Lesson video

Hello you five and six, it's Mrs. Mulligan, and I'm back to teach you for today's lesson, which I'm really excited about.

So you have finished your topic on fractions with us and well done because it's not the easiest topic in the world, and I'm sure that you worked really hard and you've learned a lot.

We're now going to move on to a new topic and that's linked to number, addition, and subtraction, and I've just heard you now say, "but I know how to add and subtract, I know how to do it." Well, this is not just about doing the maths, this is about really understanding the maths, about really understanding addition, subtraction and deepening your understanding within that topic.

By engaging in these sequence of lessons, you are going to be a much more competent mathematician, because this is all about strategies that will help you to calculate flexibly and efficiently, so it's not just about being able to do it.

Specifically, we are going to have a look at the compensation property of addition and subtraction, and what that means is that we adjust calculations, we reformulate them in order to make calculating easier.

Over the next few lessons, we are specifically going to look at equivalent and the equal sign, and we're going to deepen your understanding around that, I'm really looking forward to it, let's get going.

So, we are going to start by having a look at some key language that you're going to need in this lesson, and indeed through the next few lessons.

I'm going to be using it, I'm going to be modelling it to you, and I hope that you're going to use it too.

So the first word is distribute, which is all about giving out, and I'm sure you've heard of this word in real life and you've maybe used it, you know, we distribute food, we give things out.

And we'll also use some kind of derivatives of this, we might use redistribute, but that word distribute is going to be really important in today's lesson.

The next word that we are going to look at is to conserve, and again, there's some derivatives, you know, conservation, which you may have heard of in real life.

But to conserve something means to keep something from changing, we're going to keep it the same, is not going to change.

So, those words distribute, and conserve are going to be very, very important today in a mathematical context, and we're now going to look at a situation that really exemplifies what these words mean.

So I am now in my kitchen, I didn't want to spill water all over the computer upstairs.

I'm on the kitchen table and I'm going to use this jug with some water in and these two glasses to really help us understand these ideas of distribute, redistribute, and conserving an amount, keeping something the same.

So, here's my water, and what I want to show is that when I distribute the water into these two glasses, how that impacts the total amount of liquid that was in the jug, I'm hoping that it's going to stay the same, should we have a look? So, you can see I've got this red line, which is marking how much liquid I've got in my jug in total, and I'm going to distribute, give out the liquid between the two glasses, I'm going to fill this one right up, and this one, I'm not going to fill up so much.

I wonder if I can put a bit more in there and then I've got that out in there, so there's no liquid left in my jug now, I have distributed it between the two glasses.

Now, what I'm going to do is I'm going to redistribute the liquid between the two glasses by pouring, whoops, a bit into there, I had a little bit of a spill, and now I've got much more liquid in this glass than I have in this glass, I've really distributed the total amount of liquid.

But has the total amount of liquid stayed the same, or has it changed despite the fact that I redistributed between these two glasses? Well, apart from my little spill, if I pour it back in here, we should see that the total amount of liquid has been conserved, it has not changed, it has stayed the same.

So let's see if we conserved our liquid, I don't think that little spill should have too much of an impact, there we go, so let's, let that settle.

Okay, can you see my red line here? I've showed that even though I redistribute the liquid between the two glasses, I've conserved the total amount, that has not changed, that has stayed the same.

So, that key language is going to be really important today and over the next few lessons, and those are words that we will have heard in real life, I'm pretty sure of that, but there were two other words that are going to be very important today and they're mathematical and you may have heard of them, you may not.

But these are the two words, addend and sum, and I'm pretty sure you will have heard of some before, but we're going to look at it in this context and really be clear about how we want to use it.

So, this is a part-whole diagram and it's something that I'm sure you're really familiar with, you'll be familiar with a whole being split into parts, and then the parts recombining, being combined or added together to make whole.

And here we have, as you can see, a whole of five, and we can split it into two parts like this, we split it into one and four, and one and four make five.

But we're not going to use the words part and whole in this lesson, we're going to be more precise, we're going to be a bit more mathematical.

And, we are going to use these words addend, as I mentioned here, so we have addends of one and four which is replacing parts.

