Lesson video

Hello, welcome to today's maths lesson.

My name is Dr.

Shorrock and I'm really looking forward to learning with you today.

Let's get started.

Today's lesson is from our unit, Calculating with Decimal Fractions.

The lesson is called Predicting the Size of a Product.

As we move through the learning today, we will deepen our understanding of what a product is, so the answer to a multiplication question, and how we can predict its size and we're going to use this to sense-check any calculations that we do.

Sometimes new learning can be a little bit tricky, but I know if we work really hard together and I am here to guide you, then we can be successful.

Let's get started, shall we then? How can we predict the size of a product? The key words from our learning today are product and factor.

You may have heard those words before, but it's always useful to practise them.

My turn.

Product.

Your turn.

Nice.

My turn.

Factor.

Your turn.

Lovely, the product is a result of two or more numbers multiplied together, so it's the answer that we get.

And the factors are integers, so whole numbers that are multiplied together to get a product.

We are going to start our learning today by thinking about the scaling structure of multiplication and we have Lucas and Sofia to help us.

Let's start by looking at this equation, 2 multiplied by 6.

Well, what do we know? Well, we know that 2 multiplied by six 1's is equal to 12 1's.

But is there anything else we know, anything else we can determine? What do you think? If one factor is made 10 times the size, so if we make 6 10 times the size to convert it to 60, we know the product will be 10 times the size.

12 multiplied by 10 is 120.

So we knew 2 multiplied by 6 was 12, but from that we could determine 2 multiplied by 60 equals 120.

Is there anything else we could determine? Well, we could do 100 times the size.

So if the factor 6 was multiplied by 100 to 600, the product would also be 100 times the size.

12 multiplied by 100 is 1200 or 1,200.

So let's look at these equations.

Is there something that you notice, something that is the same, something that is different? Well Lucas has noticed that for each of these the product is greater than the factors.

Can you see 12, that's greater than 2 or 6? 120 is greater than 2 or 60 and 1,200 is greater than 2 or 600.

And Lucas is saying that's because multiplication always makes things bigger.

Hmm.

Do you agree with Lucas? What do you think? Sofia is respectfully challenging him.

But why? That means she thinks he's wrong.

I wonder why.

What about if we multiply by a decimal fraction Sofia is questioning? So 2 multiplied by a decimal fraction, 0.

6, well that equals 1.

2.

And 2 multiplied by 0.

06 is equal to 0.

12.

The answer there is not bigger than both factors, is it? 1.

2 is not bigger than 2.

0.

12 is not bigger than 2.

And Lucas tells us if one factor is made one 10th times the size, the products will be one 10th times the size.

So 6 made one 10th times smaller is 0.

6 so we need to make that product 12, one 10th times the size.

12 multiplied by 0.

1 is 1.

2.

And the same if we look at making a factor 100th times the size.

6 made 100th times the size is 0.

06.

So the product will be 100th times the size, 0.

12.

So Lucas can see now he was wrong.

These examples here have proved to him that multiplication does not always make things bigger.

I wonder if you knew that? Sofia is just clarifying that when we multiply by a decimal fraction that is less than one, the product becomes smaller, not greater.

And we can represent this on a place value grid.

We've got 2 multiplied by 6 is equal to 12.

If the factor 6 is made 10 times the size, the product will be 10 times the size.

If we make the factor 6 100 times the size, the product will be 100 times the size.

And again, if we make a factor one 10th times the size, the product will be one 10th times the size.

And if we make the factor 100th times the size, the product will be 100th times the size.

Let's summarise what we have learned so far.

We started with 2 multiplied by 6 1's was equal to 12 1's.

If one factor is made 10 times the size, the product will be 10 times the size.

If one factor is made 100 times the size, the product will be 100 times the size.

If one factor is made one 10th times the size, the product will be one 10th times the size.

And if one factor is made 100th times the size, the product will be 100th times the size.

So we could determine all of those just from our known fact, two 6's are equal to 12.

Let's check your understanding with this.

Could you use the given known fact to complete the equations? 4 multiplied by 8 is equal to 32.

And use the stem sentence that Lucas is saying to help you.

If one factor is made mm times the size, the project will be mm times the size.

Pause the video while you have a go at completing the equations.

When you are ready to go through the answers, press play.

How did you get on with those questions? Did you use the fact that you knew 4 times 8 is equal to 32? To work out 4 times 80 must be 320 because the factor was made 10 times the size so the product will be 10 times the size.

And 4 times 800 must be 3,200 because the factor is made 100 times the size so the products must be 100 times the size.

And again, 4 multiplied by 8 was equal to 32.

If we make the 8 one 10th times the size, 0.

8, the product must be one 10th times the size, 3.

2.

And 4 multiplied by 0.

08, well the 8 was made 100th times the size so the product must be 100th times the size, 0.

32.

How did you get on with those? Well done.

It's your turn to practise now.

For question one, could you complete the equations by filling in the missing numbers then explain what you notice.

For question two, could you calculate the product of 8 and 0.

06 but could you make a mistake when you do it? And I'd like you to explain the mistake that you made.

