Lesson video

Hello and welcome to today's math lesson.

I trust you are feeling well.

My name is Dr.

Shorrock, and I'm here to guide you through the learning today.

Today's lesson is from our unit: "Order, Compare, and Calculate with Numbers with up to Eight Digits".

This lesson is called "Add and Subtract Mentally Without Bridging a Boundary".

Now, as we move through the learning today, we are going to deepen our understanding of how and when we can add and subtract with seven-digit numbers, using a mental strategy.

Now, sometimes new learning can be a little bit tricky, but I know we are going to work really hard together today.

And when we work really hard we can be successful, and I'm here to guide you along the way.

Let's get started then, shall we? How do we add and subtract mentally without bridging a boundary? This is the key word that we will be using throughout our learning today.

The word is addend.

Now, it's always good to practise saying these words aloud.

So let's have a go together.

My turn, addend.

Your turn.

Nice.

An addend is a number added to another.

They're also known as the parts, and you can see I have represented these in two bar models.

An addend is equivalent to a part.

Let's get started with our learning today by looking at addition and subtraction when there is only one changing digit.

And in the lesson today, we have Aisha and Lucas to help us.

Aisha is thinking of a number, and Aisha tells us, she gives us a clue.

Her number is 4,000 more than 1,203,957.

I wonder if you can guess what her number is? Good thinking, Lucas.

It's always a good idea to represent things in a bar model.

Let's take a look.

Aisha's number is the whole, that's the number she's thinking of, and she tells us that it is 4,000 more than 1,203,957.

So they must both be the parts.

We have two parts to addends, we need to add them to find the unknown whole.

And we can represent both addends in a place value chart to help us add.

Take a look at this place value chart.

What do you notice? Can you see I represented both addends in the place value chart? Is there something that you notice? Lucas has noticed that one of the addends, the 4,000, only has one non-zero digit, the four.

The other digits are zeros, aren't they? Well, what does that mean? How can that help us to add 4,000 to 1,203,957? That's right, we can just add four 1,000s onto the 1,000s digit of 1,203,957.

The 1,000s digit is a three.

If we add four onto that three, four plus three is equal to seven so 4,000 + 3,000 = 7,000.

So the number that Aisha must be thinking of was 1,207,957, and that number is 4,000 more than the other addend we were given of 1,203,957.

Let's look at the equation in more detail.

What do you notice about this calculation? Anything? Lucas has noticed that only the digit in the 1,000s place has changed.

But why is that? Why has only the digit in the 1,000s place changed? Aha, that's because we are adding a number where the only non-zero digit is in the 1,000s place.

That second addend only had a digit four, the other digits were zeros, weren't they? So only that one 1,000s place will change.

Let's look at a different calculation.

Can you read that calculation? 2,314,523, add 30,000.

What do you notice with this calculation? Can you make a connection with the previous calculation? Lucas notices that one of the addends has only one non-zero digit.

That 30,000 has got a three as a non-zero digit, the other digits are zeros, aren't they? Well, how does that help us? And that three, that's in the 10,000s place, so what does that mean? So Aisha is predicting that only the digit in the 10,000s place will change.

Would you agree with her? Let's represent this using place value counters, and see if Aisha is correct.

Lucas has represented 2,314,523 with place value counters, and we are adding 30,000, and 30,000 is composed of three 10,000s.

So Lucas needs to add another three 10,000 counters.

There we go, what do you notice? Well, we get the sum of 2,344,523.

And let's look at that calculation in more detail.

Look at the equation.

Is there something that you notice? Think about what we noticed with the previous equation.

What happens here? That's right, only the digit in the 10,000s place changed.

Why is that? Well, that's because we are adding a number where the only non-zero digit is in the 10,000s place.

So only that 10,000s digit will change because the others are zeros, aren't they? They're not going to change anything.

Let's look at another different calculation.

4,120,368, add 500,000.

What do you notice this time? Can you make a connection with the previous two equations that we have looked at? Lucas notices that one of the addends has only one non-zero digit.

That sounds familiar, doesn't it? It's like our other ones, isn't it? Which place has got the non-zero digit this time? That's right, it's in the 100,000s place, isn't it? So what does that mean? That's right, it means that only the digit in the 100,000s place will change.

Now we know this, we can find the sum without using a representation because we know we just need to add that five, that digit five to the 100,000 onto the 100,000s place, which in our first addend is a one.

We know 1 + 5 = 6, so 100,000 + 500,000 must be equal to 600,000.

So the sum is 4,620,368.

