Lesson video

Hello everybody, welcome back to lesson 17 of addition and subtraction for Upper Key Stage 2.

In the last session, we were looking at how to transform into calculations and we're going to carry on where we left off.

So if you could have your practise examples ready, with how you solved them, and some new pen and paper for today's session, and we'll get cracking.

So far as we know, make sure you've got everything you need and I'll see you in a moment.

Let's take a look at the practise activity.

If you remember from the previous session, we were looking at linear equation, and we were writing in particular to solve it in a written method.

We found a problem didn't we?, when the minuend was a multiple of 1000.

Can you tell me what the problem was that we had to deal with? Yes, that's right.

Because my minuend is multiple 1000, I had to exchange across all those A-rows.

Do you remember it?, Took us ages, didn't it? So what did we do? How did we solve that problem? That's right, we transformed the calculation, didn't we? And we used the same difference generalisation to help us.

Can you remember what that is? Shall we say it together? If we change the minuend in the subtrahend, by the same amount, then the difference will stay the same, I'm sure you can recite that one off by heart now too.

So how did you transform this calculation to make it easier in a written method? What did you subtract? Did you subtract, one? Well done.

And if I subtract one off the minuend, it's no longer a multiple of 1000, is it? And I could subtract one, across the boundary like that really easily.

So 7,999 is one less than 8000.

And it's a really really good mental strategy to apply within this calculation to make the written method easier.

Also I have to subtract one from my subtrahend, which gave me 4,386 and I haven't written it in a vertical calculation, for this slide because I've run out of room.

I know you will have done, and did you find it easier? So basically then, we can say that the difference will be 3,613.

Because I have nine ones subtract six ones, nine 10s subtract eight 10s and nine 100s subtract three 100s.

So I didn't need to do any exchanging, did I? Really well done if you got that answer.

So how can we apply that to the second one, where we've got 42,000 subtract 34,832.

What do you notice about that calculation? Yes, that's right.

So my minuend is a multiple of 1000 again, isn't it? So if I subtract one from the minuend, and one from the subtrahend, then I will find an equivalent calculation that's much easily easier to solve vertically.

What is 42,000, subtract one? It's 41,999, excellent.

It's really easy to decrease a number by one, isn't it? And I've also got to decrease the subtrahend by one, 34,831.

And now if you've done that vertically, you will have found that your difference is 7,168.

Well done if you've got that correct.

Okay, can I reapply the same principles to the third one? Is there anything different about this one that might make it harder? A decimal fraction, haven't I? Does that make it harder for me to apply the same difference principle? The same difference generalisation? No, I can still apply that to this subtraction so I can transform the minuend and the subtrahend by the same amount, and my difference will stay the same.

I'm just thinking, when I look at this, when you were looking at mental calculations, we might perhaps have subtracted 0.

318 my way and subtracted 318,000s.

Would it be easy to subtract 318,000s from 176? That'll be quite tricky, wouldn't it? Actually, I don't think it would make the written calculation any easier.

In number one and number two, we subtracted one, didn't we? What do we have to subtract one off in number three? We can subtract 1000th, fantastic.

So if I subtract 1000th, I will have 175.

999, excellent, Because the principles are decreasing across a place value boundary are the same, whether we do that on the 100,000s boundary, the 10,000 boundary, the tens boundary, or if we do that, subtracting 1000th, excellent.

So we've got 175.

999.

And if I add, what do I need to add to that to make 176, I need to add 1000th.

So increasing it by 1000th would make a whole number and decreasing it by 1000th gets 175.

999, two part.

And we had to do the same to the subtrahend.

That was much easier to think about, isn't it? 22.

317.

But now we can put that in a written calculation.

And again, I've not got any exchanging have I? So 9000s subtract 7000s is 2000s, etcetra.

And you will have been able to work across your written calculation and get your difference as 153.

682.

Really well done if you've got those all correct.

I'm hoping that you found that transforming those written calculations made it easier for you to solve by using a written method.

I'm going to carry on looking at that in this section two.

This was an interesting problem.

An aquarium has two water tanks.

The large one has a capacity of 520,050 litres, I think that's the one with the shark in, and the smaller one has a capacity of 1,755 litres.

