Lesson video

Hi ya, my name's Ms. Lambell.

Thank you so much for popping along today to do some maths.

I hope you enjoy it.

Welcome to today's lesson.

The title of today's lesson is, checking and securing understanding of written addition and subtraction strategies.

And it's within our unit, arithmetic procedures with integers and decimals.

By the end of this lesson, you'll be able to generalise and fluently use written addition and subtraction strategies, including columnar formats with decimals.

Most of what you see today should be familiar, but don't worry if it isn't, because we are gonna take it through step by step at a time.

Here are some keywords which we'll be using throughout today's lesson, so it'd be useful to have a quick recap of what those are.

In the second learning cycle, we're going to be looking at what effect units might have on how we deal with calculations.

We're going to be using the term metric units, and these are based around the standard units of metre, gramme, and litre.

Particularly in today's session, we're going to just be concentrating on metre.

Additive inverse.

The additive inverse of a number is the number that when added to the original number gives a sum of zero.

So the two things, when we find the sum of them, it's zero.

An absolute value.

Absolute value of a number is its distance away from zero.

So for example, five and negative five both have an absolute value of five because they're both five away from zero.

Those are the words that we'll be using throughout today's lesson.

Today's lesson, we are going to split into two separate learning cycles.

In the first we will look at checking addition and subtraction strategies.

And in the second, like I said, we'll be looking at considering what do we do if we're given values which are not in the same unit.

Let's start then with addition and subtraction strategies.

You may have seen these counters before to represent numbers, there are place value counters.

What I'd like you to do is to think about what is the same and what is different about the two sets of place value counters that I've given you here.

Pause the video, have a think, remember what's same, what's different.

When you are ready, come back and we'll have a chat about it.

Okay, what did you think? What was the same and what was different? You may have said something similar to this; they both represent 23.

They do actually represent the same number, so that's what was the same, but what was different? The difference was, is that the image on the right hand side had not exchanged the 10 ones for for 10.

These 10 ones here have been exchanged for a 10 in the previous one.

The one on the left had been written in its simplest form, whereas the one on the right had not.

It's really worth remembering here about exchanging and carrying.

I can exchange those 10 ones for a 10.

Again here, what is this same and what is different about these sets of place value counters? Pause the video, come back when you've got your ideas.

What did you think this time? Well, I've written, and again, these are just examples, they both represent 32.

1.

The one on the right has not exchanged the 10 tenths for a one.

Remember, in maths often we like to represent things in their simplest form, and so the one on the left is in a simpler form than the one on the right.

These 10 0.

1s ones should really be exchanged for a one, and then we can see that the two have exactly the same value.

Three 10s, two ones, and one 0.

1, one 10th.

I'd like you to rewrite this following representation in its simplest form.

So think about what we've just done on the previous two slides, and I'd like you to make sure that this is written in its simplest form.

Pause the video.

When you've got your answer, come and check in.

Great work.

Here, if we were to write it in its simplest form, we would've exchanged the 10 ones for a 10, given us four 10s, and we would've exchanged 10 of the tenths for one, given us just an extra one, and then just one 10th left.

This represents 45.

1.

The reason I wanted to just recap this is because we're going to be looking at addition and subtraction, and it's really important we remember this idea of exchanging.

Here we have Aisha and Jacob, and they've noticed that they can spend more money than they have on their accounts in the canteen.

Dunno if you go to the school canteen, but they've noticed that they can spend more than it's on their account.

Not a lot more, I wouldn't have thought, but just to make sure they don't go hungry.

They wonder if this can happen anywhere else.

Can you think of anywhere else where you might be able to spend more money than you have? Hmm, have a think.

Aisha says that she thinks it can happen with a bank account.

Can it happen with a bank account? Can you spend more money than you've got? And then Jacob says, "Oh yes, you're right.

It's called an overdraft." I'm wondering if you've heard that term before, overdraft.

It's when the bank allows you to spend more money than you've got, and you go into something called an overdraft.

Not a very sensible thing to do though, we should only spend money that we really have.

A bank account has a balance of 75 pounds.

A 134.

95 pounds is spent.

What is the new balance? Aisha has given us the calculation, let's see.

Aisha's written, "75 subtract 134.

95 pounds." And Jacob says, "I thought the larger number always had to go on the top." Is Jacob right, does the larger number always have to go on the top? Aisha says, "But then we wouldn't be calculating 75 pounds subtract 134.

95." Jacob doesn't seem to think that that matters.

Do you agree with Jacob? Let's take a look at it together.

So the bank account had a balance of 75 pounds and 134.

95 was spent.

So what we're trying to do, is we're trying to work out a new balance.

If that person were to get a bank statement, what would the balance say? Jacob suggested that the biggest number needs to go on the top for the subtraction.

