Lesson video

Hello, it's Mrs. Mol again.

I'm back today to teach you the next lesson in the series of lessons all around number addition and subtraction, and I'm really looking forward to it.

So let's start with the practise activity that I set you to do at the end of the previous lesson.

So we're going to take a look at each part in turn and let's see how you got on.

So let's have a look at each example.

Now remember, this is not just about getting a right answer, and indeed when my jottings come up, the right answer is going to be there.

It's really very much about understanding how you can redistribute those addends in order to make the calculation easier, so that you can be a more efficient and flexible mathematician.

Whether you decided to increase the first addend or the second addend, that was up to you.

You may have done it in the same way that I did, you may not.

The really important thing though, is that you have got a clear choice as to why you chose to increase or decrease a certain addend.

Let's have a look at what I did.

So here's the first example.

So it was 53 add 39.

Now, you can see there that I chose to increase 39 as an addend, because that was really close to 40, making it a multiple of 10.

So if I increase the 39, I then decreased the 53 by one.

So that made my calculation then 52 add 40, and I could add that up very easily, and the sum was 92.

And then above, you can see that I've recorded my equivalent calculations.

53 add 39 is equal to 52 add 40, which is equal to 92.

That's just the way we were doing it through the previous lesson.

Is that the same way you did it? It doesn't really matter if it wasn't.

Again, the important thing is that you knew why you chose the method that you did.

Here's my second one.

48 add 35.

Now, I looked at that and I again, zoomed in on that 48 realising it's quite close to 50.

So I increased that addend by two.

And if I increase one addend, what do I need to do? I need to decrease the other one by the same amount.

So I decreased the 35 by two, and that gave me a redistributed calculation below of 50 add 33, which is equal to 83.

It was easy to add that up.

And then at the top I was able to record my equivalent calculations.

48 add 35 is equal to 50 add 33, which is equal to 83.

My final calculation, it was a three digit number, but I bet that was no harder for you.

236 add 198.

So my 198 was the addend that I wanted to increase to 200.

That was adding two, so I needed to subtract two from my other addend, from 236.

So that gave me a redistributed calculation of 234 add 200, which was much easier to add together.

So that gives me a sum of 434.

And then again at the top, I have recorded my equivalent calculations.

236 add 198 is equal to 234 add 200, which is equal to 434.

So how did you get on? Did you do those in the same way as me? Did you do them differently? As I said, it doesn't matter which addend you increase and decrease as long as you followed our generalisation, and therefore got the right sum.

And very importantly, you could explain your choice of which one you increased and which one you decreased.

Well done.

Let's take a look at the challenge activity as I set at the end.

So here are my jottings of that little activity that was in blue.

The calculation presented to you was 24 add 39 is equal to 63.

And then I gave you three choices, and asked you to explain which ones had been redistributed correctly or incorrectly.

So have a look at the first one.

The redistribution was 25 add 40.

So I can see that the 24 has been increased by one, but, oh, dear, they increased the second addend and we know that that's not what we do.

We know that if we increase one addend by an amount, we have to decrease the other addend by the same amount, so that's not right.

The second calculation is 23 add 40.

So it's been redistributed.

Right, I can see that 23 has been decreased by one, and 40 has been increased by one, so that's right.

When we increased one addend by an amount or decreased it, then we increased or decreased the other addend by the same amount.

And we can check that 23 add 40, isn't it equal to 63? The final one.

23 add 39.

Well, 23, that means that 24 has been decreased by one, we've subtracted one, but 39, well, actually nothing has been done to that addend.

We haven't added or subtracted anything, so that's not right.

And no, I can see that for sure.

So the middle one was the one that was right.

I'm sure you performed brilliantly there and got those all right, so well done.

We're now going to move on to today's lesson.

So just like in yesterday's lesson, we are going be deepening our understanding of equations of the equal sign.

I'm thinking how we can make adjustments, changes to calculations in order to calculate more efficiently using a mental strategy.

So here we have got a calculation, it's three digit numbers, and we can see that it is 127 add 118.

Now, do you remember those delicious caramel sweets that I had in yesterday's lesson in two bowls? Well, let's imagine this again, but this time we've got big bowls.

So we've got 127 sweets in this bowl, and we've got 118 sweets in this bowl.

We can see from the part-whole diagram, that the total number of sweets, the sum is 245.

Let's stay with this part-whole diagram for a minute and let's connect it to our abstract calculation.

Where is the 127 in the part-whole diagram? That's it, it's down here in this part.