It's got all the seahorses in it.

What is the difference between the capacities of the water tanks? We need to use a written calculation, don't we? So if we wrote it vertically like that initially, we'd have some issues, wouldn't we? What issues would we have? We've got to exchange through the zeros, which we know can be quite tricky, can't it? Unless there's a couple of zeros that we'd need to exchange for? What can we do to make that simpler? We can subtract the same amount from the minuend of the subtrahend.

Subtracting one wouldn't help us here, would it? Because if I change that to 49 at the end of the minuend it wouldn't really help us.

What could I subtract? If I subtract 51, well done, is that what you did? So I've got to do that off the minuend and the subtrahend, haven't I? Because then we know the difference will stay the same.

So I subtract 51 from 520,050, which is going through that boundary again, aren't we? And I will end up with 519 999, super.

It's really easy, isn't it? And I've got to subtract 51 of my subtrahend as well but that's quite an easy calculation.

I will get 1,704.

Now I don't have any exchanging for zero for this written calculation, do I? So much easier to do.

And that small amount of mental calculation subtracting the 51 three and five for us, isn't it? So nine subtract four is five, nine subtract zero, 9, 10 subtract zero, 10s is nine 10s.

Nine 100s subtract seven 100s is two 100s, 9000 subtract 1000 is 8000, and then I got nothing to subtract from my 51,000, have I? Fantastic.

So did you get your differences 518,295? Well done if you did.

And let's just take a minute to notice how much easier that written calculation is than the one where we have the zeros in the minuend, super.

We've got a two step problem for our challenge and sometimes it can be quite easy to get lost in a two step problem and lose focus on what we're trying to find.

So let's read this carefully and see what we need to do.

A school plans to spend 14,848 pounds on computers and 1,694 on printers.

The budget for the items is 18,000.

Estimate whether this is enough for purchasing the computers and printers.

So entire budget is 18,000 pounds.

What's the best thing to do to estimate? I'm pressed to use a mental strategy, aren't I? So I can round up or round down to do that, can't I? 14,848 are best rounded up to 15,000, aren't I? and 1,694, I'm going to run that down to 1,500 and I find altogether they are 16,500.

So the sum of those is 16,500, brilliant.

So do I have enough money? 'Cause my budget was 18,000 and the difference between what I have and with the total amount I have to spend which is 18,000 and how much I'm spending which is 16,500 would be 1500.

So I do have enough, fantastic.

Okay, let's move on to part two.

If the budget is sufficient, how much money will be left after the purchase? If the budget is sufficient.

So I know the budget is sufficient because I've estimated I'm spending around about 16,500.

How much money will be left? How will I find that out? So I've got to subtract what I've spent from my overall budget, fantastic.

So is 16,500 what I've spent? No, it's an estimation of what I've spent.

So I'm going to have to work that out actually, aren't I? So I'm going to have to work out exactly what I spent.

How can I do that? I've got to find the sum, haven't I? So I need to find the sum of 14,848 and 1,694, and I could use a written vertical calculation to do that, couldn't I? I don't have enough room on my screen here but I imagine you wrote a vertical calculation and the total amount I spent there was 16,542 pounds, fantastic.

Better finished now, haven't I? Pardon? Yeah, I think I've done an awful lot, I feel like I've done more than two steps.

You think I'm finished? No, I'm not finished, have I? Because if we look at number two again, it says, If the budget is sufficient, how much money will be left after the purchase? So so far, I only know an estimate of how much I would like to spend and the exact amount I spent.

We can see currently, our estimate was really close.

So we did really well there.

But now I want to find how much money is left over.

So I'm finding the different, aren't I? between the total amount of my budget, which is 18,000 and the amount I spent, accurate amount I spent, which is 16,542.

I'm going to need to for that, aren't I? Okay so 18,000, subtract 16,542.

I can hear you screaming at me, Oh, Ms. Heathen, there's lots of zeros in the minuend.

What are we going to do? We're going to subtract one, fantastic.

So instead of doing 18,000 subtract 16,542, I hope you wrote in your vertical method 17,999 subtract 16,541 because that would mean it's the same, wouldn't it? If we subtract the same amount from the minuend and the subtrahend.