So here are his calculations.

So he's written, "134.

95 pounds subtract 75 pounds." Right, we're just gonna double check that Jacob's done all of the things that he should, lined up his decimal points, made sure that there's no gaps where we need zeros for placeholders.

We could put one in the hundreds column, but we don't need to.

Five subtract zero is five, nine subtract zero is nine.

Four subtract five, I can't do that, so I'm going to exchange here to make that 14.

So I've exchanged, 14 subtract five is nine, I can't take seven from two, so again, I'm going to do that exchange.

And then 12 subtract seven is five.

Aisha says, "That cannot be right.

As you spent more money than was in the account, so you will have a negative balance." Do you agree with Aisha? Yes, Aisha's right, isn't she? If you didn't have enough money in your bank, you spent more than you have.

You owe the bank money, so your balance will be negative.

We have to start with the balance of 75 pounds and subtract the amount of money spent.

This is the calculation we should have been doing.

Aisha also knows that we are now writing any subtraction as an equivalent addition.

So rewrites this as 75 add the additive inverse of 134.

95 pounds, which is minus 134.

95 pounds.

What we're going to do now is we're going to take that minus 134.

95 pounds and we're gonna split it into two parts.

The first part, we are going to make sure we have an additive inverse over on the left hand side so that that becomes a zero pair, and then the minus 59.

95 pounds is what's left.

The new balance therefore will be negative 59.

95 pounds.

Effectively the person owes the bank 59.

95 pounds.

I'd like you again to consider what is the same and what is different about Aisha and Jacob's answers.

Aisha's answer was negative 59.

95 pounds, and Jacob's answer was 59.

95 pounds.

What is the same and what is different? Pause the video.

And when you've got your ideas, pop back.

What did you think? I think it's pretty obvious they both ended up with the same digits.

So Jacob's has said his method gave the correct absolute value.

Remember the absolute value is the distance from zero.

Aisha says, "Yes.

You just have to remember that the answer will be negative because you are subtracting more money than you have." So as long as Jacob remembers that his answer will be negative, then he's okay to put the bigger number at the top.

We'll now have a look at another problem.

This bank balance is 481.

24 pounds.

Somebody buys a bike for 575.

75 pounds.

We want to know what the new balance will be if they buy the bike.

This is a calculation, so we want the starting balance and the amount of money spent.

Rewrite that as an equivalent addition.

So change the subtraction to an addition of the additive inverse.

This is what it would look like.

Here again, what we're going to do is we're going to separate the minus 575.

75 pounds.

We're going to make sure that we take out a part that is going to make that zero, because they're additive inverses, remember that's what additive inverse is.

Not so obvious here, so I need to now work out what's left.

How much of that 575.

75 is left if I've already used 481.

24 pounds to make the additive inverse with the 481.

24 pounds? And I'm gonna do that by doing a subtraction, because I'm trying to find the difference.

I'm going to calculate 575.

75 pounds subtract 481.

24 pounds.

And here's my calculation on the bottom right hand side of the screen.

Five subtract four is one, seven subtract two is five, five subtract one is four.

We can't take eight from seven, so we're going to do exchange.

17 subtract eight is nine, that means that the new balance is negative 94.

51 pounds.

The person owes the bank 94.

51 pounds.

I'd like you now to have a go at very similar question.

So it's the same bike, same price, but this time the account balance is 150.

88 pounds.

Using the method that we have just done, remember I don't want you using a calculator, I'd like you please to have a go at working out what the new balance would be if the bike was bought.

Pause the video, come back when you're ready.

Remember this is quite challenging.

So if it takes you a little while, don't worry.

You just come back as soon as you're ready, I'll be here waiting for you.

Well done.

Now we're gonna have a look and we're gonna check to see that your answer's right, even though we know it will be.

So the calculation I'm going to do first is I'm going to find the difference between the two.

Which is 424.

87 pounds.

But remember it's going to be a negative balance because we spent more money than we had.

So remember Jacob's method worked, but we had to remember that it was the negative of that.

So the new balance will be negative 424.

87 pounds.

Well done for getting that right.

And remember, that was the absolute value.

You're now gonna have a go at this task.

So you're going to apply the skills that we've just looked at.

Remember if you need to go back and check anything, that's absolutely fine.

So here in the table, I've given you the current balance of a bank account.

In the middle column, I have told you what the person has bought.

On the right is a price list, and your job is to find the new balance.

So I'd like you to assume that the person has bought the item in the middle column, their current balance, and then find the new balance.

Remember no calculators, come back when you're ready to check those answers.

Good luck with that.

Well done, off you go.

Super work, quite a lot to do there, well done.

Here are your answers.

Person who bought the car, balance is negative 1080 pounds.

