Lesson video

Hello there, my name is Mr. Tilstone.

I'm a teacher.

Give me a thumbs up if you are having a good day today.

Give me two thumbs up if you are having a great day today.

For me, it's two thumbs up, and shall I tell you why.

It's because I get to teach you this math lesson, and maths is my favourite subject.

But more than that I get to teach you this lesson, which is all about fractions.

And fractions is probably my favourite part of maths.

So if you are ready for a challenge, let's begin.

The outcome of today's lesson is this: I can subtract a proper fraction from a mixed number crossing the whole.

And you might have had some recent experience of adding fractions crossing a whole.

Our key words, we've got two.

If I say them, will you say them back, please? Okay, so my turn, "mixed number." Your turn.

My turn, "improper fraction." Your turn.

They are our most important words today.

But what do they mean? Well, a mixed number is a whole number and a fraction combined.

So for example is 1 1/2.

That's how we say it and that's how we write it.

That's a mixed number.

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

So for example, 5/3 and 9/8 are improper fractions.

Can you think of a different example? Our lesson today is split into two parts or two cycles.

The first will be subtracting a proper fraction from a whole number, and the second, subtracting a proper fraction from a mixed number.

So if you're ready, let's begin by focusing on subtracting a proper fraction from a whole number.

Can you think of an example of what that might be? Hmm, a proper fraction, so for example, 1/3, or maybe 4/5, they're proper fractions.

And what about a whole number? Maybe 8 or maybe 3.

What examples could you think of? What might that look like? 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, and very good they are too.

Andeep needs to cycle four kilometres to Izzy's house.

So four kilometres, is that a fraction? Is that a whole number? Is it a mixed number? It's a whole number.

He stops for a drink after 3/4 of a kilometre.

So 3/4, what's that? Is that a whole number? No, is it a mixed number? No, is it an improper fraction? No, is it a proper fraction? Yes, it is.

But what might the question be, what do you think? See if you can predict.

The question is this, "How much further does he have to go?" Well done if you predicted that.

"Let's represent this in a bar model," says Andeep.

Here we go.

So here's what we know.

We know the whole length of the journey is four kilometres, that's our whole number.

We know that one part is 3/4 of a kilometre, that's our fractional part.

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

So here we've got 4 - 3/4 is equal to something.

Hmm, how can we go about that? Well, let's represent it using rods.

And if you've got number rods in front of you, you could do this as well.

So here we've got the whole number part, that's our 4.

How would you subtract the 3/4? Hmm, what can you do? We've got a whole number but we're subtracting a fraction from it, how? Well, 4/4 are equivalent to one whole.

We can regroup one whole, or 1, from 4 ones.

So instead of four, we can represent it as 3 4/4.

What would that look like with the rods? Just like this.

And again, if you've got the number rods, you can reenact this with us.

So instead of thinking of it as 4, we can think of it as 3 4/4.

Now, it's a little easier, isn't it? 3 4/4 - 3/4 is equal to what? Happens when we take away three of those 1/4.

Let's see, we're left with 3 1/4.

So 3 4/4 - 3/4 = 3 1/4.

So therefore, 4 - 3/4 = 3 1/4.

This subtraction can also be represented on a number line.

So you can see this number line goes from 3 to 4 in quarters.

We're starting at 4, and we're subtracting 3/4.

So 4 - 3/4 = 3 1/4.

So two different approaches there, which one did you like best? Now, what do you notice about this calculation? We've got 4 - 3/4 = 3 1/4.

What do you notice, anything? Andeep notices that the solution is a mixed number.

So we started with a whole number, we subtracted a proper fraction, and we were left with a mixed number.

That's because we've subtracted part of a whole from a whole.

So when we take away parts of a whole from a whole, we're left with a fraction.

"I notice," says Izzy "that the denominators of the fractional parts are the same: 4." That's because the denominator tells us the unit we are working with.

So in this case, we're working with quarters.

So we can think of things in terms of quarters.

Okay, calculate this: 6 - 1/4 is equal to what? You might want to do it with rods if you've got rods or a number line, or whatever you like.

Have a go at 6 - 1/4, pause the video.

Well, you could think of that not as 6 but as 5 4/4 and then take away the 1/4.