We can see our 1 hundred and our 2 tens or the 20 and our 7 ones, 127.

So this side of the part-whole diagram, where's final calculation? That's right, it's the 118.

So these are our two addends, 127 add 118.

Now, I want to add that, and at the moment it looks a little bit difficult to add that mentally.

So I'm going to consider redistributing those addends in order just to make it easier.

So having a look at the numbers, which addend do you think we should increase, and which one do you think we should decrease, and why? Remembering that generalisation from yesterday's lesson.

Hmm.

Well, I'm looking at that 118 and I think that if I could increase that to make it 120, that will make the calculation much easier.

So how many would I need to increase 118 by to make it 120? That's right.

It's just two.

And where am I going to get that two from? If I'm going to increase that addend by two, what am I going to do to the other addend? That's right.

I'm going to decrease it by the same amount, by two.

So I'm going to get that two from here.

So in order to do that, I'm going to partition 127 into 125, and two, because then I can redistribute this two to the 118, just like that.

So now that I've redistributed by increasing one addend and decreasing the other addend by the same amount, watch what happens in my animation.

That two is going to just move across to the other addend I've redistributed.

So I no longer have 127 add 18.

Have a look at our part-whole diagram.

What do I have here? I have 125, which was here in our partitioned section of 127.

What do I have here? I no longer have 118 because my two moved across, and therefore it's 120.

So our calculation now, our two addends are 125 add 120.

Imagine those bowls of sweets, we saw the two sweets move over from one bowl to the other.

That is our redistributed equivalent calculation.

And looking at that, it feels much easier to add 125 add 120.

Why? Why is it easier? Have a think about that 120.

That's it, it's a multiple of 10, and it's really quite easy to add a multiple of 10 onto a three digit number.

So we know that our sum is 245.

It's not really about getting the answer here, is it? It's about thinking deeply about that equation, about redistributing those addends.

Remember, redistributing the sweets in order to make the calculation easier.

Well done.

So we have got exactly the same calculation here, but we are just going to look at it represented in a slightly different way.

So coming down here, we are very familiar with this idea of recording our redistributed calculation below.

127 subtract two is 125, and then we know from the previous slide that we then added two to 118 to give us 120, and 125 is our redistributed calculation.

But we can also record it up here.

And we can do that with an equal sign in the middle of both of those calculations.

Why can we do that? I want you to pause the video and explain to somebody or to say it out loud as to why we can record in that horizontal way with the equal sign.

Pause it now.

Well done.

So I am sure that you said that an equal sign means that whatever is on one side is equivalent to, has the same value as whatever is on the other side.

They are equivalent calculations, and we know that the sum of those is 245.

But have a look now at how I've recorded the redistribution.

Again, I want you to pause the video.

I want you to think about what's the same and what's different between this representation here and this representation here.

Pause the video now.

Well done.

I'm sure that you noticed that I'm just recording here, my subtract two and my add two.

So that's the same.

I've recorded it here, but I've recorded it in a different way by working horizontally and not working vertically.

So to just finish off here, what has remained the same throughout all of this? That's right, it's the sum.

We have conserved the sum of 245.

And again, why can we record the 245 here with an equal sign? That's it.

It's because equals means is equivalent to, has the same value of as.

So we know 127 add 118 has the same value as, is equal to 125 add 120, which is equal to 245.

Great work.

So when we are calculating mentally, it's much easier to add in multiple of 10 or a multiple of 100, or with larger numbers, a multiple of 1,000, multiple of 10,000, et cetera, onto another addend or another number.

So when we look at calculation, what's really important to do is to look at both addends and think, which one would be easier to increase or decrease to make that a multiple of 10.

It's not always the first addend or the second addend that we need to change into a multiple of 10 by increasing and decreasing those addends.

So the first thing I want you to do here, and we're going to use that story of sweets, it seems to be quite a helpful analogy.

So imagine a really big bowl with 328 sweets in, another really big bowl with 124 sweets in.

Those are our two addends.

I want you to, first of all, pause the video, and I want you to draw a part-whole model to represent this calculation.

And while you're drawing, be having a think about which addend you'd like to increase, or you think it'd be easier to increase, and therefore which one you would like to decrease.

Pause the video now and have a go at that.

So did you have a think, which addend do you think you would increase and which would you decrease? So I'm looking at that 328, it's very close to 330, a multiple of 10.

So if I increase 328 by two, that's right.

That's going to make it 330.

