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Hi, everyone, my name is Ms. Ku, and I'm really happy to be learning with you today.

It's going to be fun, and really interesting lesson, and I'm so excited to be learning with you.

You'll come across some new keywords, and maybe some keywords you've already come across before.

Now, we're gonna work really hard today, but I am here to help, and we can learn together.

In today's lesson from the unit Arithmetic Procedures with Integers and Decimals, we'll be looking at multiplying decimals, and by the end of the lesson, you'll be able to generalise and fluently use multiplication strategies to calculate accurately with decimals.

So let's have a look at some keywords.

A factor is a term which divides exactly into another term, and factorising means to express a term as the product of its factors.

For example, 10 is a factor of 50, because 50 divided by 10 is equal to 5.

A non-example is the fact that 8 is not a factor of 10.

This is because 10 divided by 8 is equal to 1.

25.

Today's lesson on multiplying with decimals will consist of two parts.

The first will be multiplying with decimals, and the second will be using appropriate answers to calculations.

So let's make a start on multiplying with decimals.

Well, there are lots of different ways in which to multiply two integers together, and there's very little difference between the strategies used to multiply integers together and multiplying decimals together.

However, it is important to have an understanding behind each process.

So I'm going to start with a simple multiplication.

Let's look at 2 multiplied by 3, and we're going to use an area model.

So here, you can see I have 2, and we're multiplying it by 3.

So we know the area is 6.

2 times 3 is 6.

Now, I'm going to zoom in a bit.

Still using our area model, the 2 by 3, we're going to break up each 1 into 10 parts by 10 parts.

So each unit will be broken up into 10 parts.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

So you can see, I've broken up each 1 unit by 10 by 10 parts.

Now, breaking it up will allow us to see and understand multiplying by decimals a little bit more.

So let's have a look at an example of 2.

5 multiplied by 1.

Here, you can see I have my 2.

5, and I have my 1.

Well, how can this diagram help us calculate that decimal multiplication? Well, first of all, we do know each big square represents 1.

So what do you think each strip represents? Well, this represents 1/10 or 0.

Remember how we split that 1 unit into 10 parts? So each strip represents 0.

1 or 1/10.

Now, I want you to have a look at this tiny little square.

What do you think this little square represents? Well done if you spotted it.

It's 1/100 or 0.

01.

And if you look at that one unit grid, you can see there's 100 little squares in that one unit grid.

So that means each little square must be 1/100 or 0.

01.

Now, we know this, how is it going to help us work out the answer to 2.

5 multiplied by 1.

3? Well, we know that one big square represents 1, so that means here, we have 2.

We know that each of those long strips represents 0.

So I have 1, 2, 3, 4, 5, so that means I have 0.

5, five strips of that 0.

Then, I have 1, 2, 3, so three strips of that 0.

1 is 0.

I have another three strips of that 0.

1, which is 0.

3, and let's count up those little tiny squares.

Well, I have 15 of those little tiny squares, and each of those little tiny squares is 0.

01, so that means I have 0.

15.

So now, I can work out the answer to 2.

5 multiplied by 1.

3, by simply adding up all our values that you can see on our area model.

So we have 3.

25.

So 2.

5 multiplied by 1.

3 is 3.

25.

The area model does not have to be drawn to scale here, but it does help us see what's happening when you do multiply by decimals.

So let's see if we can have a look at another way.

Here, you can see on the right-hand side, I've drawn a multiplication grid.

I've split the numbers 2.

5 into the integer 2 and 0.

I've split the number 1.

3 into the numbers 1 and 0.

3, and you can see how each number appears according to that area model grid.

So here, you can see the 2, our 0.

5, two lots of our 0.

3, and our 0.

15, giving us the total answer to be 3.

25.

Let's see if there's a slightly easy way when multiplying by decimals.

So now, I'm going to compare the same method, but multiplying with integers.

So I've got 25 multiplied by 13.

On the left-hand side, you can see how I've used the multiplication grid, 20, and the 5, multiplied by the 10 and the 3, giving me an answer of 325.

And what I want you to do is have a look and see what you notice.

Well, converting your calculation to integers will make it easier to calculate.

We can use the integer multiplication strategy to find the answer to decimal multiplication by using multiplication or division of powers of 10.

So the 2.

5 multiplied by the 1.

3, well, we could work out 25 multiplied by 0.

1 because that's the same as 2.

5, and we could also work out 13 times 0.

1, which is the same as 1.

Grouping together those integers, 25 by 13, and then grouping together those 0.

1s, we have our calculation that can help us work out the answer.

So we know 25 multiplied by 13 is 325, and then the 0.