Now, is it always true that only one digit will change if one addend has only one non-zero digit? Hmm, what do you think? Well, Aisha is saying, "It's been true for all the examples we've looked at so far, hasn't it?" All our examples have had only one non-zero digit, and only one digit has changed.

So she's saying.

"Yes, it must be true." Hmm, but Lucas is respectfully challenging her.

I wonder if you know why he's challenging her.

If we have an addend where the non-zero digit is a complement to ten or more, what will happen? That's right, we would have to regroup, and then more than one digit would change.

Let's check your understanding with this.

Take a look at this equation: 5,012,431, add 70,000.

Could you tell me its sum? Is it a, 5,019,431? Is it b, 5,072,431? Is it c, 5,082,431? Or is it d, 5,712,431? Pause the video while you have a go at that, and when you are ready for the answers, press play.

How did you get on? Did you say, "Well, the second addend only has one non-zero digit of a seven, and that's in the 10,000s place so we know we're looking at the 10,000s place.

In the first addend we've got the digit one in the 10,000s place.

We know one add seven is eight, so one 10,000 add seven 10,000s must be eight 10,000s or 80,000, so it must be C." One addend has one non-zero digit, and it is not a complement to ten or more than ten, so only that one digit changed, that 10,000s digit.

How did you get on with that? Well done.

This time, Lucas thinks of a number, and his clue is that his number is 3,000 less than 3,515,104.

Can you guess what his number is? Good idea, let's represent this in a bar model.

So this time we've got the whole number, and Lucas' number is 3,000 less so this time Lucas' number is a part.

We've got the whole and we've got a part, so to find the unknown part we need to subtract.

And we can use place value counters to help us subtract.

Aisha has represented the number 3,515,104 with place value counters, and written a subtraction equation this time.

3,000 is composed of three 1,000s, and we are subtracting this time so Aisha needs to remove three 1,000 counters.

And what's she left with? That's right, she's left with two 1,000 counters or 2,000.

So Lucas' number must be 3,512,104.

And let's look at this calculation in further detail.

Look at the equation, what do you notice? That's right, only the digit in the 1,000s place changed.

And why is that? That's because we're subtracting a number where the only non-zero digit is in the 1,000s place.

We've got 3,000, it's got the digit three, the rest is zeros, aren't they? So the 100s, 10s, and ones, they won't change.

Let's look at a different calculation.

We want to subtract 20,000 from 5,238,010.

What do you notice? Could you make a connection with our prior learning? We need to subtract 20,000, so apart from our whole, 5,238,010.

Lucas has noticed that in our known part it only has one non-zero digit, that two, and it's in the 10,000s place.

What does that mean then? That's right, Aisha's going to predict that only the digit in the 10,000s place will change.

Would you agree with Aisha? And let's represent this using a place value chart, and let's see if Aisha's correct, shall we? Lucas has represented 5,238,010 in the place value chart, and we know we need to subtract 20,000.

That's taking away two from the 10,000s place, which leaves us with no 10,000s.

So the difference is 5,208,010, that's the number that Lucas was thinking of.

Let's look at a different calculation.

4,920,368 - 500,000.

Do you notice something here? Something that helps you make a connection with our prior learning, what we've already been thinking about? That's right, the part we are subtracting has only one non-zero digit.

It has a five, doesn't it? And it's in the 100,000s place, so what does that mean? That's right, it means that only the digit in the 100,000s place will change, and then we can now find the difference without using a representation, now that we've spotted that and made those connections.

Again, remember, this will only be true if no regrouping is needed, but this time we can have a look and we can see that no regrouping will be needed.

We've got nine 100,000s and we need to subtract 500,000.

9 - 5 = 4, so 900,000 - 500,000 would be equal to 400,000.

So the number Lucas was thinking of must be 4,420,368.

Let's check your understanding with this.

Have a look at this equation, and can you tell me what the difference would be? Pause the video, maybe talk to a friend about this, and when you are ready to go for the answer, press play.

How did you get on? Did you say it must be B? It must be B because the part we are subtracting has one non-zero digit, and that is in the 10,000s place.

And if we look at the 10,000s place in the whole, we have seven 10,000s or 70,000.

So there's going to be no regrouping needed because we can subtract 40,000 from 70,000, and that would be 30,000.

So the answer must be b, 5,632,431.

How did you get on with that? Well done.

Your turn to practise now.

For question one, could you just complete these equations? For question two, could you complete these equations, and is there something in our learning that could help you? And take care to notice that these are now subtractions.

For question three, could you complete these equations? (Dr.

Shorrock gasping) But stop, what do you notice? Did you notice that they were a mixture of addition and subtraction? So I need you to think about the overall difference.

