Lesson video

Hello, everybody, I'm Miss Heaton, and I'm going to be teaching your lesson today.

It's lesson 16 of addition and subtraction, through key stage two.

I hope you've got your practise activity from yesterday at hand and some pen and paper.

So you don't have all of those just quickly pause me and gather them all together.

And I'll see you in a moment.

Before we share the solutions to these subtractions, I'd really like us to just quickly review the generalisation that you used in the last session.

Let's say this together.

If the minuend and subtrahend are changed by the same amount, the difference stays the same.

Well done.

That's really important isn't it looking at these mental calculations that you've been given? We can see that the minuend subtract the subtrahend this was our difference.

So you've been using your mental strategies to see what you notice about these numbers and see if you can transform the subtraction to make it easier.

Repeat again for the first one, did you get 2,868? Well done if you did.

How did you do it? What did you notice about this subtraction? That's right.

Did you notice that the subtrahend is close to a multiple of a thousand.

Well done.

And so what do we need to, how can we transform that to a thousand? We have to add all four, well done.

And as we've said here, if the minuend or subtrahend are changed by the same amount, but if it stays the same.

So if I add four to the subtrahend hand, I get 3000 and if I add four to this, I get 5,868.

And so it's really easy then isn't it to subtract a multiple of a thousand from my minuend.

And on this gives us 2,868.

Well done.

Did you apply that to the rest of them? Let's have a see, did you get 4,741 for the second one? What did you notice about the calculation? Excellent, again, so we know to start with the subtrahend is close or multiple of a thousand, and he's really easy to subtract multiples of a thousand, isn't it? So if I subtract five from both minuend on the subtrahend, then I can find my difference of 4,741 quite simply.

What about the next one? Did you get 26,346? Well done.

How did you do this one? What did you notice about the minuend and the subtrahend for this one? Yes, well done.

Again, the subtrahend is close to a multiple of 10,000 this time.

Isn't it? What do I need to add onto that to make multiple of 10,000? I need to add 15, really well done.

So if I add 15 to this I get 30,000 and it's really easy to subtract 30,000, isn't it? Is that all I have to do though? No, well done.

I nearly made a mistake that I didn't? I nearly just transformed the subtrahend and actually I've got to sort of transform the minuend as well, haven't I, so make sure that I add 50 to both of these, and then my difference will stay the same.

Super.

What about the next one, did you get 45.

85? Well done.

What's different about this one? Yes, I got the decimal fraction haven't I? Does this make it any harder? I suppose of subtraction the same.

Yes they are.

So the minuend subtract the subtrahend is still the difference, but my place value reasoning is changing isn't it? I've got to think about my strategies within a different part of place value.

I just have a see then what do you notice about the minuend and the subtrahend? Yes so again, the subtrahend is very close here.

This time is very close to a power of 10, isn't it? It's a multiple of 10.

How can I transform the subtrahend to a power of 10? I can subtract three one hundredths, brilliant, three hundredths, so 0.

03, and if I subtract 0.

03, both the subtrahend and the minuend then my difference is 45.

85 because I can just then subtract 12 from the minuend.

Excellent.

And this last one of this group, did you get 7.

78? Brilliant.

I bet you're really pleased if he got all of those right.

How did you do this one? Did you transform the minuend and the subtrahend by the same amount? Fantastic.

What was that amount? What did you notice? Brilliant, see, I noticed this was very close to 10 and actually subtracting 10 is really easy.

So I also knew that I couldn't use my number bombs to help me make that round this up to 10.

And so I added on 0.

22, because I added on 0.

22 here.

I had to add on 0.

22 to my minuend, which again is quite easy to do isn't it.

So I got 17.

78, and then I could just subtract 10.

And subtracting 10 from 17 is easiest.

So I got 7.

78, brilliant.

Hands up if you've got all those rights.

Brilliant.

Okay, so the challenge again, really tricky for you to think about this one.

We're thinking if the minuend and subtrahend to change by the same amount, the different stays the same.

Well, perhaps this isn't quite as easy to transform as some of the mental calculations you've looked at there.

So I'd like to look at this one and a little more detail.

Your challenge was to subtract 1,658 for 20,000.

How did you get on with that? Did you find it easy? Did you transform the calculation to help you do it? I'm wondering actually, how many of you used a mental strategy and did anybody try and a written method to help solve it? I might to explore some of those ideas.

So if we think about solving this mentally first, perhaps you use a mental normal line.

