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Hi, welcome back to our series on number addition and subtraction.

I'm Mrs Furlong.

And I'm going to be taking this session today.

In the previous lesson you were left with this practise, which we're going to start the lesson by having a look at.

I know there was also another question at the bottom which was a bit more open-ended, and we'll also look at that too.

Okay, so originally you were left with the calculation 52.

13 subtract 3.

76 is equal to 48.

37.

And then you were to use this fact that you'd been given, to see if you could work out the empty boxes.

So, in the first calculation with the empty box, we have a missing number subtract 3.

76 equals 18.

37.

And if we use that original calculation, what did you notice was the same? And what did you notice was different? That's right, the subtrahend was the same, the 3.

76.

So if we go back to our generalisation, if the minuend is changed by an amount and the subtrahend is kept the same, the difference changes by the same amount.

Well in this case we don't know what the minuend has changed by but we do know what the difference has changed by.

The difference has dropped down by 30.

So therefore the minuend also needs to drop down by 30.

So that 52.

13, we need to subtract 30 or three tens from 52.

13 which would give us 22.

13.

I hope you got that one right.

So now you can see in the bottom left-hand corner that I filled in that missing number the 22.

13, and I've highlighted in pale blue the calculation that we're now going to look at.

And that calculation is something subtract 3.

76 equals 8.

37.

So you need to make a decision.

Are you going to look at the original calculation 52.

13 subtract 3.

76 is equal to 48.

37? Or are you going to use the one you've already just solved? I decided that because the one just above it, the 22.

13, here is only 10 more than the 18.

37.

I would actually use that calculation.

So I'm going to focus on this one to help me solve it.

So you can see now my top calculation has now changed to 22.

13 subtract 3.

76 equals 18.

37.

And now again, let's have a look at what's the same, what's different, that's right.

The subtrahend is the same, the 3.

76, but the 18.

37, the difference has become 8.

37.

So what's happened to that difference? It's dropped down by 10, it's decreased by 10.

If the difference is decreased by 10, then using our generalisation, the minuend must also decrease by 10.

So 22.

13 needs to decrease by 10 and one ten less than 22.

13 is? That's right, it's 12.

13, well done.

So if you look now in that bottom left-hand corner, you can see that two of our missing boxes have now been filled in.

So we now have three facts in our sequence that are all connected.

So now we're onto the missing box.

Subtract 3.

76 is equal to 88.

37.

That one that's highlighted at the bottom in blue.

Right, so let's have a think.

Do we want to use the original calculation? The 52.

13 subtract 3.

76, this one here, or do we want to use a different one? Well, I was looking really carefully here and I was thinking about what's the same and all of them have that same subtrahend.

But what's changed? Well this one's 48.

37 and the next one's 18.

37, the next one's 8.

37 and the one we're about to calculate is 88.

37.

And I noticed that was actually closer to our original calculation, but stuck with that original calculation, of 52.

13 subtract 3.

76 is equal to 48.

37.

So, what's happened to the minuend? We don't know yet, what's happened to the subtrahend? That's right, stayed the same.

What's happened to the difference? Ah, it's changed this time, hasn't it? And it's increased by 40 or 4 tens.

So therefore, what needs to happen now? Yeah, our minuend needs to increase by 40 or 4 tens as well.

And 4 tens more on to 52.

13 would give us 92.

13, well done.

Okay, so we've almost completed all of those calculations, that had that information and we've got one more to complete in this sequence.

So again, let's have a look, a careful look at what we've got to complete.

We've got something subtract 3.

76 is equal to 88.

57.

So I would now be looking at my calculations down here, that we've already filled in.

And considering which one has the difference as the closest? Did you spot it here? The second to last calculation has 88.

37 and our final calculation has 88.

57 as the difference.

They're quite close together, aren't they? So I'm not going to use the original calculation.

If help me to do this, I'm going to get rid off that.

And instead I'm going to use the 92.

13 subtract 3.

76 equals 88.

37 to make my connections.

So what do we notice? Yeah, the subtrahend stayed the same.

What's happened to the difference this time? That's right, it's increased by two tenths.

So therefore our minuend also has to increase by two tenths or 0.

2, so we get? brilliant, 92.

33, well done.

Okay, so for the last thing you were asked to do, you were asked to see if you can find a way of completing these empty boxes that it was still part of these related calculations.

So what did you need to think about first? Remember our generalised statement, the minuend changed by an amount, the subtrahend is kept the same, and the difference changes by the same amount.

