Lesson video

Hello there.

My name is Mr. Tilstone.

I'm a teacher.

I hope you're having a happy and successful day.

My favourite lesson is maths, and my favourite part of maths I would say is probably fractions, so good news, that's the lesson that we're doing today, it's all about fractions.

So if you are ready, I'm ready.

Let's begin.

The outcome of today's lesson is this, I can choose efficient approaches when subtracting mixed numbers.

You may have had some recent experience of subtracting mixed numbers.

Can you choose the most efficient one depending on the numbers? And we've got some key words.

My turn, mixed number.

Your turn.

My turn, minuend.

Your turn.

And my turn, subtrahend.

Your turn.

Now, those definitely aren't common everyday words, particularly minuend and subtrahend, so let's have a reminder about what they mean.

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

So for example, 1 1/2 is written like this.

That's a mixed number.

Could you think of a different example of a mixed number? And the minuend is the first number in a subtraction, the number from which another number is to be subtracted.

The subtrahend is the number that is to be subtracted, the second number in a subtraction.

So here's an example.

8 subtract 5 is equal 3.

8 is the minuend, 5 is a subtrahend, and 3 is the difference.

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

The first will be using partitioning to subtract mixed numbers, and the second, choosing efficient approaches to subtraction.

Let's begin by using partitioning to subtract mixed numbers.

In this lesson, you will meet Andeep and Izzy.

Have you met them before? They're here today to give us a helping hand with the maths, and very good they are too.

Andeep spends 2 1/3 hours playing hockey.

2 1/3, what kind of number is that? Well, it's got a whole number part and a fractional part, so, therefore, it's a mixed number.

Izzy spends 1 2/3 hours swimming.

They are an active bunch, aren't they? 1 2/3, what kind of number is that? Well, has it got a whole number part? Yes it has.

Has it got a fractional part? Yes it has, so, therefore, it is also a mixed number.

So here we've got a problem with two mixed numbers.

What do you wonder? What could you wonder? What could the question be? Andeep says, "I wonder how much time we spend in total doing sports." And Izzy says, "I wonder how much less time I spent exercising than you." Hmm, good questions.

When two mixed numbers are added, we can partition and treat the whole number parts and the fractional parts separately.

So in this case, we've got 2 1/3 plus 1 2/3.

We can partition them.

So our whole numbers are 2 and 1, and we can add them together to get 3.

And our fractional parts are 1/3 and 2/3, and we can add them together to get 3/3, or one whole.

So that makes 4 altogether.

Andeep and Izzy spent a total of four hours exercising.

So partitioning was really helpful there and really quick and efficient.

So Andeep wonders, "Could we use the partitioning method for subtraction?" Hmm, well, let's investigate.

So we've got our numbers again, 2 1/3, which we partitioned into 2, the whole number part, and 1/3, the fractional part, and then 1 2/3, which we partitioned into 1, the whole number part, and 2/3, the fractional part.

Andeep says two 1s subtract one 1 is one 1.

And then he says, "Two 1/3 subtract one 1/3 is one 1/3." He says, "This is 1 1/3." Hmm.

Izzy says, "Stop.

I respectfully challenge you." What a nice way to say that.

What a respectful way to say it.

"I respectfully challenge you." She doesn't think that's right.

I don't either.

Do you? Can you see what went wrong? She says, "Andeep has partitioned correctly," so well done, Andeep, you're on the road there, "but subtracted incorrectly." So he made a mistake with the subtracting part.

Let's look at this in a simpler case.

So let's look at some whole numbers, not fractions.

53 subtract 39 is equal to what? So we could subtract those two numbers, so 53 into 50 and 3, and 39 into 30 and 9.

We've partitioned them accurately, and Andeep was good at that part, wasn't he? And we can subtract 30 from 50 no problems. That leaves us with 20.

But here's where Andeep went wrong.

He says, "Then we subtract the 1s.

9 is greater than 3, so subtract 3 from 9." He's saying 9 subtract 3 is equal to 6.

Well, it is, but that's not what that question is.

So he's saying it's 20 plus 6, which is equal to 26.

Can you spot where Andeep is making his mistake? Because he made the same mistake there with the whole numbers as he made with the fractions.