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

How did you get on? For question one, you are asked to complete the equations.

So we know our known fact nine 4's are equal to 36 and we can use that because the factor 4 has become 10 times the size so the product must be 10 times the size, 360.

And here one factor has become one 10th times the size so the product must be one 10th times the size, 3.

6.

9 multiplied by 0.

04.

Well the one factor has become 100th times the size so the product must be 100th times the size, 0.

36.

7 multiplied by 0.

2.

Well we can use seven 2's are equal to 14 to help us.

One factor has become one 10th times the size so we need to make the product one 10th times the size, 1.

4.

0.

02 is 100th times the size of 2 so the product must be 100th times the size of 14, 0.

14.

7 times 200, well one factor has become 100 times larger so the product must be 100 times larger, 1,400.

5 multiplied by 0.

23 is equal to 1.

15.

So 5 multiplied by 23 must be equal to 115 because the factor has become 100 times the size so the product is 100 times the size.

2.

3 times 5.

Well the factor has become 10 times smaller, so the product must be 10 times smaller, 11.

5.

230 multiplied by 5.

One factor has become 10 times larger so the product must be 10 times larger, 1,150.

You might have said that you noticed that if one factor is made mm times the size, the product is also mm times the size.

For question two, you are asked to calculate the product of 8 and 0.

06 but to make a mistake.

You might have started by forming equations of the known facts that you were going to use.

8 multiplied by 6 is equal to 48.

So 8 multiplied by 0.

06 is? And you might have solved and calculated the product incorrectly to be 4.

8 but then explains your mistake.

You might have explained that when one factor is made 100th times smaller, then the product must become 100th times smaller.

The product was only made one 10th times smaller, the product should have been 0.

48.

How did you get on with both those questions? Brilliant.

Fantastic learning so far.

We're going to move on now and look at how we can predict the size of the product.

Let's revisit the equations that we have looked at in the first part of the lesson.

What else do you notice? Lucas has noticed that when a number is multiplied by a value greater than 1, so when 2 was multiplied by 600, 60 or 6, the product was greater than the original number.

The product was greater than 2 wasn't it? And we can summarise this here.

When a number or this case 2 is multiplied by a number greater than 1, the product was greater than the original number or 2.

So when a number is multiplied by a value greater than 1, the product is greater than the original number.

And what else do you notice? Did you notice something else about the bottom two equations? So Lucas has noticed that when a number is multiplied by a value less than 1, so we've got 2 being multiplied by 0.

6 or by 0.

06 and they are less than 1, the product is smaller than the original number, it's smaller than 2, isn't it? And we can represent that here and see that here.

When 2 is multiplied by a number that's less than 1, the answer is less than 2, less than our starting number.

So when a number is multiplied by a value less than 1, the product is smaller than the original number.

Let's look at another example.

We've got 57 multiplied by 3 is equal to 171.

So if we know this, what else do we know? Well that's right, we could determine 57 multiplied by 30 because when one factor is made 10 times the size, then the product will be 10 times the size, 1,710.

And Lucas says, "We can sense-check our answers using what we've just learned." We know that if we multiply a number, so in this case 57 by a value greater than 1, so in this case 30, then the product must be greater than the original number.

Our product 1,710 is greater than 57, isn't it? So our answer must be reasonable.

Let's look at a different example.

So we know 57 multiplied by 3 is 171.

What else do we know? Well, we know when one factor is made one 10th times the size, then the product will be one 10th times the size.

And again, we can sense-check our answers.

We know if we multiply a number, so 57 by a value less than 1, or 0.

3 in this case, then the product must be less than the original number and it is.

17.

1 is less than 57.

And that's because we've multiplied by a number that is less than 1.

Let's look at a different example.

49 multiplied by 0.

7 is equal to 34.

3.

We know when we multiply a decimal fraction, the product will always be smaller than the original number.

Oh, but Sofia is challenging him again.

She doesn't think that's always the case.

What do you think? Do you think Lucas is correct or not? Let's investigate multiplying with decimal fractions a bit more, shall we to see who is correct.

7 multiplied by 7 tenths is equal to 49 tenths.

7 multiplied by 8 tenths is equal to 56 tenths.

7 multiplied by 9 tenths is equal to 63 tenths.

Seven multiplied by 10 tenths is equal to 70 tenths.

Seven multiplied by 11 tenths is equal to 77 tenths.

And seven multiplied by 12 tenths is equal to 84 tenths.

Is there something that you've spotted there? Something that you notice? This is what I've noticed.

When our number 7 is multiplied by a number less than 1, the answer was less than our starting number or 7.

But when we multiplied 7 with a number or a decimal fraction that was greater than 1, our answer was greater than the number we started with, 7.

Did you spot that? So when a number is multiplied by a value less than 1, the product is less than the original number.

And when a number is multiplied by a value greater than 1, the product is greater than the original number.

So even when we multiply by decimal fractions, if the decimal fraction has a value of greater than 1, the product will be greater than the original number.

And we can summarise this here.

