeep.

Let's look at a different problem.

Andeep's school day is six and one-quarter hours long.

Andeep stays after school for science club, and another one and three-quarter hours on Tuesday.

What might the question be? What could we ask? What might you wonder there? How would you start to answer this? The question is, how long is Andeep at school for on Tuesday? Well, let's start by representing it as a bar model.

How long is Andeep at school for? This is a total length of time in school, so it must be the whole, and it is unknown at the moment.

So we don't know the whole.

But we do know the two parts.

One part is that his school day is six and one-quarter hours long.

Then the other part is his science club, another one and three-quarter hours.

So let's write that here.

That's our two parts.

We can use a bar model to form an equation to solve.

So we've got six and one-quarter, plus one and three-quarters.

To find the unknown whole or total time, we need to add the known parts.

So we're going to add six and one-quarter, and one and three-quarters.

What method would you use? It will be efficient to add by partitioning.

Yes it will.

Let's do that.

Let's add the whole numbers together, and then the fractional parts together.

First, add the whole number parts, and then the fractional parts.

So that gives us six and one-quarter, plus one and three-quarters, is equal to seven and four-quarters.

But we've still got an improper fraction there with the four-quarters.

So we need to turn that into a whole number.

So let's do that final conversion.

So instead of four quarters, it's going to be another one, which makes eight.

I'm at school for eight hours.

That's dedication for you.

Let's have a little check.

Which bar model represents this problem? Izzy eats three and two-sevenths of pizza.

How much more pizza does she eat than Andeep, who has eaten two and six-sevenths of pizza? Which of those bar models matches that problem? Pause the video.

Did you spot it? Which one matched up? It is this one.

This is a comparison problem, so we have one whole and a part.

It's time for some final practise.

You're ready for this, I know you are.

Represent these problems in a bar model, and then solve.

A, a tailor has three and seven-tenth metres of ribbon, she uses one and nine-tenths to complete a dress.

How much ribbon is left? B, at the beginning of break, there are 15 oranges.

After break, there are three and two-sixth oranges left.

How many oranges were eaten? So think each time are we adding, or are we subtracting? C, Andeep did a sponsored walk for charity, he walked six and four-fifth kilometres one day, and three and three-fifth kilometres the next day.

How far did he walk in all? If you can work with a partner, and your teacher is happy for you to do that, please do, because two heads are better than one, as they say.

You can share and combine strategies.

Pause the video and away you go.

Welcome back.

How are you finding this? Let's have a look.

So a tailor has three and seven-tenth metres of ribbon, she uses one and nine-tenths to complete a dress.

How much ribbon is left? Well, here's what that might look like as a bar model.

So you can see we're trying to find the difference between those.

This is the equation.

Three and seven-tenths, subtract one and nine-tenth.

But we can't use partitioning there, can you see why not? Well, look at the fractional parts.

Look at the subtrahend, nine-tenths is greater than the minuend, seven-tenths.

So that strategy won't work very well.

But other strategies will work.

Because the subtrahend is small, we can find the difference by counting on, because the numbers are not too far apart.

Or we could convert to improper fractions, it doesn't really matter.

If we do that, if we convert to improper fractions, that gives us 37-tenths, subtract 19-tenths, which gives us 18-tenths.

We're not quite there yet.

We need to convert that.

That's one and eight-tenths.

That's the difference, there's one and eight-tenth metres of ribbon left.

And B, at the beginning of break there are 15 oranges, after break there are three and two-sixth oranges left.

How many oranges were eaten? So this time we've got a whole number, and then we're subtracting a mixed number.

We're finding the difference between those.

So that's 15, subtract three and two-sixths.

You might have chosen to count back to solve this equation.

Whichever method you used, should give the same answer.

There we go.

That's what this would look like.

This is a number line, 15 on one end, and we could count back.

So starting with the whole number part, that takes us to 12, subtracting three.

And then we subtract the fractional part, That takes us to 11 and four-sixths.

11 and four-sixth oranges were eaten.

Well done if you got that.

And C, Andeep did a sponsored walk for charity.

He walked six and four-fifth kilometres one day, and three and three-fifth kilometres the next day.

So this time two mixed numbers.

How far did he walk in all? Well, as a bar model, it looks like this.

And that shows that we need to add together the two parts.

You might have chosen partitioning.

That's always good if it works to solve the equation.

So first, add the whole parts.

Pretty straightforward.

That's nine, isn't it.

And then add the fractional parts, That's seven-fifths.

Okay, that's not the answer, is it, we need to do something with that.

We need a final conversion.

So let's turn that improper fraction into a mixed number.

How many times does five fit into seven, well one, and two-fifths left over.

So we can add one and two-fifths to the nine, and that makes 10 and two-fifths.

Very well done if you got that.

Andeep walked 10 and two-fifth kilometres in all.

We've come to the end of the lesson.

My goodness, you've been great today.

Today we've been solving problems involving addition and subtraction of mixed numbers.

When interpreting data or solving problems, number-sense is needed to determine the most efficient strategy to use when adding or subtracting mixed numbers.

And I always say, look before you leap.

Have a good look at the numbers, see what you notice before you start.

You might save yourself lots of time by choosing the correct strategy.

Bar models are a useful tool to represent word problems, then that helps you to see what operations are needed, and support in forming an equation to solve.

And then you need to decide your best strategy.

Sometimes it might be partitioning, sometimes it might be counting on, sometimes it might be converting to improper fractions.

Very well done on your accomplishments and your achievements today.

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

But until then, have a wonderful day whatever you've got in store, and be the best version of you that you can possibly be.

You can't ask for more than that.

Take care and goodbye.