Andeep has subtracted incorrectly.

We should subtract 9 from 3.

So Andeep thinks subtraction is commutative, that you can swap over the numbers.

You can't.

It's not like addition.

Addition is commutative, but subtraction is not.

You can't swap the numbers over and get the same answer.

So 3 subtract 9 is equal to something.

We can't just say 9 subtract 3 is equal to something.

But the trouble is here there are insufficient 1s to perform this subtraction.

Partitioning is not efficient when the part being subtracted is greater.

So remember that.

This is a really important message in today's lesson.

It's not efficient when the part being subtracted is greater.

So let's revisit our fraction calculations.

So this is 2 1/3 subtract 1 2/3.

We can subtract the whole number parts of the mixed numbers, no problem because 2 is greater than 1.

So we can do that.

2 subtract 1 is equal to 1.

Fine.

The fractional part of the subtrahend is greater than that of the minuend.

Yes.

So let's have a look at that.

So 1/3 subtract 2/3.

So the fractional part, that 2/3 is greater, so it doesn't work.

It can't work.

So we have insufficient thirds to subtract, so we don't know the answer yet.

Andeep says, "I swapped the fractional parts around to enable me to subtract but this then meant I subtracted incorrectly." So he realises now what he's done.

You can't do that.

You can't swap them over when subtracting.

When subtracting, we need to use our number-sense superpower.

Let's represent this on a number line to prove it.

So we've got a number line with 1 2/3 on the left and 2 1/3 on the right.

What do you notice? What could you say? Well, Andeep says, "My answer of 1 1/3 must be incorrect.

The numbers are too close together." Yes they are.

"Rather than partitioning, it's more efficient to find the difference." There's quite a small difference between these two numbers.

So we can count on from the smallest number.

So 1 2/3 plus 1/3 will take us to 2, and then another 1/3 will take us from 2 to 2 1/3.

And all we've got to do is add them together.

2 1/3 subtract 1 2/3 is equal to 2/3.

Andeep spent 2/3 of an hour longer exercising.

So that finding the difference was definitely more efficient.

Look at this calculation and the workings.

Spot and then correct the error.

So we've got 4 1/6 subtract 3 5/6 is equal to 1 4/6.

Well, that's not correct.

Let's see what happened with the calculations.

We did 4 subtract 3 is equal to 1, okay, and then 5/6 subtract 1/6 is equal to 4/6.

Okay.

I can spot what's gone wrong, can you? Pause the video.

Did you spot it? Part of that was right, wasn't it? The whole number part was fine.

So you might have spotted that the mixed numbers have been partitioned and the whole number parts had been correctly subtracted, but the fractional parts were incorrectly subtracted.

5/6 should have been subtracted from 1/6 but there are insufficient sixths to subtract.

Remember, you can't simply swap them over.

You might have noted that partitioning is not efficient for this example because a fractional part of the subtrahend is greater than that of the minuend, so it didn't work.

Another strategy, such as find the difference, would've been more efficient.

Adding 1/6 will take us to 4, adding another 1/6 will take us to 4 1/6, so the difference is 2/6.

That was much more efficient, wasn't it? Let's see if you can put that into practise.

Number 1, sort these calculations into the table according to whether you could use partitioning efficiently to subtract or not.

So think about the subtrahend and the minuend.

See what you notice about the fractions each time.

Is it better to use partitioning or is it better to use a different method such as finding the difference using a number line? And number 2, Andeep has performed this calculation, 9 1/5 subtract 8 3/5 is equal to.

And then he's done this, 9 subtract 8 is equal to 1, 3/5 subtract 1/5 is equal to 2/5.

So he's given the answer 1 2/5, but that's not right.

Can you spot the mistake and can you explain it? Can you help Andeep out? Okay, pause the video and away you go.

Welcome back.

How did you get on? Are you starting to feel confident? Are you getting good at this? I hope so.

Let's give you some answers and you can see.

So let's start with this one, 2 2/6 subtract 1 5/6, a different method.

Can you see why? Let's have a look at the fractional parts, 2/6 and then 5/6.

Well, the subtrahend has a greater fractional part, so it's not efficient to partition.

It would be efficient to use a number line and find the difference by counting on.