1 times 0.

1 is simply 0.

01.

So all I need to do is multiply the 325 by 0.

01, thus giving us 3.

25.

This method of multiplying by integers and then multiplying by powers of 10 is an effective approach to work out decimal multiplication.

So let's have a quick check.

You have 3.

42, and we want to find out which calculation is it equivalent to? See if you can give it a go, and press pause if you need.

Well done if you spotted it.

It's 342 times 0.

1 times 0.

1, because we know 342 times 0.

1 is 34.

2, then multiplied by another 0.

1 gives us 3.

42.

Well done if you got that one right.

And we also have 342 times 0.

01 because these calculations are the same.

342 times 0.

01 is the same as 342 times 0.

1 times 0.

Well done if you got those right.

Now, let's have a look at another check question.

When dividing by 100, it's equivalent to which of the following? See if you can give it a go, and press pause if you need.

Well, hopefully, you spotted it's multiplying by 0.

1 and by 0.

This is because we know 0.

1 is the same as 1/10, so 0.

1 times 0.

1 is the same as 1/10 times 1/10, which is 1/100, which we know is the same as dividing by 100.

And we also have 0.

01 because 0.

01 is exactly the same as the 0.

1 times 0.

1, which is the same as our 1/100.

Well done if you got both of those right.

Let's do another check.

Looking at a calculation, which of these has the same product as 0.

52 multiplied by 3.

18? See if you can give it a go and press pause if you need.

Well, for Part A, we know 0.

0001 is the same as 0.

01 times 0.

01.

So that means 52 times 0.

01 times 318 times 0.

01 gives us 0.

52 multiplied by 3.

18.

So that means yes, it is the same.

Have a look at B.

Well, we know when we divide by 1,000, it's the same as multiplying by that 0.

001.

Well, we know we can split that 0.

001 into 0.

01 multiplied 0.

01 So applying it to our calculation, 52 times 0.

01 times 318 times 0.

1 is not the same as 0.

52 times 3.

18.

Lastly, let's have a look at C.

Well, we know 52 times 0.

01 is 0.

52.

318 multiplied by 0.

01 is 3.

18.

So yes, it is the same.

A huge well done if you got that one right.

Now, let's have a look at our last check.

Andeep starts the following working out to calculate the answer to 12.

4 multiplied by 1.

Can you help him finish finish his working out and find the correct answer? See if you can give it a go, and press pause if you need.

Well done.

So let's see how you got on.

Well, 12.

4 times 1.

We're going to split it into 124 multiplied by 0.

1, multiplied by 14 times the 0.

So you can see why Andeep chose the 100, 20, and 4, and the 10 and 4.

Then, we're going to multiply 124 by 14, then multiply it by 0.

1, and then 0.

Filling in our grid, you should have had these values.

Summing them up gives us a final answer of 1,736.

But remember, we need to multiply this by the 0.

1 and the 0.

So multiplying our 1,736 by 0.

01 gives us the final answer of 17.

36.

Massive well done if you got this one right.

Now, it's time for your task.

For question 1A, you need to work out the answers to the following.

Make sure you show all your working out and take your time.

Press pause if you need.

Well done.

So let's move on to question two.

Question two wants you to tick the correct calculations.

Take your time when working this out, and press pause if you need more time.

Well done.

Let's move on to question three.

Given that 456 times 12 is equal to 5,472, using this, work out 45.

6 times 12, work out 4.

56 times 12, and work out 4.

56 times 1.

Use the calculation rather than working out the multiplication of the decimal.

See if you can give it a go, and press pause if you need more time.

Well done.

So let's move on to question four.

Question four wants you to fill in some blanks to make the calculation correct.

2 times 8.

what gives 25.

92.

13.

6 times 6.

what gives 87.

04.

21 times something.

4 is 26.

94.

This is a great question, and really does make you think about multiplying those decimals.

See if you can give it a go, and press pause if you need more time.

Well done.

So let's go through our answers.

Well, for question one, it's all about that really structured working out.

So I'm going just to show you here that you should have got the answer to 8.

3 multiplied by 11 to be 91.

You can see how I've used the integers 83 multiplied by 11, and made sure I only multiplied by 0.

1 to convert the calculation to 8.

3 multiplied by 11.

Well done if you got that one right.

7 times 6.

You can see I've used the integers of 37 multiplied by 64.

Then, I've multiplied this by 0.

01 to get the answer to 3.

7 multiplied by 6.

4, which is 23.

68.

Well done if you got that one right.

Now, let's have a look at C.

Well, we have 2.

1 times 1.

28.

I'm going to use the integers 21 and 128.

