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Hi, 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 Simplify multiplication and Division using Multiples of 10.

This is within the unit arithmetic procedures with integers and decimals.

By the end of this lesson, you'll be able to factorised multiples of 10th power of N in order to simplify multiplication and division of both integers and decimals.

Now, I don't know about you, but that sounds pretty scary to me.

Factorised multiples of 10 to the power of N.

Don't worry, it's not scary at all.

You'll be absolutely fine and I'll be with you every step of the way to help you through.

Here are some key words that we will use in today's lesson, and they're product and partition.

We need to remember that the product is a result of two or more integers or numbers being multiplied together.

And partitioning is the act of splitting something, an object or a value down into smaller parts.

So we'll be using both of these words frequently throughout today's lesson.

The lesson will be split into two separate learning cycles.

In the first one, we will concentrate on multiplication, and in the second one we will concentrate on division.

Let's get started with simplifying multiplication using multiples of 10.

Here we go.

This is a Gattegno chart.

You may be familiar with the Gattegno chart and you may not.

If you are not, maybe just pause the video a minute and have a look to see what's happening in the chart.

Great, you're back now, good.

So this Gattegno chart, if you look every time we go down a row, we're actually dividing by 10.

So 1,000 divided by 10 is 100, 100 divided by 10 is 10, 10 divided by 10.

And so on.

The value or the product, I should say, of the pair of numbers that is highlighted is 21.

I'd like you to pause the video and see if you can find any other numbers on the chart that have a product of 21.

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

Great, you're back now, good.

Let's have a look.

Now, if I was going to do this question, I would probably start by looking at my factor pair for 21.

'Cause I prefer working with integers to start with 'cause it just makes life a little bit easier.

So I know my factor pairs or factor pair for 21 that's shown on my chart is three and seven, three multiplied by seven.

Remember we can have that either way round.

I've just written it that way round 'cause that's how they appear on the chart.

The three before the seven.

Are there other pairs on the chart then? Let's have a look.

So three multiplied by seven.

So I've just highlighted that we already knew that 0.

3 multiplied by 70 was 21.

Notice what happened.

The number in the third column went up one row whilst the number in the seventh column went down by one row.

Here are some other pairs.

So I've gone up a row, so therefore I need to go down a row.

So 30 and 0.

If I go up a row, I need to go down a row.

So 300 and 0.

07.

If I go up a row, I go down a row.

Oh, my chart doesn't go that far, but we can work that out, can't we? We know what that's gonna be.

If it's 0.

7, 0.

07, it's 0.

007.

Let's go back here now then.

So I've got 0.

3 and 70, which is what I started with.

And now I'm going to go down on the left of that.

I've got to write that one down first.

Now I'm gonna go down on the left and up on the right.

So 0.

03 and 700.

And then I'm just gonna do the last one is 0.

03 multiplied by 7,000.

So the important thing here is that if we want the product to stay the same, then we need to make those adjustments.

If I make one number 10 times bigger, then the other number needs to be 10 times smaller in order for the product to stay at 21.

We've got Laura and Jun and they're actually now, they're looking at some function machines.

So this is what we've got, three multiplied by 10, multiplied by 10, multiplied by 10.

Well that's 3000.

And then seven multiplied by 10, divided by 10, divided by 10, divided by 10 is 0.

07 and the answer is 210.

So Laura and Jun decide that they're going to rearrange the functional machines.

They're just going to put them in a different order.

Laura does this.

What calculation is Laura's function machine showing? It's showing 300 multiplied by 0.

So 300 multiplied by 0.

7 is 210.

So there's Laura's.

Now let's have a look at what Jun did.

So he's grouped the divide by 10s altogether at the beginning and the multiplied by tens altogether at the end, still gives us a result of 210.

What calculation is Jun's function machines showing? Yeah, that's right, 0.

003 multiplied by 70,000.

And we know that that's 210.

Notice that the order of the machines doesn't change the output if the same input is used.

So we could rearrange those multiply by tens and divide by tens in any order and we're still going to end up with 210 if we are using three and seven.