So we can regroup a 1 from the 6 and represent 6 as 5 4/4.

And then subtract that 1/4, that seems a lot easier now, doesn't it? Five and four one-quarters subtract the 1/4 is five and three one-quarters, or as we would tend to say, 5 3/4.

So once again, 6 - 1/4 = 5 3/4, And we could also use a number line, maybe you did that too.

Here we go, going from 5 to 6 in quarters, and we're taking away 1/4.

Subtracting 1/4, that leaves us with 5 3/4.

I think you are ready for some practise.

Number one, solve these equations, then describe the patterns that you spot in each column of equations.

So we've got 3 - 1/7, 3 - 1/6, 3 - 1/5, and 3 - 1/4.

Now, those are similar, aren't they? Think about what they've got in common and see what you notice.

B, we've got 6 - 1/4 and then 8 - 1/4.

See if you notice anything.

And then C, 5 - 1/6, 5 - 2/6, 5 - 3/6, and 5 - 4/6.

Those calculations are linked, aren't they? See what you notice.

And then D, 7 - 1/4, and then 10 - 2/4.

And number two, solve these equations: 5 - 2/3, B, 3 - 7/10, and C 12 - 5/9.

Number three, solve this problem: Andeep has five metres of rope.

He cuts off 4/10 of a metre.

Okay, now did you spot your whole number there, and did you spot your proper fraction? How much rope is left? Check your answer using a different method.

Okay, very best of luck with that.

If your teacher is happy for you to do so, I always recommend working in pairs or small groups, then you can bounce ideas off each of them.

Pause the video, and away you go.

Number one, 3 - 1/7.

How did you think of that? Did you think of it as 3 or did you think of it in a different way? You might have solved the equations by regrouping 1 and rewriting the 3 as a 2 alongside a fractional part equivalent to one whole.

So for example, we can think of 3 as two and seven one-sevenths.

Two and seven one-sevenths subtract one one-sevenths is two and six one-sevenths, or two six-sevenths as we would tend to say.

So that's the answer to that.

And then for the next one, we could think of it not as 3 but as 2 6/6, 2 6/6 - 1/6 = 2 5/6.

And then we could think of it not as 3 but as 2 5/5, 2 5/5 - 1/5 = 2 4/5 And then we could think of it not as 3, but as 2 4/4.

2 4/4 - 1/4 = 2 3/4.

You might have noticed that we were always subtracting a unit fraction, so the difference was always one unit fraction away from the whole.

So 6/7 is one unit fraction away from 7/7, that's a whole, for example.

The numerator and denominator of the fractional part of the mixed number varied by one.

B, 6 - 1/4, you might have solved the equations by regrouping 1 from the whole number to have a fractional party equivalent to one whole.

So not 6, but 5 4/4.

And 5 4/4 subtract one 1/4 is 5 3/4, and that's how we write that.

And then for the next one, you might have thought of it not as 8 but as 7 4/4.

7 4/4 - 1/4 = 7 3/4.

You might have noticed we were always subtracting a 1/4 this time.

So the mixed number difference always had a fractional part of 3/4.

Did you see that, they've got that in common? You might also have noticed that the whole number part of the mixed number difference was always one less than the whole number minuend.

So 3/4 is 1/4 away from 4/4 or 1 whole for example.

We could have used the answer to the first calculation to help us determine the answer to the second just by adding 2.

Did you do that? That was really efficient if you did that.

And C, let's have a look at C.

So this is 5 - 1/6, and you could think of that as 4 6/6 - 1/6, and that's 4 5/6.

5 - 2/6 = 4 4/6, and again you could think of it as 4 6/6, just like before, 4 6/6 - 2/6 = 4 4/6.

And then 4 3/6, and you might have spotted the pattern by now, 4 2/6, what did you notice there? You might have noticed the whole number remained the same, and each time the part being subtracted increased by 1/6.

This resulted in the difference decreasing by one 1/6 each time.

You may have predicted the next equation would be 5 - 5/6 = 4 1/6.

And D, 7 - 1/4, you might think of it not as 7, but as 6 4/4.

So that makes it 6 3/4.

And then 10 - 2/4, you might have thought of that as 9 4/4 - 2/4, and that's 9 2/4.