And then, we are going to use the word sum, we're not going to use the word whole, so we have, a sum, made up of two addends, and the two addends are equal to the sum.

So, just to be really clear that language of whole and part, we're not going to use in this lesson and throughout our sequence of lessons, we are going to use this vocabulary, a sum is equal to an addend add an addend, and an addend add an addend is equal to the sum, I hope that's really clear.

So, we are going to move on and we're going to use this very simple idea of sweets and two bowls to put the sweets into in order to help us with our understanding of the idea of distribution and conserving, keeping something the same, keeping the sum the same.

We're also going to look at how we use that very precise mathematical vocabulary of addend and sum, so let's take a look.

Now, these sweeties if you've not had them before are absolutely delicious, they're Carol Chocolates, but anyway, let's not think about that, let's focus on our mathematics.

So, as you can see, I have got 10 sweets here, that is my total, that is my sum, and I'm going to distribute them into my two bowls, and my bowls represent addends, here's one addend, here's another addend, and when I add my two addends together, it's going to equal my sum.

But watch what I do here, so I'm going to take, let's say two of my sweets, I'm going to put them into that bowl there representing one addend, I've got eight left and I'm going to take them and I'm going to put them into this bowl, so I've got two sweets and eight sweets and, the sum is 10, I've conserved the sum.

If I want to keep 10 sweets, but I want to distribute that sum in a different way, well, these are the only sweets I've got, so I've got to do something to this bowl and do something to this bowl.

I could take one out of this bowl and put it in there, so now I've got one and nine.

If I still, I can take take three out of here, and then I can put them back in there, so I've decreased that by three and increase that by three.

I could decrease again by one and I could increase this bowl, or this addend by one as well, so I'm decreasing and increasing, increasing and decreasing.

So I'm showing here that if I want to conserve my sum, conserve my total, keep it the same to not let it change, then if I increase one addend by an amount by one then I decrease the other addend by the same amount.

So that video showed us that we conserved the sum of 10 sweets in this case, even when we distributed the addends differently.

And notice how, when we increased one addend by putting one more sweet, for example, in another bowl, we have to decrease the other bowl by the same amount in order to conserve that sum in order for the 10 sweets not to change.

Now, we're going to have a look at another video using a representation that I think will be really familiar to you, a tens frame.

And I want you to really focus on how I'm redistributing the addends, what I'm doing to the red counters, and what I'm doing to the yellow counters in order to keep the sum conserved, to keep the sum at 10 so that it doesn't change.

Okay, so have a look here, we have got 10 counters on a tens frame, we've got a total of 10 or a sum of 10, and throughout this, the sum, the total is going to stay the same, it's not going to change, we are going to conserve that sum of 10.

But, I'm going to distribute this sum now across two tens frames, which are going to represent my two addends, so let's go back up here.

Now, I'm going to take six of my counters, so from my sum, which is still 10, I'm going to take six and you can see that there are four left.

And I'm going to place them on this tens frame here, so I'm distributing my sum into six and then back up here, my four counters, so I've still got the 10 in total, I'm going to put on this tens frame here.

Okay, so I have distributed my sum into two addends of six and four.

Now, I'm going to increase one of my addends by one, I'm going to do it to this one here, I'm going to increase six by one.

Watch what I do, when I increase this addend by one, this addend decreased by one because I've kept the sum the same, seven add three is equal to 10, before I had six add four, which was equal to 10.

So I'm proving again that when I increase one addend by an amount, which in this case is one, I decrease my other addend by the same amount, so I'm decreasing by one and my sum stays the same.

So in that last video, I showed you how we could have a sum of 10 and conserve that sum of 10 by redistributing the addends by increasing one addend by an amount and then decreasing the other addend by an amount, and, and this is what I worked with, was seven and three.

Now I know these numbers seem really small, but I promise you that by working with small numbers now so that you're freed up in your mind to really understand these concepts of distribution and conservation.

Then in the next few lessons, it will really help you to work confidently with larger numbers and decimal numbers, so bear with me.

