Lesson video

Hiya, my name's Miss 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 Understanding of Arithmetic Procedures, with Integers and Decimals", within the unit, Arithmetic Procedures with Integers and Decimals.

By the end of this lesson, you'll be able to add, subtract, multiply, and divide with integers and decimals.

All of this will be familiar with you, this should be just a quick recap to make sure that we are really solid with this before we move on.

Some keywords that we'll be using throughout today's lesson are here in bold, there are integer, decimal form, and digit.

An integer is any positive or negative whole number or zero.

So for example, negative two, zero or 153.

A number is in decimal form when a decimal point is shown and there are digits to the right of the decimal point.

And a digit is one of the symbols of a number system.

And the symbols that we use are the digits zero through to nine.

Today's lesson we will split into three separate learning cycles.

The first one, arithmetic procedures with integers, second one, addition and subtraction with decimals, and then finally, we'll finish up with multiplication and division of decimals.

Let's get started on that first one, arithmetic procedures with integers.

Here we've got Izzy and Lucas, and they're working on this problem.

They need to find the missing digits.

Here's the problem they're working on.

They need to find what digit is covered by each of the shapes.

So I'm going to give you a moment to have a think about this for yourself before we have a look and see what Izzy and Lucas have done when having a go at solving this problem.

Well done for having a look at that.

Let's see whether you came up with the same things that Izzy and Lucas did.

Hmm, so Izzy says, hmm, this doesn't work.

You can't get a one in the ones column as you already have seven ones in the top number.

Hmm.

She could be right, couldn't she? What do you think? Lucas says, mm, yes it does.

If you add four to seven, you get 11, which has a one in the ones column.

Oh yes, Lucas is right, isn't he? If we replaced the square with a four, we would end up with 11.

And that does have a ones digit of one.

Let's move on and see what they looked at next.

So same problem.

Izzy's now saying that she thinks the hexagon and the cloud could be any digit.

What do you think Lucas? Lucas says, "I think they could be any digits that sum to seven." Do you agree or disagree with Lucas? Well done.

I'm sure you said that you agree with Lucas.

What could they be? So thinking about them, what could they be? Let's think about what we know already.

Well, Lucas was right.

The square must be four for the ones digit to have a sum of one.

So let's put that there.

So now we can move on with our calculations.

So seven add four is 11.

So we've got the one in the ones column and then we carry over the one 10.

Now six, add five, add one, is 12.

So we can write down the units, which is two, and then we can carry the one.

Now we can see why Lucas had spotted that the hexagon and the cloud couldn't be any digits, they had to have a sum of seven because we've got a one, so they have to sum seven to make the total sum eight.

So they could have been one and six, two and five, or three and four.

And those of course could have been in any order, we could have had six and one, et cetera.

So we're at this stage of the problem, and we're now actually we've got some extra clues for this puzzle.

And those clues are these.

The larger number is on the top.

So we know that the bigger number has to be on the top.

And the difference between the hundreds digits is three.

So think back to those pairs of numbers we had.

We had 1, 6, 2 and 5, 3 and 4.

Which of those has a digit difference of three? Hmm, more than one solution.

No.

Now we know the hexagon has to be five, and we know that the cloud has to be two, because let's just check, five subtract two is three.

So that's a difference of three between the hundreds digits.

And let's just check that calculation.

Seven add four was 11, six add five add the one we carried was 12, and then five add two add the one we carried was eight.

We've solved the problem.

Maybe you came out with it yourself first.

Well done if you did.

Little bit harder now because we're now subtracting, so we're sticking with the same problem, missing digits, but this time we are looking at subtraction calculation.

So same thing, we need to find the digits that each of the shapes is covering.

I'll give you a moment, just have a think about this problem for yourself first.

Maybe you want to scribble this out and have a go at some different ideas and see if they work.

Or maybe you've got a strategy.

Have a think about it, and then come back.

So when we are working on arithmetic procedures with addition and subtraction, we always start with the ones column.

If it's just integers we're working with.

So let's have a think about the square first.

So we've got something we know that the square subtract eight has to be five.

What subtract eight is five? Wow, there brilliant, well done, you said 13.

So 13.

Hmm.

But remember, each of the shapes is covering up just one digit.

We know therefore that that is our ones digit three.

Now where is that one coming from? Well, 13 has one 10.

So we need to take this one, and give it to the ones column or exchange, I should say, to the ones column.

Meaning that I have now zero tens in my top number.

So let's just check 13 subtract eight is definitely five.

That's okay.