For the first equation, you can just add 100,000, can't you, because actually you are adding 900,000, subtracting 800,000, so the overall difference there is 100,000.

So that's the sort of strategy I need you to be using in this question.

Pause the video while you have a go at all three questions, and when you are ready to go through the answers, press play.

How did you get on? Let's have a look.

Question one, you are asked to complete the equations.

So you can see the sum for the first one is 7,514,203, only the millions column changed.

For the second one, this now would have a change in the 100,000s column, so 3,914,203.

For the third equation, there would be a change in the 10,000s column, 3,554,203.

And for the last question there would be a change in the 1,000s column, 3,518,203.

For the next equation you were asked to find the first addend, so you would need to compare the whole with the other addend.

And you see here for the first one, only the millions column has changed, and we need to add three million.

For the second equation, the 100,000s digit had changed, so we're adding 500,000.

For the third one, only the 10,000s digit changed, we needed to add 10,000.

For the fourth one only the 1,000s column changed, we needed to add 6,000.

Then we needed to add 700 because only the 100s column changed, and then we needed to add 30 because only the 10s column changed.

For question two you were asked to complete these equations, and this time there were subtractions.

But again, the similar principle applies, we just needed to note that the known part had only one non-zero digit, so only that column would change unless there's some regrouping.

So, for the first equation we needed to subtract 20,000, so the 10,000s column would change and the difference would be 104,362.

For the second equation only the 1,000s column would change, 122,362.

For the third question only the 100s column would change, 124,162.

And for the last one only the 10s digit would change, so 124,342.

For the second set of questions, you needed to find the unknown part.

Again, comparing the whole to the difference.

Here, for the first question we could see the 100,000s digit changed, we need to subtract 100,000.

For the second question the 100s digit changed, we needed to subtract 300.

And for the last question the ones digit changed, so we needed to subtract five ones.

And then the next set of questions, here you needed to find the whole.

And again, we could do this by comparing what has changed between the given part and the difference.

So then we could see that we needed a whole that was 300,000 more, so 4,843,654.

For the second question our whole needs to be 2,000 more, 4,545,654.

And then for the last one the whole needs to be 40 more, four 10s more, which would be 4,543,694.

For question three, you were asked to complete the equations.

So for the first one, and you had to think about the overall difference, if we're adding 900,000 and subtracting 800,000, actually that's got an overall difference of just adding 100,000, and that only has one non-zero digit, so the only digit that would change would be the one in the 100,000 position, 3,614,203.

For the second equation we were subtracting 700,000 but then adding 400,000, so there's an overall difference of 300,000, we would need to subtract 300,000.

Only the 100,000s digit would change, 3,214,203.

For the next question we could see we have an overall difference of adding 3,000, so only the 3,000's digit would change, 3,517,203.

And for the last question here we had an overall difference of adding 2,000, so only the 1,000's digit would change, 3,516,203.

How did you get on with those questions? Brilliant.

Fantastic learning so far, really impressed with how you moved your learning on and have a deeper understanding of addition and subtraction where only one digit changes.

Now, we're going to move on and look at addition and subtraction where multiple digits change, so that means more than one.

So here's a question for you: What is 124,000 + 342? Stop.

Before we rush to calculate, we should always see what we notice.

Is there something that you notice? Aisha has noticed that one addend is a multiple of 1,000.

Did you see that 124,000, that's a multiple of 1,000? And both addends have more than one non-zero digit, so this is different from the previous part of the learning where we had an addend that only had one non-zero digit.

So, what does this mean for us? Well, it means that more than one digit will change, and we can represent this using a place value chart or counters.

So I've represented 124,000 in my place value chart and with place value counters, and we need to add 342.

There we go, we've added 342, we can add it on the place value chart and with counters.

And we can see here that the sum would be 124,342.

Let's look at this equation in more detail.

What do you notice about it? Is there something that you notice that has changed or stayed the same? That's right, only the digits in the 100s, 10s and ones places changed.

But why is that? Can you see? Well, this is because we're adding to a number which is a multiple of 1,000.

So we had three zeros in the places that we were adding that 342 to, so they were going to stay the same at 342, weren't they? Let's check your understanding with that.

1,445,000, add 124.

Would that be A, B, C, or D? Pause the video while you have a go, maybe talk about your answer with a friend and give reasons for your answer.

And when you're ready to hear the answer, press play.

How did you get on? Did you say it must be A? We've got a multiple of 1,000 at 1,445,000, which has got zeros in the 100s, 10s and ones positions.

And we're adding a 100s number, 124, so only those digits will change.

Adding a three-digit number to a multiple of 1,000, only the 100s, 10s and ones will change.

Let's look at this calculation.