So the total left, the total amount of money left would be 1,458 pounds, brilliant.

So, we can see it might be quite easy to get lost in that problem, there were quite a few stages.

Well done if you found out, the money left over was 1,458 pounds.

And for our new learning.

So we've been spending quite a bit of time transforming calculations using the same different strategy and I want us to look at these three problems together.

We've been using the same different strategy in our previous sessions and I want you to see if you can think away of transforming these calculations that will make them easier as a written method.

We're going to do each one, one by one.

Let's have a read of this problem together.

A football stadium has a capacity of 19,000.

During a match when it is full, there are 67,899 home fans.

How many away supporters are there? Okay, I can already see you thinking, Oh, I need to find the difference Ms. Heathen and I know you've got so many tools in your toolkit now to help you do this as a written strategy.

I know that you're going to remember that you don't want to exchange across those zeros and you're going to need to transform the minuends and subtrahend, by the same amount to keep that written calculation the same and the difference the same.

Okay, so pause for now, have a go with that and we'll come back and look at it together.

How did you get on? So we have our problem, don't we? We can see it here, it's really nice in this table.

The full capacity of the football stadium is 90,000 spectators and there are 67,899 home fans.

So we need to find out how many away fans there are, don't we? So we find in the difference, aren't we? between 90,000 and 67,899.

So we've got a part pothole problem here, haven't we? We know what the whole is? The whole is 90,000.

One of the parts is 67,899.

We need to find the missing part.

How did you find the missing part? Did you subtract 67,899 from 90,000 to find the difference between them? Yes, that's what I would do as well.

Okay, then.

So we can write that out as a vertical method.

Did you write it out like that? No, why not? Why didn't you write it out like that? Because it's really inefficient, isn't it? I would have to exchange for all of those zeros so that I had a unit in the columns of the minuend to subtract from.

What we found out in the last session that was really tricky, didn't we? So what could we do instead? What did you do? Did you subtract the same amount from the minuend and the subtrahend? Fantastic.

So we can transform the calculation to make it easier can't we? And here's our calculation written vertically again.

What should I transform it by? I can subtract one, really well done.

And we know it's really easy, isn't it? to subtract coastal boundaries.

So 90,000 subtract one is 89,999.

We've got to do the same to the subtrahend, so we subtract one from there, 67,898.

And now, it's really simple, isn't it? Nine ones subtract eight ones is 1 ones.

9 tens subtract 9 tens is zero tens.

9 hundreds subtract 8 hundreds is 1 hundred.

9 thousands subtract 7 thousands is 2 thousands and 8 ten thousands subtract 6 ten thousands is 2 ten thousands.

So how many fans did we find as are missing part? 22,101.

Which equivalent calculation would be easier to solve? It's obviously the second one, isn't it? Really well done.

So there are 22,101 away supporters.

We had 67,898 home fans and that leaves a total attendance of 89,999.

Really well done.

Here is the next problem I'd like you to solve.

I think you're going to find this one a double.

Hari had saved 400 pounds.

He's really well saving that amount of money, hasn't he? And he bought a bike for 369.

99 pounds.

How much does he have left? So what do you notice about these two numbers, Initially, I can see my mind's ticking over already.

I think, again, you've got all the tools in your toolkit to be able to do this, because we looked at decimal fractions, didn't we at the end of the previous session.

So I'd like you to pause for now and have a go at these.

Remember, I want you to use a written vertical calculation and to transform your minuend and your subtrahend by the same amount, so that your different stays the same.

See you in a moment.

How did you get on.

So Hari's looking very pleased with himself that he saved 400 pounds, I'd be very pleased with myself too and he's going to get his big ambition, he's buying his bike and he wants to see how much he's going to have left.

So Hari's been really, really careful and he's written a bimodal to help him understand what he's got.

And I can see how the whole of my money is 400 pounds.

And to buy the bike, I need to spend 369.

99 pounds.

So that's really good, isn't it? It's really clearly showing us the part that we are missing there.

So how, what does that represent, the missing part? It represents how much you have left over, really well done, which is what we want to find.

How could we do that using a written calculation? So I want to subtract 369.

99 pounds from 400 pounds.

How did you do it? I can see some of you waving your papers at me there.

