Lesson video

Hello, welcome back to lesson 22, to addition and subtraction.

My name's Ms. Eaton, and we're going to continue looking at the efficient calculation strategies that you've got to help you with your mental and written calculation and start thinking about the strategies that we've got at our fingertips and which ones might be the most efficient for various calculations.

So make sure you've got your practise activity for yesterday and some pen and paper, and we'll get cracking.

This is your practise from yesterday, and Mrs. Furlong left you these.

And you have been looking at how you can identify known facts within a calculation to help you solve it.

You also had a generalisation that you were using.

You think you can remember it? Shall we say it together? If the minuend is changed by an amount, and the subtrahend is kept the same, the difference changes by the same amount.

So we can see here, can't we, the minuend.

If this is changed by an amount, and the subtrahends are kept the same, then the difference will be changed by the same amount that we have changed the minuend by.

Fantastic.

So what was the known fact that you could see? Let's have a look.

We've got 75,640 subtract 340.

So a known fact within there that you can see quite easily.

Yes, I can hear you shouting at me.

It's 600 here, isn't it? If I've got 640 subtract 340, I can find 300 quite easily, can't I? Now how did I transform the minuend to identify, to expose my known fact of 640 subtract 340? I subtracted 75,000, well done.

So if I've subtracted 75,000 from the minuend, I've kept the subtrahends the same, so that means I must have already subtracted 75,000 from the difference.

So have I finished now? Can I say that the difference between these two is 300? No, I can't, can I, because I've got to add 75,000 back on to the difference, and I will then get, shout out at me, 75,300.

Well done.

Did you all get that? Fantastic.

Let's look at number two.

Now you did a really, really good fluency session with Mrs. Furlong yesterday.

Really enjoyed watching that.

And you were looking at number bonds to 100 and number bonds to hundredths, which really, really relatable aren't they? So what do we need to do here? Can we find any number bonds? Excellent, what's the number bond that you can see? So we could maybe here, our known fact could be one subtract 0.

34.

And that's really easy to do, isn't it, because we know that 0.

34, 34/100, added to 66/100 would make one whole.

And we're finding the difference between 0.

34 and one whole, so our difference must be the 66/100, 0.

66, well done.

Okay, we're finished.

So is the difference between 136 subtract 0.

34, 0.

66? No, don't be silly, Ms. Eaton, of course it's not.

So this was the known fact that I have seen within my original subtraction, but actually to find that known fact, I transformed the minuend, didn't I? And what have I subtracted from my minuend? I've subtracted 135.

Well done.

And so what do I have to do to this difference here? I have to add 135 back on.

So my answer, my difference will be 135.

66.

Brilliant, well done if you got that one.

So let's look at number three.

How did you transform the minuend and the difference in this one? Did you find a known fact? Which known fact did you find? Can I subtract the minuend from one again? No, it doesn't, and that wouldn't be helpful, would it? So what could I have? I could have four subtract 3.

37.

That's really good.

And the same principles apply, don't they? So I've got 0.

37 here, and I know if I add another 0.

63, I will make one whole.

So the difference between four and 3.

37 is 0.

63.

Excellent.

So what have I subtracted from my minuend? I've subtracted 94, super.

And so am I done now? So 98 subtract 3.

37 is 0.

63.

No, that's right.

I've got to add 94.

I come here, haven't I? Did you all do that? So the difference between 98 and 3.

37 is 94.

63, fantastic.

So I subtracted 94 from the minuend, which meant that 94 had been subtracted from the difference.

And I had to add that 94 back on from my original subtraction, brilliant.

Now Mrs. Furlong asked you to have a look at some different ways that you could change the minuend and the difference in questions two and three.

And in both of them, we have subtracted from the minuend, haven't we? Did anyone do it a different way? That's what we're going to do today.

We're going to look at all the different ways that we can transform calculations to make them easier mentally or actually if we want to transform them to make the written calculation easier too.

Let's have a look at number two.

Is there a way we could transform this? Actually, rather than seeing the known factors one subtract 0.

34, perhaps we could round off to a multiple of 10.

So how could I transform that? I could add on four.

So if I added on four to this side, have 140 here, I wouldn't I? Is it still quite easy to subtract 0.

34 from 140? It is if I'm good at my number bonds, isn't it? Because I know I'm still going to have 0.