So one thing that everybody should have done was to think about the subtrahend is kept the same and we can see over here all of our subtrahends are 3.

76, aren't they? So therefore your subtrahends should have been 3.

76 as well.

And there're about infinite possibilities and I could be here for hours giving you examples.

I'm going to do that, you'd get very bored.

But what I am going to do is just show you a couple of solutions so you can also use my explanations, to check what you've done.

So I chose an original calculation.

I'm not going to tell you which one yet.

And I decided to increase it by 100 for the minuend.

So therefore increase the difference by 100 as well.

So can you see which original calculation I chose? Yeah, that's right, this one here.

So you can see that my 52.

13 has increased to 152.

13 for the minuend and my difference 48.

37 has increased by a 100 to 148.

37.

That's not the only solution I did, I've done one more.

This time I decided to go slightly crazy, and not just increase it or decrease it by one or a 10 or 100 et cetera.

I went to 88, I've decreased it by 88, why not? My new minuend is 4.

33, my new difference is 0.

57 So which calculation did I decrease both the minuend and the difference by 88? Well done, you're right, it's the one.

This one's going down by 88 and the minuend and the difference has also gone down by 88.

There were hundreds of solutions, just check in yours, whichever calculation you focused on, make sure that those changes are the same.

So your minuend and your difference have either increased or decreased by the same amount.

So in today's session, our generalised statement is, if the minuend is changed by an amount and the subtrahend is kept the same, the difference changes by the same amount is key.

The other thing that's key is using known related facts, things like number bonds, so they're going to help us to solve harder calculations by using those number bonds and looking for them hiding inside calculations.

So going to start our session with a bit of fluency, to make sure that your number bonds are up to speed and that your brain is awake.

Okay, I'd really like to get your math brains warmed up now.

And, so we're going to do a little bit of quick fluency.

So I'm going to ask you a question.

I want you to really quickly answer it as fast as you can using known facts.

So we're going to start with number bonds that go to a 100 and then we're going to use those known facts to answer some of the questions really quickly.

So here's your first one, 100 subtract 13, you should know that very fast.

Yep, that's right, it's 87.

Can we use that to help us with 200 subtract 13? Oh, did you spot my minuend just increased by a 100, so my difference from my previous calculation needs to increase by a 100.

Yeah, 187, you're right, well done.

Let's have a look at the next one.

100 subtract 47, know what it is? Be pretty quick because 47 is not that far from 50.

Yep, it's 53, well done.

Okay, so my related calculation, 300 subtract 47.

Have you spotted that the 300, the minuend is 200 more than the 100? That's right, it's 253.

I just want you to pause for a moment here and have a little careful look at something.

Have you noticed that if I go this way, in my calculation, that my minuends have increased and my difference has increased by 200.

But if I do the inverse and I go the other way, my minuend and has decreased by 200 and my difference has decreased by 200.

You just follow that pattern, brilliant.

So I'm going to challenge you a little bit more now.

See how quickly you can do this one, one subtract 0.

27 Well done if you've got it right, 0.

73, brilliant.

What about this calculation? 100 hundreds subtract 27 hundreds equals? Yes, that's right, it's 73 one hundreds.

And let's go back to our generalised statement.

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

Is that relevant with these two calculations? It's not, is it, why not? Ah, yes, I've got rid of it because, if you have a really powerful look, let's start with our minuend, we've got one here, and 100 hundreds is also worth one whole isn't it? So they're identical, it's just different notation.

So in the bottom calculation, we've used fractional notation and in the top calculation we've used and whole numbers and decimal notation, haven't we? And then as we have a look at the difference, the 0.

73, the difference in this first calculation, that's worth 0.

73, or seven tenths and three hundreds or 73 hundreds which in it's fractional notation is 73 over 100, 73 one hundreds.

So all that's changed there the values are identical, it's just the mathematical notation that's different, one's in decimal form and one's in fractional form.

Okay, so let's have a look at the next one.

One subtract 0.

27.

Ah, if you're awake, you'll remember we've just done this.

So what is one subtract 0.

27? That's it, 0.

73.

Okay, let's have a look at my related calculation.

10 subtract 0.

27, what have you spotted is the same? Yes, that subtrahend 0.

27 is the same.

So how can we use our original calculation to help us with this new calculation? Ah, well done, it's 9.

73.

Can we say, then we write our minuend has increased by nine and subtrahends have stayed the same.

Here we are, they're both the same.

And therefore, our difference must also increase by nine.

But if we have a look at it in the inverse way, so we go the opposite.