Now, remember our generalisation.

I have increased one addend by two, so what do I need to do with the other addend? That's right.

I need to decrease it by the same amount, by two.

So I need to decrease 124 by two.

I need to subtract two.

And our representation here shows that I'm adding two to one addend and that makes it 330.

And with my other addend, 124, I am going to subtract two, and that's going to make it 122.

So my redistributed calculation, my equivalent calculation, you can see that it has come at the top of the screen as well, it's now 330 add 122.

And that feels much easier to calculate mentally.

Why? Because I've got that multiple of 10.

Now, we know the reason that we increase and decrease addends by the same amounts is because we want to conserve the sum, we want to keep the sum the same.

So remember those sweets.

We took those of one bowl and we put them in the other bowl.

And the number of sweets hasn't changed.

In this case, what is the sum? 330 add 122 is equal to 452.

In our original calculation, 328 add 124 is equal to 452.

They are all equivalent.

So remember, when you look at a calculation, consider which addend you would like to increase and which one you would like to decrease in order to make the calculation easier.

So what is your representation of the calculation look like? You might have drawn two part-whole diagrams like I did, which is here.

So here I've got my original calculation.

My addend of 328, my addend of 124, which is represented here.

And then my sum or total of 452, which I can see represented here.

And then below, I've got my redistributed calculation with my addend, and addend, and my sum, and that is represented here.

What you might have done is you might have drawn out some deans here.

So I can see my 3 hundreds and my 2 tens, and my 8 ones.

And then on this side, I can see my 1 hundred and my 2 tens and my 4 ones.

And you might have shown with an arrow how we have redistributed the two ones.

What you might have done is this kind of working where we've worked horizontally.

And can you see that partitioning there? We've petitioned 124 into 122 and two, because we wanted to redistribute that two over here, so the 328 in order to make a much easier calculation.

330 add 122.

Whichever way you represented it, well done for thinking deeply and for getting the right answer.

So here's another example, and we are going to stick with the idea of sweets.

It seems to be quite helpful.

So we've got two bowls, one's got 146 sweets in, and the other bowl has got 339.

And we are going to redistribute these addends, redistribute the sweets to make finding the total, the sum easier.

Now, I want us to stick with this idea of examining the addends to see which one would be easy to increase and which one would be easy to decrease.

So take a moment and just have a look at them.

Is 339 really jumping out again? It is me as well.

So I know that that's really close to 340.

All I would have to do is increase 339 by one.

That's right, to make it 340.

Now, remember our generalisation.

If we increased that addend by one, what would I need to do to the other addend? That's right.

I'd need to decrease it by the same amount, by one.

So the bowls of sweets, I'm going to take one sweet away from 146, and I'm going to redistribute it and add it to the bowl that's got 339 in.

So here add one to 340, but I took it away, I subtracted it from 146, or took it from the other bowl, and that is now, there's 145 sweets left in that bowl.

So we've got our redistributed calculation.

And it feels much easier to add those two addends together now that they've been redistributed.

Why? That's it.

Because 340 is a multiple of 10, and it's much easier to add a multiple of 10 onto a three digit number in this case.

Now, because we redistributed those addends by adding and subtracting the same amount from each addend, What have we done to the sum? That's right.

We've conserved it.

It stayed the same.

And therefore 485 is our conserved sum.

Both of those calculations add up to equal 485.

Now, we increased in this case, the second addend.

Is that the only way that we could have redistributed the calculation? Redistributed the addend? Or was there another way? So I want you to now, before the next slide, pause the video.

Can you think of another way of redistributing those addends? Pause the video now before the next slide and have a think.

So I just asked you to pause the video and to consider another way to redistribute these addends in order to solve the calculation.

In the previous slide, we increased the second addend by one, and therefore decreased the first addend.

So the other way, which I'm sure you spotted to solve this calculation is that we could increase the first addend of 146 by, that's right, four, in order to make that 150, because that's the nearest next multiple of 10.

Therefore if I've added four to 146, that addend, I need to subtract four from the other addend.

I need to subtract four from 339.

And indeed that's what we can see here.

So we have now got a redistributed calculation of 150 add 335.

That's an equivalent calculation.

And just as before, we can see that that sums, that totals to 485.

So which way did you prefer? It doesn't really matter.

Both ways of redistributing the addends will make the calculation simpler, easier to solve, and so whichever way you prefer is absolutely fine.

And in fact, it's good to know that there are two ways to do it.

Well done.

So I'm going to give you a little bit of practise now.