66 here because 0.

66 and 0.

34 is a whole.

So I already know that, so I can just use that fact with a larger number.

My larger numbers are 140, so I would have 139.

66.

Really well done.

And then I'd have to subtract the four to get 135.

66.

So we can see actually that no matter how I transform the minuend, if I transform the difference by the same amount, I can then get my calculation correct again.

Well done.

And so for this one, I could do something similar here, couldn't I? I could round it up to 100 by adding two, and then I would have 100 subtract 3.

37.

So now I would have 96.

63.

Well done.

And I could add the two back, subtract the two, to find my difference of 94.

63.

So there's lots of different ways, isn't there, that we can look at that and we can, as long as we spot that known fact.

So here we could see, oh, I can subtract those 100s really easily.

Hey, we're thinking about our decimal fractions, aren't we? And here again, thinking about our decimal fractions and our number bonds to 100, 10, hundredths, and they can all help us to solve those.

Brilliant, well done.

Well, we're going to move on from there by also exploring the different ways that we can transform our calculations to make them easier to solve, also to think about which strategy is going to be the most efficient.

What I'd like us to do in this session is to really begin to analyse the calculations and think about the strategies that we have and how we can use the most efficient one to find our solutions.

I've got two calculations here, and I'd like you to just pause me for a moment and have a look at them and tell me what you notice about the two different calculations.

How did you get on, what did you find? Can you tell me anything that's the same or anything that's different about these calculations? What's the same? Yes, they're both subtractions, fantastic.

So they're both subtractions.

So they've both got a minuend and a subtrahend, and we don't know the difference.

So that's important as well.

They've both got a missing difference.

What else can you tell me? Is there anything else that's the same? Ah, that's a good idea.

So the values of the minuends are almost the same, aren't they? They've both got 100, and they both got eight 10s, but they've got a different amount of ones.

Fantastic, and what about the minuend? Okay, yeah, that's a good idea.

So the minuends, they're quite close in value if I've got four 10s, but again, they've got a different number of ones in the minuend.

And we don't know the difference, do we? But can we make some assumptions about the difference based on what you've just told me about the values of these numbers? We can assume, can't we, that the difference is, the differences are also going to be quite similar in value, brilliant.

I'm really glad that you spotted all of those things because that might help us to decide how to solve them.

Do you think we would solve these in the same way? If you were subtracting 49 from a minuend would you use the same strategy of subtracting 46? I'm not sure I would, and I think I might have a different strategy for both of those subtrahends.

But that's what I want to explore today.

I want us to think about the strategies that we're going to use and how many different strategies we have and which one will be the most efficient.

So should we go and explore those in more detail now? Come on, let's have a go.

Let's have a look at this subtraction in more detail.

So we've got 183 subtract 49, and we'd like to find the difference.

Now Ellie has said that she thinks she's going to transform the minuend so she can subtract 49 easily.

And the difference will be a multiple of 10.

So that's interesting, isn't it? I think Ellie is looking for a known fact like you were doing in your previous session.

Do you think you know what Ellie is about to do? Do you think you've worked out what Ellie's doing? So I'd like you to pause me now and decide how Ellie has transformed the minuend and how she has worked out the difference.

So you in a moment.

How did you get on? What do you think Ellie did? So she's going to transform the minuend so she can subtract 49 easily.

How many ones would she need? She would need nine ones in her minuend, wouldn't she? And how can she transform 183 so there are nine ones in the minuend? She could add six, fantastic.

Hang on a moment, though.

If she's using the strategy that you used in the last lesson, then what's the stem centres? What's the generalisation that we need to remember? If the minuend is changed by an amount, and the subtrahend is kept the same, then the difference changes by the same amount, fantastic.

So Ellie has decided to add six to her minuend, and this would make 189.

And she said, didn't she, that she wanted to find the difference, it would be really easy, and she'd get a multiple of 10.

Is that correct? It is, isn't it, because it's really easy to subtract four tens, isn't it? Eight tens subtract four tens is four tens.

So 189 subtract 49 gives us a difference of 140, fantastic.

So is she finished now? No, that's right, because the difference has changed by the same amount, hasn't it? So the difference has increased by six.

So in order to find the difference of the original calculation, we need to decrease by six.

Well done.

So if we decrease 140 by six, our difference will be 134, fantastic.