You might have noticed that whole increase by three, so the whole part of the mix number difference also increased by three.

You might have noticed the fraction being subtracted increased by one 1/4.

So the fractional part of the mixed number difference decreased by one 1/4.

Well done if you spotted that.

And solve these equations: 5 - 2/3.

Well, maybe don't think of it as 5, maybe think of it as 4 3/3.

And when we subtract the 2/3 from that, that gives us 4 1/3.

And for B, maybe don't think of it as 3, but as 2 10/10 and then subtract the 7/10, and that gives us 2 3/10.

And for C, maybe don't think of it as 12, maybe think of it as 11 9/9 and then subtract the 5/9, that gives us 11 4/9.

And then Andeep has five metres of rope, that's our whole number, he cuts off 4/10 of a metre, that's how proper fraction.

How much rope is left? Check your answer using a different method.

So what method did you use? What do you like? Maybe you used a bar model like this to represent it to start with.

So this is 5, our whole number, subtract 4/10, our fractional part.

5 - 4/10, but maybe we don't think of it as 5, maybe we think of it as 4 10/10 - 4/10.

How about that? That makes it a lot easier, 4 6/10 of a metre.

And you might have checked your answer using a number line like this going from 4 to 5 in tenths.

You could sketch that really quickly, can't you? So 5 - 4/10, take away those 4/10, that takes us back to 4 6/10.

There is 4 6/10 metres of rope left.

Well done if you've got the answer, and especially well done if you check that answer and still got the same answer.

Well, you're doing really, really well, and you are ready, I think, for the next part of the lesson, and that's subtracting a proper fraction from a mixed number this time.

Izzy has 5 2/10 metres of rope.

So this time we've got a mixed number.

It's got a whole number part, what's that? That's a 5 and it's got a fractional part, what's that? That's the 2/10.

She cuts off 4/10 of a metre.

What kind of fraction is 4/10? It's a proper fraction.

So that's what we're doing, we're subtracting a proper fraction from a mixed number.

What might the question be here though? Hmm, see if you can predict.

What could we be about to ask? The question is this, "How much rope is left?" Did you guess that? "Let's represent it in a bar model," says Izzy.

Absolutely, always a good way to understand maths.

She says, "We know the whole length of the rope is 5 2/10 metres," just like that.

"We know one part is 4/10 of a metre.

To find the missing part, we need to subtract the known part from the whole." How can we do that? Have you got any good ideas? That's our equation, 5 2/10 - 4/10 is equal to something, hmm.

Let's represent it using rods.

If you've got number rods you could do this too.

So 5 2/10, can you represent that? And then we're going to take away 4/10.

Here is 5 2/10.

What do you notice? We only have two 1/10, but we need to subtract four 1/10.

Hmm, tricky, or is it, have you got any good ideas? How would you subtract 4/10 when you can only see two? Could we make any more tenths somehow? We could count back in tenths using a number line.

Number lines are always good, let's have a look.

So this is 5 2/10, take away 1/10, take away another 1/10, and another 1/10, and another 1/10, that's 4/10, and that takes us to 4 8/10, that's correct, 4 8/10, a mixed number.

Is there a more efficient way to use that number line there? There was a lot of jumps there, wasn't there? Do we need to do that many jumps? "We could have been more efficient." Andeep, you're quite right.

We could have jumped back to the nearest whole number in just one jump.

So instead of subtracting 4/10, we could subtract 2/10 to go back to the nearest whole number, and then jumped another 2/10.

So in other words, we've partitioned that fraction and bridged.

So the 4/10 is treated as 2/10 and 2/10, and that made it a lot easier too.

So we did that in just two jumps, that was more efficient.

Let's see what subtracting 4/10 looks like using the rods.

So what could we do? We've only got 2/10 here, but we need to take away or subtract 4/10.

Can we make any more tenths somehow? Well, we can subtract the two 1/10, they're gone, but the question was 4/10.

We still need to subtract another two 1/10, but how? Well, we can regroup one and represent 5 as 4 10/10, just like you did in the previous cycle.

So that's 4 10/10, and now it's easy to take away that remaining 2/10.

Four and 10 one-tenths subtract two one-tenths is four and eight one-tenths, or four and eight-tenths.