So did you write 400 pounds subtract 369.

99 pounds.

And now we've got tens and hundreds, haven't we? because they're 100 pence in a pound.

What's difficult about this calculation again though? We don't want to exchange for those zeros, do we? So do you think we could transform it in some way? Let's hope and see if we can? Okay, so right against in the linear equation, we've got 400.

00, subtract 369.

99 and it loses a little bit of its context now that we've taken those pound signs out, doesn't it? Let's not forget that, that we're working in pounds and pence here, aren't we? And we said we could write it in a linear equation like this.

I think that looks just looks too tricky, doesn't it? and awful lot of work.

If you remember in the previous session how long it took us to exchange across all of those columns.

So what could we do? I'm thinking again, thinking about how you're working in pounds and pence here, what can I subtract from 400 pounds? To make this an easier calculation to transform, I can subtract one pence, really well done.

And if I subtract one pence, I will have 399 pounds and 99 pence, which is exactly the same as what we did with problems where we were working in pounds and pence, but we still had decimal fractions.

So I thought to remove the same amount from the subtrahend as well, haven't I? So if I take one pence off the subtrahend, I have 369 pounds and 98 pence.

So which equivalent calculation is easier to solve? Oh, it's obvious, isn't it? so much easier when I don't have to exchange.

Should we have a go at doing that? Or you've done it, haven't you? Should I check but I can do it right and that you got the same as me.

So nine one hundredths subtract eight one hundredths is 100.

Nine tenths subtract nine tenths is zero tenths.

It's interesting to note there as well.

I could say 99 one hundredths subtract 98 one hundredths is the same as one hundred.

So we could say that in both both ways, couldn't we? It's having on 9 ones subtract 9 ones is zero ones, 9 tens subtract 6 tens is 3 tens and 3 one hundredths subtract 3 one hundredths is note, 100 zero hundredths.

Fantastic.

So how much did he have left? 30 pounds and a penny, fantastic.

I would go with that penny.

So we had 30 pound and one pence left.

As you can see it's much easier when we're transformer calculation to perform that with a written method.

Well done.

This is our third example.

Are we doing well? Let's read it together.

A female African elephant has a mass of 5,050 kilogrammes.

The mass of her calf is 298 kilogrammes, oh baby elephant.

How much less is the mass of the calf than the mass of its mother? So I'd like you to have a look at that, tell me how you're going to find the difference with a written calculation.

I want you to transform it.

So pause for now and have a go with that.

How did you get on? We can see the African elephant and her beautiful calf here, can't we? So the African elephant weighs 5,050 kilogrammes and we can see there, that's a whole of the African elephant, and the calf's mass is only 298 kilogrammes.

And we can see it as considerably smaller, isn't it? when we compare it to the weight of its mother? And we want to find the difference and how many kilogrammes that is more than.

What did you write? Was your written calculation something like this? 5,050 subtract 298.

But we have some issues with that, don't we? We know we've got to exchange through the zeros and perhaps we could make this slightly easier.

There are a number of different ways we could do it I think, by looking at the minuend and the subtrahend and trying to find an easier calculation.

Let's have a look at the different ways you might have done it.

How did you get on with this one? I'm thinking that actually people will have done this in a myriad of different ways.

I think some of you might have subtracted 51, like we've been doing in previous examples, so that you got 4999 as your minuend but then you'd have to subtract 51 from 298 as well, wouldn't you? So I think some of you might have done it that way.

I've got a feeling that some of you might have looked at the value of the subtrahend and thought, that's really close to a multiple of 100.

So actually, would it be easier to transform it by adding something to the minuend and the subtrahend together? What do we think? What could we add to the minuend and the subtrahend to make the subtrahend a multiple of 100? We could add two, really well done.

So my written calculation will transform to 5,052 and subtract 300.

Fantastic.

Is that still an easy calculation to do? Yes, it is.

I mean, we've got one zero, haven't we? in the minuend, but it's quite easy to exchange across one zero.

Is when they're all zeros that we really really find it more onerous and difficult.

So let's have a look at this.

Two ones subtract zero ones is two ones.

Five tens subtract zero tens is five tens.