The next one, negative 6.

75 pounds.

Person who bought the bike was negative 149.

10 pounds.

The fridge, negative 137.

75 pounds.

The TV, negative 124.

08 pounds.

And the mobile phone, 227.

63 pounds.

And then the final one is an item of your choice.

I wonder what you decided to buy.

My example here is negative 215.

81 pounds.

I can't remember what I bought, but it certainly looks to me like I must have decided to buy the bike.

We can now move on, and we're going to start looking at values that are not measured in the same unit.

Here we have Jun, Sam, and Laura.

Now, they've been growing sunflowers.

They've been growing their sunflowers and they have just decided they're going to measure the height of their sunflowers.

Jun says, "Mine is 86.

4 centimetres." Sam, "Mine is 2.

26 metres." And Laura, "Mine is 376 millimetres." What's the total height of their sunflowers? Have a look, here's the calculation.

So the total, remember we're finding the total of something, we're going to sum all of the values, we're going to add all three of the heights of the sunflowers.

And here's a calculation.

Let's check we've lined up the decimal points.

Let's check we've got placeholders after the decimal point.

And let's just check, yes, we are adding them, 'cause we're finding the total, the sum.

So we can do that calculation.

So zero, add six, add zero is six.

Four, add two, add zero is six.

Six, add two, add six is 14, so four and carry one.

Eight, add seven, add one is 16, so six and carry one.

And then we've got three, add one is four.

"What unit is our answer in," Jun asks.

What unit do you think their answer is in? Hmm? Sam says, "I don't know because we each used a different unit." Do you agree with Sam? Laura says, "Oh, we should have changed them all into the same unit before we added them." Because otherwise what unit is their answer in? And actually, is their answer even right? Like Laura says, what they should have done is they should have changed all of their measurements first.

And we're going to change all of our measurements into metres.

Now, you will be familiar with doing this from previous, but we're going to use exactly the same method.

So if you are a little bit rusty on it, don't worry, I'm sure you'll catch us up.

So Jun says, "Mine is 86.

4 centimetres." Here's our place value chart.

So we've got metres in the middle because that is what we're going to be converting everything to.

Jun says, "Mine is 86.

4 centimetres." So we need to find the centimetres, and we're going to place our decimal point after there.

Now I can place the digit from Jun's measurement into my place value table.

So I've got a four, a six, and an eight.

So just check that does read 86.

4.

It does.

But we decided we were going to convert all of our measurements into metres.

So metres is here, we're now going to put our decimal point at the end of the metres.

Now we're going to fill in our digits, remembering that we didn't have any whole metres, so we do need that zero there.

So Jun's sunflower is 0.

864 metres.

Sam says, "Mine's already in metres, it's 2.

26 metres." So we don't need to do anything with Sam's measurement.

Laura, probably because she wanted to make hers sound bigger than it was, measured hers in millimetres.

So this time millimetres, we don't need a decimal point because Laura's measurement doesn't have a decimal point in it.

But we could put one at the end of the millilitre.

Millimetres, sorry.

And then we've got our 376.

Remember we're converting to metres, so the decimal point needs to go at the end of metres.

Put our digits in, remembering that there was zero metres.

This means that, actually Laura's sunflower was 0.

376 metres high.

Now we can find the total height of all of their sunflowers.

So this is our calculation now.

Notice decimal points all lined up, and making sure that we've got those placeholders.

We add all of these up, we can see our calculation.

We end up with 3.

500.

What units now is the answer in? Right, it's in metres.

We changed them all into metres, so our answer must be in metres.

So it's 3.

5 metres.

Notice I've left off the trailing zeros at the end.

I don't need those there.

I don't need to be able to see that there were no hundredths and no thousandths.

When working then, with values in different units, we must make sure that we convert 'em all to the same unit before working with them together.

Really, really important.

So here we changed them all into metres because that was the standard unit that we were using.

Let's have a look at this one then.

"Alex lives 2.

3 kilometres from school.

Izzy lives 875 metres from school.

"They both walk to school." Good for them.

We want to know how much further does Alex walk.

So we look at the question.

First thing we want to do is to look at the units of our measurements.

And we can see that they are not the same, so we do need to do that conversion before we think about combining those two measurements.

Alex lives 2.

3 kilometres, let's put that into our table.

So kilometres, put our decimal point in, 2.

3.

Remember we want to convert everything into metres, so metres is here, pop our decimal point in.

So we've got the two and the three.

Is my answer 23? No, I do need to show here that I have no 10s and no ones.

This means that Alex lives 2,300 metres from school.

We now have both of our measurements in the same unit, so now we can compare them and we can combine them.

We can clearly see that Alex lives a lot further away, and we can now answer the question, which was, how much further does Alex walk? Here's the calculation we need to do, 2,300 subtract 875.