And so we could perhaps start at 20,000 current ways subtract a thousand and then 600 and then 50 and then eight to find our difference.

However, would that be particularly easy do you think? It's quite a bit to it, isn't it? I got to remember quite a lot of things going along there.

So perhaps that's not particularly efficient.

Did anyone use any of the methods? Interesting isn't it? That not all subtractions lend themselves particularly well to a mental strategy and in some cases, a written method might be much more efficient.

I'm wondering if a lot of you reverted to a written method for this one.

So let's explore that in more detail.

Because I'd like us in this session to think about how we can transform calculations to make written methods easier still.

Okay, did you write to your written calculation like this? If you were trying to find the difference using a written calculation instead of mentally, what do you notice about it? I noticed that the minuend is a multiple of 10,000 now.

It's interesting, isn't it? In the mental calculations when a subtrahend was a multiple of a thousand or 10,000, it was really easy to subtract, wasn't it? Now that my minuend is a multiple of 10,000.

Does it make it easier or harder? I think it makes it harder, doesn't it? Why do we think it makes it harder? So in my minuend here, I've got an awful lot of zeroes in the columns and so it's really difficult to subtract a unit from the column if the column starts there and if it's not got any units in it, what do have to do then to solve that problem.

We usually, we exchange don't we? Fantastic.

And exchanging is needed when there's an insufficient number of a unit to subtract from in an even column.

So a unit is exchanged from the column to the left.

Really, do you all remember that? Let's do that then, that sounds easy enough.

So if I look at my one's column, I've not got a unit to subtract from.

So I'll stage from the column to the left.

Oh, a unit to exchange there either, because I've got to zero in my tens column, let's go to the next one on the left.

It's zero again, isn't it.

This is proven to be a little trickier isn't it? And my next column zero, okay.

So I'll have to start exchanging from here.

Is that what you did as well? Let's get that to go with then.

So I have two, ten thousands, excellent.

And I'm going to give one 10,000 to my thousands column, Do you agree? Fantastic, let's have a look at what that says.

So two ten thousands.

I'm going to exchange one 10,000.

So I left with one 10,000 in my ten thousands column.

And I have ten thousands now in my thousands column, because one 10,000 is the same as ten thousands super.

And I do now.

So I have ten thousands in my thousands column and I need to exchange from the column on the left.

So the talking gets a unit in my hundreds column.

So I have ten thousands.

I'm going to exchange one of those thousands into my hundreds column.

So how many hundreds do I have in my hundreds column now? Ten hundreds, so I am exchanging 1000.

So I've got nine thousands in my thousands column and 1000 is the same as ten hundreds, super.

Okay, I still need to exchange from this column on the left now.

So I have ten hundreds let's do that one.

So I go to exchange 100 to my tens column.

So what have I got in now? So ten hundreds, I exchange one of those hundreds, which left me with nine hundreds in that column.

And 100 is the same as how many tens, ten tens.

Fantastic, so I've got 10 tens now in my tens column.

And I think I've got, have I got over exchange I need to do here.

So I still can't subtract, can I here? 'Cause I've got no units in this column.

So I'm going to exchange from the left again.

So I have 10 tens, I'm going to exchange one of those tens to my one's column.

So 10 tens, if I exchange one, I have nine tens in my tens column and I now have, what do I have in my ones column, 10 ones.

Fantastic, I've got 10 ones.

Oh gosh that was an awful lot of work, wasn't it? Did you all manage to do that? Fantastic, I complete my written subtraction now.

Yes.

I just want to check has anything happened to my minuend? I've exchanged haven't I? Across each column through the zeros, but it has the value with my minuend changed? No it's not.

Did some of you think that the value of my minuend might have changed? Shall we check? Let's see what we've got here.

So we had one 10,000, should we count them all together? One 10,000 plus, 9,000, so I've got 19,000.

Plus 900, 19,900.

Plus 90 because that's nine tens.

So 19,990 and I got to add on 10 ones, 19,990 plus 10 is 20,000, fantastic.

So my minuend has stayed the same.

I've just exchanged through the zeros to ensure that I have a unit in each column that I can subtract from.

Okay, let's take quite a wireless how's that? As not as quick as a mental strategy, is it? But can I do the subtraction now? So I can say 10 ones subtract eight ones is two ones.

Fantastic.

I've got nine tens subtract five tens is four tens.

Great, I've got nine one hundreds subtract six one hundreds is three one hundreds.

If it had of me there, let's say this one together, nine thousands subtract 1000 is 8,000.