What calculation is shown here? So it's your turn now.

I'd like you to pause a video and write down what calculation is being shown here.

Great work.

You should have got 0.

6 multiplied by 1,200, which is 720.

I'm sure you've got that right and well done.

We're back now with the original function machine.

We're just going to look at how we can actually partition to help us do this calculation.

So 300 I can partition into three multiplied by 100, and 0.

7 I can partition into seven multiplied by 0.

I'm gonna collect together the three and the seven and the 100 and the 0.

1, I'm gonna multiply, three multiplied by seven is 21 and 100 multiplied by 0.

1 is 10.

And then I multiply those two together and we can see that the answer there that we've got in the box at the end is correct, is 210.

So this is where partitioning is useful to help us to be able to answer these questions.

Oh, Jun's got something to say now.

So all of those multiply by 10 and divide by 10 function machines and all we needed to do was multiply the product of three and seven by 10.

So Jun's peeling a little bit put out now because he is been made to do lots of work, but he thinks that actually all he needed to do was multiply together the three and the seven, which was 21 and then multiply that by 10.

Laura says, "Well of course 'cause multiply by 10 and divide by 10 are inverses of each other." So let's have a look at the function machines.

I've got a multiply by 10 and a divide by 10.

They're inverses, they cancel each other out.

I've got another multiplied by 10 and divide by 10.

So again they're gonna disappear and I've got another one multiplied by 10 and divide by 10.

So Jim was right, actually we just needed to find the product of three and seven and then multiply that by 10.

Well spotted Jun.

So here was how we partitioned this out.

So that's just what we had on the previous slide and we went through it of how we partitioned the 300 and the 0.

Well, so Jun's got something to say again.

He says "It was also notice that seven multiplied by 0.

1 and seven divided by 10, they're both 0.

So does that mean that multiply by 0.

1 is the same as divided by 10?" What do you think? Laura says, "Yes they are the same." And think about why that is because 0.

1 is one 10th of one and when you divide by 10 you're actually finding a 10th of something.

So the two things are the same, multiply by 0.

1 and divide by 10, that's worth knowing and worth remembering.

We're now going to have a look and see if the product of 0.

00007 and 30,000 is less than, greater than, or equal to 21.

We know that the digits two and one are gonna be in there somewhere 'cause it's the product of seven and three.

But what we want to know is it going to be more than, less than or equal to? Let's take a look at how we can think about this problem.

While 0.

00007 is a 100,000 times less than seven and 30,000 is 10,000 times greater than three, since 30,000 is only 10,000 times larger than three, the product is going to be less than 21.

Now you are ready to do a check for understanding.

The product of that number, and I'm not gonna read it out 'cause I'll probably say the wrong number of zeros, and 3 million is equal to 2.

I want you to decide if it's true or false and then also you need to pick your correct justification.

So no guessing, I want you to show me that you really understand whether the product of that is equal to 2.

Pause a video and then come back when you've got your answer and justification, remember? Okay, well done.

Let's have a look.

So you should have true.

And the justification for that is the first one that the decimal number is 10 million times smaller than seven and that 3 million is a million times larger than three.

That was our justification.

Moving on.

We know that four multiplied by six is 24.

We're going to use that fact and we're going to write some products using the digits zero, four and six and we're going to put them into the correct group.

And the groups are less than 24, equal to 24 and larger than 24.

We'll take a look at some products together.

Let's take a look at this first one, 0.

4 and 60.

Well we've made the four 10 times smaller and we've made the six 10 times bigger, therefore it's going to be equal to.

So if the number of times bigger matches the number of times smaller, it's equal to.

Let's take a look at another one.

I should have said here as well that I've said that it equals 24, but that might not necessarily be true.

We just know it's gonna have the digits two and four in there.

So if we take a look at this one, we've made the four 100 times smaller and the six 1,000 times bigger.

So this is going to be larger than 24.

Now let's have a look at this one.

We've made the four 100,000 times smaller and we've made the six 10,000 times larger.

So the number of times smaller is greater than the number of times larger, so it's less than.

And this one.