Let's do another one.

2 5/6 subtract 1 2/6, well, yes, it would be efficient to use a partitioning method there because the minuend has a greater fractional part than the subtrahend, 5 is greater than 2, so it works.

And then 5 2/9 subtract 4 6/9, a different method will be better there, and again, it's all about the minuends and the subtrahends.

The subtrahend has a greater fractional part, 6/9 rather than 2/9, so it wouldn't be efficient.

A number line would be great, it would work really well there.

That would help you to find the difference.

Partitioning, not so much.

8 6/7 subtract 3 2/7, yes, partitioning would work just fine there because 6/7 is greater than 2/7, so the minuend is greater than the subtrahend.

It works.

And then 8 2/7 subtract 3 6/7, no, it wouldn't really work to use partitioning there.

It would be better to use a different method such as counting on using a number line because again, the subtrahend has a greater fractional part than the minuend.

So 6/7 is greater than 2/7.

It's not helpful to use partitioning.

So you might have noticed that the partitioning method can be used efficiently only if both the whole number part and the fractional parts of the minuend are greater than those in the subtrahend.

If those in the subtrahend are greater, then another method will be more appropriate.

And the number line's really good, isn't it? Number 2, Andeep has performed this calculation, but he's gone wrong.

Part of it was right, part of it was wrong.

He's getting there, but he's not quite there yet.

Let's help him.

So you might have spotted that the mixed numbers had been partitioned and the whole number parts had been correctly subtracted, so well done.

But the fractional parts were incorrectly subtracted.

3/5 should have been subtracted from 1/5, but there are insufficient fifths to subtract.

You might have noted that partitioning is not efficient for this example because the fractional part of the subtrahend is greater than the fractional part of the minuend.

So 3/5 is greater than 1/5, and that's why that doesn't work.

If it was the other way round, if it was 9 3/5 subtract 8 1/5, that would've been fine, but it's not, so another method would be more efficient, and I would recommend using the number line and finding the difference.

Here's the number line.

8 3/5 add another 2/5 takes us to 9, the whole number, and then we count on from the whole number.

That's an extra 1/5.

Add them together, and we've got 3/5.

That was better, wasn't it? That was quicker.

That was more efficient.

That method worked really well.

Okay, you're doing really well, and it's time to choose some efficient approaches to subtraction.

Andeep is working on a maths puzzle.

It's a tricky math puzzle.

He's trying to fill in the missing numbers in the number sequence.

What do you notice? Have a look.

Well, Andeep says, "I notice the sequence is decreasing." It's going down.

Did you notice that? Well done if you did.

"I also notice that the unit we are working with is ninth." Did you notice that? Well done if you did.

Ninths, 8 1/9, 5 6/9, 3 2/9, and even the 2 could be thought of as ninths.

It could be thought of as 18 ninths.

Where is a good place for Andeep to start? What would you do? He's trying to find out those missing numbers.

Let's start by determining how the sequence is decreasing.

How could we do that? What would be a useful starting point here? Well, Andeep says, "Let's use the two numbers that are next to each other." That's a good idea.

That's 3 2/9 and 2.

So 3 2/9 subtract something is equal to 2.

What's the difference between those two numbers? To find the missing part, we subtract the known part from the whole.

So that's 3 2/9 subtract 2.

Let's use our number sense to determine which strategy to use.

An efficient approach would be to use a number line to count back.

So here's 3 2/9, we're taking away 1, that would take us to 2 2/9 and another 1 would take us to 1 2/9.

And perhaps you didn't even need to use a number line for that.

Perhaps you just did 3 takeaway 2.

The number line is decreasing in steps of 1 2/9.

So that's the step size.

We can use that now.

Now we know how the number sequence is decreasing, we can determine the missing numbers.

To find the first missing number, we need to subtract.

So we've got 8 1/9, and if we subtract that 1 2/9, the step that we're working with, well, let's work it out.

Let's use our number sense.

Which strategy should we use? The fractional part of the minuend is smaller than that of the subtrahend, so we can't use partitioning efficiently.

We could convert to improper fractions or we could find the difference by counting on.

But it's more efficient to count back because the subtrahend is small.

So let's have a look at that.

So we've got our 8 1/9.