We're going to do the first one together.

So we've got to find the missing numbers.

We've got some unknown values in these calculations and we need to work them out.

So let's have a look at number one here.

We've got 219 add 47 is equal to 220 add, and then we've got two unknowns.

Here I can see that my first addend, I know both of those values in each calculation.

And I can see by looking at the 220, that it's been increased from 219.

How much has it been increased by? That's right.

It's been increased by one.

So if one addend is increased, and this case by one, what do I need to do to the other addend? That's it, I need to decrease it by one.

So I'm going to decrease this 47, I'm going to subtract one from that, and that is going to give me an equivalent calculation of 220 add 46.

I've redistributed my addends in order to conserve the sum.

Now, that is much easier to work with.

220 add 46.

That's easy, It's equal to 266.

I'd like you to pause the video now.

I'd like you to read each question, considering what we know, what's known and what's unknown, and then using what you know in the first calculation to consider how the addends have been adjusted in the second, and to find those missing values.

Pause the video now and have a go yourself.

Well done.

So how did you get on? So here 716 add 57 is equal to? This first addend is unknown in the equivalent calculation, but we do know the second addend, and then we can use that once we find it out to find the total or the sum.

So I'm going to examine the 60 and the 57.

And I can see that 60 has been increased.

That's right.

I've added on from 57.

How many? Three.

So I've 57 add three is equal to 60.

So if I've added three onto that addend, what do I need to do to this first addend here to find the equivalent addend, the redistributed addend in this calculation? That's right.

I need to subtract three.

And so that gives us 713.

So here is my redistributed calculation, and it's much easier to work with 713 add 60.

That's a multiple of 10.

Much easier.

Is equal to 773.

How about this last one? I bet you're finding this really easy now.

135 add 738.

Examining the addend here that I know, and it's been redistributed.

I can see that it's been decreased.

So I'm going to subtract.

From 135, I'm going to subtract two, and that gives me 133.

See we have to have subtracted from the first addend, and I've subtracted two, what do I need to do to that second one? That's right.

I need to increase it.

I need to add two.

And that is going to give me a very friendly multiple of 10, 740.

And now I can add those together easily.

133 add 740 is equal to 873.

I'm going to put a bet on the fact that you got all of those right.

Well done.

Uh, look at the size of that number, it's really big.

But don't worry, this is going to be easy.

You really understand the strategy now, and all of those small steps that we took.

Remember, we started with numbers within 10 yesterday.

They're really going to help you now to use it with much larger numbers.

Have a look at that number, it's got lots of nines in it.

Should we have a go at reading it together? Ready? 199,999.

Well done.

Now, this is the number of sweets that a shop has sold in the first part of the year.

In the second part of the year, they sold this many sweets.

Should we read it together? 345,222.

Well done.

And our task is to find out the total number of sweets that the shop sold in one year.

And you know what we're going to do.

We're going to add them together, but we're going to make this a much friendlier calculation by redistributing the addends to make it easier to solve.

And that's right, we are going to conserve the sum.

We're going to keep that the same throughout.

So let's have a look at these numbers.

Are your eyes being drawn to that 199,999? Because mine are.

Why are yours being drawn to that number? All those nines telling me that it's going to be very close to a multiple of 10, 100, 1,000.

In fact, if I add how many? One to 199,999, I am going to make 200,000, and that's a really nice rounded number.

It's a multiple of 100,000, and that's going to be much easier to work with.

So if I have added one to the first addend, what have I got to do to the second addend? This is easy, isn't it? I'm going to subtract one.

So 345,222 subtract one.

Let's focus on that ones column, is 345,221.

And that is now much easier to add.

200,000 add 345,221 is equal to 545,221.

Why don't you pause the video and just use that STEM sentence again to verbalise what you did when we made that calculation easier in order to solve it.

Well done.

So let's have a closer look at this calculation.

Where did the one that I added to 199,999 come from? Yes, it came from the other addend.

I subtracted one from this addend in order to add it to this addend here.

So we partitioned.

Book the one, and then we had a remaining 345,221.

And we can see here that that one is redistributed to the 199,999.

And we therefore get an equivalent calculation of 200,000 add 345,221, and that's here.

So my question is, just to make sure that you really understand this.

Where is my 200,000 in this calculation? Have a look.

Ah, it's here.

It's got the red ring around it.

It's the 199,000 add the one.

Perfect.

So why is this 345,221 over here? Ah, it's here.

