Lesson video

Hello, welcome back.

It's Miss Heaton again, and we're going to continue looking at efficient strategies for calculation.

This is lesson 23 of upper key stage two, addition and subtraction.

Have you got everything you need? Have you got the questions from the last practise session that I sent you? And let's have a look at those and you'll need some fresh pen and paper for today's jottings.

All right, see you in a moment.

Let's get cracking.

In the last session, we were looking at the most efficient strategy we could find for subtractions, weren't we? And then I sent you this practise activity, which was to find as many different strategies as you could but then to think about which one was the most efficient, and perhaps you'll be able to tell Sam and Ellie which one you found to be the most efficient.

So should we have a look and see what you did? Before we have a look at the strategies you've used, shall we have a quick recap of the generalisations that we can use to help us remember some efficient strategies? So the first one, let's say it together.

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

And that was the one that Sam was using an awful lot, wasn't it? Perhaps you use that generalisation to help you solve the problem that I left you.

And the second generalisation we looked at quite a bit, let's say it together.

If the minuend is increased or decreased and the subtrahend is kept the same, the difference increases or decreases by the same amount.

And so we use that quite a bit, didn't we, to help us transform calculations and find the most efficient strategy.

So let's have a look, which of those you use, maybe you use both of them, maybe use some other strategies and see which was the most efficient.

Did anybody solve the calculation using this strategy? Can you see what I've done? This is one of the efficient ways that I could solve this calculation, isn't it? What have I done? I subtracted 17 from the minuend, well done.

So if I subtracted 17 from the minuend, what else can you see I've done here? I'll have to have subtracted 17 from the difference, won't I? And we know that the generalisation for that is that if the minuend is increased, or decreased in this case, and the subtrahend is kept the same, then the difference also decreases by the same amount.

And we can see, can't we, that the difference has stayed the same.

So how has that helped me solve this calculation? I have one subtract 0.

75, don't I? Which is really easy.

We looked at this in the last session, didn't we? I know my number bonds within hundredths, I know that 0.

75 plus 0.

25 is one whole, and that should be one that comes to my mind immediately.

So I have 0.

25 there, fantastic! But because we know that the difference has decreased by 17 as well, what do I have to do? I have to increase that difference by 17.

So that I get 17.

25 as the difference for the original calculation.

How many of you did it that way? It's quite an efficient inefficient, isn't it? Well done if you chose that method.

Did anyone solve it a different way? Perhaps you use the same difference strategy.

If the minuend and subtrahend are changed by the same amount, then the difference will stay the same, won't it? And I can look at my minuend and subtrahend there, and we said, didn't we on the previous slide, that I've got some really strong knowledge of nom facts when I look at 0.

75, my brain immediately shouts 0.

25, because I know that 25 100ths added to 75 100ths makes one whole.

And that's a fact that I'm really, really confident with.

So if I add 25 100ths to 0.

75, I get one whole and that was really easy to do it, wasn't it? What do I have to do if I'm using the same difference strategy? I have to add the same amount, don't I, to the minuend.

So if I increase the minuend by 0.

25, I get 18.

25.

That was really easy so far, wasn't it? I like this same difference strategy for this calculation.

And is 18.

25 subtract one, is that nice and easy? Yes, it's an absolute doddle, isn't it? So I would get 17.

25, super! And the difference stays the same because I transformed the minuend and the subtrahend by the same amount.

I really like the same difference strategy for that one.

But perhaps you prefer the previous strategy and that's fine because we're choosing the most efficient one.

What is the same and what is different with these calculations? What do you notice? I'd like you to pause me in a moment and think about the similarities and differences, and also to start thinking about what strategy you think is being used here to find the difference.

So just pause me, have a look at what's the same and what's different, and come back and let me know.

How did he get on? What's the same? What can you tell me is the same? Yes, that's right, the subtrahend is the same, isn't it.

We've got three ones in the subtrahend.

What can you tell me about the minuend? The minuends are different, yes.

They're both a multiple of 500, that's right.

Well, the minuend in the second subtraction has been increased by 400.

It has four more, one hundreds.

Excellent.

Does that remind you of a strategy we've been using? So if I increase the minuend by 400, yes, the difference has been increased by 400 as well.