First we jump back by the whole part of the mixed number.

And the whole part is 1.

So we're subtracting 1.

That takes us to 7 1/9.

Then we jump back by the fractional part, partitioning as needed.

So if we take away that 1/9, that leaves us with 7, and then subtract another 1/9, that takes us to 6 8/9.

So the answer is 6 8/9.

So we can add that.

Let's check our answer by finding the difference by counting on.

So let's do that.

So 1 2/9 and 8 1/9.

So if we add another 7/9, it takes us to 2, the next whole number, and if we add 6, it takes us to 8, the nearest whole number.

And then if we add another 1/9, it takes us to 8 1/9.

Then we've gotta add them all together, and we get 6 8/9.

So yeah, it checks out.

Our answers agree, we are likely to be correct.

Now we can determine the next and final missing number.

To find the next missing number, we need to subtract.

5 6/9 subtract 1 2/9.

How could we do that? What do you notice? Look at the fractional parts.

Think about the minuend, think about the subtrahend.

The minuend is 6/9, subtrahend is 2/9.

What strategy could work really well there? Let's use our number sense.

Well, the fractional part of the minuend is greater than that of the subtrahend, so we can use partitioning efficiently.

And when you can use partitioning, I do recommend that.

So that's 5 subtract 1 is equal to 4, and then 6/9 subtract 2/9 is equal to 4/9, so that's 4 4/9.

So in this case, we were able to use partitioning, and it was really and quick and efficient.

So that's 4 4/9.

You might have noticed that although we subtracted to find the missing numbers, we could also have added.

We could have gone the other way from the known fractions.

Let's do a little check.

True or false? The most efficient strategy to use to calculate this is partitioning.

So we've got 6 1/9 subtract 2 8/9.

Pause the video.

What do you think? That's one method but is it the best method? Is it the most efficient method in this particular case? That is false but why? Well, let's think about the fractional parts.

The fractional part of the minuend, that's 1/9, is smaller than that of the subtrahend, that's 8/9, so partitioning would not be efficient.

We will be trying to do 1/9 subtract 8/9, and it's not going to work.

Instead, other strategies such as converting to improper fractions and then subtracting could have been used more efficiently, or indeed using a number line.

It is time for some final practise.

I think you're ready for this.

Number 1, match the calculations to the most efficient method to use to calculate them.

Then use that method to calculate the answer.

So we've got some different equations there.

Let's look at our possibilities.

So we've got reduction, that's using a number line to count back because the subtrahend is small.

We've got partitioning, that's both the whole number part and the fractional part are greater in the minuend.

We've got find the difference, the numbers are close together.

And we've got express as improper fractions, the numbers are not close together.

Then number 2, determine how this number sequence is decreasing and then complete the missing numbers.

Think about what would be a good starting point there.

Think about how we started it before.

Okay, very best of luck with that.

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

Pause the video and away you go.

Welcome back.

How did you get on? Are you feeling good? Are you feeling confident? Do you think you've got this? Let's see.

So number 1, 12 7/8 subtract 4 1/8, partitioning is absolutely fine there.

And when you can use that, do use that.

Both the whole number part and the fractional part are greater in the minuend, so that works perfectly.

That's going to be nice and quick.

6 2/5 subtract 1 4/5, that's a reduction example, using a number line to count back because the subtrahend is small.

So 1 4/5 is quite small, and it's quite far away from 6 2/5, so reduction would work well.

9 2/5 subtract 3 4/5, well, you could express those as improper fractions because the numbers are not close together.

And then 9 3/7 subtract 8 4/7, I would find the difference because the numbers are close together.

But why would partitioning not work? Well, think about the minuend and the subtrahend.

The fractional part of the minuend is greater than the fractional part of the subtrahend, so partitioning would not be the right method here.

And what about this, 12 7/8 subtract 4 1/8? You might have noticed that the partitioning method can be used efficiently in this calculation because both the whole number part and the fractional part of the minuend are greater than those in the subtrahend.

That's 12 subtract 4 is equal to 8, and then 7/8 subtract 1/8 is equal to 6/8.

So that gives 8 6/8.

So well done if you got that.

You could also have used a number line to count back, express these mixed numbers as improper fractions or use the find the difference strategy.