So I wonder now whether you can have a look at this calculation here, seven add three is equal to 10, and I wonder whether using this stem sentence, so I really want you to verbalise this.

I wonder if you can pause the video, and whether you can consider increasing one of those addends, say the yellow counters, the yellow counters they're representing three, by an amount, and then considering what you have to do to the red counters if you've increased the yellow counters.

And I want you to use that stem sentence, I want you to say it out loud or explain it to somebody near you to really embed that understanding in your mind about how we conserve the sum of 10, how we keep that from changing, pause the video and have a go.

So I think we are really starting to understand this idea of distribution and redistribution, and how we can redistribute addends and still conserve the sum, and I think this model here, this diagram with the animations that I'm going to show you, are really going to solidify that in your mind.

So in this part-whole diagram, we have got a total of five cubes, we've got a sum of five, and when I animate it here, you can see that we distribute that into two addends, four and one.

Now I want to redistribute those addends, but I still want to conserve that sum of five, so watch what was going to happen now from the left part of the diagram to the right.

I have redistributed those addends, so now, one from the left, I have decreased that by one and it moved over to the right, so that's increased by one.

I've still got five, but my addends are no longer four and one, my addends are now three and two, and I'm proving there with our last animation that I still have the total of five, the sum has been conserved, that's still five.

So just before we introduce our generalisation, which is going to help us with this lesson and the next few lessons, I want you to have a look at this slide, and I want you to consider as I progress through the animation at what is changing and what is staying the same.

So let's take a look, here we have some red counters and we have no blue at the moment, then we have six red, and as I progress through, what do you notice is staying the same? That's right, it really shows us that the total number of counters, or the sum in this case of six is staying the same.

And what's changing? That's it, you've noticed that the number of blue counters is changing and the number of red counters.

And if you notice the pattern of how we are increasing the number of blue counters, and we are therefore decreasing the number of red counters or our addends, they could represent.

And why we're doing that? So that we can conserve that sum, that total of six counters, or our addends.

So I did promise you, didn't I? That we're going to start with those ideas of sweets in bowls and small numbers, working within 10, because that's going to help us want to move on to bigger numbers which we're going to do now.

But just to introduce our generalisation, which I know you will have understood at this point, and this is that, if one addend is increased by an amount and the other addend is decreased by the same amount, then the sum stays the same.

And this stem sentence we've used a few slides ago and we're going to use that throughout, because this is going to help you to really distribute your addends in order to calculate more easily, I added mmmh to this addend so I need to subtract mmmh from the other addend.

So I want you to pause the video now, I want you to say the generalisation a few times, I want you to have a look at that stem sentence and rehearse that a few times before we move on to looking at this approach, this strategy with bigger numbers.

So, we are now ready to use this strategy with larger numbers.

So we can see a calculation here 27 add 18, those are our two addends, and we can see from the part-whole diagram that our sum is 45, and we can see how that sum we can been distributed into the addends.

But when I look at a calculation, I don't just want to dive in with the method that I know, we don't want you to do this either, we want you to stand back, take a look at the numbers and think what's going to be most efficient way? What's going to be easiest for me to calculate? So, when I look at 27 and 18, I think that actually, if I could redistribute while still conserving the sum.

If I could make that 18 into 20, it's going to be much easier to add 20 on to a two-digit number because I'm adding a multiple of ten, and so, what would I need to do to 18 to make it into 20? And I want you to think about that language of increasing the addend, so I want to increase that addend by two, that's right, so, I have added two to this addend of 18.

So looking at our stem sentence, what do I need to do to the other addend? I need to subtract two from the other addend, I need to decrease it by two.

So, if 27 is going to be decreased by two, then, I'm going to partition 27 into 25 and two in order that I can redistribute that two, to my other addend of 18.

And watch my animation now, I'm going to move that two over to the other addend, and that is going to give me a redistribution, I have added two to 18, so I have subtracted two from 27, and that means that the calculation is now 25 add 20, which is equal to 27 add 18.

I have redistributed the addends, and I'm still going to have a sum of 45, and all the way through, I conserved that sum of 45.

