Lesson video

Hi, I'm Mrs. Furlong.

Welcome back to our next session on number addition and subtraction.

Okay, so in the last session, Mrs. Knights had left you with this practise activity.

She asked you to create your own sequence where the change in the subtrahend is balanced by a change in the difference.

And then she asked you to get someone at home to explain your pattern to you.

I wonder whether you managed to do that.

And did you manage to get them to use that generalised statement too.

Let's take a look at what I did.

So I joined in with the practise and I thought really carefully about that generalised statement.

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

And I thought about how it was going to do this and decided that in my first sequence, that I would keep the minuend the same and subtract from the subtrahend and therefore I'd need to add to the difference.

So this is my beginning of my sequence, 1,472 subtract 1,300 equals 172.

Have a look at my sequence and see if you can spot the pattern that I made.

Have you spotted it? I asked my daughter, if she could have a look at this and spot it too.

Now she's probably a little bit younger than you, but what she did say is that the minuend has been kept the same and I taught her those words and then she said the subtrahend has gone down by a 100 and she didn't use the word decreased but that's what she meant.

And she said, "And I noticed that the difference "has gone up by 100." Because again, she didn't use the word increased but that's what she meant.

I wonder whether your family members were able to describe your sequences.

Okay, I made one more pattern.

I did ask if my daughters either of them could explain it to me but this was a little bit too challenging for them.

I wonder whether you'll be able to spot what I've done this time.

So I put on our generalised statement and I put on our stem sentence and this is my pattern, have a little look at it.

5.

5 subtract 0.

25 equals 5.

25.

And the next one, 5.

5 because the minuend had to stay the same subtract 0.

5 equals five.

Can you work out what I added to the subtrahend and therefore what I subtracted from the difference? If you can, So I'll put a few more and see whether you can work it out.

My next one in the sequence was 5.

5.

subtract 0.

75 equals 4.

75 and then 5.

5 subtract one equals 4.

5 and finally 5.

5 subtract 1.

25 equals 4.

25.

So what did, how did I change the subtrahends and how did I change the difference in my sequence or in my pattern? That's right, I have changed it by not 0.

25.

So let's have a read of our STEM sentence.

I've kept them in new and the same and added 0.

25 to the subtrahend.

So I must subtract not 0.

25 from the difference.

I wonder what numbers you chose to increase or decrease your subtrahend and differences by.

Did you get experimental? Did you try decimals? Did you try large numbers? Did you be really brave and try fractions? I hope you had fun with it and I hope you are money to see if you could get some family members to describe those sequences too.

Okay, so our session today is rounding up all of this learning that we've been doing in the last few lessons.

And the generalised statement is going to be key to our learning today.

But we will still be using those STEM sentences to help us to look for how our calculations have changed.

But if you could read with me that generalised statements at the top and we'll just make sure it's really clearly in our heads before we get started.

Okay, are you ready? If the minuend is kept the same and the subtrahend is increased or decreased, the difference decreases or increases by the same amount.

Okay, so this lesson is going to have a lot more chance for you to practise.

So if you haven't already got something to write with, if you could go and get that now and pause the video and then come back and it's going to be you working a bit harder than me today, okay? Right, let's get started.

So we're going to start by using our generalised statements and just some sentences to help us to work out the relationship between these calculations.

We're going to always start at the moment with this central calculation, because you can see that it's been completed.

So let's see if you can read those big numbers with me.

Are you ready? 375,861 subtract 200,000 is equal to 175,861.

Okay, so let's have a look at the relationship then between that middle calculation, that's got that dotted line round it and the top calculation.

Can you see that in this one, subtrahend has increased by 100,000? So therefore our difference must, that's right, decreased by 100,000 or if we want to use the same sentence, we could have kept the minuend the same, that is the key bit of learning.

If our minuend hasn't been kept the same, then that relationship doesn't work.

So I've kept the minuend the same and added 100,000 to the subtrahend, so I'm a subtract 100,000 from the difference.

And what's our new difference going to be? That's right, 75,861.

Now I know you could have just worked that out by looking across those equations.

And those are there that are a little bit easier on purpose so that you can see that relationship.

So we'll have to of you probably think, well, hang on, I already know what 375,861 subtract 300,000 is, because I can use my place value.

I know that you know that, but it's just about making sure that you can see those connections before you go off and do some other bits for yourself.