And here, we can exchange, can't we? So we can exchange one of the hundreds so that we have 100 is the same as 10 tens, 10 tens subtract three tens is seven tens.

And 4000 with nothing to subtract from that so we get 4752.

So quite interesting, actually, isn't it? We tie together all the different things we've looked at, we can transform the calculation in various different ways and it's whichever makes that written calculation simpler.

Well done.

Okay, I'd like you to look at these four examples and I'd like you to decide which of these would benefit from be solved using the same difference strategy.

So pause for now, have a look at each one and you can put a tick or a cross to say which one you think would be really beneficial.

See you in a moment.

What about this first one? You got 83 subtract 49.

Which same different strategy help us with this? Yes, brilliant, we could add one, couldn't we? to the subtrahend and the minuend and then we get a really easy calculation, 84 subtract 50.

Excellent, well done if you put a tick and got that.

Did you use the same different strategy to help you solve this one? I'm thinking that we might have a few different ways of doing this.

So let's explore the different strategies.

Has anybody looked at the subtrahend and think, that's really close to 8.

3.

And how would I get 8.

3? I would have to add 0.

01, really well done.

And so that I, my subtrahend would be 8.

30, fantastic.

So I have to do the same to the minuend, don't I? It's really easy adding on 100 so I get 16.

90.

And actually, that will be a really simple subtraction now, wouldn't it? And I will try my difference is 8.

6, well done, and the difference would stay the same.

That's not the only way we could do it, though.

Did anybody do it differently? Does anyone look at the subtrahend and think, instead of adding on 100, I could subtract 29 hundredths, so I could subtract two tenths and nine hundredths, which is 29 hundredths altogether? And that will be quite easy as well, wouldn't it? and I would get eight whole months.

And can I subtract 29 hundreds from my minuend really easily? I can, can't I? And that would be really easy to do mentally as well.

16.

6, well done.

And is that one easy to find the difference? Yes, I can find the difference with that, can't I? and the difference is 8.

6 again.

So well done if used either of those methods.

What did you think about this one? 53,604 subtract 40,000? Do I need to transform the calculation? No, it won't be pointless, wouldn't it? There's absolutely no reason why I need to use the same different strategy because it's really simple to solve that four ten thousands followed by ten thousands.

So we don't need to do anything to that, we can use our mental strategies.

Well done.

Did you use the same different strategies to help you solve this with a written calculation.

I think I would too.

What would we do? We could subtract 4000 from the minuend and the subtrahend, well done.

And we've used this method quite successfully in a number of examples now, haven't we? If I subtract four thousandths from 70.

003, I would get 69.

999, excellent.

And it's just as easy, isn't it? subtracting one thousandth or one hundredth or one tenth as it is one, or 10 or 100, is the same rules of place value, isn't it? Fantastic.

So we do the same to the subtrahend and we would get 13.

387.

So would that makes it easy to solve vertically, shall we have a look at that? I've got my written calculation vertically, we've said, haven't we? we're going to subtract four thousandths from both the minuend and the subtrahend.

So 69.

999, subtract 38.

387.

Which one of those is easier to solve? It's obvious, isn't it? The second one is easier to solve, because I've got no exchanging.

So nine thousandths subtract seven thousandths is two thousandths.

Nine hundredths subtract eight hundredths is 1 hundredth.

Nine tenths subtract three tenths is six tenths, fantastic.

Nine ones subtract eight ones is one ones and six tens subtract three tens is three tens, brilliant.

So our same difference strategy worked really well in that case, didn't it? Well done for today, we've done an awful lot of thinking.

I'm really pleased.

I think that looking at the sequence of lessons that you've done over the last week, you've really started to think about your mental strategies, which is the most appropriate mental strategy.

And then also applying those mental strategies within a written calculation to make the written calculation more efficient too and we've been using that same difference application all throughout, haven't we? I'm thinking about that generalisation.

So the practise activity here for you today.

Dorota's got a mental strategy to calculate 342 subtract 96.

We need to have a read through, what Dorota does to help us solve it and see which of these options, you think, will enable her to solve that calculation.

And that's it from us today.

So I hope you've had a really good session, you've really moved forward and made a lot of connections and I look forward to seeing you again.

Bye, bye.