Well spotted! Which strategy is this? Shall we say the generalisation together? If the minuend is increased and the subtrahend is kept the same, the difference increases by the same amount.

And that's what this jotting is showing, isn't it? So I could find the difference between 500 and three, by looking at 100 and three and increasing them by 400 or decreasing them by 400.

Have a look at this subtraction.

I'd like you to have a go with this one on your own and decide which efficient strategy you're going to use.

There might be a few different ones to choose from.

And you might already have just looked at that and immediately used a mental strategy to find the solution.

But pause me now, have a moment, and then we'll come back and compare strategies.

How did you get on? What did you do? Did anybody subtract three from the minuend? If I subtract three from the minuend, then my minuend becomes 2010.

Is that helpful in any way? What could I do now? Ah, I could also subtract three from the subtrahend.

So this would be the same difference strategy, wouldn't it? And so I would subtract six.

Is it more helpful to subtract six from 10 than it is to subtract nine from 13? It might be, might it? Because our number bonds to 10 are really, really quick and easy in our mind, aren't they? And you might find it a bit more difficult to find the difference between nine and 13 just like that.

However, if you can, then that's good too.

So I've used the same difference strategy here, haven't I? And I've said if the minuend and subtrahend are change by the same amount, the difference will stay the same.

And I could subtract six from 10 really easily and I know that my difference is 2004.

So it will be for the first subtraction too, won't it? Did you do it that way or did you do it a different way? Perhaps you transformed the subtrahend.

How could I transform the subtrahend to make this calculation more simple? I could add one, couldn't I? And if I add one to the subtrahend, I also have to add one to the minuend, because that's my same difference generalisation, isn't it? And we know that the same difference generalisation means that if I increase the minuend or the subtrahend by the same amount, then the difference will stay the same.

And we can see here, can't we, that 14 subtract 10, that's really easy, isn't it? So 2014 subtract 10, that's just as easy, isn't it? And if my difference is 2004, then that's also my difference for 2030 subtract nine.

And again, some of you might be thinking, "Well, I can just subtract nine from 13." And if you can, then great, that's the first thing that you'll do.

But it might just be easier to subtract 10 and you choose the strategy that is the most efficient for you.

Let's have a quick look at this one again.

On the previous slide I said, well, some of you might just have said, "But Ms. Heaton, 13 subtract nine equals four! The difference between 13 and nine is four.

So why do I need to bother transforming my minuend and subtrahend if I know the difference between 13 and nine?" And I would say, "Fantastic! If you know the difference between 13 and nine, straight away, then you don't need to transform your minuend or your subtrahend to keep the difference the same." What strategy would you need to use then? So, you know that 13 subtract nine is four.

What do you notice about these two subtractions? The subtrahend has stayed the same, hasn't it? And which generalisation is this? This is the generalisation where I know now that if my minuend has increased, then my difference will increase by the same amount.

And how much has my minuend decreased by? If I have removed the 13 subtract nine to do that nom fact first.

It has increased by 2000, fantastic.

So that's the mental strategy you will have used, you will have looked to that and thought, "Well, 13 subtract nine, the difference is four and I need to increase my difference by 2000 again because our minuend has been increased by 2000.

And then you did it that way, you were using this generalisation, again, weren't you? If the minuend is increased or decreased and the subtrahend is kept the same, the difference increases by the same amount.

So well done if you did it that way, and you would have found that the difference was 2004.

Now let's have a look at some missing number problems and think about how we would solve them.

For each calculation I'd like you to use a known number fact to calculate the difference.

So this is the first one, I'd like you to pause me now and have a think about the methods that we've used previously.

How did you get on? What was your nom fact? Did anybody use one subtract 0.

23? Fantastic.

I was hoping you were all going to do that because when we look at decimal fractions and we can see 0.

23, we know immediately what we have to add to that to get one whole, don't we? And in this case, it's 0.

77 because 77 100ths plus 23 100ths is 100 100ths, which is one whole.

And we can use our number bonds to a hundred in just the same way as number bonds the hundredths.

So now that I found that nom fact and I picked that out, which generalisation have I used? I have kept the subtrahend the same haven't I? So I know I have decreased the minuend by 12.

So I've also decreased the difference by 12, haven't I? So what will I need to do? I will need to increase the difference by 12 and I have found my missing difference is 12.

77.