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

So what's the relationship from that middle calculation to the bottom calculation? That's right, so this time our subtrahend has decreased by a 100,000, so our difference must increased by a 100,000.

Brilliant cause our minuend is the same, isn't it? So therefore, what's my new difference going to be? So we need to add that 100,000 on to 175,861 and we get? Brilliant 275,861.

But this time, if we were to use our STEM sentence, we'd be saying I've kept minuend the same and subtracted 100,000 from the subtrahend, so I must add 100,000 to the difference, okay? I know this is a lot of what we've been doing before.

And this lesson is just reviewing everything and making sure that we can use this in practise and gradually removing all these arrows to see if you can do it by yourself.

Okay, so hopefully you've got something to write with.

I would like you to have a go at, looking for those relationships from this one, perhaps writing in some of those arrows, think about what's the same and what's different.

Think about those STEM sentences and pause the video here and see if he can fill in those arrows and those missing boxes.

Have you had a chance to do that? All right, let's have a look then.

What did you notice was the same? Yeah, those minuends were the same in every single calculation.

Can you tell me the number? 144,561, that comma helps me to break that number up into which digits are the thousands and which ones are the hundreds, tens and ones.

Right, okay, so 144,561 subtract 30,000 is equal to 114,561.

So if we're going from that middle calculation to the top calculation, what did you notice? Yeah, you're right.

If you increased, the subtrahend increased by 10,000, the difference must decreased by 10,000.

So did you get 104,561? Yeah, and we can just use our place value, can't we? We know it's gone down by 10,000.

So the only digits that are going to change is our ten thousands of the digit.

Excellent, okay.

And from the middle one to the bottom one? Yes, this time the subtrahend has decreased by 10,000 and so our difference most increased by 10,000.

Or we can use the STEM sentence, I've kept minuend the same and subtracted 10,000 from the subtrahend, so I must add 10,000 to the difference.

And again, because we're only adding 10,000, we can use our place value and we can see that 114,561 add 10,000 is 124,561.

Brilliant, let's move on.

Okay, so let's take a look at this one now.

What do you notice is different to the last example? Yes, it's got decimal and I agree with you there.

Is there anything else that you spotted? Yeah, the boxes are in different places, aren't they? So in the last one, the missing value was the difference but in this one, our missing values of the subtrahends.

So we might have to think of it in a slightly different way.

How about minuend, have they been kept the same? They have, haven't they? Okay, so let's have a look at what's happened to the difference then in the first calculation.

So from the middle calculation to the top calculation, what happened to the difference? It decreased by not 0.

3.

Can you tell me why 12.

23 subtract 0.

3 or subtract three tenths is 11.

93.

Where did those numbers come from? Ah, yes you're right.

So 12.

23 subtract 0.

3 subtract three tenths.

If I think about the 12.

2 parts of it, I need to subtract two tenth from that.

So have 12.

0 and then obviously I've got my three hundredths, which I mustn't forget to think about at the end.

And that's only three tenths here, so I need to go down by another tenths and one tenths less than 12.

2 is 11.

9 and I mustn't to get those hundreds.

So 12.

23 subtract three tenths is 11.

93.

So let's think about our STEM sentence.

Well, if I joined to the difference, I've subtracted from the difference, haven't I?.

So that top STEM sentence I can fill in that part.

Look my 0.

3, this is where it's come from, I've subtracted 0.

3 from the difference.

So therefore what must I do to subtrahend? That's right, add 0.

3.

So I've kept the minuend the same and added 0.

3 to the subtrahend, so I must subtract not 0.

3 from the difference.

So therefore my not 0.

5 from the middle calculation to the top one becomes? That's right 0.

8.

Can you have a go with the bottom one by yourself and then we'll have a quick go through it in the moment.

So pause the video if you need to.

Okay, so this time what's happened to our difference? It's increased by not 0.

3.

So we've added the different, so I'm going to use the bottom of my STEM sentences.

So I've added 0.

3 to the difference, so what must I do to the subtrahend? Subtract 0.

3, because don't forget those minuends are the same and that's really important.

So I've kept the minuend the same and subtracted 0.

3 from the subtrahend.

So I must add 0.

3 to the difference.

So therefore, to make that change, we need to look at those subtrahend and subtract 0.