When we multiplied 7 by decimal fractions, if those decimal fractions had a value less than 1, the product was less than the original number.

And then 7 multiplied by 1, well we know when we multiply by 1 the product remains the same.

And then when we multiply by our decimal fractions that had a value of greater than 1, even though they are decimal fractions, the product was greater than the original number.

When we multiply by a decimal fraction, the product will be smaller than the original number only if the decimal fraction has a value that is smaller than 1.

0.

7 is smaller than 1 so the product will be smaller than 49 and it is.

And so Sofia was correct.

Just because we multiply by a decimal fraction does not mean the product will be smaller.

The value of that decimal fraction has to be smaller than 1 for that to be true.

Let's check your understanding with this.

Can you tell me if this is true or false? 16 multiplied by 4 is equal to 64.

So 16 multiplied by 0.

4 is equal to 32.

Do you think that's true or false? Pause the video, maybe find someone to chat to about this.

When you're ready for the answer, press play.

How did you get on? Did you say that was false? But why is it false? Is it A, multiplication always makes things larger, so the product must be greater than 16? The product should be 64? Or is it B? When a number is multiplied by a value less than 1, the product is less than the original number.

The product should be one 10th times the size of 64, which is 6.

4.

Pause the video, have a chat to somebody about this, see if you think it's A or B and when you are ready for the answer, press play.

How did you get on? Did you say it must be B? It can't be A.

We now know that multiplication does not always make things larger.

If we are multiplying by a value less than 1, the product will be smaller.

So it must be B.

When a number is multiplied by a value less than 1, the product is less than the original number.

The product should be one 10th times the size of 64 because 0.

4 is one 10th times the size of 4, and it should be 6.

4.

Well done.

Your turn to practise now.

For question one, Sofia says that 14 multiplied by 0.

8 is equal to 112.

Do you agree with her? Could you convince me that you are correct? So don't just say yes or no, convince me.

For question two, could you fill in the missing symbols, smaller than, greater than or equals to? For question three, could you write a single digit in each box to make this equation correct? 2.

5 is greater than 0.

mm times mm.

Work systematically is a hint I would give you then you can ensure that you have all the possible solutions.

Maybe you can convince me that you know you have all the possible solutions.

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

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

Your first question, you were asked to say if you agreed with Sofia or not and to give reasons.

You might have said that this is false.

One number, in this case the multiplier 0.

8 is less than 1 so the product must be smaller than the other number.

112 is larger than 14 so the calculation cannot be correct.

Did you notice that you did not need to calculate 14 multiplied by 0.

8 to answer this question? You just needed to recognise that the multiplier was smaller than 1, so the product must be smaller than the other factor.

So the calculation was likely to be incorrect.

For question two, you had to fill in the missing symbols, 8 multiplied by 0.

7.

Well that must be less than 8 because 0.

7 is less than 1 so the product will be less than our starting number.

8 multiplied by 1, well that's equal to 8.

8 multiplied by 1.

7.

Well it has got to be greater than 8 because we're multiplying by a number with a value of greater than 1.

3 must be greater than 3 multiplied by 0.

09 because 0.

09 has a value less than 1.

And the same here, 3 must be greater than 3 multiplied by 0.

9 because 0.

9 still has a value less than 1.

And 3 though must be less than 3 multiplied by 9 because we're multiplying by 9, which has a value of greater than 1.

Did you notice that you did not need to calculate the answers to these? You could just use your number sense and our key learning that when a number is multiplied by a value less than 1, the product is less than the original number.

And when a number is multiplied by a value greater than 1, the product is greater than the original number.

Well done if you spotted that.

And for question three, you were asked to write a single digit in each box to make the equation correct.

And here are some of the solutions and you can see I worked in a systematic fashion.

I started off with thinking about 0.

1, multiplying by a single digit 1 to 9, and each of those would have a value that was smaller than 2.

5.

The same with 0.

2 multiplying by the digits 1 to 9 would have a product of smaller than 2.

5.

And then when I got to 0.

3, well I knew that 0.

3 when I multiplied that by 9, it would not have a value smaller than 2.

5.

And then when I multiplied with 0.

4, I knew that I could multiply by the digits 1 to 6 to have a product that was smaller than 2.

5.

And then when I multiplied with 0.

5 or my single digit that I wrote was 5, I can multiply by the digits 1 to 4.

And then with 0.

6 I can multiply with the digits 1 to 4.

0.

7, only with the digits 1 to 3.

And with 0.

8, only the digits 1 to 3.

And with 0.

9 the digits 1 and 2.

You might have noticed that there were 46 different equations that were true for this inequality and you knew you had all of them because you worked systematically like I did.

You might also have considered using 0 and found even more solutions.

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

Fantastic learning today.

You have really deepened your understanding about how we can predict the size of a product.

We know if one factor is made mm times the size and the other factor remains the same, the product will be mm times the size.

We know that when a number is multiplied by a value less than 1, the product is smaller than the original number.

And when a number is multiplied by a value greater than 1, the product is greater than the original number.

I have had a lot of fun learning with you today and I look forward to learning with you again soon.