So that was a really good method that Ellie used there, wasn't it? Do you think that was efficient? Yes, I think that was efficient.

So she added six so that she could remove 49 really easily.

And then she had to subtract six from the difference to ensure her calculation stayed the same.

Sam has decided to come and join our maths lesson today, and he has solved the same calculation as Ellie, but he has used a different strategy.

So he says, I think I will use the same difference strategy because the subtrahend is close to a multiple of 10.

That's interesting, isn't it? Is the subtrahend close to a multiple of 10? Oh, 49, yes it is.

It's very close to 50, isn't it? So what I'd like you to do is have a think about what Sam has done.

He's written down here some equivalent calculations, and he said, didn't he, that he'd use the same difference strategy.

Can you remember what that is? Can you remember the generalisation? So I'd like you to pause me now.

Remind yourself of the generalisation for the same difference strategy and see if you can work out what Sam has done.

And also more importantly, do you think this is an efficient method? So pause me now.

How did you get on? What did Sam do? Firstly, do you remember the generalisation? Shall we say it together? If the minuend and subtrahend are changed by the same amount, the difference stays the same.

Fantastic.

We used that loads in previous sessions, haven't we? So what has Sam done? So he's changing the minuend and the subtrahend by the same amount.

What amount is that? What has he changed it by? He has added one, fantastic.

So he's added one to his subtrahend here, hasn't he, so that his subtrahend is 50.

So it's a multiple of 10.

And we said, didn't we, even in the last strategy that it's dead easy to subtract multiples of 10.

I can take five 10s off a minuend quite easily, can't I? And so if he subtracted one from his subtrahend, he must have subtracted one from his minuend.

And so that will be 184, fantastic.

And you had a little clue, didn't you, here because he's written the equivalent calculations.

And they are equivalent.

So that means that the difference is equivalent too.

Do you think that was an efficient strategy? Which one would you choose? Would you prefer Sam's method or Ellie's method? I think I prefer Sam's method, actually, because as soon as I looked at 49, I thought, oh, that's really close to 50, and I can subtract 50 really easily.

But it's okay if you preferred Ellie's method because Ellie's method was a really efficient method too.

And actually it's all about thinking about all the tools that we have in our tool kit.

We have so many different strategies.

We're so lucky, aren't we, that we have so many different strategies we can use.

And actually the hard part is deciding which one's the most efficient.

And I find this the most efficient, but you might have found Ellie's.

Now some of you might actually look and think that you would do that a different way entirely, and that would be efficient for you too.

Let's speak about the two different strategies.

Which do you prefer? Which do you find the most efficient? It's interesting, isn't it, that Ellie decided to change the minuend first.

So she looked at the numbers in the subtraction, and thought, yeah, if I transform my minuend, and I add six, I could subtract 49 really easily because I just have to subtract four 10s.

But she also remembered that because she had added six on, she would have to subtract six from the difference after she had used her known fact.

And Sam, Sam decided to use the same difference principle because he thought, well, if I transform 49 to 50, I know that I can do the same to the minuend as well, and my difference will stay the same.

So they were both really efficient strategies.

I said, didn't I, that I think I prefer the same difference strategy for this one because my eye immediately is drawn to seeing that 49 can be transformed to 50.

But you may have preferred what Ellie died, and it's good, isn't it? It's great that we've got so many different strategies in our toolkit, and we can pick the one that's most efficient.

We had a second calculation, didn't we, that had values close to the last one that we just solved with Sam and Ellie.

Do you think we'd use the same strategies for this, and the same strategies that Ellie and Sam used or rather different strategies we could use that might be more efficient? Is the subtrahend still close to a multiple of 10? So for example, it was 49 in the last one, wasn't it? And so that was only one away from a multiple of 10.

Does it affect our decision-making if the subtrahend is 46 rather than 49? So what I'd like you to do is have a think about how many different strategies you have to solve this calculation.

But also, I'd like you to think about which one is most efficient.

So if you could pause me and have a go at looking at a couple of different ways that you could solve this, maybe you might use the methods that Sam and Ellie used.

And then think which one's the most efficient and come back and let me know.

See you in a moment.

Did anyone transform the minuend so they could subtract 46 more easily? What strategy were you using then? You'd be using the same strategy Ellie used, wouldn't you? And so what was the generalisation for that? Let's say it together.