3.

So 0.

5.

subtract 0.

3 is? That's right 0.

2.

So my new calculation in this part is 12.

73 subtract 0.

2 equals 12.

53, fantastic.

Okay, can you try this one? I haven't included everything.

I haven't even included the arrows this time.

So I think you're ready to do this without them.

The arrows are just starting to scaffold our learning and gradually we want to see if we can do this without them.

Of course, if you still need them, that's fine.

So pause the video here and see if you can complete those missing boxes.

And if you need to use the arrows, use them, if you don't that's fine.

Okay, it's your turn to look.

So what did you notice between that middle calculation and the top one? Aah, what about trying it? Increased subtrahend by one tenths, so subtrahend is increased by one tenths.

Our minuend is still the same, it's 23.

964 still.

So therefore I've added not 0.

1 or one tenths to subtrahend so I'm must subtract 0.

1 from the difference.

If we look at 23.

164, I need to subtract one tenth, where is that tenths column? Yes, straight after the decimal, isn't it? To the right-hand side of the decimal.

So 23.

164 subtract 0.

1 is 23.

064 and the other way around, what did you notice? Ah, you're right, this time my difference has, sorry, my subtrahend has decreased or I've subtracted not 0.

1 from the subtrahend, so I'm must add 0.

1 to the difference.

So this time again, we're adding one tenths, so it's only going to be changing this tenth column in this case.

So 23.

164 becomes 23.

264.

Okay, so my arrows have completely gone now.

I want you to have a go at filling in these missing boxes, pay really careful attention to that relationship and see what you can do.

All right, pause the video and we'll go through it in a minute.

Okay, have you had a chance to do that? Right, which calculations should we start looking at? That's right, the middle one 'cause it's complete, isn't it? 4,975 subtract 70 is equal to 4,905.

So let's go from that middle one to the top one.

You could have done it the other way and that's absolutely fine.

So this time we can see the change in the difference, can't we? The difference has increased by 10.

So the subtrahend must decrease by 10.

So our 70 becomes 60 and we know that works because our minuend is of the say, okay? What about the other one? Did you just assume that you could put 4,975 in because we've been saying all the time, I've kept the minuend the same, some of you might've done.

Did any of you check and make sure that deals or parts of the rule work? I think we should do that because sometimes it might not.

In which case our minuend isn't 4,975, let's see.

So from that middle calculation to the bottom one, the subtrahend was 70 in the middle one and it's increased by 10 to get to 80 in the bottom one.

So we've had an increase of 10.

So in our difference it should decrease by 10.

So 4,905 is the difference in the middle one and it has decreased by 10 to get 4,895.

So therefore we're in the same pattern, it's fitting this rule.

So if there's 4,907 for the minuend because it fits the rules.

So therefore anyone can be kept the same.

Okay, so it's your turn to be the teacher.

Now I would like you to explain the mistakes in these calculations.

So we've got the first set in purple and then we've got another set in blue and there are some mistakes on boxes that have been filled in.

And I'd like you to see if you can find those mistakes and explain what they've done wrong.

So pause the video here and spend a bit of time having a look at that and we'll come back together in a moment.

Have you had a chance to do that? Okay, let's have a look together then.

In the purple set, I'm going to focus on that middle calculation, that's got that dotted line around it because it's been completed.

And have you also spotted that all of their minuends are the same in all three of those calculations.

So we've got 32,201 as our minuend in all of them.

Okay, I'm going to go from the middle calculation to the top one.

And we'll see that the difference in the middle calculation is 31,201 and that difference has decreased by 500.

So therefore, the difference has decreased, subtrahend should increase by 500.

But it hasn't, 1000 hasn't gone up by 500 to get 500, has it? So they've made a mistake, it decreased both the subtrahend and the difference.

Did you notice that in the bottom one as well? So the difference increased by 500, so the subtrahend should have decreased by 500, but can you see the increase in both again? They've got themselves confused and forgotten that one must increase and the other must decrease and vice versa.

So the 500 and the 1,500 should be in the opposite boxes.

Right, okay, let's have a look at the blue one now.

Okay, we're going to start with that middle calculation because that's the one that's going to be completed.

So we're going to be able to use that and can we say, for the middle calculations to the top one that our minuends have kept the same? If we kept the minuends the same let's have a look, we should do it what did the difference? Well, can we see the difference here is 3.