I want you to think this time, where do you think it's gonna go before I put it there? Great.

It's larger than.

What about this one? Again, you decide.

That was equal to.

Next one.

Did you put it in the right place? I'm sure you did.

What about this one? That goes in less than 24.

Now I'd like you to have a go at one of these.

I know you've just done some of those independently, but have a go at this one.

True or false, the product of 0.

0004 and 60,000 is larger than 24.

As always, I don't just want a true or false, I do want that justification so that you can show me you really understand what we are doing.

Pause a video and when you've got an answer you can come back.

Super.

So you should have decided on false and the justification for that that was B.

Laura and Jun are working on this problem.

The decimal point or points have fallen out of this calculation.

The answer is correct.

So where should the decimal point or points be in the question? 24 multiplied by 79 equals 189.

So like I said, the decimal point in the answer is correct, but there are decimal point or points missing in the question.

Laura says it must be 2.

4 multiplied by 79, that equals 189.

Jun says it must be 24 multiplied by 7.

Can they both be right? Yes they can.

Laura has made the 24, 10 times smaller and Jun has made the 79, 10 times smaller.

So actually there were two answers that could be correct for that.

Well done Laura.

Well done Jun.

Here's another calculation.

"It is 2.

4 multiplied by 7.

9", Laura says.

Jun says "That can't be right because two multiplied by 80 is 160.

The answer has to be close to that." So Laura says, "It must be 2.

4 multiplied by 7.

9 equals 18.

96." That makes more sense as two multiplied by eight equals 16.

What Jun was doing there was thinking about the size of the answer by looking at the question.

Well done Jun.

Is that the only place for the decimal point? I'd like you to have a think.

No.

It could be 0.

24 multiplied by 79, for example, if the decimal point can be at the beginning of a number.

Now you are ready for a check for understanding.

Which of the following have the decimal point in the correct place.

So I'd like you to decide which of those is correct.

There may be more than one.

Good luck with it.

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

Well done.

So A was correct, B was correct and D was correct.

Well done if you managed to identify all three of those.

Now you are ready for some independent learning.

I'd like you in question one to use a Gattegno chart to write down at least four products of numbers with a product of the following.

If you don't have your own copy of a Gattegno chart, you might be able to just draw one out.

And then question two, I'd like you to go beyond that Gattegno chart and write down at least three pairs of numbers with a product of those.

Pause the video.

Good luck with this and come back when you are ready.

Great work.

Question three.

Some of the decimal points are just like that problem that Jun and Laura were looking at.

Some of the decimal points have fallen out these calculations.

Using the facts that 32 multiplied by 14 is 448 and that 241 multiplied by 19 is 4,579, I'd like you to put the decimal points in the correct place in the question to get the given products.

So just as before the answers are correct, you need to put the decimal points in the correct places to make sure that they give those answers.

And then what I'd like you to do for question four is to write some alternatives to your answers in question three.

Can you think of other calculations that would be correct? As always, good luck with this and then when you are ready, come back.

You can pause the video now.

Here we are with the answers then.

I'm not gonna read all of those out.

And remember if it says eg, those are not the only answers that are possible.

If you've got something different, then I would suggest that you pause the video now anyway to mark your answers and then you can check your own answers with a calculator.

Question three.

So here again, there are some examples of where you may have placed those decimal points.

If that is not your answer, again, pause the video and you can then check your answers with a calculator.

Great work.

We're ready to move on.

We are now gonna move on to looking at division.

So we're really confident now with multiplication using multiples of 10.

Now we're going to extend that to looking at division.

So I mentioned this word at the beginning of the lesson and we looked at words that we were going to be using, keywords, and it was partitioning.

Partitioning can also help to simplify division.

We're going to take a look at this 6,000 divided by 200.

Here we need to remember that a division can be written as a fraction.

So I can rewrite this as 6,000 over 200.

Now I'm going to write each number as factor pairs with a common factor.

And for efficiency we're going to try and use the HCF.

Remember HCF stands for highest common factor.

I've decided here to write 6,000 as six multiply by 100.