Now let's have a look at the hexagon.

So we've got something, but we know it's one's digit is zero, subtract something is four, right.

Have a think about that one for a moment.

Well, when we're exchanging, we can only exchange one.

So what we're going to do is exchange one of the hundreds.

So we've only got 600 left here, and we're going to give that exchange to the tens columns.

So we've exchanged that 100 for 10 tens.

So we've now got 10 subtract something, is four.

Yeah, you're right, that's six.

So now let's just double check, 10 subtract six, yep, it is four, isn't it? And now we can finish off the calculation.

The cloud is actually the easiest one to do now because six subtract five is one.

Is that what you got? Well done if you did, that's absolutely superb.

It's really difficult to think about those exchanges.

Here's a check for understanding then to see how well you'll get on with this independently.

So pause the video and when you've got an answer, come back, and we'll see how you got on.

Great work.

Let's have a look and see how you got on.

So again, it was challenging, wasn't it? It was a subtraction one.

I could have been kind and given you an addition one for the check for understanding, but I knew you were ready and up for the challenge.

So it looks like we might have a problem here, doesn't it? Can we go something take away seven is five? So that's going to be a two, because then what would give us the 12? And then the hexagon would then be eight, and the cloud would be two.

I'm sure you got that right, well done if you did.

If you didn't, don't worry too much, we'll carry on looking at some of problems like this a little bit later on, and you'll be able to practise some more of those.

Let's move on.

Now we have Laura and Sam.

Sam is working out the answer to 142 multiplied by five.

This is their working out.

Here we've got 142 multiplied by five, and we've lined up the ones digits and we've done the calculation.

So we've done five multiplied by two was 10.

So write down the zero, carry the one, four multiplied by five was 20, add the carried one is 21.

So write down the one carry the two, and then one multiply by five is five, add the two is seven, 710.

Maybe you might like to pause a moment and see whether you agree with Sam.

Laura says, "What are ya doing? There's a much quicker way to multiply by five." Do you know what Laura's talking about? So Sam wants to know, he's saying if there's a really easy way, he wants to know really, how do you do it? Hmm, let's have a look and see how Laura does it.

She says, "I always multiply by 10 and half it." Is that the same as multiplying by five? What do you think? Well, let's have a go, let's see.

Surely if Laura's method ends up with the same answer 710 as Sam got, then she must be right.

So let's have a see.

So 142 multiplied by 10 is 1,420.

So we've multiplied by 10.

Now she says we need to half it.

1,420 divided by two, remember, halving is the same as divided by two is most definitely 710.

So she's right.

Hmm.

Which of those methods do you prefer? Personally, I prefer Laura's method.

I'm really quite good at multiplying by 10 and I'm also quite good at halving.

So I think from now on I might use Laura's method.

Remember Sam's method is not wrong, but we might find Laura's method to be a little bit more efficient.

There are other efficient ways of doing other arithmetic procedures.

So what I'd like you to do is to pause the video for a moment and just have a think about are there any other alternative ways of doing some calculations? So pause the video, have a think, and when you've got some ideas, come back, see if they're the same.

Or you may have some different ones, or you might have some new ones that you've not thought about before.

That was the one that Laura talked about, so let's see what else I put into my table.

And like I said, you may have some more, so multiply by four.

Well, if I multiply by four, I actually just double it and double it again.

I multiply by two, I multiply by two.

I find that a really useful thing to be able to do.

What about multiplying by six? Well, I can split that up into multiply by three and multiply by two.

You might think, actually I'm not thinking that's that much easier, am I any better at multiplying by three than six? Maybe not.

But it's certainly an alternative method that some people might find more efficient and slightly easier to deal with.

What about multiplying by eight? I'm sure you've probably come across this one before.

So to multiply by four we said we could double and double.

So to multiply by eight we can double, double, double, multiply by two, multiplied by two, multiplied by two.

Often people get confused and might think that was multiplied by six, but remember two multiplied by two is four, and that multiplied by two is eight.

And what about divide by five? So if we multiply by five, we multiplied by 10, and then we halved it.

What about if we want to multi, sorry, divide by five? Well here we would divide by 10.

We know that's fairly easy to do, and then we would multiply by two, because if you think about it, there are twice as many fives in a number as there are tens, because they're half the size.

Another check, which of the following do you think is another way to divide by nine? A, divide by two and then divide by seven, B, divide by three and then divide by three, or C, divide by three and sorry, divide by six and then divide by three.

Pause the video and when you've got an answer, come back.

Work.