471 and what's happened to it to get to here 3.

271? That's right, we've subtracted 0.

2 up from the difference.

So we should add 0.

2 to the subtrahend.

And did that happen from 0.

5 to 0.

7, did it go up by two tenths? It did, so that one's correct, that's not the mistake.

Let's have a look at the bottom one.

A lots of people would just assume that the minuends must be kept the same 'cause that's what we've been looking at and put 3.

971 in the box.

And that's what this person's doing.

But I wonder how they actually looked to how the relationship works between the other numbers.

So from 0.

5 to 0.

3, subtrahend has decreased by two tenths.

So therefore our difference if we're following that rule should increase by two tenths, has it? No, because you can see here we've got four tenths in the different here and five tenths in the difference here.

So it's only increased by one tenths.

So therefore it doesn't follow the rule.

So instead we need to work out the minuends, on to work out in the minuend I'm going to use the inverse.

And I know that the inverse of subtraction is addition.

So 3.

571 add 0.

3 should be 3.

871 and that's the mistake.

So it shouldn't be 3.

971, it should be 3.

871.

If you spotted that, you're a very good math detective, well done.

Okay, so now it's time to see how fluently you can do these and how quickly.

So have a go at these ones with the missing numbers.

Notice whether the minuends are the same in the sets and then we'll come back in a moment and see whether you managed to do the missing numbers.

See how quickly you can do it, off you go.

Have you paused it? Have you managed to do it? Okay, let's go.

So in that first set, 125 subtract 36 is 89 and the one below, the minuend still the same, so what's happened to the difference? It's increased from 89 to 90, so it's increased by one.

So our subtrahend of 36 needs to decrease by one.

So we should have 35, brilliant.

The next set, 2,568 those minuends of the same in both calculations.

We don't know the subtrahend in the bottom one, but we do know the differences.

So 2,201 has decreased by one, so our subtrahend needed to increase by one.

Fantastic, hope you got those.

Okay, so you've got two more sets to have a look at now.

So see if you can complete those using our rule and then we'll come back and check them in a moment.

Have you had chance to do that? Brilliant, so let's have a look at the first one.

We need to check in this case, whether it does follow the rule and if it does, then we can put that minuend and it's the same, but we must check if it follows the rule first.

Well, I noticed here that these digits are the only ones that changed and all of these ones here stayed the same.

So all I needed to look at was the tens and the ones.

So the 19 changed into 20, so it increased, it's increased by one, hasn't it? And then in the other calculation the subtrahends increased by one, the difference should decrease by one.

So again, these digits here are identical, there and there.

So I think of those as a bit of noise that we don't really need to pay attention to in this case.

So our ones have decreased by one, so subtrahend is increased by one, our different sorry, our subtrahend has increased by one.

Our difference has decreased by one, so by that same amount.

So therefore we are following the rules, so our minuends are going to be the same.

The bottom calculation, what's happened to it? Yeah, our minuend is the same, we don't know one of our subtrahends, but our differences is, which the change in that to help us.

What's happened to our differences? Ah, so we've got 3.

339 and we've got 3.

34.

How has it changed, has increased, hasn't it? Has increased by 100.

No because we've got nine in thousandths column.

So in this first difference we've got three hundredths and nine thousandths.

In the second one we've got four hundredths.

So a few ways we can think about this, so we can have them all a thousandths, which is one way of doing it.

So three hundredths and nine thousandths is 39 thousandths and now we've got four hundredths here which 40 thousandths.

I'll write something like zero and just to help me think about it.

So from 39 thousandths to 40 thousandths, it's increased by one thousandths.

So our subtrahend had to decrease by one thousandths.

Again, that might be a little bit tricky to think about.

So I often end up popping on at zero in the thousandths column, cause it can help me because now I can look at this as 70 thousandths and 3.

87 or 3.

870 decreasing by a thousandths is 3.

869.

I hope you got that one right, that one was quite challenging.

Okay, so I'm going to leave you with some practise now to have a go at all by yourself.

And as you can see, I've got this paint all over my work, I've made a bit of a mess and I would like you to see if you can use the rule from today's session to work out the digits beneath the spots of paint and decide on the best starting point for yourselves and how the rule can help you.

Good luck with it, I hope you have a good day, take care.