Then, once I've got those answers, we're going to multiply by 0.

001 to give us the final answer of 2.

688.

A huge well done if you got those right.

Question two, we had to tick which calculation was correct.

Well, hopefully, you spotted it's A, C, and D.

Well done.

So now, let's have a look at question three.

Well, question three gives us 456 multiplied by 12 is 5,472, and we're asked to work out 45.

6 times 12.

Well, the answer is 547.

This is because we know 456 times 12 is 5,472.

So if you multiplied that 456 by 0.

1, you've got your 45.

The 12 still stays the same, so that gives us an answer of 547.

For B, we have 4.

56 times 12.

Well, this gives us 54.

72.

Because we know the 456 times 12 is our 5,472, changing that 456 into 4.

56 means we have to multiply 456 by 0.

01.

The 12 still stays the same, so our answer is 54.

72.

56 times 1.

Well, it gives us 5.

472.

Same again, we had to change our 456 into 4.

56 by multiplying by 0.

01.

The 12 needs to be changed to 01.

So we had to multiply by 0.

1 there.

So multiplying our 5,472 by 0.

01 and the 0.

1 gives us our 5.

472.

Well done if you got that one right.

For question four, you needed to fill in the blanks.

This is a great question, and really does make you think about the product of decimals.

Hopefully here, you would've got 1, 4, and 6.

Well done if you got that one right.

Great work so far.

So let's move on to appropriate answers to calculations.

Well, sometimes we know there is an error by simply looking at the reasonableness of the answer.

It is important to understand the context of the question and reflect on the reasonableness of an obtained answer.

For example, if we were looking at a cat, and we were asked, well, which weight would be most appropriate for the cat? Do you think it's 4.

2 kilogrammes, or do you think it's 42 kilogrammes? One answer is more reasonable and appropriate than the other.

Hopefully, you can spot its 4.

2 kilogrammes.

42 kilogrammes would be a huge cat.

So now, let's apply it to a decimal multiplication.

2 multiplied by 3.

Which do you think is the correct calculation? 192.

4 or 19.

24? And I'd like you to explain your reasoning.

See if you can give it a go.

Press pause if you need more time.

So let's have a look why.

Well, thinking about the place value of each number.

For A, we know we have one digit in the hundreds place.

We're in the hundreds with 192.

And the highest place value for option B is the tens, because we have one 10, and then a 9.

24.

Now, given the fact that we're multiplying numbers that have the highest place value of 5 ones and 3 ones, that means we know it can't be in the hundreds.

2 times 3.

7 won't give us a number in the hundreds.

Well done if you identified it to be 19.

24.

Now, let's have a look at a check.

Which student has an appropriate answer to the calculation, 3.

24 times 1.

3? Do you think it's Laura, who worked it out to be 42.

12, do you think it's Sam, who worked it out to be 421.

2, or do you think it's Lucas, who worked it out to be 4.

212? See if you can give it a go, and press pause if you need.

Well done.

So hopefully, you spotted it's Lucas.

24 times 1.

We have a 3.

24 times a 1.

3, so we know it's not gonna be in the hundreds.

We know it's not going to be in the tens, so the reasonable answer would be 4.

212.

Well done if you got that one right.

Now, let's have a look at another question.

We know the answer to the calculation is 4.

212, but which calculation could Laura and Alex use to make their answer? She worked out something to be 42.

12, and Sam worked out a calculation to be 421.

So using 3.

24 times 1.

3 equals 4.

212, what could you do to that calculation to make Laura's calculation and to make Sam's calculation? See if you can give it a go and press pause if you need.

Well done.

So let's have a look at Laura's first.

Well, there's lots of different answers for Laura's, and that's what's quite great about maths, because there's an infinite number of calculations, which will make the product of 42.

12.

32.

4 times 1.

3, 3.

24 times 13, 324 times 0.

13.

So now let's have a look at Sam's.

Well, there's an infinite number of calculations, which will give us a product of 421.

324 times 1.

3, 32.

4 times 13, 3.

24 times 130.

Well done if you got any one of those.

It's important to remember when place value's considered, it can still remain difficult to identify the correct magnitude of the answer.

A trick is to look at the last digits.

So let's have a look at 27 multiplied by 1.

Well, I'm gonna multiply the last digits together.

In other words, the 7 and the 3, and we know 7 multiplied by 3 is 21.

Now, how does that help us select which answer is the product 27 multiplied by 1.

3? Do you think it's 35.

2, 35.

1, or 35.

8? Well, you could use the area method or column multiplication, and you'd still end up in a situation where you're multiplying the 7 by the 3 and multiplying the 7 by the 3 gives us 21.