So, on this slide, we have got the same calculation, but it's presented in a slightly different way.

We haven't got the part-whole diagram, but I want you to think back to earlier when we had those two bowls of sweets, do you remember? And we could easily here have 27 sweets in one bowl and 18 sweets in another bowl, and I want to find the most efficient way to find the sum, to find the total, and we've already talked about that.

What I did was I decreased 27 by two, because I wanted to increase 18 by two to redistribute that.

So imagine me taking two sweets out of that bowl that had 27 in, and just putting them into the bowl that had 18 sweets in, and it's much friendlier to work with those numbers, 25 and 20.

It's easier to add a multiple of 10 onto a two-digit number than it is to add 18 on, but all the way through, I have conserved the sum, my sum is still 45.

So, here we have another example, the calculation is 35 add 49 is equal to 84.

Now, I want you to do that thing where you're going to stop, stand back, and take a look at the two addends.

How could you make that calculation easier to solve? Could you redistribute the addends while still conserving the sum and to make it easier? Pause the video, have a think, explained to somebody next to you, or say it out loud and think about the generalisation and use that stem sentence to verbalise your thinking.

Did you pause? So, did you zoom in on that 49? I mean, that's the addend that I would like to increase to 50 cause I'll only need to increase it by one.

So, if I do that to that addend, what have I got to do to the other addend? Keep an eye on that generalisation, that's right, I've got to decrease 35 by one, so I'm redistributing my addends.

And that's exactly what the animation shows us, the one from the 35 goes across to the 49 we've redistributed, and that makes it a much easier calculation because now you can see that we have got 34 add 50, and it is much easier to add a multiple of 10, such as 50 onto a two-digit number.

And watch our animation now, how we just add the multiple 10, the five tens onto a 30, and then we have our four ones as well, so 34 add 50 is equal to 84.

So here we have our calculation again, 35 add 49, and our very important stem sentence at the bottom, and we know that we want to increase that 49 by one, so we're going to add one to that addend.

So, what do we need to do? We need to subtract one from the other addend, but what this is going to highlight is how we then partition the 35 into 34 and one so that we can redistribute those addends, and the one can be redistributed to the 49 leaving 34.

And that then becomes an equivalent calculation, 35 add 49 is equal to 34 add 50, which is equal to 84, our sum has been conserved all the way through, that has not changed.

So you're definitely ready to have a go by yourself, have a look at this calculation 45 add 29.

Now, the sum, the total is given to you there in our part-whole diagram, we know it's 74.

So this is not about just getting the right answer, I mean, it rarely is with real good mathematics with understanding, this is about really visualising, seeing what's happening to those addends, understanding how you can redistribute those addends in order to make the calculations simpler.

So, which addend will you increase? Why will you increase that addend? Which one will you decrease? Why will you decrease it? And why will you decrease it by that amount? Perhaps, you could draw a picture to show how you're redistributing those addends proving that you really understand this strategy, pause the video now and have a go.

So, how did you get on? Was it that you looked at that 45 and 29, and you realised that 29, you would just need to add one to this addend in order to make it 30, which as we know is a multiple 10, that's easier to add on to a two-digit number? And, and did you then think, well, I added one onto 29, to that addend, so I need to subtract one from the other addend, from 45 in this case? And watch our animation now that's exactly what happens is that one is redistributed from the 45 to the 29, and that now makes our calculation into an equivalent calculation of 44 add 30.

All the while, our sum has been conserved, our sum of 74, but hopefully, you saw and proved that it was much easier to add 44 and 30 than it would have been to add 45 and 29, well done, great job.

Okay, so this is your practise activity that I'd like you to complete before the next lesson, I've put the generalisation on there, just so you can be reminded throughout, but this is all about increasing and decreasing addends to make the calculation easier while conserving the sum.

So there are three examples there for you to have at at go at, and then in blue, we've got a bit of a challenge for you, which calculations show that the sum has been redistributed correctly between the two addends so that the sum is conserved.

We want you to explain your thinking, so good luck with that, I'm sure you will do brilliantly.

Thank you for working so hard today and well done, I hope to see you again soon, bye for now.