So that was a really efficient strategy.

Well done, if you looked at it that way.

Let's have a look at this one.

What subtraction fact could you use to solve the calculation? Pause me now and have a go at this one and let's see if we come up with some of the same ideas.

How did you get on? Yes, one way I could solve it is by subtracting three from the minuend because actually that would then leave me with a multiple of 10, wouldn't it? And I'd have 520.

Now, of I decrease the minuend by three and I decrease the subtrahend by three, what happens to my difference? It stays the same, Mrs. Heaton, of course it does.

And so I am now subtracting six.

Is there any easier calculation? 20 subtract six.

Is that easier than 23 subtract nine? I think it is actually because I'm using my number bonds to 10 again, aren't I? So I only had to do a very small transformation, didn't I, on my minuend and subtrahend to get something that was just, just ever so slightly easier to calculate mental.

So my difference now is 514.

That's a really easy one.

And my difference has stayed the same, hasn't it? Did anybody do it differently? Perhaps you might want to have a look at some of the ones that I found my class doing.

I asked some of my pupils if they could transform this subtraction in a myriad of different ways, and these are four of the ways that they come up with.

I'd like you to pause me now and have a think about what's the same and what's different.

Also, the main point of these sessions at the moment is to think about the most efficient strategy.

And maybe some of these aren't efficient.

Maybe it's just transforming for transforming sake.

So pause me now, have a look at what's the same and what's different, and decide if the strategy is efficient or not.

How did you get on? Let's have a look at the first one.

So what have they done? They've transformed the minuend by increasing it by seven and so they've increased the difference by seven.

Does that make that an easier calculation? Is 530 subtract nine easier than 23 subtract nine? It could be.

It might not be for some children, but other children might think that was an easier calculation.

Actually, if I subtract nine from 30, I've got 521, haven't I? And I've got to remember, haven't I, that if I decrease my minuend.

Increase my minuend by seven, I'm going to have to decrease my difference by seven as well.

And actually, here I can use my multiplication knowledge here, can't I? My times tables facts, because I can see I've got 7, 14, 21.

And that helped me as well.

So actually, although I used a variety of mental strategies within there, they were all very efficient.

I like that one.

What about this one? In the previous slide we subtracted three, didn't we, using the same difference principle.

How do we know this is the same difference strategy? Because they've increased the minuend and the subtrahend by the same amount.

Fantastic, well spotted.

So in that case, does it make it easier? So I've got 24 subtract 10, instead of 23 subtract nine.

Yes, that might be easier for some children, might it? And so I know that my differences will stay the same.

Brilliant.

Perhaps over here, what's the same and what's different? So the difference is the same and the minuend and subtrahend are the same in the original calculations.

What's different over here? What principle have we used here? We've used the same difference strategy, haven't we? Have we used the same difference strategy in this one? No, we've not, have we.

We've transformed the minuend and the difference by the same amount.

How do either of them help us? Let's look at this one.

If we increase both the minuend and the subtrahend by 10.

Oh, what's going on here then? Is this one correct? No, well spotted.

So they've increased the minuend by 10 and it says here they decreased the minuend by 10, is that what they've actually done? No, they did decrease it by 10.

They should have put plus 10 here, shouldn't they? Well spotted.

Has it enabled them to work this calculation out easily? It's 33 subtract 19 efficient? I don't think 33 subtract 19 is any more efficient than 23 subtract nine.

I think we've got in a bit of a pickle here, haven't we? Although they've had a really good go at trying different strategies, they seem to have got a bit muddled up about whether they're increasing or decreasing here.

And actually it doesn't help, does it? I don't find it any easier to subtract 19 for 33, then I found it to subtract nine from 23.

So I'm not sure I'm altogether convinced that this one's an efficient strategy.

I don't think I'd do that one.

What about this one, is this efficient? What have they done? They've subtracted 13 from the minuend.

Oh, 510 subtract nine.

That's easy, isn't it? I can take nine off ten, really easily and I get 501.

I quite like that one.

What do I have to remember to do though? I have to remember to decrease the difference, don't I? Because I.

Oh, I'm increasing the difference because I decreased it by the same amount.

It's easy to get in a pickle and to muddle up your increasing and decreasing, isn't it? That's why you've got to be really, really sure of the strategy you're using.