And that's how we write this.

So 5 2/10 - 2/10 = 5, which is equal to 4 10/10.

4 10/10 - 2/10 = 4 8/10, bingo.

Okay, calculate this: 6 2/6 - 5/6 is equal to something.

You might want to use number rods, if you've got them.

You might want to use a number line.

You might want to use a different method.

Whatever method you choose, you might want to using a different method, and check with a partner, see if they've got the same answer too.

Right here, pause the video, and away you go.

Well, what did you come up with? Let's see.

Well, we could treat that 5/6, not as 5/6, but we could partition it into 2/6 and 3/6, and that will make it easier to count back to the nearest whole number.

So this becomes 6 2/6 - 2/6 = 6.

Then regroup a one and think of 6 as 5 6/6, like we did before.

So 5 6/6 - 3/6 is equal to 5 3/6.

5 3/6, well done if you got that.

You might have represented this on a number line, like this.

So just like before, we've partitioned that 5/6 into 2/6 and 3/6.

And then we've subtract to the 2/6 to take us back to the nearest whole number, that's 6.

And then we've subtracted 3/6 to take us to 5 3/6, so that's 5 3/6.

Okay, let's revisit this calculation.

That's 5 2/10 - 4/10.

Is there a different strategy that could be used? Hmm, could we think of that mixed number in a different way? That 5 2/10, is there another way to express that? The mixed number could be converted to an improper fraction.

Have you got any skills and experience of doing that? "How could I express 5 2/10 as an improper fraction?" wonders Izzy.

Can you help her? Well, first we multiply the whole number, so that's 5, by the denominator, then we add the numerator.

So why are we doing that? Why are we multiplying 5 by 10? Well, each whole number can be thought of as 10/10.

So 5 multiplied by 10 gives us 50/10.

So that's a whole number part, then adding on the extra fractional part, that's a 2/10, so that's why.

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

So that's 5 multiplied by 10 plus 2 and that's 52.

So we could say it's 52/10, we could think of it as an improper fraction.

Now, 52/10 - 4/10, that seems a lot easier, doesn't it? That's equal to 48/10.

Although 48/10 is mathematically correct, and there's nothing wrong with it as such, it's more conventional to express improper fractions as mixed numbers in a final answer.

So 48/10, we can convert back into a mixed number.

How could we do that? Think about how many wholes we can create and then what the fractional part left over will be.

How would you write 48/10 as a mixed number? Well, the unit we are working with is tenths.

We can make four full groups of ten-tenths, 10, 20, 30, 40, that's as close as we can get.

So 4 whole groups and then 8/10 remaining.

So that's 4 8/10.

48/10 is the same as 4 8/10.

Whichever strategy we use, the answer was 4 8/10.

Izzy has 4 8/10 metres of rope left.

Let's do a final check.

I'll do another example and then you do one too.

So let's do 6 2/7 - 4/7, and let's think of it, not as a mixed number as it is now, but as an improper fraction, let's see if we like that method.

So first, convert the mixed number to an improper fraction.

So can you remember the method, what we do? We multiply the whole number by the denominator and add the numerator.

So that's 6 multiplied by 7.

Me, that's the times tables factor that you know off by heart, that's 42 plus 2 is equal to 44.

So that's 44/7, and then subtract the 4/7.

What's that equal to, we've done the hard work now? That's 40/7.

Now, we're not quite there yet.

That's not incorrect, but it's not too conventional, we would tend to represent that as a mixed number.

Now, how many wholes can we make from that? How many whole groups we need to convert? There's five full groups of 7/7, 7, 14, 21, 28, 35, that's as close as we can get.

So five whole groups and then five left over, 5/7 remaining.

So 5 5/7 is the correct answer.

Okay, over to you.

Can you do the same thing with 2 3/8 - 6/8? Pause the video.

How did you get on? Did you manage to convert to an improper fraction to start with? Well, let's see.

2 3/8, so we need to multiply the whole number by the denominator and add the numerator.

That's 2 multiplied by 8, 2 times 8 is hopefully a known times tables fact for you, that's 16 plus 3, that's equal to 19.

So we've got 19/8 - 6/8, and hopefully that's pretty easy for you, that's 13/8.