So we go from the bottom calculation to the top.

Our minuend has decreased by nine, our subtrahends are still the same and our difference has decreased by nine.

So we can actually spot those relationships between those two calculations.

And just remember our generalised statements.

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

Okay, let's see how quickly you can do this one.

One subtract, 0.

82, remember that one is just a 100 hundreds.

Fantastic, it's 0.

18, well done.

By my related calculation, what's the same? What's different? Think about it and then see if you can give me the answer.

Fantastic, it is says 100.

18.

Because again, say it with me the minuend has increased by 100.

The difference has increased by 100 and our subtrahends have stayed the same.

And remember, if we did that relationship the a way around we could say our minuend has decreased by 100.

Our difference has decreased by 100 and our subtrahends are still the same, fantastic.

Okay, so now for our last bit of fluency, you'll see here, our subtrahends are different, and the reason is I just want you to think about your known facts.

So pause the video here, see how quickly you can answer these and then we'll come back together and see if you got them right.

Okay, have you managed that? Let's have a look at the first one, one subtract 0.

35.

0.

65, how did you know? Yeah, so one is worth 100 hundreds, so 0.

35 is worth 35 hundreds.

Well, I know a 100 subtract 35 is 65.

So 100 hundreds subtract 35 hundreds must be 0.

65 or 65 hundreds.

The next one, one subtract 0.

61.

We need those related facts again, don't we? So again, one is worth a 100 hundreds and I know that a 100 subtract 61 is 39.

So a 100 hundreds subtract 61 hundreds is 39 or 0.

39.

And the last one, one subtract, 0.

26.

You got it? Brilliant, 0.

74.

Because we know that a 100 subtract 26 is 74.

So a 100 hundreds or one whole, subtract 0.

26 or 26 hundreds must be 74 hundreds or 0.

74.

Fantastic, well done.

Alright, so now it's time to test how awake you've really been because this next calculation has a familiar number in it, a familiar subtrahend that we might have just used.

I'll leave you to remember that.

So it says on here, what number facts do you already know that are related to this calculation? So I don't want the answer to this calculation, that's why there's no equal sign there.

I just want you to pause the video and think about what you already know and particularly thinking about that subtrahend.

What known facts do you know to do with subtracting 0.

26.

So just take a moment to think about that.

Hopefully you've had time to do that.

Okay, I'm hoping that that 0.

26 really, really jumped out to you.

Because on the previous slide, in our fluency, we had this calculation, one subtract 0.

26 equals 0.

74.

Some of you would have noticed that and some of you would have been half asleep and not remembered but if you rewind the video, I promise you it's there.

Anyway, so this name factor is going to help us with this calculation.

If we know that one subtract 0.

26 is 0.

74, there is a one hiding inside 124.

And I mean, one, one as in one, one, not one in a 100.

So 124, this makes it easier to explain, is made up of 123 and one.

And we know as you can see at the bottom here, that's one, subtract 0.

26 is 0.

74.

We already know that but here was partitioned our 124 into 123 and one to help us.

And so we need to recombine it, so we get 123.

74.

But once we get on to larger numbers, this can get a little bit complicated sometimes to record in this way.

So we're going to simplify that a little on the next slide.

So instead, we're going to now we've got that equal sign on here which we didn't have on the last slide.

And we've got that related fact we've already talked about, we know that one subtract 0.

26 is 0.

74.

So let's think about our general statement.

If the minuend is changed by an amount and the subtrahend is kept the same, can you see it is, the 0.

26 is there in both calculations, then the difference changes by the same amount.

But we don't know what the difference has changed by, 'cause we've not been told it.

We only know the difference, our final calculation and our related facts.

But we do know that 124 has decreased by 123.

So whatever our difference must have been has also decreased by 123.

So remember earlier, when we were looking at fluency, the, if we went from the top calculation to the bottom, you might have had an increase or a decrease.

And then if you went from the bottom calculation to the top, you did the inverse.

So in this case, if we want to use a related fact to solve something that's a little bit harder, we now need to use that inverse to find our original solution.

So if one subtract 0.

26 equals 0.

74, then 124 subtract 0.

26, that's equal 123.

74.

Can you think why, can you see it on there? See if you can just pause the video and have a look at it really carefully, and I'm going to do a slightly more explanation in a moment.

Okay, so just to clarify for this last, for this part now.

You can see here our minuend decreased by 123 to get us to one, that easy related fact, the one subtract 0.

26.

So if that decreased, if we're then, we can then solve the calculation without a related fact.