What do you notice for this one? Have a look at the whole, have a look at the part.

Lucas has noticed that multiple digits will change.

Well, why is he saying that? How does he know that multiple digits will change? The part we are subtracting has more than one non-zero digit.

It's got a one, a two, a three, and a one, hasn't it, as well as those zeros.

We could use a place value chart or counters to help, but Aisha is going to challenge us to try using our reasoning skills.

So, let's compare the value of each digit to the whole.

We've got four million and we need to subtract one million, and that's three million.

We've got 800,000 and we need to subtract 200,000.

Well, that's 600,000.

We're doing okay with our reasoning skills so far, aren't we? There were no 10,000s or 1,000s to subtract, so those digits remain the same, seven and a six.

We have 900 and we need to subtract 300, and that's 600.

We have no 10s to subtract so that digit remains the same, at a nine.

Then we need to subtract one one from two ones, which is one.

We calculated this mentally by just comparing the values of the digits.

That was quite easy to do, wasn't it? When we calculate mentally it is important to use our number sense, and we need to ensure there's no regrouping.

Let's check your understanding with this.

Could you find the difference when we subtract 5,698,987 and we subtract 10,405? You could, if you want, use a place value chart or counters, or you could use your number sense, whichever is most efficient for you.

Pause the video while you do that, and when you are ready to go through the answers, press play.

How did you get on, did you say it must be C? We're subtracting 10,000, which means the five million and the 600,000 digits must stay the same, so it couldn't be any of the others.

And then we're subtracting one 10,000 from nine 10,000s, so that would be eight 10,000 or 80, so 5,688,582.

How did you get on with that? Well done.

Your turn to practise now.

For question one, could you complete these equations? And I've given you a hint there, use the answer to each calculation to help you solve the next one.

Hmm.

For question two, could you complete these equations? And again, use the answer to each calculation to help you solve the next one.

For question three, could you solve these problems? Part A: Lucas thinks of a number.

"My number is 231,000 more than 1,462.

What is it?" And for part B, Aisha thinks of a number.

"My number is 46,200 less than 1,987,653.

What is it?" Pause the video while you have a go at those three questions, and when you are ready to go through the answers, press play.

How did you get on? Let's have a look.

So, for question one you were asked to complete these equations just by noticing which digits are going to change.

So here, we can see that the 100,000s and the 10,000s digits will be the only digits that change, so we get 760,421.

Here, we can see that all of the 1,000s digits will change, won't they, the 100, the 10,000, and the 1,000 digits will change, 766,421.

And we look at this one, we can see, again, spot that pattern, this will be the same as above but we will have added an extra 200, 766,621.

And then the last one in the pattern, well, here we've got another five ones to add, 766,626.

And then we can have a look at question two where we were completing these equations, but this time they were subtractions.

So, we needed to work out the first one, and we can see that it will be the 10,000s and the 1,000s digits that will change.

The 100s, 10s and ones will stay the same because they're zeros.

So, the answer would be 7,101,364.

We can then use that answer to help with the rest, because you notice now the part we're going to subtract another 100, so the difference must be 7,101,264.

And now we're subtracting another 30, 7,101,234.

And for the last equation, you might have noticed it was slightly different.

We were subtracting 24,013, so we could see that the 10,000s and the 1,000s digit would need to change, but, also, so would the 10s and ones digits, so we could compare those and we would find the difference of 7,101,351.

For question three you were asked to solve some problems. Lucas was thinking of a number, and his number is 231,000 more than 1,462.

So, we can represent this in a bar model, and you can see we have two parts or both addends.

That means we have to add them to find the number that he was thinking of, or the whole number.

And we can see that the 1,000s digit, the 100s, 10s and ones digit would also change, and the sum would be 232,462.

So that was Lucas' number.

For part B, Aisha was thinking of a number.

Her number is 46,200 less than 1,987,653.

Again, we can represent this in a bar model but this time the number she's thinking of is a part, so we know we need to take the whole and subtract the other part.

And here we can see that the 46,000, those digits would change, the 10,000s and the 1,000s digits would change, as would the 100s digit.

And the difference would be 1,941,453, and that is the number Aisha would be thinking of.

How did you get on with those questions? Well done.

Fantastic learning today, everyone.

Really impressed with how hard you've tried and how you have deepened your understanding of how we can add and subtract mentally without bridging a boundary.

We know that when we add and subtract mentally, the language of unitizing supports us in addition to using our known facts.

And we know that an understanding of the structure of numbers and place value supports when we add and subtract mentally.

You should be really proud of how hard you have worked today, I know I am proud of you.

And I've had great fun and I look forward to learning with you again soon.