So let's have a look at how we're gonna do that.

So we're going to exchange, and then we're going to do another exchange until we've got something in the ones column that we can use.

10 subtract five is five.

Seven, sorry, nine subtract seven is two.

Bit more, exchanging.

12 subtract eight is four, and then one.

We now know that Alex walks an extra 1,425 metres than Izzy.

Now we're ready for a check for understanding.

This check I'd like you to please find the total of 4.

5 kilometres, 1,870 metres, and 782 centimetres.

Well, actually I don't want you to work out the answer.

What I'd like you to do is to tell me which is the correct calculation if I want to find my answer in metres.

Pause the video, decide which is your answer, and come back when you ready.

Well done, let's have a look and see how you got on.

The correct answer was C.

We've changed 4.

5 kilometres to 4,500 metres.

The second value is already in metres, and we've converted 800 and, sorry, 782 centimetres, centimetres, which is 7.

82.

The correct answer is C.

We're now going to have a look at this addition pyramid because this is what you're going to be doing in the next task.

This is how an addition pyramid works.

To fill this brick in, we need to sum the two bricks below it.

Now, notice here that I want all of the answers in metres, hence the Ms in each of the boxes.

So here I would first have to change 126 centimetres into metres, which is 1.

26 metres, and then add 3.

8 metres.

I total those and I get 5.

06 metres.

For this box here then, again, is going to be the sum of the two bricks below.

0.

52 kilometres, I need to convert that first into metres, which is 520 metres, and then I'm adding 16.

4 metres.

This box is going to be 536.

4 To find the middle box then, we're going to sum 3.

8 metres and 0.

52 kilometres.

Now, if you want a challenge, you could stop the video here and have a go at completing the rest of the pyramid.

But if you don't want to, if you want to wait until you do your independent learning in the next task, that's fine.

I'm gonna carry on and fill in the rest of the pyramid.

523.

8 metres, and then we're gonna sum the two below here.

Sum the two below, and we should end up at the top with 1589.

06 metres.

So just recapping, and in addition pyramid, it is the sum of the two bricks below that go in each of the boxes.

You're now ready to have a go or a practise at one of those as a check for understanding before we move on.

Little bit different here then.

What value is in the missing brick? So remember the two bricks below, the sum of those must equal the brick above them.

Pause the video and have a go at working out what the missing value is.

Great work, well done.

Let's have a look.

So at this time it was the difference, the difference between 652.

69 metres and 245 centimetres.

Remembering we want the answer in metres, so we'd need to change the 245 centimetres into metres, which is 2.

45, and that would give us a value of 650.

24.

You are now ready to have a go at some questions by yourself.

So they are addition pyramids.

We know how to work with those now, but just be careful to look at the unit that I want the answer of each block in.

Good luck with these, and when you come back I'll look forward to seeing how well you've gotten with them.

Well done on those.

I'm sure you've done really well, but we're not gonna mark 'em just yet because I have got a monster of a pyramid that I'd like you to have a go at now.

I told you it was a monster.

It's exactly the same as before.

To find the value of a brick, you sum the two bricks below it.

Good luck with this one.

Stick with it, I know you can do it.

I look forward to seeing how well you've got on when you come back.

Pause the video now and give it a go.

Whew, that was a big pyramid.

Now we're ready to check our answers.

Now, I'm not going to read all of these out because you may check them in a different order to me.

So what I'm going to do is I'm going to say, please pause the video, check your answers, and then when you're ready, come back and un-pause the video, and we'll see how you've got on with that monster of a pyramid which was question five.

Great, did you get all of that right? Or at least most of it, I'm sure you did.

Now, let's have a look at that monster pyramid.

Here you go again, pause the video, check your answer.

Well done if you've got answers for all of those, and amazing if you actually got this correct answer of 469,875 metres in that top brick.

If you've got that, surely you've got the rest of them right? Come back when you're ready.

Great work.

Hmm, I'm wondering if you did get 469,875 metres.

I'll bet you did.

Now we can summarise our learning from today's lesson.

Any subtraction can be rewritten as an addition using additive inverses, and there's an example there that we used in the lesson.

When working with decimals, the decimal point helps us to ensure that digits with the same place value are combined.

So remember when we looked at the sunflower problem to start with, that we wouldn't be comparing things with the same place value because the units were different.

And that leads me into the final point of today's lesson, which was numbers in different units must be changed into the same units before combining.

Also before comparing.

Because it might have looked that one of those sunflowers was amazingly tall, but they'd use really small units to measure it.

Thank you for joining me today for this lesson.

I hope you've enjoyed it, and I really look forward to seeing you again to do some maths.

Thanks again, bye.