But then we need to think, what do I need to do to go back to the original calculation? Well, if we think about our original generalisation, if the minuend is changed by an amount, and the subtrahend is kept the same and the difference changes by the same amount.

So if we think about it, if we're going from this calculation to this calculation, we are decreasing by 123.

So if I want to go back the other way, we need to increase by 123.

We don't need to do it this way 'cause it's already been filled in for us, but we can also see that that also increases by 123, just as a way of making sure we understand that structure.

Okay, so now I want you to think about our related facts to help you to solve this new calculation.

Have you spotted some then? Yes, that 0.

26 subtrahend is still there.

So our related fact is still one subtract 0.

26.

So in that case, we know that one subtract 0.

26 is 0.

74.

So how does that relate to that original calculation? Pause the video here and see if you can work out that missing box.

Okay, hopefully you've had time to do that.

Let's just have a quick look now at what we can notice.

So we already know the subtrahend is the same, but from the original calculation, we dropped down our minuend and therefore our difference by 1123 to get us to one.

So we've got that easy related fact.

The one subtract 0.

26 equals 0.

74.

So we dropped it down originally by 1,123.

So to use our related fact, 0.

74 here, to use that to help us with the original calculation, we now need to increase by 1,103 to get us back to help us with our original calculation.

So our answer to 1,124 subtract 0.

26 is 1,123.

74 because we've used that related fact.

Okay, so I've got another one for you to have a go at now.

Have you spotted it already? Yes, 0.

26 our subtrahend is still the same as it was before so, Our related fact of one subtract 0.

26 equals 0.

74 should help you to solve it, have a go now, and if you need to pause the video, that's great.

Did you manage it? So we have that related fact again, didn't we? And my minuend from the original calculation to the new one dropped by 110 decreased by 110, to give me one.

So my difference would also have done that, we just can't see it.

unless you can see the subtrahend has stayed the same.

So if I need to go back the other way, so I'm using my related fact.

I know here that I've got 0.

74 as my related fact that's going to help me here.

So how is that going to help me to get the answer? What do we need to do to get the box? Yeah, instead of decreasing by 110 we need to do the inverse now and increase by 110, so we get 110.

74.

Okay, so we're going to move on from decimals now.

And we're going to move on to larger numbers.

We've got 374,000 subtract 24,000.

Okay, just have a look at that calculation and just start by thinking what's the same and what's different.

Yeah, I spotted that we have zeroes in our hundreds, tens and ones in our subtrahend and zeros in our hundreds, tens and ones in our minuend.

That's telling me that actually I can just ignore that part of this for calculating because I know that zero subtract zero is zero.

I do need to remember them later and they are important 'cause they're important place holders.

Anything else is the same.

Ah, you're right, yes, the four in the thousands column, it's there in the minuend and it's there in the subtrahend, isn't? Okay, so can you spot any related facts that are going to help us with this? 74,000 subtract 24,000 will help us.

I bet a lot of you know that 74 subtract 24 is 50.

So 74,000 subtract 24,000 is 50,000.

But what did we do to use this related fact? How have we changed our original calculation? Yes, we've got the minuend and the difference by 300,000 and our subtrahend stayed the same.

We can't actually see that that difference has dropped by 300,000.

But we know from all that exploring we've already done that that would have happened because of our generalised statement and lots of other things that you've looked at.

So if we dropped it, for a related fact, we've dropped it by 300,000, then to get back to our original calculation, we need to use our difference from our related fact to help us.

And that difference instead of decreasing by 300,000, it now needs to increase by 300,000.

So I now know that 374,000 subtract 24,000 is 350,000.

Okay, so this is the end of the session now.

I'm just going to leave you with the practise and explain what you need to do.

So in the first question, I'd like you to consider the related fact that you need to use that will help you to solve the original calculation.

Part of that related fact has been completed at the bottom here, but you need to complete the rest and then use it to help you to answer the question and think about what will go in these boxes.

Then in number two, again, you've got a related fact, but it's down here and you need to decide what is the related fact that will help you with the original calculation.

So in other words, what are you going to subtract 0.

34 from that's much easier than your original calculation.

The same idea in number three.

So think about what your related fact is going to be.

If you finish that, I want you to explore whether all the related facts you could have used.

So could you have changed the minuend and the difference in a different way? And you might also want to think about how could I have made these calculations harder instead of easier.

But in that case, would you be using related facts or not? I'll leave you to think about that.

Alright, I hope you have a good day.

Take care, bye.