They all would've worked, but whichever method you used, it should have the same answer.

We're looking, really, to find the most efficient one though.

And taking a little bit of time to look at the numbers and see what you notice really pays off.

6 2/5 subtract 1 4/5 is equal to what? Well, you might have noticed that the partitioning method would not be efficient because the fractional part of the minuend, that's 2, is smaller than that of the subtrahend, and that's 4.

We needed to use a different strategy such as counting back because the subtrahend is small.

So let's do that.

Let's make a number line.

Let's put 6 2/5 on the end.

And then if we subtract the whole number parts of that mixed number, that will give us 5 2/5.

And now we've got to subtract the 4/5.

We'll partition that.

Subtract 2/5 takes us to 5, and then subtract another 2/5 takes us to 4 3/5.

You may also have used a number line to count on and find the difference or express these mixed numbers as improper fractions.

Whichever method you used should give you the same answer, 4 3/5.

And then what about this, 9 2/5 subtract 3 4/5? You might have noticed the partitioning method can't be used efficiently here because the fractional parts of the minuend is smaller than that in the subtrahend.

So 2/5 is smaller than 4/5.

We needed to use a different strategy.

The numbers are not really close, so it may be more efficient to express these as improper fractions.

That's one way to do it.

So first we multiply the whole number part by the denominator and then we add the numerator.

So that's 47/5 subtract 19/5.

The denominators are the same, so we can just subtract the numerator, and that gives us 28/5.

Then we express this improper fraction as a mixed number.

Well, 5, 10, 15, 20, 25, that's five whole groups, and then 3/5 left over, so that's 5 3/5.

You could also have used the number line to count back or used to find the difference strategy.

Once again, whichever method you used should give you the same answer.

It's all about choosing the best, quickest, most efficient one though.

9 3/7 subtract 8 4/7 is equal to what? Well, you might have noticed that the partitioning method would not be efficient because the fractional part of the minuend, that is 3, is smaller than that in the subtrahend, that's 4.

We needed to use a different strategy such as find the difference because the numbers are close.

So we can count on.

We can add 3/7 to take us to 9, and then another 3/7 will take us to 9 3/7.

Add those together and we've got 6/7.

That's the difference.

You may also have used a number line to count back or express these mixed numbers as improper fractions.

Whichever method you used should give you the same answer, and that's 6/7.

Well done if you got that.

And number 2, determine how this number sequence is decreasing and then complete the missing numbers.

Well, we've got 2 5/6 and 1 3/6 next to each other.

So that's a helpful starting point.

We need to find the difference between those.

We could use subtraction.

Partitioning would be the most efficient strategy to use here because look at the fractional parts.

5/6 is greater than 3/6, so we can do 5 subtract 3, that does work.

That's 1 2/6.

That's the difference.

The sequence is decreasing by 1 2/6 each step.

Now, you can either go one way and subtract 1 2/6 or go the other way and add 1 2/6.

Doesn't really matter, but let's go this way.

Let's subtract.

So 8 1/6 subtract 1 2/6.

Counting back would be the most efficient strategy to use here to find the difference between those two numbers.

That gives us 6 5/5.

Partitioning would be the efficient strategy to use here because the minuend has a fractional part that's greater than the fractional part of the subtrahend.

So 5 3/6 subtract 1 2/6 gives us 4 1/6.

And that's the final missing number.

We've come to the end of the lesson.

I've thoroughly enjoyed that, and I hope you have too.

I feel like you've made some great progress, so well done you.

Today we've been choosing efficient approaches when subtracting mixed numbers.

And by now, you've got a few different approaches up your sleeve.

Partitioning is a strategy that can be used to subtract mixed numbers.

However, for it to be used efficiently, both the whole number part and the fractional part of the minuend must be greater than those of the subtrahend.

And that's not always the case, as you've seen.

Using number sense can support us to determine the most efficient method to use when subtracting from mixed numbers.

So for example, in a lot of those cases, it was much better to find the difference by counting on.

Wow, you've been brilliant today, give yourself a pat on the back.

It is thoroughly deserved.

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

But until then, have a fantastic day, and whatever you do, be the best version of you.

Take care and goodbye.