So let's see, it is B, divide by three and divide by three is actually exactly the same as dividing by nine.

Divide by two divided by seven would be the same as divided by 14, and divided by six divided by three would be the same as dividing by 18.

You are now ready to have a go at some tasks on your own.

This task is like the one that Izzy and Lucas were looking at at the beginning of this lesson.

Instead of shapes, you've got boxes.

Your job is to find out what digit needs to go in each of the following boxes.

So you're going to pause the video, have a go at these, remember no calculators, and then when you're ready come back and check.

Moving on to two.

So within that learning cycle we looked at two things, finding missing digits, but also those efficient ways of doing calculations.

So question two is calculate the following, again, without a calculator please, try to use those efficient methods we talked about.

And maybe if you've written them down, you could refer back to them, if not, you could just rewind the video and have a look at those.

And I'd like you to have a go at these questions.

And like I said, no calculators.

And then when you're ready, come back.

Well done.

Let's have a look and see how you got on, brilliantly I'm sure.

Here are our answers.

So I'm not going to read them out 'cause they're quite difficult to read out.

So I'm just going to leave them there for a moment.

If you need a bit longer, obviously you can pause the video but you now need to mark your answers and see how you got on.

I'm just wondering here, did you actually decide to check your calculations before you came back? So if you'd put a one, a four and a four in those boxes, did you then decide to check that 146 add 394 was actually 540? Well I dunno if you did.

It's really good if we can, check our answers to make sure that we do that all the time.

And here we are with our answers to question number two.

So we use the efficient methods.

So A was 2,900, B 348, C 208, D 270 and E was 97.

Well done on those.

We're now going to move on then to our next learning cycle.

We're going to extend what we've just been looking at, but we're going to start looking now at addition and subtraction with decimals.

Like I said previously, this is a checking lesson, so this should be familiar to you.

But just a quick recap to make sure that we're happy before we move our learning on.

Now we've got Alex and Jun.

They are hoping to enter a team competition.

That's exciting, isn't it? Their total time must be less than 24 seconds.

Total time of what I'm not sure, haven't shared with us what the competition was.

Alex takes 14.

8 seconds and Jun takes 8.

7 seconds.

Can they enter the competition? So the total time has to be less than 24.

Alex took 14.

8 and Jun took 8.

7.

Can they enter the competition? Here's one way that Alex has set out his work and here is Jun's work.

Which is the correct way do you think, to set out the calculation, and why? The correct way was the second way.

So here's our two calculations, and I've just put a box around the one that's correct.

Why is the one in the box correct? The decimal points are in line.

And remember the decimal point determines the place value of each digit.

So we have to line them up when adding and subtracting because if we look at the first one, we would be combining one 10 with eight ones and that doesn't work.

So we need to make sure that we are combining ones with ones and we are combining tens with tens, and tenths with tenths, et cetera.

Alex and Jun now measure their heights.

They want to find the total of their heights.

So for some reason Jun is going to stand on Alex's head maybe to see whether he can reach something, I'm not sure.

So Alex is 1.

6 metres tall and Jun is 1.

48 metres tall.

Here's Alex's workings.

So we can see lined up the decimal points, brilliant, well done Alex, and added those up.

Check, do you agree with Alex? Has he got the answer right? And here's Jun's work and here's mine.

Okay, so lined at the decimal points, well done Jun.

Hmm.

So now I want you to think, what is the same and what is different about their workings? Maybe about how they've set them up, what they look like, maybe their answers, pause the video and have a think, what is this same and what is different? Maybe you said some things like this.

They both lined up the decimal points, they both got the same answer.

Jun didn't put the zero in the hundredths column.

That was the only difference I could really see.

You may have come up with some other things.

Now they want to work out who's the tallest.

Well we can see who's the tallest, it's Alex, but we want to know how much taller Alex is than Jun.

Here's Alex's working, and here's Jun's working.

Again, pause the video and think about what is the same and what is different about their workings and maybe their answers? Well done, what did you say this time? Well, you may have said some things like these.

They both lined up the decimal points, so they've done that, brilliant, well done.

But this time, last time they got the same answers, this time they've got different answers.

Wonder what answer you've got.

Jun didn't put the zero in the hundredths column.

So if we look in the hundredths column, the very last digit, Alex put a zero and Jun didn't.

So he couldn't see that actually there are zero hundredths minus eight.

So he's forgotten that there are zero hundredths and we're then subtracting eight.

So therefore his answer is correct.

I'm sure if you paused the video and had a go at this yourself, you would've ended up with 12 centimetres just like Alex did.