Now, we need to turn that into a mixed number.

So, let's do that.

How many whole groups can we make? Just the one this time and then five left over.

So that's 1 5/8.

Very, very well done if you've got that, you are definitely ready for the next part of the learning.

And the next part, the final part, is our practise.

So number one, using either rods and number line or converting the mixed number to an improper fraction, the choice is yours.

You might want to mix up the methods, you might want to use one to check, calculate these: 4 2/9 - 6/9, 5 1/3 - 2/3, 4 3/8 - 7/8.

Remember to express your answer as a mixed number.

And number two, solve these problems: A3 Andeep reads for 7 1/4 hours on Saturday.

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

Less, think about what that means.

How long does he spend reading on Sunday? And then B: Izzy has 6 2/10 litres of water in her bucket, that's our mixed number.

She uses 7/10 of a litre, that's our proper fraction, to water some plants.

How much water does she have left in her bucket? Left, so what operation are we using there? Okay, good luck with that.

If you can work with somebody else, please do.

Pause the video, and away you go.

Welcome back, how did you get on? Are you feeling confident? Would you like some answers? Let's see, so number one: 4 2/9 - 6/9, well, 4 2/9 becomes 38/9, because 4 multiplied by 9 is equal to 36 plus 2 is equal to 38.

So 38/9 as an improper fraction, subtract 6/9, gives us 32/9.

Now, we need to convert 9, 18, 27, that's as close as I can get.

So that's 3 wholes, and then what's left over, 5/9, so 3 5/9.

And then 5 1/3 - 2/3, 5 1/3, 5 multiplied by 3 is 15 plus 1 is 16.

So that's 16/3 - 2/3, nice and easy, that's 14/3.

Now, for the slightly trickier part, 3, 6, 9, 12, that is 4 2/3 left over.

And then 4 3/8 - 7/8, well that's 35/8.

Then subtract the 7/8, that's 28/8.

Now, to turn that into a mixed number, 8, 16, 24, it goes into it three times, so it can make three wholes from it, and then 4/8 left over, so 3 4/8.

You may have used a different strategy, but your answer should be the same.

And then number two: Andeep reads for 7 1/4 hours, that's our mixed number.

On Sunday, he reads for 3/4 of an hour less, that's our proper fraction, how long does he spend reading on Sunday? Well, this is what we can do to represent that using a bar model, and this is the equation.

7 1/4 - 3/4 is equal to something.

We could turn that into an improper fraction.

That's 29/4, and then subtract the 3/4, that gives us 26/4.

Now, that's not quite the answer yet, is it? We need to work out how many wholes we can make.

Now, we could use our times tables knowledge here, couldn't we? So, I know that 4 multiplied by 6 is equal to 24.

So it's 6 wholes and then 2/4 left over, so 6 2/4.

On Sunday, Andeep reads for 6 2/4 hours.

That's a lot of reading, good for you, Andeep.

And B: Izzy has 6 2/10 litres of water in a bucket.

She uses 7/10 of a litre of water to water some plants.

How much water does she have left in her bucket? So this is our bar model, and this is our equation.

This is our improper fraction, 62/10.

And then we're subtracting those 7/10, and that gives us 55/10.

Well, we can make five whole groups because 5 multiplied by 10 is equal to 50, and then 5/10 left over, so 5 5/10.

Izzy has 5 5/10 litres of water left in her bucket.

We've come to the end of the lesson.

You've done ever so well today.

Very well done to you.

Why don't you give yourself a little pat on the back? You deserve it.

Today we've been subtracting a proper fraction from a mixed number crossing the whole.

Rods or number lines can be used to support subtracting fractions from whole or mixed numbers.

Mixed numbers can be expressed as improper fractions to aid subtraction.

If an answer to a calculation is an improper fraction, this should be expressed as a mixed number, so you've gotta convert it back.

So you've used lots of different methods today.

I wonder if you've got a preference.

Personally, I really like converting it into an improper fraction and then converting back at the end, that's the way I find them most efficient.

Well done on your achievements and your accomplishments today, I hope you have a fabulous day whatever you've got in store, and that you are the best version of you that you could possibly be, you can't ask for more than that.

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

But until then, take care, and goodbye.