If the minuend is changed by an amount and the subtrahend is kept the same, the difference changes by the same amount.

Fantastic, and that's a really good strategy, isn't it? So how did you transform the minuend? Did you add, one, well done.

So if you added one to the minuend, then you also knew that the difference would change by increasing by one as well, didn't you? And why did you do that? Why would you add one to the minuend? So that it would become 186.

And how does that help me solve the subtraction? How does that help me with my mental strategies? Excellent, it helps me because I can see the 86 subtract 46 is 40.

So I only need to subtract four 10s don't I? So 186 subtract 46 will be 140.

Fantastic.

Now, have I finished there? What do I need to remember to do? I need to remember to subtract one, don't I? Because the difference has to decrease by one because I increased my minuend by one, brilliant.

So the difference would be 139.

How many of you did it that way? It's a really useful strategy, isn't it? Well done if you did.

Did anyone do it differently? Let's try a different method.

Did anyone use the same difference strategy? Yes, did you find an equivalent calculation by transforming the minuend on the subtrahend? What's the generalisation for that? Should we say it together? If the minuend and subtrahend are changed by the same amount, the difference stays the same, fantastic.

So how did you transform the minuend and the subtrahend? Which one did you look at to help you make your decision on what to transform? Did you look at the subtrahend and think, ooh, I'll make that a multiple of 10? So there's two ways we could make that a multiple of 10, isn't there? I could subtract six.

Would that be helpful? No, not really.

If I subtract six, and I have to subtract six from my minuend, don't I, eh, that's not altogether easy really, is it? It's not very helpful as a mental strategy.

So instead of subtracting six, I could add four, well done.

I could add four and make my subtrahend 50, fantastic.

So if I'm going to add four to my subtrahend, then I also need to add four to my minuend, Ms. Eaton.

Excellent, and if I add four to my minuend, I get 189, super.

So 185 subtract 46 and 189 subtract 50 are equivalent calculations, aren't they? And the difference will stay the same.

And the difference is 139.

And it's easy to do that, isn't it, because eight 10s subtract five 10s is three 10s, and none of the other digits have changed, have they? And so the difference for 185 subtract 46 is also 139 because they are equivalent calculations.

Did you find that an efficient strategy? Excellent.

I think I found the one before the most efficient strategy.

Well, that's just my personal choice because I quite enjoy just adding one and then subtracting one.

I can see how you would have rounded up to 50 here.

And if you used that, then well done.

Ellie and Sam are going to solve these subtractions, and you'll see we've got a group of subtractions there all with different values for their minuends and subtrahends.

Now, Ellie is going to have a look through that group of subtractions.

And she's looking for the ones that she thinks will be easily solved and efficiently solved by increasing or decreasing the minuend and then changing the difference by the same amount.

However, Sam, he's looking for the subtractions that he thinks will be really efficiently solved by changing the minuend and the subtrahend by the same amount so that the difference stays the same.

So he's going to look out far subtractions that he thinks will be solved really efficiently using the same difference strategy.

Do you think you can help them to solve these subtractions? Now, I don't want you to find the difference but identify the best strategy to solve them with.

And you can do this because you're mathematicians, and you can be really flexible in your approach.

Do you think that those two strategies are the only strategies that we can to solve all of that group of subtractions? No, there's got to be a range of different strategies we can use.

Sam and Ellie's aren't the only ones.

You might be able to use an efficient written method.

As you found out in your last lesson, you could use known facts to help you solve.

You might want to use your knowledge of place value.

So if you think that some of those subtractions needed different efficient strategy to the ones Sam and Elliot used, then you can put them in the bottom box.

But have a think about what that strategy will be.

Okay, so I'd like you to pause me now.

Remember, you don't need to find the difference.

Just have a look at the values of the numbers in the subtractions and think about the most efficient strategy for some of them.

How did you get on? Let's look at the first one together, 4,970 subtract 82.

What kind of decisions did you make when trying to decide the most efficient strategy? Did you look at the subtrahend first? So if I look at 82, can I subtract 82 quite easily from 970? Not very easily, can I? Is there a strategy I could use to resolve that? Yes, did you use Ellie's strategy? Fantastic, so transform the minuend and the difference by the same amount.