And if you know that you decreased the minuend by 13, you've also decreased the difference by 13, and then I had to increase it by 13, didn't I? To ensure that my difference is correct.

Some really interesting strategies on there.

I think it's really interesting.

Some of you will probably prefer one over another and some of you might think, "Oh, I wouldn't do any of those.

I do some of the other strategies that we've looked at." And that's fine.

We've said haven't we? Being a mathematician, it's about being flexible.

You've got a range of strategies at your fingertips and you use the one that helps you to find the solution quickly and easily.

Have a look at this calculation now.

They're quite large numbers, aren't they? What can we do to make this calculation easier? Pause me now, have a little think.

How would you make this one easier? How did you get on or did you notice? I noticed that 40000 subtract 17,000 is equivalent to 23,000.

And that was really easy for me to work out because I know that 40 subtract 17 is 23.

And actually I can use that calculation, can't I? No matter how large or small my place value is.

So if I know that 40 subtract 17 is 23, I know that 40,000 subtract 17,000 is 23,000.

Fantastic! Does that make it easier to solve? It does, doesn't it? Because I have decreased both the minuend and the difference by 500,000.

So what do I need to do? I need to increase the difference by 500,000 and then I will have found my missing box, fantastic.

And 540,000 subtract 17,000 is 523,000.

So it was much easier, wasn't it, pulling out 40,000 subtract 17,000 and solving that mentally first.

Have a look at this one.

What subtraction fact could you use to solve this calculation? Pause me now and have a little think.

What would you use to solve this calculation? How did you get on? Did you look at it this way? Did you subtract two? Fantastic.

And if I subtract two of my subtrahend, I've also got to subtract two off my minuend, fantastic, because I'm using the same difference strategy.

And if I subtract the same amount from the minuend and the subtrahend, then my difference will stay the same.

Why did subtracting two make this subtraction easier? Because 98 subtract 90 is really simple, isn't it? And I know that that would be eight, so I can find my difference really simply, and it will be 508.

We're make it all kinds of important decisions here, aren't we? We're looking at the size of the numbers, we're looking at the relationships, we're looking at what we know, what nom facts we know, and how we could alter the calculation, transform the calculation, just to make it easier for us to just manipulate in our minds.

Fantastic.

So what's the difference for the original calculation? It's 508, fantastic.

Did any of you do that one differently? So did anybody add eight perhaps? That would be a good way of doing it, wouldn't it? We could add eight to both the minuend and the subtrahend and then we would have 608 subtract 100 and you can see we'd get the same difference.

Lots of different ways we could do that.

Well done.

Here we can see if we have added eight that we would get the same difference, because if I transform the minuend and the subtrahend by the same amount, then 608 subtract 100 is 508.

So it doesn't matter whether we increase or we decrease, we will still get the same difference.

Fantastic.

Super duper.

So we worked really, really hard again in this session and you've been really thinking about the most efficient strategy and because you're so secure in your knowledge of subtraction and you're really secure about the minuend and the subtrahend you can now transform those subtractions using the generalisations to make your mental strategies easier and more efficient.

So are you ready for a challenge? Let's look at this question.

Stephanie wants to learn to jump higher.

She records her standing reach height, then the height she reaches when she jumps, to calculate how high she can jump.

She tests herself every week, she's really committed to this, for five weeks.

Now, she doesn't get any taller during the five weeks, so her standing reach doesn't change.

That's really important that our results don't get skewed, isn't it? So her standing reach doesn't change.

Her results are shown in the table, but some of the values are missing.

So you're going to need to fill in the missing information.

So what do you notice? Which paths can you fill out straight away? So we just heard haven't we, that the standard reach doesn't change at all.

So that's something that you could fill in in this table.

What else do you notice? That's right.

We can fill out the bottom row as well, can't we? Because we've got the jumping reach and the standing reach, so we should be able to find the difference.

And you will need to decide which of those is the minuend and which of those is the subtrahend.

Okay, so that's a really nice challenge for you there.

So there's some bullets there to get you started.

What do you notice about the difference? What do you notice about the minuend and subtrahend? Keep that in mind while you find those missing values.

So I've really enjoyed today.

It's been really exciting thinking about all those different strategies and finding the most efficient.

And I hope you find the most efficient strategy for this question too.

Take care, bye.