It's really important to make sure that we add in those zero placeholders.

1.

6 is 1.

6 and zero hundredth, so we must put that zero in.

Now another check for understanding.

Alex's sunflower is 1.

82 metres tall and Jun's is 2.

3 metres tall.

How much taller is Jun's? What I want you to do is to think about which is the correct way of setting out this calculation? And maybe you might want to work out the answer as well.

Well done if you do, pause the video, and come back when you've got an answer.

Superb work, well done.

The correct answer here was C.

If we check here, we've got the decimal points in line, the ones are aligned, the tenths are aligned, the hundredths are aligned, and there is that zero in the hundredths column.

Maybe have a think about what was wrong with A and what was wrong with B.

So A, we can see that the decimal points are not aligned.

B, we are adding there, but we wanted to find out how much taller Jun's sunflower was, so we should have been subtracting.

We're now ready to have a go at some more independent practise.

Here we've got Sophia and Andeep and they record how far they can kick a football.

So two things I'd like you to work out, remember again without a calculator, A, what is the total distance that they've kicked the ball? And B, how much further did Sophia kick the ball? Sophia kicks it 12.

73 metres, and Andeep kicks it 9.

8 metres.

Pause the video, remember no calculators, and then come back when you're done.

There's a second question in this task B, and we'll go sticking with the sporty theme and we're now bouncing a basketball.

Lucas, Aisha and Jun record how long they bounce the ball for.

Lucas bounced it for 35.

6 seconds, Aisha for 28.

45 seconds, and Jun for 27.

87 seconds.

What I'd like you to do this time again is, A, work out the total time.

So how long in total did the three of them manage to bounce the ball? And B, how much longer did Lucas manage than Jun? Pause the video, when you've got an answer, remember no calculators, come back.

Wow, there's another question in task B, question three.

Very similar to the calculations that we did in the first learning cycle where we're trying to find the digits covered by the shapes.

This is quite a challenging task but you've got all of the skills that you need to be able to do it.

So pause the video, have a real good go at it.

Remember, if you are stuck you can just rewind the video back to that first section maybe and have a look at how we did it with the integers, and then come back and have another go at this.

But anyway, pause the video, good luck, and come back when you're ready.

Wow, well done for sticking with that one.

It was quite a challenging task, but I know that you need a challenge.

Here are our answers then.

So for the first one, 1A, the total distance they kicked the football was 22.

53 metres.

Hopefully you remembered to put the metres on there.

Remember it wasn't 22.

53 bananas, or 22.

53 apples, it was metres, they were kicking the football a distance of metres.

B, how much further did Sophia kick the ball? 2.

93 metres, so roughly three metres longer than Andeep, well known Sophia.

On question two, the total time taken was 91.

92 seconds, and how much longer did Lucas manage than Jun? And that was 7.

73 seconds.

And then onto that really tricky, challenging task.

Pause the video, mark those, and then come back when you're ready.

And then we had to explain those statements.

So that's why this was challenging, so not just find the digits but actually the reasoning behind it.

Really well done if you've got these right.

A true for A but not B and C as we can subtract the bottom digit from the top.

B, true for A but not B, as the one of the ones would have been exchanged for tenths.

C, true for A and B but not C as one of the tens digits would have been exchanged for 10 ones.

And we've already marked our missing digits.

We're now going to move on then to our final learning cycle.

We're going to start thinking back to now some of those efficient methods that we talked about earlier.

So Jacob buys five boxes of chocolates, so it must be Christmas coming up or Easter or birthdays or something.

And he buys five boxes of chocolates.

We want to work out the total cost of those five boxes of chocolates.

And we can see there that one box of chocolates costs £3,86.

So can you remember Laura's method from earlier? She had a quicker method, or she thought it was quicker and I did.

And then maybe you did, of working out multiplying by five.

An alternative to multiplying by five is to multiply by 10 and then half.

Let's have a look at this one then.

So £3.

86 multiplied by five, our alternative method would be to multiply by 10 and then divide by two.

Well £3.

86 multiplied by 10 is £38.

60.

So the two bits in the blue are equivalent.

And then we can partition if we need to, so remember you can partition that £38.

60 'cause that's not easy to halve in your head.

So let's partition it up.

We've got £30, £8 and 60 pence.

And we are halving.

So to go from that line to this line, we're going to divide everything by two, we're going to halve it, half of 30 is 15, half of eight is four, and half of 60 pence is 30 pence.

So therefore when now we need to add together those three amounts, and that will give us our final answer.