Yes, and I've got 10,010 haven't time, one 10,000.

I'm not subtracting anything from that.

So I got one 10,000 and altogether my difference is 18,342.

Well done if that you got that difference correct.

Oh, I thought it was really tricky that wasn't it exchanging through all those zeros.

And I'm wondering if we can use the same difference principle that you've been using to help us transform written calculations in the same way that we've been transforming mental calculations to make that process a lot easier.

Shall we see if we can? Get the calculation out again so that we can see our minuend subtrahend and difference without all of the exchanges? Do you think that we can transform this written calculation to make it easier to solve.

We did that with the mental calculations didn't we? So now we rounded up around it down to the nearest thousand.

That was a really good mental strategy that we used is that applicable here.

So we need to, what was the biggest problem we had with the exchange? We need to be able to get a unit don't we? And so these columns that we can subtract from, so we'll look at the ones column, what unit would need to be in the ones column in the subtrahend in the minuend.

And for us to subtract eight, we would need nine would we? So perhaps I could add nine on to the minuend, nine onto the subtrahend.

Would that make it any easier in calculation.

So 20,009, so tract 1,667, isn't particularly easier, is it? Because I've still got these arrows here add on 9,999, but then I'd have to add that to this as well.

Wouldn't I, then I'm just making an awful lot of work for myself? I don't think that's going to work.

Is there any other way I could transform 20,000? I could subtract a small amounts.

I can hear you now.

If I perhaps subtract, subtract one, couldn't I, if I subtract one from 20,000 though, I got to do the same to the subtrahend because let's remember our generalisation.

I thought maybe you ended a subtrahend the change by the same amount.

The difference stays the same.

Well done.

So I've subtracted one from my minuend I can see that I've got 95,999.

So half got units in each of the columns rather than zeros.

So I won't need to do any exchanging.

Will I? And what do I need to do now? Subtract one from the subtrahend.

Well done, and so I will get 1,657.

Excellent.

Do you think this as an easier calculation to solve what I'd like you to do.

Pause me now and write that calculation out and see if you can find the difference.

See you in a moment.

How did you get, so you write out 19,999 subtract 1,657.

So nine ones subtract seven ones is two ones, nine tens subtract five tens is four tens, 900 subtract six one hundreds is three one hundreds nine one thousand subtract 1000 is eight thousands and one 10,000 and we're not subtracting anything there.

Brilliant.

So what else do you notice the differences the same? Fantastic.

And we can say count with a 20,000 subtract 1,658 is equivalent to it's the same calculation as 19,999 subtract 1,657.

And that the difference will stay the same.

So if minuend and subtrahend are changed by the same amount, the difference stays the same, really well done.

I actually mixing it as a linear equation because I know you use this method using your mental strategies, and I wanted to make some connections to that.

So I've got 20,000 subtract, 1,658.

I would just set how up you minuend and the subtrahend to change by the same amount.

The difference stays the same.

So I'm subtracting one to get 19,999.

Subtract one to get 1,657.

And we found the difference was 18,342, which was the same.

So which there's a linear equation and doing the same jottings as you use for your mental strategies still gets us the same difference because those two equations are equivalent.

I hope we represented some of our mental calculations on the number line as well, didn't we, I just want you to put off those images there.

So I've got my two recent calculations and the number lines, I just want you to pass me and take a moment to have a think about what is the same and what is different to me now and have a look at those and come back and tell me what you find.

What did you notice? Okay, so firstly, what have you noticed on the number? You've notice that the difference has stayed the same.

Fantastic.

So we've got the same difference on the number line, however, we've just moved.

Haven't we along the number line by one we've subtracted one and over here, we've subtracted one as well.

There's differences stay the same.

Brilliant.

What did you notice in the written calculations? What was the same? The difference was the same.

Well, it's different.

The minuend the subtrahend the different brilliant.

So the minuend here is a multiple of a thousand, the minuend, and here we've subtracted one so that we can get the nines so we can have the unit of nine in each of these columns to make our subtraction easier or would not have to do any exchange, excellent.

Did you find that much easier? Do you think we could apply that to some more examples? Let's have a go.

Do you think we can use the same difference generalisations to solve this problem? Let's do this together.

A famous pop star had 400,003 likes for an online performance 24,438 of the same people also like to video of a funny cat.

How many people only like the pop star.

Okay, so we're finding the difference aren't we? We're finding the difference between the number of people who liked the online performance and those who also liked the cat.