Now, 21 ends in a 1 because it's the product of 7 and 3.

So whether it's 7 multiplied by the 3, or the 7 multiplied by 0.

3, the digit is still ending in a 1, which is why the answer to 27.

13 is 35.

1, because the digit is ending in a one.

This is a nice little trick.

Let's see if we can apply it to this question, 6.

4 times 2.

Look at those last digits.

What do you think the answer would be? 14.

95, 14.

72 or 14.

58? See if you can figure it out, and explain your reasoning.

Well, hopefully, you've spotted it's 14.

72, and the reason is because the product of 4 and 3 is 12, thus ending in a 2, or the product 0.

4 and 0.

3 is 0.

12, but it's still ending in that 2.

So well done if you got that one right.

Next check.

We're going to look at a context question.

So a sports field has a length of 105.

3 metres by 68.

7 metres.

Which of the following would you expect to be the correct area of the sports field? I'd like you to justify your answer.

See if you can give it a go, and press pause if you need.

Well, for Part A, we have 7234.

This is a one decimal place answer, and we know 0.

3 times 0.

7 will give us a two decimal place answer, so it can't be A.

For B, well, 7234.

92, well, we have our two decimal places, but remember, 3 multiplied by 7, or 0.

7 multiplied by 0.

3, both cases end in a digit of 1, so it can't be B.

C, 723.

11.

Well, this is too small.

If you have a look, we have a 100.

5 metres in length and 68.

7 metres in length, so the area will definitely be greater than 723.

11.

So that leaves us with D.

D has two decimal places, and we have a digit ending in the 1, and it's in a reasonable answer as well.

Well done if you got that one right.

Next question.

We have Jun reflecting on a calculation.

4 times 5.

5 is equal to 35.

He says the answer must have two decimal places, the answer has a zero in its last digit, and the answer is greater than 30.

Can you explain if you agree or disagree with Jun? See if you can give it a go.

Press pause if you need.

Well done.

So let's have a look.

Well, we do know the product of 0.

4 and 0.

5 is 0.

20.

So this is the same as 0.

So it doesn't have to have those two decimal place because we ignore that last digit of zero.

Well, given that we have the largest place values of 6 ones and 5 ones, the product is expected to be greater than 30.

So in short, reflecting on this calculation, 35.

2 is an appropriate answer.

Now, let's have a look at your task questions.

For question one, you need to use your knowledge and identify a reasonable answer, and without working out, match the calculation to the correct answer.

See if you can give it a go and press pause if you need.

Well done.

Let's move on to question two.

Question two gives you some statements, and you need to identify if it's always true, sometimes true, or never true, and you can use a calculator to help justify your answers if it helps.

See if you can give it a go, and press pause if you need.

Great work.

So let's go through these answers.

Well, for question one, hopefully, you've paired up 2.

34 times 3.

8 is 8.

892.

Spot how we have three decimal places, and we expect it to be greater than 6.

34 times 6.

8 gives us 36.

312.

Looking at our decimal places, we have three decimal places, and also the 5 multiplied by 6, we expect it to be greater than the 30.

34 times 9.

Same, we have three decimal places, and we expect it to be around about the 90s.

Lastly, 3.

5 times 2.

5 multiplied by the 8 is 40.

So we do expect two decimal places, but remember it ends in that 0, so we can expect one decimal place.

So let's have a look at question two.

Are the statements always true, sometimes true, or never true? We can use a calculator.

Is the product of two decimal places, is it always a decimal place? Sometimes true.

For example, 1.

5 times 2.

3, yes, it gives a decimal of 3.

45, but 2.

5 multiplied by 4.

4 gives us an integer of 11.

So it's sometimes true.

So let's have a look at the second statement.

Well, the product of two decimals, each with two decimal places, must have four decimal places.

Well, it's sometimes true.

For example, 1.

36 times 1.

25 is 1.

7, but 0.

45 times 0.

89 is 0.

4005.

So it can work sometimes.

Next statement, the product of two decimals is always smaller than the largest multiplier.

It's sometimes true.

For example, 4.

6 times 1.

3, the product is greater than both numbers that we're multiplying, 5.

98.

But if you look at 34.

1 times 0.

4, it's smaller.

13.

64 is our product.

So it's sometimes true.

Well done if you got this one right.

So in summary, there's lots of different ways to multiply integers, and in this lesson, we focused using the integer multiplication strategy to find the answer to the decimal multiplication by using multiplication or division of powers of 10.

When an answer's been calculated, try to consider the magnitude or the size of the answer given the context of the question.

Great work.

Well done.

It was fun learning with you.

I've finished the video