And two multiplied by 100.

Why have I done that? Have a think.

Why have I done that? The reason I've done that is because we know that any integer over the same integer is just a different way of writing one.

And we know that anything multiplied by one stays the same.

So 60 over two multiplied by one is just 60 over two.

We can now change this back into a division or we can just simplify the fraction, whichever you prefer, whichever you find easier.

Here I've turned it back into a division statement, 60 divided by two and I get the answer 30.

I think you'll agree that's a fairly straightforward method to answer that question.

So this idea of using partitioning with those multiples of 10.

Let's take a look at another example, 3,400 divided by 170.

We're gonna write it as a fraction.

Then we're going to write the factor pairs and we're gonna try and use, if we can, the highest common factor, just because it makes the numbers a little bit easier to work with.

We'd still end up with the same answer if we didn't.

And I've chosen here to do 340, multiply by 10 and 17 multiplied by 10.

So remember that 10 over 10 is one and 340 over 17 multiplied by one is just 340 over 17.

And then here again, I can choose to simplify that fraction or to turn it back into a division.

340 divided by 17 is 20.

34 divided by 17 is two, but we were doing 340, which was 10 times bigger.

We need to make our answer 10 times bigger.

Let's have a go at one of these together on the left hand side.

And I'm really confident that you'll be ready to try the one on the right-hand side independently in just a moment.

Just remember the steps, write it as a fraction, write the factor pairs, making sure that you choose the highest power of 10 that you can.

In this case, that's 100.

Remember the 100 over 100 is a one.

So we can just rewrite this as 4,500 over 15.

If you want to, you can simplify that fraction or you can change it back into a division.

45 divided by 15 is three.

But actually we were doing 4,500 divided by 15, which was 100 times bigger than 45.

So therefore our answer needs to be 100 times bigger, which is why it's 300.

If you need to and you feel like you are not quite ready, you can go back and you can rewatch any of those examples.

But I'm confident that you are ready now to give one of these a go on your own.

Here you go.

I'd like you to use the method that we've just used to work out the answer to this question.

So remember using the method we just used, not reaching for that calculator, I'm sure you wouldn't of anyway.

Okay, pause a video and then come back when you've got your answer.

Great work.

Let's have a look then.

So what you should have had was 270,000 over 3000.

This time we could take a factor of 1,000.

I remember when we started and I said about that word factorised.

That's all we've done.

We've taken a factor of 1,000 out each of those numbers, that's why I said it wasn't scary.

And then we've got 270 over three.

27 divided by three is nine.

But remember we were doing 270, which is 10 times bigger.

So let's make our answer 10 times bigger.

Well done.

You're ready now to have a go at the next task.

Task B is answer the following questions, find and shade them in the grid to reveal a number.

So using the method we've just done, I'd like you please to have a go at working out the answers to these.

You then need to find the answers and they may appear more than once in the grid and then you are going to shade them in.

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

Superb work well done.

Now, did you manage to reveal a number? If it doesn't look like a number, maybe you've made an error, but I'm sure you've got it right.

And the number that we revealed was 380.

So the number we revealed was 380.

We are now ready to summarise our learning.

Multiples of 10 to the power of N can be used to simplify both multiplication and division.

10 to power end, just remember, it's just 10, 100, 1,000, et cetera.

And that's what we were doing.

If you think about it, we were taking out those factors.

I've given you there an example of a multiplication and an example of division that we've done during this lesson.

Something else that's really useful to remember is that dividing by 10 is equivalent to multiplying by 0.

And remember, the reason for that is when we divide by 10, we are finding a 10th and we know that a 10th is the first column after the decimal point in a place value chart.

And lastly, it is important to consider place value, particularly when we're working with decimals.

So thinking back to those questions where we were putting those decimal points back in, and I think it was Jun who was looking at them thinking, "Oh, that answer can't be right".

You need to be channelling your inner Jun and making sure that you are checking to see whether your answers are reasonable.

Thank you so much for joining me today.

I've had a great time.

I hope you've enjoyed the lesson and I look forward to seeing you soon.

I've finished the video