Do you think you can help me write this in a linear equation? So our minuend will be 400,003, and I'm going to subtract the subtrahend, which is 24,438 and find the difference.

Excellent.

Now, if I want to put this in a written method, I'm going to write it out vertically.

I'm going to ask you to do that shortly, but let's just very quickly think about how I could transform it to make my written method easier.

Maybe we've got to change the minuend into the subtrahend by the same amount.

We know that generalisation don't we? What would happen if I subtracted three, looking at that.

And I think it always really easy to subtract three from 400,003 and get 400,000 and then I'd get 24,435 for my subtrahend.

Would that be an easy written method? No.

Why not? I felt in zero seven, I still got all those zeros in my minuend what should I subtract by them to get a number that doesn't have all those arrows in the minuend? And I subtract another one.

Excellent.

So if I subtract by four altogether, my minuend will be 399,999.

And we know I don't need any exchanging then do I? Have a finished now, no.

I've got to subtract by the same amount on the subtrahend So I need to remove one 74,434.

For both those calculations the difference will be the same.

I think he's going to be an awful lot easier isn't it? To write 399,999 subtract 24,434 in a vertical method.

That's what I'd like us to do now.

I've written the original calculation out vertically.

And we can see that's quite a tricky return method isn't it? Because I'd have to exchange for all of those zeros and you quite rightly totally, well, if I subtract four for both the minuend and subtrahend I get this written calculation 399,999 is much easier to subtract from because I won't have any exchanges.

I get to have a go at that.

So pause for now, write out that vertical calculation and find the difference.

How did you get on? Did you find it easy to find a difference subtracting from 399,999? Let's have a look at your difference.

So nine one subtract four ones is five ones, nine tens subtract three tens is six tens, nine 100 subtract four one hundreds is five one hundreds, 9,000 subtract four one thousands, is five one thousands and nine then thousands subtract two ten thousands is seven ten thousands.

And we have $300,000, which we're not subtracting.

Did you get the difference as 375,565? Super.

So we know that that difference is the same for both of those calculations because we transformed the minuends and the subtrahend by the same amount.

So 375,565 people only liked the pop star and they didn't like the cat as well.

Fantastic.

Well done.

Would you like to have a go at some on your own now? That's a couple of examples here I'd like you to just transform and see if you can make them easy to solve using a written method.

Pause me now and have a look at those.

I'll see you in a moment.

Let's have a look at the first one.

I had 8,000 and I want to subtract 4,387.

So that's how that would look vertically.

I know that that would be really tricky to do because I'd have to exchange with the zeros.

So I'm going to transform both the minuend and the subtrahend to make this written calculation easier.

Now I know that if I subtract one, I will get 7,999.

That's really easy, isn't it.

So I need to subtract one from subtrahend as well.

And I get 4,386.

And so this is a much easier calculation now isn't it.

Nine ones subtract six ones is three ones, nine tens subtract eight tens is one ten, 900 subtract three one hundredths is six one hundredths, and seven one thousands subtract four one thousands is three one thousands.

Did you get 3,613 for your difference? Well done if you did.

How did you get on with the second one? I'm going to write it vertically.

So I've got 12,002 subtract 6,935.

Now I know that that will be quite inefficient to do as it is, because I've got these zeros to exchange through.

So I'm going to transfer my minuend and subtrahend by subtracting the same amount.

If I subtract two, it doesn't really help me does it.

So I'm going to subtract three.

And if I subtract three, I will get 11,999.

And so I need to subtract three from my subtrahend as well, which gets me 6,932.

Now I can subtract the subtrahend from the minuend without any exchanging.

So nine ones subtract two ones is seven ones, nine tens subtract three tens is six tens.

I've got the same unit in the hundreds column, haven't I so nine one hundreds subtract nine one hundreds is zero one hundreds.

Here rather than exchanging.

I can say, can't I? 11,000 subtract 6,000 is 5,000.

So my difference is 5,067.

Well done if you got the same.

You've worked really hard to transform your missing calculations, using the same difference generalisation today.

So Kudos, we're going to stop there and it's time for your practise activity.

You'll see there's a few calculations for you to transform there.

And also a written problem.

See if you can apply the same principles.

I'll do the challenge for you as well.

So we've got two step problem here.

So I want you to think really carefully about estimating now on your mental strategies and then transforming recent calculations to make them easier.

I hope you get on really well with those, and I'll see you in the next lesson to have a look how you got on.

Goodbye, bye.