That means that £3.

86 multiplied by five is £19.

30.

I dunno about you, but I definitely prefer that method to multiply and doing a column multiplication, particularly now we're working in decimals.

Aisha has raised some money.

So a Aisha's done a sponsored silence and well done Aisha, she's raised £122.

50.

And she wants to divide this between her five favourite charities.

How much does each charity get? As we looked at earlier, an alternative to dividing by five is divided by 10 and doubling.

Let's try that with Aisha's sponsored silence money.

£122.

50 divided by five.

So our alternative way is to divide by 10 and then multiply by two.

Notice here I've left the zero off at the end of the £122.

50.

That doesn't matter because we have no hundredths in that number.

It's okay here to omit that, to leave it out.

The calculation of the blue box and the result in the blue box of that division.

So remember when we are dividing by 10, we are moving all of our digits one place to the right.

So £122.

50 becomes £12.

25.

We still need to multiply that by two so that we're actually dividing by five rather than 10.

And again, we can partition if we need to.

So £12.

25 we can partition to £10, £2, and 25 pence.

And this time we are multiplying by two, we are doubling.

Double £10 is £20, double £2 is £4, and double 25 pence is 50 pence.

And remember then we need to put those back together, meaning that each charity is going to get £24.

50.

Lucky charities.

Check for understanding.

So a little bit different this time, I have made a mistake, so I've got a calculation, the calculation is correct, but I've made a mistake in working out my answer.

So what I'd like you to do is to find the mistake and correct it.

Pause the video and when you've got the answer, come back.

Great work.

Let's see whether you managed to correct my mistake.

I'm sure you did.

So the mistake was there, I divide by 10, and then what should I have done? I shouldn't have divided by two, 'cause actually that would've been the same as dividing by 20.

I actually needed to multiply by two.

So we can see there that the division should have been a multiplication.

Now let's correct that then.

So 362 divided by 10, I'd done that bit correctly.

That's 36.

2, but I should have multiplied by two instead of dividing by two, which would've given me 72.

4.

Well done.

The first question in this last and final task is a match question.

So on the left hand side I've given you some calculations and what I'd like you to do, is to match it to the equivalent on the right hand side.

So look at multiply by five and find which of the ones on the right hand side is equivalent to multiplying by five.

As always, when you're done, come back, so pause the video now, good luck.

Well done.

Moving on now to question number two.

So we now know what those efficient methods are.

What we're going to do, is we're going to use a couple of those.

So we can see we've got very, very similar problems here.

We've got our boxes of chocolates.

So in question two, we are finding the cost of various numbers of boxes of chocolates if they cost £4.

75 each.

And then for question three, this time the Oak students have decided to buy the box of chocolates, but they are actually going to share the cost of them.

They're going to divide the cost between them.

So what I'd like you to do this time for this question, each box of chocolates cost £5.

80.

If two students buy the box, how much do they each pay? And four students, et cetera.

Off you go, have a go, remember no calculators, I know I keep saying it, but I'd like you to not use a calculator.

I'd like you to use those efficient methods so you become really confident at them, pause the video, come back when you're ready.

Amazing.

Wow, you're back that quick already? That's superb, well done.

Here are our answers then.

So multiply by five is equivalent to multiply by 10, divide by two.

Divide by four is equivalent to divide by two divide by two.

Multiply by eight is equivalent to multiply by two, multiply by two, multiply by two.

Divide by five is equivalent to divide by 10, multiply by two.

Divide by six is equivalent to divide by two, divide by three.

And multiply by six is equivalent to multiply by two, multiply by three.

I'm sure you've got all of those right.

And then finally moving onto to the last two questions of today's lesson, question 2A, £9.

50, B, £47.

50, C, £23.

75, D, £28.

50.

And then question 3A, £2.

90, B, £1.

45, and C, £1.

16.

If you need to pause the video just to mark those you can, but if you're ready, we can move on and we can summarise our learning for today's lesson.

During today's lesson then we have been looking at understanding arithmetic procedures that allow us to make calculations easier with both integers and decimals.

So an alternative way of multiplying by five is to multiply by 10 and then divide by two.

Remember, it's always okay to go back and use your column method.

This is just thinking about efficiency and making things easier, particularly if you haven't got a pen and paper.

We also looked at the importance of the decimal point when adding and subtracting decimals, and that placeholders must be used when subtracting decimals.

You've done fantastically well today.

I've been really, really pleased with how you've worked.

Well done.

Thank you for joining me, and I hope to see you again soon.