So when we partitioned this original addend into one and this part as well, that was much easier to calculate with, wasn't it? And there's our total or sum of 545,221.

Well done.

Great.

More big numbers.

That's okay.

Let's read this calculation together.

7,656 add 89,994.

That wasn't too bad, but it does feel quite unfriendly.

Those numbers, mmh, I'm not sure that I could add those easily using a mental strategy.

So we're going to use the strategy we have learned by redistributing the addends while conserving the sum in order to make it easier to solve.

So thinking about the last example, what clues were there in the numbers to help us to make it easier? So all these nines here.

I notice, and I bet you have too, that this second addend is very, very close to 90,000.

If I add six to 89,994, that's going to make 90,000.

I've also noticed that in my ones column, my 6 ones and my 4 ones, they are going to sum to make a multiple of 10, so that's going to make it friendly as well.

So if I want to add six to this addend here, what am I going to have to do to this addend? I'm going to have to subtract six.

That's absolutely right.

So where do I get that six from? The first addend.

So I'm going to partition that into 7,650 and six, because this six here, I'm going to redistribute.

That is going to add onto my second addend to make it a much friendlier number.

So my equivalent calculation, my redistributed calculation is now 7,650 add 90,000.

And again, where's my 90,000 in this calculation? It's here.

It's made up of 89,994, and the six that I got from the first addend that I redistributed and put here.

And then we can see that 7,650 I've subtracted that six.

Okay, so that's much easier now to calculate with.

So when I add 7,650 and 90,000, I can almost just use my place value knowledge here.

I've got my tens of thousand, 90,000, and then 7,650.

So that then is 97,650.

Wasn't that much easier to calculate with this number that felt quite unfriendly? And it's all about stopping and standing back, and taking a look.

Well done.

Let's take a close look at this calculation.

And we're going to consider why it was easier to work with our redistributed calculation.

We know that we subtracted six from our first addend to give us 7,650, and then we added the six on to give us a much more friendlier number of 90,000.

It's a multiple of 10,000.

So in a minute, I want you to pause the video and think, why was it easier to work with that redistributed calculation than the first one? Why was it easy to calculate? There are lots of reasons, but just have a go, either telling somebody near you or just reason it through yourself.

Pause the video now and have a think.

Did you do that? Well done.

So there are lots of reasons, but the thing that I was thinking about is those numbers, as we've talked about are friendlier.

We can use our place value knowledge, and I'm just less likely to make a mistake.

If I was adding using the first calculation, there are so many nines in there, I'm just very, very close to a boundary of a 10,000, I just think I would be more likely to make a mistake with that than I would with the redistributed second calculation.

But still obviously get the same answer, but I'm sure you realise that that was a friendlier, and much easier calculation to work with as well.

Well done.

Okay, so here is a calculation that my friend Kaya solved earlier.

Can you help me check it? Where should we start? Well, let's read the first calculation, 4,998 add 23,527.

And she's got her equivalent redistributed calculation here.

Shall we check how it's been redistributed? Okay, so 4,998 add two would give me 5,000.

So she needs to decrease this second addend.

Mmh, something's not quite right here.

She's added two to 4,998 to make 5,000.

She should have decreased or subtracted two from 23,527.

But here I've got 23,529, that's bigger, she can't have decreased it.

What has she done? Ah, she hasn't actually decreased that addend by two, she's added another two.

Um, okay.

Yeah, 5,000, that makes sense.

Subtract two, should have been 23,525.

Notice here that's not what she's done.

That should have been 23,525.

That's right.

And therefore it's not 28,529.

The sum should have been 28,525.

We know that it's easy to make mistakes, but we must be really careful.

What advice would you give Kaya about using this strategy? Remember that she increased both addends.

Can you have a look at our generalisation and give her a top tip? Pause the video and have a go.

Okay, so this is the practise activity that I would like you to complete before the next lesson, it's got three parts to it.

The first one is about filling in the missing numbers, which I think you'll find quite easy.

The second part encourages much deeper thinking.

So which of these calculations would you use redistribution for? But explain why.

Some of them might be better solved with a written method, some of them, redistribution.

The important thing is that you explain your reasoning.

And the third part.

Are these equations correct or incorrect? And again, explain how you know.

If you think one or both of them are incorrect, could you make them correct? Take that to another level, see if you can really engage in some deep thinking there.

Well done for all of your hard work today.

I'm really proud of you, and I've really enjoyed teaching you as usual.

Hope to see you again soon.

Bye for now.