And what would that be? How could I transform the minuend? I could transfer the minuend by 12, fantastic, which would give me 4,982.

And then it's really easy to subtract the subtrahend of 82.

But I've got to remember, haven't I, that the difference will have increased by 12 as well.

So I will need to decrease the difference by 12 to ensure I have the correct difference.

Fantastic, what about the second one? 8,645 subtract 5,009.

What decisions did you make looking at this? I'm looking at 5,009, and I'm noticing that that's close to a multiple of 1,000.

Did you do that too? And so actually I could make a multiple of a 1,000 by subtracting nine.

And actually, yes, whose method would be the most appropriate there? I think it will be Sam's, wouldn't it? Fantastic, did you do that? So if I subtract nine from the subtrahend, I'd also need to subtract nine from the minuend.

Is that easy enough? Can I subtract nine from 45? I can, can't I, because I can use my number bonds.

So that would have been a really efficient strategy.

Well done.

What about the next one, 1,234 subtract 486.

Did you think that either of Sam and Ellie's methods will be helpful here? I'm not sure they would, would they? Well done.

What method would you use? That's right, I think I would too.

I think I would use a written calculation.

I would do a written subtraction there because I could subtract 486 from 1,234 quite easily.

There'd be an exchange, wouldn't there? But that wouldn't be a complicated exchange because I'm not exchanging through lots of zeros.

Fantastic, what about the next one? 6,001 subtract 2,479.

This is a bit like the other one that Sam had, isn't it? So I could transform my minuend to get the multiple of 1,000.

Does that help me to solve it? Does that help me to subtract 2,479? Not mentally.

Do you remember what we did in a lesson not long ago? We transformed the minuend by an amount to make the written calculation easier, didn't we, the written subtraction.

We could use Sam's method, couldn't we? But actually rather than calculating mentally, we would then calculate with a written subtraction.

And we'd have, we could transform 6,001 to 5,999.

Do you remember that lesson? And then I'd also have to subtract two from the subtrahend, wouldn't I? So that would be a really efficient method as long as I could write it out as well.

What about the next one, 3,150 subtract 164.

What do we think about this one? Huh, that's right.

Did you decide? Yes, go to Ellie because I can transform the minuend and the difference by the same amount.

What did you transform it by? Transform it by increasing the minuend by 14, fantastic.

So that will be 3,164.

And then I can subtract 164 really easily, can't it, as long as I remember to transform the difference by the same amount.

How about the next one, 1.

5 subtract 0.

75? Are Ellie or Sam's methods useful? I don't think they are, are they? Because I think I just know this one.

I think I just know that 0.

75 is a half of 1.

5, so I can find the difference really easily.

I can use a different efficient strategy.

It's a known fact, isn't it? And finally, 12,052 subtract 1,020.

Ooh, what have you done for this one? Would you have used Sam or Ellie's method? No, I'm not sure I would either.

I think that would just over-complicate things.

I could use my knowledge of place value, couldn't I, and partitioning.

So I can subtract 1,000 from 12,000, and I can subtract two 10s from five 10s.

Fantastic, so that's a really efficient strategy as well.

And some of you might write that in a vertical method, but you could do it just as easily mentally.

Well done.

You really helped me to sort those into an efficient strategy.

And that's a really important thing for us to do, isn't it, when we see a group of subtractions.

I'm not going to solve them using the same method every time.

And if we've got lots of different strategies in our toolkit, then we can pick the most efficient one.

And that's what makes us really good mathematicians, isn't it? So, well done.

Okay, we've worked really hard today.

We've been using our range of strategies to solve calculations in different ways.

And actually, at the heart of that, we've been thinking about which is the most efficient, haven't we? Because it's always best if we do something in the most efficient way because it's quicker and easier, isn't it? So that's what I'd like you to do for the practise activity.

If Sam and Ellie think that they can find several different ways to solve this subtraction, do you think you can too? What I'd like you to do is think about how many different strategies you can use to find the difference in this subtraction.

Also, what advice would you give Sam and Ellie about the most efficient one? Because we know, don't we, that Sam likes same difference, and Ellie really likes to transform her minuend and difference.

But perhaps you'll think that there's a more efficient strategy than that.

And you can come back and tell us all about it.

So have fun with the practise activity, and I'll look forward to seeing you next time.

Bye!.